# Properties

 Label 1.44.a.a Level $1$ Weight $44$ Character orbit 1.a Self dual yes Analytic conductor $11.711$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1$$ Weight: $$k$$ $$=$$ $$44$$ Character orbit: $$[\chi]$$ $$=$$ 1.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$11.7110395346$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ Defining polynomial: $$x^{3} - 11258260111 x - 264759545317170$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{11}\cdot 3^{4}\cdot 5\cdot 7$$ 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -736648 - \beta_{1} ) q^{2} + ( 8133812604 + 2135 \beta_{1} - \beta_{2} ) q^{3} + ( 3274206140608 + 810816 \beta_{1} - 408 \beta_{2} ) q^{4} + ( 178401793591390 - 23658932 \beta_{1} - 29844 \beta_{2} ) q^{5} + ( -30597991586147808 - 21343496316 \beta_{1} + 2270208 \beta_{2} ) q^{6} + ( 100657141888492952 - 411863382714 \beta_{1} - 60797226 \beta_{2} ) q^{7} + ( -5276954637929116160 + 136805587456 \beta_{1} + 901657152 \beta_{2} ) q^{8} + ( 159160708390355581077 + 108619653530952 \beta_{1} - 7904835576 \beta_{2} ) q^{9} +O(q^{10})$$ $$q +(-736648 - \beta_{1}) q^{2} +(8133812604 + 2135 \beta_{1} - \beta_{2}) q^{3} +(3274206140608 + 810816 \beta_{1} - 408 \beta_{2}) q^{4} +(178401793591390 - 23658932 \beta_{1} - 29844 \beta_{2}) q^{5} +(-30597991586147808 - 21343496316 \beta_{1} + 2270208 \beta_{2}) q^{6} +(100657141888492952 - 411863382714 \beta_{1} - 60797226 \beta_{2}) q^{7} +(-5276954637929116160 + 136805587456 \beta_{1} + 901657152 \beta_{2}) q^{8} +($$$$15\!\cdots\!77$$$$+ 108619653530952 \beta_{1} - 7904835576 \beta_{2}) q^{9} +($$$$14\!\cdots\!40$$$$- 566151100617822 \beta_{1} + 32102731776 \beta_{2}) q^{10} +($$$$88\!\cdots\!32$$$$- 4605494939944195 \beta_{1} + 137241701685 \beta_{2}) q^{11} +($$$$19\!\cdots\!12$$$$+ 43030582618825472 \beta_{1} - 3088365053344 \beta_{2}) q^{12} +($$$$89\!\cdots\!94$$$$- 6729074743426932 \beta_{1} + 22807920190380 \beta_{2}) q^{13} +($$$$46\!\cdots\!76$$$$- 863595024061582232 \beta_{1} - 82977158928384 \beta_{2}) q^{14} +($$$$11\!\cdots\!80$$$$+ 1376784350493064386 \beta_{1} - 37572060903438 \beta_{2}) q^{15} +(-$$$$26\!