Properties

Label 961.2.g.c
Level $961$
Weight $2$
Character orbit 961.g
Analytic conductor $7.674$
Analytic rank $0$
Dimension $8$
CM no
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: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{15}^{6} - 1) q^{2} + \zeta_{15}^{7} q^{3} + ( - \zeta_{15}^{7} - \zeta_{15}^{2} + 1) q^{4} + ( - \zeta_{15}^{7} - \zeta_{15}^{5} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} - \zeta_{15} + 1) q^{5} + ( - \zeta_{15}^{5} + \zeta_{15}^{4} + \zeta_{15} - 1) q^{6} + 3 \zeta_{15} q^{7} + ( - \zeta_{15}^{7} + \zeta_{15}^{6} - \zeta_{15}^{3} - \zeta_{15}^{2} - 1) q^{8} + (2 \zeta_{15}^{7} - 2 \zeta_{15}^{6} + 2 \zeta_{15}^{4} - 2 \zeta_{15}^{3} + 2 \zeta_{15}^{2} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{15}^{6} - 1) q^{2} + \zeta_{15}^{7} q^{3} + ( - \zeta_{15}^{7} - \zeta_{15}^{2} + 1) q^{4} + ( - \zeta_{15}^{7} - \zeta_{15}^{5} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} - \zeta_{15} + 1) q^{5} + ( - \zeta_{15}^{5} + \zeta_{15}^{4} + \zeta_{15} - 1) q^{6} + 3 \zeta_{15} q^{7} + ( - \zeta_{15}^{7} + \zeta_{15}^{6} - \zeta_{15}^{3} - \zeta_{15}^{2} - 1) q^{8} + (2 \zeta_{15}^{7} - 2 \zeta_{15}^{6} + 2 \zeta_{15}^{4} - 2 \zeta_{15}^{3} + 2 \zeta_{15}^{2} - 2) q^{9} + (\zeta_{15}^{7} - \zeta_{15}^{6} + \zeta_{15}^{4} - \zeta_{15}^{3} - 1) q^{10} + ( - 4 \zeta_{15}^{6} + 2 \zeta_{15}^{5} + 2 \zeta_{15}^{2} - 4 \zeta_{15}) q^{11} + (\zeta_{15}^{7} + \zeta_{15}^{4}) q^{12} + (3 \zeta_{15}^{7} - 3 \zeta_{15}^{6} - 3 \zeta_{15}^{5} + 3 \zeta_{15}^{4} - 3 \zeta_{15}^{3} - 3) q^{13} + ( - 3 \zeta_{15}^{7} - 3 \zeta_{15}) q^{14} + (\zeta_{15}^{7} + \zeta_{15}^{6} + \zeta_{15}^{3} + \zeta_{15}^{2}) q^{15} + (3 \zeta_{15}^{7} - 3 \zeta_{15}^{6} - 3 \zeta_{15}^{3} + 3 \zeta_{15}^{2}) q^{16} + ( - 2 \zeta_{15}^{7} + \zeta_{15}^{4} - 2 \zeta_{15}) q^{17} + ( - 2 \zeta_{15}^{7} + 2 \zeta_{15}^{6} + 2 \zeta_{15}^{5} - 2 \zeta_{15}^{4} + 2 \zeta_{15}^{3} - 2 \zeta_{15}^{2} + \cdots + 2) q^{18}+ \cdots + (8 \zeta_{15}^{5} - 4 \zeta_{15}^{4} - 4 \zeta_{15} + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{2} + q^{3} + 6 q^{4} + 6 q^{5} - 2 q^{6} + 3 q^{7} - 10 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 6 q^{2} + q^{3} + 6 q^{4} + 6 q^{5} - 2 q^{6} + 3 q^{7} - 10 q^{8} - 2 q^{9} - 2 q^{10} - 2 q^{11} + 2 q^{12} + 6 q^{13} - 6 q^{14} - 2 q^{15} + 18 q^{16} - 3 q^{17} - 6 q^{18} + 5 q^{19} + 7 q^{20} + 3 q^{21} - 6 q^{22} - 22 q^{23} + 5 q^{24} + 6 q^{25} - 12 q^{26} + 10 q^{27} + 6 q^{28} - 10 q^{29} + 4 q^{30} - 36 q^{32} - 16 q^{33} + 11 q^{34} + 24 q^{35} - 4 q^{36} - 8 q^{37} - 10 q^{38} + 18 q^{39} - 10 q^{40} - 8 q^{41} - 6 q^{42} + q^{43} + 6 q^{44} + 8 q^{45} + 14 q^{46} + 14 q^{47} + 6 q^{48} + 2 q^{49} - 12 q^{50} - 3 q^{51} - 3 q^{52} + 21 q^{53} - 20 q^{54} - 2 q^{55} - 20 q^{57} + 30 q^{58} - 5 q^{59} + 6 q^{60} - 16 q^{61} - 48 q^{63} - 14 q^{64} - 9 q^{65} + 12 q^{66} + 8 q^{67} - 6 q^{68} + 11 q^{69} - 18 q^{70} + 7 q^{71} + 10 