# Properties

 Label 117.2.b.a Level $117$ Weight $2$ Character orbit 117.b 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(64,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("117.64");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$117 = 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 117.b (of order $$2$$, degree $$1$$, 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: $$2$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$\beta = \sqrt{-3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} - q^{4} - 2 \beta q^{7} - \beta q^{8} +O(q^{10})$$ q - b * q^2 - q^4 - 2*b * q^7 - b * q^8 $$q - \beta q^{2} - q^{4} - 2 \beta q^{7} - \beta q^{8} + 2 \beta q^{11} + (2 \beta - 1) q^{13} - 6 q^{14} - 5 q^{16} + 6 q^{17} + 2 \beta q^{19} + 6 q^{22} + 5 q^{25} + (\beta + 6) q^{26} + 2 \beta q^{28} - 6 q^{29} - 2 \beta q^{31} + 3 \beta q^{32} - 6 \beta q^{34} + 4 \beta q^{37} + 6 q^{38} + 4 \beta q^{41} - 4 q^{43} - 2 \beta q^{44} - 2 \beta q^{47} - 5 q^{49} - 5 \beta q^{50} + ( - 2 \beta + 1) q^{52} - 6 q^{53} - 6 q^{56} + 6 \beta q^{58} - 6 \beta q^{59} - 2 q^{61} - 6 q^{62} - q^{64} - 6 \beta q^{67} - 6 q^{68} - 2 \beta q^{71} + 12 q^{74} - 2 \beta q^{76} + 12 q^{77} - 8 q^{79} + 12 q^{82} + 2 \beta q^{83} + 4 \beta q^{86} + 6 q^{88} + 4 \beta q^{89} + (2 \beta + 12) q^{91} - 6 q^{94} - 8 \beta q^{97} + 5 \beta q^{98} +O(q^{100})$$ q - b * q^2 - q^4 - 2*b * q^7 - b * q^8 + 2*b * q^11 + (2*b - 1) * q^13 - 6 * q^14 - 5 * q^16 + 6 * q^17 + 2*b * q^19 + 6 * q^22 + 5 * q^25 + (b + 6) * q^26 + 2*b * q^28 - 6 * q^29 - 2*b * q^31 + 3*b * q^32 - 6*b * q^34 + 4*b * q^37 + 6 * q^38 + 4*b * q^41 - 4 * q^43 - 2*b * q^44 - 2*b * q^47 - 5 * q^49 - 5*b * q^50 + (-2*b + 1) * q^52 - 6 * q^53 - 6 * q^56 + 6*b * q^58 - 6*b * q^59 - 2 * q^61 - 6 * q^62 - q^64 - 6*b * q^67 - 6 * q^68 - 2*b * q^71 + 12 * q^74 - 2*b * q^76 + 12 * q^77 - 8 * q^79 + 12 * q^82 + 2*b * q^83 + 4*b * q^86 + 6 * q^88 + 4*b * q^89 + (2*b + 12) * q^91 - 6 * q^94 - 8*b * q^97 + 5*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4}+O(q^{10})$$ 2 * q - 2 * q^4 $$2 q - 2 q^{4} - 2 q^{13} - 12 q^{14} - 10 q^{16} + 12 q^{17} + 12 q^{22} + 10 q^{25} + 12 q^{26} - 12 q^{29} + 12 q^{38} - 8 q^{43} - 10 q^{49} + 2 q^{52} - 12 q^{53} - 12 q^{56} - 4 q^{61} - 12 q^{62} - 2 q^{64} - 12 q^{68} + 24 q^{74} + 24 q^{77} - 16 q^{79} + 24 q^{82} + 12 q^{88} + 24 q^{91} - 12 q^{94}+O(q^{100})$$ 2 * q - 2 * q^4 - 2 * q^13 - 12 * q^14 - 10 * q^16 + 12 * q^17 + 12 * q^22 + 10 * q^25 + 12 * q^26 - 12 * q^29 + 12 * q^38 - 8 * q^43 - 10 * q^49 + 2 * q^52 - 12 * q^53 - 12 * q^56 - 4 * q^61 - 12 * q^62 - 2 * q^64 - 12 * q^68 + 24 * q^74 + 24 * q^77 - 16 * q^79 + 24 * q^82 + 12 * q^88 + 24 * q^91 - 12 * q^94

