Properties

 Label 1.54.a.a Level $1$ Weight $54$ Character orbit 1.a Self dual yes Analytic conductor $17.790$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1$$ Weight: $$k$$ $$=$$ $$54$$ Character orbit: $$[\chi]$$ $$=$$ 1.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$17.7903107608$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - x^{3} - 2315873743412 x^{2} - 421178019174503472 x + 612167648493870378955584$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{27}\cdot 3^{8}\cdot 5^{3}\cdot 7\cdot 13$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

 $$f(q)$$ $$=$$ $$q +(-17119080 + \beta_{1}) q^{2} +(-262102751820 + 8136 \beta_{1} - \beta_{2}) q^{3} +(1957409854949632 - 8049420 \beta_{1} + 374 \beta_{2} + \beta_{3}) q^{4} +(-1140973948323578250 + 3784860800 \beta_{1} - 97500 \beta_{2} - 600 \beta_{3}) q^{5} +(91310821379248619232 - 940459617900 \beta_{1} + 89361744 \beta_{2} + 36216 \beta_{3}) q^{6} +(-56210442687111357400 - 54402495474672 \beta_{1} + 5099906366 \beta_{2} - 909920 \beta_{3}) q^{7} +($$$$34\!\cdots\!60$$$$- 2009331196034176 \beta_{1} - 102810683328 \beta_{2} + 7340640 \beta_{3}) q^{8} +($$$$28\!\cdots\!53$$$$- 153884364943507200 \beta_{1} - 1414745890632 \beta_{2} + 216501552 \beta_{3}) q^{9} +O(q^{10})$$ $$q +(-17119080 + \beta_{1}) q^{2} +(-262102751820 + 8136 \beta_{1} - \beta_{2}) q^{3} +(1957409854949632 - 8049420 \beta_{1} + 374 \beta_{2} + \beta_{3}) q^{4} +(-1140973948323578250 + 3784860800 \beta_{1} - 97500 \beta_{2} - 600 \beta_{3}) q^{5} +(91310821379248619232 - 940459617900 \beta_{1} + 89361744 \beta_{2} + 36216 \beta_{3}) q^{6} +(-56210442687111357400 - 54402495474672 \beta_{1} + 5099906366 \beta_{2} - 909920 \beta_{3}) q^{7} +($$$$34\!\cdots\!60$$$$- 2009331196034176 \beta_{1} - 102810683328 \beta_{2} + 7340640 \beta_{3}) q^{8} +($$$$28\!\cdots\!53$$$$- 153884364943507200 \beta_{1} - 1414745890632 \beta_{2} + 216501552 \beta_{3}) q^{9} +($$$$59\!\cdots\!00$$$$- 4079277238575184650 \beta_{1} + 50341792216000 \beta_{2} - 9012527200 \beta_{3}) q^{10} +($$$$62\!\cdots\!32$$$$- 20797035850880243240 \beta_{1} - 400485890554035 \beta_{2} + 180852555840 \beta_{3}) q^{11} +(-$$$$92\!\cdots\!60$$$$+$$$$25\!\cdots\!64$$$$\beta_{1} - 1503331766739976 \beta_{2} - 2512033441740 \beta_{3}) q^{12} +($$$$97\!\cdots\!10$$$$+$$$$20\!\cdots\!08$$$$\beta_{1} + 51373303912919300 \beta_{2} + 26522640225640 \beta_{3}) q^{13} +(-$$$$57\!\cdots\!56$$$$-$$$$11\!\cdots\!60$$$$\beta_{1} - 399129725674167648 \beta_{2} - 221167496673552 \beta_{3}) q^{14} +($$$$27\!\cdots\!00$$$$-$$$$13\!\cdots\!00$$$$\beta_{1} + 1080428233798364250 \beta_{2} + 1479299373410400 \beta_{3}) q^{15} +(-$$$$39\!