# Properties

 Label 117.4.b.a Level $117$ Weight $4$ Character orbit 117.b Analytic conductor $6.903$ 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] = [117,4,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, 4, names="a")

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

chi := DirichletCharacter("117.64");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$117 = 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$6.90322347067$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 13) 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} - q^{4} - 3 i q^{5} + 5 i q^{7} + 7 i q^{8} +O(q^{10})$$ q + i * q^2 - q^4 - 3*i * q^5 + 5*i * q^7 + 7*i * q^8 $$q + i q^{2} - q^{4} - 3 i q^{5} + 5 i q^{7} + 7 i q^{8} + 27 q^{10} + 16 i q^{11} + (13 i + 26) q^{13} - 45 q^{14} - 71 q^{16} + 45 q^{17} - 2 i q^{19} + 3 i q^{20} - 144 q^{22} - 162 q^{23} + 44 q^{25} + (26 i - 117) q^{26} - 5 i q^{28} + 144 q^{29} - 88 i q^{31} - 15 i q^{32} + 45 i q^{34} + 135 q^{35} + 101 i q^{37} + 18 q^{38} + 189 q^{40} - 64 i q^{41} - 97 q^{43} - 16 i q^{44} - 162 i q^{46} - 37 i q^{47} + 118 q^{49} + 44 i q^{50} + ( - 13 i - 26) q^{52} + 414 q^{53} + 432 q^{55} - 315 q^{56} + 144 i q^{58} - 174 i q^{59} + 376 q^{61} + 792 q^{62} - 433 q^{64} + ( - 78 i + 351) q^{65} + 12 i q^{67} - 45 q^{68} + 135 i q^{70} + 119 i q^{71} - 366 i q^{73} - 909 q^{74} + 2 i q^{76} - 720 q^{77} - 830 q^{79} + 213 i q^{80} + 576 q^{82} - 146 i q^{83} - 135 i q^{85} - 97 i q^{86} - 1008 q^{88} + 146 i q^{89} + (130 i - 585) q^{91} + 162 q^{92} + 333 q^{94} - 54 q^{95} + 284 i q^{97} + 118 i q^{98} +O(q^{100})$$ q + i * q^2 - q^4 - 3*i * q^5 + 5*i * q^7 + 7*i * q^8 + 27 * q^10 + 16*i * q^11 + (13*i + 26) * q^13 - 45 * q^14 - 71 * q^16 + 45 * q^17 - 2*i * q^19 + 3*i * q^20 - 144 * q^22 - 162 * q^23 + 44 * q^25 + (26*i - 117) * q^26 - 5*i * q^28 + 144 * q^29 - 88*i * q^31 - 15*i * q^32 + 45*i * q^34 + 135 * q^35 + 101*i * q^37 + 18 * q^38 + 189 * q^40 - 64*i * q^41 - 97 * q^43 - 16*i * q^44 - 162*i * q^46 - 37*i * q^47 + 118 * q^49 + 44*i * q^50 + (-13*i - 26) * q^52 + 414 * q^53 + 432 * q^55 - 315 * q^56 + 144*i * q^58 - 174*i * q^59 + 376 * q^61 + 792 * q^62 - 433 * q^64 + (-78*i + 351) * q^65 + 12*i * q^67 - 45 * q^68 + 135*i * q^70 + 119*i * q^71 - 366*i * q^73 - 909 * q^74 + 2*i * q^76 - 720 * q^77 - 830 * q^79 + 213*i * q^80 + 576 * q^82 - 146*i * q^83 - 135*i * q^85 - 97*i * q^86 - 1008 * q^88 + 146*i * q^89 + (130*i - 585) * q^91 + 162 * q^92 + 333 * q^94 - 54 * q^95 + 284*i * q^97 + 118*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4}+O(q^{10})$$ 2 * q - 2 * q^4 $$2 q - 2 q^{4} + 54 q^{10} + 52 q^{13} - 90 q^{14} - 142 q^{16} + 90 q^{17} - 288 q^{22} - 324 q^{23} + 88 q^{25} - 234 q^{26} + 288 q^{29} + 270 q^{35} + 36 q^{38} + 378 q^{40} - 194 q^{43} + 236 q^{49} - 52 q^{52} + 828 q^{53} + 864 q^{55} - 630 q^{56} + 752 q^{61} + 1584 q^{62} - 866 q^{64} + 702 q^{65} - 90 q^{68} - 1818 q^{74} - 1440 q^{77} - 1660 q^{79} + 1152 q^{82} - 2016 q^{88} - 1170 q^{91} + 324 q^{92} + 666 q^{94} - 108 q^{95}+O(q^{100})$$ 2 * q - 2 * q^4 + 54 * q^10 + 52 * q^13 - 90 * q^14 - 142 * q^16 + 90 * q^17 - 288 * q^22 - 324 * q^23 + 88 * q^25 - 234 * q^26 + 288 * q^29 + 270 * q^35 + 36 * q^38 + 378 * q^40 - 194 * q^43 + 236 * q^49 - 52 * q^52 + 828 * q^53 + 864 * q^55 - 630 * q^56 + 752 * q^61 + 1584 * q^62 - 866 * q^64 + 702 * q^65 - 90 * q^68 - 1818 * q^74 - 1440 * q^77 - 1660 * q^79 + 1152 * q^82 - 2016 * q^88 - 1170 * q^91 + 324 * q^92 + 666 * q^94 - 108 * q^95

