# Properties

 Label 1.62.a.a Level 1 Weight 62 Character orbit 1.a Self dual yes Analytic conductor 23.566 Analytic rank 1 Dimension 4 CM no Inner twists 1

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$1$$ Weight: $$k$$ $$=$$ $$62$$ Character orbit: $$[\chi]$$ $$=$$ 1.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$23.5656183265$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - x^{3} - 180363795469121 x^{2} + 166129321978984507920 x + 2785609847439483545242446300$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{28}\cdot 3^{8}\cdot 5^{2}\cdot 7$$ 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.

 $$f(q)$$ $$=$$ $$q +(286578000 - \beta_{1}) q^{2} +(-143430899755500 + 48580 \beta_{1} + \beta_{2}) q^{3} +(1100749418957151232 - 838426474 \beta_{1} + 343 \beta_{2} + \beta_{3}) q^{4} +(-$$$$13\!\cdots\!50$$$$+ 100252966240 \beta_{1} - 48540 \beta_{2} - 120 \beta_{3}) q^{5} +(-$$$$20\!\cdots\!88$$$$+ 157027925895036 \beta_{1} - 109667352 \beta_{2} - 189864 \beta_{3}) q^{6} +(-$$$$15\!\cdots\!00$$$$+ 14513866966047176 \beta_{1} - 46993043006 \beta_{2} - 19628000 \beta_{3}) q^{7} +($$$$24\!\cdots\!00$$$$- 1373007096556182656 \beta_{1} - 7496719202112 \beta_{2} + 1190856000 \beta_{3}) q^{8} +($$$$29\!\cdots\!13$$$$+ 1326939277665425472 \beta_{1} - 362685230082504 \beta_{2} - 19284989328 \beta_{3}) q^{9} +O(q^{10})$$ $$q +(286578000 - \beta_{1}) q^{2} +(-143430899755500 + 48580 \beta_{1} + \beta_{2}) q^{3} +(1100749418957151232 - 838426474 \beta_{1} + 343 \beta_{2} + \beta_{3}) q^{4} +(-$$$$13\!\cdots\!50$$$$+ 100252966240 \beta_{1} - 48540 \beta_{2} - 120 \beta_{3}) q^{5} +(-$$$$20\!\cdots\!88$$$$+ 157027925895036 \beta_{1} - 109667352 \beta_{2} - 189864 \beta_{3}) q^{6} +(-$$$$15\!\cdots\!00$$$$+ 14513866966047176 \beta_{1} - 46993043006 \beta_{2} - 19628000 \beta_{3}) q^{7} +($$$$24\!\cdots\!00$$$$- 1373007096556182656 \beta_{1} - 7496719202112 \beta_{2} + 1190856000 \beta_{3}) q^{8} +($$$$29\!\cdots\!13$$$$+ 1326939277665425472 \beta_{1} - 362685230082504 \beta_{2} - 19284989328 \beta_{3}) q^{9} +(-$$$$37\!\cdots\!00$$$$+$$$$44\!\cdots\!30$$$$\beta_{1} + 900415543063520 \beta_{2} - 141501833440 \beta_{3}) q^{10} +(-$$$$10\!\cdots\!88$$$$+$$$$12\!\cdots\!80$$$$\beta_{1} + 130667557581888915 \beta_{2} + 11035029161280 \beta_{3}) q^{11} +(-$$$$24\!\cdots\!00$$$$+$$$$57\!\cdots\!92$$$$\beta_{1} - 877602562285349684 \beta_{2} - 217648250923500 \beta_{3}) q^{12} +($$$$26\!\cdots\!50$$$$+$$$$39\!\cdots\!32$$$$\beta_{1} - 12482522028380172860 \beta_{2} + 2435574015901000 \beta_{3}) q^{13} +(-$$$$52\!\cdots\!76$$$$+$$$$65\!\cdots\!28$$$$\beta_{1} +$$$$15\!\cdots\!04$$$$\beta_{2} - 15743169581051472 \beta_{3}) q^{14} +($$$$28\!\cdots\!00$$$$-$$$$69\!\cdots\!60$$$$\beta_{1} +$$$$10\!\cdots\!10$$$$\beta_{2} + 26260172144333280 \beta_{3}) q^{15} +($$$$27\!\cdots\!56$$$$-$$$$36\!\cdots\!88$$$$\beta_{1} -$$$$88\!\cdots\!84$$$$\beta_{2} + 603732751100608512 \beta_{3}) q^{16} +(-$$$$10\!\cdots\!50$$$$-$$$$38\!\cdots\!60$$$$\beta_{1} +$$$$33\!\cdots\!52$$$$\beta_{2} - 7263830520990282000 \beta_{3}) q^{17} +($$$$39\!