Properties

 Label 9.76.a.c Level $9$ Weight $76$ Character orbit 9.a Self dual yes Analytic conductor $320.606$ Analytic rank $1$ Dimension $6$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$9 = 3^{2}$$ Weight: $$k$$ $$=$$ $$76$$ Character orbit: $$[\chi]$$ $$=$$ 9.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$320.605553540$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Defining polynomial: $$x^{6} - 3 x^{5} - 38457853073924058692 x^{4} - 10276556354621685339901678086 x^{3} + 371187556674475060057870954681799784505 x^{2} + 52686123927652036687598761277591247931691204025 x - 675344021115865838575279495800656435684060652010336995750$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: multiple of $$2^{58}\cdot 3^{36}\cdot 5^{7}\cdot 7^{3}\cdot 11\cdot 13\cdot 19$$ Twist minimal: no (minimal twist has level 1) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q +(9513470340 - \beta_{1}) q^{2} +($$$$28\!\cdots\!88$$$$+ 9832395084 \beta_{1} + \beta_{2} + \beta_{3}) q^{4} +($$$$64\!\cdots\!90$$$$- 333300058621800 \beta_{1} - 2952 \beta_{2} - 340 \beta_{3} + \beta_{4}) q^{5} +($$$$32\!\cdots\!00$$$$+ 20646137267438940438 \beta_{1} + 896702338 \beta_{2} - 28211952 \beta_{3} - 166380 \beta_{4} - 32 \beta_{5}) q^{7} +(-$$$$73\!\cdots\!20$$$$-$$$$23\!\cdots\!76$$$$\beta_{1} - 82237429424 \beta_{2} - 81567621776 \beta_{3} + 43973360 \beta_{4} - 7416 \beta_{5}) q^{8} +O(q^{10})$$ $$q +(9513470340 - \beta_{1}) q^{2} +($$$$28\!\cdots\!88$$$$+ 9832395084 \beta_{1} + \beta_{2} + \beta_{3}) q^{4} +($$$$64\!\cdots\!90$$$$- 333300058621800 \beta_{1} - 2952 \beta_{2} - 340 \beta_{3} + \beta_{4}) q^{5} +($$$$32\!\cdots\!00$$$$+ 20646137267438940438 \beta_{1} + 896702338 \beta_{2} - 28211952 \beta_{3} - 166380 \beta_{4} - 32 \beta_{5}) q^{7} +(-$$$$73\!\cdots\!20$$$$-$$$$23\!\cdots\!76$$$$\beta_{1} - 82237429424 \beta_{2} - 81567621776 \beta_{3} + 43973360 \beta_{4} - 7416 \beta_{5}) q^{8} +($$$$22\!\cdots\!60$$$$+$$$$22\!\cdots\!50$$$$\beta_{1} + 223549665063932 \beta_{2} + 869735349916300 \beta_{3} + 38186921784 \beta_{4} + 25105060 \beta_{5}) q^{10} +($$$$15\!\cdots\!48$$$$-$$$$36\!\cdots\!85$$$$\beta_{1} + 4912569329548505 \beta_{2} + 4721554989781216 \beta_{3} + 3747958317080 \beta_{4} + 806101056 \beta_{5}) q^{11} +($$$$88\!\cdots\!70$$$$+$$$$30\!\cdots\!32$$$$\beta_{1} - 21816932345480852600 \beta_{2} + 5942218372062377876 \beta_{3} - 1816281211946385 \beta_{4} - 389448765184 \beta_{5}) q^{13} +(-$$$$13\!\cdots\!96$$$$-$$$$40\!\cdots\!62$$$$\beta_{1} +$$$$38\!\cdots\!82$$$$\beta_{2} - 96176923015363877798 \beta_{3} - 15630807774027100 \beta_{4} - 6283557820530 \beta_{5}) q^{14} +($$$$44\!\cdots\!36$$$$+$$$$34\!\cdots\!32$$$$\beta_{1} +$$$$73\!\cdots\!08$$$$\beta_{2} +$$$$15\!\cdots\!56$$$$\beta_{3} - 2536277958175820160 \beta_{4} + 960434547593408 \beta_{5}) q^{16} +(-$$$$30\!\cdots\!30$$$$-$$$$80\!\cdots\!92$$$$\beta_{1} +$$$$11\!\cdots\!16$$$$\beta_{2} -$$$$64\!\cdots\!32$$$$\beta_{3} - 41006711266916156330 \beta_{4} + 9197963260956288 \beta_{5}) q^{17} +($$$$17\!