# Properties

 Label 3.44.a.b Level $3$ Weight $44$ Character orbit 3.a Self dual yes Analytic conductor $35.133$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$3$$ Weight: $$k$$ $$=$$ $$44$$ Character orbit: $$[\chi]$$ $$=$$ 3.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$35.1331186037$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - x^{3} - 886516819907 x^{2} - 42308083143723387 x + 94580276745082867224894$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{14}\cdot 3^{10}\cdot 5^{2}\cdot 7\cdot 11$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 415003 + \beta_{1} ) q^{2} -10460353203 q^{3} + ( 7333437011374 + 1259527 \beta_{1} + \beta_{3} ) q^{4} + ( 412662665270741 + 73226956 \beta_{1} + 13 \beta_{2} - 22 \beta_{3} ) q^{5} + ( -4341077960304609 - 10460353203 \beta_{1} ) q^{6} + ( 28620462315178775 - 211738302932 \beta_{1} + 2997 \beta_{2} - 50246 \beta_{3} ) q^{7} + ( 19491643319063864324 + 7692359694018 \beta_{1} + 344576 \beta_{2} + 1836142 \beta_{3} ) q^{8} + 109418989131512359209 q^{9} +O(q^{10})$$ $$q +(415003 + \beta_{1}) q^{2} -10460353203 q^{3} +(7333437011374 + 1259527 \beta_{1} + \beta_{3}) q^{4} +(412662665270741 + 73226956 \beta_{1} + 13 \beta_{2} - 22 \beta_{3}) q^{5} +(-4341077960304609 - 10460353203 \beta_{1}) q^{6} +(28620462315178775 - 211738302932 \beta_{1} + 2997 \beta_{2} - 50246 \beta_{3}) q^{7} +(19491643319063864324 + 7692359694018 \beta_{1} + 344576 \beta_{2} + 1836142 \beta_{3}) q^{8} +$$$$10\!\cdots\!09$$$$q^{9} +($$$$13\!\cdots\!98$$$$+ 326714929810118 \beta_{1} - 9870336 \beta_{2} + 346741184 \beta_{3}) q^{10} +($$$$21\!\cdots\!02$$$$+ 1103916130966808 \beta_{1} - 92037574 \beta_{2} + 1868332564 \beta_{3}) q^{11} +(-$$$$76\!\cdots\!22$$$$- 13175097288714981 \beta_{1} - 10460353203 \beta_{3}) q^{12} +(-$$$$40\!\cdots\!24$$$$+ 106131612461552392 \beta_{1} + 7515407934 \beta_{2} - 21941631908 \beta_{3}) q^{13} +(-$$$$33\!\cdots\!48$$$$- 549787121670036528 \beta_{1} - 17841421312 \beta_{2} - 174730848320 \beta_{3}) q^{14} +(-$$$$43\!\cdots\!23$$$$- 765979823740540068 \beta_{1} - 135984591639 \beta_{2} + 230127770466 \beta_{3}) q^{15} +($$$$66\!\cdots\!48$$$$+ 30566899560284838300 \beta_{1} + 572000984064 \beta_{2} + 7540980445956 \beta_{3}) q^{16} +($$$$73\!\cdots\!52$$$$- 962490835473988264 \beta_{1} - 89546779862 \beta_{2} - 15082051169548 \beta_{3}) q^{17} +($$$$45\!\cdots\!27$$$$+$$$$10\!\cdots\!09$$$$\beta_{1}) q^{18} +(-$$$$64\!\cdots\!82$$$$+$$$$50\!\cdots\!68$$$$\beta_{1} - 9646988077278 \beta_{2} - 202970742410652 \beta_{3}) q^{19} +($$$$21\!\cdots\!16$$$$+$$$$37\!\cdots\!06$$$$\beta_{1} + 6867923468288 \beta_{2} + 502866084527878 \beta_{3}) q^{20} +(-$$$$29\!\cdots\!25$$$$+$$$$22\!\cdots\!96$$$$\beta_{1} - 31349678549391 \beta_{2} + 525590907037938 \beta_{3}) q^{21} +($$$$18\!\cdots\!04$$$$+$$$$18\!\cdots\!92$$$$\beta_{1} + 659992955406336 \beta_{2} + 154983834995584 \beta_{3}) q^{22} +(-$$$$41\!