\cdots\!64$$$$+ 9902624655975026688 \beta_{1} + 2383088864979456 \beta_{2}) q^{16} +(-$$$$13\!\cdots\!98$$$$- 4605229780369521720 \beta_{1} - 15383154249178488 \beta_{2}) q^{17} +(-$$$$13\!\cdots\!16$$$$-$$$$27\!\cdots\!45$$$$\beta_{1} + 55376695430406144 \beta_{2}) q^{18} +($$$$52\!\cdots\!00$$$$+$$$$85\!\cdots\!19$$$$\beta_{1} - 107308048565285397 \beta_{2}) q^{19} +($$$$48\!\cdots\!20$$$$+$$$$52\!\cdots\!44$$$$\beta_{1} - 13394879801587152 \beta_{2}) q^{20} +($$$$13\!\cdots\!12$$$$-$$$$49\!\cdots\!32$$$$\beta_{1} + 743940509412748816 \beta_{2}) q^{21} +($$$$46\!\cdots\!64$$$$-$$$$67\!\cdots\!52$$$$\beta_{1} - 2071060643092362240 \beta_{2}) q^{22} +(-$$$$40\!\cdots\!16$$$$+$$$$54\!\cdots\!86$$$$\beta_{1} + 1876113512966847906 \beta_{2}) q^{23} +(-$$$$37\!\cdots\!00$$$$-$$$$52\!\cdots\!88$$$$\beta_{1} + 1908534981075097344 \beta_{2}) q^{24} +(-$$$$77\!\cdots\!25$$$$-$$$$83\!\cdots\!40$$$$\beta_{1} + 1683060411116573520 \beta_{2}) q^{25} +(-$$$$57\!\cdots\!48$$$$-$$$$59\!\cdots\!58$$$$\beta_{1} - 34656662255444176896 \beta_{2}) q^{26} +($$$$42\!\cdots\!60$$$$+$$$$21\!\cdots\!74$$$$\beta_{1} + 30213151036337641158 \beta_{2}) q^{27} +($$$$56\!\cdots\!56$$$$-$$$$20\!\cdots\!76$$$$\beta_{1} +$$$$29\!\cdots\!04$$$$\beta_{2}) q^{28} +($$$$19\!\cdots\!50$$$$+$$$$17\!\cdots\!76$$$$\beta_{1} -$$$$10\!\cdots\!88$$$$\beta_{2}) q^{29} +(-$$$$24\!\cdots\!20$$$$-$$$$12\!\cdots\!44$$$$\beta_{1} +$$$$61\!\cdots\!52$$$$\beta_{2}) q^{30} +(-$$$$85\!\cdots\!08$$$$+$$$$11\!\cdots\!40$$$$\beta_{1} +$$$$33\!\cdots\!80$$$$\beta_{2}) q^{31} +(-$$$$48\!\cdots\!28$$$$+$$$$55\!\cdots\!24$$$$\beta_{1} -$$$$72\!\cdots\!80$$$$\beta_{2}) q^{32} +(-$$$$91\!\cdots\!72$$$$-$$$$78\!\cdots\!20$$$$\beta_{1} -$$$$23\!\cdots\!12$$$$\beta_{2}) q^{33} +($$$$15\!\cdots\!96$$$$-$$$$64\!\cdots\!58$$$$\beta_{1} +$$$$19\!\cdots\!04$$$$\beta_{2}) q^{34} +($$$$79\!\cdots\!40$$$$-$$$$22\!\cdots\!92$$$$\beta_{1} +$$$$14\!\cdots\!36$$$$\beta_{2}) q^{35} +($$$$27\!\cdots\!16$$$$+$$$$11\!\cdots\!68$$$$\beta_{1} -$$$$11\!\cdots\!84$$$$\beta_{2}) q^{36} +(-$$$$77\!\cdots\!98$$$$-$$$$43\!