# Properties

 Label 832.1.bb.a Level $832$ Weight $1$ Character orbit 832.bb Analytic conductor $0.415$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -4 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [832,1,Mod(191,832)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(832, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 4]))

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

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

chi := DirichletCharacter("832.191");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$832 = 2^{6} \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 832.bb (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: $$0.415222090511$$ 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: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 52) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.676.1 Artin image: $C_6\times S_3$ Artin field: Galois closure of 12.0.479174066176.4

## $q$-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.

 $$f(q)$$ $$=$$ $$q + q^{5} - \zeta_{6} q^{9} +O(q^{10})$$ q + q^5 - z * q^9 $$q + q^{5} - \zeta_{6} q^{9} - \zeta_{6}^{2} q^{13} + \zeta_{6} q^{17} + \zeta_{6}^{2} q^{29} + \zeta_{6}^{2} q^{37} - \zeta_{6}^{2} q^{41} - \zeta_{6} q^{45} + \zeta_{6}^{2} q^{49} + q^{53} - \zeta_{6} q^{61} - \zeta_{6}^{2} q^{65} - q^{73} + \zeta_{6}^{2} q^{81} + \zeta_{6} q^{85} + \zeta_{6}^{2} q^{89} - \zeta_{6} q^{97} +O(q^{100})$$ q + q^5 - z * q^9 - z^2 * q^13 + z * q^17 + z^2 * q^29 + z^2 * q^37 - z^2 * q^41 - z * q^45 + z^2 * q^49 + q^53 - z * q^61 - z^2 * q^65 - q^73 + z^2 * q^81 + z * q^85 + z^2 * q^89 - z * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} - q^{9}+O(q^{10})$$ 2 * q + 2 * q^5 - q^9 $$2 q + 2 q^{5} - q^{9} + q^{13} + q^{17} - q^{29} - q^{37} + q^{41} - q^{45} - q^{49} + 2 q^{53} - q^{61} + q^{65} - 2 q^{73} - q^{81} + q^{85} - 2 q^{89} - 2 q^{97}+O(q^{100})$$ 2 * q + 2 * q^5 - q^9 + q^13 + q^17 - q^29 - q^37 + q^41 - q^45 - q^49 + 2 * q^53 - q^61 + q^65 - 2 * q^73 - q^81 + q^85 - 2 * q^89 - 2 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/832\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$703$$ $$769$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
191.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 1.00000 0 0 0 −0.500000 0.866025i 0
575.1 0 0 0 1.00000 0 0 0 −0.500000 + 0.866025i 0
 $$n$$: 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
13.c even 3 1 inner
52.j odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 832.1.bb.a 2
4.b odd 2 1 CM 832.1.bb.a 2
8.b even 2 1 52.1.j.a 2
8.d odd 2 1 52.1.j.a 2
13.c even 3 1 inner 832.1.bb.a 2
16.e even 4 2 3328.1.v.b 4
16.f odd 4 2 3328.1.v.b 4
24.f even 2 1 468.1.br.a 2
24.h odd 2 1 468.1.br.a 2
40.e odd 2 1 1300.1.bc.a 2
40.f even 2 1 1300.1.bc.a 2
40.i odd 4 2 1300.1.w.a 4
40.k even 4 2 1300.1.w.a 4
52.j odd 6 1 inner 832.1.bb.a 2
56.e even 2 1 2548.1.bn.a 2
56.h odd 2 1 2548.1.bn.a 2
56.j odd 6 1 2548.1.q.a 2
