Properties

Label 961.2.d.j
Level $961$
Weight $2$
Character orbit 961.d
Analytic conductor $7.674$
Analytic rank $0$
Dimension $8$
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(374,961)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(961, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("961.374");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 961 = 31^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 961.d (of order \(5\), degree \(4\), 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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.64000000.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 2x^{6} + 4x^{4} + 8x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} + 2 \beta_{2} + 1) q^{2} + (\beta_{5} + \beta_1) q^{3} + (3 \beta_{6} + 3 \beta_{4} + 3 \beta_{2}) q^{4} + ( - 2 \beta_{6} - 2 \beta_{4} - 1) q^{5} + (\beta_{7} + \beta_{5} + \beta_{3}) q^{6} - \beta_{4} q^{7} + (\beta_{6} - 4 \beta_{2} - 4) q^{8} + (\beta_{6} + 2 \beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} + 2 \beta_{2} + 1) q^{2} + (\beta_{5} + \beta_1) q^{3} + (3 \beta_{6} + 3 \beta_{4} + 3 \beta_{2}) q^{4} + ( - 2 \beta_{6} - 2 \beta_{4} - 1) q^{5} + (\beta_{7} + \beta_{5} + \beta_{3}) q^{6} - \beta_{4} q^{7} + (\beta_{6} - 4 \beta_{2} - 4) q^{8} + (\beta_{6} + 2 \beta_{2} + 2) q^{9} + (3 \beta_{4} + 4 \beta_{2} + 3) q^{10} + (3 \beta_{7} + 3 \beta_{5} + \cdots + 3 \beta_1) q^{11}+ \cdots + (6 \beta_{7} + 3 \beta_{5} + 6 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} - 18 q^{4} + 2 q^{7} - 26 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} - 18 q^{4} + 2 q^{7} - 26 q^{8} + 10 q^{9} + 10 q^{10} + 8 q^{14} + 2 q^{16} - 10 q^{18} - 2 q^{19} - 30 q^{20} - 12 q^{28} + 60 q^{32} - 36 q^{33} + 10 q^{35} - 60 q^{36} - 8 q^{38} + 12 q^{39} - 50 q^{40} - 26 q^{41} + 10 q^{45} - 24 q^{47} + 12 q^{49} + 12 q^{51} - 24 q^{56} + 34 q^{59} + 26 q^{64} + 36 q^{66} + 48 q^{67} + 12 q^{69} + 20 q^{70} + 14 q^{71} - 10 q^{72} - 18 q^{76} - 12 q^{78} + 50 q^{80} + 38 q^{81} + 16 q^{82} - 32 q^{87} - 10 q^{90} - 96 q^{94} + 10 q^{95} + 14 q^{97} - 72 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 2x^{6} + 4x^{4} + 8x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{7} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/961\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
374.1
−0.437016 + 1.34500i
0.437016 1.34500i
−0.437016 1.34500i
0.437016 + 1.34500i
1.14412 + 0.831254i
−1.14412 0.831254i
1.14412 0.831254i
−1.14412 + 0.831254i
−0.309017 0.224514i −1.85123 + 1.34500i −0.572949 1.76336i −2.23607 0.874032 −0.309017 0.951057i −0.454915 + 1.40008i 0.690983 2.12663i 0.690983 + 0.502029i
374.2 −0.309017 0.224514i 1.85123 1.34500i −0.572949 1.76336i −2.23607 −0.874032 −0.309017 0.951057i −0.454915 + 1.40008i 0.690983 2.12663i 0.690983 + 0.502029i
388.1 −0.309017 + 0.224514i −1.85123 1.34500i −0.572949 + 1.76336i −2.23607 0.874032 −0.309017 + 0.951057i −0.454915 1.40008i 0.690983 + 2.12663i 0.690983 0.502029i
388.2 −0.309017 + 0.224514i 1.85123 + 1.34500i −0.572949 + 1.76336i −2.23607 −0.874032 −0.309017 + 0.951057i −0.454915 1.40008i 0.690983 + 2.12663i 0.690983 0.502029i
531.1 0.809017 + 2.48990i −0.270091 + 0.831254i −3.92705 + 2.85317i 2.23607 −2.28825 0.809017 0.587785i −6.04508 4.39201i 1.80902 + 1.31433i 1.80902 + 5.56758i
531.2 0.809017 + 2.48990i 0.270091 0.831254i −3.92705 + 2.85317i 2.23607 2.28825 0.809017 0.587785i −6.04508 4.39201i 1.80902 + 1.31433i 1.80902 + 5.56758i
628.1 0.809017 2.48990i −0.270091 0.831254i −3.92705 2.85317i 2.23607 −2.28825 0.809017 + 0.587785i −6.04508 + 4.39201i 1.80902 1.31433i 1.80902 5.56758i
628.2 0.809017 2.48990i 0.270091 + 0.831254i −3.92705 2.85317i 2.23607 2.28825 0.809017 + 0.587785i −6.04508 + 4.39201i 1.80902 1.31433i 1.80902 5.56758i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 374.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.b odd 2 1 inner
31.d even 5 1 inner
31.f odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 961.2.d.j 8
31.b odd 2 1 inner 961.2.d.j 8
31.c even 3 2 961.2.g.p 16
31.d even 5 1 961.2.a.h 4
31.d even 5 2 961.2.d.h 8
31.d even 5 1 inner 961.2.d.j 8
31.e odd 6 2 961.2.g.p 16
31.f odd 10 1 961.2.a.h 4
31.f odd 10 2 961.2.d.h 8
31.f odd 10 1 inner 961.2.d.j 8
31.g even 15 2 961.2.c.h 8
31.g even 15 4 961.2.g.i 16
31.g even 15 2 961.2.g.p 16
31.h odd 30 2 961.2.c.h 8
31.h odd 30 4 961.2.g.i 16
31.h odd 30 2 961.2.g.p 16
93.k even 10 1 8649.2.a.r 4
93.l odd 10 1 8649.2.a.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
961.2.a.h 4 31.d even 5 1
961.2.a.h 4 31.f odd 10 1
961.2.c.h 8 31.g even 15 2
961.2.c.h 8 31.h odd 30 2
961.2.d.h 8 31.d even 5 2
961.2.d.h 8 31.f odd 10 2
961.2.d.j 8 1.a even 1 1 trivial
961.2.d.j 8 31.b odd 2 1 inner
961.2.d.j 8 31.d even 5 1 inner
961.2.d.j 8 31.f odd 10 1 inner
961.2.g.i 16 31.g even 15 4
961.2.g.i 16 31.h odd 30 4
961.2.g.p 16 31.c even 3 2
961.2.g.p 16 31.e odd 6 2
961.2.g.p 16 31.g even 15 2
961.2.g.p 16 31.h odd 30 2
8649.2.a.r 4 93.k even 10 1
8649.2.a.r 4 93.l odd 10 1

