# Properties

 Label 117.2.q.d Level $117$ Weight $2$ Character orbit 117.q Analytic conductor $0.934$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 5x^{2} + 25$$ x^4 - 5*x^2 + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + 3 \beta_{2} q^{4} - \beta_{3} q^{5} + (2 \beta_{2} - 4) q^{7} + \beta_{3} q^{8}+O(q^{10})$$ q + b1 * q^2 + 3*b2 * q^4 - b3 * q^5 + (2*b2 - 4) * q^7 + b3 * q^8 $$q + \beta_1 q^{2} + 3 \beta_{2} q^{4} - \beta_{3} q^{5} + (2 \beta_{2} - 4) q^{7} + \beta_{3} q^{8} + ( - 5 \beta_{2} + 5) q^{10} - 2 \beta_1 q^{11} + (3 \beta_{2} + 1) q^{13} + (2 \beta_{3} - 4 \beta_1) q^{14} + ( - \beta_{2} + 1) q^{16} + (\beta_{3} + \beta_1) q^{17} + (2 \beta_{2} - 4) q^{19} + ( - 3 \beta_{3} + 3 \beta_1) q^{20} - 10 \beta_{2} q^{22} + ( - 4 \beta_{3} + 2 \beta_1) q^{23} + (3 \beta_{3} + \beta_1) q^{26} + ( - 6 \beta_{2} - 6) q^{28} + (2 \beta_{3} - \beta_1) q^{29} + ( - 3 \beta_{3} + 3 \beta_1) q^{32} + (10 \beta_{2} - 5) q^{34} + (2 \beta_{3} + 2 \beta_1) q^{35} + (\beta_{2} + 1) q^{37} + (2 \beta_{3} - 4 \beta_1) q^{38} + 5 q^{40} + \beta_1 q^{41} - 2 \beta_{2} q^{43} - 6 \beta_{3} q^{44} + ( - 10 \beta_{2} + 20) q^{46} + 2 \beta_{3} q^{47} + ( - 5 \beta_{2} + 5) q^{49} + (12 \beta_{2} - 9) q^{52} + (3 \beta_{3} - 6 \beta_1) q^{53} + (10 \beta_{2} - 10) q^{55} + ( - 2 \beta_{3} - 2 \beta_1) q^{56} + (5 \beta_{2} - 10) q^{58} + (4 \beta_{3} - 4 \beta_1) q^{59} + 7 \beta_{2} q^{61} + 13 q^{64} + ( - 4 \beta_{3} + 3 \beta_1) q^{65} + ( - 2 \beta_{2} - 2) q^{67} + (6 \beta_{3} - 3 \beta_1) q^{68} + (20 \beta_{2} - 10) q^{70} + ( - 2 \beta_{3} + 2 \beta_1) q^{71} + ( - 18 \beta_{2} + 9) q^{73} + (\beta_{3} + \beta_1) q^{74} + ( - 6 \beta_{2} - 6) q^{76} + ( - 4 \beta_{3} + 8 \beta_1) q^{77} + 8 q^{79} - \beta_1 q^{80} + 5 \beta_{2} q^{82} + 2 \beta_{3} q^{83} + ( - 5 \beta_{2} + 10) q^{85} - 2 \beta_{3} q^{86} + ( - 10 \beta_{2} + 10) q^{88} - 2 \beta_1 q^{89} + ( - 4 \beta_{2} - 10) q^{91} + ( - 6 \beta_{3} + 12 \beta_1) q^{92} + (10 \beta_{2} - 10) q^{94} + (2 \beta_{3} + 2 \beta_1) q^{95} + ( - 4 \beta_{2} + 8) q^{97} + ( - 5 \beta_{3} + 5 \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + 3*b2 * q^4 - b3 * q^5 + (2*b2 - 