# Properties

 Label 1.50.a.a Level $1$ Weight $50$ Character orbit 1.a Self dual yes Analytic conductor $15.207$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1$$ Weight: $$k$$ $$=$$ $$50$$ Character orbit: $$[\chi]$$ $$=$$ 1.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$15.2066205099$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ Defining polynomial: $$x^{3} - x^{2} - 27962089502 x + 71708842875120$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{11}\cdot 3^{5}\cdot 5^{2}\cdot 7^{2}$$ 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -8075056 + \beta_{1} ) q^{2} + ( -108984897468 + 4110 \beta_{1} + \beta_{2} ) q^{3} + ( 10500922661632 - 399312 \beta_{1} - 96 \beta_{2} ) q^{4} + ( 21294678572359750 - 830946040 \beta_{1} - 192420 \beta_{2} ) q^{5} + ( 2968712083082256192 - 140296680252 \beta_{1} - 24220416 \beta_{2} ) q^{6} + ( 169797159166237064 - 6197674750596 \beta_{1} + 657909714 \beta_{2} ) q^{7} + ( 4258131001838821888000 - 549479574747904 \beta_{1} + 2325616128 \beta_{2} ) q^{8} + ( 113875230170760151789773 - 11457427166565264 \beta_{1} - 286203921912 \beta_{2} ) q^{9} +O(q^{10})$$ $$q +(-8075056 + \beta_{1}) q^{2} +(-108984897468 + 4110 \beta_{1} + \beta_{2}) q^{3} +(10500922661632 - 399312 \beta_{1} - 96 \beta_{2}) q^{4} +(21294678572359750 - 830946040 \beta_{1} - 192420 \beta_{2}) q^{5} +(2968712083082256192 - 140296680252 \beta_{1} - 24220416 \beta_{2}) q^{6} +(169797159166237064 - 6197674750596 \beta_{1} + 657909714 \beta_{2}) q^{7} +($$$$42\!\cdots\!00$$$$- 549479574747904 \beta_{1} + 2325616128 \beta_{2}) q^{8} +($$$$11\!\cdots\!73$$$$- 11457427166565264 \beta_{1} - 286203921912 \beta_{2}) q^{9} +(-$$$$59\!\cdots\!00$$$$+ 27011895709376070 \beta_{1} + 4664342031360 \beta_{2}) q^{10} +(-$$$$69\!\cdots\!08$$$$+ 1063535619042334570 \beta_{1} - 33484749559965 \beta_{2}) q^{11} +(-$$$$33\!\cdots\!16$$$$+ 1100579697151435968 \beta_{1} + 27590671758976 \beta_{2}) q^{12} +(-$$$$62\!\cdots\!38$$$$- 57281740925974331832 \beta_{1} + 1632545471760060 \beta_{2}) q^{13} +(-$$$$31\!\cdots\!96$$$$- 88757573956408427896 \beta_{1} - 15080285330707968 \beta_{2}) q^{14} +(-$$$$68\!