# Properties

 Label 72.8.d.d Level $72$ Weight $8$ Character orbit 72.d Analytic conductor $22.492$ Analytic rank $0$ Dimension $14$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$72 = 2^{3} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 72.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$22.4917218349$$ Analytic rank: $$0$$ Dimension: $$14$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ Defining polynomial: $$x^{14} - 6 x^{13} - 52 x^{12} + 300 x^{11} - 1005 x^{10} - 23250 x^{9} + 349930 x^{8} + 2867784 x^{7} - 20463993 x^{6} - 78987210 x^{5} + 94608296 x^{4} + \cdots + 3813237677250$$ x^14 - 6*x^13 - 52*x^12 + 300*x^11 - 1005*x^10 - 23250*x^9 + 349930*x^8 + 2867784*x^7 - 20463993*x^6 - 78987210*x^5 + 94608296*x^4 - 6477861300*x^3 - 21982316987*x^2 + 613270661346*x + 3813237677250 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{42}\cdot 3^{18}$$ Twist minimal: no (minimal twist has level 24) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{13}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 + 1) q^{2} + ( - \beta_{2} + \beta_1 - 15) q^{4} + (\beta_{3} - \beta_{2} - 3 \beta_1) q^{5} + (\beta_{8} - 3 \beta_{2} - 4 \beta_1 + 98) q^{7} + ( - \beta_{13} - \beta_{9} - \beta_{8} - \beta_{4} - \beta_{2} - 15 \beta_1 + 30) q^{8}+O(q^{10})$$ q + (b1 + 1) * q^2 + (-b2 + b1 - 15) * q^4 + (b3 - b2 - 3*b1) * q^5 + (b8 - 3*b2 - 4*b1 + 98) * q^7 + (-b13 - b9 - b8 - b4 - b2 - 15*b1 + 30) * q^8 $$q + (\beta_1 + 1) q^{2} + ( - \beta_{2} + \beta_1 - 15) q^{4} + (\beta_{3} - \beta_{2} - 3 \beta_1) q^{5} + (\beta_{8} - 3 \beta_{2} - 4 \beta_1 + 98) q^{7} + ( - \beta_{13} - \beta_{9} - \beta_{8} - \beta_{4} - \beta_{2} - 15 \beta_1 + 30) q^{8} + ( - \beta_{13} - \beta_{12} - \beta_{11} + \beta_{10} - 2 \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + \cdots + 358) q^{10}+ \cdots + (2232 \beta_{13} - 1560 \beta_{12} + 4056 \beta_{11} + \cdots - 3783131) q^{98}+O(q^{100})$$ q + (b1 + 1) * q^2 + (-b2 + b1 - 15) * q^4 + (b3 - b2 - 3*b1) * q^5 + (b8 - 3*b2 - 4*b1 + 98) * q^7 + (-b13 - b9 - b8 - b4 - b2 - 15*b1 + 30) * q^8 + (-b13 - b12 - b11 + b10 - 2*b9 + b8 + b7 - b6 + b5 - b4 + 2*b3 - 3*b1 + 358) * q^10 + (-b13 + b12 + 2*b11 + b10 - b9 + b6 + 2*b4 - 5*b3 - 5*b2 + 70*b1) * q^11 + (2*b13 - 3*b11 - 3*b10 - 2*b9 - 2*b6 - 3*b5 + b4 - b3 - 6*b2 + 240*b1 - 2) * q^13 + (-5*b13 - 3*b12 + 3*b11 + b10 + 2*b9 - 5*b8 + b5 + b4 - 16*b3 + 9*b2 + 102*b1 - 333) * q^14 + (-7*b13 + 5*b12 + 2*b11 + b10 + 6*b9 - 6*b8 + 2*b7 - 9*b6 + 4*b5 - 11*b4 + 9*b3 + 19*b2 + 62*b1 - 3097) * q^16 + (-3*b13 + 4*b12 - 6*b11 + 12*b9 - 14*b8 + b6 + 4*b5 - 6*b4 - 6*b3 - 6*b2 + 630*b1 + 198) * q^17 + (b13 - 5*b12 - 6*b11 - b10 + 11*b9 - b6 - 19*b5 + 10*b4 - 75*b3 - 23*b2 - 470*b1 - 16) * q^19 + (-10*b13 - 2*b12 + 10*b11 - 2*b10 - 15*b9 - 12*b8 + 12*b7 - 14*b6 - 34*b5 + 2*b4 - 36*b3 + 34*b2 + 452*b1 - 12510) * q^20 + (-4*b13 + 12*b12 - 4*b11 + 12*b10 - 4*b9 - 20*b8 + 16*b6 + 36*b5 - 4*b4 - 72*b3 - 68*b2 + 80*b1 - 9204) * q^22 + (-32*b13 + 16*b12 - 32*b11 + 16*b10 - 8*b9 + 2*b8 - 32*b6 - 32*b4 - 16*b3 + 10*b2 - 1288*b1 + 10212) * q^23 + (-36*b13 + 6*b12 + 52*b11 - 30*b10 - 34*b9 - 68*b8 - 32*b7 - 32*b6 + 36*b5 - 36*b4 - 62*b3 - 10*b2 - 24*b1 - 14495) * q^25 + (8*b13 - 8*b12 - 24*b11 - 44*b10 - 10*b9 + 12*b8 - 30*b7 - 2*b6 + 56*b5 - 4*b4 - 8*b3 - 214*b2 + 266*b1 - 30382) * q^26 + (-38*b13 + 70*b12 + 20*b11 - 10*b10 - 12*b9 - 56*b8 - 20*b7 - 30*b6 + 80*b5 - 18*b4 + 30*b3 - 164*b2 - 622*b1 + 40504) * q^28 + (32*b13 - 20*b12 + 10*b11 + 30*b10 - 44*b9 - 32*b6 + 90*b5 - 46*b4 + 43*b3 - 265*b2 + 3093*b1 - 68) * q^29 + (-64*b13 - 14*b12 + 52*b11 + 14*b10 - 148*b9 - 77*b8 + 96*b7 + 4*b6 - 28*b5 - 36*b4 - 34*b3 - 163*b2 - 3244*b1 - 6418) * q^31 + (-4*b13 - 2*b12 + 32*b11 + 6*b10 - 4*b9 + 90*b8 + 60*b7 - 14*b6 - 36*b5 - 28*b4 - 262*b3 - 100*b2 - 3670*b1 + 63850) * q^32 + (44*b13 + 20*b12 - 44*b11 - 20*b10 + 12*b9 - 36*b8 + 32*b7 - 20*b5 - 96*b4 + 144*b3 - 708*b2 - 454*b1 + 79154) * q^34 + (3*b13 + 117*b12 + 82*b11 - 35*b10 + 119*b9 - 3*b6 + 37*b5 + 114*b4 + 207*b3 - 553*b2 - 7346*b1 + 32) * q^35 + (10*b13 + 48*b12 - 15*b11 - 63*b10 + 78*b9 - 10*b6 + 225*b5 + 133*b4 + 77*b3 - 656*b2 - 3638*b1 - 42) * q^37 + (-12*b13 + 100*b12 + 20*b11 - 140*b10 - 68*b9 - 236*b8 - 150*b7 + 10*b6 + 140*b5 - 42*b4 - 180*b3 + 570*b2 - 170*b1 + 58860) * q^38 + (-72*b13 + 146*b12 - 36*b11 - 134*b10 - 50*b9 + 82*b8 - 140*b7 - 74*b6 + 16*b5 - 56*b4 - 782*b3 - 244*b2 - 12002*b1 - 61590) * q^40 + (161*b13 - 80*b12 + 26*b11 + 20*b10 - 168*b9 - 126*b8 + 192*b7 + 253*b6 - 100*b5 + 170*b4 + 166*b3 + 110*b2 - 4818*b1 + 31634) * q^41 + (249*b13 + 75*b12 + 114*b11 + 39*b10 + 47*b9 - 249*b6 - 69*b5 - 158*b4 - 307*b3 - 967*b2 + 2618*b1 - 304) * q^43 + (88*b13 + 72*b12 - 16*b11 + 72*b10 + 48*b9 - 320*b8 + 144*b7 + 216*b6 - 320*b5 - 152*b4 + 168*b3 + 168*b2 - 8576*b1 - 91128) * q^44 + (54*b13 - 70*b12 + 198*b11 - 62*b10 - 252*b9 - 330*b8 + 256*b7 - 128*b6 + 322*b5 - 190*b4 - 1440*b3 + 1746*b2 + 12044*b1 - 154906) * q^46 + (-108*b13 + 14*b12 + 276*b11 - 174*b10 - 128*b9 - 154*b8 - 480*b7 - 400*b6 + 188*b5 - 84*b4 - 198*b3 + 1752*b2 - 8800*b1 + 75572) * q^47 + (-170*b13 + 302*b12 - 408*b11 - 62*b10 - 138*b9 - 32*b8 - 544*b7 - 350*b6 + 364*b5 - 368*b4 - 362*b3 - 118*b2 - 30508*b1 + 153709) * q^49 + (-416*b13 + 448*b12 + 304*b11 - 48*b10 + 88*b9 + 320*b8 - 192*b7 - 128*b6 + 1520*b5 - 296*b4 + 1088*b3 - 720*b2 - 14055*b1 - 22847) * q^50 + (-304*b13 - 40*b12 - 4*b11 - 296*b10 + 218*b9 - 264*b8 - 240*b7 - 104*b6 + 1468*b5 - 96*b4 + 1540*b3 - 672*b2 - 28840*b1 - 147008) * q^52 + (312*b13 - 324*b12 - 178*b11 + 146*b10 + 156*b9 - 312*b6 - 1282*b5 + 102*b4 - 1747*b3 - 4183*b2 + 35*b1 - 1196) * q^53 + (208*b13 - 58*b12 - 356*b11 + 250*b10 + 644*b9 - 380*b8 + 544*b7 + 444*b6 - 308*b5 + 212*b4 + 394*b3 + 870*b2 + 36272*b1 + 339840) * q^55 + (74*b13 + 24*b12 - 248*b11 + 120*b10 - 178*b9 + 242*b8 + 336*b7 + 408*b6 + 1944*b5 - 406*b4 - 1360*b3 - 14*b2 + 42534*b1 - 117500) * q^56 + (-225*b13 - 193*b12 + 671*b11 + 217*b10 - 1046*b9 + 201*b8 + 485*b7 + 251*b6 - 2399*b5 + 55*b4 + 1458*b3 - 2460*b2 + 1297*b1 - 392150) * q^58 + (16*b13 - 520*b12 - 696*b11 - 176*b10 - 228*b9 - 16*b6 - 29*b5 + 808*b4 + 976*b3 - 7928*b2 + 20224*b1 - 1184) * q^59 + (-282*b13 + 392*b12 + 507*b11 + 115*b10 - 582*b9 + 282*b6 - 53*b5 + 727*b4 + 1651*b3 - 4276*b2 + 35738*b1 + 114) * q^61 + (-811*b13 + 595*b12 - 419*b11 - 129*b10 + 486*b9 + 21*b8 - 416*b6 - 2433*b5 - 649*b4 - 2128*b3 + 4951*b2 - 3046*b1 - 411723) * q^62 + (-538*b13 + 18*b12 + 460*b11 - 246*b10 + 980*b9 - 288*b8 - 268*b7 + 246*b6 - 2496*b5 + 270*b4 - 3854*b3 + 5778*b2 + 62072*b1 + 275798) * q^64 + (-363*b13 - 728*b12 + 402*b11 + 972*b10 - 560*b9 + 826*b8 + 1344*b7 - 719*b6 - 1700*b5 + 642*b4 + 1446*b3 + 8478*b2 - 8314*b1 + 181484) * q^65 + (-260*b13 - 1044*b12 - 240*b11 + 804*b10 + 280*b9 + 260*b6 - 303*b5 - 144*b4 + 684*b3 - 5692*b2 - 43144*b1 - 1184) * q^67 + (-360*b13 - 552*b12 - 208*b11 - 104*b10 + 64*b9 + 240*b8 + 240*b7 - 24*b6 - 3296*b5 - 984*b4 + 1368*b3 + 1446*b2 + 76882*b1 + 267162) * q^68 + (44*b13 - 260*b12 - 1428*b11 + 284*b10 - 452*b9 + 892*b8 + 42*b7 + 1114*b6 + 3764*b5 - 186*b4 - 4340*b3 + 4770*b2 - 5114*b1 + 920860) * q^70 + (908*b13 - 182*b12 - 484*b11 + 182*b10 + 1816*b9 - 6*b8 - 672*b7 - 128*b6 - 364*b5 + 932*b4 + 1198*b3 + 9776*b2 + 50528*b1 - 368372) * q^71 + (110*b13 - 1010*b12 + 1176*b11 + 578*b10 - 330*b9 + 592*b8 + 480*b7 - 998*b6 - 1588*b5 + 1216*b4 + 1734*b3 + 14138*b2 + 1572*b1 - 386582) * q^73 + (-186*b13 - 10*b12 - 1498*b11 - 634*b10 - 822*b9 - 226*b8 - 84*b7 + 884*b6 + 4474*b5 + 42*b4 - 1700*b3 + 1266*b2 - 1108*b1 + 461926) * q^74 + (1096*b13 + 632*b12 - 548*b11 - 392*b10 + 1834*b9 + 1248*b8 - 656*b7 + 552*b6 + 6364*b5 - 456*b4 + 6372*b3 - 2184*b2 + 66592*b1 - 647824) * q^76 + (28*b13 + 212*b12 + 1068*b11 + 856*b10 - 1536*b9 - 28*b6 - 196*b5 + 796*b4 + 4784*b3 - 16788*b2 + 94768*b1 - 1496) * q^77 + (2020*b13 - 1174*b12 + 716*b11 + 214*b10 + 1600*b9 + 3159*b8 - 416*b7 + 216*b6 - 1388*b5 + 2644*b4 + 2910*b3 + 10617*b2 + 45884*b1 - 1023226) * q^79 + (-182*b13 + 190*b12 + 852*b11 - 1114*b10 + 2476*b9 - 1696*b8 - 1236*b7 + 826*b6 + 8192*b5 - 494*b4 - 4674*b3 + 13422*b2 - 48104*b1 - 1024838) * q^80 + (892*b13 + 260*b12 - 2908*b11 + 28*b10 + 1356*b9 + 1068*b8 - 480*b7 + 512*b6 - 8612*b5 + 208*b4 + 4560*b3 + 7148*b2 + 34470*b1 - 565874) * q^82 + (-51*b13 - 453*b12 + 1774*b11 + 2227*b10 + 1593*b9 + 51*b6 + 92*b5 - 306*b4 + 4641*b3 - 18631*b2 - 158702*b1 - 2272) * q^83 + (896*b13 - 2248*b12 - 2388*b11 - 140*b10 - 2048*b9 - 896*b6 - 3044*b5 - 1636*b4 + 2310*b3 - 3946*b2 + 143282*b1 - 2264) * q^85 + (436*b13 - 1116*b12 + 2452*b11 - 108*b10 - 3860*b9 + 3124*b8 + 294*b7 + 2790*b6 - 9588*b5 + 226*b4 - 1628*b3 + 862*b2 - 2566*b1 - 335076) * q^86 + (1744*b13 - 256*b12 - 1760*b11 + 1024*b10 + 2400*b9 + 1552*b8 + 640*b7 + 192*b6 - 12128*b5 - 2096*b4 + 5024*b3 + 11376*b2 - 101744*b1 + 901312) * q^88 + (3082*b13 + 760*b12 - 2828*b11 - 560*b10 + 728*b9 - 1372*b8 - 3840*b7 + 562*b6 + 1320*b5 + 1588*b4 + 1508*b3 + 17476*b2 - 167348*b1 + 854498) * q^89 + (-395*b13 - 25*b12 - 1422*b11 - 1397*b10 + 3471*b9 + 395*b6 - 3038*b5 + 2306*b4 - 903*b3 - 3891*b2 - 223758*b1 - 144) * q^91 + (564*b13 + 2572*b12 + 808*b11 - 2708*b10 + 3560*b9 + 3472*b8 - 1320*b7 + 2116*b6 - 13664*b5 + 1884*b4 - 580*b3 - 4872*b2 - 155548*b1 - 782544) * q^92 + (22*b13 + 1594*b12 + 4406*b11 + 50*b10 - 2964*b9 + 3798*b8 - 1280*b7 - 288*b6 + 18674*b5 + 1770*b4 + 8256*b3 + 674*b2 + 88844*b1 - 1097922) * q^94 + (2176*b13 - 1812*b12 - 1736*b11 + 2932*b10 + 1712*b9 + 3556*b8 + 4416*b7 + 1848*b6 - 4744*b5 + 3688*b4 + 6068*b3 + 14968*b2 + 61552*b1 + 4955864) * q^95 + (5870*b13 - 1460*b12 - 604*b11 - 292*b10 - 4532*b9 + 3748*b8 + 1600*b7 + 6302*b6 - 1168*b5 + 5076*b4 + 4584*b3 - 9920*b2 - 299004*b1 + 12870) * q^97 + (2232*b13 - 1560*b12 + 4056*b11 - 344*b10 - 2832*b9 - 3336*b8 + 1920*b7 + 384*b6 + 18632*b5 - 1896*b4 + 2912*b3 + 29864*b2 + 195853*b1 - 3783131) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14 q + 14 q^{2} - 208 q^{4} + 1372 q^{7} + 428 q^{8}+O(q^{10})$$ 14 * q + 14 * q^2 - 208 * q^4 + 1372 * q^7 + 428 * q^8 $$14 q + 14 q^{2} - 208 q^{4} + 1372 q^{7} + 428 q^{8} + 5020 q^{10} - 4636 q^{14} - 43336 q^{16} + 2908 q^{17} - 175096 q^{20} - 128480 q^{22} + 143416 q^{23} - 202626 q^{25} - 424984 q^{26} + 567520 q^{28} - 89468 q^{31} + 893944 q^{32} + 1109820 q^{34} + 823816 q^{38} - 860888 q^{40} + 441284 q^{41} - 1275264 q^{44} - 2167992 q^{46} + 1056408 q^{47} + 2158134 q^{49} - 324610 q^{50} - 2059248 q^{52} + 4757504 q^{55} - 1643704 