# Properties

 Label 117.2.g.a Level $117$ Weight $2$ Character orbit 117.g Analytic conductor $0.934$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,2,Mod(55,117)]

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, 2]))

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

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

chi := DirichletCharacter("117.55");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$117 = 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 117.g (of order $$3$$, 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: yes Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{4} + \zeta_{6} q^{7}+O(q^{10})$$ q + 2*z * q^4 + z * q^7 $$q + 2 \zeta_{6} q^{4} + \zeta_{6} q^{7} + ( - 3 \zeta_{6} + 4) q^{13} + (4 \zeta_{6} - 4) q^{16} - 8 \zeta_{6} q^{19} - 5 q^{25} + (2 \zeta_{6} - 2) q^{28} - 7 q^{31} + ( - 10 \zeta_{6} + 10) q^{37} + 13 \zeta_{6} q^{43} + ( - 6 \zeta_{6} + 6) q^{49} + (2 \zeta_{6} + 6) q^{52} + 13 \zeta_{6} q^{61} - 8 q^{64} + (11 \zeta_{6} - 11) q^{67} + 17 q^{73} + ( - 16 \zeta_{6} + 16) q^{76} - 13 q^{79} + (\zeta_{6} + 3) q^{91} - 5 \zeta_{6} q^{97} +O(q^{100})$$ q + 2*z * q^4 + z * q^7 + (-3*z + 4) * q^13 + (4*z - 4) * q^16 - 8*z * q^19 - 5 * q^25 + (2*z - 2) * q^28 - 7 * q^31 + (-10*z + 10) * q^37 + 13*z * q^43 + (-6*z + 6) * q^49 + (2*z + 6) * q^52 + 13*z * q^61 - 8 * q^64 + (11*z - 11) * q^67 + 17 * q^73 + (-16*z + 16) * q^76 - 13 * q^79 + (z + 3) * q^91 - 5*z * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} + q^{7}+O(q^{10})$$ 2 * q + 2 * q^4 + q^7 $$2 q + 2 q^{4} + q^{7} + 5 q^{13} - 4 q^{16} - 8 q^{19} - 10 q^{25} - 2 q^{28} - 14 q^{31} + 10 q^{37} + 13 q^{43} + 6 q^{49} + 14 q^{52} + 13 q^{61} - 16 q^{64} - 11 q^{67} + 34 q^{73} + 16 q^{76} - 26 q^{79} + 7 q^{91} - 5 q^{97}+O(q^{100})$$ 2 * q + 2 * q^4 + q^7 + 5 * q^13 - 4 * q^16 - 8 * q^19 - 10 * q^25 - 2 * q^28 - 14 * q^31 + 10 * q^37 + 13 * q^43 + 6 * q^49 + 14 * q^52 + 13 * q^61 - 16 * q^64 - 11 * q^67 + 34 * q^73 + 16 * q^76 - 26 * q^79 + 7 * q^91 - 5 * 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 1.00000 1.73205i 0 0 0.500000 0.866025i 0 0 0
100.1 0 0 1.00000 + 1.73205i 0 0 0.500000 + 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
13.c even 3 1 inner
39.i odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.2.g.a 2
3.b odd 2 1 CM 117.2.g.a 2
4.b odd 2 1 1872.2.t.g 2
12.b even 2 1 1872.2.t.g 2
13.c even 3 1 inner 117.2.g.a 2
13.c even 3 1 1521.2.a.b 1
13.e even 6 1 1521.2.a.c 1
13.f odd 12 2 1521.2.b.e 2
39.h odd 6 1 1521.2.a.c 1
39.i odd 6 1 inner 117.2.g.a 2
39.i odd 6 1 1521.2.a.b 1
39.k even 12 2 1521.2.b.e 2
52.j odd 6 1 1872.2.t.g 2
156.p even 6 1 1872.2.t.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.2.g.a 2 1.a even 1 1 trivial
117.2.g.a 2 3.b odd 2 1 CM
117.2.g.a 2 13.c even 3 1 inner
117.2.g.a 2 39.i odd 6 1 inner
1521.2.a.b 1 13.c even 3 1
1521.2.a.b 1 39.i odd 6 1
1521.2.a.c 1 13.e even 6 1
1521.2.a.c 1 39.h odd 6 1
1521.2.b.e 2 13.f odd 12 2
1521.2.b.e 2 39.k even 12 2
1872.2.t.g 2 4.b odd 2 1
1872.2.t.g 2 12.b even 2 1
1872.2.t.g 2 52.j odd 6 1
1872.2.t.g 2 156.p even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{2}^{\mathrm{new}}(117, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - T + 1$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 5T + 13$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 8T + 64$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$(T + 7)^{2}$$
$37$ $$T^{2} - 10T + 100$$
$41$ $$T^{2}$$
$43$ $$T^{2} - 13T + 169$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 13T + 169$$
$67$ $$T^{2} + 11T + 121$$
$71$ $$T^{2}$$
$73$ $$(T - 17)^{2}$$
$79$ $$(T + 13)^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 5T + 25$$