\cdots\!00$$$$+$$$$17\!\cdots\!15$$$$\beta_{1} +$$$$18\!\cdots\!36$$$$\beta_{2} + 42183522593278488000 \beta_{3}) q^{18} +(-$$$$89\!\cdots\!80$$$$+$$$$48\!\cdots\!64$$$$\beta_{1} -$$$$16\!\cdots\!23$$$$\beta_{2} -$$$$10\!\cdots\!36$$$$\beta_{3}) q^{19} +(-$$$$12\!\cdots\!00$$$$+$$$$66\!\cdots\!80$$$$\beta_{1} +$$$$97\!\cdots\!70$$$$\beta_{2} -$$$$34\!\cdots\!90$$$$\beta_{3}) q^{20} +(-$$$$13\!\cdots\!68$$$$-$$$$11\!\cdots\!48$$$$\beta_{1} +$$$$31\!\cdots\!36$$$$\beta_{2} +$$$$50\!\cdots\!52$$$$\beta_{3}) q^{21} +(-$$$$43\!\cdots\!00$$$$-$$$$79\!\cdots\!92$$$$\beta_{1} -$$$$10\!\cdots\!60$$$$\beta_{2} -$$$$26\!\cdots\!00$$$$\beta_{3}) q^{22} +(-$$$$10\!\cdots\!00$$$$+$$$$24\!\cdots\!24$$$$\beta_{1} -$$$$24\!\cdots\!66$$$$\beta_{2} +$$$$75\!\cdots\!00$$$$\beta_{3}) q^{23} +(-$$$$15\!\cdots\!40$$$$+$$$$67\!\cdots\!92$$$$\beta_{1} +$$$$18\!\cdots\!56$$$$\beta_{2} -$$$$10\!\cdots\!08$$$$\beta_{3}) q^{24} +(-$$$$41\!\cdots\!25$$$$-$$$$79\!\cdots\!00$$$$\beta_{1} -$$$$11\!\cdots\!00$$$$\beta_{2} +$$$$41\!\cdots\!00$$$$\beta_{3}) q^{25} +(-$$$$12\!\cdots\!08$$$$-$$$$54\!\cdots\!42$$$$\beta_{1} -$$$$19\!\cdots\!56$$$$\beta_{2} -$$$$11\!\cdots\!92$$$$\beta_{3}) q^{26} +(-$$$$32\!\cdots\!00$$$$-$$$$18\!\cdots\!96$$$$\beta_{1} +$$$$19\!\cdots\!58$$$$\beta_{2} +$$$$11\!\cdots\!00$$$$\beta_{3}) q^{27} +(-$$$$19\!\cdots\!00$$$$+$$$$86\!\cdots\!96$$$$\beta_{1} +$$$$19\!\cdots\!76$$$$\beta_{2} -$$$$48\!\cdots\!00$$$$\beta_{3}) q^{28} +($$$$16\!\cdots\!30$$$$+$$$$32\!\cdots\!76$$$$\beta_{1} -$$$$54\!\cdots\!32$$$$\beta_{2} +$$$$93\!\cdots\!76$$$$\beta_{3}) q^{29} +($$$$23\!\cdots\!00$$$$-$$$$12\!\cdots\!20$$$$\beta_{1} -$$$$18\!\cdots\!80$$$$\beta_{2} +$$$$64\!\cdots\!60$$$$\beta_{3}) q^{30} +($$$$81\!\cdots\!32$$$$-$$$$32\!\cdots\!60$$$$\beta_{1} +$$$$47\!\cdots\!20$$$$\beta_{2} -$$$$92\!\cdots\!60$$$$\beta_{3}) q^{31} +($$$$73\!\cdots\!00$$$$-$$$$27\!\cdots\!56$$$$\beta_{1} +$$$$14\!\cdots\!80$$$$\beta_{2} +$$$$24\!\cdots\!00$$$$\beta_{3}) q^{32} +($$$$20\!\cdots\!00$$$$+$$$$41\!\cdots\!20$$$$\beta_{1} -$$$$54\!\cdots\!68$$$$\beta_{2} -$$$$19\!\cdots\!00$$$$\beta_{3}) q^{33} +($$$$99\!\cdots\!04$$$$+$$$$22\!\cdots\!42$$$$\beta_{1} +$$$$54\!\cdots\!56$$$$\beta_{2} -$$$$38\!\cdots\!08$$$$\beta_{3}) q^{34} +($$$$23\!\cdots\!00$$$$-$$$$10\!\cdots\!20$$$$\beta_{1} -$$$$23\!\cdots\!80$$$$\beta_{2} +$$$$57\!\cdots\!60$$$$\beta_{3}) q^{35} +(-$$$$12\!\cdots\!84$$$$-$$$$89\!\cdots\!58$$$$\beta_{1} +$$$$48\!\cdots\!31$$$$\beta_{2} +$$$$18\!\cdots\!17$$$$\beta_{3}) q^{36} +(-$$$$17\!\cdots\!50$$$$-$$$$22\!\cdots\!72$$$$\beta_{1} -$$$$16\!\cdots\!12$$$$\beta_{2} -$$$$11\!\cdots\!00$$$$\beta_{3}) q^{37} +(-$$$$16\!\cdots\!00$$$$+$$$$61\!\cdots\!16$$$$\beta_{1} +$$$$83\!\cdots\!32$$$$\beta_{2} -$$$$29\!\cdots\!00$$$$\beta_{3}) q^{38} +(-$$$$13\!\cdots\!44$$$$+$$$$35\!\cdots\!28$$$$\beta_{1} +$$$$34\!\cdots\!54$$$$\beta_{2} +$$$$98\!\cdots\!28$$$$\beta_{3}) q^{39} +(-$$$$17\!\cdots\!00$$$$+$$$$13\!\cdots\!00$$$$\beta_{1} +$$$$29\!\cdots\!00$$$$\beta_{2} -$$$$62\!\cdots\!00$$$$\beta_{3}) q^{40} +($$$$40\!\cdots\!42$$$$-$$$$27\!