q^{56} - 5494676 q^{58} - 5767172 q^{62} + 3852224 q^{64} + 2520464 q^{65} + 3735840 q^{68} + 12890312 q^{70} - 5172696 q^{71} - 5446196 q^{73} + 6468800 q^{74} - 9084624 q^{76} - 14373548 q^{79} - 14369088 q^{80} - 7935708 q^{82} - 4738312 q^{86} + 12598720 q^{88} + 11952620 q^{89} - 11004480 q^{92} - 15440088 q^{94} + 69327376 q^{95} + 133732 q^{97} - 53030538 q^{98}+O(q^{100})$$ 14 * q + 14 * q^2 - 208 * q^4 + 1372 * q^7 + 428 * q^8 + 5020 * q^10 - 4636 * q^14 - 43336 * q^16 + 2908 * q^17 - 175096 * q^20 - 128480 * q^22 + 143416 * q^23 - 202626 * q^25 - 424984 * q^26 + 567520 * q^28 - 89468 * q^31 + 893944 * q^32 + 1109820 * q^34 + 823816 * q^38 - 860888 * q^40 + 441284 * q^41 - 1275264 * q^44 - 2167992 * q^46 + 1056408 * q^47 + 2158134 * q^49 - 324610 * q^50 - 2059248 * q^52 + 4757504 * q^55 - 1643704 * q^56 - 5494676 * q^58 - 5767172 * q^62 + 3852224 * q^64 + 2520464 * q^65 + 3735840 * q^68 + 12890312 * q^70 - 5172696 * q^71 - 5446196 * q^73 + 6468800 * q^74 - 9084624 * q^76 - 14373548 * q^79 - 14369088 * q^80 - 7935708 * q^82 - 4738312 * q^86 + 12598720 * q^88 + 11952620 * q^89 - 11004480 * q^92 - 15440088 * q^94 + 69327376 * q^95 + 133732 * q^97 - 53030538 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{14} - 6 x^{13} - 52 x^{12} + 300 x^{11} - 1005 x^{10} - 23250 x^{9} + 349930 x^{8} + 2867784 x^{7} - 20463993 x^{6} - 78987210 x^{5} + 94608296 x^{4} + \cdots + 3813237677250$$ :

 $$\beta_{1}$$ $$=$$ $$( 21691079 \nu^{13} - 270184387 \nu^{12} + 782417201 \nu^{11} + 1515938117 \nu^{10} - 53036710294 \nu^{9} - 206804744724 \nu^{8} + \cdots + 95\!\cdots\!30 ) / 41\!\cdots\!72$$ (21691079*v^13 - 270184387*v^12 + 782417201*v^11 + 1515938117*v^10 - 53036710294*v^9 - 206804744724*v^8 + 7877907621970*v^7 + 8698312701450*v^6 - 340683702304837*v^5 + 661165852971349*v^4 - 5100033620544691*v^3 - 107949855496936511*v^2 - 5342334081538492*v + 9525580676280950130) / 418271540514127872 $$\beta_{2}$$ $$=$$ $$( - 166890813 \nu^{13} + 2670091729 \nu^{12} - 17159192555 \nu^{11} + 71557375481 \nu^{10} + 211250383698 \nu^{9} + \cdots - 94\!\cdots\!82 ) / 41\!\cdots\!72$$ (-166890813*v^13 + 2670091729*v^12 - 17159192555*v^11 + 71557375481*v^10 + 211250383698*v^9 - 1753019889892*v^8 - 67422232273958*v^7 + 147038286693746*v^6 + 2281992874077703*v^5 - 11972977799330231*v^4 + 124047037053921825*v^3 + 591285133340081637*v^2 - 3930090647392449068*v - 94834434510735643782) / 418271540514127872 $$\beta_{3}$$ $$=$$ $$( 492629525 \nu^{13} - 5330686601 \nu^{12} + 12047622611 \nu^{11} - 39149525617 \nu^{10} + 861347903166 \nu^{9} + \cdots + 29\!\cdots\!38 ) / 41\!\cdots\!72$$ (492629525*v^13 - 5330686601*v^12 + 12047622611*v^11 - 39149525617*v^10 + 861347903166*v^9 - 20892987746620*v^8 + 249468178251126*v^7 + 206777229578654*v^6 - 8779842761025551*v^5 + 10671182396335807*v^4 - 60368663605491929*v^3 - 1541321821434843645*v^2 - 7427960645991195444*v + 297111538244625838038) / 418271540514127872 $$\beta_{4}$$ $$=$$ $$( - 26782385 \nu^{13} + 628599829 \nu^{12} - 4700265751 \nu^{11} + 28473483741 \nu^{10} - 108938811238 \nu^{9} + \cdots - 53\!\cdots\!22 ) / 16\!\cdots\!72$$ (-26782385*v^13 + 628599829*v^12 - 4700265751*v^11 + 28473483741*v^10 - 108938811238*v^9 - 65918650484*v^8 - 4384489495486*v^7 + 57657990547258*v^6 + 74577948397987*v^5 - 2102481401376147*v^4 + 26648181139568421*v^3 - 239944707218326151*v^2 + 1077689134009102884*v - 5304610005333105822) / 16087366942851072 $$\beta_{5}$$ $$=$$ $$( - 136458837 \nu^{13} + 1839828681 \nu^{12} - 4871646099 \nu^{11} - 27635439951 \nu^{10} + 295776184770 \nu^{9} + \cdots - 55\!