\cdots\!00$$$$+$$$$22\!\cdots\!60$$$$\beta_{1} + 56042589608616630 \beta_{2}) q^{15} +(-$$$$31\!\cdots\!04$$$$+$$$$11\!\cdots\!96$$$$\beta_{1} + 51383439727239168 \beta_{2}) q^{16} +(-$$$$11\!\cdots\!46$$$$-$$$$46\!\cdots\!40$$$$\beta_{1} - 1609842257538727608 \beta_{2}) q^{17} +(-$$$$67\!\cdots\!48$$$$+$$$$43\!\cdots\!45$$$$\beta_{1} + 7918966438100822016 \beta_{2}) q^{18} +(-$$$$15\!\cdots\!20$$$$+$$$$51\!\cdots\!22$$$$\beta_{1} - 16168497468226428699 \beta_{2}) q^{19} +($$$$65\!\cdots\!00$$$$-$$$$21\!\cdots\!80$$$$\beta_{1} - 5402253524702091840 \beta_{2}) q^{20} +($$$$20\!\cdots\!72$$$$-$$$$62\!\cdots\!24$$$$\beta_{1} + 27296567013048957008 \beta_{2}) q^{21} +($$$$59\!\cdots\!48$$$$+$$$$33\!\cdots\!32$$$$\beta_{1} +$$$$69\!\cdots\!20$$$$\beta_{2}) q^{22} +($$$$15\!\cdots\!92$$$$+$$$$41\!\cdots\!36$$$$\beta_{1} -$$$$46\!\cdots\!26$$$$\beta_{2}) q^{23} +(-$$$$83\!\cdots\!60$$$$+$$$$51\!\cdots\!76$$$$\beta_{1} +$$$$12\!\cdots\!08$$$$\beta_{2}) q^{24} +(-$$$$46\!\cdots\!25$$$$-$$$$42\!\cdots\!00$$$$\beta_{1} -$$$$10\!\cdots\!00$$$$\beta_{2}) q^{25} +(-$$$$28\!\cdots\!88$$$$-$$$$60\!\cdots\!86$$$$\beta_{1} -$$$$33\!\cdots\!88$$$$\beta_{2}) q^{26} +(-$$$$10\!\cdots\!00$$$$+$$$$37\!\cdots\!56$$$$\beta_{1} +$$$$91\!\cdots\!58$$$$\beta_{2}) q^{27} +(-$$$$19\!\cdots\!32$$$$+$$$$60\!\cdots\!64$$$$\beta_{1} -$$$$25\!\cdots\!44$$$$\beta_{2}) q^{28} +($$$$37\!\cdots\!70$$$$-$$$$49\!\cdots\!92$$$$\beta_{1} -$$$$22\!\cdots\!36$$$$\beta_{2}) q^{29} +($$$$16\!\cdots\!00$$$$-$$$$54\!\cdots\!80$$$$\beta_{1} -$$$$15\!\cdots\!40$$$$\beta_{2}) q^{30} +($$$$31\!\cdots\!72$$$$+$$$$12\!\cdots\!60$$$$\beta_{1} +$$$$52\!\cdots\!80$$$$\beta_{2}) q^{31} +($$$$24\!\cdots\!64$$$$-$$$$12\!\cdots\!64$$$$\beta_{1} -$$$$25\!\cdots\!60$$$$\beta_{2}) q^{32} +(-$$$$81\!\cdots\!56$$$$+$$$$21\!\cdots\!40$$$$\beta_{1} -$$$$20\!\cdots\!48$$$$\beta_{2}) q^{33} +(-$$$$14\!\cdots\!76$$$$-$$$$13\!\cdots\!14$$$$\beta_{1} +$$$$42\!\cdots\!88$$$$\beta_{2}) q^{34} +(-$$$$39\!\cdots\!00$$$$+$$$$12\!\cdots\!20$$$$\beta_{1} -$$$$46\!\cdots\!40$$$$\beta_{2}) q^{35} +($$$$12\!\cdots\!36$$$$-$$$$45\!\cdots\!84$$$$\beta_{1} -$$$$31\!