\cdots\!54$$$$+$$$$12\!\cdots\!56$$$$\beta_{1} - 1440843693152086 \beta_{2} - 124021677613580 \beta_{3}) q^{23} +(-$$$$20\!\cdots\!72$$$$-$$$$80\!\cdots\!54$$$$\beta_{1} - 3604386665276928 \beta_{2} - 19206693850862826 \beta_{3}) q^{24} +($$$$54\!\cdots\!95$$$$-$$$$10\!\cdots\!80$$$$\beta_{1} + 15204023814482460 \beta_{2} - 46618766772767240 \beta_{3}) q^{25} +($$$$15\!\cdots\!50$$$$-$$$$47\!\cdots\!70$$$$\beta_{1} - 8884233524930560 \beta_{2} + 258934194685234816 \beta_{3}) q^{26} -$$$$11\!\cdots\!27$$$$q^{27} +(-$$$$10\!\cdots\!16$$$$-$$$$34\!\cdots\!48$$$$\beta_{1} - 83427573725429760 \beta_{2} - 601356479924090352 \beta_{3}) q^{28} +($$$$12\!\cdots\!83$$$$+$$$$22\!\cdots\!72$$$$\beta_{1} + 127100696144575287 \beta_{2} + 947731908380650206 \beta_{3}) q^{29} +(-$$$$14\!\cdots\!94$$$$-$$$$34\!\cdots\!54$$$$\beta_{1} + 103247200792286208 \beta_{2} - 3627035254666412352 \beta_{3}) q^{30} +($$$$38\!\cdots\!51$$$$+$$$$21\!\cdots\!32$$$$\beta_{1} - 10998884783113695 \beta_{2} - 2991683904270408878 \beta_{3}) q^{31} +($$$$34\!\cdots\!48$$$$+$$$$86\!\cdots\!16$$$$\beta_{1} - 533227060395833344 \beta_{2} + 31357077182675843128 \beta_{3}) q^{32} +(-$$$$22\!\cdots\!06$$$$-$$$$11\!\cdots\!24$$$$\beta_{1} + 962745531987249522 \beta_{2} - 19543418520106602492 \beta_{3}) q^{33} +($$$$15\!\cdots\!66$$$$-$$$$49\!\cdots\!30$$$$\beta_{1} - 5181141168554637312 \beta_{2} - 11630432903386267776 \beta_{3}) q^{34} +($$$$19\!\cdots\!70$$$$-$$$$22\!\cdots\!80$$$$\beta_{1} + 9258237867206759310 \beta_{2} -$$$$10\!\cdots\!40$$$$\beta_{3}) q^{35} +($$$$80\!\cdots\!66$$$$+$$$$13\!\cdots\!43$$$$\beta_{1} +$$$$10\!\cdots\!09$$$$\beta_{3}) q^{36} +($$$$48\!\cdots\!62$$$$-$$$$74\!\cdots\!56$$$$\beta_{1} + 20544301497367946160 \beta_{2} + 16927790591022379104 \beta_{3}) q^{37} +($$$$77\!\cdots\!76$$$$-$$$$18\!\cdots\!84$$$$\beta_{1} - 68239741820818003968 \beta_{2} +$$$$17\!\cdots\!40$$$$\beta_{3}) q^{38} +($$$$42\!\cdots\!72$$$$-$$$$11\!\cdots\!76$$$$\beta_{1} - 78613821454268512602 \beta_{2} +$$$$22\!\cdots\!24$$$$\beta_{3}) q^{39} +($$$$49\!\cdots\!64$$$$+$$$$65\!\cdots\!24$$$$\beta_{1} +$$$$25\!\cdots\!52$$$$\beta_{2} +$$$$11\!\cdots\!12$$$$\beta_{3}) q^{40} +(-$$$$22\!\cdots\!16$$$$+$$$$23\!\cdots\!72$$$$\beta_{1} + 55181475299084350110 \beta_{2} -$$$$59\!\cdots\!32$$$$\beta_{3}) q^{41} +($$$$35\!\cdots\!44$$$$+$$$$57\!\cdots\!84$$$$\beta_{1} +$$$$18\!\cdots\!36$$$$\beta_{2} +$$$$18\!\cdots\!60$$$$\beta_{3}) q^{42} +($$$$14\!\cdots\!58$$$$+$$$$68\!\cdots\!92$$$$\beta_{1} -$$$$15\!\cdots\!42$$$$\beta_{2} +$$$$11\!\cdots\!08$$$$\beta_{3}) q^{43} +($$$$27\!\cdots\!16$$$$+$$$$26\!\cdots\!96$$$$\beta_{1} +$$$$74\!\cdots\!68$$$$\beta_{2} +$$$$16\!\cdots\!16$$$$\beta_{3}) q^{44} +($$$$45\!\cdots\!69$$$$+$$$$80\!\cdots\!04$$$$\beta_{1} +$$$$14\!\cdots\!17$$$$\beta_{2} -$$$$24\!\cdots\!98$$$$\beta_{3}) q^{45} +($$$$17\!\cdots\!92$$$$-$$$$35\!\cdots\!92$$$$\beta_{1} +$$$$21\!