# Properties

 Label 15.8.b.a Level $15$ Weight $8$ Character orbit 15.b Analytic conductor $4.686$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$15 = 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 15.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.68577538226$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 162x^{6} + 7361x^{4} + 87300x^{2} + 160000$$ x^8 + 162*x^6 + 7361*x^4 + 87300*x^2 + 160000 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}\cdot 3^{12}\cdot 5^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + \beta_{2} q^{3} + ( - \beta_{4} - 83) q^{4} + ( - \beta_{3} + 3 \beta_{2} - 5 \beta_1 - 55) q^{5} + ( - \beta_{7} + \beta_{4} + 61) q^{6} + ( - \beta_{6} - 2 \beta_{5} + \beta_{3} - 11 \beta_{2} - 5 \beta_1) q^{7} + ( - 2 \beta_{6} + \beta_{5} - 3 \beta_{3} + 9 \beta_{2} + 99 \beta_1) q^{8} - 729 q^{9}+O(q^{10})$$ q - b1 * q^2 + b2 * q^3 + (-b4 - 83) * q^4 + (-b3 + 3*b2 - 5*b1 - 55) * q^5 + (-b7 + b4 + 61) * q^6 + (-b6 - 2*b5 + b3 - 11*b2 - 5*b1) * q^7 + (-2*b6 + b5 - 3*b3 + 9*b2 + 99*b1) * q^8 - 729 * q^9 $$q - \beta_1 q^{2} + \beta_{2} q^{3} + ( - \beta_{4} - 83) q^{4} + ( - \beta_{3} + 3 \beta_{2} - 5 \beta_1 - 55) q^{5} + ( - \beta_{7} + \beta_{4} + 61) q^{6} + ( - \beta_{6} - 2 \beta_{5} + \beta_{3} - 11 \beta_{2} - 5 \beta_1) q^{7} + ( - 2 \beta_{6} + \beta_{5} - 3 \beta_{3} + 9 \beta_{2} + 99 \beta_1) q^{8} - 729 q^{9} + ( - 6 \beta_{7} - 2 \beta_{6} + 5 \beta_{5} + 5 \beta_{4} - \beta_{3} + 23 \beta_{2} + \cdots - 837) q^{10}+ \cdots + ( - 729 \beta_{6} + 1458 \beta_{5} + 8748 \beta_{4} + \cdots - 985608) q^{99}+O(q^{100})$$ q - b1 * q^2 + b2 * q^3 + (-b4 - 83) * q^4 + (-b3 + 3*b2 - 5*b1 - 55) * q^5 + (-b7 + b4 + 61) * q^6 + (-b6 - 2*b5 + b3 - 11*b2 - 5*b1) * q^7 + (-2*b6 + b5 - 3*b3 + 9*b2 + 99*b1) * q^8 - 729 * q^9 + (-6*b7 - 2*b6 + 5*b5 + 5*b4 - b3 + 23*b2 + 80*b1 - 837) * q^10 + (b6 - 2*b5 - 12*b4 - 7*b3 - 3*b2 - b1 + 1352) * q^11 + (-9*b5 + 9*b3 - 75*b2 - 81*b1) * q^12 + (7*b6 + 2*b5 + 5*b3 + 125*b2 - 505*b1) * q^13 + (36*b7 + 2*b6 - 4*b5 - 18*b4 - 14*b3 - 6*b2 - 2*b1 - 1694) * q^14 + (-8*b7 + 9*b6 - 10*b4 - 45*b2 - 90*b1 - 2176) * q^15 + (-24*b7 - 6*b6 + 12*b5 + 61*b4 + 42*b3 + 18*b2 + 6*b1 + 10783) * q^16 + (21*b6 + 30*b5 - 9*b3 - 45*b2 + 125*b1) * q^17 + 729*b1 * q^18 + (96*b7 - 12*b6 + 24*b5 - 112*b4 + 84*b3 + 36*b2 + 12*b1 - 3888) * q^19 + (-72*b7 - 4*b6 - 55*b5 + 210*b4 + 27*b3 - 921*b2 + 85*b1 + 10926) * q^20 + (9*b6 - 18*b5 + 144*b4 - 63*b3 - 27*b2 - 9*b1 + 8028) * q^21 + (-38*b6 + 50*b5 - 88*b3 + 320*b2 + 620*b1) * q^22 + (12*b6 - 72*b5 + 84*b3 + 1032*b2 + 92*b1) * q^23 + (64*b7 + 18*b6 - 36*b5 - 181*b4 - 126*b3 - 54*b2 - 18*b1 - 13687) * q^24 + (-96*b7 - 17*b6 + 50*b5 - 120*b4 + 47*b3 - 1081*b2 - 2695*b1 + 15973) * q^25 + (-144*b7 + 10*b6 - 20*b5 - 570*b4 - 70*b3 - 30*b2 - 10*b1 - 98926) * q^26 - 729*b2 * q^27 + (-120*b6 + 126*b5 - 246*b3 + 6162*b2 - 330*b1) * q^28 + (13*b6 - 26*b5 + 336*b4 - 91*b3 - 39*b2 - 13*b1 + 29990) * q^29 + (27*b7 - 36*b6 - 45*b5 - 360*b4 - 117*b3 - 1309*b2 + 4815*b1 - 21366) * q^30 + (96*b7 - 30*b6 + 60*b5 + 536*b4 + 210*b3 + 90*b2 + 30*b1 + 28884) * q^31 + (-98*b6 - 353*b5 + 255*b3 - 5325*b2 - 5383*b1) * q^32 + (81*b6 - 90*b5 + 171*b3 + 1575*b2 - 1863*b1) * q^33 + (-324*b7 - 18*b6 + 36*b5 - 82*b4 + 126*b3 + 54*b2 + 18*b1 + 23186) * q^34 + (288*b7 + 91*b6 + 370*b5 + 660*b4 - 349*b3 - 4293*b2 + 8405*b1 + 73916) * q^35 + (729*b4 + 60507) * q^36 + (255*b6 - 78*b5 + 333*b3 - 7299*b2 + 16287*b1) * q^37 + (136*b6 + 424*b5 - 288*b3 + 17088*b2 + 5184*b1) * q^38 + (-432*b7 - 63*b6 + 126*b5 + 396*b4 + 441*b3 + 189*b2 + 63*b1 - 57060) * q^39 + (792*b7 + 74*b6 - 230*b5 - 835*b4 + 196*b3 - 12668*b2 - 21500*b1 - 143521) * q^40 + (1152*b7 - 10*b6 + 20*b5 - 1824*b4 + 70*b3 + 30*b2 + 10*b1 + 63286) * q^41 + (162*b6 + 198*b5 - 36*b3 + 612*b2 - 26568*b1) * q^42 + (344*b6 - 608*b5 + 952*b3 + 11740*b2 - 15128*b1) * q^43 + (-1008*b7 - 48*b6 + 96*b5 + 1470*b4 + 336*b3 + 144*b2 + 48*b1 + 322218) * q^44 + (729*b3 - 2187*b2 + 3645*b1 + 40095) * q^45 + (-84*b7 + 168*b6 - 336*b5 - 484*b4 - 1176*b3 - 504*b2 - 168*b1 + 83804) * q^46 + (-122*b6 + 1180*b5 - 1302*b3 + 750*b2 - 27218*b1) * q^47 + (-486*b6 + 225*b5 - 711*b3 + 8261*b2 + 25839*b1) * q^48 + (-1344*b7 + 294*b6 - 588*b5 + 2352*b4 - 2058*b3 - 882*b2 - 294*b1 - 406841) * q^49 + (756*b7 - 338*b6 - 950*b5 - 3180*b4 + 1268*b3 - 19764*b2 + 10495*b1 - 639628) * q^50 + (128*b7 - 189*b6 + 378*b5 - 2180*b4 + 1323*b3 + 567*b2 + 189*b1 + 32584) * q^51 + (-672*b6 + 54*b5 - 726*b3 - 4686*b2 + 136662*b1) * q^52 + (-1035*b6 - 906*b5 - 129*b3 + 16563*b2 + 15365*b1) * q^53 + (729*b7 - 729*b4 - 44469) * q^54 + (-1344*b7 - 133*b6 - 410*b5 + 1920*b4 - 2339*b3 - 7763*b2 - 26945*b1 + 442212) * q^55 + (-3312*b7 - 236*b6 + 472*b5 + 9510*b4 + 1652*b3 + 708*b2 + 236*b1 + 86306) * q^56 + (-972*b6 - 360*b5 - 612*b3 + 572*b2 - 60588*b1) * q^57 + (490*b6 + 158*b5 + 332*b3 - 268*b2 - 75202*b1) * q^58 + (1152*b7 - 217*b6 + 434*b5 - 7428*b4 + 1519*b3 + 651*b2 + 217*b1 + 136624) * q^59 + (456*b7 + 252*b6 + 1170*b5 + 3495*b4 - 1494*b3 + 5562*b2 + 61470*b1 + 662637) * q^60 + (1536*b7 + 384*b6 - 768*b5 - 6688*b4 - 2688*b3 - 1152*b2 - 384*b1 - 801366) * q^61 + (1684*b6 - 908*b5 + 2592*b3 + 7440*b2 - 125292*b1) * q^62 + (729*b6 + 1458*b5 - 729*b3 + 8019*b2 + 3645*b1) * q^63 + (6744*b7 - 258*b6 + 516*b5 - 5765*b4 + 1806*b3 + 774*b2 + 258*b1 - 74295) * q^64 + (-5184*b7 + 167*b6 + 1130*b5 + 720*b4 + 19*b3 + 40743*b2 + 44365*b1 - 317528) * q^65 + (-324*b7 + 342*b6 - 684*b5 - 4410*b4 - 2394*b3 - 1026*b2 - 342*b1 - 294822) * q^66 + (1186*b6 - 3148*b5 + 4334*b3 - 24670*b2 + 89306*b1) * q^67 + (2128*b6 + 646*b5 + 1482*b3 - 71694*b2 + 40406*b1) * q^68 + (896*b7 - 108*b6 + 216*b5 + 5152*b4 + 756*b3 + 324*b2 + 108*b1 - 744884) * q^69 + (-636*b7 + 1198*b6 + 3110*b5 + 7980*b4 + 4088*b3 + 53216*b2 - 202360*b1 + 1507548) * q^70 + (4032*b7 + 258*b6 - 516*b5 + 9000*b4 - 1806*b3 - 774*b2 - 258*b1 - 175704) * q^71 + (1458*b6 - 729*b5 + 2187*b3 - 6561*b2 - 72171*b1) * q^72 + (-2190*b6 + 2940*b5 - 5130*b3 - 59922*b2 + 82578*b1) * q^73 + (8568*b7 + 666*b6 - 1332*b5 - 498*b4 - 4662*b3 - 1998*b2 - 666*b1 + 2994330) * q^74 + (-3024*b7 - 873*b6 - 2250*b5 + 720*b4 + 873*b3 + 10246*b2 + 62595*b1 + 972612) * q^75 + (-10176*b7 - 2112*b6 + 4224*b5 + 10792*b4 + 14784*b3 + 6336*b2 + 2112*b1 + 1631288) * q^76 + (-3498*b6 - 444*b5 - 3054*b3 - 654*b2 + 65670*b1) * q^77 + (810*b6 - 6246*b5 + 7056*b3 - 100728*b2 + 38232*b1) * q^78 + (-3168*b7 - 414*b6 + 828*b5 + 5240*b4 + 2898*b3 + 1242*b2 + 414*b1 - 1799316) * q^79 + (6192*b7 - 206*b6 - 545*b5 - 12810*b4 - 8165*b3 + 43935*b2 + 178525*b1 - 3902126) * q^80 + 531441 * q^81 + (-1204*b6 + 10660*b5 - 11864*b3 + 237784*b2 + 54082*b1) * q^82 + (-480*b6 + 720*b5 - 1200*b3 - 117420*b2 - 58768*b1) * q^83 + (-3024*b7 + 1080*b6 - 2160*b5 - 9342*b4 - 7560*b3 - 3240*b2 - 1080*b1 - 4543722) * q^84 + (-3264*b7 - 2903*b6 - 6850*b5 - 8080*b4 + 