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Properties

Label 912.2.bv.c
Level $912$
Weight $2$
Character orbit 912.bv
Analytic conductor $7.282$
Analytic rank $0$
Dimension $608$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [912,2,Mod(11,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 3, 6, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.bv (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: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(608\)
Relative dimension: \(152\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 608 q - 4 q^{4} - 2 q^{6} - 16 q^{12} - 4 q^{13} - 32 q^{16} - 40 q^{18} + 36 q^{19} - 12 q^{21} - 28 q^{22} - 10 q^{24} + 24 q^{27} - 20 q^{28} + 40 q^{30} - 20 q^{33} + 4 q^{34} - 18 q^{36} + 48 q^{37}+ \cdots - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −1.41325 − 0.0521599i 0.699316 + 1.58460i 1.99456 + 0.147430i 0.577533 + 2.15538i −0.905656 − 2.27591i 4.44372 −2.81112 − 0.312392i −2.02192 + 2.21627i −0.703775 − 3.07622i
11.2 −1.41247 − 0.0701125i −1.69999 + 0.331725i 1.99017 + 0.198064i 0.0854603 + 0.318942i 2.42445 − 0.349363i −3.04578 −2.79718 − 0.419297i 2.77992 − 1.12786i −0.0983486 − 0.456490i
11.3 −1.41174 + 0.0836355i 1.03275 − 1.39048i 1.98601 − 0.236143i −0.452278 − 1.68793i −1.34167 + 2.04937i −5.09195 −2.78398 + 0.499473i −0.866869 − 2.87203i 0.779669 + 2.34508i
11.4 −1.41062 − 0.100751i 1.64144 + 0.552886i 1.97970 + 0.284244i −0.347290 − 1.29610i −2.25974 − 0.945289i −1.18614 −2.76396 − 0.600417i 2.38863 + 1.81506i 0.359310 + 1.86330i
11.5 −1.41027 + 0.105527i −0.726126 + 1.57250i 1.97773 − 0.297644i −0.0937034 − 0.349706i 0.858093 − 2.29427i −2.52744 −2.75772 + 0.628463i −1.94548 − 2.28366i 0.169051 + 0.483292i
11.6 −1.39959 + 0.202878i −0.897802 − 1.48120i 1.91768 − 0.567890i 0.234322 + 0.874501i 1.55705 + 1.89092i 0.735407 −2.56875 + 1.18387i −1.38790 + 2.65965i −0.505370 − 1.17640i
11.7 −1.39869 + 0.208982i −1.73093 − 0.0622327i 1.91265 − 0.584602i −0.531816 − 1.98477i 2.43404 − 0.274690i 3.80304 −2.55303 + 1.21739i 2.99225 + 0.215441i 1.15863 + 2.66493i
11.8 −1.39760 − 0.216167i 1.55969 − 0.753246i 1.90654 + 0.604227i 0.679508 + 2.53596i −2.34264 + 0.715581i 1.30191 −2.53396 − 1.25660i 1.86524 − 2.34965i −0.401487 − 3.69113i
11.9 −1.39600 + 0.226261i 1.38779 + 1.03636i 1.89761 − 0.631719i 0.788098 + 2.94122i −2.17184 − 1.13275i −0.530811 −2.50613 + 1.31123i 0.851927 + 2.87649i −1.76566 − 3.92762i
11.10 −1.38446 + 0.288582i 0.263043 + 1.71196i 1.83344 − 0.799059i −1.10737 − 4.13275i −0.858213 − 2.29423i 0.620227 −2.30772 + 1.63536i −2.86162 + 0.900637i 2.72574 + 5.40205i
11.11 −1.38088 − 0.305222i −0.716688 + 1.57682i 1.81368 + 0.842952i 0.335957 + 1.25381i 1.47094 − 1.95865i 0.956736 −2.24719 − 1.71759i −1.97272 − 2.26018i −0.0812276 − 1.83391i