\cdots\!28$$$$\beta_{2} -$$$$19\!\cdots\!20$$$$\beta_{3}) q^{46} +($$$$12\!\cdots\!10$$$$-$$$$20\!\cdots\!64$$$$\beta_{1} +$$$$60\!\cdots\!14$$$$\beta_{2} +$$$$24\!\cdots\!36$$$$\beta_{3}) q^{47} +(-$$$$69\!\cdots\!44$$$$-$$$$31\!\cdots\!00$$$$\beta_{1} -$$$$59\!\cdots\!92$$$$\beta_{2} -$$$$78\!\cdots\!68$$$$\beta_{3}) q^{48} +(-$$$$10\!\cdots\!95$$$$+$$$$12\!\cdots\!28$$$$\beta_{1} +$$$$99\!\cdots\!36$$$$\beta_{2} +$$$$17\!\cdots\!48$$$$\beta_{3}) q^{49} +(-$$$$13\!\cdots\!15$$$$+$$$$11\!\cdots\!35$$$$\beta_{1} -$$$$18\!\cdots\!20$$$$\beta_{2} +$$$$20\!\cdots\!80$$$$\beta_{3}) q^{50} +(-$$$$76\!\cdots\!56$$$$+$$$$10\!\cdots\!92$$$$\beta_{1} +$$$$93\!\cdots\!86$$$$\beta_{2} +$$$$15\!\cdots\!44$$$$\beta_{3}) q^{51} +(-$$$$33\!\cdots\!28$$$$+$$$$22\!\cdots\!78$$$$\beta_{1} +$$$$24\!\cdots\!24$$$$\beta_{2} -$$$$32\!\cdots\!26$$$$\beta_{3}) q^{52} +(-$$$$10\!\cdots\!29$$$$+$$$$66\!\cdots\!68$$$$\beta_{1} +$$$$43\!\cdots\!95$$$$\beta_{2} -$$$$21\!\cdots\!58$$$$\beta_{3}) q^{53} +(-$$$$47\!\cdots\!81$$$$-$$$$11\!\cdots\!27$$$$\beta_{1}) q^{54} +(-$$$$11\!\cdots\!84$$$$+$$$$64\!\cdots\!56$$$$\beta_{1} -$$$$18\!\cdots\!12$$$$\beta_{2} +$$$$11\!\cdots\!28$$$$\beta_{3}) q^{55} +(-$$$$29\!\cdots\!68$$$$-$$$$13\!\cdots\!72$$$$\beta_{1} -$$$$35\!\cdots\!76$$$$\beta_{2} -$$$$40\!\cdots\!28$$$$\beta_{3}) q^{56} +($$$$67\!\cdots\!46$$$$-$$$$52\!\cdots\!04$$$$\beta_{1} +$$$$10\!\cdots\!34$$$$\beta_{2} +$$$$21\!\cdots\!56$$$$\beta_{3}) q^{57} +($$$$41\!\cdots\!38$$$$+$$$$22\!\cdots\!14$$$$\beta_{1} +$$$$30\!\cdots\!20$$$$\beta_{2} +$$$$56\!\cdots\!04$$$$\beta_{3}) q^{58} +(-$$$$78\!\cdots\!04$$$$-$$$$23\!\cdots\!40$$$$\beta_{1} +$$$$31\!\cdots\!52$$$$\beta_{2} +$$$$43\!\cdots\!68$$$$\beta_{3}) q^{59} +(-$$$$22\!\cdots\!48$$$$-$$$$39\!\cdots\!18$$$$\beta_{1} -$$$$71\!\cdots\!64$$$$\beta_{2} -$$$$52\!\cdots\!34$$$$\beta_{3}) q^{60} +(-$$$$61\!\cdots\!30$$$$-$$$$43\!\cdots\!88$$$$\beta_{1} -$$$$13\!\cdots\!36$$$$\beta_{2} -$$$$27\!\cdots\!68$$$$\beta_{3}) q^{61} +($$$$36\!\cdots\!44$$$$+$$$$32\!\cdots\!92$$$$\beta_{1} -$$$$10\!\cdots\!68$$$$\beta_{2} +$$$$19\!\cdots\!92$$$$\beta_{3}) q^{62} +($$$$31\!\cdots\!75$$$$-$$$$23\!\cdots\!88$$$$\beta_{1} +$$$$32\!\cdots\!73$$$$\beta_{2} -$$$$54\!\cdots\!14$$$$\beta_{3}) q^{63} +($$$$94\!\cdots\!32$$$$+$$$$40\!\cdots\!96$$$$\beta_{1} +$$$$58\!\cdots\!48$$$$\beta_{2} +$$$$26\!\cdots\!84$$$$\beta_{3}) q^{64} +($$$$80\!\cdots\!34$$$$-$$$$11\!\cdots\!56$$$$\beta_{1} +$$$$80\!\cdots\!62$$$$\beta_{2} +$$$$79\!\cdots\!72$$$$\beta_{3}) q^{65} +(-$$$$19\!\cdots\!12$$$$-$$$$18\!\cdots\!76$$$$\beta_{1} -$$$$69\!\cdots\!08$$$$\beta_{2} -$$$$16\!\cdots\!52$$$$\beta_{3}) q^{66} +($$$$25\!\cdots\!84$$$$+$$$$12\!\cdots\!16$$$$\beta_{1} +$$$$42\!\cdots\!68$$$$\beta_{2} -$$$$96\!\cdots\!04$$$$\beta_{3}) q^{67} +(-$$$$14\!\cdots\!16$$$$-$$$$12\!\cdots\!50$$$$\beta_{1} -$$$$23\!\cdots\!44$$$$\beta_{2} -$$$$37\!\cdots\!78$$$$\beta_{3}) q^{68} +($$$$43\!