\cdots\!72$$$$\beta_{1} +$$$$46\!\cdots\!08$$$$\beta_{2}) q^{37} +(-$$$$10\!\cdots\!40$$$$-$$$$19\!\cdots\!96$$$$\beta_{1} +$$$$49\!\cdots\!68$$$$\beta_{2}) q^{38} +(-$$$$13\!\cdots\!76$$$$+$$$$65\!\cdots\!78$$$$\beta_{1} -$$$$96\!\cdots\!14$$$$\beta_{2}) q^{39} +(-$$$$10\!\cdots\!00$$$$-$$$$86\!\cdots\!00$$$$\beta_{1} -$$$$48\!\cdots\!00$$$$\beta_{2}) q^{40} +($$$$85\!\cdots\!22$$$$+$$$$87\!\cdots\!40$$$$\beta_{1} +$$$$23\!\cdots\!80$$$$\beta_{2}) q^{41} +($$$$47\!\cdots\!44$$$$-$$$$30\!\cdots\!24$$$$\beta_{1} -$$$$30\!\cdots\!04$$$$\beta_{2}) q^{42} +($$$$80\!\cdots\!64$$$$-$$$$12\!\cdots\!83$$$$\beta_{1} +$$$$17\!\cdots\!73$$$$\beta_{2}) q^{43} +(-$$$$34\!\cdots\!44$$$$-$$$$32\!\cdots\!48$$$$\beta_{1} -$$$$10\!\cdots\!76$$$$\beta_{2}) q^{44} +($$$$85\!\cdots\!30$$$$+$$$$62\!\cdots\!36$$$$\beta_{1} -$$$$40\!\cdots\!88$$$$\beta_{2}) q^{45} +(-$$$$63\!\cdots\!88$$$$+$$$$20\!\cdots\!60$$$$\beta_{1} +$$$$19\!\cdots\!20$$$$\beta_{2}) q^{46} +($$$$10\!\cdots\!52$$$$+$$$$31\!\cdots\!32$$$$\beta_{1} -$$$$25\!\cdots\!84$$$$\beta_{2}) q^{47} +(-$$$$85\!\cdots\!16$$$$+$$$$22\!\cdots\!16$$$$\beta_{1} +$$$$29\!\cdots\!16$$$$\beta_{2}) q^{48} +($$$$11\!\cdots\!93$$$$-$$$$77\!\cdots\!00$$$$\beta_{1} -$$$$30\!\cdots\!00$$$$\beta_{2}) q^{49} +($$$$66\!\cdots\!00$$$$+$$$$79\!\cdots\!85$$$$\beta_{1} -$$$$57\!\cdots\!80$$$$\beta_{2}) q^{50} +($$$$44\!\cdots\!52$$$$+$$$$37\!\cdots\!26$$$$\beta_{1} +$$$$19\!\cdots\!62$$$$\beta_{2}) q^{51} +(-$$$$57\!\cdots\!68$$$$+$$$$23\!\cdots\!76$$$$\beta_{1} -$$$$39\!\cdots\!16$$$$\beta_{2}) q^{52} +($$$$50\!\cdots\!54$$$$+$$$$47\!\cdots\!80$$$$\beta_{1} -$$$$90\!\cdots\!36$$$$\beta_{2}) q^{53} +(-$$$$28\!\cdots\!00$$$$-$$$$39\!\cdots\!36$$$$\beta_{1} +$$$$84\!\cdots\!68$$$$\beta_{2}) q^{54} +($$$$13\!\cdots\!80$$$$-$$$$29\!\cdots\!74$$$$\beta_{1} -$$$$24\!\cdots\!58$$$$\beta_{2}) q^{55} +(-$$$$21\!\cdots\!00$$$$+$$$$60\!\cdots\!96$$$$\beta_{1} -$$$$53\!\cdots\!48$$$$\beta_{2}) q^{56} +($$$$64\!\cdots\!20$$$$+$$$$23\!\cdots\!28$$$$\beta_{1} -$$$$14\!\cdots\!24$$$$\beta_{2}) q^{57} +(-$$$$34\!\cdots\!60$$$$-$$$$32\!\cdots\!34$$$$\beta_{1} +$$$$21\!