# Properties

 Label 18.5.d.a Level $18$ Weight $5$ Character orbit 18.d Analytic conductor $1.861$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$18 = 2 \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 18.d (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.86065933551$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.221456830464.4 Defining polynomial: $$x^{8} - 4x^{7} + 38x^{6} - 100x^{5} + 449x^{4} - 736x^{3} + 1900x^{2} - 1548x + 2307$$ x^8 - 4*x^7 + 38*x^6 - 100*x^5 + 449*x^4 - 736*x^3 + 1900*x^2 - 1548*x + 2307 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}\cdot 3^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{2} + ( - \beta_{5} - 3 \beta_{2} + \beta_1 + 2) q^{3} + ( - 8 \beta_{2} + 8) q^{4} + ( - \beta_{6} - 3 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} + 4 \beta_{2} - \beta_1) q^{5} + (\beta_{7} + \beta_{6} - \beta_{4} - 2 \beta_{3} + 4 \beta_{2} - 3 \beta_1 + 4) q^{6} + ( - \beta_{7} - \beta_{6} + 4 \beta_{5} + 5 \beta_{3} - 9 \beta_{2} + 6 \beta_1 + 2) q^{7} + ( - 8 \beta_{3} - 8 \beta_1) q^{8} + ( - \beta_{7} + \beta_{6} + 2 \beta_{5} - 6 \beta_{4} + 23 \beta_{3} + 24 \beta_{2} + \cdots - 23) q^{9}+O(q^{10})$$ q - b3 * q^2 + (-b5 - 3*b2 + b1 + 2) * q^3 + (-8*b2 + 8) * q^4 + (-b6 - 3*b5 + 2*b4 - 3*b3 + 4*b2 - b1) * q^5 + (b7 + b6 - b4 - 2*b3 + 4*b2 - 3*b1 + 4) * q^6 + (-b7 - b6 + 4*b5 + 5*b3 - 9*b2 + 6*b1 + 2) * q^7 + (-8*b3 - 8*b1) * q^8 + (-b7 + b6 + 2*b5 - 6*b4 + 23*b3 + 24*b2 + 6*b1 - 23) * q^9 $$q - \beta_{3} q^{2} + ( - \beta_{5} - 3 \beta_{2} + \beta_1 + 2) q^{3} + ( - 8 \beta_{2} + 8) q^{4} + ( - \beta_{6} - 3 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} + 4 \beta_{2} - \beta_1) q^{5} + (\beta_{7} + \beta_{6} - \beta_{4} - 2 \beta_{3} + 4 \beta_{2} - 3 \beta_1 + 4) q^{6} + ( - \beta_{7} - \beta_{6} + 4 \beta_{5} + 5 \beta_{3} - 9 \beta_{2} + 6 \beta_1 + 2) q^{7} + ( - 8 \beta_{3} - 8 \beta_1) q^{8} + ( - \beta_{7} + \beta_{6} + 2 \beta_{5} - 6 \beta_{4} + 23 \beta_{3} + 24 \beta_{2} + \cdots - 23) q^{9}+ \cdots + (93 \beta_{7} - 372 \beta_{6} - 33 \beta_{5} + 369 \beta_{4} - 1950 \beta_{3} + \cdots - 2181) q^{99}+O(q^{100})$$ q - b3 * q^2 + (-b5 - 3*b2 + b1 + 2) * q^3 + (-8*b2 + 8) * q^4 + (-b6 - 3*b5 + 2*b4 - 3*b3 + 