\cdots\!62$$$$-$$$$12\!\cdots\!68$$$$\beta_{1} +$$$$15\!\cdots\!58$$$$\beta_{2} +$$$$12\!\cdots\!40$$$$\beta_{3}) q^{69} +(-$$$$34\!\cdots\!40$$$$-$$$$78\!\cdots\!40$$$$\beta_{1} -$$$$36\!\cdots\!20$$$$\beta_{2} -$$$$78\!\cdots\!20$$$$\beta_{3}) q^{70} +(-$$$$17\!\cdots\!86$$$$-$$$$13\!\cdots\!84$$$$\beta_{1} +$$$$24\!\cdots\!78$$$$\beta_{2} -$$$$24\!\cdots\!20$$$$\beta_{3}) q^{71} +($$$$21\!\cdots\!16$$$$+$$$$84\!\cdots\!62$$$$\beta_{1} +$$$$37\!\cdots\!84$$$$\beta_{2} +$$$$20\!\cdots\!78$$$$\beta_{3}) q^{72} +($$$$13\!\cdots\!70$$$$-$$$$74\!\cdots\!32$$$$\beta_{1} -$$$$27\!\cdots\!20$$$$\beta_{2} +$$$$84\!\cdots\!60$$$$\beta_{3}) q^{73} +(-$$$$98\!\cdots\!42$$$$+$$$$44\!\cdots\!54$$$$\beta_{1} +$$$$22\!\cdots\!24$$$$\beta_{2} -$$$$28\!\cdots\!36$$$$\beta_{3}) q^{74} +(-$$$$56\!\cdots\!85$$$$+$$$$10\!\cdots\!40$$$$\beta_{1} -$$$$15\!\cdots\!80$$$$\beta_{2} +$$$$48\!\cdots\!20$$$$\beta_{3}) q^{75} +(-$$$$21\!\cdots\!40$$$$+$$$$29\!\cdots\!20$$$$\beta_{1} +$$$$15\!\cdots\!68$$$$\beta_{2} -$$$$15\!\cdots\!56$$$$\beta_{3}) q^{76} +(-$$$$17\!\cdots\!28$$$$-$$$$24\!\cdots\!00$$$$\beta_{1} -$$$$22\!\cdots\!84$$$$\beta_{2} +$$$$13\!\cdots\!56$$$$\beta_{3}) q^{77} +(-$$$$15\!\cdots\!50$$$$+$$$$49\!\cdots\!10$$$$\beta_{1} +$$$$92\!\cdots\!80$$$$\beta_{2} -$$$$27\!\cdots\!48$$$$\beta_{3}) q^{78} +(-$$$$46\!\cdots\!93$$$$-$$$$21\!\cdots\!92$$$$\beta_{1} +$$$$40\!\cdots\!89$$$$\beta_{2} -$$$$68\!\cdots\!98$$$$\beta_{3}) q^{79} +($$$$10\!\cdots\!64$$$$+$$$$31\!\cdots\!24$$$$\beta_{1} +$$$$28\!\cdots\!52$$$$\beta_{2} +$$$$84\!\cdots\!12$$$$\beta_{3}) q^{80} +$$$$11\!\cdots\!81$$$$q^{81} +($$$$36\!\cdots\!38$$$$-$$$$48\!\cdots\!46$$$$\beta_{1} -$$$$20\!\cdots\!12$$$$\beta_{2} +$$$$14\!\cdots\!52$$$$\beta_{3}) q^{82} +($$$$51\!\cdots\!86$$$$-$$$$34\!\cdots\!68$$$$\beta_{1} +$$$$19\!\cdots\!70$$$$\beta_{2} -$$$$58\!\cdots\!00$$$$\beta_{3}) q^{83} +($$$$10\!\cdots\!48$$$$+$$$$35\!\cdots\!44$$$$\beta_{1} +$$$$87\!\cdots\!80$$$$\beta_{2} +$$$$62\!\cdots\!56$$$$\beta_{3}) q^{84} +($$$$55\!\cdots\!14$$$$-$$$$37\!\cdots\!76$$$$\beta_{1} +$$$$19\!\cdots\!02$$$$\beta_{2} -$$$$48\!\cdots\!88$$$$\beta_{3}) q^{85} +($$$$11\!\cdots\!32$$$$+$$$$25\!\cdots\!84$$$$\beta_{1} +$$$$65\!\cdots\!84$$$$\beta_{2} -$$$$27\!\cdots\!60$$$$\beta_{3}) q^{86} +(-$$$$13\!\cdots\!49$$$$-$$$$23\!\cdots\!16$$$$\beta_{1} -$$$$13\!\cdots\!61$$$$\beta_{2} -$$$$99\!\cdots\!18$$$$\beta_{3}) q^{87} +($$$$37\!\cdots\!96$$$$+$$$$27\!\cdots\!04$$$$\beta_{1} -$$$$35\!\cdots\!76$$$$\beta_{2} +$$$$51\!\cdots\!52$$$$\beta_{3}) q^{88} +(-$$$$11\!\cdots\!78$$$$+$$$$17\!\cdots\!60$$$$\beta_{1} -$$$$40\!\cdots\!32$$$$\beta_{2} +$$$$12\!\cdots\!88$$$$\beta_{3}) q^{89} +($$$$14\!\cdots\!82$$$$+$$$$35\!\cdots\!62$$$$\beta_{1} -$$$$10\!\cdots\!24$$$$\beta_{2} +$$$$37\!\cdots\!56$$$$\beta_{3}) q^{90} +(-$$$$60\!\cdots\!82$$$$-$$$$10\!\cdots\!52$$$$\beta_{1} +$$$$29\!\cdots\!98$$$$\beta_{2} -$$$$35\!