\cdots\!84$$$$+$$$$47\!\cdots\!80$$$$\beta_{1} + 4258703104277157888 \beta_{2} - 7939542758857728 \beta_{3}) q^{16} +(-$$$$21\!\cdots\!90$$$$+$$$$31\!\cdots\!92$$$$\beta_{1} - 40747405116440959752 \beta_{2} + 33675934369779120 \beta_{3}) q^{17} +(-$$$$16\!\cdots\!20$$$$+$$$$14\!\cdots\!21$$$$\beta_{1} + 49949006167583512704 \beta_{2} - 108552646480207680 \beta_{3}) q^{18} +(-$$$$63\!\cdots\!40$$$$-$$$$60\!\cdots\!60$$$$\beta_{1} +$$$$75\!\cdots\!11$$$$\beta_{2} + 243892004379685824 \beta_{3}) q^{19} +(-$$$$34\!\cdots\!00$$$$-$$$$16\!\cdots\!00$$$$\beta_{1} -$$$$43\!\cdots\!00$$$$\beta_{2} - 321907385167725450 \beta_{3}) q^{20} +(-$$$$11\!\cdots\!08$$$$+$$$$62\!\cdots\!60$$$$\beta_{1} +$$$$75\!\cdots\!48$$$$\beta_{2} + 475413659635414752 \beta_{3}) q^{21} +(-$$$$23\!\cdots\!60$$$$+$$$$10\!\cdots\!92$$$$\beta_{1} +$$$$14\!\cdots\!20$$$$\beta_{2} - 4868761285364777240 \beta_{3}) q^{22} +($$$$17\!\cdots\!40$$$$-$$$$64\!\cdots\!20$$$$\beta_{1} -$$$$37\!\cdots\!94$$$$\beta_{2} + 31701177037073551200 \beta_{3}) q^{23} +($$$$20\!\cdots\!80$$$$-$$$$11\!\cdots\!20$$$$\beta_{1} -$$$$41\!\cdots\!12$$$$\beta_{2} - 97319344971076633728 \beta_{3}) q^{24} +($$$$91\!\cdots\!75$$$$+$$$$26\!\cdots\!00$$$$\beta_{1} +$$$$24\!\cdots\!00$$$$\beta_{2} + 5532490760056300000 \beta_{3}) q^{25} +($$$$22\!\cdots\!72$$$$+$$$$19\!\cdots\!50$$$$\beta_{1} -$$$$54\!\cdots\!28$$$$\beta_{2} +$$$$13\!\cdots\!08$$$$\beta_{3}) q^{26} +($$$$21\!\cdots\!60$$$$-$$$$17\!\cdots\!96$$$$\beta_{1} +$$$$22\!\cdots\!62$$$$\beta_{2} -$$$$61\!\cdots\!60$$$$\beta_{3}) q^{27} +(-$$$$14\!\cdots\!40$$$$-$$$$14\!\cdots\!00$$$$\beta_{1} +$$$$30\!\cdots\!84$$$$\beta_{2} +$$$$12\!\cdots\!60$$$$\beta_{3}) q^{28} +(-$$$$35\!\cdots\!10$$$$-$$$$20\!\cdots\!80$$$$\beta_{1} +$$$$57\!\cdots\!04$$$$\beta_{2} +$$$$63\!\cdots\!16$$$$\beta_{3}) q^{29} +(-$$$$14\!\cdots\!00$$$$+$$$$94\!\cdots\!00$$$$\beta_{1} -$$$$24\!\cdots\!00$$$$\beta_{2} -$$$$12\!\cdots\!00$$$$\beta_{3}) q^{30} +(-$$$$99\!\cdots\!08$$$$+$$$$77\!\cdots\!80$$$$\beta_{1} +$$$$15\!\cdots\!20$$$$\beta_{2} +$$$$36\!\cdots\!20$$$$\beta_{3}) q^{31} +($$$$86\!\cdots\!20$$$$-$$$$56\!\cdots\!44$$$$\beta_{1} +$$$$11\!\cdots\!40$$$$\beta_{2} -$$$$34\!\cdots\!80$$$$\beta_{3}) q^{32} +($$$$66\!\cdots\!60$$$$-$$$$47\!\cdots\!68$$$$\beta_{1} -$$$$29\!\cdots\!72$$$$\beta_{2} -$$$$11\!\cdots\!20$$$$\beta_{3}) q^{33} +($$$$37\!\cdots\!84$$$$-$$$$56\!\cdots\!70$$$$\beta_{1} +$$$$24\!\cdots\!08$$$$\beta_{2} +$$$$51\!\cdots\!52$$$$\beta_{3}) q^{34} +($$$$15\!\cdots\!00$$$$+$$$$80\!\cdots\!00$$$$\beta_{1} -$$$$27\!\cdots\!00$$$$\beta_{2} -$$$$75\!\cdots\!00$$$$\beta_{3}) q^{35} +($$$$18\!\cdots\!96$$$$-$$$$77\!\cdots\!20$$$$\beta_{1} +$$$$16\!