\cdots\!72$$$$\beta_{2}) q^{36} +($$$$80\!\cdots\!34$$$$+$$$$78\!\cdots\!12$$$$\beta_{1} -$$$$16\!\cdots\!32$$$$\beta_{2}) q^{37} +($$$$38\!\cdots\!00$$$$-$$$$10\!\cdots\!76$$$$\beta_{1} +$$$$33\!\cdots\!32$$$$\beta_{2}) q^{38} +($$$$43\!\cdots\!16$$$$-$$$$11\!\cdots\!76$$$$\beta_{1} +$$$$68\!\cdots\!42$$$$\beta_{2}) q^{39} +($$$$17\!\cdots\!00$$$$-$$$$99\!\cdots\!00$$$$\beta_{1} -$$$$24\!\cdots\!00$$$$\beta_{2}) q^{40} +(-$$$$20\!\cdots\!38$$$$+$$$$44\!\cdots\!60$$$$\beta_{1} +$$$$16\!\cdots\!80$$$$\beta_{2}) q^{41} +(-$$$$48\!\cdots\!92$$$$+$$$$15\!\cdots\!24$$$$\beta_{1} -$$$$46\!\cdots\!44$$$$\beta_{2}) q^{42} +(-$$$$25\!\cdots\!48$$$$-$$$$26\!\cdots\!18$$$$\beta_{1} +$$$$53\!\cdots\!07$$$$\beta_{2}) q^{43} +($$$$78\!\cdots\!44$$$$-$$$$20\!\cdots\!64$$$$\beta_{1} +$$$$19\!\cdots\!88$$$$\beta_{2}) q^{44} +($$$$25\!\cdots\!50$$$$-$$$$91\!\cdots\!20$$$$\beta_{1} -$$$$63\!\cdots\!60$$$$\beta_{2}) q^{45} +($$$$89\!\cdots\!32$$$$+$$$$21\!\cdots\!00$$$$\beta_{1} +$$$$10\!\cdots\!00$$$$\beta_{2}) q^{46} +($$$$24\!\cdots\!24$$$$+$$$$97\!\cdots\!08$$$$\beta_{1} +$$$$65\!\cdots\!16$$$$\beta_{2}) q^{47} +($$$$52\!\cdots\!92$$$$-$$$$18\!\cdots\!24$$$$\beta_{1} -$$$$32\!\cdots\!56$$$$\beta_{2}) q^{48} +(-$$$$93\!\cdots\!43$$$$-$$$$33\!\cdots\!80$$$$\beta_{1} +$$$$15\!\cdots\!60$$$$\beta_{2}) q^{49} +(-$$$$17\!\cdots\!00$$$$-$$$$72\!\cdots\!25$$$$\beta_{1} +$$$$30\!\cdots\!00$$$$\beta_{2}) q^{50} +(-$$$$51\!\cdots\!68$$$$+$$$$17\!\cdots\!32$$$$\beta_{1} -$$$$17\!\cdots\!94$$$$\beta_{2}) q^{51} +(-$$$$41\!\cdots\!56$$$$+$$$$10\!\cdots\!24$$$$\beta_{1} -$$$$65\!\cdots\!36$$$$\beta_{2}) q^{52} +(-$$$$51\!\cdots\!18$$$$+$$$$37\!\cdots\!60$$$$\beta_{1} +$$$$57\!\cdots\!76$$$$\beta_{2}) q^{53} +($$$$27\!\cdots\!80$$$$-$$$$82\!\cdots\!28$$$$\beta_{1} -$$$$25\!\cdots\!24$$$$\beta_{2}) q^{54} +($$$$15\!\cdots\!00$$$$-$$$$40\!\cdots\!80$$$$\beta_{1} +$$$$38\!\cdots\!10$$$$\beta_{2}) q^{55} +($$$$22\!\cdots\!80$$$$+$$$$35\!\cdots\!92$$$$\beta_{1} +$$$$84\!\cdots\!36$$$$\beta_{2}) q^{56} +(-$$$$26\!\cdots\!00$$$$+$$$$52\!\cdots\!12$$$$\beta_{1} -$$$$21\!\cdots\!84$$$$\beta_{2}) q^{57} +(-$$$$55\!\cdots\!00$$$$+$$$$33\!