\cdots\!48$$$$\beta_{3}) q^{91} +(-$$$$12\!\cdots\!76$$$$-$$$$11\!\cdots\!68$$$$\beta_{1} +$$$$59\!\cdots\!84$$$$\beta_{2} -$$$$40\!\cdots\!04$$$$\beta_{3}) q^{92} +(-$$$$40\!\cdots\!53$$$$-$$$$22\!\cdots\!96$$$$\beta_{1} +$$$$11\!\cdots\!85$$$$\beta_{2} +$$$$31\!\cdots\!34$$$$\beta_{3}) q^{93} +(-$$$$31\!\cdots\!96$$$$-$$$$27\!\cdots\!44$$$$\beta_{1} +$$$$73\!\cdots\!44$$$$\beta_{2} -$$$$18\!\cdots\!00$$$$\beta_{3}) q^{94} +(-$$$$21\!\cdots\!84$$$$-$$$$40\!\cdots\!44$$$$\beta_{1} -$$$$90\!\cdots\!12$$$$\beta_{2} +$$$$19\!\cdots\!28$$$$\beta_{3}) q^{95} +(-$$$$35\!\cdots\!44$$$$-$$$$90\!\cdots\!48$$$$\beta_{1} +$$$$55\!\cdots\!32$$$$\beta_{2} -$$$$32\!\cdots\!84$$$$\beta_{3}) q^{96} +(-$$$$12\!\cdots\!78$$$$+$$$$11\!\cdots\!84$$$$\beta_{1} -$$$$10\!\cdots\!68$$$$\beta_{2} +$$$$24\!\cdots\!60$$$$\beta_{3}) q^{97} +($$$$15\!\cdots\!83$$$$-$$$$81\!\cdots\!19$$$$\beta_{1} +$$$$41\!\cdots\!40$$$$\beta_{2} +$$$$35\!\cdots\!24$$$$\beta_{3}) q^{98} +($$$$23\!\cdots\!18$$$$+$$$$12\!\cdots\!72$$$$\beta_{1} -$$$$10\!\cdots\!66$$$$\beta_{2} +$$$$20\!\cdots\!76$$$$\beta_{3}) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 1660014q^{2} - 41841412812q^{3} + 29333750564548q^{4} + 1650650807536920q^{5} - 17364332761924842q^{6} + 114481425784209728q^{7} + 77966588660971173048q^{8} + 437675956526049436836q^{9} + O(q^{10})$$ $$4q + 1660014q^{2} - 41841412812q^{3} + 29333750564548q^{4} + 1650650807536920q^{5} - 17364332761924842q^{6} + 114481425784209728q^{7} + 77966588660971173048q^{8} +$$$$43\!\cdots\!36$$$$q^{9} +$$$$53\!\cdots\!60$$$$q^{10} +$$$$87\!\cdots\!96$$$$q^{11} -$$$$30\!\cdots\!44$$$$q^{12} -$$$$16\!\cdots\!96$$$$q^{13} -$$$$13\!\cdots\!08$$$$q^{14} -$$$$17\!\cdots\!60$$$$q^{15} +$$$$26\!\cdots\!80$$$$q^{16} +$$$$29\!\cdots\!76$$$$q^{17} +$$$$18\!\cdots\!26$$$$q^{18} -$$$$25\!\cdots\!88$$$$q^{19} +$$$$85\!\cdots\!20$$$$q^{20} -$$$$11\!\cdots\!84$$$$q^{21} +$$$$74\!\cdots\!32$$$$q^{22} -$$$$16\!\cdots\!44$$$$q^{23} -$$$$81\!\cdots\!44$$$$q^{24} +$$$$21\!\cdots\!00$$$$q^{25} +$$$$61\!\cdots\!28$$$$q^{26} -$$$$45\!\cdots\!08$$$$q^{27} -$$$$41\!\cdots\!56$$$$q^{28} +$$$$50\!\cdots\!64$$$$q^{29} -$$$$56\!\cdots\!80$$$$q^{30} +$$$$15\!\cdots\!24$$$$q^{31} +$$$$13\!\cdots\!68$$$$q^{32} -$$$$91\!\cdots\!88$$$$q^{33} +$$$$60\!\cdots\!56$$$$q^{34} +$$$$79\!\cdots\!00$$$$q^{35} +$$$$32\!\cdots\!32$$$$q^{36} +$$$$19\!\cdots\!28$$$$q^{37} +$$$$30\!\cdots\!56$$$$q^{38} +$$$$16\!\cdots\!88$$$$q^{39} +$$$$19\!\cdots\!80$$$$q^{40} -$$$$90\!\cdots\!56$$$$q^{41} +$$$$14\!\cdots\!24$$$$q^{42} +$$$$57\!\cdots\!00$$$$q^{43} +$$$$11\!\cdots\!24$$$$q^{44} +$$$$18\!\cdots\!80$$$$q^{45} +$$$$71\!\cdots\!24$$$$q^{46} +$$$$48\!\cdots\!40$$$$q^{47} -$$$$27\!\cdots\!40$$$$q^{48} -$$$$43\!\cdots\!20$$$$q^{49} -$$$$55\!\cdots\!50$$$$q^{50} -$$$$30\!\cdots\!28$$$$q^{51} -$$$$13\!\cdots\!04$$$$q^{52} -$$$$40\!\cdots\!64$$$$q^{53} -$$$$18\!\cdots\!78$$$$q^{54} -$$$$47\!\cdots\!80$$$$q^{55} -$$$$11\!