# Properties

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

# Related objects

## 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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} - 1) q^{2} - 5 \zeta_{6} q^{4} + (2 \zeta_{6} - 1) q^{5} + (8 \zeta_{6} - 16) q^{7} + (26 \zeta_{6} - 13) q^{8} +O(q^{10})$$ q + (-z - 1) * q^2 - 5*z * q^4 + (2*z - 1) * q^5 + (8*z - 16) * q^7 + (26*z - 13) * q^8 $$q + ( - \zeta_{6} - 1) q^{2} - 5 \zeta_{6} q^{4} + (2 \zeta_{6} - 1) q^{5} + (8 \zeta_{6} - 16) q^{7} + (26 \zeta_{6} - 13) q^{8} + ( - 3 \zeta_{6} + 3) q^{10} + ( - 8 \zeta_{6} - 8) q^{11} + (13 \zeta_{6} + 39) q^{13} + 24 q^{14} + (\zeta_{6} - 1) q^{16} + 117 \zeta_{6} q^{17} + (66 \zeta_{6} - 132) q^{19} + ( - 5 \zeta_{6} + 10) q^{20} + 24 \zeta_{6} q^{22} + (78 \zeta_{6} - 78) q^{23} + 122 q^{25} + ( - 65 \zeta_{6} - 26) q^{26} + (40 \zeta_{6} + 40) q^{28} + (141 \zeta_{6} - 141) q^{29} + (180 \zeta_{6} - 90) q^{31} + ( - 105 \zeta_{6} + 210) q^{32} + ( - 234 \zeta_{6} + 117) q^{34} - 24 \zeta_{6} q^{35} + ( - 83 \zeta_{6} - 83) q^{37} + 198 q^{38} - 39 q^{40} + ( - 157 \zeta_{6} - 157) q^{41} - 104 \zeta_{6} q^{43} + (80 \zeta_{6} - 40) q^{44} + ( - 78 \zeta_{6} + 156) q^{46} + ( - 348 \zeta_{6} + 174) q^{47} + (151 \zeta_{6} - 151) q^{49} + ( - 122 \zeta_{6} - 122) q^{50} + ( - 260 \zeta_{6} + 65) q^{52} - 93 q^{53} + ( - 24 \zeta_{6} + 24) q^{55} - 312 \zeta_{6} q^{56} + ( - 141 \zeta_{6} + 282) q^{58} + ( - 164 \zeta_{6} + 328) q^{59} - 145 \zeta_{6} q^{61} + ( - 270 \zeta_{6} + 270) q^{62} - 307 q^{64} + (91 \zeta_{6} - 65) q^{65} + ( - 454 \zeta_{6} - 454) q^{67} + ( - 585 \zeta_{6} + 585) q^{68} + (48 \zeta_{6} - 24) q^{70} + (610 \zeta_{6} - 1220) q^{71} + (530 \zeta_{6} - 265) q^{73} + 249 \zeta_{6} q^{74} + (330 \zeta_{6} + 330) q^{76} + 192 q^{77} + 1276 q^{79} + ( - \zeta_{6} - 1) q^{80} + 471 \zeta_{6} q^{82} + (912 \zeta_{6} - 456) q^{83} + (117 \zeta_{6} - 234) q^{85} + (208 \zeta_{6} - 104) q^{86} + ( - 312 \zeta_{6} + 312) q^{88} + (564 \zeta_{6} + 564) q^{89} + (208 \zeta_{6} - 728) q^{91} + 390 q^{92} + (522 \zeta_{6} - 522) q^{94} - 198 \zeta_{6} q^{95} + ( - 116 \zeta_{6} + 232) q^{97} + ( - 151 \zeta_{6} + 302) q^{98} +O(q^{100})$$ q + (-z - 1) * q^2 - 5*z * q^4 + (2*z - 1) * q^5 + (8*z - 16) * q^7 + (26*z - 13) * q^8 + (-3*z + 3) * q^10 + (-8*z - 8) * q^11 + (13*z + 39) * q^13 + 24 * q^14 + (z - 1) * q^16 + 117*z * q^17 + (66*z - 132) * q^19 + (-5*z + 10) * q^20 + 24*z * q^22 + (78*z - 78) * q^23 + 122 * q^25 + (-65*z - 26) * q^26 + (40*z + 40) * q^28 + (141*z - 141) * q^29 + (180*z - 90) * q^31 + (-105*z + 210) * q^32 + (-234*z + 117) * q^34 - 24*z * q^35 + (-83*z - 83) * q^37 + 198 * q^38 - 39 * q^40 + (-157*z - 157) * q^41 - 104*z * q^43 + (80*z - 40) * q^44 + (-78*z + 156) * q^46 + (-348*z + 174) * q^47 + (151*z - 151) * q^49 + (-122*z - 122) * q^50 + (-260*z + 65) * q^52 - 93 * q^53 + (-24*z + 24) * q^55 - 312*z * q^56 + (-141*z + 282) * q^58 + (-164*z + 328) * q^59 - 145*z * q^61 + (-270*z + 270) * q^62 - 307 * q^64 + (91*z - 65) * q^65 + (-454*z - 454) * q^67 + (-585*z + 585) * q^68 + (48*z - 24) * q^70 + (610*z - 1220) * q^71 + (530*z - 265) * q^73 + 249*z * q^74 + (330*z + 330) * q^76 + 192 * q^77 + 1276 * q^79 + (-z - 1) * q^80 + 471*z * q^82 + (912*z - 456) * q^83 + (117*z - 234) * q^85 + (208*z - 104) * q^86 + (-312*z + 312) * q^88 + (564*z + 564) * q^89 + (208*z - 728) * q^91 + 390 * q^92 + (522*z - 522) * q^94 - 198*z * q^95 + (-116*z + 232) * q^97 + (-151*z + 302) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} - 5 q^{4} - 24 q^{7}+O(q^{10})$$ 2 * q - 3 * q^2 - 5 * q^4 - 24 * q^7 $$2 q - 3 q^{2} - 5 q^{4} - 24 q^{7} + 3 q^{10} - 24 q^{11} + 91 q^{13} + 48 q^{14} - q^{16} + 117 q^{17} - 198 q^{19} + 15 q^{20} + 24 q^{22} - 78 q^{23} + 244 q^{25} - 117 q^{26} + 120 q^{28} - 141 q^{29} + 315 q^{32} - 24 q^{35} - 249 q^{37} + 396 q^{38} - 78 q^{40} - 471 q^{41} - 104 q^{43} + 234 q^{46} - 151 q^{49} - 366 q^{50} - 130 q^{52} - 186 q^{53} + 24 q^{55} - 312 q^{56} + 423 q^{58} + 492 q^{59} - 145 q^{61} + 270 q^{62} - 614 q^{64} - 39 q^{65} - 1362 q^{67} + 585 q^{68} - 1830 q^{71} + 249 q^{74} + 990 q^{76} + 384 q^{77} + 2552 q^{79} - 3 q^{80} + 471 q^{82} - 351 q^{85} + 312 q^{88} + 1692 q^{89} - 1248 q^{91} + 780 q^{92} - 522 q^{94} - 198 q^{95} + 348 q^{97} + 453 q^{98}+O(q^{100})$$ 2 * q - 3 * q^2 - 5 * q^4 - 24 * q^7 + 3 * q^10 - 24 * q^11 + 91 * q^13 + 48 * q^14 - q^16 + 117 * q^17 - 198 * q^19 + 15 * q^20 + 24 * q^22 - 78 * q^23 + 244 * q^25 - 117 * q^26 + 120 * q^28 - 141 * q^29 + 315 * q^32 - 24 * q^35 - 249 * q^37 + 396 * q^38 - 78 * q^40 - 471 * q^41 - 104 * q^43 + 234 * q^46 - 151 * q^49 - 366 * q^50 - 130 * q^52 - 186 * q^53 + 24 * q^55 - 312 * q^56 + 423 * q^58 + 492 * q^59 - 145 * q^61 + 270 * q^62 - 614 * q^64 - 39 * q^65 - 1362 * q^67 + 585 * q^68 - 1830 * q^71 + 249 * q^74 + 990 * q^76 + 384 * q^77 + 2552 * q^79 - 3 * q^80 + 471 * q^82 - 351 * q^85 + 312 * q^88 + 1692 * q^89 - 1248 * q^91 + 780 * q^92 - 522 * q^94 - 198 * q^95 + 348 * q^97 + 453 * 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
−1.50000 + 0.866025i 0 −2.50000 + 4.33013i 1.73205i 0 −12.0000 6.92820i 22.5167i 0 1.50000 + 2.59808i
82.1 −1.50000 0.866025i 0 −2.50000 4.33013i 1.73205i 0 −12.0000 + 6.92820i 22.5167i 0 1.50000 2.59808i
 $$n$$: 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.a 2
3.b odd 2 1 13.4.e.b 2
12.b even 2 1 208.4.w.b 2
13.e even 6 1 inner 117.4.q.a 2
13.f odd 12 2 1521.4.a.o 2
39.d odd 2 1 169.4.e.a 2
39.f even 4 2 169.4.c.h 4
39.h odd 6 1 13.4.e.b 2
39.h odd 6 1 169.4.b.d 2
39.i odd 6 1 169.4.b.d 2
39.i odd 6 1 169.4.e.a 2
39.k even 12 2 169.4.a.i 2
39.k even 12 2 169.4.c.h 4
156.r even 6 1 208.4.w.b 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 3T_{2} + 3$$ acting on $$S_{4}^{\mathrm{new}}(117, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 3T + 3$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 3$$
$7$ $$T^{2} + 24T + 192$$
$11$ $$T^{2} + 24T + 192$$
$13$ $$T^{2} - 91T + 2197$$
$17$ $$T^{2} - 117T + 13689$$
$19$ $$T^{2} + 198T + 13068$$
$23$ $$T^{2} + 78T + 6084$$
$29$ $$T^{2} + 141T + 19881$$
$31$ $$T^{2} + 24300$$
$37$ $$T^{2} + 249T + 20667$$
$41$ $$T^{2} + 471T + 73947$$
$43$ $$T^{2} + 104T + 10816$$
$47$ $$T^{2} + 90828$$
$53$ $$(T + 93)^{2}$$
$59$ $$T^{2} - 492T + 80688$$
$61$ $$T^{2} + 145T + 21025$$
$67$ $$T^{2} + 1362 T + 618348$$
$71$ $$T^{2} + 1830 T + 1116300$$
$73$ $$T^{2} + 210675$$
$79$ $$(T - 1276)^{2}$$
$83$ $$T^{2} + 623808$$
$89$ $$T^{2} - 1692 T + 954288$$
$97$ $$T^{2} - 348T + 40368$$