\cdots\!60$$$$\beta_{1} -$$$$10\!\cdots\!80$$$$\beta_{2} -$$$$34\!\cdots\!60$$$$\beta_{3}) q^{41} +($$$$37\!\cdots\!00$$$$-$$$$15\!\cdots\!84$$$$\beta_{1} -$$$$38\!\cdots\!24$$$$\beta_{2} +$$$$89\!\cdots\!00$$$$\beta_{3}) q^{42} +($$$$18\!\cdots\!00$$$$+$$$$70\!\cdots\!68$$$$\beta_{1} +$$$$15\!\cdots\!67$$$$\beta_{2} -$$$$33\!\cdots\!00$$$$\beta_{3}) q^{43} +($$$$38\!\cdots\!84$$$$+$$$$67\!\cdots\!72$$$$\beta_{1} -$$$$86\!\cdots\!04$$$$\beta_{2} -$$$$13\!\cdots\!28$$$$\beta_{3}) q^{44} +($$$$15\!\cdots\!50$$$$+$$$$10\!\cdots\!20$$$$\beta_{1} -$$$$59\!\cdots\!20$$$$\beta_{2} -$$$$23\!\cdots\!60$$$$\beta_{3}) q^{45} +(-$$$$37\!\cdots\!28$$$$-$$$$52\!\cdots\!80$$$$\beta_{1} -$$$$56\!\cdots\!40$$$$\beta_{2} +$$$$62\!\cdots\!20$$$$\beta_{3}) q^{46} +(-$$$$54\!\cdots\!00$$$$-$$$$53\!\cdots\!48$$$$\beta_{1} +$$$$90\!\cdots\!16$$$$\beta_{2} +$$$$28\!\cdots\!00$$$$\beta_{3}) q^{47} +(-$$$$21\!\cdots\!00$$$$+$$$$76\!\cdots\!64$$$$\beta_{1} +$$$$24\!\cdots\!64$$$$\beta_{2} -$$$$47\!\cdots\!00$$$$\beta_{3}) q^{48} +($$$$38\!\cdots\!57$$$$-$$$$12\!\cdots\!80$$$$\beta_{1} -$$$$50\!\cdots\!40$$$$\beta_{2} +$$$$74\!\cdots\!20$$$$\beta_{3}) q^{49} +(-$$$$93\!\cdots\!00$$$$+$$$$40\!\cdots\!25$$$$\beta_{1} -$$$$28\!\cdots\!00$$$$\beta_{2} +$$$$11\!\cdots\!00$$$$\beta_{3}) q^{50} +($$$$51\!\cdots\!72$$$$-$$$$23\!\cdots\!36$$$$\beta_{1} -$$$$84\!\cdots\!98$$$$\beta_{2} -$$$$13\!\cdots\!36$$$$\beta_{3}) q^{51} +($$$$85\!\cdots\!00$$$$+$$$$29\!\cdots\!36$$$$\beta_{1} +$$$$41\!\cdots\!74$$$$\beta_{2} +$$$$20\!\cdots\!50$$$$\beta_{3}) q^{52} +($$$$21\!\cdots\!50$$$$-$$$$18\!\cdots\!80$$$$\beta_{1} -$$$$98\!\cdots\!44$$$$\beta_{2} -$$$$81\!\cdots\!00$$$$\beta_{3}) q^{53} +($$$$51\!\cdots\!20$$$$-$$$$25\!\cdots\!76$$$$\beta_{1} -$$$$84\!\cdots\!68$$$$\beta_{2} +$$$$20\!\cdots\!24$$$$\beta_{3}) q^{54} +(-$$$$47\!\cdots\!00$$$$-$$$$80\!\cdots\!20$$$$\beta_{1} +$$$$10\!\cdots\!70$$$$\beta_{2} +$$$$16\!\cdots\!60$$$$\beta_{3}) q^{55} +(-$$$$22\!\cdots\!80$$$$+$$$$19\!\cdots\!64$$$$\beta_{1} -$$$$22\!\cdots\!48$$$$\beta_{2} -$$$$97\!\cdots\!36$$$$\beta_{3}) q^{56} +(-$$$$12\!\cdots\!00$$$$-$$$$14\!\cdots\!52$$$$\beta_{1} +$$$$67\!\cdots\!96$$$$\beta_{2} +$$$$14\!\cdots\!00$$$$\beta_{3}) q^{57} +(-$$$$10\!\cdots\!00$$$$-$$$$17\!\cdots\!06$$$$\beta_{1} -$$$$68\!\cdots\!12$$$$\beta_{2} +$$$$82\!\cdots\!00$$$$\beta_{3}) q^{58} +(-$$$$53\!\cdots\!40$$$$-$$$$31\!\cdots\!88$$$$\beta_{1} -$$$$66\!\cdots\!09$$$$\beta_{2} -$$$$38\!\cdots\!88$$$$\beta_{3}) q^{59} +($$$$41\!\cdots\!00$$$$-$$$$28\!\cdots\!20$$$$\beta_{1} -$$$$69\!\cdots\!80$$$$\beta_{2} +$$$$11\!\cdots\!60$$$$\beta_{3}) q^{60} +($$$$10\!\cdots\!62$$$$+$$$$63\!\cdots\!00$$$$\beta_{1} +$$$$25\!\cdots\!00$$$$\beta_{2} +$$$$50\!\cdots\!00$$$$\beta_{3}) q^{61} +($$$$12\!\cdots\!00$$$$+$$$$94\!\cdots\!28$$$$\beta_{1} +$$$$72\!\cdots\!20$$$$\beta_{2} -$$$$11\!\cdots\!00$$$$\beta_{3}) q^{62} +($$$$44\!\cdots\!00$$$$+$$$$22\!\cdots\!96$$$$\beta_{1} -$$$$15\!\cdots\!82$$$$\beta_{2} -$$$$82\!\cdots\!00$$$$\beta_{3}) q^{63} +($$$$49\!\cdots\!52$$$$-$$$$57\!\cdots\!88$$$$\beta_{1} +$$$$11\!\cdots\!16$$$$\beta_{2} +$$$$25\!