# Properties

 Label 961.1.f.a Level $961$ Weight $1$ Character orbit 961.f Analytic conductor $0.480$ Analytic rank $0$ Dimension $4$ Projective image $D_{3}$ CM discriminant -31 Inner twists $8$

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Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [961,1,Mod(333,961)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(961, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([9]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("961.333");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$961 = 31^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 961.f (of order $$10$$, degree $$4$$, 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: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ 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: $C_5\times S_3$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{15} - \cdots)$$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{10}^{3} q^{2} - q^{5} + \zeta_{10} q^{7} + \zeta_{10}^{4} q^{8} + \zeta_{10}^{4} q^{9}+O(q^{10})$$ q + z^3 * q^2 - q^5 + z * q^7 + z^4 * q^8 + z^4 * q^9 $$q + \zeta_{10}^{3} q^{2} - q^{5} + \zeta_{10} q^{7} + \zeta_{10}^{4} q^{8} + \zeta_{10}^{4} q^{9} - \zeta_{10}^{3} q^{10} + \zeta_{10}^{4} q^{14} - \zeta_{10}^{2} q^{16} - \zeta_{10}^{2} q^{18} + \zeta_{10}^{3} q^{19} - q^{32} - \zeta_{10} q^{35} - \zeta_{10} q^{38} - \zeta_{10}^{4} q^{40} + \zeta_{10}^{3} q^{41} - \zeta_{10}^{4} q^{45} + \zeta_{10}^{2} q^{47} - q^{56} - \zeta_{10}^{2} q^{59} - q^{63} - \zeta_{10}^{3} q^{64} + q^{67} - \zeta_{10}^{4} q^{70} - \zeta_{10}^{4} q^{71} - \zeta_{10}^{3} q^{72} + \zeta_{10}^{2} q^{80} - \zeta_{10}^{3} q^{81} - \zeta_{10} q^{82} + \zeta_{10}^{2} q^{90} - 2 q^{94} - \zeta_{10}^{3} q^{95} + \zeta_{10} q^{97} +O(q^{100})$$ q + z^3 * q^2 - q^5 + z * q^7 + z^4 * q^8 + z^4 * q^9 - z^3 * q^10 + z^4 * q^14 - z^2 * q^16 - z^2 * q^18 + z^3 * q^19 - q^32 - z * q^35 - z * q^38 - z^4 * q^40 + z^3 * q^41 - z^4 * q^45 + z^2 * q^47 - q^56 - z^2 * q^59 - q^63 - z^3 * q^64 + q^67 - z^4 * q^70 - z^4 * q^71 - z^3 * q^72 + z^2 * q^80 - z^3 * q^81 - z * q^82 + z^2 * q^90 - 2 * q^94 - z^3 * q^95 + z * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{2} - 4 q^{5} + q^{7} - q^{8} - q^{9}+O(q^{10})$$ 4 * q + q^2 - 4 * q^5 + q^7 - q^8 - q^9 $$4 q + q^{2} - 4 q^{5} + q^{7} - q^{8} - q^{9} - q^{10} - q^{14} + q^{16} + q^{18} + q^{19} - q^{35} - q^{38} + q^{40} + q^{41} + q^{45} - 2 q^{47} - 4 q^{56} + q^{59} - 4 q^{63} - q^{64} + 8 q^{67} + q^{70} + q^{71} - q^{72} - q^{80} - q^{81} - q^{82} - q^{90} - 8 q^{94} - q^{95} + q^{97}+O(q^{100})$$ 4 * q + q^2 - 4 * q^5 + q^7 - q^8 - q^9 - q^10 - q^14 + q^16 + q^18 + q^19 - q^35 - q^38 + q^40 + q^41 + q^45 - 2 * q^47 - 4 * q^56 + q^59 - 4 * q^63 - q^64 + 8 * q^67 + q^70 + q^71 - q^72 - q^80 - q^81 - q^82 - q^90 - 8 * q^94 - q^95 + q^97

