Simon Cowell and Lauren Silverman are "settling into their new life" with baby Eric.
The couple welcomed their son into the world last month and they're beginning to adapt to parenthood and having a young tot around the house.
A source told E! News: "It's a happy time. They're settling into their new life."
Meanwhile, five-week-old Eric - who was named after Simon's late father - doesn't seem to be giving his parents a difficult job as he's already a "good sleeper".
The source explained: "The baby's doing well, he's very strong. He can already hold his head up! He's not fussy and and he's a good sleeper."
Lauren's eight-year-old son Adam - from her previous marriage with Andrew Silverman - is also besotted with his new half-brother and recently jetted over from New York to spend the holidays with him.
The source added: "Adam came to London for his spring break from school. He adores his little brother and loves helping with him."
Simon, 54, and Lauren are currently living in London due to the music mogul's busy schedule, but they are planning on dividing their time between the UK and the US as Adam lives in New York on a permanent basis.

Kevin Connolly has broken his leg in two places.
The 40-year-old actor suffered the painful injury while shooting the 'Entourage' movie last week while playing football with Seattle Seahawks quarterback Russell Wilson, who has a cameo role in the film.
Kevin tripped on a sprinkler while trying to make the play, resulting in co-star Jerry Ferrara falling on his leg.
The actor - who reprises his role of Eric 'E' Murphy in the movie - immediately iced the injury and went back to work on his next scene shortly afterwards, Deadline reports and filming wasn't interrupted at all.
Meanwhile, 'Entourage' star Jeremy Piven - who plays agent Ari Gold - has promised fans of the original HBO show, which ended in 2011, will feel "satisfied" with the new movie.
He told E! News: "I think for the people that watch the series and are fans, I think they're going to be really satisfied because everyone's back, and we picked up where we left off, and then we have incredible cameos."
And one cameo, featuring Liam Neeson, is likely to be a hit as even paparazzi watching filming in Los Angeles couldn't hide their laughter.
Jeremy said: "I had a moment with Liam Neeson, I pull up in a very under-the-radar red Ferrari, and he's in a Bentley and I hear like a laugh track but
I look out of the corner of my eye and the paparazzi are laughing.
"The fact that they've let their cameras down for even a second to laugh means we're onto something."

Kim Kardashian loves "dressing up" her daughter.
The 'Keeping Up with the Kardashians' star admits to borrowing items from fiance Kanye West's wardrobe to create costumes for their nine-month-old baby North.
She told the new issue of America's Vogue magazine: "I take pictures of her all the time and dress her up. I put Kanye's big chains around her, and I put a little Louis bag and some Jordans, and I was like, 'What up, Daddy?' "
And Kanye also loves to have fun with his little girl, having made a stop-frame video of her natural movements which he has edited to make it seem like she is breakdancing.
Meanwhile, Kim admitted she and her sister Kourtney once went through their mother Kris Jenner's wardrobe to decide who would inherit which items when she dies.
The brunette beauty said: "[Mom] would always wear really tight - she was into Vicky Tiel, Moschino, or Chanel - with huge Moschino heart earrings. She had an amazing body and was always so in shape.
"One time Kourtney and I sat in my mom's closet - we must've been eight or nine - and we were like, 'If Mom dies, we're going to write down who gets what.' We went through every last piece of her wardrobe. We were so mean then!"

Mila Kunis and Ashton Kutcher are planning to get married in a registry office.
The couple, who got engaged earlier this month, are expecting their first child together and are keen to tie the knot in April before their new bundle of joy arrives.
A source told The Sun newspaper: "They are planning to get married in LA in April. They just want it to be made legal so are happy with a registry office.
"Mila, especially, wants it to be very low-key, with just their family there and a BBQ in their garden afterwards. She doesn't want to do anything showy or over the top."
Though they are planning low-key nuptials, Ashton - who was previously wed to Demi Moore - wants to mark their marriage with a big celebration later in the year.
The source added: "Ashton wants a party for all their friends later in the year in the south of France, which Mila is up for, depending on how they cope with having a baby."
The loved-up pair are adamant they will be "hands-on" parents when their baby arrives and are said to have ruled out hiring nannies, despite them both having busy acting careers.
The duo sparked engagement rumours a few weeks ago when Mila, 30, was spotted wearing a diamond sparkler on her ring finger while out shopping in Los Angeles.
Ashton, 36, began dating the brunette beauty in 2012, following his split from Demi, but they have known each other for years after playing the lead roles in hit 1990s sitcom 'That '70s Show'.

