# Properties

 Label 117.4.q.d Level $117$ Weight $4$ Character orbit 117.q Analytic conductor $6.903$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-17})$$ Defining polynomial: $$x^{4} - 17x^{2} + 289$$ x^4 - 17*x^2 + 289 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + 9 \beta_{2} q^{4} + (2 \beta_{3} + 6 \beta_{2} - 3) q^{5} + (3 \beta_{3} + 11 \beta_{2} - 3 \beta_1 - 22) q^{7} + \beta_{3} q^{8}+O(q^{10})$$ q + b1 * q^2 + 9*b2 * q^4 + (2*b3 + 6*b2 - 3) * q^5 + (3*b3 + 11*b2 - 3*b1 - 22) * q^7 + b3 * q^8 $$q + \beta_1 q^{2} + 9 \beta_{2} q^{4} + (2 \beta_{3} + 6 \beta_{2} - 3) q^{5} + (3 \beta_{3} + 11 \beta_{2} - 3 \beta_1 - 22) q^{7} + \beta_{3} q^{8} + (6 \beta_{3} + 34 \beta_{2} - 3 \beta_1 - 34) q^{10} + (21 \beta_{2} + \beta_1 + 21) q^{11} + ( - 12 \beta_{3} + 12 \beta_{2} + 9 \beta_1 + 4) q^{13} + (11 \beta_{3} - 22 \beta_1 - 51) q^{14} + ( - 55 \beta_{2} + 55) q^{16} + (\beta_{3} + 36 \beta_{2} + \beta_1) q^{17} + (9 \beta_{3} - 37 \beta_{2} - 9 \beta_1 + 74) q^{19} + (18 \beta_{3} + 27 \beta_{2} - 18 \beta_1 - 54) q^{20} + (21 \beta_{3} + 17 \beta_{2} + 21 \beta_1) q^{22} + (14 \beta_{3} - 69 \beta_{2} - 7 \beta_1 + 69) q^{23} + (12 \beta_{3} - 24 \beta_1 + 30) q^{25} + (12 \beta_{3} - 51 \beta_{2} + 4 \beta_1 + 204) q^{26} + ( - 99 \beta_{2} - 27 \beta_1 - 99) q^{28} + (44 \beta_{3} - 3 \beta_{2} - 22 \beta_1 + 3) q^{29} + ( - 42 \beta_{3} - 156 \beta_{2} + 78) q^{31} + ( - 63 \beta_{3} + 63 \beta_1) q^{32} + (36 \beta_{3} + 34 \beta_{2} - 17) q^{34} + ( - 31 \beta_{3} - 201 \beta_{2} - 31 \beta_1) q^{35} + (82 \beta_{2} - 45 \beta_1 + 82) q^{37} + ( - 37 \beta_{3} + 74 \beta_1 - 153) q^{38} + (3 \beta_{3} - 6 \beta_1 - 34) q^{40} + ( - 30 \beta_{2} + \beta_1 - 30) q^{41} + (15 \beta_{3} + 235 \beta_{2} + 15 \beta_1) q^{43} + (9 \beta_{3} + 378 \beta_{2} - 189) q^{44} + ( - 69 \beta_{3} + 119 \beta_{2} + 69 \beta_1 - 238) q^{46} + ( - 79 \beta_{3} + 222 \beta_{2} - 111) q^{47} + ( - 132 \beta_{3} - 173 \beta_{2} + 66 \beta_1 + 173) q^{49} + ( - 204 \beta_{2} + 30 \beta_1 - 204) q^{50} + ( - 27 \beta_{3} + 144 \beta_{2} + 108 \beta_1 - 108) q^{52} + (18 \beta_{3} - 36 \beta_1 + 567) q^{53} + (90 \beta_{3} + 223 \beta_{2} - 45 \beta_1 - 223) q^{55} + ( - 11 \beta_{3} - 51 \beta_{2} - 11 \beta_1) q^{56} + ( - 3 \beta_{3} + 374 \beta_{2} + 3 \beta_1 - 748) q^{58} + ( - 8 \beta_{3} + 360 \beta_{2} + 8 \beta_1 - 720) q^{59} + (87 \beta_{3} - 80 \beta_{2} + 87 \beta_1) q^{61} + ( - 156 \beta_{3} - 714 \beta_{2} + 78 \beta_1 + 714) q^{62} + 631 q^{64} + (50 \beta_{3} + 366 \beta_{2} + 21 \beta_1 + 18) q^{65} + ( - 83 \beta_{2} - 21 \beta_1 - 83) q^{67} + (18 \beta_{3} + 324 \beta_{2} - 9 \beta_1 - 324) q^{68} + ( - 201 \beta_{3} - 1054 \beta_{2} + 527) q^{70} + ( - 17 \beta_{3} - 219 \beta_{2} + 17 \beta_1 + 438) q^{71} + ( - 450 \beta_{2} + 225) q^{73} + (82 \beta_{3} - 765 \beta_{2} + 82 \beta_1) q^{74} + (333 \beta_{2} - 81 \beta_1 + 333) q^{76} + (74 \beta_{3} - 148 \beta_1 - 744) q^{77} + (126 \beta_{3} - 252 \beta_1 + 2) q^{79} + (165 \beta_{2} + 110 \beta_1 + 165) q^{80} + ( - 30 \beta_{3} + 17 \beta_{2} - 30 \beta_1) q^{82} + ( - 241 \beta_{3} - 354 \beta_{2} + 177) q^{83} + (81 \beta_{3} + 142 \beta_{2} - 81 \beta_1 - 284) q^{85} + (235 \beta_{3} + 510 \beta_{2} - 255) q^{86} + (42 \beta_{3} + 17 \beta_{2} - 21 \beta_1 - 17) q^{88} + (42 \beta_{2} - 242 \beta_1 + 42) q^{89} + (243 \beta_{3} + 524 \beta_{2} - 114 \beta_1 - 679) q^{91} + (63 \beta_{3} - 126 \beta_1 + 621) q^{92} + (222 \beta_{3} - 1343 \beta_{2} - 111 \beta_1 + 1343) q^{94} + (47 \beta_{3} + 27 \beta_{2} + 47 \beta_1) q^{95} + ( - 402 \beta_{3} + 56 \beta_{2} + 402 \beta_1 - 112) q^{97} + ( - 173 \beta_{3} - 1122 \beta_{2} + 173 \beta_1 + 2244) q^{98}+O(q^{100})$$ q + b1 * q^2 + 9*b2 * q^4 + (2*b3 + 6*b2 - 3) * q^5 + (3*b3 + 11*b2 - 3*b1 - 22) * q^7 + b3 * q^8 + (6*b3 + 34*b2 - 3*b1 - 34) * q^10 + (21*b2 + b1 + 21) * q^11 + (-12*b3 + 12*b2 + 9*b1 + 4) * q^13 + (11*b3 - 22*b1 - 51) * q^14 + (-55*b2 + 55) * q^16 + (b3 + 36*b2 + b1) * q^17 + (9*b3 - 37*b2 - 9*b1 + 74) * q^19 + (18*b3 + 27*b2 - 18*b1 - 54) * q^20 + (21*b3 + 17*b2 + 21*b1) * q^22 + (14*b3 - 69*b2 - 7*b1 + 69) * q^23 + (12*b3 - 24*b1 + 30) * q^25 + (12*b3 - 51*b2 + 4*b1 + 204) * q^26 + (-99*b2 - 27*b1 - 99) * q^28 + (44*b3 - 3*b2 - 22*b1 + 3) * q^29 + (-42*b3 - 156*b2 + 78) * q^31 + (-63*b3 + 63*b1) * q^32 + (36*b3 + 34*b2 - 17) * q^34 + (-31*b3 - 201*b2 - 31*b1) * q^35 + (82*b2 - 45*b1 + 82) * q^37 + (-37*b3 + 74*b1 - 153) * q^38 + (3*b3 - 6*b1 - 34) * q^40 + (-30*b2 + b1 - 30) * q^41 + (15*b3 + 235*b2 + 15*b1) * q^43 + (9*b3 + 378*b2 - 189) * q^44 + (-69*b3 + 119*b2 + 69*b1 - 238) * q^46 + (-79*b3 + 222*b2 - 111) * q^47 + (-132*b3 - 173*b2 + 66*b1 + 173) * q^49 + (-204*b2 + 30*b1 - 204) * q^50 + (-27*b3 + 144*b2 + 108*b1 - 108) * q^52 + (18*b3 - 36*b1 + 567) * q^53 + (90*b3 + 223*b2 - 45*b1 - 223) * q^55 + (-11*b3 - 51*b2 - 11*b1) * q^56 + (-3*b3 + 374*b2 + 3*b1 - 748) * q^58 + (-8*b3 + 360*b2 + 8*b1 - 720) * q^59 + (87*b3 - 80*b2 + 87*b1) * q^61 + (-156*b3 - 714*b2 + 78*b1 + 714) * q^62 + 631 * q^64 + (50*b3 + 366*b2 + 21*b1 + 18) * q^65 + (-83*b2 - 21*b1 - 83) * q^67 + (18*b3 + 324*b2 - 9*b1 - 324) * q^68 + (-201*b3 - 1054*b2 + 527) * q^70 + (-17*b3 - 219*b2 + 17*b1 + 438) * q^71 + (-450*b2 + 225) * q^73 + (82*b3 - 765*b2 + 82*b1) * q^74 + (333*b2 - 81*b1 + 333) * q^76 + (74*b3 - 148*b1 - 744) * q^77 + (126*b3 - 252*b1 + 2) * q^79 + (165*b2 + 110*b1 + 165) * q^80 + (-30*b3 + 17*b2 - 30*b1) * q^82 + (-241*b3 - 354*b2 + 177) * q^83 + (81*b3 + 142*b2 - 81*b1 - 284) * q^85 + (235*b3 + 510*b2 - 255) * q^86 + (42*b3 + 17*b2 - 21*b1 - 17) * q^88 + (42*b2 - 242*b1 + 42) * q^89 + (243*b3 + 524*b2 - 114*b1 - 679) * q^91 + (63*b3 - 126*b1 + 621) * q^92 + (222*b3 - 1343*b2 - 111*b1 + 1343) * q^94 + (47*b3 + 27*b2 + 47*b1) * q^95 + (-402*b3 + 56*b2 + 402*b1 - 112) * q^97 + (-173*b3 - 1122*b2 + 173*b1 + 2244) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 18 q^{4} - 66 q^{7}+O(q^{10})$$ 4 * q + 18 * q^4 - 66 * q^7 $$4 q + 18 q^{4} - 66 q^{7} - 68 q^{10} + 126 q^{11} + 40 q^{13} - 204 q^{14} + 110 q^{16} + 72 q^{17} + 222 q^{19} - 162 q^{20} + 34 q^{22} + 138 q^{23} + 120 q^{25} + 714 q^{26} - 594 q^{28} + 6 q^{29} - 402 q^{35} + 492 q^{37} - 612 q^{38} - 136 q^{40} - 180 q^{41} + 470 q^{43} - 714 q^{46} + 346 q^{49} - 1224 q^{50} - 144 q^{52} + 2268 q^{53} - 446 q^{55} - 102 q^{56} - 2244 q^{58} - 2160 q^{59} - 160 q^{61} + 1428 q^{62} + 2524 q^{64} + 804 q^{65} - 498 q^{67} - 648 q^{68} + 1314 q^{71} - 1530 q^{74} + 1998 q^{76} - 2976 q^{77} + 8 q^{79} + 990 q^{80} + 34 q^{82} - 852 q^{85} - 34 q^{88} + 252 q^{89} - 1668 q^{91} + 2484 q^{92} + 2686 q^{94} + 54 q^{95} - 336 q^{97} + 6732 q^{98}+O(q^{100})$$ 4 * q + 18 * q^4 - 66 * q^7 - 68 * q^10 + 126 * q^11 + 40 * q^13 - 204 * q^14 + 110 * q^16 + 72 * q^17 + 222 * q^19 - 162 * q^20 + 34 * q^22 + 138 * q^23 + 120 * q^25 + 714 * q^26 - 594 * q^28 + 6 * q^29 - 402 * q^35 + 492 * q^37 - 612 * q^38 - 136 * q^40 - 180 * q^41 + 470 * q^43 - 714 * q^46 + 346 * q^49 - 1224 * q^50 - 144 * q^52 + 2268 * q^53 - 446 * q^55 - 102 * q^56 - 2244 * q^58 - 2160 * q^59 - 160 * q^61 + 1428 * q^62 + 2524 * q^64 + 804 * q^65 - 498 * q^67 - 648 * q^68 + 1314 * q^71 - 1530 * q^74 + 1998 * q^76 - 2976 * q^77 + 8 * q^79 + 990 * q^80 + 34 * q^82 - 852 * q^85 - 34 * q^88 + 252 * q^89 - 1668 * q^91 + 2484 * q^92 + 2686 * q^94 + 54 * q^95 - 336 * q^97 + 6732 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 17x^{2} + 289$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 17$$ (v^2) / 17 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 17$$ (v^3) / 17
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$17\beta_{2}$$ 17*b2 $$\nu^{3}$$ $$=$$ $$17\beta_{3}$$ 17*b3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$$.

