# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$117 = 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 117.g (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.934249703649$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ 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

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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{2} + \zeta_{6} q^{4} + q^{5} - 2 \zeta_{6} q^{7} + 3 q^{8} +O(q^{10})$$ q + (-z + 1) * q^2 + z * q^4 + q^5 - 2*z * q^7 + 3 * q^8 $$q + ( - \zeta_{6} + 1) q^{2} + \zeta_{6} q^{4} + q^{5} - 2 \zeta_{6} q^{7} + 3 q^{8} + ( - \zeta_{6} + 1) q^{10} + (2 \zeta_{6} - 2) q^{11} + ( - \zeta_{6} - 3) q^{13} - 2 q^{14} + ( - \zeta_{6} + 1) q^{16} - 7 \zeta_{6} q^{17} + 6 \zeta_{6} q^{19} + \zeta_{6} q^{20} + 2 \zeta_{6} q^{22} + (6 \zeta_{6} - 6) q^{23} - 4 q^{25} + (3 \zeta_{6} - 4) q^{26} + ( - 2 \zeta_{6} + 2) q^{28} + (\zeta_{6} - 1) q^{29} + 4 q^{31} + 5 \zeta_{6} q^{32} - 7 q^{34} - 2 \zeta_{6} q^{35} + (\zeta_{6} - 1) q^{37} + 6 q^{38} + 3 q^{40} + ( - 9 \zeta_{6} + 9) q^{41} - 6 \zeta_{6} q^{43} - 2 q^{44} + 6 \zeta_{6} q^{46} - 6 q^{47} + ( - 3 \zeta_{6} + 3) q^{49} + (4 \zeta_{6} - 4) q^{50} + ( - 4 \zeta_{6} + 1) q^{52} + 9 q^{53} + (2 \zeta_{6} - 2) q^{55} - 6 \zeta_{6} q^{56} + \zeta_{6} q^{58} - \zeta_{6} q^{61} + ( - 4 \zeta_{6} + 4) q^{62} + 7 q^{64} + ( - \zeta_{6} - 3) q^{65} + ( - 2 \zeta_{6} + 2) q^{67} + ( - 7 \zeta_{6} + 7) q^{68} - 2 q^{70} + 6 \zeta_{6} q^{71} + 11 q^{73} + \zeta_{6} q^{74} + (6 \zeta_{6} - 6) q^{76} + 4 q^{77} - 4 q^{79} + ( - \zeta_{6} + 1) q^{80} - 9 \zeta_{6} q^{82} + 14 q^{83} - 7 \zeta_{6} q^{85} - 6 q^{86} + (6 \zeta_{6} - 6) q^{88} + (14 \zeta_{6} - 14) q^{89} + (8 \zeta_{6} - 2) q^{91} - 6 q^{92} + (6 \zeta_{6} - 6) q^{94} + 6 \zeta_{6} q^{95} + 2 \zeta_{6} q^{97} - 3 \zeta_{6} q^{98} +O(q^{100})$$ q + (-z + 1) * q^2 + z * q^4 + q^5 - 2*z * q^7 + 3 * q^8 + (-z + 1) * q^10 + (2*z - 2) * q^11 + (-z - 3) * q^13 - 2 * q^14 + (-z + 1) * q^16 - 7*z * q^17 + 6*z * q^19 + z * q^20 + 2*z * q^22 + (6*z - 6) * q^23 - 4 * q^25 + (3*z - 4) * q^26 + (-2*z + 2) * q^28 + (z - 1) * q^29 + 4 * q^31 + 5*z * q^32 - 7 * q^34 - 2*z * q^35 + (z - 1) * q^37 + 6 * q^38 + 3 * q^40 + (-9*z + 9) * q^41 - 6*z * q^43 - 2 * q^44 + 6*z * q^46 - 6 * q^47 + (-3*z + 3) * q^49 + (4*z - 4) * q^50 + (-4*z + 1) * q^52 + 9 * q^53 + (2*z - 2) * q^55 - 6*z * q^56 + z * q^58 - z * q^61 + (-4*z + 4) * q^62 + 7 * q^64 + (-z - 3) * q^65 + (-2*z + 2) * q^67 + (-7*z + 7) * q^68 - 2 * q^70 + 6*z * q^71 + 11 * q^73 + z * q^74 + (6*z - 6) * q^76 + 4 * q^77 - 4 * q^79 + (-z + 1) * q^80 - 9*z * q^82 + 14 * q^83 - 7*z * q^85 - 6 * q^86 + (6*z - 6) * q^88 + (14*z - 14) * q^89 + (8*z - 2) * q^91 - 6 * q^92 + (6*z - 6) * q^94 + 6*z * q^95 + 2*z * q^97 - 3*z * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8}+O(q^{10})$$ 2 * q + q^2 + q^4 + 2 * q^5 - 2 * q^7 + 6 * q^8 $$2 q + q^{2} + q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8} + q^{10} - 2 q^{11} - 7 q^{13} - 4 q^{14} + q^{16} - 7 q^{17} + 6 q^{19} + q^{20} + 2 q^{22} - 6 q^{23} - 8 q^{25} - 5 q^{26} + 2 q^{28} - q^{29} + 8 q^{31} + 5 q^{32} - 14 q^{34} - 2 q^{35} - q^{37} + 12 q^{38} + 6 q^{40} + 9 q^{41} - 6 q^{43} - 4 q^{44} + 6 q^{46} - 12 q^{47} + 3 q^{49} - 4 