\cdots\!80$$$$+$$$$31\!\cdots\!11$$$$\beta_{1} +$$$$61\!\cdots\!69$$$$\beta_{2} +$$$$41\!\cdots\!56$$$$\beta_{3} +$$$$23\!\cdots\!60$$$$\beta_{4} - 482175934113126848 \beta_{5}) q^{19} +(-$$$$15\!\cdots\!80$$$$-$$$$34\!\cdots\!00$$$$\beta_{1} -$$$$29\!\cdots\!26$$$$\beta_{2} -$$$$54\!\cdots\!70$$$$\beta_{3} +$$$$68\!\cdots\!88$$$$\beta_{4} - 2380295833270272000 \beta_{5}) q^{20} +($$$$25\!\cdots\!20$$$$-$$$$30\!\cdots\!83$$$$\beta_{1} -$$$$83\!\cdots\!15$$$$\beta_{2} +$$$$11\!\cdots\!21$$$$\beta_{3} +$$$$54\!\cdots\!90$$$$\beta_{4} - 27991143699221201689 \beta_{5}) q^{22} +(-$$$$25\!\cdots\!80$$$$+$$$$20\!\cdots\!30$$$$\beta_{1} -$$$$12\!\cdots\!62$$$$\beta_{2} -$$$$11\!\cdots\!80$$$$\beta_{3} +$$$$36\!\cdots\!00$$$$\beta_{4} -$$$$54\!\cdots\!80$$$$\beta_{5}) q^{23} +($$$$32\!\cdots\!75$$$$+$$$$83\!\cdots\!00$$$$\beta_{1} -$$$$15\!\cdots\!80$$$$\beta_{2} +$$$$22\!\cdots\!00$$$$\beta_{3} -$$$$19\!\cdots\!60$$$$\beta_{4} +$$$$28\!\cdots\!00$$$$\beta_{5}) q^{25} +(-$$$$19\!\cdots\!92$$$$-$$$$15\!\cdots\!38$$$$\beta_{1} +$$$$12\!\cdots\!88$$$$\beta_{2} -$$$$61\!\cdots\!76$$$$\beta_{3} +$$$$10\!\cdots\!80$$$$\beta_{4} +$$$$13\!\cdots\!56$$$$\beta_{5}) q^{26} +($$$$24\!\cdots\!20$$$$+$$$$15\!\cdots\!20$$$$\beta_{1} +$$$$80\!\cdots\!48$$$$\beta_{2} -$$$$30\!\cdots\!76$$$$\beta_{3} +$$$$68\!\cdots\!60$$$$\beta_{4} -$$$$12\!\cdots\!16$$$$\beta_{5}) q^{28} +(-$$$$24\!\cdots\!70$$$$+$$$$40\!\cdots\!36$$$$\beta_{1} -$$$$12\!\cdots\!76$$$$\beta_{2} -$$$$83\!\cdots\!80$$$$\beta_{3} +$$$$18\!\cdots\!05$$$$\beta_{4} -$$$$74\!\cdots\!84$$$$\beta_{5}) q^{29} +(-$$$$69\!\cdots\!48$$$$-$$$$73\!\cdots\!20$$$$\beta_{1} +$$$$14\!\cdots\!60$$$$\beta_{2} -$$$$70\!\cdots\!28$$$$\beta_{3} -$$$$45\!\cdots\!40$$$$\beta_{4} -$$$$72\!\cdots\!48$$$$\beta_{5}) q^{31} +(-$$$$19\!\cdots\!60$$$$-$$$$40\!\cdots\!56$$$$\beta_{1} -$$$$74\!\cdots\!20$$$$\beta_{2} -$$$$46\!\cdots\!72$$$$\beta_{3} -$$$$95\!\cdots\!80$$$$\beta_{4} -$$$$24\!\cdots\!52$$$$\beta_{5}) q^{32} +($$$$50\!\cdots\!76$$$$+$$$$49\!\cdots\!38$$$$\beta_{1} -$$$$83\!\cdots\!28$$$$\beta_{2} -$$$$13\!\cdots\!36$$$$\beta_{3} -$$$$34\!\cdots\!40$$$$\beta_{4} -$$$$38\!\cdots\!68$$$$\beta_{5}) q^{34} +(-$$$$45\!\cdots\!60$$$$+$$$$80\!\cdots\!00$$$$\beta_{1} +$$$$16\!\cdots\!88$$$$\beta_{2} -$$$$21\!\cdots\!00$$$$\beta_{3} -$$$$83\!\cdots\!44$$$$\beta_{4} -$$$$58\!\cdots\!60$$$$\beta_{5}) q^{35} +($$$$16\!\cdots\!90$$$$-$$$$58\!\cdots\!40$$$$\beta_{1} -$$$$93\!\cdots\!84$$$$\beta_{2} -$$$$12\!\cdots\!52$$$$\beta_{3} -$$$$29\!\cdots\!05$$$$\beta_{4} +$$$$37\!\cdots\!68$$$$\beta_{5}) q^{37} +(-$$$$20\!\cdots\!80$$$$-$$$$38\!\cdots\!79$$$$\beta_{1} +$$$$20\!\cdots\!29$$$$\beta_{2} -$$$$21\!\cdots\!27$$$$\beta_{3} +$$$$22\!\cdots\!70$$$$\beta_{4} -$$$$23\!\cdots\!57$$$$\beta_{5}) q^{38} +($$$$14\!\cdots\!00$$$$+$$$$41\!\cdots\!00$$$$\beta_{1} +$$$$38\!\cdots\!40$$$$\beta_{2} +$$$$16\!\cdots\!00$$$$\beta_{3} -$$$$38\!\cdots\!20$$$$\beta_{4} +$$$$10\!\cdots\!00$$$$\beta_{5}) q^{40} +(-$$$$83\!