# Properties

 Label 117.4.q.c Level $117$ Weight $4$ Character orbit 117.q Analytic conductor $6.903$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$117 = 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 117.q (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.90322347067$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} + 2) q^{2} + 4 \zeta_{6} q^{4} + ( - 16 \zeta_{6} + 8) q^{5} + ( - 13 \zeta_{6} + 26) q^{7} + ( - 16 \zeta_{6} + 8) q^{8}+O(q^{10})$$ q + (2*z + 2) * q^2 + 4*z * q^4 + (-16*z + 8) * q^5 + (-13*z + 26) * q^7 + (-16*z + 8) * q^8 $$q + (2 \zeta_{6} + 2) q^{2} + 4 \zeta_{6} q^{4} + ( - 16 \zeta_{6} + 8) q^{5} + ( - 13 \zeta_{6} + 26) q^{7} + ( - 16 \zeta_{6} + 8) q^{8} + ( - 48 \zeta_{6} + 48) q^{10} + (13 \zeta_{6} + 13) q^{11} + (52 \zeta_{6} - 39) q^{13} + 78 q^{14} + ( - 80 \zeta_{6} + 80) q^{16} + 27 \zeta_{6} q^{17} + (51 \zeta_{6} - 102) q^{19} + ( - 32 \zeta_{6} + 64) q^{20} + 78 \zeta_{6} q^{22} + ( - 57 \zeta_{6} + 57) q^{23} - 67 q^{25} + (130 \zeta_{6} - 182) q^{26} + (52 \zeta_{6} + 52) q^{28} + (69 \zeta_{6} - 69) q^{29} + ( - 84 \zeta_{6} + 42) q^{31} + ( - 96 \zeta_{6} + 192) q^{32} + (108 \zeta_{6} - 54) q^{34} - 312 \zeta_{6} q^{35} + ( - 23 \zeta_{6} - 23) q^{37} - 306 q^{38} - 192 q^{40} + (227 \zeta_{6} + 227) q^{41} + 85 \zeta_{6} q^{43} + (104 \zeta_{6} - 52) q^{44} + ( - 114 \zeta_{6} + 228) q^{46} + (396 \zeta_{6} - 198) q^{47} + ( - 164 \zeta_{6} + 164) q^{49} + ( - 134 \zeta_{6} - 134) q^{50} + (52 \zeta_{6} - 208) q^{52} - 426 q^{53} + ( - 312 \zeta_{6} + 312) q^{55} - 312 \zeta_{6} q^{56} + (138 \zeta_{6} - 276) q^{58} + ( - 11 \zeta_{6} + 22) q^{59} + 17 \zeta_{6} q^{61} + ( - 252 \zeta_{6} + 252) q^{62} - 64 q^{64} + (208 \zeta_{6} + 520) q^{65} + (95 \zeta_{6} + 95) q^{67} + (108 \zeta_{6} - 108) q^{68} + ( - 1248 \zeta_{6} + 624) q^{70} + (337 \zeta_{6} - 674) q^{71} + (1160 \zeta_{6} - 580) q^{73} - 138 \zeta_{6} q^{74} + ( - 204 \zeta_{6} - 204) q^{76} + 507 q^{77} - 1244 q^{79} + ( - 640 \zeta_{6} - 640) q^{80} + 1362 \zeta_{6} q^{82} + (492 \zeta_{6} - 246) q^{83} + ( - 216 \zeta_{6} + 432) q^{85} + (340 \zeta_{6} - 170) q^{86} + ( - 312 \zeta_{6} + 312) q^{88} + ( - 177 \zeta_{6} - 177) q^{89} + (1183 \zeta_{6} - 338) q^{91} + 228 q^{92} + (1188 \zeta_{6} - 1188) q^{94} + 1224 \zeta_{6} q^{95} + ( - 713 \zeta_{6} + 1426) q^{97} + ( - 328 \zeta_{6} + 656) q^{98} +O(q^{100})$$ q + (2*z + 2) * q^2 + 4*z * q^4 + (-16*z + 8) * q^5 + (-13*z + 26) * q^7 + (-16*z + 8) * q^8 + (-48*z + 48) * q^10 + (13*z + 13) * q^11 + (52*z - 39) * q^13 + 78 * q^14 + (-80*z + 80) * q^16 + 27*z * q^17 + (51*z - 102) * q^19 + (-32*z + 64) * q^20 + 78*z * q^22 + (-57*z + 57) * q^23 - 67 * q^25 + (130*z - 182) * q^26 + (52*z + 52) * q^28 + (69*z - 69) * q^29 + (-84*z + 42) * q^31 + (-96*z + 192) * q^32 + (108*z - 54) * q^34 - 312*z * q^35 + (-23*z - 23) * q^37 - 306 * q^38 - 192 * q^40 + (227*z + 227) * q^41 + 85*z * q^43 + (104*z - 52) * q^44 + (-114*z + 228) * q^46 + (396*z - 198) * q^47 + (-164*z + 164) * q^49 + (-134*z - 134) * q^50 + (52*z - 208) * q^52 - 426 * q^53 + (-312*z + 312) * q^55 - 312*z * q^56 + (138*z - 276) * q^58 + (-11*z + 22) * q^59 + 17*z * q^61 + (-252*z + 252) * q^62 - 64 * q^64 + (208*z + 520) * q^65 + (95*z + 95) * q^67 + (108*z - 108) * q^68 + (-1248*z + 624) * q^70 + (337*z - 674) * q^71 + (1160*z - 580) * q^73 - 138*z * q^74 + (-204*z - 204) * q^76 + 507 * q^77 - 1244 * q^79 + (-640*z - 640) * q^80 + 