Properties

 Label 3.38.a.b Level 3 Weight 38 Character orbit 3.a Self dual yes Analytic conductor 26.014 Analytic rank 0 Dimension 4 CM no Inner twists 1

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$3$$ Weight: $$k$$ $$=$$ $$38$$ Character orbit: $$[\chi]$$ $$=$$ 3.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$26.0142114374$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - x^{3} - 11777633936 x^{2} - 35120319927360 x + 11967042111800832000$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{15}\cdot 3^{10}\cdot 5\cdot 7$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 109391 - \beta_{1} ) q^{2} + 387420489 q^{3} + ( 86524834843 - 191939 \beta_{1} + \beta_{3} ) q^{4} + ( -1024957187485 - 503509 \beta_{1} + \beta_{2} - 28 \beta_{3} ) q^{5} + ( 42380314712199 - 387420489 \beta_{1} ) q^{6} + ( 1651450994333561 + 2985430279 \beta_{1} + 405 \beta_{2} + 20404 \beta_{3} ) q^{7} + ( 35121079860249342 - 103049124558 \beta_{1} - 5440 \beta_{2} + 299578 \beta_{3} ) q^{8} + 150094635296999121 q^{9} +O(q^{10})$$ $$q +(109391 - \beta_{1}) q^{2} +387420489 q^{3} +(86524834843 - 191939 \beta_{1} + \beta_{3}) q^{4} +(-1024957187485 - 503509 \beta_{1} + \beta_{2} - 28 \beta_{3}) q^{5} +(42380314712199 - 387420489 \beta_{1}) q^{6} +(1651450994333561 + 2985430279 \beta_{1} + 405 \beta_{2} + 20404 \beta_{3}) q^{7} +(35121079860249342 - 103049124558 \beta_{1} - 5440 \beta_{2} + 299578 \beta_{3}) q^{8} +150094635296999121 q^{9} +(-5380189282634790 + 4867048785914 \beta_{1} + 290304 \beta_{2} - 15949312 \beta_{3}) q^{10} +(5238439959104483494 - 25492139539658 \beta_{1} - 1387870 \beta_{2} - 26831288 \beta_{3}) q^{11} +(33521493825519298227 - 74361101238171 \beta_{1} + 387420489 \beta_{3}) q^{12} +(12957972542954522468 - 498690534219638 \beta_{1} + 14789790 \beta_{2} + 865098424 \beta_{3}) q^{13} +(-$$$$45\!\cdots\!20$$$$- 4216580839272024 \beta_{1} - 55114240 \beta_{2} - 6231930368 \beta_{3}) q^{14} +(-$$$$39\!\cdots\!65$$$$- 195069702995901 \beta_{1} + 387420489 \beta_{2} - 10847773692 \beta_{3}) q^{15} +($$$$13\!\cdots\!28$$$$- 58713905237284428 \beta_{1} - 2380337280 \beta_{2} + 70964221188 \beta_{3}) q^{16} +($$$$20\!\cdots\!68$$$$- 26353299406777634 \beta_{1} + 6152361370 \beta_{2} - 41466582232 \beta_{3}) q^{17} +($$$$16\!\cdots\!11$$$$- 150094635296999121 \beta_{1}) q^{18} +(-$$$$13\!