 $$n$$ $$28$$ $$92$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$

## 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
 −3.57071 + 2.06155i 3.57071 − 2.06155i −3.57071 − 2.06155i 3.57071 + 2.06155i
−3.57071 + 2.06155i 0 4.50000 7.79423i 3.05006i 0 −5.78786 3.34162i 4.12311i 0 −6.28786 10.8909i
10.2 3.57071 2.06155i 0 4.50000 7.79423i 13.4424i 0 −27.2121 15.7109i 4.12311i 0 −27.7121 47.9988i
82.1 −3.57071 2.06155i 0 4.50000 + 7.79423i 3.05006i 0 −5.78786 + 3.34162i 4.12311i 0 −6.28786 + 10.8909i
82.2 3.57071 + 2.06155i 0 4.50000 + 7.79423i 13.4424i 0 −27.2121 + 15.7109i 4.12311i 0 −27.7121 + 47.9988i
 $$n$$: 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.d 4
3.b odd 2 1 39.4.j.b 4
12.b even 2 1 624.4.bv.c 4
13.e even 6 1 inner 117.4.q.d 4
13.f odd 12 2 1521.4.a.z 4
39.h odd 6 1 39.4.j.b 4
39.h odd 6 1 507.4.b.e 4
39.i odd 6 1 507.4.b.e 4
39.k even 12 2 507.4.a.k 4
156.r even 6 1 624.4.bv.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.j.b 4 3.b odd 2 1
39.4.j.b 4 39.h odd 6 1
117.4.q.d 4 1.a even 1 1 trivial
117.4.q.d 4 13.e even 6 1 inner
507.4.a.k 4 39.k even 12 2
507.4.b.e 4 39.h odd 6 1
507.4.b.e 4 39.i odd 6 1
624.4.bv.c 4 12.b even 2 1
624.4.bv.c 4 156.r even 6 1
1521.4.a.z 4 13.f odd 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 17T_{2}^{2} + 289$$ acting on $$S_{4}^{\mathrm{new}}(117, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 17T^{2} + 289$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 190T^{2} + 1681$$
$7$ $$T^{4} + 66 T^{3} + 1662 T^{2} + \cdots + 44100$$
$11$ $$T^{4} - 126 T^{3} + 6598 T^{2} + \cdots + 1705636$$
$13$ $$T^{4} - 40 T^{3} + 663 T^{2} + \cdots + 4826809$$
$17$ $$T^{4} - 72 T^{3} + 3939 T^{2} + \cdots + 1550025$$
$19$ $$T^{4} - 222 T^{3} + 19158 T^{2} + \cdots + 7452900$$
$23$ $$T^{4} - 138 T^{3} + 16782 T^{2} + \cdots + 5116644$$
$29$ $$T^{4} - 6 T^{3} + 24711 T^{2} + \cdots + 608855625$$
$31$ $$T^{4} + 96480 T^{2} + \cdots + 137733696$$
$37$ $$T^{4} - 492 T^{3} + \cdots + 203148009$$
$41$ $$T^{4} + 180 T^{3} + 13483 T^{2} + \cdots + 7198489$$
$43$ $$T^{4} - 470 T^{3} + \cdots + 1914062500$$
$47$ $$T^{4} + 286120 T^{2} + \cdots + 4779509956$$
$53$ $$(T^{2} - 1134 T + 304965)^{2}$$
$59$ $$T^{4} + 2160 T^{3} + \cdots + 150320594944$$
$61$ $$T^{4} + 160 T^{3} + \cdots + 144110585161$$
$67$ $$T^{4} + 498 T^{3} + \cdots + 173448900$$
$71$ $$T^{4} - 1314 T^{3} + \cdots + 19312660900$$
$73$ $$(T^{2} + 151875)^{2}$$
$79$ $$(T^{2} - 4 T - 809672)^{2}$$
$83$ $$T^{4} + 2162728 T^{2} + \cdots + 798145692100$$
$89$ $$T^{4} - 252 T^{3} + \cdots + 980686167616$$
$97$ $$T^{4} + 336 T^{3} + \cdots + 7495877379600$$