4*b2 - b1) * q^5 + (b7 + b6 - b4 - 2*b3 + 4*b2 - 3*b1 + 4) * q^6 + (-b7 - b6 + 4*b5 + 5*b3 - 9*b2 + 6*b1 + 2) * q^7 + (-8*b3 - 8*b1) * q^8 + (-b7 + b6 + 2*b5 - 6*b4 + 23*b3 + 24*b2 + 6*b1 - 23) * q^9 + (b7 + 3*b6 + 8*b5 - b4 + 4*b3 - 4*b2 + 2*b1 + 4) * q^10 + (-2*b7 + b6 - 4*b5 + 12*b4 - 17*b3 + 67*b2 - 6*b1 - 122) * q^11 + (-8*b4 - 24*b2 + 8*b1 - 8) * q^12 + (6*b7 - b6 + 15*b5 - 10*b4 + 33*b3 - 12*b2 + 17*b1 + 10) * q^13 + (-3*b7 - 4*b6 + 11*b4 - 6*b3 + 28*b2 - 12*b1 + 24) * q^14 + (-6*b7 - 3*b6 - 3*b5 - 6*b4 + 3*b3 + 36*b2 - 51*b1 + 120) * q^15 - 64*b2 * q^16 + (7*b7 + 6*b6 - 25*b5 + 20*b4 - 26*b3 - 205*b2 - 13*b1 + 103) * q^17 + (5*b7 - 2*b6 - 16*b5 + 3*b4 + 11*b3 + 156*b2 + 27*b1 - 128) * q^18 + (2*b7 - 3*b6 - 11*b5 - 11*b4 - 73*b3 + b2 + 76*b1 + 7) * q^19 + (-8*b7 - 8*b6 - 16*b5 - 8*b3 - 8*b2 + 16) * q^20 + (-3*b7 + 3*b6 + 14*b5 + 4*b4 - 93*b3 - 243*b2 - 12*b1 + 180) * q^21 + (-6*b7 + 4*b6 - 24*b5 + 22*b4 + 102*b3 - 64*b2 + 49*b1 + 72) * q^22 + (6*b7 + 13*b6 + 15*b5 - 32*b4 + 27*b3 + 86*b2 + 49*b1 + 114) * q^23 + (8*b7 - 8*b4 + 8*b3 - 32*b2 - 16*b1 + 64) * q^24 + (3*b7 - 3*b6 - 12*b5 + 12*b4 - 177*b3 + 212*b2 - 330*b1 - 6) * q^25 + (-11*b7 - 15*b6 + 56*b5 - 37*b4 + 42*b3 + 140*b2 + 16*b1 - 68) * q^26 + (-9*b7 - 3*b6 + 18*b5 + 39*b4 + 189*b3 - 114*b2 + 189*b1 - 93) * q^27 + (-8*b7 + 8*b5 + 32*b4 - 32*b3 + 8*b2 + 8*b1 - 48) * q^28 + (31*b7 + 13*b6 + 62*b5 - 72*b4 + 133*b3 + 103*b2 + 36*b1 - 278) * q^29 + (15*b7 + 3*b6 - 24*b5 + 33*b4 - 156*b3 + 36*b2 + 24*b1 - 444) * q^30 + (-30*b7 - 9*b6 - 33*b5 - 6*b4 + 285*b3 + 368*b2 + 147*b1 - 386) * q^31 - 64*b1 * q^32 + (9*b7 + 12*b6 + 33*b5 + 78*b4 - 108*b3 + 96*b2 - 393*b1 - 441) * q^33 + (-2*b7 + 25*b6 + 8*b5 - 54*b4 - 71*b3 - 56*b2 - 204*b1 + 4) * q^34 + (-23*b7 - 6*b6 + 41*b5 - 52*b4 + 244*b3 - 361*b2 + 215*b1 + 172) * q^35 + (8*b7 + 16*b6 + 56*b5 - 48*b4 + 176*b3 + 144*b2 + 168*b1 + 64) * q^36 + (-14*b7 - 30*b6 - 76*b5 + 26*b4 - 218*b3 + 44*b2 + 146*b1 - 42) * q^37 + (20*b7 + 11*b6 + 40*b5 - 36*b4 + 21*b3 - 592*b2 + 18*b1 + 1148) * q^38 + (-24*b7 - 39*b6 - 21*b5 + 32*b4 - 393*b3 + 636*b2 + 181*b1 + 338) * q^39 + (24*b7 + 16*b6 + 40*b4 - 48*b3 + 32*b2 - 32*b1) * q^40 + (12*b7 + 8*b6 - 24*b5 - 28*b4 + b2 + 32*b1 - 15) * q^41 + (-15*b7 - 14*b6 - 