Properties

 Label 117.4.g.d Level $117$ Weight $4$ Character orbit 117.g 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.g (of order $$3$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$6.90322347067$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{17})$$ Defining polynomial: $$x^{4} - x^{3} + 5x^{2} + 4x + 16$$ x^4 - x^3 + 5*x^2 + 4*x + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{2} - 5 \beta_1 q^{4} + (5 \beta_{3} + 10) q^{5} + ( - 7 \beta_{2} - \beta_1) q^{7} + ( - 7 \beta_{3} + 4) q^{8}+O(q^{10})$$ q + (b3 + 2*b2 + b1 - 2) * q^2 - 5*b1 * q^4 + (5*b3 + 10) * q^5 + (-7*b2 - b1) * q^7 + (-7*b3 + 4) * q^8 $$q + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{2} - 5 \beta_1 q^{4} + (5 \beta_{3} + 10) q^{5} + ( - 7 \beta_{2} - \beta_1) q^{7} + ( - 7 \beta_{3} + 4) q^{8} + ( - 5 \beta_{3} - 5 \beta_1) q^{10} + ( - 15 \beta_{3} - \beta_{2} - 15 \beta_1 + 1) q^{11} + ( - 2 \beta_{3} - 40 \beta_{2} + 5 \beta_1 + 49) q^{13} + ( - 10 \beta_{3} + 18) q^{14} + ( - 15 \beta_{3} + 36 \beta_{2} - 15 \beta_1 - 36) q^{16} + (43 \beta_{2} - 16 \beta_1) q^{17} + (73 \beta_{2} - 5 \beta_1) q^{19} + (100 \beta_{2} - 25 \beta_1) q^{20} + (62 \beta_{2} + 46 \beta_1) q^{22} + (33 \beta_{3} - 89 \beta_{2} + 33 \beta_1 + 89) q^{23} + (75 \beta_{3} + 75) q^{25} + (30 \beta_{3} + 106 \beta_{2} + 55 \beta_1 - 46) q^{26} + (40 \beta_{3} + 20 \beta_{2} + 40 \beta_1 - 20) q^{28} + ( - 60 \beta_{3} + 13 \beta_{2} - 60 \beta_1 - 13) q^{29} + ( - 100 \beta_{3} - 120) q^{31} + (20 \beta_{2} + 65 \beta_1) q^{32} + ( - 5 \beta_{3} - 22) q^{34} + ( - 50 \beta_{2} + 30 \beta_1) q^{35} + ( - 44 \beta_{3} - 73 \beta_{2} - 44 \beta_1 + 73) q^{37} + (58 \beta_{3} - 126) q^{38} + ( - 15 \beta_{3} - 100) q^{40} + ( - 20 \beta_{3} - 259 \beta_{2} - 20 \beta_1 + 259) q^{41} + ( - 179 \beta_{2} - 97 \beta_1) q^{43} + (80 \beta_{3} - 300) q^{44} + (46 \beta_{2} - 10 \beta_1) q^{46} + ( - 140 \beta_{3} - 100) q^{47} + (15 \beta_{3} - 290 \beta_{2} + 15 \beta_1 + 290) q^{49} + ( - 150 \beta_{3} - 150 \beta_{2} - 150 \beta_1 + 150) q^{50} + (175 \beta_{3} - 140 \beta_{2} - 80 \beta_1 + 100) q^{52} + (165 \beta_{3} - 190) q^{53} + ( - 70 \beta_{3} + 290 \beta_{2} - 70 \beta_1 - 290) q^{55} + ( - 56 \beta_{2} - 60 \beta_1) q^{56} + (214 \beta_{2} + 167 \beta_1) q^{58} + ( - 377 \beta_{2} - 55 \beta_1) q^{59} + ( - 351 \beta_{2} + 200 \beta_1) q^{61} + (180 \beta_{3} + 160 \beta_{2} + 180 \beta_1 - 160) q^{62} + (95 \beta_{3} - 588) q^{64} + (235 \beta_{3} - 500 \beta_{2} + 225 \beta_1 + 450) q^{65} + (91 \beta_{3} - 283 \beta_{2} + 91 \beta_1 + 283) q^{67} + ( - 135 \beta_{3} + 320 \beta_{2} - 135 \beta_1 - 320) q^{68} + (40 \beta_{3} - 20) q^{70} + (11 \beta_{2} + 105 \beta_1) q^{71} + ( - 85 \beta_{3} + 250) q^{73} + (322 \beta_{2} + 205 \beta_1) q^{74} + ( - 340 \beta_{3} + 100 \beta_{2} - 340 \beta_1 - 100) q^{76} + (121 \beta_{3} - 67) q^{77} + (40 \beta_{3} + 140) q^{79} + ( - 255 \beta_{3} + 660 \beta_{2} - 255 \beta_1 - 660) q^{80} + (598 \beta_{2} + 319 \beta_1) q^{82} + ( - 100 \beta_{3} - 180) q^{83} + (750 \beta_{2} - 295 \beta_1) q^{85} + ( - 470 \beta_{3} + 746) q^{86} + ( - 172 \beta_{3} - 424 \beta_{2} - 172 \beta_1 + 424) q^{88} + (125 \beta_{3} - 523 \beta_{2} + 125 \beta_1 + 523) q^{89} + ( - 91 \beta_{2} - 65 \beta_1 - 260) q^{91} + (280 \beta_{3} + 660) q^{92} + (320 \beta_{3} + 360 \beta_{2} + 320 \beta_1 - 360) q^{94} + (830 \beta_{2} - 390 \beta_1) q^{95} + ( - 27 \beta_{2} + 469 \beta_1) q^{97} + (520 \beta_{2} + 245 \beta_1) q^{98}+O(q^{100})$$ q + (b3 + 2*b2 + b1 - 2) * q^2 - 5*b1 * q^4 + (5*b3 + 10) * q^5 + (-7*b2 - b1) * q^7 + (-7*b3 + 4) * q^8 + (-5*b3 - 5*b1) * q^10 + (-15*b3 - b2 - 15*b1 + 1) * q^11 + (-2*b3 - 40*b2 + 5*b1 + 49) * q^13 + (-10*b3 + 18) * q^14 + (-15*b3 + 36*b2 - 15*b1 - 36) * q^16 + (43*b2 - 16*b1) * q^17 + (73*b2 - 5*b1) * q^19 + (100*b2 - 25*b1) * q^20 + (62*b2 + 46*b1) * q^22 + (33*b3 - 89*b2 + 33*b1 + 89) * q^23 + (75*b3 + 75) * q^25 + (30*b3 + 106*b2 + 55*b1 - 46) * q^26 + (40*b3 + 20*b2 + 40*b1 - 20) * q^28 + (-60*b3 + 13*b2 - 60*b1 - 13) * q^29 + (-100*b3 - 120) * q^31 + (20*b2 + 65*b1) * q^32 + (-5*b3 - 22) * q^34 + (-50*b2 + 30*b1) * q^35 + (-44*b3 - 73*b2 - 44*b1 + 73) * q^37 + (58*b3 - 126) * q^38 + (-15*b3 - 100) * q^40 + (-20*b3 - 259*b2 - 20*b1 + 259) * q^41 + (-179*b2 - 97*b1) * q^43 + (80*b3 - 300) * q^44 + (46*b2 - 10*b1) * q^46 + (-140*b3 - 100) * q^47 + (15*b3 - 290*b2 + 15*b1 + 290) * q^49 + (-150*b3 - 150*b2 - 150*b1 + 150) * q^50 + (175*b3 - 140*b2 - 80*b1 + 100) * q^52 + (165*b3 - 190) * q^53 + (-70*b3 + 290*b2 - 70*b1 - 290) * q^55 + (-56*b2 - 60*b1) * q^56 + (214*b2 + 167*b1) * q^58 + (-377*b2 - 55*b1) * q^59 + (-351*b2 + 200*b1) * q^61 + (180*b3 + 160*b2 + 180*b1 - 160) * q^62 + (95*b3 - 588) * q^64 + (235*b3 - 500*b2 + 225*b1 + 450) * q^65 + (91*b3 - 283*b2 + 91*b1 + 283) * q^67 + (-135*b3 + 320*b2 - 135*b1 - 320) * q^68 + (40*b3 - 20) * q^70 + (11*b2 + 105*b1) * q^71 + (-85*b3 + 250) * q^73 + (322*b2 + 205*b1) * q^74 + (-340*b3 + 100*b2 - 340*b1 - 100) * q^76 + (121*b3 - 67) * q^77 + (40*b3 + 140) * q^79 + (-255*b3 + 660*b2 - 255*b1 - 660) * q^80 + (598*b2 + 319*b1) * q^82 + (-100*b3 - 180) * q^83 + (750*b2 - 295*b1) * q^85 + (-470*b3 + 746) * q^86 + (-172*b3 - 424*b2 - 