\cdots\!86$$$$\beta_{1} +$$$$10\!\cdots\!48$$$$\beta_{2}) q^{58} +(-$$$$20\!\cdots\!60$$$$-$$$$21\!\cdots\!14$$$$\beta_{1} +$$$$21\!\cdots\!63$$$$\beta_{2}) q^{59} +(-$$$$29\!\cdots\!00$$$$+$$$$10\!\cdots\!20$$$$\beta_{1} +$$$$10\!\cdots\!60$$$$\beta_{2}) q^{60} +(-$$$$80\!\cdots\!58$$$$-$$$$20\!\cdots\!00$$$$\beta_{1} +$$$$35\!\cdots\!00$$$$\beta_{2}) q^{61} +($$$$62\!\cdots\!68$$$$+$$$$96\!\cdots\!92$$$$\beta_{1} -$$$$13\!\cdots\!40$$$$\beta_{2}) q^{62} +(-$$$$26\!\cdots\!88$$$$+$$$$27\!\cdots\!84$$$$\beta_{1} +$$$$14\!\cdots\!98$$$$\beta_{2}) q^{63} +($$$$17\!\cdots\!12$$$$+$$$$24\!\cdots\!36$$$$\beta_{1} +$$$$32\!\cdots\!88$$$$\beta_{2}) q^{64} +(-$$$$81\!\cdots\!00$$$$+$$$$22\!\cdots\!60$$$$\beta_{1} -$$$$12\!\cdots\!20$$$$\beta_{2}) q^{65} +($$$$17\!\cdots\!64$$$$-$$$$52\!\cdots\!44$$$$\beta_{1} +$$$$46\!\cdots\!48$$$$\beta_{2}) q^{66} +(-$$$$33\!\cdots\!96$$$$-$$$$14\!\cdots\!10$$$$\beta_{1} -$$$$96\!\cdots\!43$$$$\beta_{2}) q^{67} +($$$$49\!\cdots\!48$$$$-$$$$16\!\cdots\!24$$$$\beta_{1} +$$$$16\!\cdots\!12$$$$\beta_{2}) q^{68} +(-$$$$16\!\cdots\!24$$$$+$$$$50\!\cdots\!72$$$$\beta_{1} +$$$$13\!\cdots\!76$$$$\beta_{2}) q^{69} +($$$$93\!\cdots\!00$$$$-$$$$29\!\cdots\!60$$$$\beta_{1} -$$$$57\!\cdots\!80$$$$\beta_{2}) q^{70} +(-$$$$10\!\cdots\!68$$$$+$$$$40\!\cdots\!00$$$$\beta_{1} -$$$$15\!\cdots\!50$$$$\beta_{2}) q^{71} +($$$$34\!\cdots\!00$$$$-$$$$13\!\cdots\!72$$$$\beta_{1} -$$$$36\!\cdots\!96$$$$\beta_{2}) q^{72} +($$$$18\!\cdots\!42$$$$-$$$$22\!\cdots\!04$$$$\beta_{1} +$$$$14\!\cdots\!04$$$$\beta_{2}) q^{73} +($$$$33\!\cdots\!64$$$$+$$$$15\!\cdots\!30$$$$\beta_{1} +$$$$32\!\cdots\!40$$$$\beta_{2}) q^{74} +(-$$$$40\!\cdots\!00$$$$+$$$$14\!\cdots\!50$$$$\beta_{1} +$$$$34\!\cdots\!75$$$$\beta_{2}) q^{75} +($$$$25\!\cdots\!60$$$$-$$$$50\!\cdots\!76$$$$\beta_{1} +$$$$20\!\cdots\!92$$$$\beta_{2}) q^{76} +(-$$$$10\!\cdots\!12$$$$+$$$$10\!\cdots\!08$$$$\beta_{1} -$$$$19\!\cdots\!92$$$$\beta_{2}) q^{77} +(-$$$$92\!\cdots\!36$$$$+$$$$30\!\cdots\!64$$$$\beta_{1} -$$$$15\!\cdots\!56$$$$\beta_{2}) q^{78} +(-$$$$26\!\cdots\!80$$$$-$$$$14\!\cdots\!52$$$$\beta_{1} +$$$$39\!