# Properties

 Label 117.2.q.c Level $117$ Weight $2$ Character orbit 117.q Analytic conductor $0.934$ 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] = [117,2,Mod(10,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([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("117.10");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$117 = 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 117.q (of order $$6$$, degree $$2$$, 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: $$0.934249703649$$ 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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} + 1) q^{2} + \zeta_{6} q^{4} + (2 \zeta_{6} - 1) q^{5} + ( - 2 \zeta_{6} + 1) q^{8}+O(q^{10})$$ q + (z + 1) * q^2 + z * q^4 + (2*z - 1) * q^5 + (-2*z + 1) * q^8 $$q + (\zeta_{6} + 1) q^{2} + \zeta_{6} q^{4} + (2 \zeta_{6} - 1) q^{5} + ( - 2 \zeta_{6} + 1) q^{8} + (3 \zeta_{6} - 3) q^{10} + ( - 3 \zeta_{6} - 1) q^{13} + ( - 5 \zeta_{6} + 5) q^{16} - 3 \zeta_{6} q^{17} + (2 \zeta_{6} - 4) q^{19} + (\zeta_{6} - 2) q^{20} + (6 \zeta_{6} - 6) q^{23} + 2 q^{25} + ( - 7 \zeta_{6} + 2) q^{26} + ( - 3 \zeta_{6} + 3) q^{29} + (4 \zeta_{6} - 2) q^{31} + ( - 3 \zeta_{6} + 6) q^{32} + ( - 6 \zeta_{6} + 3) q^{34} + (5 \zeta_{6} + 5) q^{37} - 6 q^{38} + 3 q^{40} + (3 \zeta_{6} + 3) q^{41} - 8 \zeta_{6} q^{43} + (6 \zeta_{6} - 12) q^{46} + (4 \zeta_{6} - 2) q^{47} + (7 \zeta_{6} - 7) q^{49} + (2 \zeta_{6} + 2) q^{50} + ( - 4 \zeta_{6} + 3) q^{52} + 3 q^{53} + ( - 3 \zeta_{6} + 6) q^{58} + (4 \zeta_{6} - 8) q^{59} - \zeta_{6} q^{61} + (6 \zeta_{6} - 6) q^{62} - q^{64} + ( - 5 \zeta_{6} + 7) q^{65} + (2 \zeta_{6} + 2) q^{67} + ( - 3 \zeta_{6} + 3) q^{68} + (2 \zeta_{6} - 4) q^{71} + (2 \zeta_{6} - 1) q^{73} + 15 \zeta_{6} q^{74} + ( - 2 \zeta_{6} - 2) q^{76} + 4 q^{79} + (5 \zeta_{6} + 5) q^{80} + 9 \zeta_{6} q^{82} + ( - 16 \zeta_{6} + 8) q^{83} + ( - 3 \zeta_{6} + 6) q^{85} + ( - 16 \zeta_{6} + 8) q^{86} + (4 \zeta_{6} + 4) q^{89} - 6 q^{92} + (6 \zeta_{6} - 6) q^{94} - 6 \zeta_{6} q^{95} + ( - 4 \zeta_{6} + 8) q^{97} + (7 \zeta_{6} - 