\cdots\!02$$$$+$$$$62\!\cdots\!20$$$$\beta_{1} -$$$$13\!\cdots\!60$$$$\beta_{2} +$$$$66\!\cdots\!88$$$$\beta_{3} +$$$$12\!\cdots\!40$$$$\beta_{4} +$$$$12\!\cdots\!08$$$$\beta_{5}) q^{41} +($$$$46\!\cdots\!00$$$$+$$$$18\!\cdots\!21$$$$\beta_{1} -$$$$49\!\cdots\!89$$$$\beta_{2} +$$$$16\!\cdots\!60$$$$\beta_{3} -$$$$77\!\cdots\!00$$$$\beta_{4} +$$$$14\!\cdots\!60$$$$\beta_{5}) q^{43} +($$$$14\!\cdots\!24$$$$-$$$$35\!\cdots\!48$$$$\beta_{1} -$$$$36\!\cdots\!12$$$$\beta_{2} +$$$$53\!\cdots\!36$$$$\beta_{3} -$$$$63\!\cdots\!60$$$$\beta_{4} +$$$$26\!\cdots\!08$$$$\beta_{5}) q^{44} +(-$$$$13\!\cdots\!48$$$$+$$$$59\!\cdots\!10$$$$\beta_{1} +$$$$48\!\cdots\!30$$$$\beta_{2} +$$$$77\!\cdots\!42$$$$\beta_{3} -$$$$42\!\cdots\!40$$$$\beta_{4} +$$$$11\!\cdots\!02$$$$\beta_{5}) q^{46} +($$$$23\!\cdots\!80$$$$+$$$$77\!\cdots\!76$$$$\beta_{1} +$$$$13\!\cdots\!48$$$$\beta_{2} -$$$$61\!\cdots\!20$$$$\beta_{3} -$$$$95\!\cdots\!00$$$$\beta_{4} +$$$$24\!\cdots\!80$$$$\beta_{5}) q^{47} +(-$$$$95\!\cdots\!07$$$$-$$$$43\!\cdots\!80$$$$\beta_{1} -$$$$25\!\cdots\!80$$$$\beta_{2} +$$$$11\!\cdots\!32$$$$\beta_{3} +$$$$39\!\cdots\!60$$$$\beta_{4} +$$$$67\!\cdots\!52$$$$\beta_{5}) q^{49} +(-$$$$52\!\cdots\!00$$$$-$$$$89\!\cdots\!75$$$$\beta_{1} -$$$$35\!\cdots\!20$$$$\beta_{2} -$$$$41\!\cdots\!00$$$$\beta_{3} +$$$$13\!\cdots\!60$$$$\beta_{4} +$$$$30\!\cdots\!00$$$$\beta_{5}) q^{50} +($$$$69\!\cdots\!00$$$$+$$$$22\!\cdots\!08$$$$\beta_{1} -$$$$26\!\cdots\!82$$$$\beta_{2} -$$$$10\!\cdots\!30$$$$\beta_{3} +$$$$69\!\cdots\!00$$$$\beta_{4} +$$$$58\!\cdots\!20$$$$\beta_{5}) q^{52} +(-$$$$10\!\cdots\!10$$$$-$$$$47\!\cdots\!84$$$$\beta_{1} +$$$$18\!\cdots\!52$$$$\beta_{2} +$$$$32\!\cdots\!44$$$$\beta_{3} +$$$$16\!\cdots\!85$$$$\beta_{4} -$$$$18\!\cdots\!96$$$$\beta_{5}) q^{53} +($$$$72\!\cdots\!20$$$$-$$$$37\!\cdots\!50$$$$\beta_{1} -$$$$41\!\cdots\!46$$$$\beta_{2} -$$$$84\!\cdots\!20$$$$\beta_{3} +$$$$25\!\cdots\!48$$$$\beta_{4} +$$$$88\!\cdots\!00$$$$\beta_{5}) q^{55} +(-$$$$48\!\cdots\!20$$$$-$$$$40\!\cdots\!84$$$$\beta_{1} +$$$$36\!\cdots\!44$$$$\beta_{2} +$$$$10\!\cdots\!80$$$$\beta_{3} +$$$$23\!\cdots\!80$$$$\beta_{4} -$$$$26\!\cdots\!44$$$$\beta_{5}) q^{56} +(-$$$$29\!\cdots\!80$$$$+$$$$52\!\cdots\!46$$$$\beta_{1} +$$$$82\!\cdots\!04$$$$\beta_{2} +$$$$11\!\cdots\!12$$$$\beta_{3} -$$$$28\!\cdots\!20$$$$\beta_{4} +$$$$98\!\cdots\!92$$$$\beta_{5}) q^{58} +($$$$41\!\cdots\!60$$$$-$$$$28\!\cdots\!53$$$$\beta_{1} -$$$$58\!\cdots\!67$$$$\beta_{2} -$$$$31\!\cdots\!52$$$$\beta_{3} -$$$$87\!\cdots\!00$$$$\beta_{4} -$$$$20\!\cdots\!60$$$$\beta_{5}) q^{59} +(-$$$$42\!\cdots\!98$$$$-$$$$26\!\cdots\!00$$$$\beta_{1} +$$$$94\!\cdots\!00$$$$\beta_{2} +$$$$60\!\cdots\!00$$$$\beta_{3} -$$$$40\!\cdots\!25$$$$\beta_{4} +$$$$64\!\cdots\!00$$$$\beta_{5}) q^{61} +($$$$48\!\cdots\!80$$$$+$$$$21\!\cdots\!28$$$$\beta_{1} +$$$$10\!\cdots\!20$$$$\beta_{2} -$$$$85\!\cdots\!68$$$$\beta_{3} -$$$$67\!\cdots\!20$$$$\beta_{4} -$$$$10\!\cdots\!88$$$$\beta_{5}) q^{62} +($$$$79\!\cdots\!08$$$$+$$$$23\!