4) * q^7 + b3 * q^8 + (-5*b2 + 5) * q^10 - 2*b1 * q^11 + (3*b2 + 1) * q^13 + (2*b3 - 4*b1) * q^14 + (-b2 + 1) * q^16 + (b3 + b1) * q^17 + (2*b2 - 4) * q^19 + (-3*b3 + 3*b1) * q^20 - 10*b2 * q^22 + (-4*b3 + 2*b1) * q^23 + (3*b3 + b1) * q^26 + (-6*b2 - 6) * q^28 + (2*b3 - b1) * q^29 + (-3*b3 + 3*b1) * q^32 + (10*b2 - 5) * q^34 + (2*b3 + 2*b1) * q^35 + (b2 + 1) * q^37 + (2*b3 - 4*b1) * q^38 + 5 * q^40 + b1 * q^41 - 2*b2 * q^43 - 6*b3 * q^44 + (-10*b2 + 20) * q^46 + 2*b3 * q^47 + (-5*b2 + 5) * q^49 + (12*b2 - 9) * q^52 + (3*b3 - 6*b1) * q^53 + (10*b2 - 10) * q^55 + (-2*b3 - 2*b1) * q^56 + (5*b2 - 10) * q^58 + (4*b3 - 4*b1) * q^59 + 7*b2 * q^61 + 13 * q^64 + (-4*b3 + 3*b1) * q^65 + (-2*b2 - 2) * q^67 + (6*b3 - 3*b1) * q^68 + (20*b2 - 10) * q^70 + (-2*b3 + 2*b1) * q^71 + (-18*b2 + 9) * q^73 + (b3 + b1) * q^74 + (-6*b2 - 6) * q^76 + (-4*b3 + 8*b1) * q^77 + 8 * q^79 - b1 * q^80 + 5*b2 * q^82 + 2*b3 * q^83 + (-5*b2 + 10) * q^85 - 2*b3 * q^86 + (-10*b2 + 10) * q^88 - 2*b1 * q^89 + (-4*b2 - 10) * q^91 + (-6*b3 + 12*b1) * q^92 + (10*b2 - 10) * q^94 + (2*b3 + 2*b1) * q^95 + (-4*b2 + 8) * q^97 + (-5*b3 + 5*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{4} - 12 q^{7}+O(q^{10})$$ 4 * q + 6 * q^4 - 12 * q^7 $$4 q + 6 q^{4} - 12 q^{7} + 10 q^{10} + 10 q^{13} + 2 q^{16} - 12 q^{19} - 20 q^{22} - 36 q^{28} + 6 q^{37} + 20 q^{40} - 4 q^{43} + 60 q^{46} + 10 q^{49} - 12 q^{52} - 20 q^{55} - 30 q^{58} + 14 q^{61} + 52 q^{64} - 12 q^{67} - 36 q^{76} + 32 q^{79} + 10 q^{82} + 30 q^{85} + 20 q^{88} - 48 q^{91} - 20 q^{94} + 24 q^{97}+O(q^{100})$$ 4 * q + 6 * q^4 - 12 * q^7 + 10 * q^10 + 10 * q^13 + 2 * q^16 - 12 * q^19 - 20 * q^22 - 36 * q^28 + 6 * q^37 + 20 * q^40 - 4 * q^43 + 60 * q^46 + 10 * q^49 - 12 * q^52 - 20 * q^55 - 30 * q^58 + 14 * q^61 + 52 * q^64 - 12 * q^67 - 36 * q^76 + 32 * q^79 + 10 * q^82 + 30 * q^85 + 20 * q^88 - 48 * q^91 - 20 * q^94 + 24 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5x^{2} + 25$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 5$$ (v^2) / 5 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 5$$ (v^3) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$5\beta_{2}$$ 5*b2 $$\nu^{3}$$ $$=$$ $$5\beta_{3}$$ 5*b3

