Properties

Label 961.1.h.a
Level $961$
Weight $1$
Character orbit 961.h
Analytic conductor $0.480$
Analytic rank $0$
Dimension $8$
Projective image $D_{3}$
CM discriminant -31
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [961,1,Mod(115,961)]
 
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([17]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("961.115");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 961 = 31^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 961.h (of order \(30\), 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: \(0.479601477140\)
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)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.31.1
Artin image: $S_3\times C_{15}$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{45} - \cdots)\)

$q$-expansion

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

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{30}^{3} q^{2} - \zeta_{30}^{10} q^{5} + \zeta_{30}^{11} q^{7} - \zeta_{30}^{9} q^{8} + \zeta_{30}^{4} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{30}^{3} q^{2} - \zeta_{30}^{10} q^{5} + \zeta_{30}^{11} q^{7} - \zeta_{30}^{9} q^{8} + \zeta_{30}^{4} q^{9} - \zeta_{30}^{13} q^{10} + \zeta_{30}^{14} q^{14} - \zeta_{30}^{12} q^{16} + \zeta_{30}^{7} q^{18} - \zeta_{30}^{8} q^{19} + \zeta_{30}^{6} q^{35} - \zeta_{30}^{11} q^{38} - \zeta_{30}^{4} q^{40} + \zeta_{30}^{13} q^{41} - \zeta_{30}^{14} q^{45} + 2 \zeta_{30}^{12} q^{47} + \zeta_{30}^{5} q^{56} - \zeta_{30}^{2} q^{59} - q^{63} - \zeta_{30}^{3} q^{64} + 2 \zeta_{30}^{10} q^{67} + \zeta_{30}^{9} q^{70} - \zeta_{30}^{4} q^{71} - \zeta_{30}^{13} q^{72} - \zeta_{30}^{7} q^{80} + \zeta_{30}^{8} q^{81} - \zeta_{30} q^{82} + \zeta_{30}^{2} q^{90} - 2 q^{94} - \zeta_{30}^{3} q^{95} - \zeta_{30}^{6} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} + 4 q^{5} - q^{7} - 2 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} + 4 q^{5} - q^{7} - 2 q^{8} + q^{9} + q^{10} + q^{14} + 2 q^{16} - q^{18} - q^{19} - 2 q^{35} + q^{38} - q^{40} - q^{41} - q^{45} - 4 q^{47} + 4 q^{56} - q^{59} - 8 q^{63} - 2 q^{64} - 8 q^{67} + 2 q^{70} - q^{71} + q^{72} + q^{80} + q^{81} + q^{82} + q^{90} - 16 q^{94} - 2 q^{95} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(-\zeta_{30}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
115.1
−0.104528 + 0.994522i
−0.104528 0.994522i
0.669131 0.743145i
−0.978148 0.207912i
0.913545 + 0.406737i
0.913545 0.406737i
0.669131 + 0.743145i
−0.978148 + 0.207912i
−0.309017 + 0.951057i 0 0 0.500000 + 0.866025i 0 −0.913545 + 0.406737i −0.809017 + 0.587785i 0.913545 + 0.406737i −0.978148 + 0.207912i
117.1 −0.309017 0.951057i 0 0 0.500000 0.866025i 0 −0.913545 0.406737i −0.809017 0.587785i 0.913545 0.406737i −0.978148 0.207912i