## 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
 − 1.00000i 1.00000i
3.00000i 0 −1.00000 9.00000i 0 15.0000i 21.0000i 0 27.0000
64.2 3.00000i 0 −1.00000 9.00000i 0 15.0000i 21.0000i 0 27.0000
 $$n$$: 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 117.4.b.a 2
3.b odd 2 1 13.4.b.a 2
12.b even 2 1 208.4.f.b 2
13.b even 2 1 inner 117.4.b.a 2
13.d odd 4 1 1521.4.a.d 1
13.d odd 4 1 1521.4.a.i 1
15.d odd 2 1 325.4.c.b 2
15.e even 4 1 325.4.d.a 2
15.e even 4 1 325.4.d.b 2
24.f even 2 1 832.4.f.c 2
24.h odd 2 1 832.4.f.e 2
39.d odd 2 1 13.4.b.a 2
39.f even 4 1 169.4.a.b 1
39.f even 4 1 169.4.a.c 1
39.h odd 6 2 169.4.e.d 4
39.i odd 6 2 169.4.e.d 4
39.k even 12 2 169.4.c.b 2
39.k even 12 2 169.4.c.c 2
156.h even 2 1 208.4.f.b 2
195.e odd 2 1 325.4.c.b 2
195.s even 4 1 325.4.d.a 2
195.s even 4 1 325.4.d.b 2
312.b odd 2 1 832.4.f.e 2
312.h even 2 1 832.4.f.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.b.a 2 3.b odd 2 1
13.4.b.a 2 39.d odd 2 1
117.4.b.a 2 1.a even 1 1 trivial
117.4.b.a 2 13.b even 2 1 inner
169.4.a.b 1 39.f even 4 1
169.4.a.c 1 39.f even 4 1
169.4.c.b 2 39.k even 12 2
169.4.c.c 2 39.k even 12 2
169.4.e.d 4 39.h odd 6 2
169.4.e.d 4 39.i odd 6 2
208.4.f.b 2 12.b even 2 1
208.4.f.b 2 156.h even 2 1
325.4.c.b 2 15.d odd 2 1
325.4.c.b 2 195.e odd 2 1
325.4.d.a 2 15.e even 4 1
325.4.d.a 2 195.s even 4 1
325.4.d.b 2 15.e even 4 1
325.4.d.b 2 195.s even 4 1
832.4.f.c 2 24.f even 2 1
832.4.f.c 2 312.h even 2 1
832.4.f.e 2 24.h odd 2 1
832.4.f.e 2 312.b odd 2 1
1521.4.a.d 1 13.d odd 4 1
1521.4.a.i 1 13.d odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 9$$ acting on $$S_{4}^{\mathrm{new}}(117, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 9$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 81$$
$7$ $$T^{2} + 225$$
$11$ $$T^{2} + 2304$$
$13$ $$T^{2} - 52T + 2197$$
$17$ $$(T - 45)^{2}$$
$19$ $$T^{2} + 36$$
$23$ $$(T + 162)^{2}$$
$29$ $$(T - 144)^{2}$$
$31$ $$T^{2} + 69696$$
$37$ $$T^{2} + 91809$$
$41$ $$T^{2} + 36864$$
$43$ $$(T + 97)^{2}$$
$47$ $$T^{2} + 12321$$
$53$ $$(T - 414)^{2}$$
$59$ $$T^{2} + 272484$$
$61$ $$(T - 376)^{2}$$
$67$ $$T^{2} + 1296$$
$71$ $$T^{2} + 127449$$
$73$ $$T^{2} + 1205604$$
$79$ $$(T + 830)^{2}$$
$83$ $$T^{2} + 191844$$
$89$ $$T^{2} + 191844$$
$97$ $$T^{2} + 725904$$