Hecke kernels

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

\( T_{2}^{4} - T_{2}^{3} + 6T_{2}^{2} + 4T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{8} - 2T_{3}^{6} + 24T_{3}^{4} + 32T_{3}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{3} + 6 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( (T^{2} - 5)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 18 T^{6} + \cdots + 104976 \) Copy content Toggle raw display
$13$ \( T^{8} - 18 T^{6} + \cdots + 104976 \) Copy content Toggle raw display
$17$ \( T^{8} - 8 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 72 T^{6} + \cdots + 104976 \) Copy content Toggle raw display
$29$ \( T^{8} - 8 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{2} - 18)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 13 T^{3} + \cdots + 121)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 152 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$47$ \( (T^{4} + 12 T^{3} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} - 72 T^{6} + \cdots + 26873856 \) Copy content Toggle raw display
$59$ \( (T^{4} - 17 T^{3} + \cdots + 5041)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 214 T^{2} + 3844)^{2} \) Copy content Toggle raw display
$67$ \( (T - 6)^{8} \) Copy content Toggle raw display
$71$ \( (T^{4} - 7 T^{3} + \cdots + 121)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 18 T^{6} + \cdots + 104976 \) Copy content Toggle raw display
$79$ \( T^{8} + 198 T^{6} + \cdots + 104976 \) Copy content Toggle raw display
$83$ \( T^{8} + 10 T^{6} + \cdots + 10000 \) Copy content Toggle raw display
$89$ \( T^{8} - 90 T^{6} + \cdots + 65610000 \) Copy content Toggle raw display
$97$ \( (T^{4} - 7 T^{3} + \cdots + 2401)^{2} \) Copy content Toggle raw display
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