48*b5 + 39*b4 - 216*b3 - 772*b2 - 256*b1 + 704) * q^42 + (30*b7 + 3*b6 - 120*b5 + 54*b4 - 231*b3 + 85*b2 - 288*b1 - 60) * q^43 + (8*b7 + 24*b6 - 80*b5 + 40*b4 - 136*b3 + 1024*b2 - 104*b1 - 520) * q^44 + (-27*b6 - 81*b5 + 567*b3 - 1674*b2 + 351*b1 + 1242) * q^45 + (11*b7 - 15*b6 - 56*b5 - 59*b4 - 118*b3 + 4*b2 + 136*b1 + 236) * q^46 + (-63*b7 - 21*b6 - 126*b5 + 168*b4 - 147*b3 - 69*b2 - 84*b1 + 306) * q^47 + (64*b5 - 64*b4 - 192) * q^48 + (30*b7 + b6 + 57*b5 - 26*b4 + 435*b3 - 637*b2 + 217*b1 + 639) * q^49 + (-3*b7 + 12*b6 + 48*b5 - 21*b4 + 42*b3 - 1332*b2 + 209*b1 - 1272) * q^50 + (-9*b7 - 24*b6 - 6*b5 - 222*b4 - 18*b3 + 1320*b2 - 288*b1 - 2220) * q^51 + (-8*b7 - 56*b6 + 32*b5 + 96*b4 + 40*b3 + 8*b2 + 144*b1 + 16) * q^52 + (56*b7 - 56*b5 + 112*b4 - 532*b3 + 280*b2 - 476*b1 - 112) * q^53 + (-48*b7 - 18*b6 - 48*b5 + 120*b4 + 33*b3 + 1632*b2 - 180*b1 + 72) * q^54 + (63*b7 + 150*b6 + 387*b5 - 102*b4 + 90*b3 - 213*b2 + 249*b1 - 306) * q^55 + (-32*b7 - 8*b6 - 64*b5 + 96*b4 - 16*b3 - 128*b2 - 48*b1 + 352) * q^56 + (57*b7 + 141*b6 + 34*b5 - 150*b4 + 39*b3 + 1452*b2 - 34*b1 + 622) * q^57 + (-21*b7 - 62*b6 + 144*b5 - 227*b4 + 474*b3 + 404*b2 + 268*b1 - 528) * q^58 + (-87*b7 - 104*b6 + 36*b5 + 295*b4 - 138*b3 + 845*b2 - 515*b1 + 777) * q^59 + (-24*b7 + 24*b6 + 96*b5 - 96*b4 + 552*b3 - 1080*b2 + 48*b1 + 1344) * q^60 + (-135*b7 - 15*b6 + 540*b5 - 240*b4 + 513*b3 - 1577*b2 + 246*b1 + 270) * q^61 + (69*b7 + 33*b6 - 168*b5 + 171*b4 + 182*b3 + 2508*b2 + 284*b1 - 1236) * q^62 + (121*b7 + 101*b6 + 46*b5 - 270*b4 - 425*b3 - 1011*b2 - 186*b1 - 352) * q^63 - 512 * q^64 + (-31*b7 - 55*b6 - 62*b5 - 96*b4 - 697*b3 + 1847*b2 + 48*b1 - 3790) * q^65 + (-120*b7 - 33*b6 - 24*b5 + 48*b4 + 465*b3 - 720*b2 + 45*b1 - 2148) * q^66 + (99*b7 + 96*b6 - 90*b5 + 285*b4 + 18*b3 - 1639*b2 - 39*b1 + 1831) * q^67 + (48*b7 - 8*b6 - 216*b5 - 32*b4 - 120*b3 - 808*b2 - 8*b1 - 1032) * q^68 + (60*b7 + 63*b6 - 213*b5 + 114*b4 + 357*b3 + 1038*b2 + 909*b1 - 1278) * q^69 + (34*b7 - 41*b6 - 136*b5 + 150*b4 - 332*b3 + 1672*b2 - 378*b1 - 68) * q^70 + (-98*b7 - 48*b6 + 242*b5 - 244*b4 - 392*b3 - 3430*b2 - 538*b1 + 1690) * q^71 + (-16*b7 - 56*b6 - 64*b5 - 48*b4 - 64*b3 + 960*b2 + 216*b1 + 160) * q^72 + (-135*b7 - 168*b6 - 369*b5 + 372*b4 + 108*b3 + 