\cdots\!28$$$$\beta_{1} +$$$$31\!\cdots\!72$$$$\beta_{2} -$$$$57\!\cdots\!64$$$$\beta_{3} -$$$$20\!\cdots\!80$$$$\beta_{4} +$$$$15\!\cdots\!44$$$$\beta_{5}) q^{64} +(-$$$$20\!\cdots\!20$$$$-$$$$68\!\cdots\!00$$$$\beta_{1} -$$$$16\!\cdots\!24$$$$\beta_{2} +$$$$12\!\cdots\!00$$$$\beta_{3} -$$$$72\!\cdots\!88$$$$\beta_{4} -$$$$16\!\cdots\!20$$$$\beta_{5}) q^{65} +($$$$15\!\cdots\!80$$$$-$$$$88\!\cdots\!13$$$$\beta_{1} -$$$$35\!\cdots\!11$$$$\beta_{2} +$$$$11\!\cdots\!12$$$$\beta_{3} -$$$$36\!\cdots\!20$$$$\beta_{4} -$$$$10\!\cdots\!08$$$$\beta_{5}) q^{67} +(-$$$$20\!\cdots\!60$$$$+$$$$44\!\cdots\!88$$$$\beta_{1} -$$$$58\!\cdots\!06$$$$\beta_{2} -$$$$10\!\cdots\!74$$$$\beta_{3} +$$$$73\!\cdots\!40$$$$\beta_{4} +$$$$19\!\cdots\!16$$$$\beta_{5}) q^{68} +(-$$$$57\!\cdots\!40$$$$+$$$$42\!\cdots\!00$$$$\beta_{1} +$$$$69\!\cdots\!92$$$$\beta_{2} -$$$$11\!\cdots\!60$$$$\beta_{3} -$$$$24\!\cdots\!96$$$$\beta_{4} -$$$$10\!\cdots\!00$$$$\beta_{5}) q^{70} +($$$$42\!\cdots\!48$$$$+$$$$12\!\cdots\!50$$$$\beta_{1} +$$$$11\!\cdots\!50$$$$\beta_{2} -$$$$41\!\cdots\!00$$$$\beta_{3} -$$$$35\!\cdots\!00$$$$\beta_{4} +$$$$19\!\cdots\!00$$$$\beta_{5}) q^{71} +(-$$$$51\!\cdots\!70$$$$-$$$$10\!\cdots\!60$$$$\beta_{1} -$$$$12\!\cdots\!08$$$$\beta_{2} -$$$$21\!\cdots\!88$$$$\beta_{3} -$$$$39\!\cdots\!70$$$$\beta_{4} -$$$$24\!\cdots\!08$$$$\beta_{5}) q^{73} +($$$$40\!\cdots\!64$$$$+$$$$24\!\cdots\!70$$$$\beta_{1} -$$$$22\!\cdots\!80$$$$\beta_{2} +$$$$10\!\cdots\!92$$$$\beta_{3} -$$$$51\!\cdots\!40$$$$\beta_{4} +$$$$65\!\cdots\!12$$$$\beta_{5}) q^{74} +($$$$16\!\cdots\!40$$$$+$$$$14\!\cdots\!28$$$$\beta_{1} +$$$$89\!\cdots\!52$$$$\beta_{2} +$$$$39\!\cdots\!00$$$$\beta_{3} -$$$$17\!\cdots\!60$$$$\beta_{4} +$$$$84\!\cdots\!08$$$$\beta_{5}) q^{76} +(-$$$$26\!\cdots\!00$$$$+$$$$63\!\cdots\!84$$$$\beta_{1} +$$$$10\!\cdots\!64$$$$\beta_{2} -$$$$45\!\cdots\!92$$$$\beta_{3} -$$$$82\!\cdots\!80$$$$\beta_{4} -$$$$16\!\cdots\!72$$$$\beta_{5}) q^{77} +($$$$19\!\cdots\!20$$$$-$$$$22\!\cdots\!76$$$$\beta_{1} -$$$$29\!\cdots\!24$$$$\beta_{2} -$$$$19\!\cdots\!72$$$$\beta_{3} +$$$$21\!\cdots\!60$$$$\beta_{4} +$$$$52\!\cdots\!92$$$$\beta_{5}) q^{79} +(-$$$$20\!\cdots\!60$$$$-$$$$85\!\cdots\!00$$$$\beta_{1} -$$$$45\!\cdots\!72$$$$\beta_{2} -$$$$62\!\cdots\!40$$$$\beta_{3} +$$$$43\!\cdots\!36$$$$\beta_{4} -$$$$21\!\cdots\!00$$$$\beta_{5}) q^{80} +(-$$$$42\!\cdots\!80$$$$-$$$$10\!\cdots\!78$$$$\beta_{1} -$$$$30\!\cdots\!20$$$$\beta_{2} -$$$$98\!\cdots\!72$$$$\beta_{3} +$$$$31\!\cdots\!20$$$$\beta_{4} +$$$$15\!\cdots\!48$$$$\beta_{5}) q^{82} +($$$$13\!\cdots\!60$$$$+$$$$11\!\cdots\!97$$$$\beta_{1} +$$$$35\!\cdots\!71$$$$\beta_{2} -$$$$13\!\cdots\!00$$$$\beta_{3} +$$$$12\!\cdots\!00$$$$\beta_{4} -$$$$39\!\cdots\!00$$$$\beta_{5}) q^{83} +(-$$$$60\!\cdots\!40$$$$+$$$$10\!\cdots\!00$$$$\beta_{1} +$$$$86\!\cdots\!72$$$$\beta_{2} +$$$$12\!\cdots\!00$$$$\beta_{3} +$$$$46\!\cdots\!14$$$$\beta_{4} +$$$$69\!\cdots\!60$$$$\beta_{5}) q^{85} +(-$$$$12\!\cdots\!72$$$$-$$$$73\!\cdots\!29$$$$\beta_{1} -$$$$24\!