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Properties

Label 961.2.c.k
Level $961$
Weight $2$
Character orbit 961.c
Analytic conductor $7.674$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [961,2,Mod(439,961)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(961, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("961.439");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 961 = 31^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 961.c (of order \(3\), degree \(2\), not 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: \(7.67362363425\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 32 q^{4} - 8 q^{5} - 8 q^{7} + 24 q^{8} - 20 q^{9} - 20 q^{10} - 28 q^{14} - 32 q^{16} + 8 q^{18} - 16 q^{19} + 20 q^{20} - 12 q^{25} + 20 q^{28} + 48 q^{32} - 80 q^{33} + 112 q^{35} - 40 q^{36}+ \cdots - 4 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
439.1 −2.13825 −1.45242 + 2.51567i 2.57209 0.648062 + 1.12248i 3.10564 − 5.37913i 1.35517 − 2.34722i −1.22328 −2.71907 − 4.70958i −1.38572 − 2.40013i
439.2 −2.13825 1.45242 − 2.51567i 2.57209 0.648062 + 1.12248i −3.10564 + 5.37913i 1.35517 − 2.34722i −1.22328 −2.71907 − 4.70958i −1.38572 − 2.40013i
439.3 −1.80264 −0.0954837 + 0.165383i 1.24951 1.23318 + 2.13593i 0.172123 − 0.298125i 0.526075 − 0.911189i 1.35286 1.48177 + 2.56649i −2.22298 − 3.85032i
439.4 −1.80264 0.0954837 − 0.165383i 1.24951 1.23318 + 2.13593i −0.172123 + 0.298125i 0.526075 − 0.911189i 1.35286 1.48177 + 2.56649i −2.22298 − 3.85032i
439.5 −1.24512 −1.33932 + 2.31977i −0.449667 −1.30938 − 2.26791i 1.66762 − 2.88840i −0.602271 + 1.04316i 3.05014 −2.08754 − 3.61573i 1.63034 + 2.82383i
439.6 −1.24512 1.33932 − 2.31977i −0.449667 −1.30938 − 2.26791i −1.66762 + 2.88840i −0.602271 + 1.04316i 3.05014 −2.08754 − 3.61573i 1.63034 + 2.82383i
439.7 0.720616 −0.949328 + 1.64428i −1.48071 −1.39356 − 2.41371i −0.684101 + 1.18490i −2.10066 + 3.63846i −2.50826 −0.302446 − 0.523851i −1.00422 − 1.73936i
439.8 0.720616 0.949328 − 1.64428i −1.48071 −1.39356 − 2.41371i 0.684101 − 1.18490i −2.10066 + 3.63846i −2.50826 −0.302446 − 0.523851i −1.00422 − 1.73936i
439.9 1.96916 −0.810772 + 1.40430i 1.87757 −1.75290 − 3.03611i −1.59654 + 2.76528i −1.04579 + 1.81136i −0.241077 0.185296 + 0.320943i −3.45173 − 5.97857i
439.10 1.96916 0.810772 − 1.40430i 1.87757 −1.75290 − 3.03611i 1.59654 − 2.76528i −1.04579 + 1.81136i −0.241077 0.185296 + 0.320943i −3.45173 − 5.97857i
439.11 2.49624 −1.23653 + 2.14173i 4.23120 0.574589 + 0.995217i −3.08667 + 5.34626i −0.132518 + 0.229528i 5.56961 −1.55800 − 2.69853i 1.43431 + 2.48430i
439.12 2.49624 1.23653 − 2.14173i 4.23120 0.574589 + 0.995217i 3.08667 − 5.34626i −0.132518 + 0.229528i 5.56961 −1.55800 − 2.69853i 1.43431 + 2.48430i
521.1 −2.13825 −1.45242 − 2.51567i 2.57209 0.648062 − 1.12248i 3.10564 + 5.37913i 1.35517 + 2.34722i −1.22328 −2.71907 + 4.70958i −1.38572 + 2.40013i
521.2 −2.13825 1.45242 + 2.51567i 2.57209 0.648062 − 1.12248i −3.10564 − 5.37913i 1.35517 + 2.34722i −1.22328 −2.71907 + 4.70958i −1.38572 + 2.40013i
521.3 −1.80264 −0.0954837 − 0.165383i 1.24951 1.23318 − 2.13593i 0.172123 + 0.298125i 0.526075 + 0.911189i 1.35286 1.48177 − 2.56649i −2.22298 + 3.85032i
521.4 −1.80264 0.0954837 + 0.165383i 1.24951 1.23318 − 2.13593i −0.172123 − 0.298125i 0.526075 + 0.911189i 1.35286 1.48177 − 2.56649i −2.22298 + 3.85032i
521.5 −1.24512 −1.33932 − 2.31977i −0.449667 −1.30938 + 2.26791i 1.66762 + 2.88840i −0.602271 − 1.04316i 3.05014 −2.08754 + 3.61573i 1.63034 − 2.82383i
521.6 −1.24512 1.33932 + 2.31977i −0.449667 −1.30938 + 2.26791i −1.66762 − 2.88840i −0.602271 − 1.04316i 3.05014 −2.08754 + 3.61573i 1.63034 − 2.82383i
521.7 0.720616 −0.949328 − 1.64428i −1.48071 −1.39356 + 2.41371i −0.684101 − 1.18490i −2.10066 − 3.63846i −2.50826 −0.302446 + 0.523851i −1.00422 + 1.73936i
521.8 0.720616 0.949328 + 1.64428i −1.48071 −1.39356 + 2.41371i 0.684101 + 1.18490i −2.10066 − 3.63846i −2.50826 −0.302446 + 0.523851i −1.00422 + 1.73936i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 439.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.b odd 2 1 inner
31.c even 3 1 inner
31.e odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 961.2.c.k 24
31.b odd 2 1 inner 961.2.c.k 24
31.c even 3 1 961.2.a.k ✓ 12
31.c even 3 1 inner 961.2.c.k 24
31.d even 5 4 961.2.g.v 96
31.e odd 6 1 961.2.a.k ✓ 12
31.e odd 6 1 inner 961.2.c.k 24
31.f odd 10 4 961.2.g.v 96
31.g even 15 4 961.2.d.r 48
31.g even 15 4 961.2.g.v 96
31.h odd 30 4 961.2.d.r 48
31.h odd 30 4 961.2.g.v 96
93.g even 6 1 8649.2.a.bp 12
93.h odd 6 1 8649.2.a.bp 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
961.2.a.k ✓ 12 31.c even 3 1
961.2.a.k ✓ 12 31.e odd 6 1
961.2.c.k 24 1.a even 1 1 trivial
961.2.c.k 24 31.b odd 2 1 inner
961.2.c.k 24 31.c even 3 1 inner
961.2.c.k 24 31.e odd 6 1 inner
961.2.d.r 48 31.g even 15 4
961.2.d.r 48 31.h odd 30 4
961.2.g.v 96 31.d even 5 4
961.2.g.v 96 31.f odd 10 4
961.2.g.v 96 31.g even 15 4
961.2.g.v 96 31.h odd 30 4
8649.2.a.bp 12 93.g even 6 1
8649.2.a.bp 12 93.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(961, [\chi])\):

\( T_{2}^{6} - 10T_{2}^{4} - 2T_{2}^{3} + 28T_{2}^{2} + 8T_{2} - 17 \) Copy content Toggle raw display
\( T_{3}^{24} + 28 T_{3}^{22} + 482 T_{3}^{20} + 5336 T_{3}^{18} + 43680 T_{3}^{16} + 259504 T_{3}^{14} + \cdots + 16384 \) Copy content Toggle raw display