\cdots\!18$$$$\beta_{2} -$$$$47\!\cdots\!83$$$$\beta_{3}) q^{36} +(-$$$$70\!\cdots\!70$$$$-$$$$17\!\cdots\!80$$$$\beta_{1} -$$$$13\!\cdots\!68$$$$\beta_{2} +$$$$42\!\cdots\!80$$$$\beta_{3}) q^{37} +(-$$$$53\!\cdots\!60$$$$-$$$$51\!\cdots\!24$$$$\beta_{1} -$$$$10\!\cdots\!92$$$$\beta_{2} -$$$$75\!\cdots\!20$$$$\beta_{3}) q^{38} +(-$$$$92\!\cdots\!04$$$$+$$$$10\!\cdots\!80$$$$\beta_{1} +$$$$26\!\cdots\!82$$$$\beta_{2} +$$$$19\!\cdots\!88$$$$\beta_{3}) q^{39} +(-$$$$13\!\cdots\!00$$$$-$$$$25\!\cdots\!00$$$$\beta_{1} -$$$$59\!\cdots\!00$$$$\beta_{2} +$$$$17\!\cdots\!00$$$$\beta_{3}) q^{40} +($$$$20\!\cdots\!22$$$$+$$$$37\!\cdots\!80$$$$\beta_{1} -$$$$31\!\cdots\!80$$$$\beta_{2} -$$$$40\!\cdots\!80$$$$\beta_{3}) q^{41} +($$$$86\!\cdots\!60$$$$-$$$$10\!\cdots\!40$$$$\beta_{1} -$$$$44\!\cdots\!56$$$$\beta_{2} +$$$$42\!\cdots\!80$$$$\beta_{3}) q^{42} +($$$$11\!\cdots\!00$$$$+$$$$24\!\cdots\!44$$$$\beta_{1} +$$$$11\!\cdots\!73$$$$\beta_{2} -$$$$40\!\cdots\!00$$$$\beta_{3}) q^{43} +($$$$91\!\cdots\!24$$$$-$$$$48\!\cdots\!20$$$$\beta_{1} +$$$$30\!\cdots\!48$$$$\beta_{2} -$$$$11\!\cdots\!88$$$$\beta_{3}) q^{44} +(-$$$$13\!\cdots\!50$$$$+$$$$46\!\cdots\!00$$$$\beta_{1} -$$$$76\!\cdots\!00$$$$\beta_{2} +$$$$36\!\cdots\!00$$$$\beta_{3}) q^{45} +(-$$$$71\!\cdots\!88$$$$+$$$$23\!\cdots\!80$$$$\beta_{1} -$$$$13\!\cdots\!60$$$$\beta_{2} -$$$$45\!\cdots\!00$$$$\beta_{3}) q^{46} +(-$$$$11\!\cdots\!60$$$$-$$$$10\!\cdots\!16$$$$\beta_{1} -$$$$97\!\cdots\!16$$$$\beta_{2} -$$$$39\!\cdots\!00$$$$\beta_{3}) q^{47} +(-$$$$76\!\cdots\!60$$$$-$$$$11\!\cdots\!60$$$$\beta_{1} +$$$$51\!\cdots\!56$$$$\beta_{2} +$$$$19\!\cdots\!80$$$$\beta_{3}) q^{48} +($$$$15\!\cdots\!57$$$$-$$$$20\!\cdots\!80$$$$\beta_{1} -$$$$36\!\cdots\!80$$$$\beta_{2} -$$$$10\!\cdots\!60$$$$\beta_{3}) q^{49} +($$$$13\!\cdots\!00$$$$+$$$$11\!\cdots\!75$$$$\beta_{1} -$$$$20\!\cdots\!00$$$$\beta_{2} -$$$$43\!\cdots\!00$$$$\beta_{3}) q^{50} +($$$$12\!\cdots\!12$$$$-$$$$43\!\cdots\!60$$$$\beta_{1} +$$$$36\!\cdots\!06$$$$\beta_{2} +$$$$35\!\cdots\!04$$$$\beta_{3}) q^{51} +($$$$16\!\cdots\!00$$$$+$$$$73\!\cdots\!08$$$$\beta_{1} -$$$$90\!\cdots\!84$$$$\beta_{2} +$$$$14\!\cdots\!50$$$$\beta_{3}) q^{52} +($$$$97\!\cdots\!30$$$$-$$$$35\!\cdots\!64$$$$\beta_{1} -$$$$22\!\cdots\!16$$$$\beta_{2} -$$$$19\!\cdots\!60$$$$\beta_{3}) q^{53} +(-$$$$22\!\cdots\!40$$$$-$$$$84\!\cdots\!60$$$$\beta_{1} +$$$$15\!\cdots\!16$$$$\beta_{2} -$$$$40\!\cdots\!56$$$$\beta_{3}) q^{54} +(-$$$$63\!\cdots\!00$$$$+$$$$29\!\cdots\!00$$$$\beta_{1} -$$$$14\!\cdots\!50$$$$\beta_{2} +$$$$84\!\cdots\!00$$$$\beta_{3}) q^{55} +(-$$$$10\!\cdots\!40$$$$+$$$$58\!\cdots\!80$$$$\beta_{1} +$$$$19\!\cdots\!16$$$$\beta_{2} +$$$$76\!\cdots\!64$$$$\beta_{3}) q^{56} +(-$$$$19\!