# Properties

 Label 961.1.e.a Level $961$ Weight $1$ Character orbit 961.e Analytic conductor $0.480$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -31 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [961,1,Mod(440,961)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(961, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5]))

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

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

chi := DirichletCharacter("961.440");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$961 = 31^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 961.e (of order $$6$$, degree $$2$$, 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: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - 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: $C_3\times S_3$ Artin field: Galois closure of 6.0.28629151.2

## $q$-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.

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

## Character values

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

 $$n$$ $$3$$ $$\chi(n)$$ $$\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
440.1
 0.5 − 0.866025i 0.5 + 0.866025i
−1.00000 0 0 0.500000 0.866025i 0 0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i −0.500000 + 0.866025i
522.1 −1.00000 0 0 0.500000 + 0.866025i 0 0.500000 0.866025i 1.00000 −0.500000 0.866025i −0.500000 0.866025i
 $$n$$: 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.c even 3 1 inner
31.e odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 961.1.e.a 2
31.b odd 2 1 CM 961.1.e.a 2
31.c even 3 1 31.1.b.a 1
31.c even 3 1 inner 961.1.e.a 2
31.d even 5 4 961.1.h.a 8
31.e odd 6 1 31.1.b.a 1
31.e odd 6 1 inner 961.1.e.a 2
31.f odd 10 4 961.1.h.a 8
31.g even 15 4 961.1.f.a 4
31.g even 15 4 961.1.h.a 8
31.h odd 30 4 961.1.f.a 4
31.h odd 30 4 961.1.h.a 8
93.g even 6 1 279.1.d.b 1
93.h odd 6 1 279.1.d.b 1
124.g even 6 1 496.1.e.a 1
124.i odd 6 1 496.1.e.a 1
155.i odd 6 1 775.1.d.b 1
155.j even 6 1 775.1.d.b 1
155.o odd 12 2 775.1.c.a 2
155.p even 12 2 775.1.c.a 2
217.e even 3 1 1519.1.n.b 2
217.g even 3 1 1519.1.n.b 2
217.j odd 6 1 1519.1.n.a 2
217.k odd 6 1 1519.1.n.b 2
217.l even 6 1 1519.1.n.a 2
217.o odd 6 1 1519.1.n.b 2
217.q odd 6 1 1519.1.n.a 2
217.r even 6 1 1519.1.n.a 2
217.s even 6 1 1519.1.c.a 1
217.u odd 6 1 1519.1.c.a 1
248.l odd 6 1 1984.1.e.a 1
248.m odd 6 1 1984.1.e.b 1
248.p even 6 1 1984.1.e.a 1
248.q even 6 1 1984.1.e.b 1
279.e even 3 1 2511.1.m.e 2
279.g even 3 1 2511.1.m.e 2
279.l odd 6 1 2511.1.m.e 2
279.n odd 6 1 2511.1.m.e 2
279.o even 6 1 2511.1.m.a 2
279.p odd 6 1 2511.1.m.a 2
279.q odd 6 1 2511.1.m.a 2
279.r even 6 1 2511.1.m.a 2
341.l odd 6 1 3751.1.d.b 1
341.m even 6 1 3751.1.d.b 1
341.bi even 15 4 3751.1.t.c 4
341.bp odd 30 4 3751.1.t.c 4
341.bx even 30 4 3751.1.t.a 4
341.cb odd 30 4 3751.1.t.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.1.b.a 1 31.c even 3 1
31.1.b.a 1 31.e odd 6 1
279.1.d.b 1 93.g even 6 1
279.1.d.b 1 93.h odd 6 1
496.1.e.a 1 124.g even 6 1
496.1.e.a 1 124.i odd 6 1
775.1.c.a 2 155.o odd 12 2
775.1.c.a 2 155.p even 12 2
775.1.d.b 1 155.i odd 6 1
775.1.d.b 1 155.j even 6 1
961.1.e.a 2 1.a even 1 1 trivial
961.1.e.a 2 31.b odd 2 1 CM
961.1.e.a 2 31.c even 3 1 inner
961.1.e.a 2 31.e odd 6 1 inner
961.1.f.a 4 31.g even 15 4
961.1.f.a 4 31.h odd 30 4
961.1.h.a 8 31.d even 5 4
961.1.h.a 8 31.f odd 10 4
961.1.h.a 8 31.g even 15 4
961.1.h.a 8 31.h odd 30 4
1519.1.c.a 1 217.s even 6 1
1519.1.c.a 1 217.u odd 6 1
1519.1.n.a 2 217.j odd 6 1
1519.1.n.a 2 217.l even 6 1
1519.1.n.a 2 217.q odd 6 1
1519.1.n.a 2 217.r even 6 1
1519.1.n.b 2 217.e even 3 1
1519.1.n.b 2 217.g even 3 1
1519.1.n.b 2 217.k odd 6 1
1519.1.n.b 2 217.o odd 6 1
1984.1.e.a 1 248.l odd 6 1
1984.1.e.a 1 248.p even 6 1
1984.1.e.b 1 248.m odd 6 1
1984.1.e.b 1 248.q even 6 1
2511.1.m.a 2 279.o even 6 1
2511.1.m.a 2 279.p odd 6 1
2511.1.m.a 2 279.q odd 6 1
2511.1.m.a 2 279.r even 6 1
2511.1.m.e 2 279.e even 3 1
2511.1.m.e 2 279.g even 3 1
2511.1.m.e 2 279.l odd 6 1
2511.1.m.e 2 279.n odd 6 1
3751.1.d.b 1 341.l odd 6 1
3751.1.d.b 1 341.m even 6 1
3751.1.t.a 4 341.bx even 30 4
3751.1.t.a 4 341.cb odd 30 4
3751.1.t.c 4 341.bi even 15 4
3751.1.t.c 4 341.bp odd 30 4

## Hecke kernels

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

## Hecke characteristic polynomials

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