\cdots\!12$$$$\beta_{3}) q^{64} +(-$$$$10\!\cdots\!00$$$$-$$$$36\!\cdots\!60$$$$\beta_{1} -$$$$50\!\cdots\!40$$$$\beta_{2} -$$$$25\!\cdots\!20$$$$\beta_{3}) q^{65} +(-$$$$81\!\cdots\!56$$$$-$$$$12\!\cdots\!08$$$$\beta_{1} +$$$$18\!\cdots\!56$$$$\beta_{2} +$$$$26\!\cdots\!92$$$$\beta_{3}) q^{66} +(-$$$$37\!\cdots\!00$$$$+$$$$19\!\cdots\!60$$$$\beta_{1} +$$$$59\!\cdots\!17$$$$\beta_{2} +$$$$26\!\cdots\!00$$$$\beta_{3}) q^{67} +(-$$$$49\!\cdots\!00$$$$+$$$$18\!\cdots\!24$$$$\beta_{1} -$$$$61\!\cdots\!58$$$$\beta_{2} -$$$$15\!\cdots\!50$$$$\beta_{3}) q^{68} +(-$$$$16\!\cdots\!04$$$$+$$$$21\!\cdots\!04$$$$\beta_{1} -$$$$16\!\cdots\!28$$$$\beta_{2} +$$$$37\!\cdots\!04$$$$\beta_{3}) q^{69} +($$$$41\!\cdots\!00$$$$-$$$$41\!\cdots\!40$$$$\beta_{1} -$$$$38\!\cdots\!60$$$$\beta_{2} +$$$$15\!\cdots\!20$$$$\beta_{3}) q^{70} +($$$$66\!\cdots\!72$$$$-$$$$92\!\cdots\!00$$$$\beta_{1} +$$$$71\!\cdots\!50$$$$\beta_{2} +$$$$40\!\cdots\!00$$$$\beta_{3}) q^{71} +($$$$25\!\cdots\!00$$$$-$$$$11\!\cdots\!88$$$$\beta_{1} -$$$$57\!\cdots\!76$$$$\beta_{2} -$$$$69\!\cdots\!00$$$$\beta_{3}) q^{72} +($$$$10\!\cdots\!50$$$$+$$$$79\!\cdots\!44$$$$\beta_{1} +$$$$17\!\cdots\!24$$$$\beta_{2} -$$$$19\!\cdots\!00$$$$\beta_{3}) q^{73} +($$$$68\!\cdots\!64$$$$+$$$$10\!\cdots\!30$$$$\beta_{1} +$$$$32\!\cdots\!40$$$$\beta_{2} +$$$$45\!\cdots\!80$$$$\beta_{3}) q^{74} +($$$$57\!\cdots\!00$$$$-$$$$17\!\cdots\!00$$$$\beta_{1} -$$$$42\!\cdots\!25$$$$\beta_{2} -$$$$13\!\cdots\!00$$$$\beta_{3}) q^{75} +(-$$$$22\!\cdots\!60$$$$+$$$$14\!\cdots\!68$$$$\beta_{1} +$$$$58\!\cdots\!24$$$$\beta_{2} -$$$$59\!\cdots\!32$$$$\beta_{3}) q^{76} +(-$$$$15\!\cdots\!00$$$$-$$$$14\!\cdots\!68$$$$\beta_{1} +$$$$38\!\cdots\!68$$$$\beta_{2} +$$$$34\!\cdots\!00$$$$\beta_{3}) q^{77} +(-$$$$15\!\cdots\!00$$$$-$$$$58\!\cdots\!84$$$$\beta_{1} -$$$$80\!\cdots\!36$$$$\beta_{2} -$$$$44\!\cdots\!00$$$$\beta_{3}) q^{78} +(-$$$$54\!\cdots\!20$$$$-$$$$32\!\cdots\!24$$$$\beta_{1} -$$$$14\!\cdots\!32$$$$\beta_{2} +$$$$16\!\cdots\!76$$$$\beta_{3}) q^{79} +(-$$$$20\!\cdots\!00$$$$+$$$$22\!\cdots\!40$$$$\beta_{1} +$$$$20\!\cdots\!60$$$$\beta_{2} -$$$$83\!\cdots\!20$$$$\beta_{3}) q^{80} +($$$$41\!\cdots\!21$$$$+$$$$57\!\cdots\!56$$$$\beta_{1} -$$$$15\!\cdots\!92$$$$\beta_{2} -$$$$21\!\cdots\!44$$$$\beta_{3}) q^{81} +($$$$10\!\cdots\!00$$$$+$$$$18\!\cdots\!18$$$$\beta_{1} +$$$$29\!\cdots\!20$$$$\beta_{2} +$$$$29\!\cdots\!00$$$$\beta_{3}) q^{82} +(-$$$$48\!\cdots\!00$$$$+$$$$14\!\cdots\!56$$$$\beta_{1} +$$$$64\!\cdots\!33$$$$\beta_{2} -$$$$66\!\cdots\!00$$$$\beta_{3}) q^{83} +($$$$67\!\cdots\!24$$$$-$$$$38\!\cdots\!04$$$$\beta_{1} -$$$$13\!\cdots\!72$$$$\beta_{2} +$$$$13\!\cdots\!96$$$$\beta_{3}) q^{84} +($$$$59\!\cdots\!00$$$$-$$$$22\!\cdots\!20$$$$\beta_{1} +$$$$74\!\cdots\!20$$$$\beta_{2} +$$$$18\!\cdots\!60$$$$\beta_{3}) q^{85} +(-$$$$17\!\cdots\!68$$$$-$$$$99\!\cdots\!96$$$$\beta_{1} +$$$$95\!\cdots\!72$$$$\beta_{2} -$$$$29\!\cdots\!96$$$$\beta_{3}) q^{86} +(-$$$$64\!\cdots\!00$$$$+$$$$24\!\cdots\!32$$$$\beta_{1} -$$$$16\!\cdots\!86$$$$\beta_{2} +$$$$17\!\cdots\!00$$$$\beta_{3}) q^{87} +(-$$$$11\!