14) q^{98} +O(q^{100})$$ q + (z + 1) * q^2 + z * q^4 + (2*z - 1) * q^5 + (-2*z + 1) * q^8 + (3*z - 3) * q^10 + (-3*z - 1) * q^13 + (-5*z + 5) * q^16 - 3*z * q^17 + (2*z - 4) * q^19 + (z - 2) * q^20 + (6*z - 6) * q^23 + 2 * q^25 + (-7*z + 2) * q^26 + (-3*z + 3) * q^29 + (4*z - 2) * q^31 + (-3*z + 6) * q^32 + (-6*z + 3) * q^34 + (5*z + 5) * q^37 - 6 * q^38 + 3 * q^40 + (3*z + 3) * q^41 - 8*z * q^43 + (6*z - 12) * q^46 + (4*z - 2) * q^47 + (7*z - 7) * q^49 + (2*z + 2) * q^50 + (-4*z + 3) * q^52 + 3 * q^53 + (-3*z + 6) * q^58 + (4*z - 8) * q^59 - z * q^61 + (6*z - 6) * q^62 - q^64 + (-5*z + 7) * q^65 + (2*z + 2) * q^67 + (-3*z + 3) * q^68 + (2*z - 4) * q^71 + (2*z - 1) * q^73 + 15*z * q^74 + (-2*z - 2) * q^76 + 4 * q^79 + (5*z + 5) * q^80 + 9*z * q^82 + (-16*z + 8) * q^83 + (-3*z + 6) * q^85 + (-16*z + 8) * q^86 + (4*z + 4) * q^89 - 6 * q^92 + (6*z - 6) * q^94 - 6*z * q^95 + (-4*z + 8) * q^97 + (7*z - 14) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} + q^{4}+O(q^{10})$$ 2 * q + 3 * q^2 + q^4 $$2 q + 3 q^{2} + q^{4} - 3 q^{10} - 5 q^{13} + 5 q^{16} - 3 q^{17} - 6 q^{19} - 3 q^{20} - 6 q^{23} + 4 q^{25} - 3 q^{26} + 3 q^{29} + 9 q^{32} + 15 q^{37} - 12 q^{38} + 6 q^{40} + 9 q^{41} - 8 q^{43} - 18 q^{46} - 7 q^{49} + 6 q^{50} + 2 q^{52} + 6 q^{53} + 9 q^{58} - 12 q^{59} - q^{61} - 6 q^{62} - 2 q^{64} + 9 q^{65} + 6 q^{67} + 3 q^{68} - 6 q^{71} + 15 q^{74} - 6 q^{76} + 8 q^{79} + 15 q^{80} + 9 q^{82} + 9 q^{85} + 12 q^{89} - 12 q^{92} - 6 q^{94} - 6 q^{95} + 12 q^{97} - 21 q^{98}+O(q^{100})$$ 2 * q + 3 * q^2 + q^4 - 3 * q^10 - 5 * q^13 + 5 * q^16 - 3 * q^17 - 6 * q^19 - 3 * q^20 - 6 * q^23 + 4 * q^25 - 3 * q^26 + 3 * q^29 + 9 * q^32 + 15 * q^37 - 12 * q^38 + 6 * q^40 + 9 * q^41 - 8 * q^43 - 18 * q^46 - 7 * q^49 + 6 * q^50 + 2 * q^52 + 6 * q^53 + 9 * q^58 - 12 * q^59 - q^61 - 6 * q^62 - 2 * q^64 + 9 * q^65 + 6 * q^67 + 3 * q^68 - 6 * q^71 + 15 * q^74 - 6 * q^76 + 8 * q^79 + 15 * q^80 + 9 * q^82 + 9 * q^85 + 12 * q^89 - 12 * q^92 - 6 * q^94 - 6 * q^95 + 12 * q^97 - 21 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$$.

