Properties

Label 961.2.g.d
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{15}^{6} - 1) q^{2} + (2 \zeta_{15}^{5} - 2 \zeta_{15}^{4} + 2) q^{3} + ( - \zeta_{15}^{7} - \zeta_{15}^{2} + 1) q^{4} + \zeta_{15}^{5} q^{5} + ( - 4 \zeta_{15}^{5} + 2 \zeta_{15}^{4} + \cdots - 4) q^{6}+ \cdots + ( - \zeta_{15}^{7} + 5 \zeta_{15}^{6} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{15}^{6} - 1) q^{2} + (2 \zeta_{15}^{5} - 2 \zeta_{15}^{4} + 2) q^{3} + ( - \zeta_{15}^{7} - \zeta_{15}^{2} + 1) q^{4} + \zeta_{15}^{5} q^{5} + ( - 4 \zeta_{15}^{5} + 2 \zeta_{15}^{4} + \cdots - 4) q^{6}+ \cdots + (10 \zeta_{15}^{5} - 8 \zeta_{15}^{4} + \cdots + 10) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{2} + 6 q^{3} + 6 q^{4} - 4 q^{5} - 12 q^{6} - 7 q^{7} - 10 q^{8} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 6 q^{2} + 6 q^{3} + 6 q^{4} - 4 q^{5} - 12 q^{6} - 7 q^{7} - 10 q^{8} - 7 q^{9} + 3 q^{10} - 2 q^{11} + 2 q^{12} + 6 q^{13} + 4 q^{14} - 12 q^{15} + 18 q^{16} - 8 q^{17} + 19 q^{18} - 5 q^{19} - 3 q^{20} - 2 q^{21} + 4 q^{22} - 32 q^{23} + 16 q^{25} - 12 q^{26} - 4 q^{28} + 24 q^{30} - 36 q^{32} + 24 q^{33} - 4 q^{34} + 14 q^{35} - 14 q^{36} - 8 q^{37} + 8 q^{39} + 5 q^{40} + 7 q^{41} - 6 q^{42} - 4 q^{43} - 4 q^{44} - 7 q^{45} + 24 q^{46} - 16 q^{47} + 6 q^{48} - 18 q^{49} - 12 q^{50} + 12 q^{51} + 2 q^{52} - 4 q^{53} + 40 q^{54} - 2 q^{55} + 20 q^{56} + 20 q^{57} - 20 q^{58} + 5 q^{59} - 4 q^{60} + 24 q^{61} + 32 q^{63} - 14 q^{64} + 6 q^{65} - 8 q^{66} - 32 q^{67} - 16 q^{68} - 4 q^{69} - 8 q^{70} + 27 q^{71} - 25 q^{72} + 6 q^{73} + 6 q^{74} - 24 q^{75} - 5 q^{76} + 12 q^{77} - 56 q^{78} - 10 q^{79} - 9 q^{80} + 41 q^{81} - 14 q^{82} + 26 q^{83} - 14 q^{84} + 16 q^{85} - 2 q^{86} - 20 q^{89} + 19 q^{90} + 4 q^{91} - 64 q^{92} + 32 q^{94} + 10 q^{95} - 22 q^{96} - 26 q^{97} + 36 q^{98} + 24 q^{99}+O(q^{100}) \) 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
−1.30902 + 0.951057i 2.95630 1.31623i 0.190983 0.587785i −0.500000 0.866025i −2.61803 + 4.53457i 0.157960 + 0.175433i −0.690983 2.12663i 4.99983 5.55288i 1.47815 + 0.658114i
338.1 −1.30902 + 0.951057i −0.338261 + 3.21834i 0.190983 0.587785i −0.500000 + 0.866025i −2.61803 4.53457i −0.230909 + 0.0490813i −0.690983 2.12663i −7.30885 1.55354i −0.169131 1.60917i
448.1 −0.190983 + 0.587785i −0.827091 + 0.918578i 1.30902 + 0.951057i −0.500000 + 0.866025i −0.381966 0.661585i 0.442790 4.21286i −1.80902 + 1.31433i 0.153880 + 1.46407i −0.413545 0.459289i
547.1 −0.190983 0.587785i −0.827091 0.918578i 1.30902 0.951057i −0.500000 0.866025i −0.381966 + 0.661585i 0.442790 + 4.21286i −1.80902 1.31433i 0.153880 1.46407i −0.413545 + 0.459289i
732.1 −1.30902 0.951057i 2.95630 + 1.31623i 0.190983 + 0.587785i −0.500000 + 0.866025i −2.61803 4.53457i 0.157960 0.175433i −0.690983 + 2.12663i 4.99983 + 5.55288i 1.47815 0.658114i
816.1 −1.30902 0.951057i −0.338261 3.21834i 0.190983 + 0.587785i −0.500000 0.866025i −2.61803 + 4.53457i −0.230909 0.0490813i −0.690983 + 2.12663i −7.30885 + 1.55354i −0.169131 + 1.60917i
844.1 −0.190983 0.587785i 1.20906 0.256993i 1.30902 0.951057i −0.500000 + 0.866025i −0.381966 0.661585i −3.86984 1.72296i −1.80902 1.31433i −1.34486 + 0.598772i 0.604528 + 0.128496i
