Properties

Label 961.2.g.e
Level $961$
Weight $2$
Character orbit 961.g
Analytic conductor $7.674$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [961,2,Mod(235,961)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(961, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([26]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("961.235");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 961 = 31^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 961.g (of order \(15\), degree \(8\), 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: \(8\)
Coefficient field: \(\Q(\zeta_{15})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 31)
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-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 primitive root of unity \(\zeta_{15}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{15}^{6} + \zeta_{15}^{3} + 1) q^{2} + (2 \zeta_{15}^{7} - 2 \zeta_{15}^{5} + \cdots - 2) q^{3} + (\zeta_{15}^{7} - \zeta_{15}^{6} + \cdots - 1) q^{4} + \zeta_{15}^{5} q^{5} + ( - 2 \zeta_{15}^{5} - 2 \zeta_{15}^{4} + \cdots - 2) q^{6}+ \cdots + (2 \zeta_{15}^{5} + 8 \zeta_{15}^{4} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 12 q^{6} + 3 q^{7} + 10 q^{8} + 13 q^{9} - 2 q^{10} - 2 q^{11} + 2 q^{12} - 4 q^{13} - q^{14} + 8 q^{15} - 12 q^{16} + 2 q^{17} - 6 q^{18} + 5 q^{19} + 2 q^{20}+ \cdots + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(1 - \zeta_{15}^{2} + \zeta_{15}^{3} - \zeta_{15}^{4} + \zeta_{15}^{6} - \zeta_{15}^{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
235.1
0.669131 + 0.743145i
−0.978148 + 0.207912i
−0.104528 + 0.994522i
−0.104528 0.994522i
0.669131 0.743145i
−0.978148 0.207912i
0.913545 + 0.406737i
0.913545 0.406737i
0.500000 0.363271i −1.12920 + 0.502754i −0.500000 + 1.53884i −0.500000 0.866025i −0.381966 + 0.661585i −2.83448 3.14801i 0.690983 + 2.12663i −0.985051 + 1.09401i −0.564602 0.251377i
338.1 0.500000 0.363271i 0.129204 1.22930i −0.500000 + 1.53884i −0.500000 + 0.866025i −0.381966 0.661585i 4.14350 0.880728i 0.690983 + 2.12663i 1.43997 + 0.306074i 0.0646021 + 0.614648i
448.1 0.500000 1.53884i 2.16535 2.40487i −0.500000 0.363271i −0.500000 + 0.866025i −2.61803 4.53457i −0.0246758 + 0.234775i 1.80902 1.31433i −0.781051 7.43120i 1.08268 + 1.20243i
547.1 0.500000 + 1.53884i 2.16535 + 2.40487i −0.500000 + 0.363271i −0.500000 0.866025i −2.61803 + 4.53457i −0.0246758 0.234775i 1.80902 + 1.31433i −0.781051 + 7.43120i 1.08268 1.20243i
732.1 0.500000 + 0.363271i −1.12920 0.502754i −0.500000 1.53884i −0.500000 + 0.866025i −0.381966 0.661585i −2.83448 + 3.14801i 0.690983 2.12663i −0.985051 1.09401i −0.564602 + 0.251377i
816.1 0.500000 + 0.363271i 0.129204 + 1.22930i −0.500000 1.53884i −0.500000 0.866025i −0.381966 + 0.661585i 4.14350 + 0.880728i 0.690983 2.12663i 1.43997 0.306074i 0.0646021 0.614648i
844.1 0.500000 + 1.53884i −3.16535 + 0.672816i −0.500000 + 0.363271i −0.500000 + 0.866025i −2.61803 4.53457i 0.215659 + 0.0960175i 1.80902 + 1.31433i 6.82614 3.03919i −1.58268 0.336408i
846.1 0.500000 1.53884i −3.16535 0.672816i −0.500000 0.363271i −0.500000 0.866025i −2.61803 + 4.53457i 0.215659 0.0960175i 1.80902 1.31433i 6.82614 + 3.03919i −1.58268 + 0.336408i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 235.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.c even 3 1 inner
31.d even 5 1 inner
31.g even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 961.2.g.e 8
31.b odd 2 1 961.2.g.h 8
31.c even 3 1 961.2.d.g 4
31.c even 3 1 inner 961.2.g.e 8
31.d even 5 1 961.2.c.c 4
31.d even 5 2 961.2.g.d 8
31.d even 5 1 inner 961.2.g.e 8
31.e odd 6 1 961.2.d.d 4
31.e odd 6 1 961.2.g.h 8
31.f odd 10 1 961.2.c.e 4
31.f odd 10 2 961.2.g.a 8
31.f odd 10 1 961.2.g.h 8
31.g even 15 1 961.2.a.f 2
31.g even 15 1 961.2.c.c 4
31.g even 15 2 961.2.d.a 4
31.g even 15 1 961.2.d.g 4
31.g even 15 2 961.2.g.d 8
31.g even 15 1 inner 961.2.g.e 8
31.h odd 30 1 31.2.a.a 2
31.h odd 30 1 961.2.c.e 4
31.h odd 30 2 961.2.d.c 4
31.h odd 30 1 961.2.d.d 4
31.h odd 30 2 961.2.g.a 8
31.h odd 30 1 961.2.g.h 8
93.o odd 30 1 8649.2.a.c 2
93.p even 30 1 279.2.a.a 2
124.p even 30 1 496.2.a.i 2
155.v odd 30 1 775.2.a.d 2
155.x even 60 2 775.2.b.d 4
217.be even 30 1 1519.2.a.a 2
248.bb even 30 1 1984.2.a.n 2
248.bf odd 30 1 1984.2.a.r 2
341.bu even 30 1 3751.2.a.b 2
372.bc odd 30 1 4464.2.a.bf 2
403.by odd 30 1 5239.2.a.f 2
465.bm even 30 1 6975.2.a.y 2
527.bb odd 30 1 8959.2.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.2.a.a 2 31.h odd 30 1
279.2.a.a 2 93.p even 30 1
496.2.a.i 2 124.p even 30 1
775.2.a.d 2 155.v odd 30 1
775.2.b.d 4 155.x even 60 2
961.2.a.f 2 31.g even 15 1
961.2.c.c 4 31.d even 5 1
961.2.c.c 4 31.g even 15 1
961.2.c.e 4 31.f odd 10 1
961.2.c.e 4 31.h odd 30 1
961.2.d.a 4 31.g even 15 2
961.2.d.c 4 31.h odd 30 2
961.2.d.d 4 31.e odd 6 1
961.2.d.d 4 31.h odd 30 1
961.2.d.g 4 31.c even 3 1
961.2.d.g 4 31.g even 15 1
961.2.g.a 8 31.f odd 10 2
961.2.g.a 8 31.h odd 30 2
961.2.g.d 8 31.d even 5 2
961.2.g.d 8 31.g even 15 2
961.2.g.e 8 1.a even 1 1 trivial
961.2.g.e 8 31.c even 3 1 inner
961.2.g.e 8 31.d even 5 1 inner
961.2.g.e 8 31.g even 15 1 inner
961.2.g.h 8 31.b odd 2 1
961.2.g.h 8 31.e odd 6 1
961.2.g.h 8 31.f odd 10 1
961.2.g.h 8 31.h odd 30 1
1519.2.a.a 2 217.be even 30 1
1984.2.a.n 2 248.bb even 30 1
1984.2.a.r 2 248.bf odd 30 1
3751.2.a.b 2 341.bu even 30 1
4464.2.a.bf 2 372.bc odd 30 1
5239.2.a.f 2 403.by odd 30 1
6975.2.a.y 2 465.bm even 30 1
8649.2.a.c 2 93.o odd 30 1
8959.2.a.b 2 527.bb odd 30 1

