# Properties

 Label 117.4.g.a Level $117$ Weight $4$ Character orbit 117.g 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(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, 4, names="a")

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

chi := DirichletCharacter("117.55");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$117 = 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$6.90322347067$$ 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: $$1$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{2} + 7 \zeta_{6} q^{4} - 7 q^{5} + 10 \zeta_{6} q^{7} - 15 q^{8}+O(q^{10})$$ q + (z - 1) * q^2 + 7*z * q^4 - 7 * q^5 + 10*z * q^7 - 15 * q^8 $$q + (\zeta_{6} - 1) q^{2} + 7 \zeta_{6} q^{4} - 7 q^{5} + 10 \zeta_{6} q^{7} - 15 q^{8} + ( - 7 \zeta_{6} + 7) q^{10} + (22 \zeta_{6} - 22) q^{11} + ( - 13 \zeta_{6} - 39) q^{13} - 10 q^{14} + (41 \zeta_{6} - 41) q^{16} + 37 \zeta_{6} q^{17} - 30 \zeta_{6} q^{19} - 49 \zeta_{6} q^{20} - 22 \zeta_{6} q^{22} + (162 \zeta_{6} - 162) q^{23} - 76 q^{25} + ( - 39 \zeta_{6} + 52) q^{26} + (70 \zeta_{6} - 70) q^{28} + (113 \zeta_{6} - 113) q^{29} + 196 q^{31} - 161 \zeta_{6} q^{32} - 37 q^{34} - 70 \zeta_{6} q^{35} + (13 \zeta_{6} - 13) q^{37} + 30 q^{38} + 105 q^{40} + ( - 285 \zeta_{6} + 285) q^{41} + 246 \zeta_{6} q^{43} - 154 q^{44} - 162 \zeta_{6} q^{46} + 462 q^{47} + ( - 243 \zeta_{6} + 243) q^{49} + ( - 76 \zeta_{6} + 76) q^{50} + ( - 364 \zeta_{6} + 91) q^{52} + 537 q^{53} + ( - 154 \zeta_{6} + 154) q^{55} - 150 \zeta_{6} q^{56} - 113 \zeta_{6} q^{58} + 576 \zeta_{6} q^{59} + 635 \zeta_{6} q^{61} + (196 \zeta_{6} - 196) q^{62} - 167 q^{64} + (91 \zeta_{6} + 273) q^{65} + (202 \zeta_{6} - 202) q^{67} + (259 \zeta_{6} - 259) q^{68} + 70 q^{70} - 1086 \zeta_{6} q^{71} - 805 q^{73} - 13 \zeta_{6} q^{74} + ( - 210 \zeta_{6} + 210) q^{76} - 220 q^{77} + 884 q^{79} + ( - 287 \zeta_{6} + 287) q^{80} + 285 \zeta_{6} q^{82} - 518 q^{83} - 259 \zeta_{6} q^{85} - 246 q^{86} + ( - 330 \zeta_{6} + 330) q^{88} + ( - 194 \zeta_{6} + 194) q^{89} + ( - 520 \zeta_{6} + 130) q^{91} - 1134 q^{92} + (462 \zeta_{6} - 462) q^{94} + 210 \zeta_{6} q^{95} + 1202 \zeta_{6} q^{97} + 243 \zeta_{6} q^{98} +O(q^{100})$$ q + (z - 1) * q^2 + 7*z * q^4 - 7 * q^5 + 10*z * q^7 - 15 * q^8 + (-7*z + 7) * q^10 + (22*z - 22) * q^11 + (-13*z - 39) * q^13 - 10 * q^14 + (41*z - 41) * q^16 + 37*z * q^17 - 30*z * q^19 - 49*z * q^20 - 22*z * q^22 + (162*z - 162) * q^23 - 76 * q^25 + (-39*z + 52) * q^26 + (70*z - 70) * q^28 + (113*z - 113) * q^29 + 196 * q^31 - 161*z * q^32 - 37 * q^34 - 70*z * q^35 + (13*z - 13) * q^37 + 30 * q^38 + 105 * q^40 + (-285*z + 285) * q^41 + 246*z * q^43 - 154 * q^44 - 162*z * q^46 + 462 * q^47 + (-243*z + 243) * q^49 + (-76*z + 76) * q^50 + (-364*z + 91) * q^52 + 537 * q^53 + (-154*z + 154) * q^55 - 150*z * q^56 - 113*z * q^58 + 576*z * q^59 + 635*z * q^61 + (196*z - 196) * q^62 - 167 * q^64 + (91*z + 273) * q^65 + (202*z - 202) * q^67 + (259*z - 259) * q^68 + 70 * q^70 - 1086*z * q^71 - 805 * q^73 - 13*z * q^74 + (-210*z + 210) * q^76 - 220 * q^77 + 884 * q^79 + (-287*z + 287) * q^80 + 285*z * q^82 - 518 * q^83 - 259*z * q^85 - 246 * q^86 + (-330*z + 330) * q^88 + (-194*z + 194) * q^89 + (-520*z + 130) * q^91 - 1134 * q^92 + (462*z - 462) * q^94 + 210*z * q^95 + 1202*z * q^97 + 243*z * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 7 q^{4} - 14 q^{5} + 10 q^{7} - 30 q^{8}+O(q^{10})$$ 2 * q - q^2 + 7 * q^4 - 14 * q^5 + 10 * q^7 - 30 * q^8 $$2 q - q^{2} + 7 q^{4} - 14 q^{5} + 10 q^{7} - 30 q^{8} + 7 q^{10} - 22 q^{11} - 91 q^{13} - 20 q^{14} - 41 q^{16} + 37 q^{17} - 30 q^{19} - 49 q^{20} - 22 q^{22} - 162 q^{23} - 152 q^{25} + 65 q^{26} - 70 q^{28} - 113 q^{29} + 392 q^{31} - 161 q^{32} - 74 q^{34} - 70 q^{35} - 13 q^{37} + 60 q^{38} + 210 q^{40} + 285 q^{41} + 246 q^{43} - 308 q^{44} - 162 q^{46} + 924 q^{47} + 243 q^{49} + 76 q^{50} - 182 q^{52} + 1074 q^{53} + 154 q^{55} - 150 q^{56} - 113 q^{58} + 576 q^{59} + 635 q^{61} - 196 q^{62} - 334 q^{64} + 637 q^{65} - 202 q^{67} - 259 q^{68} + 140 q^{70} - 1086 q^{71} - 1610 q^{73} - 13 q^{74} + 210 q^{76} - 440 q^{77} + 1768 q^{79} + 287 q^{80} + 285 q^{82} - 1036 q^{83} - 259 q^{85} - 492 q^{86} + 330 q^{88} + 194 q^{89} - 260 q^{91} - 2268 q^{92} - 462 q^{94} + 210 q^{95} + 1202 q^{97} + 243 q^{98}+O(q^{100})$$ 2 * q - q^2 + 7 * q^4 - 14 * q^5 + 10 * q^7 - 30 * q^8 + 7 * q^10 - 22 * q^11 - 91 * q^13 - 20 * q^14 - 41 * q^16 + 37 * q^17 - 30 * q^19 - 49 * q^20 - 22 * q^22 - 162 * q^23 - 152 * q^25 + 65 * q^26 - 70 * q^28 - 113 * q^29 + 392 * q^31 - 161 * q^32 - 74 * q^34 - 70 * q^35 - 13 * q^37 + 60 * q^38 + 210 * q^40 + 285 * q^41 + 246 * q^43 - 308 * q^44 - 162 * q^46 + 924 * q^47 + 243 * q^49 + 76 * q^50 - 182 * q^52 + 1074 * q^53 + 154 * q^55 - 150 * q^56 - 113 * q^58 + 576 * q^59 + 635 * q^61 - 196 * q^62 - 334 * q^64 + 637 * q^65 - 202 * q^67 - 259 * q^68 + 140 * q^70 - 1086 * q^71 - 1610 * q^73 - 13 * q^74 + 210 * q^76 - 440 * q^77 + 1768 * q^79 + 287 * q^80 + 285 * q^82 - 1036 * q^83 - 259 * q^85 - 492 * q^86 + 330 * q^88 + 194 * q^89 - 260 * q^91 - 2268 * q^92 - 462 * q^94 + 210 * q^95 + 1202 * q^97 + 243 * q^98