\cdots\!00$$$$+$$$$47\!\cdots\!28$$$$\beta_{1} +$$$$32\!\cdots\!56$$$$\beta_{2} -$$$$24\!\cdots\!00$$$$\beta_{3}) q^{88} +(-$$$$15\!\cdots\!10$$$$+$$$$39\!\cdots\!28$$$$\beta_{1} -$$$$20\!\cdots\!96$$$$\beta_{2} +$$$$35\!\cdots\!28$$$$\beta_{3}) q^{89} +(-$$$$31\!\cdots\!00$$$$-$$$$35\!\cdots\!10$$$$\beta_{1} +$$$$19\!\cdots\!60$$$$\beta_{2} -$$$$32\!\cdots\!20$$$$\beta_{3}) q^{90} +(-$$$$11\!\cdots\!88$$$$-$$$$72\!\cdots\!16$$$$\beta_{1} -$$$$95\!\cdots\!88$$$$\beta_{2} -$$$$80\!\cdots\!16$$$$\beta_{3}) q^{91} +($$$$40\!\cdots\!00$$$$-$$$$12\!\cdots\!40$$$$\beta_{1} +$$$$15\!\cdots\!92$$$$\beta_{2} -$$$$16\!\cdots\!00$$$$\beta_{3}) q^{92} +(-$$$$10\!\cdots\!00$$$$-$$$$27\!\cdots\!60$$$$\beta_{1} +$$$$20\!\cdots\!92$$$$\beta_{2} -$$$$29\!\cdots\!00$$$$\beta_{3}) q^{93} +($$$$16\!\cdots\!44$$$$+$$$$17\!\cdots\!04$$$$\beta_{1} -$$$$12\!\cdots\!28$$$$\beta_{2} +$$$$41\!\cdots\!04$$$$\beta_{3}) q^{94} +($$$$27\!\cdots\!00$$$$-$$$$17\!\cdots\!00$$$$\beta_{1} -$$$$70\!\cdots\!50$$$$\beta_{2} +$$$$70\!\cdots\!00$$$$\beta_{3}) q^{95} +($$$$37\!\cdots\!52$$$$+$$$$19\!\cdots\!76$$$$\beta_{1} -$$$$10\!\cdots\!32$$$$\beta_{2} -$$$$10\!\cdots\!24$$$$\beta_{3}) q^{96} +(-$$$$20\!\cdots\!50$$$$+$$$$47\!\cdots\!52$$$$\beta_{1} -$$$$30\!\cdots\!84$$$$\beta_{2} -$$$$42\!\cdots\!00$$$$\beta_{3}) q^{97} +($$$$42\!\cdots\!00$$$$-$$$$25\!\cdots\!77$$$$\beta_{1} -$$$$48\!\cdots\!40$$$$\beta_{2} +$$$$22\!\cdots\!00$$$$\beta_{3}) q^{98} +(-$$$$78\!\cdots\!44$$$$-$$$$27\!\cdots\!96$$$$\beta_{1} +$$$$21\!\cdots\!47$$$$\beta_{2} +$$$$10\!\cdots\!04$$$$\beta_{3}) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 1146312000q^{2} - 573723599022000q^{3} + 4402997675828604928q^{4} -$$$$52\!\cdots\!00$$$$q^{5} -$$$$81\!\cdots\!52$$$$q^{6} -$$$$63\!\cdots\!00$$$$q^{7} +$$$$97\!\cdots\!00$$$$q^{8} +$$$$11\!\cdots\!52$$$$q^{9} + O(q^{10})$$ $$4q + 1146312000q^{2} - 573723599022000q^{3} + 4402997675828604928q^{4} -$$$$52\!\cdots\!00$$$$q^{5} -$$$$81\!\cdots\!52$$$$q^{6} -$$$$63\!\cdots\!00$$$$q^{7} +$$$$97\!\cdots\!00$$$$q^{8} +$$$$11\!\cdots\!52$$$$q^{9} -$$$$14\!\cdots\!00$$$$q^{10} -$$$$41\!\cdots\!52$$$$q^{11} -$$$$99\!\cdots\!00$$$$q^{12} +$$$$10\!\cdots\!00$$$$q^{13} -$$$$21\!\cdots\!04$$$$q^{14} +$$$$11\!\cdots\!00$$$$q^{15} +$$$$10\!\cdots\!24$$$$q^{16} -$$$$40\!\cdots\!00$$$$q^{17} +$$$$15\!\cdots\!00$$$$q^{18} -$$$$35\!\cdots\!20$$$$q^{19} -$$$$50\!\cdots\!00$$$$q^{20} -$$$$55\!\cdots\!72$$$$q^{21} -$$$$17\!\cdots\!00$$$$q^{22} -$$$$41\!\cdots\!00$$$$q^{23} -$$$$61\!\cdots\!60$$$$q^{24} -$$$$16\!\cdots\!00$$$$q^{25} -$$$$49\!\cdots\!32$$$$q^{26} -$$$$12\!\cdots\!00$$$$q^{27} -$$$$79\!\cdots\!00$$$$q^{28} +$$$$66\!\cdots\!20$$$$q^{29} +$$$$95\!\cdots\!00$$$$q^{30} +$$$$32\!\cdots\!28$$$$q^{31} +$$$$29\!\cdots\!00$$$$q^{32} +$$$$80\!\cdots\!00$$$$q^{33} +$$$$39\!\cdots\!16$$$$q^{34} +$$$$94\!\cdots\!00$$$$q^{35} -$$$$49\!\cdots\!36$$$$q^{36} -$$$$71\!\cdots\!00$$$$q^{37} -$$$$65\!\cdots\!00$$$$q^{38} -$$$$53\!\cdots\!76$$$$q^{39} -$$$$69\!\cdots\!00$$$$q^{40} +$$$$16\!\cdots\!68$$$$q^{41} +$$$$15\!\cdots\!00$$$$q^{42} +$$$$75\!\cdots\!00$$$$q^{43} +$$$$15\!