5465*b3 + 102545*b2 - 88945*b1 - 178608) * q^85 + (-3492*b7 + 1904*b6 - 3808*b5 - 24732*b4 - 13328*b3 - 5712*b2 - 1904*b1 - 2462252) * q^86 + (1053*b6 + 3258*b5 - 2205*b3 + 28953*b2 + 15633*b1) * q^87 + (-3268*b6 - 4958*b5 + 1690*b3 - 177998*b2 - 341258*b1) * q^88 + (-16128*b7 + 312*b6 - 624*b5 - 672*b4 - 2184*b3 - 936*b2 - 312*b1 + 5732154) * q^89 + (4374*b7 + 1458*b6 - 3645*b5 - 3645*b4 + 729*b3 - 16767*b2 - 58320*b1 + 610173) * q^90 + (20928*b7 + 1578*b6 - 3156*b5 - 15384*b4 - 11046*b3 - 4734*b2 - 1578*b1 + 3010728) * q^91 + (-1952*b6 - 3020*b5 + 1068*b3 + 155772*b2 + 49076*b1) * q^92 + (-2430*b6 + 5148*b5 - 7578*b3 + 25874*b2 + 7938*b1) * q^93 + (-16212*b7 - 2604*b6 + 5208*b5 - 2504*b4 + 18228*b3 + 7812*b2 + 2604*b1 - 5720400) * q^94 + (2016*b7 + 4412*b6 + 440*b5 + 45120*b4 + 12148*b3 + 165876*b2 - 282660*b1 - 3238208) * q^95 + (-3480*b7 + 882*b6 - 1764*b5 + 30309*b4 - 6174*b3 - 2646*b2 - 882*b1 + 4196823) * q^96 + (-736*b6 - 1424*b5 + 688*b3 - 163304*b2 - 30032*b1) * q^97 + (-2100*b6 - 1932*b5 - 168*b3 - 219576*b2 + 306545*b1) * q^98 + (-729*b6 + 1458*b5 + 8748*b4 + 5103*b3 + 2187*b2 + 729*b1 - 985608) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 666 q^{4} - 444 q^{5} + 486 q^{6} - 5832 q^{9}+O(q^{10})$$ 8 * q - 666 * q^4 - 444 * q^5 + 486 * q^6 - 5832 * q^9 $$8 q - 666 q^{4} - 444 q^{5} + 486 q^{6} - 5832 q^{9} - 6686 q^{10} + 10752 q^{11} - 13524 q^{14} - 17496 q^{15} + 86530 q^{16} - 30464 q^{19} + 87444 q^{20} + 64152 q^{21} - 110322 q^{24} + 127616 q^{25} - 793524 q^{26} + 240072 q^{29} - 172044 q^{30} + 233728 q^{31} + 184748 q^{34} + 593520 q^{35} + 485514 q^{36} - 454896 q^{39} - 1147102 q^{40} + 507648 q^{41} + 2578572 q^{44} + 323676 q^{45} + 662408 q^{46} - 3267160 q^{49} - 5117736 q^{50} + 264384 q^{51} - 354294 q^{54} + 3525696 q^{55} + 705660 q^{56} + 1091424 q^{59} + 5307606 q^{60} - 6433520 q^{61} - 568594 q^{64} - 2555592 q^{65} - 2382372 q^{66} - 5940864 q^{69} + 12097800 q^{70} - 1381824 q^{71} + 23961276 q^{74} + 7768224 q^{75} + 13115664 q^{76} - 14380160 q^{79} - 31251876 q^{80} + 4251528 q^{81} - 36423756 q^{84} - 1452008 q^{85} - 19837608 q^{86} + 45778896 q^{89} + 4874094 q^{90} + 24075648 q^{91} - 45728896 q^{94} - 25774656 q^{95} + 33586002 q^{96} - 7838208 q^{99}+O(q^{100})$$ 8 * q - 666 * q^4 - 444 * q^5 + 486 * q^6 - 5832 * q^9 - 6686 * q^10 + 10752 * q^11 - 13524 * q^14 - 17496 * q^15 + 86530 * q^16 - 30464 * q^19 + 87444 * q^20 + 64152 * q^21 - 110322 * q^24 + 127616 * q^25 - 793524 * q^26 + 240072 * q^29 - 172044 * q^30 + 233728 * q^31 + 184748 * q^34 + 593520 * q^35 + 485514 * q^36 - 454896 * q^39 - 1147102 * q^40 + 507648 * q^41 + 2578572 * q^44 + 323676 * q^45 + 662408 * q^46 - 3267160 * q^49 - 5117736 * q^50 + 264384 * q^51 - 354294 * q^54 + 3525696 * q^55 + 705660 * q^56 + 1091424 * q^59 + 5307606 * q^60 - 6433520 * q^61 - 568594 * q^64 - 2555592 * q^65 - 2382372 * q^66 - 5940864 * q^69 + 12097800 * q^70 - 1381824 * q^71 + 23961276 * q^74 + 7768224 * q^75 + 13115664 * q^76 - 14380160 * q^79 - 31251876 * q^80 + 4251528 * q^81 - 36423756 * q^84 - 1452008 * q^85 - 19837608 * q^86 + 45778896 * q^89 + 4874094 * q^90 + 24075648 * q^91 - 45728896 * q^94 - 25774656 * q^95 + 33586002 * q^96 - 7838208 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 162x^{6} + 7361x^{4} + 87300x^{2} + 160000$$ :

 $$\beta_{1}$$ $$=$$ $$( -3\nu^{7} - 486\nu^{5} - 22883\nu^{3} - 286700\nu ) / 20000$$ (-3*v^7 - 486*v^5 - 22883*v^3 - 286700*v) / 20000 $$\beta_{2}$$ $$=$$ $$( 9\nu^{7} + 1458\nu^{5} + 62649\nu^{3} + 494100\nu ) / 20000$$ (9*v^7 + 1458*v^5 + 62649*v^3 + 494100*v) / 20000 $$\beta_{3}$$ $$=$$ $$( 3\nu^{7} + 160\nu^{6} + 86\nu^{5} + 21920\nu^{4} - 29517\nu^{3} + 609760\nu^{2} - 1453300\nu - 996000 ) / 20000$$ (3*v^7 + 160*v^6 + 86*v^5 + 21920*v^4 - 29517*v^3 + 609760*v^2 - 1453300*v - 996000) / 20000 $$\beta_{4}$$ $$=$$ $$( \nu^{6} + 242\nu^{4} + 13441\nu^{2} + 81400 ) / 500$$ (v^6 + 242*v^4 + 13441*v^2 + 81400) / 500 $$\beta_{5}$$ $$=$$ $$( 9\nu^{7} + 20\nu^{6} + 1408\nu^{5} + 2740\nu^{4} + 60849\nu^{3} + 76220\nu^{2} + 791350\nu - 124500 ) / 2500$$ (9*v^7 + 20*v^6 + 1408*v^5 + 2740*v^4 + 60849*v^3 + 76220*v^2 + 791350*v - 124500) / 2500 $$\beta_{6}$$ $$=$$ $$( 189 \nu^{7} - 160 \nu^{6} + 27018 \nu^{5} - 21920 \nu^{4} + 932029 \nu^{3} - 609760 \nu^{2} + 3684100 \nu + 996000 ) / 20000$$ (189*v^7 - 160*v^6 + 27018*v^5 - 21920*v^4 + 932029*v^3 - 609760*v^2 + 3684100*v + 996000) / 20000 $$\beta_{7}$$ $$=$$ $$( \nu^{6} + 152\nu^{4} + 6151\nu^{2} + 45300 ) / 50$$ (v^6 + 152*v^4 + 6151*v^2 + 45300) / 50