\cdots\!72$$$$\beta_{2}) q^{58} +($$$$75\!\cdots\!00$$$$-$$$$62\!\cdots\!43$$$$\beta_{1} +$$$$37\!\cdots\!09$$$$\beta_{2}) q^{59} +($$$$57\!\cdots\!40$$$$+$$$$21\!\cdots\!88$$$$\beta_{1} -$$$$56\!\cdots\!04$$$$\beta_{2}) q^{60} +(-$$$$31\!\cdots\!18$$$$-$$$$68\!\cdots\!00$$$$\beta_{1} -$$$$30\!\cdots\!00$$$$\beta_{2}) q^{61} +(-$$$$69\!\cdots\!16$$$$+$$$$12\!\cdots\!48$$$$\beta_{1} -$$$$20\!\cdots\!20$$$$\beta_{2}) q^{62} +(-$$$$32\!\cdots\!16$$$$+$$$$28\!\cdots\!14$$$$\beta_{1} +$$$$11\!\cdots\!62$$$$\beta_{2}) q^{63} +(-$$$$37\!\cdots\!12$$$$-$$$$13\!\cdots\!68$$$$\beta_{1} +$$$$11\!\cdots\!84$$$$\beta_{2}) q^{64} +(-$$$$90\!\cdots\!20$$$$-$$$$32\!\cdots\!84$$$$\beta_{1} -$$$$31\!\cdots\!28$$$$\beta_{2}) q^{65} +($$$$97\!\cdots\!44$$$$+$$$$67\!\cdots\!48$$$$\beta_{1} -$$$$28\!\cdots\!24$$$$\beta_{2}) q^{66} +(-$$$$24\!\cdots\!48$$$$-$$$$23\!\cdots\!25$$$$\beta_{1} +$$$$46\!\cdots\!27$$$$\beta_{2}) q^{67} +($$$$18\!\cdots\!56$$$$+$$$$14\!\cdots\!36$$$$\beta_{1} +$$$$81\!\cdots\!28$$$$\beta_{2}) q^{68} +($$$$65\!\cdots\!44$$$$+$$$$99\!\cdots\!24$$$$\beta_{1} -$$$$90\!\cdots\!12$$$$\beta_{2}) q^{69} +($$$$19\!\cdots\!40$$$$-$$$$58\!\cdots\!32$$$$\beta_{1} -$$$$11\!\cdots\!44$$$$\beta_{2}) q^{70} +(-$$$$61\!\cdots\!88$$$$-$$$$47\!\cdots\!50$$$$\beta_{1} +$$$$97\!\cdots\!50$$$$\beta_{2}) q^{71} +(-$$$$32\!\cdots\!20$$$$-$$$$19\!\cdots\!68$$$$\beta_{1} +$$$$15\!\cdots\!44$$$$\beta_{2}) q^{72} +(-$$$$67\!\cdots\!66$$$$+$$$$18\!\cdots\!36$$$$\beta_{1} -$$$$13\!\cdots\!44$$$$\beta_{2}) q^{73} +($$$$55\!\cdots\!36$$$$+$$$$14\!\cdots\!50$$$$\beta_{1} -$$$$24\!\cdots\!00$$$$\beta_{2}) q^{74} +(-$$$$70\!\cdots\!00$$$$-$$$$18\!\cdots\!55$$$$\beta_{1} +$$$$77\!\cdots\!65$$$$\beta_{2}) q^{75} +($$$$25\!\cdots\!00$$$$+$$$$93\!\cdots\!92$$$$\beta_{1} -$$$$56\!\cdots\!96$$$$\beta_{2}) q^{76} +($$$$19\!\cdots\!64$$$$-$$$$84\!\cdots\!68$$$$\beta_{1} -$$$$12\!\cdots\!72$$$$\beta_{2}) q^{77} +(-$$$$65\!\cdots\!32$$$$-$$$$11\!\cdots\!76$$$$\beta_{1} +$$$$16\!\cdots\!16$$$$\beta_{2}) q^{78} +($$$$50\!\cdots\!00$$$$+$$$$26\!\cdots\!96$$$$\beta_{1} +$$$$11\!\cdots\!52$$$$\beta_{2}) q^{79} +(-$$$$33\!