56.j odd 6 1 2548.1.bi.a 2
56.k odd 6 1 2548.1.q.b 2
56.k odd 6 1 2548.1.bi.b 2
56.m even 6 1 2548.1.q.a 2
56.m even 6 1 2548.1.bi.a 2
56.p even 6 1 2548.1.q.b 2
56.p even 6 1 2548.1.bi.b 2
104.e even 2 1 676.1.j.a 2
104.h odd 2 1 676.1.j.a 2
104.j odd 4 2 676.1.i.a 4
104.m even 4 2 676.1.i.a 4
104.n odd 6 1 52.1.j.a 2
104.n odd 6 1 676.1.c.b 1
104.p odd 6 1 676.1.c.a 1
104.p odd 6 1 676.1.j.a 2
104.r even 6 1 52.1.j.a 2
104.r even 6 1 676.1.c.b 1
104.s even 6 1 676.1.c.a 1
104.s even 6 1 676.1.j.a 2
104.u even 12 2 676.1.b.a 2
104.u even 12 2 676.1.i.a 4
104.x odd 12 2 676.1.b.a 2
104.x odd 12 2 676.1.i.a 4
208.bg odd 12 2 3328.1.v.b 4
208.bj even 12 2 3328.1.v.b 4
312.bh odd 6 1 468.1.br.a 2
312.bn even 6 1 468.1.br.a 2
520.bv even 6 1 1300.1.bc.a 2
520.bx odd 6 1 1300.1.bc.a 2
520.cm even 12 2 1300.1.w.a 4
520.cq odd 12 2 1300.1.w.a 4
728.bf even 6 1 2548.1.bi.a 2
728.bg even 6 1 2548.1.bi.b 2
728.bq odd 6 1 2548.1.q.a 2
728.bx odd 6 1 2548.1.q.b 2
728.ce odd 6 1 2548.1.bn.a 2
728.cl even 6 1 2548.1.q.a 2
728.cw even 6 1 2548.1.q.b 2
728.da even 6 1 2548.1.bn.a 2
728.de odd 6 1 2548.1.bi.a 2
728.di odd 6 1 2548.1.bi.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.1.j.a 2 8.b even 2 1
52.1.j.a 2 8.d odd 2 1
52.1.j.a 2 104.n odd 6 1
52.1.j.a 2 104.r even 6 1
468.1.br.a 2 24.f even 2 1
468.1.br.a 2 24.h odd 2 1
468.1.br.a 2 312.bh odd 6 1
468.1.br.a 2 312.bn even 6 1
676.1.b.a 2 104.u even 12 2
676.1.b.a 2 104.x odd 12 2
676.1.c.a 1 104.p odd 6 1
676.1.c.a 1 104.s even 6 1
676.1.c.b 1 104.n odd 6 1
676.1.c.b 1 104.r even 6 1
676.1.i.a 4 104.j odd 4 2
676.1.i.a 4 104.m even 4 2
676.1.i.a 4 104.u even 12 2
676.1.i.a 4 104.x odd 12 2
676.1.j.a 2 104.e even 2 1
676.1.j.a 2 104.h odd 2 1
676.1.j.a 2 104.p odd 6 1
676.1.j.a 2 104.s even 6 1
832.1.bb.a 2 1.a even 1 1 trivial
832.1.bb.a 2 4.b odd 2 1 CM
832.1.bb.a 2 13.c even 3 1 inner
832.1.bb.a 2 52.j odd 6 1 inner
1300.1.w.a 4 40.i odd 4 2
1300.1.w.a 4 40.k even 4 2
1300.1.w.a 4 520.cm even 12 2
1300.1.w.a 4 520.cq odd 12 2
1300.1.bc.a 2 40.e odd 2 1
1300.1.bc.a 2 40.f even 2 1
1300.1.bc.a 2 520.bv even 6 1
1300.1.bc.a 2 520.bx odd 6 1
2548.1.q.a 2 56.j odd 6 1
2548.1.q.a 2 56.m even 6 1
2548.1.q.a 2 728.bq odd 6 1
2548.1.q.a 2 728.cl even 6 1
2548.1.q.b 2 56.k odd 6 1
2548.1.q.b 2 56.p even 6 1
2548.1.q.b 2 728.bx odd 6 1
2548.1.q.b 2 728.cw even 6 1
2548.1.bi.a 2 56.j odd 6 1
2548.1.bi.a 2 56.m even 6 1
2548.1.bi.a 2 728.bf even 6 1
2548.1.bi.a 2 728.de odd 6 1
2548.1.bi.b 2 56.k odd 6 1
2548.1.bi.b 2 56.p even 6 1
2548.1.bi.b 2 728.bg even 6 1
2548.1.bi.b 2 728.di odd 6 1
2548.1.bn.a 2 56.e even 2 1
2548.1.bn.a 2 56.h odd 2 1
2548.1.bn.a 2 728.ce odd 6 1
2548.1.bn.a 2 728.da even 6 1
3328.1.v.b 4 16.e even 4 2
3328.1.v.b 4 16.f odd 4 2
3328.1.v.b 4 208.bg odd 12 2
3328.1.v.b 4 208.bj even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{1}^{\mathrm{new}}(832, [\chi])$$.

## Hecke characteristic polynomials

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