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Properties

Label 1224.2.j.a
Level $1224$
Weight $2$
Character orbit 1224.j
Analytic conductor $9.774$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1224,2,Mod(35,1224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1224, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1224.35");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1224.j (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.77368920740\)
Analytic rank: \(0\)
Dimension: \(64\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 64 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 16 q^{10} + 24 q^{16} + 8 q^{22} + 64 q^{25} + 8 q^{28} + 24 q^{40} - 64 q^{43} - 32 q^{46} - 32 q^{49} + 56 q^{52} + 40 q^{58} + 72 q^{64} + 32 q^{67} + 40 q^{70} + 32 q^{73} - 40 q^{76} - 40 q^{82} - 80 q^{88} + 8 q^{94} - 64 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1 −1.40515 − 0.159827i 0 1.94891 + 0.449163i 0.0703373 0 − 4.02347i −2.66673 − 0.942632i 0 −0.0988346 − 0.0112418i
35.2 −1.40515 + 0.159827i 0 1.94891 − 0.449163i 0.0703373 0 4.02347i −2.66673 + 0.942632i 0 −0.0988346 + 0.0112418i
35.3 −1.40053 − 0.196264i 0 1.92296 + 0.549747i 3.84894 0 − 0.134897i −2.58527 − 1.14734i 0 −5.39055 − 0.755409i
35.4 −1.40053 + 0.196264i 0 1.92296 − 0.549747i 3.84894 0 0.134897i −2.58527 + 1.14734i 0 −5.39055 + 0.755409i
35.5 −1.39701 − 0.219909i 0 1.90328 + 0.614429i −4.13342 0 − 3.97831i −2.52379 − 1.27691i 0 5.77443 + 0.908974i
35.6 −1.39701 + 0.219909i 0 1.90328 − 0.614429i −4.13342 0 3.97831i −2.52379 + 1.27691i 0 5.77443 − 0.908974i
35.7 −1.39040 − 0.258436i 0 1.86642 + 0.718660i −1.44941 0 0.677387i −2.40934 − 1.48157i 0 2.01526 + 0.374580i
35.8 −1.39040 + 0.258436i 0 1.86642 − 0.718660i −1.44941 0 − 0.677387i −2.40934 + 1.48157i 0 2.01526 − 0.374580i
35.9 −1.31565 − 0.518724i 0 1.46185 + 1.36491i −1.17018 0 1.76028i −1.21526 − 2.55404i 0 1.53954 + 0.607001i
35.10 −1.31565 + 0.518724i 0 1.46185 − 1.36491i −1.17018 0 − 1.76028i −1.21526 + 2.55404i 0 1.53954 − 0.607001i
35.11 −1.24540 − 0.670061i 0 1.10204 + 1.66899i 0.596180 0 2.01866i −0.254152 − 2.81699i 0 −0.742482 − 0.399477i
35.12 −1.24540 + 0.670061i 0 1.10204 − 1.66899i 0.596180 0 − 2.01866i −0.254152 + 2.81699i 0 −0.742482 + 0.399477i
35.13 −1.12089 − 0.862321i 0 0.512805 + 1.93314i −3.57638 0 − 3.75208i 1.09219 − 2.60905i 0 4.00875 + 3.08399i
35.14 −1.12089 + 0.862321i 0 0.512805 − 1.93314i −3.57638 0 3.75208i 1.09219 + 2.60905i 0 4.00875 − 3.08399i
35.15 −1.02328 − 0.976168i 0 0.0941927 + 1.99778i 1.37455 0 4.53929i 1.85378 − 2.13623i 0 −1.40654 − 1.34179i
35.16 −1.02328 + 0.976168i 0 0.0941927 − 1.99778i 1.37455 0 − 4.53929i 1.85378 + 2.13623i 0 −1.40654 + 1.34179i
35.17 −0.891020 − 1.09822i 0 −0.412166 + 1.95707i 3.64894 0 − 1.71621i 2.51654 − 1.29114i 0 −3.25128 − 4.00733i
35.18 −0.891020 + 1.09822i 0 −0.412166 − 1.95707i 3.64894 0 1.71621i 2.51654 + 1.29114i 0 −3.25128 + 4.00733i
35.19 −0.769703 − 1.18641i 0 −0.815114 + 1.82636i 1.43325 0 0.898362i 2.79420 − 0.438699i 0 −1.10318 − 1.70042i
35.20 −0.769703 + 1.18641i 0 −0.815114 − 1.82636i 1.43325 0 − 0.898362i 2.79420 + 0.438699i 0 −1.10318 + 1.70042i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.64
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1224.2.j.a ✓ 64
3.b odd 2 1 inner 1224.2.j.a ✓ 64
4.b odd 2 1 4896.2.j.a 64
8.b even 2 1 4896.2.j.a 64
8.d odd 2 1 inner 1224.2.j.a ✓ 64
12.b even 2 1 4896.2.j.a 64
24.f even 2 1 inner 1224.2.j.a ✓ 64
24.h odd 2 1 4896.2.j.a 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1224.2.j.a ✓ 64 1.a even 1 1 trivial
1224.2.j.a ✓ 64 3.b odd 2 1 inner
1224.2.j.a ✓ 64 8.d odd 2 1 inner
1224.2.j.a ✓ 64 24.f even 2 1 inner
4896.2.j.a 64 4.b odd 2 1
4896.2.j.a 64 8.b even 2 1
4896.2.j.a 64 12.b even 2 1
4896.2.j.a 64 24.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1224, [\chi])\).