Simon Cowell is trying to hire Rihanna for 'The X Factor' judging panel.
The 54-year-old music mogul - who is set to return to the show with Cheryl Cole - is hoping to lure the 'Diamonds' hitmaker to join them because she draws lots of media attention and Simon is desperate to make the show more popular than BBC One rival 'Strictly Come Dancing'.
A source told the Daily Star newspaper: "Simon and Cheryl have two months to find someone to sit in that chair and they want to get the right person.
"This year is an important one for 'The X Factor'. Simon can't be complacent and has to come back with a real A-Team.
"Rihanna would be a shock selection because she is not an obvious choice for a family show - but that's what he wants."
Simon previously revealed he will include Cheryl, 30, in any discussions regarding who will join them on the panel and she will have final say on new appointments.
It's thought the pair are at odds over whether to rehire the show's longest serving judge, Louis Walsh, with Cheryl eager to axe him from the show.
Other names being discussed for 'The X Factor' panel include LMFAO rapper Redfoo - who is a judge on the Australian version of the singing competition - and Black Eyed Peas frontwoman Fergie.

Corey Stoll is in talks to join Marvel's 'Ant-Man'. The 'House of Cards' actor - who played Peter Russo in the first series of the political revenge drama - is negotiating a deal to join the superhero movie in an unspecified role, reports Variety. Hi...

Ferne McCann doesn't think 'The Only Way is Essex' has ruined her relationship with Charlie Sims.
The reality star broke up with her boyfriend last year on the show after he was photographed smooching another woman last summer, but the brunette beauty doesn't believe the programme was to blame for their split.
She told NEW! magazine: "'TOWIE' hasn't ruined mine and Charlie's relationship or our lives. If anything it's benefited it and it's accelerated our break up. We would have broken up 'TOWIE' or no 'TOWIE'."
Ferne also hasn't ruled out getting back with Charlie, who she has dated on/off for three years, because she'll always love him.
She said: "I love him and always will - he's my first love."
However, the 21-year-old hunk may not feel the same after he found out that Ferne had cheated on him back in December with Frank Major.
He said previously: Every excuse she has given is bulls**t. She says in every interview that she wasn't happy and couldn't trust me because of the cheating. She cheated on me and then dumped me the next day, a week before Christmas. That girl has no f***ing heart. I had Christmas presents under the tree for her - I'm pretty sure she didn't have them for me."

Kirk Norcross wanted to release a single so he could listen to it in his car.
The former 'The Only Way is Essex' star is trying his hand at music and has brought out his own track, but he insists he didn't create a song to storm onto the charts.
He explained: "It's always been a dream of mine. I'm not worried about the charts, I just want to listen to it in my car! That's why I'm giving the proceeds to the ADHD Foundation [Kirk was diagnosed with Attention Deficit Hyperactivity Disorder when he was younger].
"I always get knocked for what I do, 'Celebrity Big Brother' and 'TOWIE' and stuff like that, but this is the only thing that's had positive comments. I really want to keep on singing."
The 25-year-old hunk admits he knows his song will go down well with his fans because N Dubz singer Dappy told him it was "amazing" when he played it to him.
Kirk said: "Dappy said it was amazing, which was enough for me because I think he's so talented."
Despite his admiration for the cheeky rapper, Kirk insists he'd never collaborate with him because he believes Dappy would put him to shame.
He told NEW! magazine: "Do you know what, I wouldn't, because he's so good, I just want him to do his thing."
Kirk's single 'Ain't No Sunshine' is out now.