\cdots\!80$$$$+$$$$14\!\cdots\!48$$$$\beta_{1} +$$$$37\!\cdots\!24$$$$\beta_{2} -$$$$26\!\cdots\!00$$$$\beta_{3}) q^{57} +($$$$57\!\cdots\!60$$$$-$$$$27\!\cdots\!66$$$$\beta_{1} -$$$$53\!\cdots\!88$$$$\beta_{2} -$$$$14\!\cdots\!20$$$$\beta_{3}) q^{58} +($$$$40\!\cdots\!80$$$$+$$$$34\!\cdots\!60$$$$\beta_{1} -$$$$12\!\cdots\!27$$$$\beta_{2} +$$$$87\!\cdots\!52$$$$\beta_{3}) q^{59} +($$$$10\!\cdots\!00$$$$-$$$$95\!\cdots\!00$$$$\beta_{1} +$$$$23\!\cdots\!00$$$$\beta_{2} -$$$$22\!\cdots\!00$$$$\beta_{3}) q^{60} +($$$$16\!\cdots\!82$$$$+$$$$12\!\cdots\!00$$$$\beta_{1} +$$$$35\!\cdots\!00$$$$\beta_{2} -$$$$23\!\cdots\!00$$$$\beta_{3}) q^{61} +($$$$99\!\cdots\!40$$$$+$$$$97\!\cdots\!72$$$$\beta_{1} -$$$$35\!\cdots\!40$$$$\beta_{2} +$$$$12\!\cdots\!80$$$$\beta_{3}) q^{62} +(-$$$$75\!\cdots\!80$$$$+$$$$79\!\cdots\!92$$$$\beta_{1} +$$$$77\!\cdots\!02$$$$\beta_{2} +$$$$39\!\cdots\!80$$$$\beta_{3}) q^{63} +(-$$$$25\!\cdots\!08$$$$-$$$$89\!\cdots\!00$$$$\beta_{1} -$$$$13\!\cdots\!12$$$$\beta_{2} -$$$$24\!\cdots\!68$$$$\beta_{3}) q^{64} +(-$$$$85\!\cdots\!00$$$$-$$$$57\!\cdots\!00$$$$\beta_{1} -$$$$27\!\cdots\!00$$$$\beta_{2} -$$$$88\!\cdots\!00$$$$\beta_{3}) q^{65} +(-$$$$16\!\cdots\!76$$$$-$$$$14\!\cdots\!80$$$$\beta_{1} +$$$$33\!\cdots\!88$$$$\beta_{2} +$$$$48\!\cdots\!92$$$$\beta_{3}) q^{66} +(-$$$$27\!\cdots\!40$$$$-$$$$59\!\cdots\!88$$$$\beta_{1} -$$$$10\!\cdots\!17$$$$\beta_{2} +$$$$25\!\cdots\!20$$$$\beta_{3}) q^{67} +($$$$71\!\cdots\!80$$$$+$$$$35\!\cdots\!88$$$$\beta_{1} -$$$$21\!\cdots\!92$$$$\beta_{2} -$$$$29\!\cdots\!90$$$$\beta_{3}) q^{68} +($$$$19\!\cdots\!16$$$$+$$$$67\!\cdots\!00$$$$\beta_{1} -$$$$71\!\cdots\!84$$$$\beta_{2} -$$$$30\!\cdots\!76$$$$\beta_{3}) q^{69} +($$$$82\!\cdots\!00$$$$-$$$$16\!\cdots\!00$$$$\beta_{1} +$$$$10\!\cdots\!00$$$$\beta_{2} +$$$$68\!\cdots\!00$$$$\beta_{3}) q^{70} +($$$$89\!\cdots\!12$$$$-$$$$66\!\cdots\!00$$$$\beta_{1} +$$$$11\!\cdots\!50$$$$\beta_{2} -$$$$60\!\cdots\!00$$$$\beta_{3}) q^{71} +($$$$66\!\cdots\!80$$$$-$$$$11\!\cdots\!08$$$$\beta_{1} -$$$$18\!\cdots\!64$$$$\beta_{2} -$$$$38\!\cdots\!40$$$$\beta_{3}) q^{72} +(-$$$$17\!\cdots\!10$$$$-$$$$19\!\cdots\!80$$$$\beta_{1} +$$$$54\!\cdots\!96$$$$\beta_{2} -$$$$58\!\cdots\!20$$$$\beta_{3}) q^{73} +(-$$$$61\!\cdots\!36$$$$+$$$$12\!\cdots\!30$$$$\beta_{1} -$$$$17\!\cdots\!20$$$$\beta_{2} +$$$$12\!\cdots\!60$$$$\beta_{3}) q^{74} +(-$$$$53\!\cdots\!00$$$$+$$$$41\!\cdots\!00$$$$\beta_{1} -$$$$15\!\cdots\!75$$$$\beta_{2} +$$$$11\!\cdots\!00$$$$\beta_{3}) q^{75} +($$$$11\!\cdots\!20$$$$-$$$$48\!\cdots\!40$$$$\beta_{1} +$$$$53\!\cdots\!32$$$$\beta_{2} -$$$$64\!\cdots\!72$$$$\beta_{3}) q^{76} +(-$$$$40\!\cdots\!00$$$$+$$$$52\!