\cdots\!60$$$$+$$$$56\!\cdots\!48$$$$\beta_{1} +$$$$15\!\cdots\!16$$$$\beta_{2}) q^{80} +($$$$24\!\cdots\!21$$$$+$$$$14\!\cdots\!44$$$$\beta_{1} -$$$$50\!\cdots\!72$$$$\beta_{2}) q^{81} +(-$$$$10\!\cdots\!56$$$$+$$$$20\!\cdots\!18$$$$\beta_{1} +$$$$32\!\cdots\!80$$$$\beta_{2}) q^{82} +(-$$$$29\!\cdots\!76$$$$-$$$$22\!\cdots\!81$$$$\beta_{1} +$$$$66\!\cdots\!87$$$$\beta_{2}) q^{83} +(-$$$$11\!\cdots\!04$$$$-$$$$43\!\cdots\!84$$$$\beta_{1} -$$$$34\!\cdots\!08$$$$\beta_{2}) q^{84} +($$$$14\!\cdots\!40$$$$+$$$$55\!\cdots\!68$$$$\beta_{1} +$$$$76\!\cdots\!56$$$$\beta_{2}) q^{85} +($$$$83\!\cdots\!32$$$$-$$$$55\!\cdots\!84$$$$\beta_{1} -$$$$75\!\cdots\!08$$$$\beta_{2}) q^{86} +($$$$57\!\cdots\!80$$$$+$$$$12\!\cdots\!62$$$$\beta_{1} -$$$$18\!\cdots\!46$$$$\beta_{2}) q^{87} +(-$$$$85\!\cdots\!20$$$$+$$$$82\!\cdots\!92$$$$\beta_{1} +$$$$64\!\cdots\!64$$$$\beta_{2}) q^{88} +($$$$67\!\cdots\!50$$$$-$$$$45\!\cdots\!52$$$$\beta_{1} +$$$$31\!\cdots\!76$$$$\beta_{2}) q^{89} +(-$$$$77\!\cdots\!20$$$$-$$$$14\!\cdots\!94$$$$\beta_{1} +$$$$31\!\cdots\!52$$$$\beta_{2}) q^{90} +(-$$$$38\!\cdots\!28$$$$-$$$$28\!\cdots\!64$$$$\beta_{1} -$$$$77\!\cdots\!68$$$$\beta_{2}) q^{91} +($$$$22\!\cdots\!52$$$$+$$$$40\!\cdots\!60$$$$\beta_{1} -$$$$35\!\cdots\!28$$$$\beta_{2}) q^{92} +(-$$$$16\!\cdots\!32$$$$-$$$$11\!\cdots\!00$$$$\beta_{1} +$$$$55\!\cdots\!68$$$$\beta_{2}) q^{93} +(-$$$$44\!\cdots\!44$$$$-$$$$43\!\cdots\!16$$$$\beta_{1} +$$$$48\!\cdots\!08$$$$\beta_{2}) q^{94} +($$$$10\!\cdots\!00$$$$+$$$$52\!\cdots\!50$$$$\beta_{1} +$$$$12\!\cdots\!50$$$$\beta_{2}) q^{95} +($$$$36\!\cdots\!12$$$$+$$$$13\!\cdots\!44$$$$\beta_{1} -$$$$11\!\cdots\!72$$$$\beta_{2}) q^{96} +(-$$$$12\!\cdots\!98$$$$-$$$$11\!\cdots\!28$$$$\beta_{1} +$$$$21\!\cdots\!96$$$$\beta_{2}) q^{97} +($$$$80\!\cdots\!36$$$$-$$$$10\!\cdots\!93$$$$\beta_{1} -$$$$31\!\cdots\!00$$$$\beta_{2}) q^{98} +(-$$$$47\!\cdots\!36$$$$-$$$$12\!\cdots\!51$$$$\beta_{1} +$$$$17\!