\cdots\!30$$$$+ 796715087148526350 \beta_{1} - 23805860310 \beta_{2} + 691505159016 \beta_{3}) q^{19} +(-$$$$89\!\cdots\!90$$$$+ 2683989303798924654 \beta_{1} - 10617389056 \beta_{2} - 6636900927482 \beta_{3}) q^{20} +($$$$63\!\cdots\!29$$$$+ 1156616858565586431 \beta_{1} + 156905298045 \beta_{2} + 7904927657556 \beta_{3}) q^{21} +($$$$59\!\cdots\!56$$$$- 3659504875238983628 \beta_{1} - 45541647360 \beta_{2} + 41255532921856 \beta_{3}) q^{22} +(-$$$$15\!\cdots\!30$$$$+ 16703658020546342574 \beta_{1} - 178310885110 \beta_{2} - 114016465995032 \beta_{3}) q^{23} +($$$$13\!\cdots\!38$$$$- 39923342227282268862 \beta_{1} - 2107567460160 \beta_{2} + 116062655253642 \beta_{3}) q^{24} +($$$$23\!\cdots\!15$$$$-$$$$10\!\cdots\!64$$$$\beta_{1} + 6346460995596 \beta_{2} - 254265946315088 \beta_{3}) q^{25} +($$$$10\!\cdots\!10$$$$-$$$$17\!\cdots\!62$$$$\beta_{1} - 2665381043200 \beta_{2} + 393049925745664 \beta_{3}) q^{26} +$$$$58\!\cdots\!69$$$$q^{27} +($$$$61\!\cdots\!04$$$$+$$$$55\!\cdots\!60$$$$\beta_{1} - 29365958246400 \beta_{2} + 1482154037997144 \beta_{3}) q^{28} +($$$$10\!\cdots\!69$$$$+$$$$14\!\cdots\!09$$$$\beta_{1} + 23642784642315 \beta_{2} - 2583712585520436 \beta_{3}) q^{29} +(-$$$$20\!\cdots\!10$$$$+$$$$18\!\cdots\!46$$$$\beta_{1} + 112469717638656 \beta_{2} - 6179090254253568 \beta_{3}) q^{30} +($$$$22\!\cdots\!29$$$$+$$$$56\!\cdots\!67$$$$\beta_{1} - 144867143474775 \beta_{2} - 6432129197675132 \beta_{3}) q^{31} +($$$$91\!\cdots\!96$$$$-$$$$14\!\cdots\!96$$$$\beta_{1} + 33174084381440 \beta_{2} + 57167919940477864 \beta_{3}) q^{32} +($$$$20\!\cdots\!66$$$$-$$$$98\!\cdots\!62$$$$\beta_{1} - 537689274068430 \beta_{2} - 10394990717459832 \beta_{3}) q^{33} +($$$$78\!\cdots\!14$$$$-$$$$16\!\cdots\!50$$$$\beta_{1} + 1074505638620160 \beta_{2} - 60791269671865344 \beta_{3}) q^{34} +($$$$10\!\cdots\!50$$$$-$$$$42\!\cdots\!10$$$$\beta_{1} - 126089450324010 \beta_{2} - 175002471366658920 \beta_{3}) q^{35} +($$$$12\!\cdots\!03$$$$-$$$$28\!\cdots\!19$$$$\beta_{1} + 150094635296999121 \beta_{3}) q^{36} +($$$$13\!\cdots\!14$$$$+$$$$11\!\cdots\!12$$$$\beta_{1} + 556310835696600 \beta_{2} + 508236733739231328 \beta_{3}) q^{37} +(-$$$$18\!\cdots\!40$$$$+$$$$10\!\cdots\!80$$$$\beta_{1} - 7046615894062080 \beta_{2} - 402356896847195136 \beta_{3}) q^{38} +($$$$50\!\cdots\!52$$$$-$$$$19\!