# 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 $16$

# Related objects

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: $$x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1$$ x^8 - x^7 + x^5 - x^4 + x^3 - x + 1 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.

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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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