# Properties

 Label 832.2.w.a Level $832$ Weight $2$ Character orbit 832.w Analytic conductor $6.644$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [832,2,Mod(257,832)]

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([0, 0, 5]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("832.257");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$832 = 2^{6} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 832.w (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: $$6.64355344817$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ 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 13) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 2) q^{3} + (2 \zeta_{6} - 1) q^{5} - \zeta_{6} q^{9} +O(q^{10})$$ q + (2*z - 2) * q^3 + (2*z - 1) * q^5 - z * q^9 $$q + (2 \zeta_{6} - 2) q^{3} + (2 \zeta_{6} - 1) q^{5} - \zeta_{6} q^{9} + (3 \zeta_{6} + 1) q^{13} + ( - 2 \zeta_{6} - 2) q^{15} + 3 \zeta_{6} q^{17} + (2 \zeta_{6} - 4) q^{19} + (6 \zeta_{6} - 6) q^{23} + 2 q^{25} - 4 q^{27} + ( - 3 \zeta_{6} + 3) q^{29} + ( - 4 \zeta_{6} + 2) q^{31} + ( - 5 \zeta_{6} - 5) q^{37} + (2 \zeta_{6} - 8) q^{39} + ( - 3 \zeta_{6} - 3) q^{41} - 8 \zeta_{6} q^{43} + ( - \zeta_{6} + 2) q^{45} + (4 \zeta_{6} - 2) q^{47} + (7 \zeta_{6} - 7) q^{49} - 6 q^{51} + 3 q^{53} + ( - 8 \zeta_{6} + 4) q^{57} + ( - 4 \zeta_{6} + 8) q^{59} + \zeta_{6} q^{61} + (5 \zeta_{6} - 7) q^{65} + (2 \zeta_{6} + 2) q^{67} - 12 \zeta_{6} q^{69} + (2 \zeta_{6} - 4) q^{71} + (2 \zeta_{6} - 1) q^{73} + (4 \zeta_{6} - 4) q^{75} - 4 q^{79} + ( - 11 \zeta_{6} + 11) q^{81} + (16 \zeta_{6} - 8) q^{83} + (3 \zeta_{6} - 6) q^{85} + 6 \zeta_{6} q^{87} + ( - 4 \zeta_{6} - 4) q^{89} + (4 \zeta_{6} + 4) q^{93} - 6 \zeta_{6} q^{95} + ( - 4 \zeta_{6} + 8) q^{97} +O(q^{100})$$ q + (2*z - 2) * q^3 + (2*z - 1) * q^5 - z * q^9 + (3*z + 1) * q^13 + (-2*z - 2) * q^15 + 3*z * q^17 + (2*z - 4) * q^19 + (6*z - 6) * q^23 + 2 * q^25 - 4 * q^27 + (-3*z + 3) * q^29 + (-4*z + 2) * q^31 + (-5*z - 5) * q^37 + (2*z - 8) * q^39 + (-3*z - 3) * q^41 - 8*z * q^43 + (-z + 2) * q^45 + (4*z - 2) * q^47 + (7*z - 7) * q^49 - 6 * q^51 + 3 * q^53 + (-8*z + 4) * q^57 + (-4*z + 8) * q^59 + z * q^61 + (5*z - 7) * q^65 + (2*z + 2) * q^67 - 12*z * q^69 + (2*z - 4) * q^71 + (2*z - 1) * q^73 + (4*z - 4) * q^75 - 4 * q^79 + (-11*z + 11) * q^81 + (16*z - 8) * q^83 + (3*z - 6) * q^85 + 6*z * q^87 + (-4*z - 4) * q^89 + (4*z + 4) * q^93 - 6*z * q^95 + (-4*z + 8) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - q^9 $$2 q - 2 q^{3} - q^{9} + 5 q^{13} - 6 q^{15} + 3 q^{17} - 6 q^{19} - 6 q^{23} + 4 q^{25} - 8 q^{27} + 3 q^{29} - 15 q^{37} - 14 q^{39} - 9 q^{41} - 8 q^{43} + 3 q^{45} - 7 q^{49} - 12 q^{51} + 6 q^{53} + 12 q^{59} + q^{61} - 9 q^{65} + 6 q^{67} - 12 q^{69} - 6 q^{71} - 4 q^{75} - 8 q^{79} + 11 q^{81} - 9 q^{85} + 6 q^{87} - 12 q^{89} + 12 q^{93} - 6 q^{95} + 12 q^{97}+O(q^{100})$$ 2 * q - 2 * q^3 - q^9 + 5 * q^13 - 6 * q^15 + 3 * q^17 - 6 * q^19 - 6 * q^23 + 4 * q^25 - 8 * q^27 + 3 * q^29 - 15 * q^37 - 14 * q^39 - 9 * q^41 - 8 * q^43 + 3 * q^45 - 7 * q^49 - 12 * q^51 + 6 * q^53 + 12 * q^59 + q^61 - 9 * q^65 + 6 * q^67 - 12 * q^69 - 6 * q^71 - 4 * q^75 - 8 * q^79 + 11 * q^81 - 9 * q^85 + 6 * q^87 - 12 * q^89 + 12 * q^93 - 6 * q^95 + 12 * 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