846.1 −0.190983 + 0.587785i 1.20906 + 0.256993i 1.30902 + 0.951057i −0.500000 0.866025i −0.381966 + 0.661585i −3.86984 + 1.72296i −1.80902 + 1.31433i −1.34486 0.598772i 0.604528 0.128496i
\(n\): 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.d 8
31.b odd 2 1 961.2.g.a 8
31.c even 3 1 961.2.d.a 4
31.c even 3 1 inner 961.2.g.d 8
31.d even 5 1 961.2.c.c 4
31.d even 5 1 inner 961.2.g.d 8
31.d even 5 2 961.2.g.e 8
31.e odd 6 1 961.2.d.c 4
31.e odd 6 1 961.2.g.a 8
31.f odd 10 1 961.2.c.e 4
31.f odd 10 1 961.2.g.a 8
31.f odd 10 2 961.2.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 1 961.2.d.a 4
31.g even 15 2 961.2.d.g 4
31.g even 15 1 inner 961.2.g.d 8
31.g even 15 2 961.2.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 1 961.2.d.c 4
31.h odd 30 2 961.2.d.d 4
31.h odd 30 1 961.2.g.a 8
31.h odd 30 2 961.2.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.c even 3 1
961.2.d.a 4 31.g even 15 1
961.2.d.c 4 31.e odd 6 1
961.2.d.c 4 31.h odd 30 1
961.2.d.d 4 31.h odd 30 2
961.2.d.g 4 31.g even 15 2
961.2.g.a 8 31.b odd 2 1
961.2.g.a 8 31.e odd 6 1
961.2.g.a 8 31.f odd 10 1
961.2.g.a 8 31.h odd 30 1
961.2.g.d 8 1.a even 1 1 trivial
961.2.g.d 8 31.c even 3 1 inner
961.2.g.d 8 31.d even 5 1 inner
961.2.g.d 8 31.g even 15 1 inner
961.2.g.e 8 31.d even 5 2
961.2.g.e 8 31.g even 15 2
961.2.g.h 8 31.f odd 10 2
961.2.g.h 8 31.h odd 30 2
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 \(S_{2}^{\mathrm{new}}(961, [\chi])\):

\( T_{2}^{4} + 3T_{2}^{3} + 4T_{2}^{2} + 2T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{8} - 6T_{3}^{7} + 20T_{3}^{6} - 64T_{3}^{5} + 144T_{3}^{4} - 64T_{3}^{3} - 256T_{3} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 3 T^{3} + 4 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 6 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} + 7 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{8} + 2 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( T^{8} - 6 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{8} + 8 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( T^{8} + 5 T^{7} + \cdots + 625 \) Copy content Toggle raw display
$23$ \( (T^{4} + 16 T^{3} + \cdots + 1936)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 40 T^{2} + \cdots + 400)^{2} \) 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
$43$ \( T^{8} + 4 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$47$ \( (T^{4} + 8 T^{3} + \cdots + 256)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 4 T^{7} + \cdots + 65536 \) 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
$71$ \( T^{8} - 27 T^{7} + \cdots + 214358881 \) Copy content Toggle raw display
$73$ \( T^{8} - 6 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$79$ \( T^{8} + 10 T^{7} + \cdots + 160000 \) Copy content Toggle raw display
$83$ \( T^{8} - 26 T^{7} + \cdots + 3748096 \) Copy content Toggle raw display
$89$ \( (T^{4} + 10 T^{3} + \cdots + 400)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 13 T^{3} + \cdots + 961)^{2} \) Copy content Toggle raw display
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