\cdots\!82$$$$\beta_{1} + 5729867674007310 \beta_{2} + 335156854459209336 \beta_{3}) q^{39} +(-$$$$66\!\cdots\!80$$$$+$$$$13\!\cdots\!08$$$$\beta_{1} - 5259366714736512 \beta_{2} - 1063635595164787364 \beta_{3}) q^{40} +(-$$$$21\!\cdots\!68$$$$-$$$$50\!\cdots\!30$$$$\beta_{1} + 46408136639113950 \beta_{2} - 101394467803539528 \beta_{3}) q^{41} +(-$$$$17\!\cdots\!80$$$$-$$$$16\!\cdots\!36$$$$\beta_{1} - 21352385811663360 \beta_{2} - 2414377510584509952 \beta_{3}) q^{42} +(-$$$$12\!\cdots\!18$$$$+$$$$36\!\cdots\!66$$$$\beta_{1} - 106767768332859570 \beta_{2} + 3642447885337803640 \beta_{3}) q^{43} +($$$$70\!\cdots\!20$$$$-$$$$84\!\cdots\!24$$$$\beta_{1} - 39966717409034240 \beta_{2} + 12399904291854550988 \beta_{3}) q^{44} +(-$$$$15\!\cdots\!85$$$$-$$$$75\!\cdots\!89$$$$\beta_{1} + 150094635296999121 \beta_{2} - 4202649788315975388 \beta_{3}) q^{45} +(-$$$$37\!\cdots\!96$$$$+$$$$18\!\cdots\!52$$$$\beta_{1} + 595645525841955840 \beta_{2} - 26579969078865144832 \beta_{3}) q^{46} +($$$$10\!\cdots\!42$$$$+$$$$12\!\cdots\!70$$$$\beta_{1} - 580153156762703090 \beta_{2} - 13645452734161870216 \beta_{3}) q^{47} +($$$$53\!\cdots\!92$$$$-$$$$22\!\cdots\!92$$$$\beta_{1} - 922191433002529920 \beta_{2} + 27492993274159120932 \beta_{3}) q^{48} +($$$$10\!\cdots\!93$$$$+$$$$55\!\cdots\!36$$$$\beta_{1} - 335873647667195280 \beta_{2} + 40861585860796998592 \beta_{3}) q^{49} +($$$$25\!\cdots\!85$$$$+$$$$23\!\cdots\!69$$$$\beta_{1} + 2258916821970397184 \beta_{2} - 3778457216840394752 \beta_{3}) q^{50} +($$$$78\!\cdots\!52$$$$-$$$$10\!\cdots\!26$$$$\beta_{1} + 2383550850470109930 \beta_{2} - 16065003565480151448 \beta_{3}) q^{51} +($$$$46\!\cdots\!42$$$$-$$$$10\!\cdots\!10$$$$\beta_{1} - 4538664793591971840 \beta_{2} +$$$$13\!\cdots\!30$$$$\beta_{3}) q^{52} +(-$$$$31\!\cdots\!59$$$$-$$$$45\!\cdots\!55$$$$\beta_{1} - 1922334687103569825 \beta_{2} -$$$$40\!\cdots\!04$$$$\beta_{3}) q^{53} +($$$$63\!\cdots\!79$$$$-$$$$58\!\cdots\!69$$$$\beta_{1}) q^{54} +(-$$$$81\!\cdots\!60$$$$+$$$$16\!\cdots\!96$$$$\beta_{1} + 8408678824795228056 \beta_{2} + 22892580198991444832 \beta_{3}) q^{55} +($$$$12\!\cdots\!84$$$$-$$$$19\!\cdots\!56$$$$\beta_{1} - 4540106882371079680 \beta_{2} +$$$$85\!\cdots\!52$$$$\beta_{3}) q^{56} +(-$$$$52\!\cdots\!70$$$$+$$$$30\!\cdots\!50$$$$\beta_{1} - 9222878042365891590 \beta_{2} +$$$$26\!