q^{72} + 21 q^{73} + 11 q^{74} - 9 q^{75} - 15 q^{76} - 48 q^{77} - 6 q^{78} + 6 q^{80} + q^{81} - 4 q^{82} - 14 q^{83} - 9 q^{84} - 4 q^{85} - 7 q^{86} - 30 q^{87} + 20 q^{88} - 10 q^{89} + 4 q^{90} - 36 q^{91} - 44 q^{92} - 28 q^{94} - 10 q^{95} - 2 q^{96} - 6 q^{97} - 4 q^{98} + 24 q^{99}+O(q^{100}) \) 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 0.913545 0.406737i 0.190983 0.587785i 0.190983 + 0.330792i −0.809017 + 1.40126i 2.00739 + 2.22943i −0.690983 2.12663i −1.33826 + 1.48629i −0.564602 0.251377i
338.1 −1.30902 + 0.951057i −0.104528 + 0.994522i 0.190983 0.587785i 0.190983 0.330792i −0.809017 1.40126i −2.93444 + 0.623735i −0.690983 2.12663i 1.95630 + 0.415823i 0.0646021 + 0.614648i
448.1 −0.190983 + 0.587785i 0.669131 0.743145i 1.30902 + 0.951057i 1.30902 2.26728i 0.309017 + 0.535233i −0.313585 + 2.98357i −1.80902 + 1.31433i 0.209057 + 1.98904i 1.08268 + 1.20243i
547.1 −0.190983 0.587785i 0.669131 + 0.743145i 1.30902 0.951057i 1.30902 + 2.26728i 0.309017 0.535233i −0.313585 2.98357i −1.80902 1.31433i 0.209057 1.98904i 1.08268 1.20243i
732.1 −1.30902 0.951057i 0.913545 + 0.406737i 0.190983 + 0.587785i 0.190983 0.330792i −0.809017 1.40126i 2.00739 2.22943i −0.690983 + 2.12663i −1.33826 1.48629i −0.564602 + 0.251377i
816.1 −1.30902 0.951057i −0.104528 0.994522i 0.190983 + 0.587785i 0.190983 + 0.330792i −0.809017 + 1.40126i −2.93444 0.623735i −0.690983 + 2.12663i 1.95630 0.415823i 0.0646021 0.614648i
844.1 −0.190983 0.587785i −0.978148 + 0.207912i 1.30902 0.951057i 1.30902 2.26728i 0.309017 + 0.535233i 2.74064 + 1.22021i −1.80902 1.31433i −1.82709 + 0.813473i −1.58268 0.336408i
846.1 −0.190983 + 0.587785i −0.978148 0.207912i 1.30902 + 0.951057i 1.30902 + 2.26728i 0.309017 0.535233i 2.74064 1.22021i −1.80902 + 1.31433i −1.82709 0.813473i −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.c 8
31.b odd 2 1 961.2.g.b 8
31.c even 3 1 961.2.d.b 4
31.c even 3 1 inner 961.2.g.c 8
31.d even 5 1 961.2.c.d 4
31.d even 5 1 inner 961.2.g.c 8
31.d even 5 2 961.2.g.g 8
31.e odd 6 1 31.2.d.a 4
31.e odd 6 1 961.2.g.b 8
31.f odd 10 1 961.2.c.f 4
31.f odd 10 1 961.2.g.b 8
31.f odd 10 2 961.2.g.f 8
31.g even 15 1 961.2.a.e 2
31.g even 15 1 961.2.c.d 4
31.g even 15 1 961.2.d.b 4
31.g even 15 2 961.2.d.e 4
31.g even 15 1 inner 961.2.g.c 8
31.g even 15 2 961.2.g.g 8
31.h odd 30 1 31.2.d.a 4
31.h odd 30 1 961.2.a.d 2
31.h odd 30 1 961.2.c.f 4
31.h odd 30 2 961.2.d.f 4
31.h odd 30 1 961.2.g.b 8
31.h odd 30 2 961.2.g.f 8
93.g even 6 1 279.2.i.a 4
93.o odd 30 1 8649.2.a.f 2
93.p even 30 1 279.2.i.a 4
93.p even 30 1 8649.2.a.g 2
124.g even 6 1 496.2.n.b 4
124.p even 30 1 496.2.n.b 4
155.i odd 6 1 775.2.k.c 4
155.p even 12 2 775.2.bf.a 8
155.v odd 30 1 775.2.k.c 4
155.x even 60 2 775.2.bf.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.2.d.a 4 31.e odd 6 1
31.2.d.a 4 31.h odd 30 1
279.2.i.a 4 93.g even 6 1
279.2.i.a 4 93.p even 30 1
496.2.n.b 4 124.g even 6 1
496.2.n.b 4 124.p even 30 1
775.2.k.c 4 155.i odd 6 1