## 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
−0.500000 0.866025i 0 3.50000 6.06218i −7.00000 0 5.00000 8.66025i −15.0000 0 3.50000 + 6.06218i
100.1 −0.500000 + 0.866025i 0 3.50000 + 6.06218i −7.00000 0 5.00000 + 8.66025i −15.0000 0 3.50000 6.06218i
 $$n$$: 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.4.g.a 2
3.b odd 2 1 39.4.e.b 2
12.b even 2 1 624.4.q.c 2
13.c even 3 1 inner 117.4.g.a 2
13.c even 3 1 1521.4.a.h 1
13.e even 6 1 1521.4.a.e 1
39.h odd 6 1 507.4.a.d 1
39.i odd 6 1 39.4.e.b 2
39.i odd 6 1 507.4.a.b 1
39.k even 12 2 507.4.b.d 2
156.p even 6 1 624.4.q.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.e.b 2 3.b odd 2 1
39.4.e.b 2 39.i odd 6 1
117.4.g.a 2 1.a even 1 1 trivial
117.4.g.a 2 13.c even 3 1 inner
507.4.a.b 1 39.i odd 6 1
507.4.a.d 1 39.h odd 6 1
507.4.b.d 2 39.k even 12 2
624.4.q.c 2 12.b even 2 1
624.4.q.c 2 156.p even 6 1
1521.4.a.e 1 13.e even 6 1
1521.4.a.h 1 13.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + T_{2} + 1$$ acting on $$S_{4}^{\mathrm{new}}(117, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$(T + 7)^{2}$$
$7$ $$T^{2} - 10T + 100$$
$11$ $$T^{2} + 22T + 484$$
$13$ $$T^{2} + 91T + 2197$$
$17$ $$T^{2} - 37T + 1369$$
$19$ $$T^{2} + 30T + 900$$
$23$ $$T^{2} + 162T + 26244$$
$29$ $$T^{2} + 113T + 12769$$
$31$ $$(T - 196)^{2}$$
$37$ $$T^{2} + 13T + 169$$
$41$ $$T^{2} - 285T + 81225$$
$43$ $$T^{2} - 246T + 60516$$
$47$ $$(T - 462)^{2}$$
$53$ $$(T - 537)^{2}$$
$59$ $$T^{2} - 576T + 331776$$
$61$ $$T^{2} - 635T + 403225$$
$67$ $$T^{2} + 202T + 40804$$
$71$ $$T^{2} + 1086 T + 1179396$$
$73$ $$(T + 805)^{2}$$
$79$ $$(T - 884)^{2}$$
$83$ $$(T + 518)^{2}$$
$89$ $$T^{2} - 194T + 37636$$
$97$ $$T^{2} - 1202 T + 1444804$$