\cdots\!84$$$$\beta_{2}) q^{79} +(-$$$$10\!\cdots\!00$$$$+$$$$37\!\cdots\!60$$$$\beta_{1} +$$$$63\!\cdots\!80$$$$\beta_{2}) q^{80} +($$$$22\!\cdots\!81$$$$+$$$$94\!\cdots\!48$$$$\beta_{1} -$$$$10\!\cdots\!16$$$$\beta_{2}) q^{81} +($$$$39\!\cdots\!28$$$$-$$$$18\!\cdots\!18$$$$\beta_{1} -$$$$44\!\cdots\!40$$$$\beta_{2}) q^{82} +($$$$31\!\cdots\!72$$$$+$$$$45\!\cdots\!34$$$$\beta_{1} +$$$$10\!\cdots\!53$$$$\beta_{2}) q^{83} +($$$$25\!\cdots\!04$$$$-$$$$11\!\cdots\!92$$$$\beta_{1} -$$$$29\!\cdots\!36$$$$\beta_{2}) q^{84} +($$$$99\!\cdots\!00$$$$-$$$$33\!\cdots\!80$$$$\beta_{1} +$$$$31\!\cdots\!60$$$$\beta_{2}) q^{85} +(-$$$$11\!\cdots\!28$$$$-$$$$50\!\cdots\!08$$$$\beta_{1} -$$$$10\!\cdots\!64$$$$\beta_{2}) q^{86} +(-$$$$58\!\cdots\!00$$$$+$$$$23\!\cdots\!68$$$$\beta_{1} +$$$$45\!\cdots\!74$$$$\beta_{2}) q^{87} +(-$$$$35\!\cdots\!00$$$$-$$$$13\!\cdots\!68$$$$\beta_{1} -$$$$43\!\cdots\!24$$$$\beta_{2}) q^{88} +($$$$12\!\cdots\!10$$$$+$$$$21\!\cdots\!64$$$$\beta_{1} -$$$$11\!\cdots\!88$$$$\beta_{2}) q^{89} +(-$$$$66\!\cdots\!00$$$$+$$$$22\!\cdots\!10$$$$\beta_{1} +$$$$16\!\cdots\!80$$$$\beta_{2}) q^{90} +($$$$53\!\cdots\!92$$$$-$$$$21\!\cdots\!68$$$$\beta_{1} +$$$$74\!\cdots\!56$$$$\beta_{2}) q^{91} +($$$$15\!\cdots\!04$$$$-$$$$48\!\cdots\!00$$$$\beta_{1} -$$$$13\!\cdots\!88$$$$\beta_{2}) q^{92} +($$$$19\!\cdots\!04$$$$-$$$$68\!\cdots\!20$$$$\beta_{1} -$$$$23\!\cdots\!48$$$$\beta_{2}) q^{93} +($$$$30\!\cdots\!84$$$$+$$$$27\!\cdots\!92$$$$\beta_{1} -$$$$16\!\cdots\!64$$$$\beta_{2}) q^{94} +($$$$49\!\cdots\!00$$$$-$$$$97\!\cdots\!00$$$$\beta_{1} +$$$$41\!\cdots\!50$$$$\beta_{2}) q^{95} +(-$$$$89\!\cdots\!48$$$$+$$$$29\!\cdots\!68$$$$\beta_{1} +$$$$72\!\cdots\!44$$$$\beta_{2}) q^{96} +(-$$$$34\!\cdots\!26$$$$+$$$$19\!\cdots\!48$$$$\beta_{1} +$$$$59\!\cdots\!36$$$$\beta_{2}) q^{97} +(-$$$$94\!\cdots\!92$$$$-$$$$12\!\cdots\!03$$$$\beta_{1} -$$$$33\!\cdots\!80$$$$\beta_{2}) q^{98} +(-$$$$37\!\cdots\!84$$$$-$$$$95\!\cdots\!78$$$$\beta_{1} -$$$$13\!