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
64.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.73205i 0 −1.00000 0 0 3.46410i 1.73205i 0 0
64.2 1.73205i 0 −1.00000 0 0 3.46410i 1.73205i 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.2.b.a 2
3.b odd 2 1 39.2.b.a 2
4.b odd 2 1 1872.2.c.e 2
12.b even 2 1 624.2.c.e 2
13.b even 2 1 inner 117.2.b.a 2
13.d odd 4 2 1521.2.a.l 2
15.d odd 2 1 975.2.b.d 2
15.e even 4 2 975.2.h.f 4
21.c even 2 1 1911.2.c.d 2
24.f even 2 1 2496.2.c.d 2
24.h odd 2 1 2496.2.c.k 2
39.d odd 2 1 39.2.b.a 2
39.f even 4 2 507.2.a.f 2
39.h odd 6 1 507.2.j.a 2
39.h odd 6 1 507.2.j.c 2
39.i odd 6 1 507.2.j.a 2
39.i odd 6 1 507.2.j.c 2
39.k even 12 4 507.2.e.e 4
52.b odd 2 1 1872.2.c.e 2
156.h even 2 1 624.2.c.e 2
156.l odd 4 2 8112.2.a.bv 2
195.e odd 2 1 975.2.b.d 2
195.s even 4 2 975.2.h.f 4
273.g even 2 1 1911.2.c.d 2
312.b odd 2 1 2496.2.c.k 2
312.h even 2 1 2496.2.c.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.b.a 2 3.b odd 2 1
39.2.b.a 2 39.d odd 2 1
117.2.b.a 2 1.a even 1 1 trivial
117.2.b.a 2 13.b even 2 1 inner
507.2.a.f 2 39.f even 4 2
507.2.e.e 4 39.k even 12 4
507.2.j.a 2 39.h odd 6 1
507.2.j.a 2 39.i odd 6 1
507.2.j.c 2 39.h odd 6 1
507.2.j.c 2 39.i odd 6 1
624.2.c.e 2 12.b even 2 1
624.2.c.e 2 156.h even 2 1
975.2.b.d 2 15.d odd 2 1
975.2.b.d 2 195.e odd 2 1
975.2.h.f 4 15.e even 4 2
975.2.h.f 4 195.s even 4 2
1521.2.a.l 2 13.d odd 4 2
1872.2.c.e 2 4.b odd 2 1
1872.2.c.e 2 52.b odd 2 1
1911.2.c.d 2 21.c even 2 1
1911.2.c.d 2 273.g even 2 1
2496.2.c.d 2 24.f even 2 1
2496.2.c.d 2 312.h even 2 1
2496.2.c.k 2 24.h odd 2 1
2496.2.c.k 2 312.b odd 2 1
8112.2.a.bv 2 156.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 3$$ acting on $$S_{2}^{\mathrm{new}}(117, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 3$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 12$$
$11$ $$T^{2} + 12$$
$13$ $$T^{2} + 2T + 13$$
$17$ $$(T - 6)^{2}$$
$19$ $$T^{2} + 12$$
$23$ $$T^{2}$$
$29$ $$(T + 6)^{2}$$
$31$ $$T^{2} + 12$$
$37$ $$T^{2} + 48$$
$41$ $$T^{2} + 48$$
$43$ $$(T + 4)^{2}$$
$47$ $$T^{2} + 12$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2} + 108$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} + 108$$
$71$ $$T^{2} + 12$$
$73$ $$T^{2}$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2} + 12$$
$89$ $$T^{2} + 48$$
$97$ $$T^{2} + 192$$