\cdots\!34 ) / 69\!\cdots\!12$$ (-136458837*v^13 + 1839828681*v^12 - 4871646099*v^11 - 27635439951*v^10 + 295776184770*v^9 + 417110421564*v^8 - 51793529810166*v^7 + 79737812279010*v^6 + 2290922466223311*v^5 - 6592518699840639*v^4 + 31777295062177305*v^3 + 488892118317291837*v^2 - 2829353467017697740*v - 55592758119569467734) / 69711923419021312 $$\beta_{6}$$ $$=$$ $$( - 907013959 \nu^{13} + 7107775043 \nu^{12} + 14703813455 \nu^{11} - 626741066437 \nu^{10} + 4316482827926 \nu^{9} + \cdots - 59\!\cdots\!14 ) / 41\!\cdots\!72$$ (-907013959*v^13 + 7107775043*v^12 + 14703813455*v^11 - 626741066437*v^10 + 4316482827926*v^9 + 40959103074324*v^8 - 645804083056466*v^7 + 412054139991798*v^6 + 18208825101956549*v^5 - 65294508415789781*v^4 - 1127607716292560077*v^3 + 14564682816687558847*v^2 - 4006132970964798532*v - 593457742882227072114) / 418271540514127872 $$\beta_{7}$$ $$=$$ $$( - 1068982365 \nu^{13} + 23519404401 \nu^{12} - 172128800203 \nu^{11} + 271435164313 \nu^{10} + 4040669859986 \nu^{9} + \cdots - 41\!\cdots\!02 ) / 41\!\cdots\!72$$ (-1068982365*v^13 + 23519404401*v^12 - 172128800203*v^11 + 271435164313*v^10 + 4040669859986*v^9 - 13270169512804*v^8 - 432166085296614*v^7 + 2720114252070578*v^6 + 20485589300046183*v^5 - 209631055225254679*v^4 + 1497392427477004225*v^3 - 50347203861011963*v^2 - 68356703591373355436*v - 415809877340742405702) / 418271540514127872 $$\beta_{8}$$ $$=$$ $$( - 1197811243 \nu^{13} + 11406180407 \nu^{12} - 29346275821 \nu^{11} + 4316955983 \nu^{10} + 2168133588542 \nu^{9} + \cdots - 51\!\cdots\!50 ) / 41\!\cdots\!72$$ (-1197811243*v^13 + 11406180407*v^12 - 29346275821*v^11 + 4316955983*v^10 + 2168133588542*v^9 + 5280359777220*v^8 - 393639118494218*v^7 - 51689868387042*v^6 + 20589636476679857*v^5 - 63675591055007873*v^4 + 576415206273195623*v^3 + 4585102734777636547*v^2 - 13589658096873408820*v - 513774345453268065450) / 418271540514127872 $$\beta_{9}$$ $$=$$ $$( - 149795513 \nu^{13} + 1775346621 \nu^{12} - 1910006095 \nu^{11} - 26079690555 \nu^{10} + 399012730218 \nu^{9} + \cdots - 75\!\cdots\!38 ) / 52\!\cdots\!84$$ (-149795513*v^13 + 1775346621*v^12 - 1910006095*v^11 - 26079690555*v^10 + 399012730218*v^9 + 453876647916*v^8 - 69583694081198*v^7 + 94818021029130*v^6 + 2572196887404603*v^5 - 6935864044025643*v^4 + 58136499118743245*v^3 + 828039329087708481*v^2 - 7932669704260375484*v - 75799269214339172238) / 52283942564265984 $$\beta_{10}$$ $$=$$ $$( 172051509 \nu^{13} - 2404852809 \nu^{12} + 7188831539 \nu^{11} + 65785727119 \nu^{10} - 1183451299138 \nu^{9} + \cdots + 88\!\cdots\!18 ) / 52\!\cdots\!84$$ (172051509*v^13 - 2404852809*v^12 + 7188831539*v^11 + 65785727119*v^10 - 1183451299138*v^9 + 1732351672772*v^8 + 106222818156342*v^7 - 428922343776610*v^6 - 2971941593001711*v^5 + 18744536723320319*v^4 - 148692741427404281*v^3 - 385977066843797501*v^2 + 7432883771286962572*v + 88196075771050853718) / 52283942564265984 $$\beta_{11}$$ $$=$$ $$( - 1513426675 \nu^{13} + 17693523359 \nu^{12} + 18463120539 \nu^{11} - 613756439561 \nu^{10} + 1692810357390 \nu^{9} + \cdots - 10\!\cdots\!22 ) / 41\!\cdots\!72$$ (-1513426675*v^13 + 17693523359*v^12 + 18463120539*v^11 - 613756439561*v^10 + 1692810357390*v^9 + 38297255912740*v^8 - 813252529283450*v^7 - 779662428284306*v^6 + 51075768350251913*v^5 - 28661712426074713*v^4 - 583094441890359857*v^3 + 8119892215556506443*v^2 - 36141653170953603092*v - 1097231587839181943322) / 418271540514127872 $$\beta_{12}$$ $$=$$ $$( - 59960501 \nu^{13} + 905146077 \nu^{12} - 3961879171 \nu^{11} + 12082154925 \nu^{10} - 16745842566 \nu^{9} + \cdots - 44\!