\cdots\!01$$$$\beta_{2} -$$$$58\!\cdots\!17$$$$\beta_{3} +$$$$58\!\cdots\!70$$$$\beta_{4} +$$$$24\!\cdots\!89$$$$\beta_{5}) q^{86} +(-$$$$80\!\cdots\!60$$$$-$$$$16\!\cdots\!48$$$$\beta_{1} -$$$$84\!\cdots\!52$$$$\beta_{2} -$$$$10\!\cdots\!48$$$$\beta_{3} +$$$$45\!\cdots\!80$$$$\beta_{4} +$$$$22\!\cdots\!32$$$$\beta_{5}) q^{88} +(-$$$$89\!\cdots\!10$$$$+$$$$17\!\cdots\!48$$$$\beta_{1} +$$$$16\!\cdots\!32$$$$\beta_{2} -$$$$12\!\cdots\!60$$$$\beta_{3} +$$$$25\!\cdots\!90$$$$\beta_{4} +$$$$93\!\cdots\!68$$$$\beta_{5}) q^{89} +($$$$57\!\cdots\!72$$$$+$$$$40\!\cdots\!84$$$$\beta_{1} -$$$$39\!\cdots\!44$$$$\beta_{2} -$$$$30\!\cdots\!60$$$$\beta_{3} +$$$$23\!\cdots\!20$$$$\beta_{4} -$$$$55\!\cdots\!36$$$$\beta_{5}) q^{91} +(-$$$$31\!\cdots\!80$$$$+$$$$56\!\cdots\!68$$$$\beta_{1} -$$$$11\!\cdots\!44$$$$\beta_{2} -$$$$56\!\cdots\!48$$$$\beta_{3} -$$$$10\!\cdots\!20$$$$\beta_{4} -$$$$16\!\cdots\!68$$$$\beta_{5}) q^{92} +(-$$$$49\!\cdots\!84$$$$-$$$$14\!\cdots\!04$$$$\beta_{1} -$$$$13\!\cdots\!16$$$$\beta_{2} -$$$$10\!\cdots\!04$$$$\beta_{3} -$$$$24\!\cdots\!40$$$$\beta_{4} -$$$$90\!\cdots\!48$$$$\beta_{5}) q^{94} +(-$$$$32\!\cdots\!00$$$$-$$$$15\!\cdots\!50$$$$\beta_{1} -$$$$10\!\cdots\!10$$$$\beta_{2} -$$$$34\!\cdots\!00$$$$\beta_{3} -$$$$34\!\cdots\!20$$$$\beta_{4} -$$$$78\!\cdots\!00$$$$\beta_{5}) q^{95} +(-$$$$12\!\cdots\!30$$$$+$$$$94\!\cdots\!04$$$$\beta_{1} +$$$$13\!\cdots\!72$$$$\beta_{2} +$$$$16\!\cdots\!44$$$$\beta_{3} -$$$$53\!\cdots\!90$$$$\beta_{4} -$$$$13\!\cdots\!96$$$$\beta_{5}) q^{97} +($$$$27\!\cdots\!20$$$$+$$$$84\!\cdots\!27$$$$\beta_{1} -$$$$63\!\cdots\!20$$$$\beta_{2} +$$$$57\!\cdots\!72$$$$\beta_{3} +$$$$83\!\cdots\!80$$$$\beta_{4} +$$$$17\!\cdots\!52$$$$\beta_{5}) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 57080822040q^{2} +$$$$17\!\cdots\!28$$$$q^{4} +$$$$38\!\cdots\!40$$$$q^{5} +$$$$19\!\cdots\!00$$$$q^{7} -$$$$44\!\cdots\!20$$$$q^{8} + O(q^{10})$$ $$6q + 57080822040q^{2} +$$$$17\!\cdots\!28$$$$q^{4} +$$$$38\!\cdots\!40$$$$q^{5} +$$$$19\!\cdots\!00$$$$q^{7} -$$$$44\!\cdots\!20$$$$q^{8} +$$$$13\!\cdots\!60$$$$q^{10} +$$$$94\!\cdots\!88$$$$q^{11} +$$$$53\!\cdots\!20$$$$q^{13} -$$$$82\!\cdots\!76$$$$q^{14} +$$$$26\!\cdots\!16$$$$q^{16} -$$$$18\!\cdots\!80$$$$q^{17} +$$$$10\!\cdots\!80$$$$q^{19} -$$$$92\!\cdots\!80$$$$q^{20} +$$$$15\!\cdots\!20$$$$q^{22} -$$$$15\!\cdots\!80$$$$q^{23} +$$$$19\!\cdots\!50$$$$q^{25} -$$$$11\!\cdots\!52$$$$q^{26} +$$$$14\!\cdots\!20$$$$q^{28} -$$$$14\!\cdots\!20$$$$q^{29} -$$$$41\!\cdots\!88$$$$q^{31} -$$$$11\!\cdots\!60$$$$q^{32} +$$$$30\!\cdots\!56$$$$q^{34} -$$$$27\!\cdots\!60$$$$q^{35} +$$$$98\!\cdots\!40$$$$q^{37} -$$$$12\!\cdots\!80$$$$q^{38} +$$$$88\!\cdots\!00$$$$q^{40} -$$$$50\!\cdots\!12$$$$q^{41} +$$$$27\!\cdots\!00$$$$q^{43} +$$$$86\!\cdots\!44$$$$q^{44} -$$$$82\!\cdots\!88$$$$q^{46} +$$$$13\!\cdots\!80$$$$q^{47} -$$$$57\!\cdots\!42$$$$q^{49} -$$$$31\!\cdots\!00$$$$q^{50} +$$$$41\!\cdots\!00$$$$q^{52} -$$$$64\!\cdots\!60$$$$q^{53} +$$$$43\!\cdots\!20$$$$q^{55} -$$$$28\!\cdots\!20$$$$q^{56} -$$$$17\!