\cdots\!60$$$$q^{56} +$$$$27\!\cdots\!64$$$$q^{57} +$$$$16\!\cdots\!72$$$$q^{58} -$$$$31\!\cdots\!32$$$$q^{59} -$$$$89\!\cdots\!60$$$$q^{60} -$$$$24\!\cdots\!60$$$$q^{61} +$$$$14\!\cdots\!76$$$$q^{62} +$$$$12\!\cdots\!52$$$$q^{63} +$$$$37\!\cdots\!52$$$$q^{64} +$$$$32\!\cdots\!80$$$$q^{65} -$$$$77\!\cdots\!96$$$$q^{66} +$$$$10\!\cdots\!76$$$$q^{67} -$$$$57\!\cdots\!08$$$$q^{68} +$$$$17\!\cdots\!32$$$$q^{69} -$$$$13\!\cdots\!00$$$$q^{70} -$$$$71\!\cdots\!72$$$$q^{71} +$$$$85\!\cdots\!32$$$$q^{72} +$$$$54\!\cdots\!96$$$$q^{73} -$$$$39\!\cdots\!88$$$$q^{74} -$$$$22\!\cdots\!00$$$$q^{75} -$$$$84\!\cdots\!08$$$$q^{76} -$$$$70\!\cdots\!24$$$$q^{77} -$$$$63\!\cdots\!84$$$$q^{78} -$$$$18\!\cdots\!60$$$$q^{79} +$$$$42\!\cdots\!80$$$$q^{80} +$$$$47\!\cdots\!24$$$$q^{81} +$$$$14\!\cdots\!56$$$$q^{82} +$$$$20\!\cdots\!08$$$$q^{83} +$$$$43\!\cdots\!68$$$$q^{84} +$$$$22\!\cdots\!80$$$$q^{85} +$$$$45\!\cdots\!16$$$$q^{86} -$$$$52\!\cdots\!92$$$$q^{87} +$$$$15\!\cdots\!88$$$$q^{88} -$$$$44\!\cdots\!68$$$$q^{89} +$$$$58\!\cdots\!40$$$$q^{90} -$$$$24\!\cdots\!36$$$$q^{91} -$$$$50\!\cdots\!32$$$$q^{92} -$$$$16\!\cdots\!72$$$$q^{93} -$$$$12\!\cdots\!72$$$$q^{94} -$$$$87\!\cdots\!80$$$$q^{95} -$$$$14\!\cdots\!04$$$$q^{96} -$$$$50\!\cdots\!64$$$$q^{97} +$$$$62\!\cdots\!46$$$$q^{98} +$$$$95\!\cdots\!64$$$$q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 886516819907 x^{2} - 42308083143723387 x + 94580276745082867224894$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$6 \nu - 1$$ $$\beta_{2}$$ $$=$$ $$($$$$27 \nu^{3} - 2660112 \nu^{2} - 17984400403617 \nu + 322364875892989014$$$$)/43072$$ $$\beta_{3}$$ $$=$$ $$36 \nu^{2} - 2577138 \nu - 15957302114051$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1} + 1$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 429523 \beta_{1} + 15957302543574$$$$)/36$$ $$\nu^{3}$$ $$=$$ $$($$$$147784 \beta_{3} + 86144 \beta_{2} + 6058276761571 \beta_{1} + 1713510242113696527$$$$)/54$$

## 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
 −837306. −388484. 321562. 904229.
−4.60883e6 −1.04604e10 1.24452e13 −9.11231e14 4.82100e16 3.55193e17 −1.68183e19 1.09419e20 4.19971e21
1.2 −1.91590e6 −1.04604e10 −5.12540e12 2.05854e15 2.00410e16 1.37114e18 2.66723e19 1.09419e20 −3.94396e21
1.3 2.34437e6 −1.04604e10 −3.30001e12 −6.18875e14 −2.45230e16 −6.01464e16 −2.83578e19 1.09419e20 −1.45087e21
1.4 5.84038e6 −1.04604e10 2.53139e13 1.12222e15 −6.10924e16 −1.55171e18 9.64704e19 1.09419e20 6.55417e21
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$

## 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 3.44.a.b 4
3.b odd 2 1 9.44.a.c 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 1660014 T_{2}^{3} -$$30881238086592

'>$$30\!\cdots\!92$$$$T_{2}^{2} +$$17064803544699174912
