# Properties

 Label 117.2.q.a Level $117$ Weight $2$ Character orbit 117.q Analytic conductor $0.934$ Analytic rank $0$ Dimension $2$ CM no 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) 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.

 $$f(q)$$ $$=$$ $$q - 2 \zeta_{6} q^{4} + ( - 4 \zeta_{6} + 2) q^{5} + (\zeta_{6} - 2) q^{7} +O(q^{10})$$ q - 2*z * q^4 + (-4*z + 2) * q^5 + (z - 2) * q^7 $$q - 2 \zeta_{6} q^{4} + ( - 4 \zeta_{6} + 2) q^{5} + (\zeta_{6} - 2) q^{7} + (2 \zeta_{6} + 2) q^{11} + ( - \zeta_{6} + 4) q^{13} + (4 \zeta_{6} - 4) q^{16} + ( - 2 \zeta_{6} + 4) q^{19} + (4 \zeta_{6} - 8) q^{20} + (6 \zeta_{6} - 6) q^{23} - 7 q^{25} + (2 \zeta_{6} + 2) q^{28} + ( - 6 \zeta_{6} + 6) q^{29} + (2 \zeta_{6} - 1) q^{31} + 6 \zeta_{6} q^{35} + (4 \zeta_{6} + 4) q^{41} + \zeta_{6} q^{43} + ( - 8 \zeta_{6} + 4) q^{44} + ( - 4 \zeta_{6} + 2) q^{47} + (4 \zeta_{6} - 4) q^{49} + ( - 6 \zeta_{6} - 2) q^{52} - 12 q^{53} + ( - 12 \zeta_{6} + 12) q^{55} + ( - 2 \zeta_{6} + 4) q^{59} - \zeta_{6} q^{61} + 8 q^{64} + ( - 14 \zeta_{6} + 4) q^{65} + (5 \zeta_{6} + 5) q^{67} + ( - 6 \zeta_{6} + 12) q^{71} + (2 \zeta_{6} - 1) q^{73} + ( - 4 \zeta_{6} - 4) q^{76} - 6 q^{77} - 11 q^{79} + (8 \zeta_{6} + 8) q^{80} + (16 \zeta_{6} - 8) q^{83} + ( - 4 \zeta_{6} - 4) q^{89} + (5 \zeta_{6} - 7) q^{91} + 12 q^{92} - 12 \zeta_{6} q^{95} + (3 \zeta_{6} - 6) q^{97} +O(q^{100})$$ q - 2*z * q^4 + (-4*z + 2) * q^5 + (z - 2) * q^7 + (2*z + 2) * q^11 + (-z + 4) * q^13 + (4*z - 4) * q^16 + (-2*z + 4) * q^19 + (4*z - 8) * q^20 + (6*z - 6) * q^23 - 7 * q^25 + (2*z + 2) * q^28 + (-6*z + 6) * q^29 + (2*z - 1) * q^31 + 6*z * q^35 + (4*z + 4) * q^41 + z * q^43 + (-8*z + 4) * q^44 + (-4*z + 2) * q^47 + (4*z - 4) * q^49 + (-6*z - 2) * q^52 - 12 * q^53 + (-12*z + 12) * q^55 + (-2*z + 4) * q^59 - z * q^61 + 8 * q^64 + (-14*z + 4) * q^65 + (5*z + 5) * q^67 + (-6*z + 12) * q^71 + (2*z - 1) * q^73 + (-4*z - 4) * q^76 - 6 * q^77 - 11 * q^79 + (8*z + 8) * q^80 + (16*z - 8) * q^83 + (-4*z - 4) * q^89 + (5*z - 7) * q^91 + 12 * q^92 - 12*z * q^95 + (3*z - 6) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 3 q^{7}+O(q^{10})$$ 2 * q - 2 * q^4 - 3 * q^7 $$2 q - 2 q^{4} - 3 q^{7} + 6 q^{11} + 7 q^{13} - 4 q^{16} + 6 q^{19} - 12 q^{20} - 6 q^{23} - 14 q^{25} + 6 q^{28} + 6 q^{29} + 6 q^{35} + 12 q^{41} + q^{43} - 4 q^{49} - 10 q^{52} - 24 q^{53} + 12 q^{55} + 6 q^{59} - q^{61} + 16 q^{64} - 6 q^{65} + 15 q^{67} + 18 q^{71} - 12 q^{76} - 12 q^{77} - 22 q^{79} + 24 q^{80} - 12 q^{89} - 9 q^{91} + 24 q^{92} - 12 q^{95} - 9 q^{97}+O(q^{100})$$ 2 * q - 2 * q^4 - 3 * q^7 + 6 * q^11 + 7 * q^13 - 4 * q^16 + 6 * q^19 - 12 * q^20 - 6 * q^23 - 14 * q^25 + 6 * q^28 + 6 * q^29 + 6 * q^35 + 12 * q^41 + q^43 - 4 * q^49 - 10 * q^52 - 24 * q^53 + 12 * q^55 + 6 * q^59 - q^61 + 16 * q^64 - 6 * q^65 + 15 * q^67 + 18 * q^71 - 12 * q^76 - 12 * q^77 - 22 * q^79 + 24 * q^80 - 12 * q^89 - 9 * q^91 + 24 * q^92 - 12 * q^95 - 9 * q^97