\cdots\!10 ) / 13\!\cdots\!96$$ (-59960501*v^13 + 905146077*v^12 - 3961879171*v^11 + 12082154925*v^10 - 16745842566*v^9 + 1682766149100*v^8 - 34146929420918*v^7 + 53626341412986*v^6 + 1276084613712255*v^5 - 3307297499966955*v^4 + 28485599636046665*v^3 + 487589208644078889*v^2 - 6749383360885652*v - 44743803933251838510) / 13070985641066496 $$\beta_{13}$$ $$=$$ $$( 2715351137 \nu^{13} - 33639911173 \nu^{12} + 83405896423 \nu^{11} + 519337332915 \nu^{10} - 9612896802170 \nu^{9} + \cdots + 11\!\cdots\!06 ) / 41\!\cdots\!72$$ (2715351137*v^13 - 33639911173*v^12 + 83405896423*v^11 + 519337332915*v^10 - 9612896802170*v^9 - 8017380726220*v^8 + 1151055790567006*v^7 - 1411489544949274*v^6 - 48465275197458355*v^5 + 299027351567474211*v^4 - 957779343047620245*v^3 - 15443687196181462057*v^2 + 36413201493649256988*v + 1171772180068753260606) / 418271540514127872
 $$\nu$$ $$=$$ $$( - 9 \beta_{13} + 9 \beta_{12} - 9 \beta_{10} - 36 \beta_{9} - 16 \beta_{7} - 7 \beta_{6} + 18 \beta_{5} + 2 \beta_{4} + 13 \beta_{3} + 7 \beta_{2} - 74 \beta _1 + 2226 ) / 5184$$ (-9*b13 + 9*b12 - 9*b10 - 36*b9 - 16*b7 - 7*b6 + 18*b5 + 2*b4 + 13*b3 + 7*b2 - 74*b1 + 2226) / 5184 $$\nu^{2}$$ $$=$$ $$( - 27 \beta_{13} + 135 \beta_{12} - 72 \beta_{11} + 81 \beta_{10} + 9 \beta_{9} - 32 \beta_{7} - 5 \beta_{6} + 214 \beta_{5} - 86 \beta_{4} + 323 \beta_{3} - 103 \beta_{2} + 1058 \beta _1 + 51810 ) / 5184$$ (-27*b13 + 135*b12 - 72*b11 + 81*b10 + 9*b9 - 32*b7 - 5*b6 + 214*b5 - 86*b4 + 323*b3 - 103*b2 + 1058*b1 + 51810) / 5184 $$\nu^{3}$$ $$=$$ $$( - 135 \beta_{13} + 891 \beta_{12} - 792 \beta_{11} - 243 \beta_{10} - 387 \beta_{9} + 112 \beta_{7} - 1481 \beta_{6} + 1614 \beta_{5} - 1346 \beta_{4} - 1297 \beta_{3} - 5971 \beta_{2} - 20686 \beta _1 + 91374 ) / 5184$$ (-135*b13 + 891*b12 - 792*b11 - 243*b10 - 387*b9 + 112*b7 - 1481*b6 + 1614*b5 - 1346*b4 - 1297*b3 - 5971*b2 - 20686*b1 + 91374) / 5184 $$\nu^{4}$$ $$=$$ $$( 4533 \beta_{13} + 1551 \beta_{12} + 168 \beta_{11} - 1767 \beta_{10} - 393 \beta_{9} + 2592 \beta_{8} - 672 \beta_{7} + 267 \beta_{6} + 3638 \beta_{5} - 54 \beta_{4} + 6363 \beta_{3} - 5343 \beta_{2} + \cdots + 1360866 ) / 1728$$ (4533*b13 + 1551*b12 + 168*b11 - 1767*b10 - 393*b9 + 2592*b8 - 672*b7 + 267*b6 + 3638*b5 - 54*b4 + 6363*b3 - 5343*b2 - 256974*b1 + 1360866) / 1728 $$\nu^{5}$$ $$=$$ $$( 28233 \beta_{13} + 35811 \beta_{12} + 79128 \beta_{11} - 13131 \beta_{10} - 67005 \beta_{9} + 137376 \beta_{8} + 18672 \beta_{7} - 57 \beta_{6} - 99794 \beta_{5} + 20958 \beta_{4} + \cdots + 59148270 ) / 5184$$ (28233*b13 + 35811*b12 + 79128*b11 - 13131*b10 - 67005*b9 + 137376*b8 + 18672*b7 - 57*b6 - 99794*b5 + 20958*b4 + 228135*b3 - 236139*b2 + 697602*b1 + 59148270) / 5184 $$\nu^{6}$$ $$=$$ $$( 375219 \beta_{13} + 427977 \beta_{12} + 10008 \beta_{11} - 224721 \beta_{10} + 188289 \beta_{9} + 1218240 \beta_{8} - 84064 \beta_{7} + 441005 \beta_{6} + 193274 \beta_{5} + \cdots - 134979234 ) / 5184$$ (375219*b13 + 427977*b12 + 10008*b11 - 224721*b10 + 188289*b9 + 1218240*b8 - 84064*b7 + 441005*b6 + 193274*b5 + 772214*b4 + 2171197*b3 - 5761625*b2 + 36263038*b1 - 134979234) / 5184 $$\nu^{7}$$ $$=$$ $$( 2045673 \beta_{13} + 1855827 \beta_{12} - 165384 \beta_{11} + 2623365 \beta_{10} + 3234879 \beta_{9} + 8999424 \beta_{8} + 2969328 \beta_{7} + 1948359 \beta_{6} + \cdots - 5879743938 ) / 5184$$ (2045673*b13 + 1855827*b12 - 