q^{50} - 2 q^{52} + 18 q^{53} - 2 q^{55} - 6 q^{56} + q^{58} - q^{61} + 4 q^{62} + 14 q^{64} - 7 q^{65} + 2 q^{67} + 7 q^{68} - 4 q^{70} + 6 q^{71} + 22 q^{73} + q^{74} - 6 q^{76} + 8 q^{77} - 8 q^{79} + q^{80} - 9 q^{82} + 28 q^{83} - 7 q^{85} - 12 q^{86} - 6 q^{88} - 14 q^{89} + 4 q^{91} - 12 q^{92} - 6 q^{94} + 6 q^{95} + 2 q^{97} - 3 q^{98}+O(q^{100})$$ 2 * q + q^2 + q^4 + 2 * q^5 - 2 * q^7 + 6 * q^8 + q^10 - 2 * q^11 - 7 * q^13 - 4 * q^14 + q^16 - 7 * q^17 + 6 * q^19 + q^20 + 2 * q^22 - 6 * q^23 - 8 * q^25 - 5 * q^26 + 2 * q^28 - q^29 + 8 * q^31 + 5 * q^32 - 14 * q^34 - 2 * q^35 - q^37 + 12 * q^38 + 6 * q^40 + 9 * q^41 - 6 * q^43 - 4 * q^44 + 6 * q^46 - 12 * q^47 + 3 * q^49 - 4 * q^50 - 2 * q^52 + 18 * q^53 - 2 * q^55 - 6 * q^56 + q^58 - q^61 + 4 * q^62 + 14 * q^64 - 7 * q^65 + 2 * q^67 + 7 * q^68 - 4 * q^70 + 6 * q^71 + 22 * q^73 + q^74 - 6 * q^76 + 8 * q^77 - 8 * q^79 + q^80 - 9 * q^82 + 28 * q^83 - 7 * q^85 - 12 * q^86 - 6 * q^88 - 14 * q^89 + 4 * q^91 - 12 * q^92 - 6 * q^94 + 6 * q^95 + 2 * q^97 - 3 * 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.

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 0.500000 0.866025i 1.00000 0 −1.00000 + 1.73205i 3.00000 0 0.500000 + 0.866025i
100.1 0.500000 0.866025i 0 0.500000 + 0.866025i 1.00000 0 −1.00000 1.73205i 3.00000 0 0.500000 0.866025i
 $$n$$: 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 117.2.g.b 2
3.b odd 2 1 39.2.e.a 2
4.b odd 2 1 1872.2.t.j 2
12.b even 2 1 624.2.q.c 2
13.c even 3 1 inner 117.2.g.b 2
13.c even 3 1 1521.2.a.a 1
13.e even 6 1 1521.2.a.d 1
13.f odd 12 2 1521.2.b.c 2
15.d odd 2 1 975.2.i.f 2
15.e even 4 2 975.2.bb.d 4
39.d odd 2 1 507.2.e.c 2
39.f even 4 2 507.2.j.d 4
39.h odd 6 1 507.2.a.b 1
39.h odd 6 1 507.2.e.c 2
39.i odd 6 1 39.2.e.a 2
39.i odd 6 1 507.2.a.c 1
39.k even 12 2 507.2.b.b 2
39.k even 12 2 507.2.j.d 4
52.j odd 6 1 1872.2.t.j 2
156.p even 6 1 624.2.q.c 2
156.p even 6 1 8112.2.a.w 1
156.r even 6 1 8112.2.a.bc 1
195.x odd 6 1 975.2.i.f 2
195.bl even 12 2 975.2.bb.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.a 2 3.b odd 2 1
39.2.e.a 2 39.i odd 6 1
117.2.g.b 2 1.a even 1 1 trivial
117.2.g.b 2 13.c even 3 1 inner
507.2.a.b 1 39.h odd 6 1
507.2.a.c 1 39.i odd 6 1
507.2.b.b 2 39.k even 12 2
507.2.e.c 2 39.d odd 2 1
507.2.e.c 2 39.h odd 6 1
507.2.j.d 4 39.f even 4 2
507.2.j.d 4 39.k even 12 2
624.2.q.c 2 12.b even 2 1
624.2.q.c 2 156.p even 6 1
975.2.i.f 2 15.d odd 2 1
975.2.i.f 2 195.x odd 6 1
975.2.bb.d 4 15.e even 4 2
975.2.bb.d 4 195.bl even 12 2
1521.2.a.a 1 13.c even 3 1
1521.2.a.d 1 13.e even 6 1
1521.2.b.c 2 13.f odd 12 2
1872.2.t.j 2 4.b odd 2 1
1872.2.t.j 2 52.j odd 6 1
8112.2.a.w 1 156.p even 6 1
8112.2.a.bc 1 156.r even 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} + 2T + 4$$
$11$ $$T^{2} + 2T + 4$$
$13$ $$T^{2} + 7T + 13$$
$17$ $$T^{2} + 7T + 49$$
$19$ $$T^{2} - 6T + 36$$
$23$ $$T^{2} + 6T + 36$$
$29$ $$T^{2} + T + 1$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + T + 1$$
$41$ $$T^{2} - 9T + 81$$
$43$ $$T^{2} + 6T + 36$$
$47$ $$(T + 6)^{2}$$
$53$ $$(T - 9)^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + T + 1$$
$67$ $$T^{2} - 2T + 4$$
$71$ $$T^{2} - 6T + 36$$
$73$ $$(T - 11)^{2}$$
$79$ $$(T + 4)^{2}$$
$83$ $$(T - 14)^{2}$$
$89$ $$T^{2} + 14T + 196$$
$97$ $$T^{2} - 2T + 4$$