\cdots\!56$$$$\beta_{1} +$$$$13\!\cdots\!32$$$$\beta_{2} +$$$$76\!\cdots\!20$$$$\beta_{3}) q^{77} +($$$$12\!\cdots\!00$$$$-$$$$53\!\cdots\!12$$$$\beta_{1} -$$$$20\!\cdots\!04$$$$\beta_{2} +$$$$36\!\cdots\!60$$$$\beta_{3}) q^{78} +(-$$$$39\!\cdots\!60$$$$-$$$$55\!\cdots\!00$$$$\beta_{1} -$$$$23\!\cdots\!36$$$$\beta_{2} -$$$$16\!\cdots\!04$$$$\beta_{3}) q^{79} +($$$$28\!\cdots\!00$$$$+$$$$81\!\cdots\!00$$$$\beta_{1} +$$$$31\!\cdots\!00$$$$\beta_{2} +$$$$66\!\cdots\!00$$$$\beta_{3}) q^{80} +(-$$$$12\!\cdots\!39$$$$+$$$$25\!\cdots\!80$$$$\beta_{1} -$$$$28\!\cdots\!16$$$$\beta_{2} +$$$$33\!\cdots\!16$$$$\beta_{3}) q^{81} +($$$$39\!\cdots\!40$$$$-$$$$16\!\cdots\!98$$$$\beta_{1} +$$$$68\!\cdots\!60$$$$\beta_{2} +$$$$35\!\cdots\!80$$$$\beta_{3}) q^{82} +(-$$$$65\!\cdots\!80$$$$+$$$$62\!\cdots\!32$$$$\beta_{1} -$$$$71\!\cdots\!13$$$$\beta_{2} -$$$$47\!\cdots\!00$$$$\beta_{3}) q^{83} +(-$$$$19\!\cdots\!56$$$$+$$$$38\!\cdots\!20$$$$\beta_{1} -$$$$95\!\cdots\!36$$$$\beta_{2} -$$$$81\!\cdots\!44$$$$\beta_{3}) q^{84} +(-$$$$58\!\cdots\!00$$$$-$$$$20\!\cdots\!00$$$$\beta_{1} +$$$$82\!\cdots\!00$$$$\beta_{2} +$$$$17\!\cdots\!00$$$$\beta_{3}) q^{85} +(-$$$$19\!\cdots\!08$$$$+$$$$12\!\cdots\!60$$$$\beta_{1} -$$$$97\!\cdots\!84$$$$\beta_{2} -$$$$33\!\cdots\!96$$$$\beta_{3}) q^{86} +(-$$$$11\!\cdots\!20$$$$+$$$$65\!\cdots\!32$$$$\beta_{1} +$$$$48\!\cdots\!86$$$$\beta_{2} -$$$$10\!\cdots\!20$$$$\beta_{3}) q^{87} +($$$$14\!\cdots\!20$$$$-$$$$36\!\cdots\!32$$$$\beta_{1} -$$$$33\!\cdots\!96$$$$\beta_{2} -$$$$12\!\cdots\!20$$$$\beta_{3}) q^{88} +(-$$$$94\!\cdots\!30$$$$+$$$$31\!\cdots\!60$$$$\beta_{1} -$$$$31\!\cdots\!08$$$$\beta_{2} -$$$$24\!\cdots\!32$$$$\beta_{3}) q^{89} +($$$$52\!\cdots\!00$$$$+$$$$28\!\cdots\!50$$$$\beta_{1} +$$$$58\!\cdots\!00$$$$\beta_{2} +$$$$77\!\cdots\!00$$$$\beta_{3}) q^{90} +($$$$31\!\cdots\!32$$$$-$$$$67\!\cdots\!20$$$$\beta_{1} -$$$$13\!\cdots\!44$$$$\beta_{2} -$$$$26\!\cdots\!76$$$$\beta_{3}) q^{91} +($$$$22\!\cdots\!60$$$$-$$$$34\!\cdots\!48$$$$\beta_{1} +$$$$84\!\cdots\!68$$$$\beta_{2} -$$$$12\!\cdots\!20$$$$\beta_{3}) q^{92} +(-$$$$24\!\cdots\!40$$$$+$$$$73\!\cdots\!52$$$$\beta_{1} +$$$$19\!\cdots\!88$$$$\beta_{2} -$$$$53\!\cdots\!60$$$$\beta_{3}) q^{93} +(-$$$$92\!\cdots\!96$$$$-$$$$14\!\cdots\!60$$$$\beta_{1} -$$$$11\!\cdots\!04$$$$\beta_{2} -$$$$11\!\cdots\!36$$$$\beta_{3}) q^{94} +(-$$$$42\!\cdots\!00$$$$+$$$$26\!\cdots\!00$$$$\beta_{1} -$$$$36\!\cdots\!50$$$$\beta_{2} +$$$$41\!\cdots\!00$$$$\beta_{3}) q^{95} +(-$$$$28\!\cdots\!48$$$$+$$$$15\!\cdots\!40$$$$\beta_{1} -$$$$24\!\cdots\!56$$$$\beta_{2} -$$$$11\!\cdots\!64$$$$\beta_{3}) q^{96} +($$$$25\!\cdots\!90$$$$-$$$$24\!\cdots\!56$$$$\beta_{1} +$$$$59\!