775.2.k.c 4 155.v odd 30 1
775.2.bf.a 8 155.p even 12 2
775.2.bf.a 8 155.x even 60 2
961.2.a.d 2 31.h odd 30 1
961.2.a.e 2 31.g even 15 1
961.2.c.d 4 31.d even 5 1
961.2.c.d 4 31.g even 15 1
961.2.c.f 4 31.f odd 10 1
961.2.c.f 4 31.h odd 30 1
961.2.d.b 4 31.c even 3 1
961.2.d.b 4 31.g even 15 1
961.2.d.e 4 31.g even 15 2
961.2.d.f 4 31.h odd 30 2
961.2.g.b 8 31.b odd 2 1
961.2.g.b 8 31.e odd 6 1
961.2.g.b 8 31.f odd 10 1
961.2.g.b 8 31.h odd 30 1
961.2.g.c 8 1.a even 1 1 trivial
961.2.g.c 8 31.c even 3 1 inner
961.2.g.c 8 31.d even 5 1 inner
961.2.g.c 8 31.g even 15 1 inner
961.2.g.f 8 31.f odd 10 2
961.2.g.f 8 31.h odd 30 2
961.2.g.g 8 31.d even 5 2
961.2.g.g 8 31.g even 15 2
8649.2.a.f 2 93.o odd 30 1
8649.2.a.g 2 93.p even 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
\( T_{3}^{8} - T_{3}^{7} + T_{3}^{5} - T_{3}^{4} + T_{3}^{3} - T_{3} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 3 T^{3} + 4 T^{2} + 2 T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - T^{7} + T^{5} - T^{4} + T^{3} - T + 1 \) Copy content Toggle raw display
$5$ \( (T^{4} - 3 T^{3} + 8 T^{2} - 3 T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 3 T^{7} + 27 T^{5} + \cdots + 6561 \) Copy content Toggle raw display
$11$ \( T^{8} + 2 T^{7} - 20 T^{6} + 112 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( T^{8} - 6 T^{7} - 54 T^{5} + \cdots + 6561 \) Copy content Toggle raw display
$17$ \( T^{8} + 3 T^{7} - 10 T^{6} + 43 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{8} - 5 T^{7} + 125 T^{5} + \cdots + 390625 \) Copy content Toggle raw display
$23$ \( (T^{4} + 11 T^{3} + 61 T^{2} + 171 T + 361)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 5 T^{3} + 60 T^{2} + 550 T + 3025)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} + 4 T^{3} + 17 T^{2} - 4 T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 8 T^{7} + 128 T^{5} + \cdots + 65536 \) Copy content Toggle raw display
$43$ \( T^{8} - T^{7} - 15 T^{6} + 116 T^{5} + \cdots + 14641 \) Copy content Toggle raw display
$47$ \( (T^{4} - 7 T^{3} + 24 T^{2} - 38 T + 361)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} - 21 T^{7} + 270 T^{6} + \cdots + 6561 \) Copy content Toggle raw display
$59$ \( T^{8} + 5 T^{7} - 60 T^{6} + 575 T^{5} + \cdots + 625 \) Copy content Toggle raw display
$61$ \( (T^{2} + 4 T - 76)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 4 T^{3} + 17 T^{2} + 4 T + 1)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 7 T^{7} - 75 T^{6} - 832 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$73$ \( T^{8} - 21 T^{7} + 135 T^{6} + \cdots + 96059601 \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} + 14 T^{7} + 100 T^{6} + \cdots + 707281 \) Copy content Toggle raw display
$89$ \( (T^{4} + 5 T^{3} + 60 T^{2} + 550 T + 3025)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 3 T^{3} + 279 T^{2} - 2673 T + 9801)^{2} \) Copy content Toggle raw display
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