# Properties

 Label 117.1.j.a Level $117$ Weight $1$ Character orbit 117.j Analytic conductor $0.058$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -3 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,1,Mod(73,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("117.73");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$117 = 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 117.j (of order $$4$$, 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: $$0.0583906064781$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.6591.1 Artin image: $C_4\wr C_2$ Artin field: Galois closure of 8.0.1601613.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q + i q^{4} + ( - i - 1) q^{7}+O(q^{10})$$ q + z * q^4 + (-z - 1) * q^7 $$q + i q^{4} + ( - i - 1) q^{7} - i q^{13} - q^{16} + (i - 1) q^{19} + i q^{25} + ( - i + 1) q^{28} + ( - i + 1) q^{31} + (i + 1) q^{37} + i q^{49} + q^{52} - i q^{64} + (i - 1) q^{67} + ( - i - 1) q^{73} + ( - i - 1) q^{76} + (i - 1) q^{91} + ( - i + 1) q^{97} +O(q^{100})$$ q + z * q^4 + (-z - 1) * q^7 - z * q^13 - q^16 + (z - 1) * q^19 + z * q^25 + (-z + 1) * q^28 + (-z + 1) * q^31 + (z + 1) * q^37 + z * q^49 + q^52 - z * q^64 + (z - 1) * q^67 + (-z - 1) * q^73 + (-z - 1) * q^76 + (z - 1) * q^91 + (-z + 1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{7}+O(q^{10})$$ 2 * q - 2 * q^7 $$2 q - 2 q^{7} - 2 q^{16} - 2 q^{19} + 2 q^{28} + 2 q^{31} + 2 q^{37} + 2 q^{52} - 2 q^{67} - 2 q^{73} - 2 q^{76} - 2 q^{91} + 2 q^{97}+O(q^{100})$$ 2 * q - 2 * q^7 - 2 * q^16 - 2 * q^19 + 2 * q^28 + 2 * q^31 + 2 * q^37 + 2 * q^52 - 2 * q^67 - 2 * q^73 - 2 * q^76 - 2 * q^91 + 2 * q^97

## Character values

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

 $$n$$ $$28$$ $$92$$ $$\chi(n)$$ $$-i$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
73.1
 − 1.00000i 1.00000i
0 0 1.00000i 0 0 −1.00000 + 1.00000i 0 0 0
109.1 0 0 1.00000i 0 0 −1.00000 1.00000i 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.d odd 4 1 inner
39.f even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.1.j.a 2
3.b odd 2 1 CM 117.1.j.a 2
4.b odd 2 1 1872.1.bd.a 2
5.b even 2 1 2925.1.s.a 2
5.c odd 4 1 2925.1.t.a 2
5.c odd 4 1 2925.1.t.b 2
9.c even 3 2 1053.1.bb.a 4
9.d odd 6 2 1053.1.bb.a 4
12.b even 2 1 1872.1.bd.a 2
13.b even 2 1 1521.1.j.b 2
13.c even 3 2 1521.1.bd.c 4
13.d odd 4 1 inner 117.1.j.a 2
13.d odd 4 1 1521.1.j.b 2
13.e even 6 2 1521.1.bd.b 4
13.f odd 12 2 1521.1.bd.b 4
13.f odd 12 2 1521.1.bd.c 4
15.d odd 2 1 2925.1.s.a 2
15.e even 4 1 2925.1.t.a 2
15.e even 4 1 2925.1.t.b 2
39.d odd 2 1 1521.1.j.b 2
39.f even 4 1 inner 117.1.j.a 2
39.f even 4 1 1521.1.j.b 2
39.h odd 6 2 1521.1.bd.b 4
39.i odd 6 2 1521.1.bd.c 4
39.k even 12 2 1521.1.bd.b 4
39.k even 12 2 1521.1.bd.c 4
52.f even 4 1 1872.1.bd.a 2
65.f even 4 1 2925.1.t.a 2
65.g odd 4 1 2925.1.s.a 2
65.k even 4 1 2925.1.t.b 2
117.y odd 12 2 1053.1.bb.a 4
117.z even 12 2 1053.1.bb.a 4
156.l odd 4 1 1872.1.bd.a 2
195.j odd 4 1 2925.1.t.b 2
195.n even 4 1 2925.1.s.a 2
195.u odd 4 1 2925.1.t.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.1.j.a 2 1.a even 1 1 trivial
117.1.j.a 2 3.b odd 2 1 CM
117.1.j.a 2 13.d odd 4 1 inner
117.1.j.a 2 39.f even 4 1 inner
1053.1.bb.a 4 9.c even 3 2
1053.1.bb.a 4 9.d odd 6 2
1053.1.bb.a 4 117.y odd 12 2
1053.1.bb.a 4 117.z even 12 2
1521.1.j.b 2 13.b even 2 1
1521.1.j.b 2 13.d odd 4 1
1521.1.j.b 2 39.d odd 2 1
1521.1.j.b 2 39.f even 4 1
1521.1.bd.b 4 13.e even 6 2
1521.1.bd.b 4 13.f odd 12 2
1521.1.bd.b 4 39.h odd 6 2
1521.1.bd.b 4 39.k even 12 2
1521.1.bd.c 4 13.c even 3 2
1521.1.bd.c 4 13.f odd 12 2
1521.1.bd.c 4 39.i odd 6 2
1521.1.bd.c 4 39.k even 12 2
1872.1.bd.a 2 4.b odd 2 1
1872.1.bd.a 2 12.b even 2 1
1872.1.bd.a 2 52.f even 4 1
1872.1.bd.a 2 156.l odd 4 1
2925.1.s.a 2 5.b even 2 1
2925.1.s.a 2 15.d odd 2 1
2925.1.s.a 2 65.g odd 4 1
2925.1.s.a 2 195.n even 4 1
2925.1.t.a 2 5.c odd 4 1
2925.1.t.a 2 15.e even 4 1
2925.1.t.a 2 65.f even 4 1
2925.1.t.a 2 195.u odd 4 1
2925.1.t.b 2 5.c odd 4 1
2925.1.t.b 2 15.e even 4 1
2925.1.t.b 2 65.k even 4 1
2925.1.t.b 2 195.j odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(117, [\chi])$$.

## Hecke characteristic polynomials

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