\cdots\!24$$$$\beta_{2} -$$$$29\!\cdots\!60$$$$\beta_{3}) q^{97} +(-$$$$22\!\cdots\!60$$$$-$$$$79\!\cdots\!83$$$$\beta_{1} +$$$$30\!\cdots\!60$$$$\beta_{2} -$$$$12\!\cdots\!80$$$$\beta_{3}) q^{98} +($$$$50\!\cdots\!96$$$$-$$$$18\!\cdots\!20$$$$\beta_{1} -$$$$62\!\cdots\!79$$$$\beta_{2} -$$$$12\!\cdots\!16$$$$\beta_{3}) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 68476320q^{2} - 1048411007280q^{3} + 7829639419798528q^{4} - 4563895793294313000q^{5} +$$$$36\!\cdots\!28$$$$q^{6} -$$$$22\!\cdots\!00$$$$q^{7} +$$$$13\!\cdots\!40$$$$q^{8} +$$$$11\!\cdots\!12$$$$q^{9} + O(q^{10})$$ $$4q - 68476320q^{2} - 1048411007280q^{3} + 7829639419798528q^{4} - 4563895793294313000q^{5} +$$$$36\!\cdots\!28$$$$q^{6} -$$$$22\!\cdots\!00$$$$q^{7} +$$$$13\!\cdots\!40$$$$q^{8} +$$$$11\!\cdots\!12$$$$q^{9} +$$$$23\!\cdots\!00$$$$q^{10} +$$$$24\!\cdots\!28$$$$q^{11} -$$$$36\!\cdots\!40$$$$q^{12} +$$$$38\!\cdots\!40$$$$q^{13} -$$$$23\!\cdots\!24$$$$q^{14} +$$$$10\!\cdots\!00$$$$q^{15} -$$$$15\!\cdots\!36$$$$q^{16} -$$$$86\!\cdots\!60$$$$q^{17} -$$$$67\!\cdots\!80$$$$q^{18} -$$$$25\!\cdots\!60$$$$q^{19} -$$$$13\!\cdots\!00$$$$q^{20} -$$$$45\!\cdots\!32$$$$q^{21} -$$$$93\!\cdots\!40$$$$q^{22} +$$$$69\!\cdots\!60$$$$q^{23} +$$$$80\!\cdots\!20$$$$q^{24} +$$$$36\!\cdots\!00$$$$q^{25} +$$$$88\!\cdots\!88$$$$q^{26} +$$$$85\!\cdots\!40$$$$q^{27} -$$$$56\!\cdots\!60$$$$q^{28} -$$$$14\!\cdots\!40$$$$q^{29} -$$$$57\!\cdots\!00$$$$q^{30} -$$$$39\!\cdots\!32$$$$q^{31} +$$$$34\!\cdots\!80$$$$q^{32} +$$$$26\!\cdots\!40$$$$q^{33} +$$$$14\!\cdots\!36$$$$q^{34} +$$$$61\!\cdots\!00$$$$q^{35} +$$$$74\!\cdots\!84$$$$q^{36} -$$$$28\!\cdots\!80$$$$q^{37} -$$$$21\!\cdots\!40$$$$q^{38} -$$$$36\!\cdots\!16$$$$q^{39} -$$$$53\!\cdots\!00$$$$q^{40} +$$$$81\!\cdots\!88$$$$q^{41} +$$$$34\!\cdots\!40$$$$q^{42} +$$$$45\!\cdots\!00$$$$q^{43} +$$$$36\!\cdots\!96$$$$q^{44} -$$$$52\!\cdots\!00$$$$q^{45} -$$$$28\!\cdots\!52$$$$q^{46} -$$$$45\!\cdots\!40$$$$q^{47} -$$$$30\!\cdots\!40$$$$q^{48} +$$$$61\!\cdots\!28$$$$q^{49} +$$$$52\!\cdots\!00$$$$q^{50} +$$$$48\!\cdots\!48$$$$q^{51} +$$$$64\!\cdots\!00$$$$q^{52} +$$$$38\!\cdots\!20$$$$q^{53} -$$$$90\!\cdots\!60$$$$q^{54} -$$$$25\!\cdots\!00$$$$q^{55} -$$$$41\!\cdots\!60$$$$q^{56} -$$$$78\!\cdots\!20$$$$q^{57} +$$$$23\!\cdots\!40$$$$q^{58} +$$$$16\!\cdots\!20$$$$q^{59} +$$$$40\!\cdots\!00$$$$q^{60} +$$$$65\!\cdots\!28$$$$q^{61} +$$$$39\!\cdots\!60$$$$q^{62} -$$$$30\!\cdots\!20$$$$q^{63} -$$$$10\!\cdots\!32$$$$q^{64} -$$$$34\!\cdots\!00$$$$q^{65} -$$$$65\!\cdots\!04$$$$q^{66} -$$$$11\!\cdots\!60$$$$q^{67} +$$$$28\!\cdots\!20$$$$q^{68} +$$$$77\!\cdots\!64$$$$q^{69} +$$$$33\!\cdots\!00$$$$q^{70} +$$$$35\!\cdots\!48$$$$q^{71} +$$$$26\!\cdots\!20$$$$q^{72} -$$$$70\!