\cdots\!13$$$$\beta_{2}) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 2209944q^{2} + 24401437812q^{3} + 9822618421824q^{4} + 535205380774170q^{5} - 91793974758443424q^{6} + 301971425665478856q^{7} - 15830863913787348480q^{8} + 477482125171066743231q^{9} + O(q^{10})$$ $$3q - 2209944q^{2} + 24401437812q^{3} + 9822618421824q^{4} + 535205380774170q^{5} - 91793974758443424q^{6} + 301971425665478856q^{7} - 15830863913787348480q^{8} +$$$$47\!\cdots\!31$$$$q^{9} +$$$$42\!\cdots\!20$$$$q^{10} +$$$$26\!\cdots\!96$$$$q^{11} +$$$$59\!\cdots\!36$$$$q^{12} +$$$$26\!\cdots\!82$$$$q^{13} +$$$$14\!\cdots\!28$$$$q^{14} +$$$$35\!\cdots\!40$$$$q^{15} -$$$$79\!\cdots\!92$$$$q^{16} -$$$$40\!\cdots\!94$$$$q^{17} -$$$$41\!\cdots\!48$$$$q^{18} +$$$$15\!\cdots\!00$$$$q^{19} +$$$$14\!\cdots\!60$$$$q^{20} +$$$$39\!\cdots\!36$$$$q^{21} +$$$$13\!\cdots\!92$$$$q^{22} -$$$$12\!\cdots\!48$$$$q^{23} -$$$$11\!\cdots\!00$$$$q^{24} -$$$$23\!\cdots\!75$$$$q^{25} -$$$$17\!\cdots\!44$$$$q^{26} +$$$$12\!\cdots\!80$$$$q^{27} +$$$$16\!\cdots\!68$$$$q^{28} +$$$$57\!\cdots\!50$$$$q^{29} -$$$$73\!\cdots\!60$$$$q^{30} -$$$$25\!\cdots\!24$$$$q^{31} -$$$$14\!\cdots\!84$$$$q^{32} -$$$$27\!\cdots\!16$$$$q^{33} +$$$$46\!\cdots\!88$$$$q^{34} +$$$$23\!\cdots\!20$$$$q^{35} +$$$$81\!\cdots\!48$$$$q^{36} -$$$$23\!\cdots\!94$$$$q^{37} -$$$$30\!\cdots\!20$$$$q^{38} -$$$$39\!\cdots\!28$$$$q^{39} -$$$$32\!\cdots\!00$$$$q^{40} +$$$$25\!\cdots\!66$$$$q^{41} +$$$$14\!\cdots\!32$$$$q^{42} +$$$$24\!\cdots\!92$$$$q^{43} -$$$$10\!\cdots\!32$$$$q^{44} +$$$$25\!\cdots\!90$$$$q^{45} -$$$$18\!\cdots\!64$$$$q^{46} +$$$$30\!\cdots\!56$$$$q^{47} -$$$$25\!\cdots\!48$$$$q^{48} +$$$$34\!\cdots\!79$$$$q^{49} +$$$$19\!\cdots\!00$$$$q^{50} +$$$$13\!\cdots\!56$$$$q^{51} -$$$$17\!\cdots\!04$$$$q^{52} +$$$$15\!\cdots\!62$$$$q^{53} -$$$$84\!\cdots\!00$$$$q^{54} +$$$$39\!\cdots\!40$$$$q^{55} -$$$$64\!\cdots\!00$$$$q^{56} +$$$$19\!\cdots\!60$$$$q^{57} -$$$$10\!\cdots\!80$$$$q^{58} +$$$$22\!\cdots\!00$$$$q^{59} +$$$$17\!\cdots\!20$$$$q^{60} -$$$$93\!\cdots\!54$$$$q^{61} -$$$$20\!\cdots\!48$$$$q^{62} -$$$$96\!\cdots\!48$$$$q^{63} -$$$$11\!\cdots\!36$$$$q^{64} -$$$$27\!\cdots\!60$$$$q^{65} +$$$$29\!\cdots\!32$$$$q^{66} -$$$$73\!\cdots\!44$$$$q^{67} +$$$$54\!\cdots\!68$$$$q^{68} +$$$$19\!\cdots\!32$$$$q^{69} +$$$$59\!\cdots\!20$$$$q^{70} -$$$$18\!