 $$\nu$$ $$=$$ $$( 3\beta_{5} - 3\beta_{3} - 5\beta_{2} + 54\beta_1 ) / 270$$ (3*b5 - 3*b3 - 5*b2 + 54*b1) / 270 $$\nu^{2}$$ $$=$$ $$( 14\beta_{7} + 3\beta_{6} - 6\beta_{5} - 20\beta_{4} - 21\beta_{3} - 9\beta_{2} - 3\beta _1 - 10922 ) / 270$$ (14*b7 + 3*b6 - 6*b5 - 20*b4 - 21*b3 - 9*b2 - 3*b1 - 10922) / 270 $$\nu^{3}$$ $$=$$ $$( -183\beta_{5} + 183\beta_{3} - 595\beta_{2} - 5994\beta_1 ) / 270$$ (-183*b5 + 183*b3 - 595*b2 - 5994*b1) / 270 $$\nu^{4}$$ $$=$$ $$( -428\beta_{7} - 81\beta_{6} + 162\beta_{5} + 1040\beta_{4} + 567\beta_{3} + 243\beta_{2} + 81\beta _1 + 258794 ) / 90$$ (-428*b7 - 81*b6 + 162*b5 + 1040*b4 + 567*b3 + 243*b2 + 81*b1 + 258794) / 90 $$\nu^{5}$$ $$=$$ $$( -1350\beta_{6} + 13623\beta_{5} - 14973\beta_{3} + 103745\beta_{2} + 538164\beta_1 ) / 270$$ (-1350*b6 + 13623*b5 - 14973*b3 + 103745*b2 + 538164*b1) / 270 $$\nu^{6}$$ $$=$$ $$( 122554 \beta_{7} + 18483 \beta_{6} - 36966 \beta_{5} - 351220 \beta_{4} - 129381 \beta_{3} - 55449 \beta_{2} - 18483 \beta _1 - 63059842 ) / 270$$ (122554*b7 + 18483*b6 - 36966*b5 - 351220*b4 - 129381*b3 - 55449*b2 - 18483*b1 - 63059842) / 270 $$\nu^{7}$$ $$=$$ $$( 218700\beta_{6} - 1097763\beta_{5} + 1316463\beta_{3} - 11790395\beta_{2} - 48422934\beta_1 ) / 270$$ (218700*b6 - 1097763*b5 + 1316463*b3 - 11790395*b2 - 48422934*b1) / 270

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/15\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$11$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 9.60626i − 1.49423i − 3.83609i 7.26440i − 7.26440i 3.83609i 1.49423i − 9.60626i
21.0486i 27.0000i −315.042 −238.301 + 146.073i 568.311 923.886i 3936.97i −729.000 3074.63 + 5015.90i
4.2 17.7812i 27.0000i −188.170 57.4967 273.531i −480.092 1233.23i 1069.90i −729.000 −4863.70 1022.36i
4.3 8.75511i 27.0000i 51.3480 231.605 156.474i 236.388 536.160i 1570.21i −729.000 −1369.95 2027.73i
4.4 3.02250i 27.0000i 118.865 −272.800 60.8703i −81.6074 1505.28i 746.147i −729.000 −183.980 + 824.537i
4.5 3.02250i 27.0000i 118.865 −272.800 + 60.8703i −81.6074 1505.28i 746.147i −729.000 −183.980 824.537i
4.6 8.75511i 27.0000i 51.3480 231.605 + 156.474i 236.388 536.160i 1570.21i −729.000 −1369.95 + 2027.73i
4.7 17.7812i 27.0000i −188.170 57.4967 + 273.531i −480.092 1233.23i 1069.90i −729.000 −4863.70 + 1022.36i