165384*b11 + 2623365*b10 + 3234879*b9 + 8999424*b8 + 2969328*b7 + 1948359*b6 - 1809298*b5 + 4630062*b4 + 17503575*b3 - 68887515*b2 - 213646206*b1 - 5879743938) / 5184 $$\nu^{8}$$ $$=$$ $$( 8231313 \beta_{13} + 2397795 \beta_{12} - 1354488 \beta_{11} + 990309 \beta_{10} + 17113947 \beta_{9} + 11042784 \beta_{8} + 13734368 \beta_{7} + 8425967 \beta_{6} + \cdots - 5898236454 ) / 1728$$ (8231313*b13 + 2397795*b12 - 1354488*b11 + 990309*b10 + 17113947*b9 + 11042784*b8 + 13734368*b7 + 8425967*b6 - 41708802*b5 + 11190818*b4 + 3902719*b3 - 204140723*b2 - 1265559926*b1 - 5898236454) / 1728 $$\nu^{9}$$ $$=$$ $$( 154797201 \beta_{13} - 42135381 \beta_{12} + 133399224 \beta_{11} - 169790643 \beta_{10} + 350444007 \beta_{9} + 255817440 \beta_{8} + 143369456 \beta_{7} + \cdots - 89174752386 ) / 5184$$ (154797201*b13 - 42135381*b12 + 133399224*b11 - 169790643*b10 + 350444007*b9 + 255817440*b8 + 143369456*b7 + 224794175*b6 - 1800498018*b5 + 391251182*b4 + 917505775*b3 - 2724378899*b2 - 50271127502*b1 - 89174752386) / 5184 $$\nu^{10}$$ $$=$$ $$( 90051723 \beta_{13} - 30935439 \beta_{12} + 1785733272 \beta_{11} - 219360825 \beta_{10} + 5588196705 \beta_{9} + 2239042176 \beta_{8} - 311056736 \beta_{7} + \cdots + 741190189038 ) / 5184$$ (90051723*b13 - 30935439*b12 + 1785733272*b11 - 219360825*b10 + 5588196705*b9 + 2239042176*b8 - 311056736*b7 + 1549437205*b6 - 18976581302*b5 + 5045887174*b4 + 7946208773*b3 - 9847963777*b2 - 204204793234*b1 + 741190189038) / 5184 $$\nu^{11}$$ $$=$$ $$( - 16746813639 \beta_{13} + 5288240547 \beta_{12} + 8044070328 \beta_{11} + 452918709 \beta_{10} + 51171467895 \beta_{9} - 13418955072 \beta_{8} + \cdots - 4174643792514 ) / 5184$$ (-16746813639*b13 + 5288240547*b12 + 8044070328*b11 + 452918709*b10 + 51171467895*b9 - 13418955072*b8 - 14490979088*b7 - 7610976329*b6 - 89836284978*b5 + 32956208878*b4 + 47384773991*b3 - 208042477483*b2 - 859464707422*b1 - 4174643792514) / 5184 $$\nu^{12}$$ $$=$$ $$( - 14070738661 \beta_{13} + 6573049441 \beta_{12} + 5758500056 \beta_{11} + 5460206711 \beta_{10} + 35427013849 \beta_{9} - 51469850016 \beta_{8} + \cdots - 4268598731138 ) / 576$$ (-14070738661*b13 + 6573049441*b12 + 5758500056*b11 + 5460206711*b10 + 35427013849*b9 - 51469850016*b8 - 3675756384*b7 - 12014994011*b6 - 41496998870*b5 + 20721643286*b4 + 35450215381*b3 - 156168745489*b2 - 2078212306834*b1 - 4268598731138) / 576 $$\nu^{13}$$ $$=$$ $$( - 1398515170815 \beta_{13} + 972388573371 \beta_{12} + 1256975130168 \beta_{11} - 339276246147 \beta_{10} + 759639983895 \beta_{9} + \cdots + 230969302740414 ) / 5184$$ (-1398515170815*b13 + 972388573371*b12 + 1256975130168*b11 - 339276246147*b10 + 759639983895*b9 - 6337448032608*b8 + 48904836336*b7 - 1674634659057*b6 - 5552142556226*b5 - 342804913362*b4 + 1143839547135*b3 + 379560440349*b2 - 148964776326702*b1 + 230969302740414) / 5184

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/72\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$55$$ $$65$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 8.85262 + 1.52851i 8.85262 − 1.52851i 1.24645 − 7.99620i 1.24645 + 7.99620i 7.97707 − 3.91414i 7.97707 + 3.91414i −5.80663 − 4.20354i −5.80663 + 4.20354i 2.71713 − 7.81354i 2.71713 + 7.81354i −6.99438 − 0.299706i −6.99438 + 0.299706i −4.99225 + 5.30027i −4.99225 − 5.30027i
−9.38113 6.32411i 0 48.0111 + 118.655i 425.308i 0 1664.03 299.987 1416.74i 0 2689.70 3989.87i
37.2 −9.38113 + 6.32411i 0 48.0111 118.655i 425.308i 0 1664.03 299.987 + 1416.74i 0 2689.70 + 3989.87i