'>$$17\!\cdots\!12$$$$T_{2} +$$120901670049507055201419264'>$$12\!\cdots\!64$$ acting on $$S_{44}^{\mathrm{new}}(\Gamma_0(3))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 1660014 T + 4303134002240 T^{2} - 26740109141803597824 T^{3} +$$$$41\!\cdots\!76$$$$T^{4} -$$$$23\!\cdots\!92$$$$T^{5} +$$$$33\!\cdots\!60$$$$T^{6} -$$$$11\!\cdots\!68$$$$T^{7} +$$$$59\!\cdots\!96$$$$T^{8}$$
$3$ $$( 1 + 10460353203 T )^{4}$$
$5$ $$1 - 1650650807536920 T +$$$$25\!\cdots\!00$$$$T^{2} -$$$$38\!\cdots\!00$$$$T^{3} +$$$$45\!\cdots\!50$$$$T^{4} -$$$$44\!\cdots\!00$$$$T^{5} +$$$$33\!\cdots\!00$$$$T^{6} -$$$$24\!\cdots\!00$$$$T^{7} +$$$$16\!\cdots\!25$$$$T^{8}$$
$7$ $$1 - 114481425784209728 T +$$$$65\!\cdots\!88$$$$T^{2} -$$$$12\!\cdots\!92$$$$T^{3} +$$$$19\!\cdots\!70$$$$T^{4} -$$$$27\!\cdots\!56$$$$T^{5} +$$$$31\!\cdots\!12$$$$T^{6} -$$$$11\!\cdots\!96$$$$T^{7} +$$$$22\!\cdots\!01$$$$T^{8}$$
$11$ $$1 -$$$$87\!\cdots\!96$$$$T +$$$$13\!\cdots\!12$$$$T^{2} -$$$$69\!\cdots\!60$$$$T^{3} +$$$$11\!\cdots\!86$$$$T^{4} -$$$$41\!\cdots\!60$$$$T^{5} +$$$$50\!\cdots\!32$$$$T^{6} -$$$$19\!\cdots\!36$$$$T^{7} +$$$$13\!\cdots\!21$$$$T^{8}$$
$13$ $$1 +$$$$16\!\cdots\!96$$$$T +$$$$27\!\cdots\!36$$$$T^{2} +$$$$32\!\cdots\!48$$$$T^{3} +$$$$32\!\cdots\!50$$$$T^{4} +$$$$25\!\cdots\!56$$$$T^{5} +$$$$17\!\cdots\!24$$$$T^{6} +$$$$81\!\cdots\!08$$$$T^{7} +$$$$39\!\cdots\!81$$$$T^{8}$$
$17$ $$1 -$$$$29\!\cdots\!76$$$$T +$$$$29\!\cdots\!68$$$$T^{2} -$$$$64\!\cdots\!00$$$$T^{3} +$$$$35\!\cdots\!26$$$$T^{4} -$$$$52\!\cdots\!00$$$$T^{5} +$$$$19\!\cdots\!92$$$$T^{6} -$$$$15\!\cdots\!72$$$$T^{7} +$$$$43\!\cdots\!61$$$$T^{8}$$
$19$ $$1 +$$$$25\!\cdots\!88$$$$T +$$$$20\!\cdots\!32$$$$T^{2} +$$$$42\!\cdots\!16$$$$T^{3} +$$$$20\!\cdots\!14$$$$T^{4} +$$$$40\!\cdots\!44$$$$T^{5} +$$$$19\!\cdots\!92$$$$T^{6} +$$$$23\!\cdots\!52$$$$T^{7} +$$$$88\!\cdots\!61$$$$T^{8}$$
$23$ $$1 +$$$$16\!\cdots\!44$$$$T +$$$$11\!\cdots\!12$$$$T^{2} +$$$$15\!\cdots\!64$$$$T^{3} +$$$$56\!\cdots\!30$$$$T^{4} +$$$$55\!\cdots\!88$$$$T^{5} +$$$$14\!\cdots\!68$$$$T^{6} +$$$$76\!\cdots\!72$$$$T^{7} +$$$$16\!\cdots\!21$$$$T^{8}$$
$29$ $$1 -$$$$50\!\cdots\!64$$$$T +$$$$33\!\cdots\!00$$$$T^{2} -$$$$10\!\cdots\!28$$$$T^{3} +$$$$38\!\cdots\!58$$$$T^{4} -$$$$83\!\cdots\!92$$$$T^{5} +$$$$19\!\cdots\!00$$$$T^{6} -$$$$22\!\cdots\!16$$$$T^{7} +$$$$34\!\cdots\!41$$$$T^{8}$$
$31$ $$1 -$$$$15\!\cdots\!24$$$$T +$$$$45\!\cdots\!08$$$$T^{2} -$$$$44\!\cdots\!52$$$$T^{3} +$$$$83\!\cdots\!94$$$$T^{4} -$$$$60\!\cdots\!32$$$$T^{5} +$$$$81\!\cdots\!48$$$$T^{6} -$$$$37\!\cdots\!04$$$$T^{7} +$$$$32\!\cdots\!61$$$$T^{8}$$
$37$ $$1 -$$$$19\!\cdots\!28$$$$T +$$$$22\!\cdots\!48$$$$T^{2} -$$$$18\!\cdots\!12$$$$T^{3} +$$$$10\!\cdots\!70$$$$T^{4} -$$$$49\!\cdots\!36$$$$T^{5} +$$$$16\!\cdots\!32$$$$T^{6} -$$$$38\!\cdots\!56$$$$T^{7} +$$$$53\!\cdots\!81$$$$T^{8}$$
$41$ $$1 +$$$$90\!\cdots\!56$$$$T -$$$$37\!\cdots\!32$$$$T^{2} +$$$$41\!\cdots\!28$$$$T^{3} +$$$$71\!\cdots\!74$$$$T^{4} +$$$$92\!\cdots\!88$$$$T^{5} -$$$$18\!\cdots\!12$$$$T^{6} +$$$$10\!\cdots\!16$$$$T^{7} +$$$$25\!\cdots\!81$$$$T^{8}$$