1362*z * q^82 + (492*z - 246) * q^83 + (-216*z + 432) * q^85 + (340*z - 170) * q^86 + (-312*z + 312) * q^88 + (-177*z - 177) * q^89 + (1183*z - 338) * q^91 + 228 * q^92 + (1188*z - 1188) * q^94 + 1224*z * q^95 + (-713*z + 1426) * q^97 + (-328*z + 656) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{2} + 4 q^{4} + 39 q^{7}+O(q^{10})$$ 2 * q + 6 * q^2 + 4 * q^4 + 39 * q^7 $$2 q + 6 q^{2} + 4 q^{4} + 39 q^{7} + 48 q^{10} + 39 q^{11} - 26 q^{13} + 156 q^{14} + 80 q^{16} + 27 q^{17} - 153 q^{19} + 96 q^{20} + 78 q^{22} + 57 q^{23} - 134 q^{25} - 234 q^{26} + 156 q^{28} - 69 q^{29} + 288 q^{32} - 312 q^{35} - 69 q^{37} - 612 q^{38} - 384 q^{40} + 681 q^{41} + 85 q^{43} + 342 q^{46} + 164 q^{49} - 402 q^{50} - 364 q^{52} - 852 q^{53} + 312 q^{55} - 312 q^{56} - 414 q^{58} + 33 q^{59} + 17 q^{61} + 252 q^{62} - 128 q^{64} + 1248 q^{65} + 285 q^{67} - 108 q^{68} - 1011 q^{71} - 138 q^{74} - 612 q^{76} + 1014 q^{77} - 2488 q^{79} - 1920 q^{80} + 1362 q^{82} + 648 q^{85} + 312 q^{88} - 531 q^{89} + 507 q^{91} + 456 q^{92} - 1188 q^{94} + 1224 q^{95} + 2139 q^{97} + 984 q^{98}+O(q^{100})$$ 2 * q + 6 * q^2 + 4 * q^4 + 39 * q^7 + 48 * q^10 + 39 * q^11 - 26 * q^13 + 156 * q^14 + 80 * q^16 + 27 * q^17 - 153 * q^19 + 96 * q^20 + 78 * q^22 + 57 * q^23 - 134 * q^25 - 234 * q^26 + 156 * q^28 - 69 * q^29 + 288 * q^32 - 312 * q^35 - 69 * q^37 - 612 * q^38 - 384 * q^40 + 681 * q^41 + 85 * q^43 + 342 * q^46 + 164 * q^49 - 402 * q^50 - 364 * q^52 - 852 * q^53 + 312 * q^55 - 312 * q^56 - 414 * q^58 + 33 * q^59 + 17 * q^61 + 252 * q^62 - 128 * q^64 + 1248 * q^65 + 285 * q^67 - 108 * q^68 - 1011 * q^71 - 138 * q^74 - 612 * q^76 + 1014 * q^77 - 2488 * q^79 - 1920 * q^80 + 1362 * q^82 + 648 * q^85 + 312 * q^88 - 531 * q^89 + 507 * q^91 + 456 * q^92 - 1188 * q^94 + 1224 * q^95 + 2139 * q^97 + 984 * 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
10.1
 0.5 − 0.866025i 0.5 + 0.866025i
3.00000 1.73205i 0 2.00000 3.46410i 13.8564i 0 19.5000 + 11.2583i 13.8564i 0 24.0000 + 41.5692i
82.1 3.00000 + 1.73205i 0 2.00000 + 3.46410i 13.8564i 0 19.5000 11.2583i 13.8564i 0 24.0000 41.5692i
 $$n$$: 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.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.4.q.c 2
3.b odd 2 1 13.4.e.a 2
12.b even 2 1 208.4.w.a 2
13.e even 6 1 inner 117.4.q.c 2
13.f odd 12 2 1521.4.a.q 2
39.d odd 2 1 169.4.e.b 2
39.f even 4 2 169.4.c.i 4
39.h odd 6 1 13.4.e.a 2
39.h odd 6 1 169.4.b.b 2
39.i odd 6 1 169.4.b.b 2
39.i odd 6 1 169.4.e.b 2
39.k even 12 2 169.4.a.h 2
39.k even 12 2 169.4.c.i 4
156.r even 6 1 208.4.w.a 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 6T_{2} + 12$$ acting on $$S_{4}^{\mathrm{new}}(117, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 6T + 12$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 192$$
$7$ $$T^{2} - 39T + 507$$
$11$ $$T^{2} - 39T + 507$$
$13$ $$T^{2} + 26T + 2197$$
$17$ $$T^{2} - 27T + 729$$
$19$ $$T^{2} + 153T + 7803$$
$23$ $$T^{2} - 57T + 3249$$
$29$ $$T^{2} + 69T + 4761$$
$31$ $$T^{2} + 5292$$
$37$ $$T^{2} + 69T + 1587$$
$41$ $$T^{2} - 681T + 154587$$
$43$ $$T^{2} - 85T + 7225$$
$47$ $$T^{2} + 117612$$
$53$ $$(T + 426)^{2}$$
$59$ $$T^{2} - 33T + 363$$
$61$ $$T^{2} - 17T + 289$$
$67$ $$T^{2} - 285T + 27075$$
$71$ $$T^{2} + 1011 T + 340707$$
$73$ $$T^{2} + 1009200$$
$79$ $$(T + 1244)^{2}$$
$83$ $$T^{2} + 181548$$
$89$ $$T^{2} + 531T + 93987$$
$97$ $$T^{2} - 2139 T + 1525107$$
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