172*b1 + 424) * q^88 + (125*b3 - 523*b2 + 125*b1 + 523) * q^89 + (-91*b2 - 65*b1 - 260) * q^91 + (280*b3 + 660) * q^92 + (320*b3 + 360*b2 + 320*b1 - 360) * q^94 + (830*b2 - 390*b1) * q^95 + (-27*b2 + 469*b1) * q^97 + (520*b2 + 245*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 5 q^{2} - 5 q^{4} + 30 q^{5} - 15 q^{7} + 30 q^{8}+O(q^{10})$$ 4 * q - 5 * q^2 - 5 * q^4 + 30 * q^5 - 15 * q^7 + 30 * q^8 $$4 q - 5 q^{2} - 5 q^{4} + 30 q^{5} - 15 q^{7} + 30 q^{8} + 5 q^{10} + 17 q^{11} + 125 q^{13} + 92 q^{14} - 57 q^{16} + 70 q^{17} + 141 q^{19} + 175 q^{20} + 170 q^{22} + 145 q^{23} + 150 q^{25} + 23 q^{26} - 80 q^{28} + 34 q^{29} - 280 q^{31} + 105 q^{32} - 78 q^{34} - 70 q^{35} + 190 q^{37} - 620 q^{38} - 370 q^{40} + 538 q^{41} - 455 q^{43} - 1360 q^{44} + 82 q^{46} - 120 q^{47} + 565 q^{49} + 450 q^{50} - 310 q^{52} - 1090 q^{53} - 510 q^{55} - 172 q^{56} + 595 q^{58} - 809 q^{59} - 502 q^{61} - 500 q^{62} - 2542 q^{64} + 555 q^{65} + 475 q^{67} - 505 q^{68} - 160 q^{70} + 127 q^{71} + 1170 q^{73} + 849 q^{74} + 140 q^{76} - 510 q^{77} + 480 q^{79} - 1065 q^{80} + 1515 q^{82} - 520 q^{83} + 1205 q^{85} + 3924 q^{86} + 1020 q^{88} + 921 q^{89} - 1287 q^{91} + 2080 q^{92} - 1040 q^{94} + 1270 q^{95} + 415 q^{97} + 1285 q^{98}+O(q^{100})$$ 4 * q - 5 * q^2 - 5 * q^4 + 30 * q^5 - 15 * q^7 + 30 * q^8 + 5 * q^10 + 17 * q^11 + 125 * q^13 + 92 * q^14 - 57 * q^16 + 70 * q^17 + 141 * q^19 + 175 * q^20 + 170 * q^22 + 145 * q^23 + 150 * q^25 + 23 * q^26 - 80 * q^28 + 34 * q^29 - 280 * q^31 + 105 * q^32 - 78 * q^34 - 70 * q^35 + 190 * q^37 - 620 * q^38 - 370 * q^40 + 538 * q^41 - 455 * q^43 - 1360 * q^44 + 82 * q^46 - 120 * q^47 + 565 * q^49 + 450 * q^50 - 310 * q^52 - 1090 * q^53 - 510 * q^55 - 172 * q^56 + 595 * q^58 - 809 * q^59 - 502 * q^61 - 500 * q^62 - 2542 * q^64 + 555 * q^65 + 475 * q^67 - 505 * q^68 - 160 * q^70 + 127 * q^71 + 1170 * q^73 + 849 * q^74 + 140 * q^76 - 510 * q^77 + 480 * q^79 - 1065 * q^80 + 1515 * q^82 - 520 * q^83 + 1205 * q^85 + 3924 * q^86 + 1020 * q^88 + 921 * q^89 - 1287 * q^91 + 2080 * q^92 - 1040 * q^94 + 1270 * q^95 + 415 * q^97 + 1285 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 5x^{2} + 4x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20$$ (-v^3 + 5*v^2 - 5*v + 16) / 20 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 4 ) / 5$$ (v^3 + 4) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 4\beta_{2} + \beta _1 - 4$$ b3 + 4*b2 + b1 - 4 $$\nu^{3}$$ $$=$$ $$5\beta_{3} - 4$$ 5*b3 - 4

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 1.28078 − 2.21837i −0.780776 + 1.35234i 1.28078 + 2.21837i −0.780776 − 1.35234i