145.1 0.809017 + 0.587785i 0 0 0.500000 + 0.866025i 0 0.978148 + 0.207912i 0.309017 0.951057i −0.978148 + 0.207912i −0.104528 + 0.994522i
229.1 0.809017 + 0.587785i 0 0 0.500000 0.866025i 0 −0.669131 + 0.743145i 0.309017 0.951057i 0.669131 + 0.743145i 0.913545 0.406737i
414.1 −0.309017 0.951057i 0 0 0.500000 + 0.866025i 0 0.104528 + 0.994522i −0.809017 0.587785i −0.104528 + 0.994522i 0.669131 0.743145i
513.1 −0.309017 + 0.951057i 0 0 0.500000 0.866025i 0 0.104528 0.994522i −0.809017 + 0.587785i −0.104528 0.994522i 0.669131 + 0.743145i
623.1 0.809017 0.587785i 0 0 0.500000 0.866025i 0 0.978148 0.207912i 0.309017 + 0.951057i −0.978148 0.207912i −0.104528 0.994522i
726.1 0.809017 0.587785i 0 0 0.500000 + 0.866025i 0 −0.669131 0.743145i 0.309017 + 0.951057i 0.669131 0.743145i 0.913545 + 0.406737i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 115.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.b odd 2 1 CM by \(\Q(\sqrt{-31}) \)
31.c even 3 1 inner
31.d even 5 3 inner
31.e odd 6 1 inner
31.f odd 10 3 inner
31.g even 15 3 inner
31.h odd 30 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 961.1.h.a 8
31.b odd 2 1 CM 961.1.h.a 8
31.c even 3 1 961.1.f.a 4
31.c even 3 1 inner 961.1.h.a 8
31.d even 5 1 961.1.e.a 2
31.d even 5 3 inner 961.1.h.a 8
31.e odd 6 1 961.1.f.a 4
31.e odd 6 1 inner 961.1.h.a 8
31.f odd 10 1 961.1.e.a 2
31.f odd 10 3 inner 961.1.h.a 8
31.g even 15 1 31.1.b.a 1
31.g even 15 1 961.1.e.a 2
31.g even 15 3 961.1.f.a 4
31.g even 15 3 inner 961.1.h.a 8
31.h odd 30 1 31.1.b.a 1
31.h odd 30 1 961.1.e.a 2
31.h odd 30 3 961.1.f.a 4
31.h odd 30 3 inner 961.1.h.a 8
93.o odd 30 1 279.1.d.b 1
93.p even 30 1 279.1.d.b 1
124.n odd 30 1 496.1.e.a 1
124.p even 30 1 496.1.e.a 1
155.u even 30 1 775.1.d.b 1
155.v odd 30 1 775.1.d.b 1
155.w odd 60 2 775.1.c.a 2
155.x even 60 2 775.1.c.a 2
217.z even 15 1 1519.1.n.b 2
217.ba even 15 1 1519.1.n.b 2
217.bd odd 30 1 1519.1.c.a 1
217.be even 30 1 1519.1.c.a 1
217.bf even 30 1 1519.1.n.a 2
217.bh odd 30 1 1519.1.n.b 2
217.bj odd 30 1 1519.1.n.a 2
217.bk even 30 1 1519.1.n.a 2
217.bm odd 30 1 1519.1.n.a 2
217.bn odd 30 1 1519.1.n.b 2
248.bb even 30 1 1984.1.e.b 1
248.bc even 30 1 1984.1.e.a 1
248.be odd 30 1 1984.1.e.b 1
248.bf odd 30 1 1984.1.e.a 1
279.ba even 15 1 2511.1.m.e 2
279.bb even 15 1 2511.1.m.e 2
279.bd odd 30 1 2511.1.m.a 2
279.be even 30 1 2511.1.m.a 2
279.bh even 30 1 2511.1.m.a 2
279.bi odd 30 1 2511.1.m.a 2
279.bk odd 30 1 2511.1.m.e 2
279.bl odd 30 1 2511.1.m.e 2
341.bg even 15 1 3751.1.t.c 4
341.bj even 15 1 3751.1.t.c 4
341.bk even 15 1 3751.1.t.c 4
341.bl even 15 1 3751.1.t.c 4
341.bm even 30 1 3751.1.t.a 4
341.bn odd 30 1 3751.1.t.a 4
341.bq odd 30 1 3751.1.t.c 4
341.br odd 30 1 3751.1.t.c 4
341.bs odd 30 1 3751.1.t.c 4
341.bt odd 30 1 3751.1.t.a 4
341.bu even 30 1 3751.1.d.b 1