303*b2 - 681*b1 + 2405) * q^73 + (64*b7 + 76*b6 + 128*b5 + 48*b4 + 50*b3 - 1280*b2 - 24*b1 + 2608) * q^74 + (135*b7 - 189*b6 - 239*b5 + 23*b4 + 459*b3 - 1323*b2 - 486*b1 + 2715) * q^75 + (-24*b7 - 40*b6 + 72*b5 - 136*b4 - 1032*b3 - 216*b2 - 496*b1 + 136) * q^76 + (66*b7 - 17*b6 - 315*b5 - 32*b4 - 183*b3 + 3044*b2 + 589*b1 + 2712) * q^77 + (13*b7 + 21*b6 + 120*b5 + 179*b4 - 374*b3 - 2836*b2 + 532*b1 + 4508) * q^78 + (151*b7 + 145*b6 - 604*b5 + 12*b4 + 379*b3 - 1129*b2 + 1374*b1 - 302) * q^79 + (-64*b7 + 64*b5 - 128*b4 + 128*b3 - 320*b2 + 64*b1 + 128) * q^80 + (-204*b7 - 3*b6 + 147*b5 + 234*b4 - 1329*b3 - 1107*b2 - 657*b1 - 2046) * q^81 + (40*b7 + 24*b6 + 32*b5 - 136*b4 + 79*b3 - 64*b2 + 65*b1 + 160) * q^82 + (57*b7 + 45*b6 + 114*b5 - 48*b4 + 1413*b3 + 267*b2 + 24*b1 - 582) * q^83 + (24*b7 + 48*b6 - 8*b5 + 96*b4 - 768*b3 - 1320*b2 - 856*b1 - 512) * q^84 + (-210*b7 - 90*b6 - 150*b5 - 150*b4 - 2298*b3 + 3918*b2 - 1104*b1 - 4098) * q^85 + (36*b7 + 120*b6 + 216*b5 - 276*b4 + 288*b3 - 1272*b2 + 121*b1 - 936) * q^86 + (-9*b7 - 141*b6 + 60*b5 + 258*b4 - 639*b3 + 4857*b2 + 1308*b1 - 1080) * q^87 + (32*b7 + 80*b6 - 128*b5 - 96*b4 + 488*b3 - 640*b2 + 1008*b1 - 64) * q^88 + (-44*b7 - 12*b6 + 80*b5 - 100*b4 + 2380*b3 - 6544*b2 + 2324*b1 + 3256) * q^89 + (81*b7 + 81*b6 + 216*b5 - 81*b4 - 1134*b3 + 4860*b2 - 1674*b1 - 2052) * q^90 + (93*b7 - 93*b5 - 372*b4 + 1506*b3 - 93*b2 - 1227*b1 - 5780) * q^91 + (104*b7 + 56*b6 + 208*b5 - 192*b4 - 88*b3 - 1000*b2 + 96*b1 + 1808) * q^92 + (-288*b7 - 9*b6 + 315*b5 + 500*b4 + 2169*b3 - 2928*b2 + 787*b1 + 230) * q^93 + (21*b7 + 126*b6 - 336*b5 + 483*b4 - 726*b3 + 252*b2 - 426*b1) * q^94 + (-60*b7 + 268*b6 + 1044*b5 - 476*b4 + 924*b3 + 950*b2 - 788*b1 + 2262) * q^95 + (-64*b6 + 192*b3 - 512*b2 + 64*b1 + 256) * q^96 + (10*b7 - 128*b6 - 40*b5 + 276*b4 + 436*b3 + 7997*b2 + 1188*b1 - 20) * q^97 + (-61*b7 - 57*b6 + 232*b5 - 179*b4 - 403*b3 + 3028*b2 - 521*b1 - 1516) * q^98 + (93*b7 - 372*b6 - 33*b5 + 369*b4 - 1950*b3 - 1008*b2 - 2268*b1 - 2181) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 6 q^{3} + 32 q^{4} + 18 q^{5} + 48 q^{6} - 26 q^{7} - 78 q^{9}+O(q^{10})$$ 8 * q + 6 * q^3 + 32 * q^4 + 18 * q^5 + 48 * q^6 - 26 * q^7 - 78 * q^9 $$8 q + 6 q^{3} + 32 q^{4} + 18 q^{5} + 48 