\cdots\!49$$$$\beta_{2}) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 24225168q^{2} - 326954692404q^{3} + 31502767984896q^{4} + 63884035717079250q^{5} + 8906136249246768576q^{6} + 509391477498711192q^{7} + 12774393005516465664000q^{8} + 341625690512280455369319q^{9} + O(q^{10})$$ $$3q - 24225168q^{2} - 326954692404q^{3} + 31502767984896q^{4} + 63884035717079250q^{5} + 8906136249246768576q^{6} + 509391477498711192q^{7} +$$$$12\!\cdots\!00$$$$q^{8} +$$$$34\!\cdots\!19$$$$q^{9} -$$$$17\!\cdots\!00$$$$q^{10} -$$$$20\!\cdots\!24$$$$q^{11} -$$$$10\!\cdots\!48$$$$q^{12} -$$$$18\!\cdots\!14$$$$q^{13} -$$$$94\!\cdots\!88$$$$q^{14} -$$$$20\!\cdots\!00$$$$q^{15} -$$$$95\!\cdots\!12$$$$q^{16} -$$$$33\!\cdots\!38$$$$q^{17} -$$$$20\!\cdots\!44$$$$q^{18} -$$$$45\!\cdots\!60$$$$q^{19} +$$$$19\!\cdots\!00$$$$q^{20} +$$$$61\!\cdots\!16$$$$q^{21} +$$$$17\!\cdots\!44$$$$q^{22} +$$$$45\!\cdots\!76$$$$q^{23} -$$$$25\!\cdots\!80$$$$q^{24} -$$$$13\!\cdots\!75$$$$q^{25} -$$$$85\!\cdots\!64$$$$q^{26} -$$$$31\!\cdots\!00$$$$q^{27} -$$$$59\!\cdots\!96$$$$q^{28} +$$$$11\!\cdots\!10$$$$q^{29} +$$$$50\!\cdots\!00$$$$q^{30} +$$$$93\!\cdots\!16$$$$q^{31} +$$$$73\!\cdots\!92$$$$q^{32} -$$$$24\!\cdots\!68$$$$q^{33} -$$$$44\!\cdots\!28$$$$q^{34} -$$$$11\!\cdots\!00$$$$q^{35} +$$$$37\!\cdots\!08$$$$q^{36} +$$$$24\!\cdots\!02$$$$q^{37} +$$$$11\!\cdots\!00$$$$q^{38} +$$$$12\!\cdots\!48$$$$q^{39} +$$$$52\!\cdots\!00$$$$q^{40} -$$$$62\!\cdots\!14$$$$q^{41} -$$$$14\!\cdots\!76$$$$q^{42} -$$$$77\!\cdots\!44$$$$q^{43} +$$$$23\!\cdots\!32$$$$q^{44} +$$$$76\!\cdots\!50$$$$q^{45} +$$$$26\!\cdots\!96$$$$q^{46} +$$$$72\!\cdots\!72$$$$q^{47} +$$$$15\!\cdots\!76$$$$q^{48} -$$$$28\!\cdots\!29$$$$q^{49} -$$$$53\!\cdots\!00$$$$q^{50} -$$$$15\!\cdots\!04$$$$q^{51} -$$$$12\!\cdots\!68$$$$q^{52} -$$$$15\!\cdots\!54$$$$q^{53} +$$$$82\!\cdots\!40$$$$q^{54} +$$$$46\!\cdots\!00$$$$q^{55} +$$$$67\!\cdots\!40$$$$q^{56} -$$$$79\!\cdots\!00$$$$q^{57} -$$$$16\!\cdots\!00$$$$q^{58} -$$$$62\!\cdots\!80$$$$q^{59} -$$$$88\!\cdots\!00$$$$q^{60} -$$$$24\!\cdots\!74$$$$q^{61} +$$$$18\!\cdots\!04$$$$q^{62} -$$$$79\!\cdots\!64$$$$q^{63} +$$$$51\!\cdots\!36$$$$q^{64} -$$$$24\!\cdots\!00$$$$q^{65} +$$$$52\!\cdots\!92$$$$q^{66} -$$$$10\!\cdots\!88$$$$q^{67} +$$$$14\!\cdots\!44$$$$q^{68} -$$$$48\!\cdots\!72$$$$q^{69} +$$$$28\!