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Properties

Label 945.2.ch.a
Level $945$
Weight $2$
Character orbit 945.ch
Analytic conductor $7.546$
Analytic rank $0$
Dimension $128$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [945,2,Mod(53,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 9, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.53");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.ch (of order \(12\), degree \(4\), 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: \(7.54586299101\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(32\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 128 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q - 8 q^{7} - 8 q^{10} + 64 q^{16} - 8 q^{25} + 8 q^{28} + 16 q^{31} - 8 q^{37} + 40 q^{40} - 32 q^{43} + 80 q^{52} + 32 q^{55} + 16 q^{58} - 24 q^{61} - 16 q^{67} - 80 q^{70} + 64 q^{73} - 160 q^{76} - 40 q^{82} - 64 q^{85} - 48 q^{88} - 136 q^{91} - 208 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 −0.717338 + 2.67714i 0 −4.92046 − 2.84083i 1.18392 + 1.89693i 0 0.381154 + 2.61815i 7.21531 − 7.21531i 0 −5.92761 + 1.80879i
53.2 −0.650762 + 2.42868i 0 −3.74293 − 2.16098i −1.11718 − 1.93699i 0 −2.56343 + 0.654856i 4.12826 − 4.12826i 0 5.43133 − 1.45275i
53.3 −0.627864 + 2.34322i 0 −3.36441 − 1.94244i −1.79819 + 1.32910i 0 2.58407 + 0.567951i 3.23325 − 3.23325i 0 −1.98535 − 5.04805i
53.4 −0.623462 + 2.32679i 0 −3.29320 − 1.90133i 2.01489 + 0.969657i 0 −1.20080 − 2.35756i 3.07052 − 3.07052i 0 −3.51239 + 4.08367i
53.5 −0.565842 + 2.11175i 0 −2.40727 − 1.38984i 1.70473 − 1.44703i 0 2.32045 − 1.27104i 1.20531 − 1.20531i 0 2.09117 + 4.41875i
53.6 −0.531900 + 1.98508i 0 −1.92556 − 1.11172i 0.139150 − 2.23173i 0 −1.92995 + 1.80977i 0.324703 − 0.324703i 0 4.35615 + 1.46328i
53.7 −0.514993 + 1.92198i 0 −1.69674 − 0.979613i −1.29351 + 1.82396i 0 −1.89802 − 1.84323i −0.0573689 + 0.0573689i 0 −2.83947 − 3.42543i
53.8 −0.421942 + 1.57471i 0 −0.569622 − 0.328872i 2.16427 − 0.562093i 0 1.53488 + 2.15502i −1.54731 + 1.54731i 0 −0.0280624 + 3.64526i
53.9 −0.358510 + 1.33798i 0 0.0703965 + 0.0406434i −2.12920 − 0.683012i 0 2.63508 + 0.237385i −2.03855 + 2.03855i 0 1.67719 − 2.60396i
53.10 −0.356878 + 1.33189i 0 0.0854880 + 0.0493565i −1.93075 − 1.12792i 0 −0.915760 − 2.48221i −2.04627 + 2.04627i 0 2.19130 − 2.16901i
53.11 −0.245318 + 0.915538i 0 0.954021 + 0.550805i 2.02471 + 0.948966i 0 −2.27390 + 1.35255i −2.07876 + 2.07876i 0 −1.36551 + 1.62090i
53.12 −0.238087 + 0.888552i 0 0.999211 + 0.576895i −0.916077 − 2.03980i 0 0.919860 + 2.48070i −2.05143 + 2.05143i 0 2.03058 − 0.328332i
53.13 −0.196967 + 0.735092i 0 1.23049 + 0.710422i 0.232890 + 2.22391i 0 2.20081 − 1.46848i −1.84084 + 1.84084i 0 −1.68065 − 0.266841i
53.14 −0.136653 + 0.509995i 0 1.49063 + 0.860615i −1.13030 + 1.92936i 0 0.574059 + 2.58272i −1.38929 + 1.38929i 0 −0.829507 − 0.840098i
53.15 −0.132004 + 0.492647i 0 1.50677 + 0.869937i 2.16562 + 0.556843i 0 1.00545 − 2.44726i −1.34876 + 1.34876i 0 −0.560199 + 0.993383i
53.16 −0.0202033 + 0.0753998i 0 1.72677 + 0.996953i −1.91068 + 1.16160i 0 −2.64190 + 0.142720i −0.220450 + 0.220450i 0 −0.0489822 − 0.167533i
53.17 0.0202033 − 0.0753998i 0 1.72677 + 0.996953i 1.91068 − 1.16160i 0 −2.64190 + 0.142720i 0.220450 − 0.220450i 0 −0.0489822 − 0.167533i
53.18 0.132004 − 0.492647i 0 1.50677 + 0.869937i −2.16562 − 0.556843i 0 1.00545 − 2.44726i 1.34876 − 1.34876i 0 −0.560199 + 0.993383i
53.19 0.136653 − 0.509995i 0 1.49063 + 0.860615i 1.13030 − 1.92936i 0 0.574059 + 2.58272i 1.38929 − 1.38929i 0 −0.829507 − 0.840098i
53.20 0.196967 − 0.735092i 0 1.23049 + 0.710422i −0.232890 − 2.22391i 0 2.20081 − 1.46848i 1.84084 − 1.84084i 0 −1.68065 − 0.266841i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
7.c even 3 1 inner
15.e even 4 1 inner
21.h odd 6 1 inner
35.l odd 12 1 inner
105.x even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.ch.a ✓ 128
3.b odd 2 1 inner 945.2.ch.a ✓ 128
5.c odd 4 1 inner 945.2.ch.a ✓ 128
7.c even 3 1 inner 945.2.ch.a ✓ 128
15.e even 4 1 inner 945.2.ch.a ✓ 128
21.h odd 6 1 inner 945.2.ch.a ✓ 128
35.l odd 12 1 inner 945.2.ch.a ✓ 128
105.x even 12 1 inner 945.2.ch.a ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.ch.a ✓ 128 1.a even 1 1 trivial
945.2.ch.a ✓ 128 3.b odd 2 1 inner
945.2.ch.a ✓ 128 5.c odd 4 1 inner
945.2.ch.a ✓ 128 7.c even 3 1 inner
945.2.ch.a ✓ 128 15.e even 4 1 inner
945.2.ch.a ✓ 128 21.h odd 6 1 inner
945.2.ch.a ✓ 128 35.l odd 12 1 inner
945.2.ch.a ✓ 128 105.x even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{128} - 240 T_{2}^{124} + 33088 T_{2}^{120} - 3085844 T_{2}^{116} + 216040644 T_{2}^{112} + \cdots + 3906250000 \) acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\). Copy content Toggle raw display