q^{6} - 26 q^{7} - 78 q^{9} - 720 q^{11} - 144 q^{12} + 10 q^{13} + 288 q^{14} + 1134 q^{15} - 256 q^{16} - 384 q^{18} + 100 q^{19} + 144 q^{20} + 438 q^{21} + 336 q^{22} + 1278 q^{23} + 384 q^{24} + 794 q^{25} - 1296 q^{27} - 416 q^{28} - 1854 q^{29} - 3456 q^{30} - 1478 q^{31} - 3384 q^{33} - 96 q^{34} + 1056 q^{36} - 32 q^{37} + 6768 q^{38} + 5274 q^{39} - 36 q^{41} + 2592 q^{42} - 68 q^{43} + 3402 q^{45} + 2112 q^{46} + 2214 q^{47} - 1536 q^{48} + 2442 q^{49} - 15552 q^{50} - 12006 q^{51} - 80 q^{52} + 7056 q^{54} - 3996 q^{55} + 2304 q^{56} + 10902 q^{57} - 2400 q^{58} + 9108 q^{59} + 6480 q^{60} - 4478 q^{61} - 6654 q^{63} - 4096 q^{64} - 22554 q^{65} - 19872 q^{66} + 7504 q^{67} - 11088 q^{68} - 5994 q^{69} + 6048 q^{70} + 5376 q^{72} + 20716 q^{73} + 15264 q^{74} + 16590 q^{75} + 400 q^{76} + 34434 q^{77} + 24096 q^{78} - 6050 q^{79} - 21150 q^{81} + 1152 q^{82} - 3834 q^{83} - 9600 q^{84} - 16092 q^{85} - 12528 q^{86} + 10170 q^{87} - 2688 q^{88} + 2592 q^{90} - 45868 q^{91} + 10224 q^{92} - 10926 q^{93} + 672 q^{94} + 20880 q^{95} + 31336 q^{97} - 22338 q^{99}+O(q^{100})$$ 8 * q + 6 * q^3 + 32 * q^4 + 18 * q^5 + 48 * q^6 - 26 * q^7 - 78 * q^9 - 720 * q^11 - 144 * q^12 + 10 * q^13 + 288 * q^14 + 1134 * q^15 - 256 * q^16 - 384 * q^18 + 100 * q^19 + 144 * q^20 + 438 * q^21 + 336 * q^22 + 1278 * q^23 + 384 * q^24 + 794 * q^25 - 1296 * q^27 - 416 * q^28 - 1854 * q^29 - 3456 * q^30 - 1478 * q^31 - 3384 * q^33 - 96 * q^34 + 1056 * q^36 - 32 * q^37 + 6768 * q^38 + 5274 * q^39 - 36 * q^41 + 2592 * q^42 - 68 * q^43 + 3402 * q^45 + 2112 * q^46 + 2214 * q^47 - 1536 * q^48 + 2442 * q^49 - 15552 * q^50 - 12006 * q^51 - 80 * q^52 + 7056 * q^54 - 3996 * q^55 + 2304 * q^56 + 10902 * q^57 - 2400 * q^58 + 9108 * q^59 + 6480 * q^60 - 4478 * q^61 - 6654 * q^63 - 4096 * q^64 - 22554 * q^65 - 19872 * q^66 + 7504 * q^67 - 11088 * q^68 - 5994 * q^69 + 6048 * q^70 + 5376 * q^72 + 20716 * q^73 + 15264 * q^74 + 16590 * q^75 + 400 * q^76 + 34434 * q^77 + 24096 * q^78 - 6050 * q^79 - 21150 * q^81 + 1152 * q^82 - 3834 * q^83 - 9600 * q^84 - 16092 * q^85 - 12528 * q^86 + 10170 * q^87 - 2688 * q^88 + 2592 * q^90 - 45868 * q^91 + 10224 * q^92 - 10926 * q^93 + 672 * q^94 + 20880 * q^95 + 31336 * q^97 - 22338 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{7} + 38x^{6} - 100x^{5} + 449x^{4} - 736x^{3} + 1900x^{2} - 1548x + 2307$$ :