\cdots\!40$$$$q^{73} -$$$$24\!\cdots\!44$$$$q^{74} -$$$$21\!\cdots\!00$$$$q^{75} +$$$$45\!\cdots\!80$$$$q^{76} -$$$$16\!\cdots\!00$$$$q^{77} +$$$$51\!\cdots\!00$$$$q^{78} -$$$$15\!\cdots\!40$$$$q^{79} +$$$$11\!\cdots\!00$$$$q^{80} -$$$$49\!\cdots\!56$$$$q^{81} +$$$$15\!\cdots\!60$$$$q^{82} -$$$$26\!\cdots\!20$$$$q^{83} -$$$$76\!\cdots\!24$$$$q^{84} -$$$$23\!\cdots\!00$$$$q^{85} -$$$$77\!\cdots\!32$$$$q^{86} -$$$$45\!\cdots\!80$$$$q^{87} +$$$$56\!\cdots\!80$$$$q^{88} -$$$$37\!\cdots\!20$$$$q^{89} +$$$$20\!\cdots\!00$$$$q^{90} +$$$$12\!\cdots\!28$$$$q^{91} +$$$$89\!\cdots\!40$$$$q^{92} -$$$$99\!\cdots\!60$$$$q^{93} -$$$$37\!\cdots\!84$$$$q^{94} -$$$$17\!\cdots\!00$$$$q^{95} -$$$$11\!\cdots\!92$$$$q^{96} +$$$$10\!\cdots\!60$$$$q^{97} -$$$$88\!\cdots\!40$$$$q^{98} +$$$$20\!\cdots\!84$$$$q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2315873743412 x^{2} - 421178019174503472 x + 612167648493870378955584$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$96 \nu - 24$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 611437 \nu^{2} + 1620402960440 \nu - 392120600840009328$$$$)/119308$$ $$\beta_{3}$$ $$=$$ $$($$$$187 \nu^{3} + 435432545 \nu^{2} - 452992885700072 \nu - 563273827738686130416$$$$)/59654$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1} + 24$$$$)/96$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 374 \beta_{2} + 26188788 \beta_{1} + 10671546209644800$$$$)/9216$$ $$\nu^{3}$$ $$=$$ $$($$$$611437 \beta_{3} - 870865090 \beta_{2} + 171571478170596 \beta_{1} + 2911198475853482464512$$$$)/9216$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.26509e6 −708531. 447512. 1.52611e6
−1.38568e8 −5.95414e12 1.01938e16 −5.35900e18 8.25051e20 2.55363e22 −1.64427e23 1.60685e25 7.42585e26
1.2 −8.51381e7 6.54009e12 −1.75871e15 2.26338e17 −5.56810e20 −3.24923e22 9.16589e23 2.33895e25 −1.92700e25
1.3 2.58420e7 −2.97907e12 −8.33939e15 5.38136e18 −7.69850e19 2.33436e22 −4.48271e23 −1.05084e25 1.39065e26
1.4 1.29387e8 1.34470e12 7.73392e15 −4.81259e18 1.73988e20 −1.66125e22 −1.64746e23 −1.75750e25 −6.22689e26
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1.54.a.a 4
3.b odd 2 1 9.54.a.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.54.a.a 4 1.a even 1 1 trivial
9.54.a.b 4 3.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace $$S_{54}^{\mathrm{new}}(\Gamma_0(1))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$39\!\cdots\!76$$$$-$$$$10\!\cdots\!80$$$$T - 19584715019010048 T^{2} + 68476320 T^{3} + T^{4}$$
$3$ $$15\!\cdots\!76$$$$-$$$$61\!\cdots\!80$$$$T -$$$$43\!\cdots\!52$$$$T^{2} + 1048411007280 T^{3} + T^{4}$$
$5$ $$31\!\cdots\!00$$$$-$$$$13\!\cdots\!00$$$$T -$$$$30\!\cdots\!00$$$$T^{2} + 4563895793294313000 T^{3} + T^{4}$$