\cdots\!24$$$$\beta_{3}) q^{57} +(-$$$$18\!\cdots\!22$$$$-$$$$56\!\cdots\!82$$$$\beta_{1} + 17317722461316364800 \beta_{2} -$$$$20\!\cdots\!48$$$$\beta_{3}) q^{58} +(-$$$$33\!\cdots\!36$$$$+$$$$15\!\cdots\!44$$$$\beta_{1} + 3830895234533430040 \beta_{2} +$$$$24\!\cdots\!76$$$$\beta_{3}) q^{59} +(-$$$$34\!\cdots\!10$$$$+$$$$10\!\cdots\!06$$$$\beta_{1} - 4113394059978768384 \beta_{2} -$$$$25\!\cdots\!98$$$$\beta_{3}) q^{60} +(-$$$$30\!\cdots\!62$$$$-$$$$70\!\cdots\!84$$$$\beta_{1} - 56929543616748524580 \beta_{2} +$$$$14\!\cdots\!52$$$$\beta_{3}) q^{61} +(-$$$$94\!\cdots\!24$$$$-$$$$89\!\cdots\!08$$$$\beta_{1} + 15001434910129364480 \beta_{2} -$$$$43\!\cdots\!40$$$$\beta_{3}) q^{62} +($$$$24\!\cdots\!81$$$$+$$$$44\!\cdots\!59$$$$\beta_{1} + 60788327295284644005 \beta_{2} +$$$$30\!\cdots\!84$$$$\beta_{3}) q^{63} +($$$$21\!\cdots\!64$$$$-$$$$10\!\cdots\!08$$$$\beta_{1} + 20735073056676072960 \beta_{2} +$$$$10\!\cdots\!96$$$$\beta_{3}) q^{64} +($$$$39\!\cdots\!70$$$$+$$$$30\!\cdots\!78$$$$\beta_{1} + 42166817812091905458 \beta_{2} -$$$$13\!\cdots\!24$$$$\beta_{3}) q^{65} +($$$$23\!\cdots\!84$$$$-$$$$14\!\cdots\!92$$$$\beta_{1} - 17643767290076759040 \beta_{2} +$$$$15\!\cdots\!84$$$$\beta_{3}) q^{66} +($$$$40\!\cdots\!44$$$$+$$$$87\!\cdots\!32$$$$\beta_{1} -$$$$14\!\cdots\!80$$$$\beta_{2} -$$$$11\!\cdots\!24$$$$\beta_{3}) q^{67} +($$$$15\!\cdots\!98$$$$+$$$$28\!\cdots\!54$$$$\beta_{1} -$$$$36\!\cdots\!40$$$$\beta_{2} +$$$$13\!\cdots\!98$$$$\beta_{3}) q^{68} +(-$$$$59\!\cdots\!70$$$$+$$$$64\!\cdots\!86$$$$\beta_{1} - 69081290303339018790 \beta_{2} -$$$$44\!\cdots\!48$$$$\beta_{3}) q^{69} +($$$$20\!\cdots\!00$$$$+$$$$13\!\cdots\!60$$$$\beta_{1} +$$$$93\!\cdots\!60$$$$\beta_{2} -$$$$12\!\cdots\!80$$$$\beta_{3}) q^{70} +($$$$27\!\cdots\!38$$$$+$$$$10\!\cdots\!26$$$$\beta_{1} +$$$$11\!\cdots\!90$$$$\beta_{2} +$$$$95\!\cdots\!68$$$$\beta_{3}) q^{71} +($$$$52\!\cdots\!82$$$$-$$$$15\!\cdots\!18$$$$\beta_{1} -$$$$81\!\cdots\!40$$$$\beta_{2} +$$$$44\!\cdots\!38$$$$\beta_{3}) q^{72} +(-$$$$48\!\cdots\!86$$$$+$$$$22\!\cdots\!88$$$$\beta_{1} -$$$$19\!\cdots\!00$$$$\beta_{2} -$$$$24\!\cdots\!60$$$$\beta_{3}) q^{73} +(-$$$$22\!\cdots\!94$$$$-$$$$74\!\cdots\!58$$$$\beta_{1} -$$$$26\!\cdots\!20$$$$\beta_{2} -$$$$65\!\cdots\!20$$$$\beta_{3}) q^{74} +($$$$92\!\cdots\!35$$$$-$$$$42\!