37.3 −8.24265 7.74976i 0 7.88255 + 127.757i 76.0929i 0 −222.735 925.113 1114.14i 0 −589.701 + 627.207i
37.4 −8.24265 + 7.74976i 0 7.88255 127.757i 76.0929i 0 −222.735 925.113 + 1114.14i 0 −589.701 627.207i
37.5 −3.06293 10.8912i 0 −109.237 + 66.7181i 23.0228i 0 −1547.56 1061.23 + 985.368i 0 250.747 70.5175i
37.6 −3.06293 + 10.8912i 0 −109.237 66.7181i 23.0228i 0 −1547.56 1061.23 985.368i 0 250.747 + 70.5175i
37.7 2.60309 11.0102i 0 −114.448 57.3210i 468.400i 0 81.2421 −929.033 + 1110.88i 0 5157.17 + 1219.29i
37.8 2.60309 + 11.0102i 0 −114.448 + 57.3210i 468.400i 0 81.2421 −929.033 1110.88i 0 5157.17 1219.29i
37.9 6.09641 9.53067i 0 −53.6675 116.206i 137.155i 0 808.153 −1434.70 196.951i 0 −1307.18 836.155i
37.10 6.09641 + 9.53067i 0 −53.6675 + 116.206i 137.155i 0 808.153 −1434.70 + 196.951i 0 −1307.18 + 836.155i
37.11 7.69468 8.29409i 0 −9.58387 127.641i 455.347i 0 −743.502 −1132.41 902.665i 0 −3776.69 3503.75i
37.12 7.69468 + 8.29409i 0 −9.58387 + 127.641i 455.347i 0 −743.502 −1132.41 + 902.665i 0 −3776.69 + 3503.75i
37.13 11.2925 0.691979i 0 127.042 15.6284i 124.215i 0 646.373 1423.81 264.394i 0 85.9543 + 1402.70i
37.14 11.2925 + 0.691979i 0 127.042 + 15.6284i 124.215i 0 646.373 1423.81 + 264.394i 0 85.9543 1402.70i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 37.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.8.d.d 14
3.b odd 2 1 24.8.d.a 14
4.b odd 2 1 288.8.d.d 14
8.b even 2 1 inner 72.8.d.d 14
8.d odd 2 1 288.8.d.d 14
12.b even 2 1 96.8.d.a 14
24.f even 2 1 96.8.d.a 14
24.h odd 2 1 24.8.d.a 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.8.d.a 14 3.b odd 2 1
24.8.d.a 14 24.h odd 2 1
72.8.d.d 14 1.a even 1 1 trivial
72.8.d.d 14 8.b even 2 1 inner
96.8.d.a 14 12.b even 2 1
96.8.d.a 14 24.f even 2 1
288.8.d.d 14 4.b odd 2 1
288.8.d.d 14 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{14} + 648188 T_{5}^{12} + 147837846096 T_{5}^{10} + \cdots + 73\!\cdots\!00$$ acting on $$S_{8}^{\mathrm{new}}(72, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{14} + \cdots + 562949953421312$$
$3$ $$T^{14}$$
$5$ $$T^{14} + 648188 T^{12} + \cdots + 73\!\cdots\!00$$
$7$ $$(T^{7} - 686 T^{6} + \cdots + 18\!\cdots\!56)^{2}$$
$11$ $$T^{14} + 146207744 T^{12} + \cdots + 92\!\cdots\!00$$
$13$ $$T^{14} + 491467248 T^{12} + \cdots + 82\!\cdots\!64$$
$17$ $$(T^{7} - 1454 T^{6} + \cdots - 20\!\cdots\!36)^{2}$$
$19$ $$T^{14} + 5540192496 T^{12} + \cdots + 23\!\cdots\!76$$
$23$ $$(T^{7} - 71708 T^{6} + \cdots + 23\!\cdots\!68)^{2}$$
$29$ $$T^{14} + 109525673372 T^{12} + \cdots + 30\!\cdots\!76$$
$31$ $$(T^{7} + 44734 T^{6} + \cdots - 33\!\cdots\!12)^{2}$$
$37$ $$T^{14} + 473007845568 T^{12} + \cdots + 88\!\cdots\!76$$
$41$ $$(T^{7} - 220642 T^{6} + \cdots - 38\!\cdots\!28)^{2}$$
$43$ $$T^{14} + 2638324171248 T^{12} + \cdots + 14\!\cdots\!64$$
$47$ $$(T^{7} - 528204 T^{6} + \cdots - 43\!\cdots\!56)^{2}$$
$53$ $$T^{14} + 9574301782364 T^{12} + \cdots + 98\!\cdots\!24$$
$59$ $$T^{14} + 25057896090608 T^{12} + \cdots + 33\!\cdots\!36$$
$61$ $$T^{14} + 17284536099072 T^{12} + \cdots + 54\!\cdots\!00$$
$67$ $$T^{14} + 41923367956080 T^{12} + \cdots + 45\!\cdots\!96$$
$71$ $$(T^{7} + 2586348 T^{6} + \cdots + 12\!\cdots\!68)^{2}$$
$73$ $$(T^{7} + 2723098 T^{6} + \cdots - 49\!\cdots\!48)^{2}$$
$79$ $$(T^{7} + 7186774 T^{6} + \cdots + 64\!\cdots\!80)^{2}$$
$83$ $$T^{14} + 231376940860160 T^{12} + \cdots + 31\!\cdots\!64$$
$89$ $$(T^{7} - 5976310 T^{6} + \cdots + 78\!\cdots\!80)^{2}$$
$97$ $$(T^{7} - 66866 T^{6} + \cdots - 18\!\cdots\!04)^{2}$$