341.bv even 30 1 3751.1.t.a 4
341.bw even 30 1 3751.1.t.a 4
341.by odd 30 1 3751.1.d.b 1
341.bz odd 30 1 3751.1.t.a 4
341.ca odd 30 1 3751.1.t.a 4
341.cc even 30 1 3751.1.t.a 4
341.cd odd 30 1 3751.1.t.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.1.b.a 1 31.g even 15 1
31.1.b.a 1 31.h odd 30 1
279.1.d.b 1 93.o odd 30 1
279.1.d.b 1 93.p even 30 1
496.1.e.a 1 124.n odd 30 1
496.1.e.a 1 124.p even 30 1
775.1.c.a 2 155.w odd 60 2
775.1.c.a 2 155.x even 60 2
775.1.d.b 1 155.u even 30 1
775.1.d.b 1 155.v odd 30 1
961.1.e.a 2 31.d even 5 1
961.1.e.a 2 31.f odd 10 1
961.1.e.a 2 31.g even 15 1
961.1.e.a 2 31.h odd 30 1
961.1.f.a 4 31.c even 3 1
961.1.f.a 4 31.e odd 6 1
961.1.f.a 4 31.g even 15 3
961.1.f.a 4 31.h odd 30 3
961.1.h.a 8 1.a even 1 1 trivial
961.1.h.a 8 31.b odd 2 1 CM
961.1.h.a 8 31.c even 3 1 inner
961.1.h.a 8 31.d even 5 3 inner
961.1.h.a 8 31.e odd 6 1 inner
961.1.h.a 8 31.f odd 10 3 inner
961.1.h.a 8 31.g even 15 3 inner
961.1.h.a 8 31.h odd 30 3 inner
1519.1.c.a 1 217.bd odd 30 1
1519.1.c.a 1 217.be even 30 1
1519.1.n.a 2 217.bf even 30 1
1519.1.n.a 2 217.bj odd 30 1
1519.1.n.a 2 217.bk even 30 1
1519.1.n.a 2 217.bm odd 30 1
1519.1.n.b 2 217.z even 15 1
1519.1.n.b 2 217.ba even 15 1
1519.1.n.b 2 217.bh odd 30 1
1519.1.n.b 2 217.bn odd 30 1
1984.1.e.a 1 248.bc even 30 1
1984.1.e.a 1 248.bf odd 30 1
1984.1.e.b 1 248.bb even 30 1
1984.1.e.b 1 248.be odd 30 1
2511.1.m.a 2 279.bd odd 30 1
2511.1.m.a 2 279.be even 30 1
2511.1.m.a 2 279.bh even 30 1
2511.1.m.a 2 279.bi odd 30 1
2511.1.m.e 2 279.ba even 15 1
2511.1.m.e 2 279.bb even 15 1
2511.1.m.e 2 279.bk odd 30 1
2511.1.m.e 2 279.bl odd 30 1
3751.1.d.b 1 341.bu even 30 1
3751.1.d.b 1 341.by odd 30 1
3751.1.t.a 4 341.bm even 30 1
3751.1.t.a 4 341.bn odd 30 1
3751.1.t.a 4 341.bt odd 30 1
3751.1.t.a 4 341.bv even 30 1
3751.1.t.a 4 341.bw even 30 1
3751.1.t.a 4 341.bz odd 30 1
3751.1.t.a 4 341.ca odd 30 1
3751.1.t.a 4 341.cc even 30 1
3751.1.t.c 4 341.bg even 15 1
3751.1.t.c 4 341.bj even 15 1
3751.1.t.c 4 341.bk even 15 1
3751.1.t.c 4 341.bl even 15 1
3751.1.t.c 4 341.bq odd 30 1
3751.1.t.c 4 341.br odd 30 1
3751.1.t.c 4 341.bs odd 30 1
3751.1.t.c 4 341.cd odd 30 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(961, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} + T^{7} - T^{5} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} + T^{7} - T^{5} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} + T^{7} - T^{5} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} + T^{7} - T^{5} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T^{2} + 2 T + 4)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} + T^{7} - T^{5} + \cdots + 1 \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
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