\cdots\!36$$$$q^{44} +$$$$60\!\cdots\!00$$$$q^{45} -$$$$15\!\cdots\!12$$$$q^{46} -$$$$21\!\cdots\!00$$$$q^{47} -$$$$84\!\cdots\!00$$$$q^{48} +$$$$15\!\cdots\!28$$$$q^{49} -$$$$37\!\cdots\!00$$$$q^{50} +$$$$20\!\cdots\!88$$$$q^{51} +$$$$34\!\cdots\!00$$$$q^{52} +$$$$84\!\cdots\!00$$$$q^{53} +$$$$20\!\cdots\!80$$$$q^{54} -$$$$18\!\cdots\!00$$$$q^{55} -$$$$89\!\cdots\!20$$$$q^{56} -$$$$48\!\cdots\!00$$$$q^{57} -$$$$42\!\cdots\!00$$$$q^{58} -$$$$21\!\cdots\!60$$$$q^{59} +$$$$16\!\cdots\!00$$$$q^{60} +$$$$42\!\cdots\!48$$$$q^{61} +$$$$51\!\cdots\!00$$$$q^{62} +$$$$17\!\cdots\!00$$$$q^{63} +$$$$19\!\cdots\!08$$$$q^{64} -$$$$40\!\cdots\!00$$$$q^{65} -$$$$32\!\cdots\!24$$$$q^{66} -$$$$15\!\cdots\!00$$$$q^{67} -$$$$19\!\cdots\!00$$$$q^{68} -$$$$65\!\cdots\!16$$$$q^{69} +$$$$16\!\cdots\!00$$$$q^{70} +$$$$26\!\cdots\!88$$$$q^{71} +$$$$10\!\cdots\!00$$$$q^{72} +$$$$43\!\cdots\!00$$$$q^{73} +$$$$27\!\cdots\!56$$$$q^{74} +$$$$22\!\cdots\!00$$$$q^{75} -$$$$91\!\cdots\!40$$$$q^{76} -$$$$62\!\cdots\!00$$$$q^{77} -$$$$62\!\cdots\!00$$$$q^{78} -$$$$21\!\cdots\!80$$$$q^{79} -$$$$81\!\cdots\!00$$$$q^{80} +$$$$16\!\cdots\!84$$$$q^{81} +$$$$41\!\cdots\!00$$$$q^{82} -$$$$19\!\cdots\!00$$$$q^{83} +$$$$26\!\cdots\!96$$$$q^{84} +$$$$23\!\cdots\!00$$$$q^{85} -$$$$71\!\cdots\!72$$$$q^{86} -$$$$25\!\cdots\!00$$$$q^{87} -$$$$45\!\cdots\!00$$$$q^{88} -$$$$60\!\cdots\!40$$$$q^{89} -$$$$12\!\cdots\!00$$$$q^{90} -$$$$45\!\cdots\!52$$$$q^{91} +$$$$16\!\cdots\!00$$$$q^{92} -$$$$43\!\cdots\!00$$$$q^{93} +$$$$64\!\cdots\!76$$$$q^{94} +$$$$11\!\cdots\!00$$$$q^{95} +$$$$15\!\cdots\!08$$$$q^{96} -$$$$80\!\cdots\!00$$$$q^{97} +$$$$17\!\cdots\!00$$$$q^{98} -$$$$31\!\cdots\!76$$$$q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 180363795469121 x^{2} + 166129321978984507920 x + 2785609847439483545242446300$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$192 \nu - 48$$ $$\beta_{2}$$ $$=$$ $$($$$$12 \nu^{3} + 20695116 \nu^{2} - 1765108515928884 \nu - 371162118903041598888$$$$)/13402613$$ $$\beta_{3}$$ $$=$$ $$($$$$-588 \nu^{3} + 69567928692 \nu^{2} + 184007562081366900 \nu - 6347030828361347486393976$$$$)/1914659$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1} + 48$$$$)/192$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 343 \beta_{2} - 265270378 \beta_{1} + 3324465478086847488$$$$)/36864$$ $$\nu^{3}$$ $$=$$ $$($$$$-1724593 \beta_{3} + 40581291737 \beta_{2} + 28699219691868298 \beta_{1} - 4593138507376746544705536$$$$)/36864$$

## 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
 1.21599e7 4.76281e6 −3.61406e6 −1.33086e7
−2.04812e9 1.78996e14 1.88894e18 −2.27994e20 −3.66606e23 −4.42559e25 8.53860e26 −9.51338e28 4.66958e29
1.2 −6.27881e8 −6.22194e14 −1.91161e18 2.33985e20 3.90664e23 6.25792e25 2.64806e27 2.59952e29 −1.46915e29
1.3 9.80478e8 2.49037e14 −1.34451e18 1.59503e20 2.44176e23 1.63507e25 −3.57909e27 −6.51539e28 1.56389e29
1.4 2.84183e9 −3.79563e14 5.77017e18 −6.89620e20 −1.07865e24 −9.80127e25 9.84504e27 1.68945e28 −1.95979e30
 $$n$$: 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 1.62.a.a 4
3.b odd 2 1 9.62.a.a 4