\cdots\!64$$$$q^{71} -$$$$98\!\cdots\!60$$$$q^{72} -$$$$20\!\cdots\!98$$$$q^{73} +$$$$16\!\cdots\!08$$$$q^{74} -$$$$21\!\cdots\!00$$$$q^{75} +$$$$77\!\cdots\!00$$$$q^{76} +$$$$59\!\cdots\!92$$$$q^{77} -$$$$19\!\cdots\!96$$$$q^{78} +$$$$15\!\cdots\!00$$$$q^{79} -$$$$10\!\cdots\!80$$$$q^{80} +$$$$73\!\cdots\!63$$$$q^{81} -$$$$32\!\cdots\!68$$$$q^{82} -$$$$89\!\cdots\!28$$$$q^{83} -$$$$34\!\cdots\!12$$$$q^{84} +$$$$43\!\cdots\!20$$$$q^{85} +$$$$25\!\cdots\!96$$$$q^{86} +$$$$17\!\cdots\!40$$$$q^{87} -$$$$25\!\cdots\!60$$$$q^{88} +$$$$20\!\cdots\!50$$$$q^{89} -$$$$23\!\cdots\!60$$$$q^{90} -$$$$11\!\cdots\!84$$$$q^{91} +$$$$68\!\cdots\!56$$$$q^{92} -$$$$49\!\cdots\!96$$$$q^{93} -$$$$13\!\cdots\!32$$$$q^{94} +$$$$31\!\cdots\!00$$$$q^{95} +$$$$10\!\cdots\!36$$$$q^{96} -$$$$38\!\cdots\!94$$$$q^{97} +$$$$24\!\cdots\!08$$$$q^{98} -$$$$14\!\cdots\!08$$$$q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 11258260111 x - 264759545317170$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-6 \nu^{2} + 343074 \nu + 45033040444$$$$)/8369$$ $$\beta_{2}$$ $$=$$ $$($$$$14838 \nu^{2} + 1176206478 \nu - 111366709018012$$$$)/8369$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 2473 \beta_{1}$$$$)/241920$$ $$\nu^{2}$$ $$=$$ $$($$$$57179 \beta_{2} - 196034413 \beta_{1} + 1815732190702080$$$$)/241920$$

## 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
 −24885.9 116336. −91450.2
−4.65343e6 3.22027e10 1.28583e13 5.54482e14 −1.49853e17 −5.57604e17 −1.89031e19 7.08758e20 −2.58024e21
1.2 −1.18359e6 −1.79507e10 −7.39521e12 −6.39116e14 2.12463e16 −1.72730e18 1.91639e19 −6.02939e18 7.56451e20
1.3 3.62707e6 1.01494e10 4.35955e12 6.19839e14 3.68127e16 2.58688e18 −1.60917e19 −2.25246e20 2.24820e21
 $$n$$: 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 1.44.a.a 3
3.b odd 2 1 9.44.a.b 3
4.b odd 2 1 16.44.a.c 3

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{44}^{\mathrm{new}}(\Gamma_0(1))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-19976984434430705664 - 15663522502656 T + 2209944 T^{2} + T^{3}$$
$3$ $$58\!\cdots\!08$$$$-$$$$43\!\cdots\!84$$$$T - 24401437812 T^{2} + T^{3}$$