\cdots\!00$$$$q^{70} -$$$$32\!\cdots\!04$$$$q^{71} +$$$$10\!\cdots\!00$$$$q^{72} +$$$$56\!\cdots\!26$$$$q^{73} +$$$$10\!\cdots\!92$$$$q^{74} -$$$$12\!\cdots\!00$$$$q^{75} +$$$$75\!\cdots\!80$$$$q^{76} -$$$$32\!\cdots\!36$$$$q^{77} -$$$$27\!\cdots\!08$$$$q^{78} -$$$$78\!\cdots\!40$$$$q^{79} -$$$$30\!\cdots\!00$$$$q^{80} +$$$$67\!\cdots\!43$$$$q^{81} +$$$$11\!\cdots\!84$$$$q^{82} +$$$$94\!\cdots\!16$$$$q^{83} +$$$$77\!\cdots\!12$$$$q^{84} +$$$$29\!\cdots\!00$$$$q^{85} -$$$$34\!\cdots\!84$$$$q^{86} -$$$$17\!\cdots\!00$$$$q^{87} -$$$$10\!\cdots\!00$$$$q^{88} +$$$$36\!\cdots\!30$$$$q^{89} -$$$$20\!\cdots\!00$$$$q^{90} +$$$$16\!\cdots\!76$$$$q^{91} +$$$$46\!\cdots\!12$$$$q^{92} +$$$$59\!\cdots\!12$$$$q^{93} +$$$$90\!\cdots\!52$$$$q^{94} +$$$$14\!\cdots\!00$$$$q^{95} -$$$$26\!\cdots\!44$$$$q^{96} -$$$$10\!\cdots\!78$$$$q^{97} -$$$$28\!\cdots\!76$$$$q^{98} -$$$$11\!\cdots\!52$$$$q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 27962089502 x + 71708842875120$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-24 \nu^{2} + 58056 \nu + 447393412688$$$$)/14051$$ $$\beta_{2}$$ $$=$$ $$($$$$39888 \nu^{2} + 59389824528 \nu - 743587680658656$$$$)/14051$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 1662 \beta_{1} + 1411200$$$$)/4233600$$ $$\nu^{2}$$ $$=$$ $$($$$$2419 \beta_{2} - 2474576022 \beta_{1} + 78920201411856000$$$$)/4233600$$

## 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
 −168486. 165922. 2565.11
−2.54182e7 −8.64745e11 8.31363e13 1.67413e17 2.19803e19 −3.42668e20 1.21960e22 5.08484e23 −4.25535e24
1.2 −2.25719e7 5.57972e11 −5.34581e13 −1.06460e17 −1.25945e19 5.68014e20 1.39135e22 7.20337e22 2.40300e24
1.3 2.37650e7 −2.01823e10 1.82457e12 2.93049e15 −4.79631e17 −2.24836e20 −1.33351e22 −2.38892e23 6.96431e22
 $$n$$: 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 1.50.a.a 3
3.b odd 2 1 9.50.a.a 3
4.b odd 2 1 16.50.a.c 3

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{50}^{\mathrm{new}}(\Gamma_0(1))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-$$$$13\!\cdots\!72$$$$- 566746931810304 T + 24225168 T^{2} + T^{3}$$
$3$ $$-$$$$97\!\cdots\!04$$$$-$$$$47\!\cdots\!76$$$$T + 326954692404 T^{2} + T^{3}$$