Rita Ora has likened her new single to Whitney Houston.
The 23-year-old singer will launch her second studio album with comeback track 'I Will Never Let You Down' - which was produced by her boyfriend Calvin Harris - and she claims the song is reminiscent of some of the late 'I Will Always Love You' legend's earlier work.
She explained: "'I Will Never Let You Down' is obviously a very important song to me because it's a love song, but it reminds you of Whitney, and that was the vibe we were on. We just wanted to feel good. You know that feeling of 'How Will I Know' with the synth? It just makes you feel happy ... that's what I wanted to create."
Rita also confirmed she has collaborated with 'Thrift Shop' hitmaker Macklemore on her new album.
She gushed to Paper magazine: "We have this amazing song, 'Young Dreamers' with Macklemore and it's so cool. It's for the kids, and it's liberating and so fun."
The 'R.I.P.' hitmaker is determined to crack America with her new album, but admits she is "scared" about taking on the challenge.
She added: "Conquering America for a UK artist is incredible because that's what everybody wants and dreams about. The accomplishment is dominating a market which you aren't familiar with. I am a bit nervous. I am a bit scared, I'll be honest."

Ariana Grande broke three toes after slipping in urine.
The 20-year-old songbird suffered an agonising injury over the weekend when her pet dog Ophelia urinated on the floor and she accidentally walked right into the puddle.
Ariana's manager Scooter Braun - who also oversees Justin Bieber's career - shared the star's embarrassing moment by posting a screenshot of her text message on Instagram.
She wrote in the text: "Had a party last night and in front of a lot of people I slipped on a huge surprise pile of Ophelia piss, slammed into a wall and broke 3 toes. Have a great day."
Scooter captioned the screenshot, writing: "Monday morning texts from @arianagrande are always the best. Lol. Gotta love this girl. Feel better Ari (sic)"
Despite breaking her toes, 'The Way' hitmaker said she had an amazing weekend and even treated herself to a new tattoo.
Ariana wrote on Twitter: "had the best weekend I've had in forever minus my minor foot injury lol...
"a lot of my friends got their 1st tats last night at my house, it was such a bonding experience thank u @romeolacoste for the beautiful art (sic)"
Quizzed if she got a new tattoo, Ariana wrote: "yes I got 'bellissima' small on my side. it's what grandpa's called me my whole life (sic)"

Kelly Brook has celebrated her engagement to "the most wonderful man".
The 34-year-old beauty is set to walk down the aisle with former 'Gladiators' star David McIntosh following a whirlwind romance, which began on New Year's Eve, and she has shared the big news with fans by posting a picture of the pair holding a balloon reading 'Engaged' on Instagram.
Kelly and David - a former Royal Marines commando - are all smiles in the snap as David puts his arm around his stunning bride-to-be.
In the caption, Kelly gushed about meeting her beau's family during a recent trip to his hometown of Wigan.
She wrote: "Lovely Weekend in Wigan meeting my New extended Family Such happy times and excited to share my life with the most wonderful man I've ever met (sic)"
Kelly and David have had an on/off relationship since they met on New Year's Eve three months ago.
The burly hunk is thought to have spent large amounts of time at Kelly's north London home in the early stages of their romance, but the pair reportedly split shortly after Valentine's Day in February - only to get back together.
A source recently said of the engagement: "It's still a closely guarded secret and came to Kelly as a massive shock.
"But she is at that time in her life now where she wants to make a commitment and she has fallen head over heals for David.
"He has also had enough of being a bachelor and can't believe he has landed someone like her."

Amy Poehler watches a new film with her kids every Thanksgiving.
The 42-year-old actress - who has five-year-old son Archibald and three-year-old Abel with actor Will Arnett - celebrated the last festive holiday, which fell November 28, while watching 1971's 'Willy Wonka & the Chocolate Factory' and has already started planning what the children will watch this year.
She said: "Well I started a new thing with my kids that we would watch a movie on thanksgiving that we've never seen before. That's always fun to show your kids movies they've never seen. Last year we watched the original 'Willie Wonka' on Thanksgiving, which is great. And so maybe this year we will watch 'Free Birds'."
The actress voices Jenny, a turkey, in 2013's 'Free Birds' and where she stars alongside Woody Harrelson who plays fellow bird Reggie.
Harrelson - who has daughters 20-year-old Deni, 17-year-old Zoe and seven-year-old Makani with Laura Louie - insists the animated comedy has become a favourite for his youngest child.
He added: "This the first movie I've done that my kids can watch, so uh ... even the 20-year-old. The weird thing was the other day I was with my seven-year-old rascal, Makani and asked her what her favourite movie was, not tryna lead her toward anything but really wanting to know and she goes 'Free Bird!' "
'Free Birds' is released on DVD, Blu-ray and Digital Download in the UK on March 24.