$5$ $$21\!\cdots\!00$$$$-$$$$40\!\cdots\!00$$$$T - 535205380774170 T^{2} + T^{3}$$
$7$ $$-$$$$24\!\cdots\!84$$$$-$$$$49\!\cdots\!36$$$$T - 301971425665478856 T^{2} + T^{3}$$
$11$ $$32\!\cdots\!32$$$$-$$$$14\!\cdots\!28$$$$T -$$$$26\!\cdots\!96$$$$T^{2} + T^{3}$$
$13$ $$-$$$$50\!\cdots\!72$$$$+$$$$20\!\cdots\!96$$$$T -$$$$26\!\cdots\!82$$$$T^{2} + T^{3}$$
$17$ $$28\!\cdots\!76$$$$-$$$$75\!\cdots\!96$$$$T +$$$$40\!\cdots\!94$$$$T^{2} + T^{3}$$
$19$ $$-$$$$21\!\cdots\!00$$$$-$$$$18\!\cdots\!00$$$$T -$$$$15\!\cdots\!00$$$$T^{2} + T^{3}$$
$23$ $$36\!\cdots\!48$$$$-$$$$54\!\cdots\!24$$$$T +$$$$12\!\cdots\!48$$$$T^{2} + T^{3}$$
$29$ $$73\!\cdots\!00$$$$+$$$$46\!\cdots\!00$$$$T -$$$$57\!\cdots\!50$$$$T^{2} + T^{3}$$
$31$ $$-$$$$17\!\cdots\!88$$$$+$$$$13\!\cdots\!92$$$$T +$$$$25\!\cdots\!24$$$$T^{2} + T^{3}$$
$37$ $$56\!\cdots\!96$$$$-$$$$26\!\cdots\!16$$$$T +$$$$23\!\cdots\!94$$$$T^{2} + T^{3}$$
$41$ $$27\!\cdots\!52$$$$-$$$$40\!\cdots\!48$$$$T -$$$$25\!\cdots\!66$$$$T^{2} + T^{3}$$
$43$ $$-$$$$49\!\cdots\!12$$$$+$$$$14\!\cdots\!36$$$$T -$$$$24\!\cdots\!92$$$$T^{2} + T^{3}$$
$47$ $$86\!\cdots\!56$$$$-$$$$33\!\cdots\!76$$$$T -$$$$30\!\cdots\!56$$$$T^{2} + T^{3}$$
$53$ $$-$$$$90\!\cdots\!92$$$$+$$$$66\!\cdots\!16$$$$T -$$$$15\!\cdots\!62$$$$T^{2} + T^{3}$$
$59$ $$78\!\cdots\!00$$$$+$$$$87\!\cdots\!00$$$$T -$$$$22\!\cdots\!00$$$$T^{2} + T^{3}$$
$61$ $$-$$$$87\!\cdots\!68$$$$-$$$$87\!\cdots\!28$$$$T +$$$$93\!\cdots\!54$$$$T^{2} + T^{3}$$
$67$ $$42\!\cdots\!76$$$$-$$$$19\!\cdots\!96$$$$T +$$$$73\!\cdots\!44$$$$T^{2} + T^{3}$$
$71$ $$18\!\cdots\!72$$$$+$$$$10\!\cdots\!32$$$$T +$$$$18\!\cdots\!64$$$$T^{2} + T^{3}$$
$73$ $$-$$$$31\!\cdots\!52$$$$+$$$$69\!\cdots\!76$$$$T +$$$$20\!\cdots\!98$$$$T^{2} + T^{3}$$
$79$ $$-$$$$26\!\cdots\!00$$$$-$$$$83\!\cdots\!00$$$$T -$$$$15\!\cdots\!00$$$$T^{2} + T^{3}$$
$83$ $$-$$$$17\!\cdots\!32$$$$-$$$$30\!\cdots\!44$$$$T +$$$$89\!\cdots\!28$$$$T^{2} + T^{3}$$
$89$ $$-$$$$10\!\cdots\!00$$$$-$$$$52\!\cdots\!00$$$$T -$$$$20\!\cdots\!50$$$$T^{2} + T^{3}$$
$97$ $$-$$$$26\!\cdots\!44$$$$-$$$$17\!\cdots\!76$$$$T +$$$$38\!\cdots\!94$$$$T^{2} + T^{3}$$