 $$n$$ $$28$$ $$92$$ $$\chi(n)$$ $$\zeta_{6}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
10.1
 0.5 − 0.866025i 0.5 + 0.866025i
1.50000 0.866025i 0 0.500000 0.866025i 1.73205i 0 0 1.73205i 0 −1.50000 2.59808i
82.1 1.50000 + 0.866025i 0 0.500000 + 0.866025i 1.73205i 0 0 1.73205i 0 −1.50000 + 2.59808i
 $$n$$: 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 117.2.q.c 2
3.b odd 2 1 13.2.e.a 2
4.b odd 2 1 1872.2.by.d 2
12.b even 2 1 208.2.w.b 2
13.c even 3 1 1521.2.b.a 2
13.e even 6 1 inner 117.2.q.c 2
13.e even 6 1 1521.2.b.a 2
13.f odd 12 2 1521.2.a.k 2
15.d odd 2 1 325.2.n.a 2
15.e even 4 2 325.2.m.a 4
21.c even 2 1 637.2.q.a 2
21.g even 6 1 637.2.k.c 2
21.g even 6 1 637.2.u.b 2
21.h odd 6 1 637.2.k.a 2
21.h odd 6 1 637.2.u.c 2
24.f even 2 1 832.2.w.a 2
24.h odd 2 1 832.2.w.d 2
39.d odd 2 1 169.2.e.a 2
39.f even 4 2 169.2.c.a 4
39.h odd 6 1 13.2.e.a 2
39.h odd 6 1 169.2.b.a 2
39.i odd 6 1 169.2.b.a 2
39.i odd 6 1 169.2.e.a 2
39.k even 12 2 169.2.a.a 2
39.k even 12 2 169.2.c.a 4
52.i odd 6 1 1872.2.by.d 2
156.p even 6 1 2704.2.f.b 2
156.r even 6 1 208.2.w.b 2
156.r even 6 1 2704.2.f.b 2
156.v odd 12 2 2704.2.a.o 2
195.y odd 6 1 325.2.n.a 2
195.bf even 12 2 325.2.m.a 4
195.bh even 12 2 4225.2.a.v 2
273.u even 6 1 637.2.q.a 2
273.x odd 6 1 637.2.k.a 2
273.y even 6 1 637.2.k.c 2
273.bp odd 6 1 637.2.u.c 2
273.br even 6 1 637.2.u.b 2
273.ca odd 12 2 8281.2.a.q 2
312.ba even 6 1 832.2.w.a 2
312.bg odd 6 1 832.2.w.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.2.e.a 2 3.b odd 2 1
13.2.e.a 2 39.h odd 6 1
117.2.q.c 2 1.a even 1 1 trivial
117.2.q.c 2 13.e even 6 1 inner
169.2.a.a 2 39.k even 12 2
169.2.b.a 2 39.h odd 6 1
169.2.b.a 2 39.i odd 6 1
169.2.c.a 4 39.f even 4 2
169.2.c.a 4 39.k even 12 2
169.2.e.a 2 39.d odd 2 1
169.2.e.a 2 39.i odd 6 1
208.2.w.b 2 12.b even 2 1
208.2.w.b 2 156.r even 6 1
325.2.m.a 4 15.e even 4 2
325.2.m.a 4 195.bf even 12 2
325.2.n.a 2 15.d odd 2 1
325.2.n.a 2 195.y odd 6 1
637.2.k.a 2 21.h odd 6 1
637.2.k.a 2 273.x odd 6 1
637.2.k.c 2 21.g even 6 1
637.2.k.c 2 273.y even 6 1
637.2.q.a 2 21.c even 2 1
637.2.q.a 2 273.u even 6 1
637.2.u.b 2 21.g even 6 1
637.2.u.b 2 273.br even 6 1
637.2.u.c 2 21.h odd 6 1
637.2.u.c 2 273.bp odd 6 1
832.2.w.a 2 24.f even 2 1
832.2.w.a 2 312.ba even 6 1
832.2.w.d 2 24.h odd 2 1
832.2.w.d 2 312.bg odd 6 1
1521.2.a.k 2 13.f odd 12 2
1521.2.b.a 2 13.c even 3 1
1521.2.b.a 2 13.e even 6 1
1872.2.by.d 2 4.b odd 2 1
1872.2.by.d 2 52.i odd 6 1
2704.2.a.o 2 156.v odd 12 2
2704.2.f.b 2 156.p even 6 1
2704.2.f.b 2 156.r even 6 1
4225.2.a.v 2 195.bh even 12 2
8281.2.a.q 2 273.ca odd 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(117, [\chi])$$:

 $$T_{2}^{2} - 3T_{2} + 3$$ T2^2 - 3*T2 + 3 $$T_{5}^{2} + 3$$ T5^2 + 3

## Hecke characteristic polynomials

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