 $$\beta_{1}$$ $$=$$ $$( -19\nu^{7} - 286\nu^{6} + 545\nu^{5} - 8755\nu^{4} + 14939\nu^{3} - 71959\nu^{2} + 65748\nu - 147099 ) / 4230$$ (-19*v^7 - 286*v^6 + 545*v^5 - 8755*v^4 + 14939*v^3 - 71959*v^2 + 65748*v - 147099) / 4230 $$\beta_{2}$$ $$=$$ $$( -8\nu^{7} + 28\nu^{6} - 290\nu^{5} + 655\nu^{4} - 2912\nu^{3} + 3727\nu^{2} - 7344\nu + 3777 ) / 1410$$ (-8*v^7 + 28*v^6 - 290*v^5 + 655*v^4 - 2912*v^3 + 3727*v^2 - 7344*v + 3777) / 1410 $$\beta_{3}$$ $$=$$ $$( -19\nu^{7} + 419\nu^{6} - 1570\nu^{5} + 10985\nu^{4} - 21016\nu^{3} + 78911\nu^{2} - 67497\nu + 146886 ) / 4230$$ (-19*v^7 + 419*v^6 - 1570*v^5 + 10985*v^4 - 21016*v^3 + 78911*v^2 - 67497*v + 146886) / 4230 $$\beta_{4}$$ $$=$$ $$( 179\nu^{7} - 979\nu^{6} + 6665\nu^{5} - 23380\nu^{4} + 70091\nu^{3} - 149926\nu^{2} + 205212\nu - 201981 ) / 4230$$ (179*v^7 - 979*v^6 + 6665*v^5 - 23380*v^4 + 70091*v^3 - 149926*v^2 + 205212*v - 201981) / 4230 $$\beta_{5}$$ $$=$$ $$( -217\nu^{7} + 407\nu^{6} - 5575\nu^{5} + 5870\nu^{4} - 40213\nu^{3} + 18698\nu^{2} - 73716\nu + 843 ) / 4230$$ (-217*v^7 + 407*v^6 - 5575*v^5 + 5870*v^4 - 40213*v^3 + 18698*v^2 - 73716*v + 843) / 4230 $$\beta_{6}$$ $$=$$ $$( 467\nu^{7} - 1987\nu^{6} + 14990\nu^{5} - 36385\nu^{4} + 122048\nu^{3} - 129703\nu^{2} + 188301\nu + 78702 ) / 4230$$ (467*v^7 - 1987*v^6 + 14990*v^5 - 36385*v^4 + 122048*v^3 - 129703*v^2 + 188301*v + 78702) / 4230 $$\beta_{7}$$ $$=$$ $$( 532\nu^{7} - 1157\nu^{6} + 14350\nu^{5} - 13595\nu^{4} + 101998\nu^{3} + 17587\nu^{2} + 197916\nu + 235632 ) / 4230$$ (532*v^7 - 1157*v^6 + 14350*v^5 - 13595*v^4 + 101998*v^3 + 17587*v^2 + 197916*v + 235632) / 4230
 $$\nu$$ $$=$$ $$( \beta_{7} - \beta_{6} + 2\beta_{5} + \beta_{4} - 5\beta_{3} - 5\beta _1 + 10 ) / 18$$ (b7 - b6 + 2*b5 + b4 - 5*b3 - 5*b1 + 10) / 18 $$\nu^{2}$$ $$=$$ $$( \beta_{5} + \beta_{4} - 2\beta _1 - 22 ) / 3$$ (b5 + b4 - 2*b1 - 22) / 3 $$\nu^{3}$$ $$=$$ $$( -8\beta_{7} + 11\beta_{6} - 16\beta_{5} + 4\beta_{4} + 103\beta_{3} + 60\beta_{2} + 88\beta _1 - 242 ) / 18$$ (-8*b7 + 11*b6 - 16*b5 + 4*b4 + 103*b3 + 60*b2 + 88*b1 - 242) / 18 $$\nu^{4}$$ $$=$$ $$( \beta_{7} + 2\beta_{6} - 15\beta_{5} - 