11.12 −1.37426 + 0.333786i −1.60788 + 0.643978i 1.77717 − 0.917416i 1.09127 + 4.07268i 1.99470 − 1.42168i 2.40582 −2.13608 + 1.85396i 2.17058 − 2.07089i −2.85909 − 5.23267i
11.13 −1.36641 − 0.364580i −0.317697 − 1.70267i 1.73416 + 0.996333i −0.589439 − 2.19982i −0.186653 + 2.44237i −0.229056 −2.00634 − 1.99364i −2.79814 + 1.08186i 0.00340737 + 3.22075i
11.14 −1.36520 − 0.369090i 0.940671 − 1.45435i 1.72755 + 1.00776i −0.244185 − 0.911310i −1.82099 + 1.63829i 0.114597 −1.98649 − 2.01342i −1.23028 − 2.73613i −0.00299425 + 1.33425i
11.15 −1.36133 + 0.383106i 0.339860 − 1.69838i 1.70646 − 1.04307i 0.873190 + 3.25879i 0.187997 + 2.44226i −0.637912 −1.92345 + 2.07372i −2.76899 − 1.15442i −2.43716 − 4.10177i
11.16 −1.34949 − 0.422926i −1.52192 − 0.826889i 1.64227 + 1.14147i −0.957603 − 3.57382i 1.70412 + 1.75954i −1.67607 −1.73347 − 2.23497i 1.63251 + 2.51693i −0.219182 + 5.22785i
11.17 −1.34935 − 0.423382i −1.45472 − 0.940095i 1.64149 + 1.14258i 0.855012 + 3.19095i 1.56491 + 1.88442i −3.25979 −1.73120 − 2.23672i 1.23244 + 2.73516i 0.197280 − 4.66771i
11.18 −1.31577 + 0.518414i 1.65487 − 0.511284i 1.46249 − 1.36423i −0.517548 − 1.93152i −1.91237 + 1.53064i −0.0616876 −1.21707 + 2.55318i 2.47718 − 1.69222i 1.68230 + 2.27312i
11.19 −1.30480 − 0.545442i 1.63145 + 0.581694i 1.40499 + 1.42338i −0.871534 − 3.25261i −1.81143 − 1.64885i 4.63516 −1.05685 − 2.62356i 2.32326 + 1.89801i −0.636935 + 4.71936i
11.20 −1.30307 − 0.549560i −0.977266 − 1.43002i 1.39597 + 1.43223i 0.370929 + 1.38433i 0.487562 + 2.40048i 4.60728 −1.03194 − 2.63346i −1.08990 + 2.79502i 0.277426 − 2.00772i
See next 80 embeddings (of 608 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.152
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.f odd 4 1 inner
19.c even 3 1 inner
48.k even 4 1 inner
57.h odd 6 1 inner
304.y odd 12 1 inner
912.bv even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.bv.c ✓ 608
3.b odd 2 1 inner 912.2.bv.c ✓ 608
16.f odd 4 1 inner 912.2.bv.c ✓ 608
19.c even 3 1 inner 912.2.bv.c ✓ 608
48.k even 4 1 inner 912.2.bv.c ✓ 608
57.h odd 6 1 inner 912.2.bv.c ✓ 608
304.y odd 12 1 inner 912.2.bv.c ✓ 608
912.bv even 12 1 inner 912.2.bv.c ✓ 608
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.bv.c ✓ 608 1.a even 1 1 trivial
912.2.bv.c ✓ 608 3.b odd 2 1 inner
912.2.bv.c ✓ 608 16.f odd 4 1 inner
912.2.bv.c ✓ 608 19.c even 3 1 inner
912.2.bv.c ✓ 608 48.k even 4 1 inner
912.2.bv.c ✓ 608 57.h odd 6 1 inner
912.2.bv.c ✓ 608 304.y odd 12 1 inner
912.2.bv.c ✓ 608 912.bv even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{608} - 4850 T_{5}^{604} + 12224137 T_{5}^{600} - 21107987326 T_{5}^{596} + 27888840230520 T_{5}^{592} + \cdots + 52\!\cdots\!56 \) acting on \(S_{2}^{\mathrm{new}}(912, [\chi])\). Copy content Toggle raw display