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{62}^{\mathrm{new}}(\Gamma_0(1))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 1146312000 T + 3067202781185085440 T^{2} -$$$$53\!\cdots\!00$$$$T^{3} +$$$$70\!\cdots\!08$$$$T^{4} -$$$$12\!\cdots\!00$$$$T^{5} +$$$$16\!\cdots\!60$$$$T^{6} -$$$$14\!\cdots\!00$$$$T^{7} +$$$$28\!\cdots\!16$$$$T^{8}$$
$3$ $$1 + 573723599022000 T +$$$$36\!\cdots\!80$$$$T^{2} +$$$$16\!\cdots\!00$$$$T^{3} +$$$$69\!\cdots\!18$$$$T^{4} +$$$$20\!\cdots\!00$$$$T^{5} +$$$$58\!\cdots\!20$$$$T^{6} +$$$$11\!\cdots\!00$$$$T^{7} +$$$$26\!\cdots\!81$$$$T^{8}$$
$5$ $$1 +$$$$52\!\cdots\!00$$$$T +$$$$17\!\cdots\!00$$$$T^{2} +$$$$67\!\cdots\!00$$$$T^{3} +$$$$11\!\cdots\!50$$$$T^{4} +$$$$29\!\cdots\!00$$$$T^{5} +$$$$32\!\cdots\!00$$$$T^{6} +$$$$42\!\cdots\!00$$$$T^{7} +$$$$35\!\cdots\!25$$$$T^{8}$$
$7$ $$1 +$$$$63\!\cdots\!00$$$$T +$$$$83\!\cdots\!00$$$$T^{2} +$$$$47\!\cdots\!00$$$$T^{3} +$$$$38\!\cdots\!98$$$$T^{4} +$$$$17\!\cdots\!00$$$$T^{5} +$$$$10\!\cdots\!00$$$$T^{6} +$$$$28\!\cdots\!00$$$$T^{7} +$$$$15\!\cdots\!01$$$$T^{8}$$
$11$ $$1 +$$$$41\!\cdots\!52$$$$T +$$$$70\!\cdots\!08$$$$T^{2} +$$$$24\!\cdots\!04$$$$T^{3} +$$$$24\!\cdots\!70$$$$T^{4} +$$$$81\!\cdots\!44$$$$T^{5} +$$$$78\!\cdots\!68$$$$T^{6} +$$$$15\!\cdots\!12$$$$T^{7} +$$$$12\!\cdots\!41$$$$T^{8}$$
$13$ $$1 -$$$$10\!\cdots\!00$$$$T +$$$$17\!\cdots\!60$$$$T^{2} -$$$$12\!\cdots\!00$$$$T^{3} +$$$$14\!\cdots\!38$$$$T^{4} -$$$$10\!\cdots\!00$$$$T^{5} +$$$$13\!\cdots\!40$$$$T^{6} -$$$$76\!\cdots\!00$$$$T^{7} +$$$$63\!\cdots\!61$$$$T^{8}$$
$17$ $$1 +$$$$40\!\cdots\!00$$$$T +$$$$40\!\cdots\!20$$$$T^{2} +$$$$10\!\cdots\!00$$$$T^{3} +$$$$63\!\cdots\!78$$$$T^{4} +$$$$11\!\cdots\!00$$$$T^{5} +$$$$52\!\cdots\!80$$$$T^{6} +$$$$60\!\cdots\!00$$$$T^{7} +$$$$16\!\cdots\!21$$$$T^{8}$$
$19$ $$1 +$$$$35\!\cdots\!20$$$$T +$$$$16\!\cdots\!76$$$$T^{2} +$$$$29\!\cdots\!40$$$$T^{3} +$$$$19\!\cdots\!66$$$$T^{4} +$$$$29\!\cdots\!60$$$$T^{5} +$$$$16\!\cdots\!36$$$$T^{6} +$$$$36\!\cdots\!80$$$$T^{7} +$$$$10\!\cdots\!21$$$$T^{8}$$
$23$ $$1 +$$$$41\!\cdots\!00$$$$T +$$$$43\!\cdots\!40$$$$T^{2} +$$$$13\!\cdots\!00$$$$T^{3} +$$$$73\!\cdots\!58$$$$T^{4} +$$$$15\!\cdots\!00$$$$T^{5} +$$$$58\!\cdots\!60$$$$T^{6} +$$$$64\!\cdots\!00$$$$T^{7} +$$$$18\!\cdots\!41$$$$T^{8}$$
$29$ $$1 -$$$$66\!\cdots\!20$$$$T +$$$$43\!\cdots\!16$$$$T^{2} +$$$$46\!\cdots\!60$$$$T^{3} +$$$$85\!\cdots\!46$$$$T^{4} +$$$$74\!\cdots\!40$$$$T^{5} +$$$$11\!\cdots\!56$$$$T^{6} -$$$$27\!\cdots\!80$$$$T^{7} +$$$$66\!\cdots\!81$$$$T^{8}$$
$31$ $$1 -$$$$32\!\cdots\!28$$$$T +$$$$28\!\cdots\!68$$$$T^{2} -$$$$75\!\cdots\!76$$$$T^{3} +$$$$38\!\cdots\!70$$$$T^{4} -$$$$70\!\cdots\!56$$$$T^{5} +$$$$25\!\cdots\!48$$$$T^{6} -$$$$27\!\cdots\!48$$$$T^{7} +$$$$78\!\cdots\!21$$$$T^{8}$$
$37$ $$1 +$$$$71\!\cdots\!00$$$$T +$$$$98\!\cdots\!60$$$$T^{2} +$$$$68\!\cdots\!00$$$$T^{3} +$$$$69\!\cdots\!38$$$$T^{4} +$$$$31\!\cdots\!00$$$$T^{5} +$$$$20\!\cdots\!40$$$$T^{6} +$$$$67\!\cdots\!00$$$$T^{7} +$$$$43\!\cdots\!61$$$$T^{8}$$