\cdots\!80$$$$q^{58} +$$$$24\!\cdots\!60$$$$q^{59} -$$$$25\!\cdots\!88$$$$q^{61} +$$$$29\!\cdots\!80$$$$q^{62} +$$$$47\!\cdots\!48$$$$q^{64} -$$$$12\!\cdots\!20$$$$q^{65} +$$$$95\!\cdots\!80$$$$q^{67} -$$$$12\!\cdots\!60$$$$q^{68} -$$$$34\!\cdots\!40$$$$q^{70} +$$$$25\!\cdots\!88$$$$q^{71} -$$$$30\!\cdots\!20$$$$q^{73} +$$$$24\!\cdots\!84$$$$q^{74} +$$$$10\!\cdots\!40$$$$q^{76} -$$$$15\!\cdots\!00$$$$q^{77} +$$$$11\!\cdots\!20$$$$q^{79} -$$$$12\!\cdots\!60$$$$q^{80} -$$$$25\!\cdots\!80$$$$q^{82} +$$$$79\!\cdots\!60$$$$q^{83} -$$$$36\!\cdots\!40$$$$q^{85} -$$$$72\!\cdots\!32$$$$q^{86} -$$$$48\!\cdots\!60$$$$q^{88} -$$$$53\!\cdots\!60$$$$q^{89} +$$$$34\!\cdots\!32$$$$q^{91} -$$$$18\!\cdots\!80$$$$q^{92} -$$$$29\!\cdots\!04$$$$q^{94} -$$$$19\!\cdots\!00$$$$q^{95} -$$$$74\!\cdots\!80$$$$q^{97} +$$$$16\!\cdots\!20$$$$q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3 x^{5} - 38457853073924058692 x^{4} - 10276556354621685339901678086 x^{3} + 371187556674475060057870954681799784505 x^{2} + 52686123927652036687598761277591247931691204025 x - 675344021115865838575279495800656435684060652010336995750$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$72 \nu - 36$$ $$\beta_{2}$$ $$=$$ $$($$$$-4948663505321853 \nu^{5} + 55881811387913650437514590 \nu^{4} + 52606012732895043205010735272635258 \nu^{3} - 1201592828386165762149652689242435079026451552 \nu^{2} + 319255507039165145569757904502642123052905054955529291 \nu + 2824371663419232055800468842793976703776935082646788469007152058$$$$)/$$$$10\!\cdots\!92$$ $$\beta_{3}$$ $$=$$ $$($$$$4948663505321853 \nu^{5} - 55881811387913650437514590 \nu^{4} - 52606012732895043205010735272635258 \nu^{3} + 1755095654489252908059771761513699488141354080 \nu^{2} - 541112784048666521961353184744658440761253271177159755 \nu - 9919881784066391320337103830604500724241879350737223725145748410$$$$)/$$$$10\!\cdots\!92$$ $$\beta_{4}$$ $$=$$ $$($$$$28511749716630153831 \nu^{5} - 38083656874945883578519397562 \nu^{4} - 963750039419722142343400597053740315214 \nu^{3} + 676707927180689783081282528077970424187899981600 \nu^{2} + 6997720090731681654577112441812622855800067824107978126735 \nu - 2510088276770176763088355182310893932211704380215188959388288448270$$$$)/$$$$33\!\cdots\!60$$ $$\beta_{5}$$ $$=$$ $$($$$$135360654096714045236379 \nu^{5} - 181916208099474888046078592692146 \nu^{4} - 3229398604947863711716280584882203213740214 \nu^{3} + 1182663053764971608085531729909002820674984623978656 \nu^{2} + 8277972932132865971256201289051763726161866888924340410394275 \nu + 7463683837539256728882869193632480747348831952586787915716598591246730$$$$)/$$$$26\!