## Character values

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

 $$n$$ $$28$$ $$92$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$

## 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
 −1.93649 + 1.11803i 1.93649 − 1.11803i −1.93649 − 1.11803i 1.93649 + 1.11803i
−1.93649 + 1.11803i 0 1.50000 2.59808i 2.23607i 0 −3.00000 1.73205i 2.23607i 0 2.50000 + 4.33013i
10.2 1.93649 1.11803i 0 1.50000 2.59808i 2.23607i 0 −3.00000 1.73205i 2.23607i 0 2.50000 + 4.33013i
82.1 −1.93649 1.11803i 0 1.50000 + 2.59808i 2.23607i 0 −3.00000 + 1.73205i 2.23607i 0 2.50000 4.33013i
82.2 1.93649 + 1.11803i 0 1.50000 + 2.59808i 2.23607i 0 −3.00000 + 1.73205i 2.23607i 0 2.50000 4.33013i
 $$n$$: 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 inner
13.e even 6 1 inner
39.h 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.q.d 4
3.b odd 2 1 inner 117.2.q.d 4
4.b odd 2 1 1872.2.by.l 4
12.b even 2 1 1872.2.by.l 4
13.c even 3 1 1521.2.b.g 4
13.e even 6 1 inner 117.2.q.d 4
13.e even 6 1 1521.2.b.g 4
13.f odd 12 2 1521.2.a.u 4
39.h odd 6 1 inner 117.2.q.d 4
39.h odd 6 1 1521.2.b.g 4
39.i odd 6 1 1521.2.b.g 4
39.k even 12 2 1521.2.a.u 4
52.i odd 6 1 1872.2.by.l 4
156.r even 6 1 1872.2.by.l 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.2.q.d 4 1.a even 1 1 trivial
117.2.q.d 4 3.b odd 2 1 inner
117.2.q.d 4 13.e even 6 1 inner
117.2.q.d 4 39.h odd 6 1 inner
1521.2.a.u 4 13.f odd 12 2
1521.2.a.u 4 39.k even 12 2
1521.2.b.g 4 13.c even 3 1
1521.2.b.g 4 13.e even 6 1
1521.2.b.g 4 39.h odd 6 1
1521.2.b.g 4 39.i odd 6 1
1872.2.by.l 4 4.b odd 2 1
1872.2.by.l 4 12.b even 2 1
1872.2.by.l 4 52.i odd 6 1
1872.2.by.l 4 156.r even 6 1

## Hecke kernels

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

 $$T_{2}^{4} - 5T_{2}^{2} + 25$$ T2^4 - 5*T2^2 + 25 $$T_{5}^{2} + 5$$ T5^2 + 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 5T^{2} + 25$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 5)^{2}$$
$7$ $$(T^{2} + 6 T + 12)^{2}$$
$11$ $$T^{4} - 20T^{2} + 400$$
$13$ $$(T^{2} - 5 T + 13)^{2}$$
$17$ $$T^{4} + 15T^{2} + 225$$
$19$ $$(T^{2} + 6 T + 12)^{2}$$
$23$ $$T^{4} + 60T^{2} + 3600$$
$29$ $$T^{4} + 15T^{2} + 225$$
$31$ $$T^{4}$$
$37$ $$(T^{2} - 3 T + 3)^{2}$$
$41$ $$T^{4} - 5T^{2} + 25$$
$43$ $$(T^{2} + 2 T + 4)^{2}$$
$47$ $$(T^{2} + 20)^{2}$$
$53$ $$(T^{2} - 135)^{2}$$
$59$ $$T^{4} - 80T^{2} + 6400$$
$61$ $$(T^{2} - 7 T + 49)^{2}$$
$67$ $$(T^{2} + 6 T + 12)^{2}$$
$71$ $$T^{4} - 20T^{2} + 400$$
$73$ $$(T^{2} + 243)^{2}$$
$79$ $$(T - 8)^{4}$$
$83$ $$(T^{2} + 20)^{2}$$
$89$ $$T^{4} - 20T^{2} + 400$$
$97$ $$(T^{2} - 12 T + 48)^{2}$$