## Character values

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

 $$n$$ $$3$$ $$\chi(n)$$ $$-\zeta_{10}^{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
333.1
 0.809017 + 0.587785i 0.809017 − 0.587785i −0.309017 + 0.951057i −0.309017 − 0.951057i
−0.309017 + 0.951057i 0 0 −1.00000 0 0.809017 + 0.587785i −0.809017 + 0.587785i −0.809017 + 0.587785i 0.309017 0.951057i
430.1 −0.309017 0.951057i 0 0 −1.00000 0 0.809017 0.587785i −0.809017 0.587785i −0.809017 0.587785i 0.309017 + 0.951057i
573.1 0.809017 0.587785i 0 0 −1.00000 0 −0.309017 + 0.951057i 0.309017 + 0.951057i 0.309017 + 0.951057i −0.809017 + 0.587785i
587.1 0.809017 + 0.587785i 0 0 −1.00000 0 −0.309017 0.951057i 0.309017 0.951057i 0.309017 0.951057i −0.809017 0.587785i
 $$n$$: e.g. 2-40 or 990-1000 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.d even 5 3 inner
31.f odd 10 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 961.1.f.a 4
31.b odd 2 1 CM 961.1.f.a 4
31.c even 3 2 961.1.h.a 8
31.d even 5 1 31.1.b.a 1
31.d even 5 3 inner 961.1.f.a 4
31.e odd 6 2 961.1.h.a 8
31.f odd 10 1 31.1.b.a 1
31.f odd 10 3 inner 961.1.f.a 4
31.g even 15 2 961.1.e.a 2
31.g even 15 6 961.1.h.a 8
31.h odd 30 2 961.1.e.a 2
31.h odd 30 6 961.1.h.a 8
93.k even 10 1 279.1.d.b 1
93.l odd 10 1 279.1.d.b 1
124.j even 10 1 496.1.e.a 1
124.l odd 10 1 496.1.e.a 1
155.m odd 10 1 775.1.d.b 1
155.n even 10 1 775.1.d.b 1
155.r even 20 2 775.1.c.a 2
155.s odd 20 2 775.1.c.a 2
217.v even 10 1 1519.1.c.a 1
217.w odd 10 1 1519.1.c.a 1
217.bb even 15 2 1519.1.n.b 2
217.bg odd 30 2 1519.1.n.b 2
217.bi odd 30 2 1519.1.n.a 2
217.bl even 30 2 1519.1.n.a 2
248.r odd 10 1 1984.1.e.a 1
248.s odd 10 1 1984.1.e.b 1
248.u even 10 1 1984.1.e.a 1
248.v even 10 1 1984.1.e.b 1
279.z even 15 2 2511.1.m.e 2
279.bf odd 30 2 2511.1.m.a 2
279.bg even 30 2 2511.1.m.a 2
279.bj odd 30 2 2511.1.m.e 2
341.f even 5 1 3751.1.t.c 4
341.g even 5 1 3751.1.t.c 4
341.j even 5 1 3751.1.t.c 4
341.k even 5 1 3751.1.t.c 4
341.o odd 10 1 3751.1.t.a 4
341.p even 10 1 3751.1.t.a 4
341.q odd 10 1 3751.1.t.c 4
341.r odd 10 1 3751.1.t.c 4
341.s odd 10 1 3751.1.t.c 4
341.v even 10 1 3751.1.t.a 4
341.x odd 10 1 3751.1.t.a 4
341.y odd 10 1 3751.1.t.a 4
341.z odd 10 1 3751.1.d.b 1
341.bb even 10 1 3751.1.t.a 4
341.bc even 10 1 3751.1.t.a 4
341.bd even 10 1 3751.1.d.b 1
341.be odd 10 1 3751.1.t.a 4
341.bf odd 10 1 3751.1.t.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.1.b.a 1 31.d even 5 1
31.1.b.a 1 31.f odd 10 1
279.1.d.b 1 93.k even 10 1
279.1.d.b 1 93.l odd 10 1
496.1.e.a 1 124.j even 10 1
496.1.e.a 1 124.l odd 10 1
775.1.c.a 2 155.r even 20 2
775.1.c.a 2 155.s odd 20 2
775.1.d.b 1 155.m odd 10 1
775.1.d.b 1 155.n even 10 1
961.1.e.a 2 31.g even 15 2
961.1.e.a 2 31.h odd 30 2
961.1.f.a 4 1.a even 1 1 trivial
961.1.f.a 4 31.b odd 2 1 CM
961.1.f.a 4 31.d even 5 3 inner
961.1.f.a 4 31.f odd 10 3 inner
961.1.h.a 8 31.c even 3 2
961.1.h.a 8 31.e odd 6 2
961.1.h.a 8 31.g even 15 6
961.1.h.a 8 31.h odd 30 6
1519.1.c.a 1 217.v even 10 1
1519.1.c.a 1 217.w odd 10 1
1519.1.n.a 2 217.bi odd 30 2
1519.1.n.a 2 217.bl even 30 2
1519.1.n.b 2 217.bb even 15 2
1519.1.n.b 2 217.bg odd 30 2
1984.1.e.a 1 248.r odd 10 1
1984.1.e.a 1 248.u even 10 1
1984.1.e.b 1 248.s odd 10 1
1984.1.e.b 1 248.v even 10 1
2511.1.m.a 2 279.bf odd 30 2
2511.1.m.a 2 279.bg even 30 2
2511.1.m.e 2 279.z even 15 2
2511.1.m.e 2 279.bj odd 30 2
3751.1.d.b 1 341.z odd 10 1
3751.1.d.b 1 341.bd even 10 1
3751.1.t.a 4 341.o odd 10 1
3751.1.t.a 4 341.p even 10 1
3751.1.t.a 4 341.v even 10 1
3751.1.t.a 4 341.x odd 10 1
3751.1.t.a 4 341.y odd 10 1
3751.1.t.a 4 341.bb even 10 1
3751.1.t.a 4 341.bc even 10 1
3751.1.t.a 4 341.be odd 10 1
3751.1.t.c 4 341.f even 5 1
3751.1.t.c 4 341.g even 5 1
3751.1.t.c 4 341.j even 5 1
3751.1.t.c 4 341.k even 5 1
3751.1.t.c 4 341.q odd 10 1
3751.1.t.c 4 341.r odd 10 1
3751.1.t.c 4 341.s odd 10 1
3751.1.t.c 4 341.bf odd 10 1

## Hecke kernels

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

## Hecke characteristic polynomials

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