$7$ $$32\!\cdots\!96$$$$+$$$$28\!\cdots\!00$$$$T -$$$$12\!\cdots\!28$$$$T^{2} +$$$$22\!\cdots\!00$$$$T^{3} + T^{4}$$
$11$ $$36\!\cdots\!76$$$$+$$$$23\!\cdots\!28$$$$T -$$$$17\!\cdots\!56$$$$T^{2} -$$$$24\!\cdots\!28$$$$T^{3} + T^{4}$$
$13$ $$14\!\cdots\!56$$$$+$$$$27\!\cdots\!40$$$$T -$$$$27\!\cdots\!32$$$$T^{2} -$$$$38\!\cdots\!40$$$$T^{3} + T^{4}$$
$17$ $$-$$$$32\!\cdots\!64$$$$-$$$$22\!\cdots\!40$$$$T -$$$$11\!\cdots\!88$$$$T^{2} +$$$$86\!\cdots\!60$$$$T^{3} + T^{4}$$
$19$ $$-$$$$62\!\cdots\!00$$$$-$$$$80\!\cdots\!00$$$$T +$$$$13\!\cdots\!00$$$$T^{2} +$$$$25\!\cdots\!60$$$$T^{3} + T^{4}$$
$23$ $$24\!\cdots\!36$$$$+$$$$33\!\cdots\!60$$$$T -$$$$86\!\cdots\!12$$$$T^{2} -$$$$69\!\cdots\!60$$$$T^{3} + T^{4}$$
$29$ $$28\!\cdots\!00$$$$+$$$$87\!\cdots\!00$$$$T +$$$$59\!\cdots\!00$$$$T^{2} +$$$$14\!\cdots\!40$$$$T^{3} + T^{4}$$
$31$ $$31\!\cdots\!96$$$$-$$$$31\!\cdots\!52$$$$T -$$$$10\!\cdots\!16$$$$T^{2} +$$$$39\!\cdots\!32$$$$T^{3} + T^{4}$$
$37$ $$39\!\cdots\!16$$$$+$$$$10\!\cdots\!80$$$$T -$$$$16\!\cdots\!08$$$$T^{2} +$$$$28\!\cdots\!80$$$$T^{3} + T^{4}$$
$41$ $$-$$$$72\!\cdots\!44$$$$+$$$$15\!\cdots\!08$$$$T -$$$$51\!\cdots\!96$$$$T^{2} -$$$$81\!\cdots\!88$$$$T^{3} + T^{4}$$
$43$ $$79\!\cdots\!96$$$$-$$$$44\!\cdots\!00$$$$T +$$$$72\!\cdots\!28$$$$T^{2} -$$$$45\!\cdots\!00$$$$T^{3} + T^{4}$$
$47$ $$82\!\cdots\!56$$$$+$$$$14\!\cdots\!40$$$$T +$$$$50\!\cdots\!32$$$$T^{2} +$$$$45\!\cdots\!40$$$$T^{3} + T^{4}$$
$53$ $$-$$$$80\!\cdots\!24$$$$+$$$$12\!\cdots\!20$$$$T -$$$$27\!\cdots\!52$$$$T^{2} -$$$$38\!\cdots\!20$$$$T^{3} + T^{4}$$
$59$ $$-$$$$37\!\cdots\!00$$$$+$$$$14\!\cdots\!00$$$$T -$$$$46\!\cdots\!00$$$$T^{2} -$$$$16\!\cdots\!20$$$$T^{3} + T^{4}$$
$61$ $$18\!\cdots\!76$$$$-$$$$18\!\cdots\!72$$$$T -$$$$87\!\cdots\!56$$$$T^{2} -$$$$65\!\cdots\!28$$$$T^{3} + T^{4}$$
$67$ $$-$$$$18\!\cdots\!64$$$$+$$$$70\!\cdots\!60$$$$T -$$$$75\!\cdots\!88$$$$T^{2} +$$$$11\!\cdots\!60$$$$T^{3} + T^{4}$$
$71$ $$83\!\cdots\!36$$$$-$$$$62\!\cdots\!12$$$$T -$$$$14\!\cdots\!36$$$$T^{2} -$$$$35\!\cdots\!48$$$$T^{3} + T^{4}$$
$73$ $$80\!\cdots\!36$$$$+$$$$20\!\cdots\!60$$$$T +$$$$18\!\cdots\!88$$$$T^{2} +$$$$70\!\cdots\!40$$$$T^{3} + T^{4}$$
$79$ $$39\!\cdots\!00$$$$-$$$$58\!\cdots\!00$$$$T -$$$$49\!\cdots\!00$$$$T^{2} +$$$$15\!\cdots\!40$$$$T^{3} + T^{4}$$
$83$ $$36\!\cdots\!16$$$$+$$$$55\!\cdots\!80$$$$T +$$$$21\!\cdots\!08$$$$T^{2} +$$$$26\!\cdots\!20$$$$T^{3} + T^{4}$$
$89$ $$-$$$$28\!\cdots\!00$$$$+$$$$73\!\cdots\!00$$$$T -$$$$31\!\cdots\!00$$$$T^{2} +$$$$37\!\cdots\!20$$$$T^{3} + T^{4}$$
$97$ $$21\!\cdots\!56$$$$+$$$$47\!\cdots\!40$$$$T -$$$$36\!\cdots\!68$$$$T^{2} -$$$$10\!\cdots\!60$$$$T^{3} + T^{4}$$