16\beta_{4} + 22\beta_{3} + 18\beta_{2} + 53\beta _1 + 177 ) / 3$$ (b7 + 2*b6 - 15*b5 - 16*b4 + 22*b3 + 18*b2 + 53*b1 + 177) / 3 $$\nu^{5}$$ $$=$$ $$( 97\beta_{7} - 118\beta_{6} + 122\beta_{5} - 239\beta_{4} - 1250\beta_{3} - 1392\beta_{2} - 821\beta _1 + 3940 ) / 18$$ (97*b7 - 118*b6 + 122*b5 - 239*b4 - 1250*b3 - 1392*b2 - 821*b1 + 3940) / 18 $$\nu^{6}$$ $$=$$ $$( -16\beta_{7} - 53\beta_{6} + 194\beta_{5} + 180\beta_{4} - 505\beta_{3} - 690\beta_{2} - 894\beta _1 - 1271 ) / 3$$ (-16*b7 - 53*b6 + 194*b5 + 180*b4 - 505*b3 - 690*b2 - 894*b1 - 1271) / 3 $$\nu^{7}$$ $$=$$ $$( - 1367 \beta_{7} + 1061 \beta_{6} - 934 \beta_{5} + 5005 \beta_{4} + 12613 \beta_{3} + 19800 \beta_{2} + 3991 \beta _1 - 56654 ) / 18$$ (-1367*b7 + 1061*b6 - 934*b5 + 5005*b4 + 12613*b3 + 19800*b2 + 3991*b1 - 56654) / 18

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/18\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$\chi(n)$$ $$1 - \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 0.5 − 1.74753i 0.5 + 3.16175i 0.5 + 2.20403i 0.5 − 3.61825i 0.5 + 1.74753i 0.5 − 3.16175i 0.5 − 2.20403i 0.5 + 3.61825i
−2.44949 + 1.41421i −7.80760 4.47676i 4.00000 6.92820i −32.5033 18.7658i 25.4557 0.0758456i −1.35458 2.34620i 22.6274i 40.9173 + 69.9055i 106.155
5.2 −2.44949 + 1.41421i 6.85811 5.82806i 4.00000 6.92820i 37.0033 + 21.3638i −8.55675 + 23.9746i −19.8424 34.3680i 22.6274i 13.0674 79.9390i −120.852
5.3 2.44949 1.41421i −1.67960 8.84189i 4.00000 6.92820i 6.41371 + 3.70296i −16.6185 19.2828i 30.1882 + 52.2875i 22.6274i −75.3579 + 29.7016i 20.9471
5.4 2.44949 1.41421i 5.62909 + 7.02235i 4.00000 6.92820i −1.91371 1.10488i 23.7195 + 9.24044i −21.9913 38.0900i 22.6274i −17.6268 + 79.0588i −6.25016
11.1 −2.44949 1.41421i −7.80760 + 4.47676i 4.00000 + 6.92820i −32.5033 + 18.7658i 25.4557 + 0.0758456i −1.35458 + 2.34620i 22.6274i 40.9173 69.9055i 106.155
11.2 −2.44949 1.41421i 6.85811 + 5.82806i 4.00000 + 6.92820i 37.0033 21.3638i −8.55675 23.9746i −19.8424 + 34.3680i 22.6274i 13.0674 + 79.9390i −120.852
11.3 2.44949 + 1.41421i −1.67960 + 8.84189i 4.00000 + 6.92820i 6.41371 3.70296i −16.6185 + 19.2828i 30.1882 52.2875i 22.6274i −75.3579 29.7016i 20.9471