Hecke kernels

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

\( T_{2}^{4} - 2T_{2}^{3} + 4T_{2}^{2} - 3T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{8} + 4T_{3}^{7} + 16T_{3}^{5} + 144T_{3}^{4} + 256T_{3}^{3} + 320T_{3}^{2} + 384T_{3} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + 4 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} - 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{8} + 2 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( T^{8} + 4 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{8} - 2 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( T^{8} - 5 T^{7} + \cdots + 625 \) Copy content Toggle raw display
$23$ \( (T^{4} - 14 T^{3} + \cdots + 1936)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 10 T^{3} + \cdots + 400)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{2} + 2 T + 4)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} - 7 T^{7} + \cdots + 5764801 \) Copy content Toggle raw display
$43$ \( T^{8} - 6 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$47$ \( (T^{4} - 12 T^{3} + \cdots + 256)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} - 16 T^{7} + \cdots + 65536 \) Copy content Toggle raw display
$59$ \( T^{8} + 5 T^{7} + \cdots + 625 \) Copy content Toggle raw display
$61$ \( (T^{2} - 6 T - 116)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 8 T + 64)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} + 23 T^{7} + \cdots + 214358881 \) Copy content Toggle raw display
$73$ \( T^{8} + 14 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$79$ \( T^{8} - 20 T^{7} + \cdots + 160000 \) Copy content Toggle raw display
$83$ \( T^{8} + 14 T^{7} + \cdots + 3748096 \) Copy content Toggle raw display
$89$ \( (T^{4} - 20 T^{3} + \cdots + 400)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 27 T^{3} + \cdots + 961)^{2} \) Copy content Toggle raw display
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