## 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
0 0 −1.00000 + 1.73205i 3.46410i 0 −1.50000 0.866025i 0 0 0
82.1 0 0 −1.00000 1.73205i 3.46410i 0 −1.50000 + 0.866025i 0 0 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 117.2.q.a 2
3.b odd 2 1 39.2.j.a 2
4.b odd 2 1 1872.2.by.f 2
12.b even 2 1 624.2.bv.b 2
13.c even 3 1 1521.2.b.f 2
13.e even 6 1 inner 117.2.q.a 2
13.e even 6 1 1521.2.b.f 2
13.f odd 12 2 1521.2.a.h 2
15.d odd 2 1 975.2.bc.c 2
15.e even 4 2 975.2.w.d 4
39.d odd 2 1 507.2.j.b 2
39.f even 4 2 507.2.e.f 4
39.h odd 6 1 39.2.j.a 2
39.h odd 6 1 507.2.b.c 2
39.i odd 6 1 507.2.b.c 2
39.i odd 6 1 507.2.j.b 2
39.k even 12 2 507.2.a.e 2
39.k even 12 2 507.2.e.f 4
52.i odd 6 1 1872.2.by.f 2
156.r even 6 1 624.2.bv.b 2
156.v odd 12 2 8112.2.a.bu 2
195.y odd 6 1 975.2.bc.c 2
195.bf even 12 2 975.2.w.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.j.a 2 3.b odd 2 1
39.2.j.a 2 39.h odd 6 1
117.2.q.a 2 1.a even 1 1 trivial
117.2.q.a 2 13.e even 6 1 inner
507.2.a.e 2 39.k even 12 2
507.2.b.c 2 39.h odd 6 1
507.2.b.c 2 39.i odd 6 1
507.2.e.f 4 39.f even 4 2
507.2.e.f 4 39.k even 12 2
507.2.j.b 2 39.d odd 2 1
507.2.j.b 2 39.i odd 6 1
624.2.bv.b 2 12.b even 2 1
624.2.bv.b 2 156.r even 6 1
975.2.w.d 4 15.e even 4 2
975.2.w.d 4 195.bf even 12 2
975.2.bc.c 2 15.d odd 2 1
975.2.bc.c 2 195.y odd 6 1
1521.2.a.h 2 13.f odd 12 2
1521.2.b.f 2 13.c even 3 1
1521.2.b.f 2 13.e even 6 1
1872.2.by.f 2 4.b odd 2 1
1872.2.by.f 2 52.i odd 6 1
8112.2.a.bu 2 156.v odd 12 2

## Hecke kernels

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

 $$T_{2}$$ T2 $$T_{5}^{2} + 12$$ T5^2 + 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 12$$
$7$ $$T^{2} + 3T + 3$$
$11$ $$T^{2} - 6T + 12$$
$13$ $$T^{2} - 7T + 13$$
$17$ $$T^{2}$$
$19$ $$T^{2} - 6T + 12$$
$23$ $$T^{2} + 6T + 36$$
$29$ $$T^{2} - 6T + 36$$
$31$ $$T^{2} + 3$$
$37$ $$T^{2}$$
$41$ $$T^{2} - 12T + 48$$
$43$ $$T^{2} - T + 1$$
$47$ $$T^{2} + 12$$
$53$ $$(T + 12)^{2}$$
$59$ $$T^{2} - 6T + 12$$
$61$ $$T^{2} + T + 1$$
$67$ $$T^{2} - 15T + 75$$
$71$ $$T^{2} - 18T + 108$$
$73$ $$T^{2} + 3$$
$79$ $$(T + 11)^{2}$$
$83$ $$T^{2} + 192$$
$89$ $$T^{2} + 12T + 48$$
$97$ $$T^{2} + 9T + 27$$