11.4 2.44949 + 1.41421i 5.62909 7.02235i 4.00000 + 6.92820i −1.91371 + 1.10488i 23.7195 9.24044i −21.9913 + 38.0900i 22.6274i −17.6268 79.0588i −6.25016
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 18.5.d.a 8
3.b odd 2 1 54.5.d.a 8
4.b odd 2 1 144.5.q.b 8
9.c even 3 1 54.5.d.a 8
9.c even 3 1 162.5.b.c 8
9.d odd 6 1 inner 18.5.d.a 8
9.d odd 6 1 162.5.b.c 8
12.b even 2 1 432.5.q.b 8
36.f odd 6 1 432.5.q.b 8
36.f odd 6 1 1296.5.e.e 8
36.h even 6 1 144.5.q.b 8
36.h even 6 1 1296.5.e.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.5.d.a 8 1.a even 1 1 trivial
18.5.d.a 8 9.d odd 6 1 inner
54.5.d.a 8 3.b odd 2 1
54.5.d.a 8 9.c even 3 1
144.5.q.b 8 4.b odd 2 1
144.5.q.b 8 36.h even 6 1
162.5.b.c 8 9.c even 3 1
162.5.b.c 8 9.d odd 6 1
432.5.q.b 8 12.b even 2 1
432.5.q.b 8 36.f odd 6 1
1296.5.e.e 8 36.f odd 6 1
1296.5.e.e 8 36.h even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(18, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - 8 T^{2} + 64)^{2}$$
$3$ $$T^{8} - 6 T^{7} + 57 T^{6} + \cdots + 43046721$$
$5$ $$T^{8} - 18 T^{7} + \cdots + 688747536$$
$7$ $$T^{8} + 26 T^{7} + \cdots + 81510250000$$
$11$ $$T^{8} + 720 T^{7} + \cdots + 53\!\cdots\!89$$
$13$ $$T^{8} - 10 T^{7} + \cdots + 54\!\cdots\!16$$
$17$ $$T^{8} + 418950 T^{6} + \cdots + 33\!\cdots\!00$$
$19$ $$(T^{4} - 50 T^{3} - 295131 T^{2} + \cdots + 8154234820)^{2}$$
$23$ $$T^{8} - 1278 T^{7} + \cdots + 44\!\cdots\!64$$
$29$ $$T^{8} + 1854 T^{7} + \cdots + 46\!\cdots\!04$$
$31$ $$T^{8} + 1478 T^{7} + \cdots + 52\!\cdots\!24$$
$37$ $$(T^{4} + 16 T^{3} + \cdots - 305304165104)^{2}$$
$41$ $$T^{8} + 36 T^{7} + \cdots + 80\!\cdots\!61$$
$43$ $$T^{8} + 68 T^{7} + \cdots + 22\!\cdots\!25$$
$47$ $$T^{8} - 2214 T^{7} + \cdots + 16\!\cdots\!64$$
$53$ $$T^{8} + 15974784 T^{6} + \cdots + 11\!\cdots\!56$$
$59$ $$T^{8} - 9108 T^{7} + \cdots + 62\!\cdots\!25$$
$61$ $$T^{8} + 4478 T^{7} + \cdots + 10\!\cdots\!76$$
$67$ $$T^{8} - 7504 T^{7} + \cdots + 78\!\cdots\!25$$
$71$ $$T^{8} + 78003288 T^{6} + \cdots + 55\!\cdots\!24$$
$73$ $$(T^{4} - 10358 T^{3} + \cdots - 13267734129308)^{2}$$
$79$ $$T^{8} + 6050 T^{7} + \cdots + 72\!\cdots\!00$$
$83$ $$T^{8} + 3834 T^{7} + \cdots + 16\!\cdots\!04$$
$89$ $$T^{8} + 295741440 T^{6} + \cdots + 74\!\cdots\!00$$
$97$ $$T^{8} - 31336 T^{7} + \cdots + 63\!\cdots\!01$$