4.8 21.0486i 27.0000i −315.042 −238.301 146.073i 568.311 923.886i 3936.97i −729.000 3074.63 5015.90i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 4.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.8.b.a 8
3.b odd 2 1 45.8.b.d 8
4.b odd 2 1 240.8.f.e 8
5.b even 2 1 inner 15.8.b.a 8
5.c odd 4 1 75.8.a.i 4
5.c odd 4 1 75.8.a.j 4
15.d odd 2 1 45.8.b.d 8
15.e even 4 1 225.8.a.z 4
15.e even 4 1 225.8.a.bb 4
20.d odd 2 1 240.8.f.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.8.b.a 8 1.a even 1 1 trivial
15.8.b.a 8 5.b even 2 1 inner
45.8.b.d 8 3.b odd 2 1
45.8.b.d 8 15.d odd 2 1
75.8.a.i 4 5.c odd 4 1
75.8.a.j 4 5.c odd 4 1
225.8.a.z 4 15.e even 4 1
225.8.a.bb 4 15.e even 4 1
240.8.f.e 8 4.b odd 2 1
240.8.f.e 8 20.d odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{8}^{\mathrm{new}}(15, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 845 T^{6} + \cdots + 98089216$$
$3$ $$(T^{2} + 729)^{4}$$
$5$ $$T^{8} + 444 T^{7} + \cdots + 37\!\cdots\!25$$
$7$ $$T^{8} + 4927752 T^{6} + \cdots + 84\!\cdots\!00$$
$11$ $$(T^{4} - 5376 T^{3} + \cdots - 36924996384576)^{2}$$
$13$ $$T^{8} + 317382120 T^{6} + \cdots + 56\!\cdots\!96$$
$17$ $$T^{8} + 1326477704 T^{6} + \cdots + 49\!\cdots\!76$$
$19$ $$(T^{4} + 15232 T^{3} + \cdots + 23\!\cdots\!00)^{2}$$
$23$ $$T^{8} + 9454242368 T^{6} + \cdots + 38\!\cdots\!00$$
$29$ $$(T^{4} - 120036 T^{3} + \cdots - 98\!\cdots\!00)^{2}$$
$31$ $$(T^{4} - 116864 T^{3} + \cdots + 64\!\cdots\!00)^{2}$$
$37$ $$T^{8} + 481863497832 T^{6} + \cdots + 29\!\cdots\!00$$
$41$ $$(T^{4} - 253824 T^{3} + \cdots + 11\!\cdots\!00)^{2}$$
$43$ $$T^{8} + 1276413690816 T^{6} + \cdots + 34\!\cdots\!56$$
$47$ $$T^{8} + 2227457267744 T^{6} + \cdots + 40\!\cdots\!16$$
$53$ $$T^{8} + 3413297274728 T^{6} + \cdots + 40\!\cdots\!00$$
$59$ $$(T^{4} - 545712 T^{3} + \cdots + 69\!\cdots\!00)^{2}$$
$61$ $$(T^{4} + 3216760 T^{3} + \cdots - 76\!\cdots\!44)^{2}$$
$67$ $$T^{8} + 23875474859424 T^{6} + \cdots + 30\!\cdots\!96$$
$71$ $$(T^{4} + 690912 T^{3} + \cdots + 64\!\cdots\!56)^{2}$$
$73$ $$T^{8} + 36893474997408 T^{6} + \cdots + 20\!\cdots\!00$$
$79$ $$(T^{4} + 7190080 T^{3} + \cdots - 18\!\cdots\!00)^{2}$$
$83$ $$T^{8} + 47430475444160 T^{6} + \cdots + 87\!\cdots\!36$$
$89$ $$(T^{4} - 22889448 T^{3} + \cdots - 29\!\cdots\!00)^{2}$$
$97$ $$T^{8} + 82955023329024 T^{6} + \cdots + 13\!\cdots\!96$$