−2.28078 3.95042i 0 −6.40388 + 11.0918i −2.80776 0 −4.78078 + 8.28055i 21.9309 0 6.40388 + 11.0918i
55.2 −0.219224 0.379706i 0 3.90388 6.76172i 17.8078 0 −2.71922 + 4.70983i −6.93087 0 −3.90388 6.76172i
100.1 −2.28078 + 3.95042i 0 −6.40388 11.0918i −2.80776 0 −4.78078 8.28055i 21.9309 0 6.40388 11.0918i
100.2 −0.219224 + 0.379706i 0 3.90388 + 6.76172i 17.8078 0 −2.71922 4.70983i −6.93087 0 −3.90388 + 6.76172i
 $$n$$: 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.d 4
3.b odd 2 1 13.4.c.b 4
12.b even 2 1 208.4.i.e 4
13.c even 3 1 inner 117.4.g.d 4
13.c even 3 1 1521.4.a.t 2
13.e even 6 1 1521.4.a.l 2
39.d odd 2 1 169.4.c.f 4
39.f even 4 2 169.4.e.g 8
39.h odd 6 1 169.4.a.j 2
39.h odd 6 1 169.4.c.f 4
39.i odd 6 1 13.4.c.b 4
39.i odd 6 1 169.4.a.f 2
39.k even 12 2 169.4.b.e 4
39.k even 12 2 169.4.e.g 8
156.p even 6 1 208.4.i.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.c.b 4 3.b odd 2 1
13.4.c.b 4 39.i odd 6 1
117.4.g.d 4 1.a even 1 1 trivial
117.4.g.d 4 13.c even 3 1 inner
169.4.a.f 2 39.i odd 6 1
169.4.a.j 2 39.h odd 6 1
169.4.b.e 4 39.k even 12 2
169.4.c.f 4 39.d odd 2 1
169.4.c.f 4 39.h odd 6 1
169.4.e.g 8 39.f even 4 2
169.4.e.g 8 39.k even 12 2
208.4.i.e 4 12.b even 2 1
208.4.i.e 4 156.p even 6 1
1521.4.a.l 2 13.e even 6 1
1521.4.a.t 2 13.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 5T_{2}^{3} + 23T_{2}^{2} + 10T_{2} + 4$$ acting on $$S_{4}^{\mathrm{new}}(117, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 5 T^{3} + 23 T^{2} + 10 T + 4$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - 15 T - 50)^{2}$$
$7$ $$T^{4} + 15 T^{3} + 173 T^{2} + \cdots + 2704$$
$11$ $$T^{4} - 17 T^{3} + 1173 T^{2} + \cdots + 781456$$
$13$ $$T^{4} - 125 T^{3} + 7956 T^{2} + \cdots + 4826809$$
$17$ $$T^{4} - 70 T^{3} + 4763 T^{2} + \cdots + 18769$$
$19$ $$T^{4} - 141 T^{3} + \cdots + 23658496$$
$23$ $$T^{4} - 145 T^{3} + 20397 T^{2} + \cdots + 394384$$
$29$ $$T^{4} - 34 T^{3} + \cdots + 225330121$$
$31$ $$(T^{2} + 140 T - 37600)^{2}$$
$37$ $$T^{4} - 190 T^{3} + 35303 T^{2} + \cdots + 635209$$
$41$ $$T^{4} - 538 T^{3} + \cdots + 4992976921$$
$43$ $$T^{4} + 455 T^{3} + \cdots + 138485824$$
$47$ $$(T^{2} + 60 T - 82400)^{2}$$
$53$ $$(T^{2} + 545 T - 41450)^{2}$$
$59$ $$T^{4} + 809 T^{3} + \cdots + 22729783696$$
$61$ $$T^{4} + 502 T^{3} + \cdots + 11448786001$$
$67$ $$T^{4} - 475 T^{3} + \cdots + 449948944$$
$71$ $$T^{4} - 127 T^{3} + \cdots + 1833894976$$
$73$ $$(T^{2} - 585 T + 54850)^{2}$$
$79$ $$(T^{2} - 240 T + 7600)^{2}$$
$83$ $$(T^{2} + 260 T - 25600)^{2}$$
$89$ $$T^{4} - 921 T^{3} + \cdots + 21215087716$$
$97$ $$T^{4} - 415 T^{3} + \cdots + 795268001284$$