$5$ $$52\!\cdots\!00$$$$-$$$$17\!\cdots\!00$$$$T - 63884035717079250 T^{2} + T^{3}$$
$7$ $$-$$$$43\!\cdots\!12$$$$-$$$$24\!\cdots\!64$$$$T - 509391477498711192 T^{2} + T^{3}$$
$11$ $$-$$$$32\!\cdots\!88$$$$-$$$$12\!\cdots\!08$$$$T +$$$$20\!\cdots\!24$$$$T^{2} + T^{3}$$
$13$ $$-$$$$71\!\cdots\!24$$$$-$$$$38\!\cdots\!16$$$$T +$$$$18\!\cdots\!14$$$$T^{2} + T^{3}$$
$17$ $$-$$$$32\!\cdots\!92$$$$+$$$$69\!\cdots\!16$$$$T +$$$$33\!\cdots\!38$$$$T^{2} + T^{3}$$
$19$ $$-$$$$81\!\cdots\!00$$$$+$$$$36\!\cdots\!00$$$$T +$$$$45\!\cdots\!60$$$$T^{2} + T^{3}$$
$23$ $$29\!\cdots\!56$$$$-$$$$52\!\cdots\!56$$$$T -$$$$45\!\cdots\!76$$$$T^{2} + T^{3}$$
$29$ $$-$$$$43\!\cdots\!00$$$$+$$$$39\!\cdots\!00$$$$T -$$$$11\!\cdots\!10$$$$T^{2} + T^{3}$$
$31$ $$52\!\cdots\!52$$$$-$$$$26\!\cdots\!48$$$$T -$$$$93\!\cdots\!16$$$$T^{2} + T^{3}$$
$37$ $$31\!\cdots\!48$$$$-$$$$41\!\cdots\!24$$$$T -$$$$24\!\cdots\!02$$$$T^{2} + T^{3}$$
$41$ $$42\!\cdots\!72$$$$+$$$$98\!\cdots\!32$$$$T +$$$$62\!\cdots\!14$$$$T^{2} + T^{3}$$
$43$ $$-$$$$92\!\cdots\!84$$$$-$$$$49\!\cdots\!36$$$$T +$$$$77\!\cdots\!44$$$$T^{2} + T^{3}$$
$47$ $$11\!\cdots\!68$$$$-$$$$11\!\cdots\!44$$$$T -$$$$72\!\cdots\!72$$$$T^{2} + T^{3}$$
$53$ $$-$$$$46\!\cdots\!04$$$$-$$$$10\!\cdots\!76$$$$T +$$$$15\!\cdots\!54$$$$T^{2} + T^{3}$$
$59$ $$17\!\cdots\!00$$$$+$$$$10\!\cdots\!00$$$$T +$$$$62\!\cdots\!80$$$$T^{2} + T^{3}$$
$61$ $$43\!\cdots\!12$$$$-$$$$29\!\cdots\!08$$$$T +$$$$24\!\cdots\!74$$$$T^{2} + T^{3}$$
$67$ $$-$$$$36\!\cdots\!92$$$$-$$$$28\!\cdots\!84$$$$T +$$$$10\!\cdots\!88$$$$T^{2} + T^{3}$$
$71$ $$-$$$$67\!\cdots\!68$$$$+$$$$21\!\cdots\!72$$$$T +$$$$32\!\cdots\!04$$$$T^{2} + T^{3}$$
$73$ $$16\!\cdots\!56$$$$-$$$$29\!\cdots\!56$$$$T -$$$$56\!\cdots\!26$$$$T^{2} + T^{3}$$
$79$ $$-$$$$10\!\cdots\!00$$$$-$$$$23\!\cdots\!00$$$$T +$$$$78\!\cdots\!40$$$$T^{2} + T^{3}$$
$83$ $$88\!\cdots\!36$$$$-$$$$18\!\cdots\!96$$$$T -$$$$94\!\cdots\!16$$$$T^{2} + T^{3}$$
$89$ $$15\!\cdots\!00$$$$-$$$$61\!\cdots\!00$$$$T -$$$$36\!\cdots\!30$$$$T^{2} + T^{3}$$
$97$ $$-$$$$12\!\cdots\!32$$$$+$$$$63\!\cdots\!56$$$$T +$$$$10\!\cdots\!78$$$$T^{2} + T^{3}$$