\cdots\!48$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1} + 36$$$$)/72$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + 28859335836 \beta_{1} + 66455170111740773427552$$$$)/5184$$ $$\nu^{3}$$ $$=$$ $$($$$$927 \beta_{5} - 5496670 \beta_{4} + 13763504113 \beta_{3} + 13847230069 \beta_{2} + 12411954295199886478449 \beta_{1} + 239731509332049065134727670101088$$$$)/46656$$ $$\nu^{4}$$ $$=$$ $$($$$$19416283325423 \beta_{5} - 65775546702321150 \beta_{4} + 2070543718263884927080 \beta_{3} + 2982224271853793103276 \beta_{2} + 129858023070497361753663350041341 \beta_{1} + 103104816764360930778502571288626452423490272$$$$)/419904$$ $$\nu^{5}$$ $$=$$ $$($$$$1385746010729527810284396 \beta_{5} - 5708884777145460848148822360 \beta_{4} + 22636204558832382529314688604361 \beta_{3} + 28230689814524467376000755560217 \beta_{2} + 11081391647423040002963344026763817931453572 \beta_{1} + 539358563573136492249987437118564971890421060743009584$$$$)/1889568$$

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
 5.23553e9 3.19848e9 1.46671e9 −1.60397e9 −3.89081e9 −4.40594e9
−3.67444e11 0 9.72365e22 −2.43619e26 0 1.21530e31 −2.18473e34 0 8.95165e37
1.2 −2.20777e11 0 1.09637e22 2.30711e25 0 1.96027e31 5.92020e33 0 −5.09358e36
1.3 −9.60898e10 0 −2.85457e22 1.49650e26 0 −3.16100e31 6.37312e33 0 −1.43799e37
1.4 1.25000e11 0 −2.21540e22 −1.80927e26 0 2.76229e31 −7.49160e33 0 −2.26158e37
1.5 2.89652e11 0 4.61192e22 2.47463e26 0 −6.83079e31 2.41577e33 0 7.16781e37
1.6 3.26741e11 0 6.89808e22 4.33451e25 0 4.24638e31 1.01949e34 0 1.41626e37
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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 9.76.a.c 6
3.b odd 2 1 1.76.a.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.76.a.a 6 3.b odd 2 1
9.76.a.c 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} - 57080822040 T_{2}^{5} -$$$$19\!\cdots\!68$$$$T_{2}^{4} +$$$$11\!\cdots\!80$$$$T_{2}^{3} +$$$$97\!\cdots\!08$$$$T_{2}^{2} -$$$$29\!\cdots\!40$$$$T_{2} -$$$$92\!\cdots\!16$$ acting on $$S_{76}^{\mathrm{new}}(\Gamma_0(9))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-$$$$92\!\cdots\!16$$$$-$$$$29\!\cdots\!40$$$$T +$$$$97\!\cdots\!08$$$$T^{2} +$$$$11\!\cdots\!80$$$$T^{3} -$$$$19\!\cdots\!68$$$$T^{4} - 57080822040 T^{5} + T^{6}$$
$3$ $$T^{6}$$
$5$ $$16\!\cdots\!00$$$$-$$$$11\!\cdots\!00$$$$T +$$$$16\!\cdots\!00$$$$T^{2} +$$$$40\!\cdots\!00$$$$T^{3} -$$$$88\!\cdots\!00$$$$T^{4} -$$$$38\!\cdots\!40$$$$T^{5} + T^{6}$$
$7$ $$60\!\cdots\!44$$$$-$$$$88\!\cdots\!00$$$$T +$$$$27\!\cdots\!88$$$$T^{2} +$$$$89\!\cdots\!00$$$$T^{3} -$$$$43\!\cdots\!08$$$$T^{4} -$$$$19\!\cdots\!00$$$$T^{5} + T^{6}$$
$11$ $$-$$$$76\!\cdots\!36$$$$-$$$$39\!\cdots\!08$$$$T +$$$$84\!\cdots\!40$$$$T^{2} +$$$$12\!\cdots\!60$$$$T^{3} -$$$$17\!\cdots\!40$$$$T^{4} -$$$$94\!\cdots\!88$$$$T^{5} + T^{6}$$
$13$ $$87\!\cdots\!16$$$$-$$$$18\!\cdots\!20$$$$T +$$$$79\!\cdots\!08$$$$T^{2} +$$$$50\!\cdots\!60$$$$T^{3} -$$$$11\!\cdots\!32$$$$T^{4} -$$$$53\!\cdots\!20$$$$T^{5} + T^{6}$$
$17$ $$-$$$$35\!\cdots\!36$$$$-$$$$12\!\cdots\!20$$$$T -$$$$11\!\cdots\!52$$$$T^{2} -$$$$33\!\cdots\!60$$$$T^{3} -$$$$14\!\cdots\!88$$$$T^{4} +$$$$18\!\cdots\!80$$$$T^{5} + T^{6}$$
$19$ $$-$$$$12\!\cdots\!00$$$$-$$$$71\!\cdots\!00$$$$T +$$$$15\!\cdots\!00$$$$T^{2} +$$$$19\!\cdots\!00$$$$T^{3} -$$$$24\!\cdots\!00$$$$T^{4} -$$$$10\!\cdots\!80$$$$T^{5} + T^{6}$$