$41$ $$1 -$$$$16\!\cdots\!68$$$$T +$$$$80\!\cdots\!48$$$$T^{2} -$$$$85\!\cdots\!16$$$$T^{3} +$$$$26\!\cdots\!70$$$$T^{4} -$$$$20\!\cdots\!56$$$$T^{5} +$$$$46\!\cdots\!88$$$$T^{6} -$$$$22\!\cdots\!28$$$$T^{7} +$$$$33\!\cdots\!61$$$$T^{8}$$
$43$ $$1 -$$$$75\!\cdots\!00$$$$T +$$$$13\!\cdots\!00$$$$T^{2} -$$$$79\!\cdots\!00$$$$T^{3} +$$$$84\!\cdots\!98$$$$T^{4} -$$$$34\!\cdots\!00$$$$T^{5} +$$$$25\!\cdots\!00$$$$T^{6} -$$$$63\!\cdots\!00$$$$T^{7} +$$$$36\!\cdots\!01$$$$T^{8}$$
$47$ $$1 +$$$$21\!\cdots\!00$$$$T +$$$$36\!\cdots\!80$$$$T^{2} +$$$$46\!\cdots\!00$$$$T^{3} +$$$$54\!\cdots\!18$$$$T^{4} +$$$$45\!\cdots\!00$$$$T^{5} +$$$$36\!\cdots\!20$$$$T^{6} +$$$$21\!\cdots\!00$$$$T^{7} +$$$$98\!\cdots\!81$$$$T^{8}$$
$53$ $$1 -$$$$84\!\cdots\!00$$$$T +$$$$55\!\cdots\!80$$$$T^{2} -$$$$20\!\cdots\!00$$$$T^{3} +$$$$89\!\cdots\!18$$$$T^{4} -$$$$31\!\cdots\!00$$$$T^{5} +$$$$12\!\cdots\!20$$$$T^{6} -$$$$29\!\cdots\!00$$$$T^{7} +$$$$52\!\cdots\!81$$$$T^{8}$$
$59$ $$1 +$$$$21\!\cdots\!60$$$$T +$$$$37\!\cdots\!36$$$$T^{2} +$$$$50\!\cdots\!20$$$$T^{3} +$$$$62\!\cdots\!86$$$$T^{4} +$$$$53\!\cdots\!80$$$$T^{5} +$$$$41\!\cdots\!16$$$$T^{6} +$$$$24\!\cdots\!40$$$$T^{7} +$$$$12\!\cdots\!61$$$$T^{8}$$
$61$ $$1 -$$$$42\!\cdots\!48$$$$T +$$$$31\!\cdots\!08$$$$T^{2} -$$$$10\!\cdots\!96$$$$T^{3} +$$$$37\!\cdots\!70$$$$T^{4} -$$$$81\!\cdots\!56$$$$T^{5} +$$$$20\!\cdots\!68$$$$T^{6} -$$$$22\!\cdots\!88$$$$T^{7} +$$$$41\!\cdots\!41$$$$T^{8}$$
$67$ $$1 +$$$$15\!\cdots\!00$$$$T +$$$$14\!\cdots\!20$$$$T^{2} +$$$$10\!\cdots\!00$$$$T^{3} +$$$$56\!\cdots\!78$$$$T^{4} +$$$$24\!\cdots\!00$$$$T^{5} +$$$$89\!\cdots\!80$$$$T^{6} +$$$$22\!\cdots\!00$$$$T^{7} +$$$$36\!\cdots\!21$$$$T^{8}$$
$71$ $$1 -$$$$26\!\cdots\!88$$$$T +$$$$15\!\cdots\!88$$$$T^{2} +$$$$22\!\cdots\!64$$$$T^{3} +$$$$64\!\cdots\!70$$$$T^{4} +$$$$18\!\cdots\!44$$$$T^{5} +$$$$10\!\cdots\!08$$$$T^{6} -$$$$16\!\cdots\!68$$$$T^{7} +$$$$50\!\cdots\!81$$$$T^{8}$$
$73$ $$1 -$$$$43\!\cdots\!00$$$$T +$$$$13\!\cdots\!40$$$$T^{2} -$$$$45\!\cdots\!00$$$$T^{3} +$$$$84\!\cdots\!58$$$$T^{4} -$$$$20\!\cdots\!00$$$$T^{5} +$$$$28\!\cdots\!60$$$$T^{6} -$$$$42\!\cdots\!00$$$$T^{7} +$$$$44\!\cdots\!41$$$$T^{8}$$
$79$ $$1 +$$$$21\!\cdots\!80$$$$T +$$$$11\!\cdots\!16$$$$T^{2} -$$$$36\!\cdots\!40$$$$T^{3} +$$$$56\!\cdots\!46$$$$T^{4} -$$$$20\!\cdots\!60$$$$T^{5} +$$$$38\!\cdots\!56$$$$T^{6} +$$$$40\!\cdots\!20$$$$T^{7} +$$$$10\!\cdots\!81$$$$T^{8}$$
$83$ $$1 +$$$$19\!\cdots\!00$$$$T +$$$$16\!\cdots\!20$$$$T^{2} +$$$$86\!\cdots\!00$$$$T^{3} +$$$$18\!\cdots\!78$$$$T^{4} +$$$$10\!\cdots\!00$$$$T^{5} +$$$$22\!\cdots\!80$$$$T^{6} +$$$$29\!\cdots\!00$$$$T^{7} +$$$$17\!\cdots\!21$$$$T^{8}$$
$89$ $$1 +$$$$60\!\cdots\!40$$$$T +$$$$42\!\cdots\!56$$$$T^{2} +$$$$15\!\cdots\!80$$$$T^{3} +$$$$56\!\cdots\!26$$$$T^{4} +$$$$12\!\cdots\!20$$$$T^{5} +$$$$28\!\cdots\!76$$$$T^{6} +$$$$33\!\cdots\!60$$$$T^{7} +$$$$44\!\cdots\!41$$$$T^{8}$$
$97$ $$1 +$$$$80\!\cdots\!00$$$$T +$$$$80\!\cdots\!80$$$$T^{2} +$$$$38\!\cdots\!00$$$$T^{3} +$$$$20\!\cdots\!18$$$$T^{4} +$$$$59\!\cdots\!00$$$$T^{5} +$$$$19\!\cdots\!20$$$$T^{6} +$$$$30\!\cdots\!00$$$$T^{7} +$$$$59\!\cdots\!81$$$$T^{8}$$
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