$43$ $$1 -$$$$57\!\cdots\!00$$$$T +$$$$28\!\cdots\!40$$$$T^{2} -$$$$91\!\cdots\!00$$$$T^{3} +$$$$40\!\cdots\!98$$$$T^{4} -$$$$15\!\cdots\!00$$$$T^{5} +$$$$85\!\cdots\!60$$$$T^{6} -$$$$30\!\cdots\!00$$$$T^{7} +$$$$90\!\cdots\!01$$$$T^{8}$$
$47$ $$1 -$$$$48\!\cdots\!40$$$$T +$$$$19\!\cdots\!80$$$$T^{2} -$$$$82\!\cdots\!20$$$$T^{3} +$$$$20\!\cdots\!58$$$$T^{4} -$$$$65\!\cdots\!60$$$$T^{5} +$$$$12\!\cdots\!20$$$$T^{6} -$$$$24\!\cdots\!80$$$$T^{7} +$$$$39\!\cdots\!41$$$$T^{8}$$
$53$ $$1 +$$$$40\!\cdots\!64$$$$T +$$$$11\!\cdots\!12$$$$T^{2} +$$$$19\!\cdots\!44$$$$T^{3} +$$$$27\!\cdots\!90$$$$T^{4} +$$$$27\!\cdots\!88$$$$T^{5} +$$$$21\!\cdots\!48$$$$T^{6} +$$$$10\!\cdots\!12$$$$T^{7} +$$$$37\!\cdots\!41$$$$T^{8}$$
$59$ $$1 +$$$$31\!\cdots\!32$$$$T +$$$$32\!\cdots\!72$$$$T^{2} +$$$$69\!\cdots\!84$$$$T^{3} +$$$$57\!\cdots\!94$$$$T^{4} +$$$$97\!\cdots\!36$$$$T^{5} +$$$$63\!\cdots\!52$$$$T^{6} +$$$$86\!\cdots\!48$$$$T^{7} +$$$$38\!\cdots\!81$$$$T^{8}$$
$61$ $$1 +$$$$24\!\cdots\!60$$$$T +$$$$18\!\cdots\!76$$$$T^{2} -$$$$57\!\cdots\!60$$$$T^{3} -$$$$20\!\cdots\!34$$$$T^{4} -$$$$33\!\cdots\!60$$$$T^{5} +$$$$65\!\cdots\!36$$$$T^{6} +$$$$50\!\cdots\!60$$$$T^{7} +$$$$11\!\cdots\!21$$$$T^{8}$$
$67$ $$1 -$$$$10\!\cdots\!76$$$$T +$$$$13\!\cdots\!68$$$$T^{2} -$$$$10\!\cdots\!00$$$$T^{3} +$$$$65\!\cdots\!26$$$$T^{4} -$$$$33\!\cdots\!00$$$$T^{5} +$$$$14\!\cdots\!92$$$$T^{6} -$$$$37\!\cdots\!72$$$$T^{7} +$$$$12\!\cdots\!61$$$$T^{8}$$
$71$ $$1 +$$$$71\!\cdots\!72$$$$T +$$$$94\!\cdots\!88$$$$T^{2} +$$$$62\!\cdots\!04$$$$T^{3} +$$$$46\!\cdots\!70$$$$T^{4} +$$$$25\!\cdots\!44$$$$T^{5} +$$$$15\!\cdots\!48$$$$T^{6} +$$$$46\!\cdots\!32$$$$T^{7} +$$$$26\!\cdots\!41$$$$T^{8}$$
$73$ $$1 -$$$$54\!\cdots\!96$$$$T +$$$$34\!\cdots\!32$$$$T^{2} -$$$$18\!\cdots\!76$$$$T^{3} +$$$$60\!\cdots\!10$$$$T^{4} -$$$$24\!\cdots\!92$$$$T^{5} +$$$$59\!\cdots\!48$$$$T^{6} -$$$$12\!\cdots\!48$$$$T^{7} +$$$$31\!\cdots\!21$$$$T^{8}$$
$79$ $$1 +$$$$18\!\cdots\!60$$$$T -$$$$12\!\cdots\!44$$$$T^{2} +$$$$59\!\cdots\!20$$$$T^{3} +$$$$25\!\cdots\!26$$$$T^{4} +$$$$23\!\cdots\!80$$$$T^{5} -$$$$19\!\cdots\!24$$$$T^{6} +$$$$11\!\cdots\!40$$$$T^{7} +$$$$24\!\cdots\!41$$$$T^{8}$$
$83$ $$1 -$$$$20\!\cdots\!08$$$$T +$$$$10\!\cdots\!80$$$$T^{2} -$$$$13\!\cdots\!52$$$$T^{3} +$$$$43\!\cdots\!46$$$$T^{4} -$$$$44\!\cdots\!24$$$$T^{5} +$$$$11\!\cdots\!20$$$$T^{6} -$$$$75\!\cdots\!24$$$$T^{7} +$$$$12\!\cdots\!61$$$$T^{8}$$
$89$ $$1 +$$$$44\!\cdots\!68$$$$T +$$$$24\!\cdots\!92$$$$T^{2} +$$$$88\!\cdots\!16$$$$T^{3} +$$$$23\!\cdots\!74$$$$T^{4} +$$$$58\!\cdots\!04$$$$T^{5} +$$$$10\!\cdots\!12$$$$T^{6} +$$$$13\!\cdots\!12$$$$T^{7} +$$$$19\!\cdots\!21$$$$T^{8}$$
$97$ $$1 +$$$$50\!\cdots\!64$$$$T +$$$$60\!\cdots\!28$$$$T^{2} +$$$$93\!\cdots\!00$$$$T^{3} +$$$$13\!\cdots\!66$$$$T^{4} +$$$$25\!\cdots\!00$$$$T^{5} +$$$$44\!\cdots\!12$$$$T^{6} +$$$$99\!\cdots\!88$$$$T^{7} +$$$$53\!\cdots\!41$$$$T^{8}$$
show more
show less