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Properties

Label 945.2.ch.b
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.

\(\operatorname{Tr}(f)(q) = \) \( 128 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q + 4 q^{7} + 4 q^{10} + 64 q^{16} + 24 q^{22} + 16 q^{25} - 52 q^{28} - 32 q^{31} + 16 q^{37} - 20 q^{40} - 32 q^{43} - 40 q^{52} - 16 q^{55} - 8 q^{58} + 32 q^{67} - 80 q^{70} - 56 q^{73} + 320 q^{76} - 148 q^{82} + 56 q^{85} - 12 q^{88} + 128 q^{91} + 104 q^{97}+O(q^{100}) \) 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 −0.708589 + 2.64449i 0 −4.75918 − 2.74771i 0.698486 − 2.12417i 0 2.13379 + 1.56426i 6.76681 − 6.76681i 0 5.12242 + 3.35231i
53.2 −0.694812 + 2.59308i 0 −4.50922 − 2.60340i −2.22843 + 0.184614i 0 −2.60561 + 0.459107i 6.08736 − 6.08736i 0 1.06963 − 5.90677i
53.3 −0.628565 + 2.34583i 0 −3.37580 − 1.94902i 2.02495 + 0.948457i 0 1.75355 − 1.98118i 3.25944 − 3.25944i 0 −3.49774 + 4.15404i
53.4 −0.622968 + 2.32495i 0 −3.28524 − 1.89673i −1.77418 + 1.36099i 0 1.53705 − 2.15348i 3.05245 − 3.05245i 0 −2.05897 − 4.97272i
53.5 −0.506990 + 1.89211i 0 −1.59100 − 0.918562i −0.473760 + 2.18530i 0 0.979667 + 2.45769i −0.225603 + 0.225603i 0 −3.89465 − 2.00433i
53.6 −0.506852 + 1.89160i 0 −1.58919 − 0.917522i −0.841797 − 2.07156i 0 0.501356 − 2.59781i −0.228422 + 0.228422i 0 4.34523 − 0.542364i
53.7 −0.477798 + 1.78316i 0 −1.21934 − 0.703984i 1.47398 − 1.68148i 0 −2.41909 − 1.07144i −0.772820 + 0.772820i 0 2.29409 + 3.43176i
53.8 −0.452182 + 1.68757i 0 −0.911360 − 0.526174i 1.98667 + 1.02622i 0 1.39810 + 2.24618i −1.17071 + 1.17071i 0 −2.63016 + 2.88860i
53.9 −0.428557 + 1.59940i 0 −0.642353 − 0.370863i −2.18849 − 0.458818i 0 −1.25029 + 2.33169i −1.47324 + 1.47324i 0 1.67172 − 3.30363i
53.10 −0.317174 + 1.18371i 0 0.431485 + 0.249118i 1.81951 − 1.29976i 0 −2.59807 − 0.500031i −2.16481 + 2.16481i 0 0.961438 + 2.56602i
53.11 −0.298189 + 1.11286i 0 0.582520 + 0.336318i 0.366384 + 2.20585i 0 −2.64573 + 0.0113310i −2.17731 + 2.17731i 0 −2.56404 − 0.250027i
53.12 −0.279918 + 1.04467i 0 0.719075 + 0.415158i −0.367526 + 2.20566i 0 0.107546 − 2.64356i −2.16448 + 2.16448i 0 −2.20130 − 1.00134i
53.13 −0.173627 + 0.647984i 0 1.34231 + 0.774985i −0.451349 − 2.19004i 0 2.54132 − 0.736014i −1.68395 + 1.68395i 0 1.49748 + 0.0877829i
53.14 −0.128796 + 0.480674i 0 1.51759 + 0.876182i 2.01956 − 0.959880i 0 2.11715 + 1.58672i −1.32037 + 1.32037i 0 0.201278 + 1.09438i
53.15 −0.0460483 + 0.171854i 0 1.70464 + 0.984173i −2.22262 + 0.244826i 0 −0.529315 − 2.59226i −0.499242 + 0.499242i 0 0.0602735 − 0.393242i
53.16 −0.0162119 + 0.0605037i 0 1.72865 + 0.998038i −1.53448 − 1.62646i 0 −1.38744 + 2.25278i −0.176993 + 0.176993i 0 0.123284 − 0.0664740i
53.17 0.0162119 − 0.0605037i 0 1.72865 + 0.998038i 1.53448 + 1.62646i 0 −1.38744 + 2.25278i 0.176993 − 0.176993i 0 0.123284 − 0.0664740i
53.18 0.0460483 − 0.171854i 0 1.70464 + 0.984173i 2.22262 − 0.244826i 0 −0.529315 − 2.59226i 0.499242 − 0.499242i 0 0.0602735 − 0.393242i
53.19 0.128796 − 0.480674i 0 1.51759 + 0.876182i −2.01956 + 0.959880i 0 2.11715 + 1.58672i 1.32037 − 1.32037i 0 0.201278 + 1.09438i
53.20 0.173627 − 0.647984i 0 1.34231 + 0.774985i 0.451349 + 2.19004i 0 2.54132 − 0.736014i 1.68395 − 1.68395i 0 1.49748 + 0.0877829i
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.b ✓ 128
3.b odd 2 1 inner 945.2.ch.b ✓ 128
5.c odd 4 1 inner 945.2.ch.b ✓ 128
7.c even 3 1 inner 945.2.ch.b ✓ 128
15.e even 4 1 inner 945.2.ch.b ✓ 128
21.h odd 6 1 inner 945.2.ch.b ✓ 128
35.l odd 12 1 inner 945.2.ch.b ✓ 128
105.x even 12 1 inner 945.2.ch.b ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.ch.b ✓ 128 1.a even 1 1 trivial
945.2.ch.b ✓ 128 3.b odd 2 1 inner
945.2.ch.b ✓ 128 5.c odd 4 1 inner
945.2.ch.b ✓ 128 7.c even 3 1 inner
945.2.ch.b ✓ 128 15.e even 4 1 inner
945.2.ch.b ✓ 128 21.h odd 6 1 inner
945.2.ch.b ✓ 128 35.l odd 12 1 inner
945.2.ch.b ✓ 128 105.x even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{128} - 240 T_{2}^{124} + 33256 T_{2}^{120} - 3102416 T_{2}^{116} + 216478932 T_{2}^{112} + \cdots + 390625 \) acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\). Copy content Toggle raw display