257.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 −1.00000 1.73205i 0 1.73205i 0 0 0 −0.500000 + 0.866025i 0
641.1 0 −1.00000 + 1.73205i 0 1.73205i 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
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 832.2.w.a 2
4.b odd 2 1 832.2.w.d 2
8.b even 2 1 208.2.w.b 2
8.d odd 2 1 13.2.e.a 2
13.e even 6 1 inner 832.2.w.a 2
24.f even 2 1 117.2.q.c 2
24.h odd 2 1 1872.2.by.d 2
40.e odd 2 1 325.2.n.a 2
40.k even 4 2 325.2.m.a 4
52.i odd 6 1 832.2.w.d 2
56.e even 2 1 637.2.q.a 2
56.k odd 6 1 637.2.k.a 2
56.k odd 6 1 637.2.u.c 2
56.m even 6 1 637.2.k.c 2
56.m even 6 1 637.2.u.b 2
104.h odd 2 1 169.2.e.a 2
104.m even 4 2 169.2.c.a 4
104.n odd 6 1 169.2.b.a 2
104.n odd 6 1 169.2.e.a 2
104.p odd 6 1 13.2.e.a 2
104.p odd 6 1 169.2.b.a 2
104.r even 6 1 2704.2.f.b 2
104.s even 6 1 208.2.w.b 2
104.s even 6 1 2704.2.f.b 2
104.u even 12 2 169.2.a.a 2
104.u even 12 2 169.2.c.a 4
104.x odd 12 2 2704.2.a.o 2
312.ba even 6 1 117.2.q.c 2
312.ba even 6 1 1521.2.b.a 2
312.bg odd 6 1 1872.2.by.d 2
312.bn even 6 1 1521.2.b.a 2
312.bq odd 12 2 1521.2.a.k 2
520.cd odd 6 1 325.2.n.a 2
520.cs even 12 2 325.2.m.a 4
520.cz even 12 2 4225.2.a.v 2
728.bh even 6 1 637.2.k.c 2
728.br odd 6 1 637.2.u.c 2
728.ci even 6 1 637.2.q.a 2
728.cv even 6 1 637.2.u.b 2
728.dd odd 6 1 637.2.k.a 2
728.ec odd 12 2 8281.2.a.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.2.e.a 2 8.d odd 2 1
13.2.e.a 2 104.p odd 6 1
117.2.q.c 2 24.f even 2 1
117.2.q.c 2 312.ba even 6 1
169.2.a.a 2 104.u even 12 2
169.2.b.a 2 104.n odd 6 1
169.2.b.a 2 104.p odd 6 1
169.2.c.a 4 104.m even 4 2
169.2.c.a 4 104.u even 12 2
169.2.e.a 2 104.h odd 2 1
169.2.e.a 2 104.n odd 6 1
208.2.w.b 2 8.b even 2 1
208.2.w.b 2 104.s even 6 1
325.2.m.a 4 40.k even 4 2
325.2.m.a 4 520.cs even 12 2
325.2.n.a 2 40.e odd 2 1
325.2.n.a 2 520.cd odd 6 1
637.2.k.a 2 56.k odd 6 1
637.2.k.a 2 728.dd odd 6 1
637.2.k.c 2 56.m even 6 1
637.2.k.c 2 728.bh even 6 1
637.2.q.a 2 56.e even 2 1
637.2.q.a 2 728.ci even 6 1
637.2.u.b 2 56.m even 6 1
637.2.u.b 2 728.cv even 6 1
637.2.u.c 2 56.k odd 6 1
637.2.u.c 2 728.br odd 6 1
832.2.w.a 2 1.a even 1 1 trivial
832.2.w.a 2 13.e even 6 1 inner
832.2.w.d 2 4.b odd 2 1
832.2.w.d 2 52.i odd 6 1
1521.2.a.k 2 312.bq odd 12 2
1521.2.b.a 2 312.ba even 6 1
1521.2.b.a 2 312.bn even 6 1
1872.2.by.d 2 24.h odd 2 1
1872.2.by.d 2 312.bg odd 6 1
2704.2.a.o 2 104.x odd 12 2
2704.2.f.b 2 104.r even 6 1
2704.2.f.b 2 104.s even 6 1
4225.2.a.v 2 520.cz even 12 2
8281.2.a.q 2 728.ec odd 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 2T_{3} + 4$$ acting on $$S_{2}^{\mathrm{new}}(832, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2T + 4$$
$5$ $$T^{2} + 3$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 5T + 13$$
$17$ $$T^{2} - 3T + 9$$
$19$ $$T^{2} + 6T + 12$$
$23$ $$T^{2} + 6T + 36$$
$29$ $$T^{2} - 3T + 9$$
$31$ $$T^{2} + 12$$
$37$ $$T^{2} + 15T + 75$$
$41$ $$T^{2} + 9T + 27$$
$43$ $$T^{2} + 8T + 64$$
$47$ $$T^{2} + 12$$
$53$ $$(T - 3)^{2}$$
$59$ $$T^{2} - 12T + 48$$
$61$ $$T^{2} - T + 1$$
$67$ $$T^{2} - 6T + 12$$
$71$ $$T^{2} + 6T + 12$$
$73$ $$T^{2} + 3$$
$79$ $$(T + 4)^{2}$$
$83$ $$T^{2} + 192$$
$89$ $$T^{2} + 12T + 48$$
$97$ $$T^{2} - 12T + 48$$