$23$ $$10\!\cdots\!96$$$$+$$$$26\!\cdots\!80$$$$T +$$$$91\!\cdots\!68$$$$T^{2} -$$$$40\!\cdots\!40$$$$T^{3} -$$$$20\!\cdots\!52$$$$T^{4} +$$$$15\!\cdots\!80$$$$T^{5} + T^{6}$$
$29$ $$99\!\cdots\!00$$$$+$$$$26\!\cdots\!00$$$$T -$$$$16\!\cdots\!00$$$$T^{2} -$$$$54\!\cdots\!00$$$$T^{3} +$$$$14\!\cdots\!00$$$$T^{4} +$$$$14\!\cdots\!20$$$$T^{5} + T^{6}$$
$31$ $$-$$$$14\!\cdots\!36$$$$+$$$$46\!\cdots\!08$$$$T +$$$$16\!\cdots\!40$$$$T^{2} -$$$$74\!\cdots\!60$$$$T^{3} -$$$$26\!\cdots\!40$$$$T^{4} +$$$$41\!\cdots\!88$$$$T^{5} + T^{6}$$
$37$ $$-$$$$18\!\cdots\!96$$$$-$$$$32\!\cdots\!40$$$$T +$$$$33\!\cdots\!68$$$$T^{2} +$$$$59\!\cdots\!80$$$$T^{3} -$$$$89\!\cdots\!48$$$$T^{4} -$$$$98\!\cdots\!40$$$$T^{5} + T^{6}$$
$41$ $$16\!\cdots\!64$$$$+$$$$46\!\cdots\!92$$$$T -$$$$70\!\cdots\!60$$$$T^{2} -$$$$16\!\cdots\!40$$$$T^{3} -$$$$25\!\cdots\!40$$$$T^{4} +$$$$50\!\cdots\!12$$$$T^{5} + T^{6}$$
$43$ $$52\!\cdots\!56$$$$-$$$$59\!\cdots\!00$$$$T +$$$$16\!\cdots\!88$$$$T^{2} +$$$$27\!\cdots\!00$$$$T^{3} -$$$$92\!\cdots\!92$$$$T^{4} -$$$$27\!\cdots\!00$$$$T^{5} + T^{6}$$
$47$ $$11\!\cdots\!24$$$$-$$$$22\!\cdots\!80$$$$T -$$$$53\!\cdots\!72$$$$T^{2} +$$$$35\!\cdots\!60$$$$T^{3} +$$$$11\!\cdots\!72$$$$T^{4} -$$$$13\!\cdots\!80$$$$T^{5} + T^{6}$$
$53$ $$83\!\cdots\!36$$$$-$$$$12\!\cdots\!40$$$$T -$$$$66\!\cdots\!52$$$$T^{2} -$$$$38\!\cdots\!80$$$$T^{3} -$$$$48\!\cdots\!12$$$$T^{4} +$$$$64\!\cdots\!60$$$$T^{5} + T^{6}$$
$59$ $$-$$$$41\!\cdots\!00$$$$+$$$$42\!\cdots\!00$$$$T +$$$$10\!\cdots\!00$$$$T^{2} +$$$$20\!\cdots\!00$$$$T^{3} -$$$$18\!\cdots\!00$$$$T^{4} -$$$$24\!\cdots\!60$$$$T^{5} + T^{6}$$
$61$ $$57\!\cdots\!64$$$$+$$$$28\!\cdots\!08$$$$T -$$$$54\!\cdots\!60$$$$T^{2} -$$$$16\!\cdots\!60$$$$T^{3} +$$$$78\!\cdots\!60$$$$T^{4} +$$$$25\!\cdots\!88$$$$T^{5} + T^{6}$$
$67$ $$21\!\cdots\!64$$$$+$$$$86\!\cdots\!20$$$$T -$$$$20\!\cdots\!52$$$$T^{2} +$$$$63\!\cdots\!60$$$$T^{3} +$$$$18\!\cdots\!12$$$$T^{4} -$$$$95\!\cdots\!80$$$$T^{5} + T^{6}$$
$71$ $$14\!\cdots\!64$$$$+$$$$81\!\cdots\!92$$$$T +$$$$82\!\cdots\!40$$$$T^{2} -$$$$70\!\cdots\!40$$$$T^{3} -$$$$17\!\cdots\!40$$$$T^{4} -$$$$25\!\cdots\!88$$$$T^{5} + T^{6}$$
$73$ $$-$$$$66\!\cdots\!04$$$$-$$$$55\!\cdots\!80$$$$T -$$$$13\!\cdots\!32$$$$T^{2} -$$$$46\!\cdots\!60$$$$T^{3} +$$$$25\!\cdots\!48$$$$T^{4} +$$$$30\!\cdots\!20$$$$T^{5} + T^{6}$$
$79$ $$15\!\cdots\!00$$$$-$$$$18\!\cdots\!00$$$$T +$$$$19\!\cdots\!00$$$$T^{2} +$$$$31\!\cdots\!00$$$$T^{3} -$$$$33\!\cdots\!00$$$$T^{4} -$$$$11\!\cdots\!20$$$$T^{5} + T^{6}$$
$83$ $$13\!\cdots\!76$$$$-$$$$50\!\cdots\!60$$$$T +$$$$25\!\cdots\!28$$$$T^{2} +$$$$85\!\cdots\!80$$$$T^{3} -$$$$13\!\cdots\!72$$$$T^{4} -$$$$79\!\cdots\!60$$$$T^{5} + T^{6}$$
$89$ $$-$$$$16\!\cdots\!00$$$$-$$$$10\!\cdots\!00$$$$T -$$$$14\!\cdots\!00$$$$T^{2} -$$$$20\!\cdots\!00$$$$T^{3} +$$$$80\!\cdots\!00$$$$T^{4} +$$$$53\!\cdots\!60$$$$T^{5} + T^{6}$$
$97$ $$36\!\cdots\!24$$$$+$$$$64\!\cdots\!80$$$$T -$$$$19\!\cdots\!72$$$$T^{2} -$$$$54\!\cdots\!60$$$$T^{3} +$$$$25\!\cdots\!72$$$$T^{4} +$$$$74\!\cdots\!80$$$$T^{5} + T^{6}$$