\cdots\!96$$$$\beta_{1} +$$$$24\!\cdots\!44$$$$\beta_{2} -$$$$98\!\cdots\!32$$$$\beta_{3}) q^{75} +(-$$$$23\!\cdots\!00$$$$+$$$$13\!\cdots\!40$$$$\beta_{1} +$$$$44\!\cdots\!40$$$$\beta_{2} -$$$$14\!\cdots\!16$$$$\beta_{3}) q^{76} +(-$$$$64\!\cdots\!64$$$$-$$$$17\!\cdots\!28$$$$\beta_{1} +$$$$20\!\cdots\!40$$$$\beta_{2} +$$$$10\!\cdots\!00$$$$\beta_{3}) q^{77} +($$$$41\!\cdots\!90$$$$-$$$$67\!\cdots\!18$$$$\beta_{1} -$$$$10\!\cdots\!00$$$$\beta_{2} +$$$$15\!\cdots\!96$$$$\beta_{3}) q^{78} +($$$$10\!\cdots\!01$$$$-$$$$96\!\cdots\!09$$$$\beta_{1} -$$$$13\!\cdots\!95$$$$\beta_{2} +$$$$69\!\cdots\!28$$$$\beta_{3}) q^{79} +(-$$$$23\!\cdots\!40$$$$+$$$$55\!\cdots\!44$$$$\beta_{1} +$$$$65\!\cdots\!84$$$$\beta_{2} -$$$$49\!\cdots\!52$$$$\beta_{3}) q^{80} +$$$$22\!\cdots\!41$$$$q^{81} +($$$$83\!\cdots\!42$$$$+$$$$19\!\cdots\!98$$$$\beta_{1} +$$$$69\!\cdots\!20$$$$\beta_{2} -$$$$12\!\cdots\!12$$$$\beta_{3}) q^{82} +(-$$$$11\!\cdots\!62$$$$-$$$$11\!\cdots\!98$$$$\beta_{1} -$$$$74\!\cdots\!50$$$$\beta_{2} -$$$$12\!\cdots\!60$$$$\beta_{3}) q^{83} +($$$$23\!\cdots\!56$$$$+$$$$21\!\cdots\!40$$$$\beta_{1} -$$$$11\!\cdots\!00$$$$\beta_{2} +$$$$57\!\cdots\!16$$$$\beta_{3}) q^{84} +($$$$46\!\cdots\!30$$$$-$$$$33\!\cdots\!98$$$$\beta_{1} +$$$$52\!\cdots\!22$$$$\beta_{2} -$$$$28\!\cdots\!16$$$$\beta_{3}) q^{85} +(-$$$$90\!\cdots\!12$$$$-$$$$34\!\cdots\!84$$$$\beta_{1} -$$$$34\!\cdots\!80$$$$\beta_{2} +$$$$14\!\cdots\!24$$$$\beta_{3}) q^{86} +($$$$40\!\cdots\!41$$$$+$$$$55\!\cdots\!01$$$$\beta_{1} +$$$$91\!\cdots\!35$$$$\beta_{2} -$$$$10\!\cdots\!04$$$$\beta_{3}) q^{87} +($$$$10\!\cdots\!84$$$$-$$$$26\!\cdots\!16$$$$\beta_{1} -$$$$66\!\cdots\!60$$$$\beta_{2} +$$$$46\!\cdots\!84$$$$\beta_{3}) q^{88} +(-$$$$78\!\cdots\!22$$$$-$$$$10\!\cdots\!72$$$$\beta_{1} +$$$$24\!\cdots\!60$$$$\beta_{2} +$$$$46\!\cdots\!00$$$$\beta_{3}) q^{89} +(-$$$$80\!\cdots\!90$$$$+$$$$73\!\cdots\!94$$$$\beta_{1} +$$$$43\!\cdots\!84$$$$\beta_{2} -$$$$23\!\cdots\!52$$$$\beta_{3}) q^{90} +($$$$65\!\cdots\!10$$$$-$$$$71\!\cdots\!62$$$$\beta_{1} -$$$$97\!\cdots\!10$$$$\beta_{2} -$$$$44\!\cdots\!44$$$$\beta_{3}) q^{91} +(-$$$$41\!\cdots\!44$$$$+$$$$66\!\cdots\!84$$$$\beta_{1} +$$$$25\!\cdots\!60$$$$\beta_{2} -$$$$13\!\cdots\!56$$$$\beta_{3}) q^{92} +($$$$87\!\cdots\!81$$$$+$$$$21\!\cdots\!63$$$$\beta_{1} -$$$$56\!\cdots\!75$$$$\beta_{2} -$$$$24\!\cdots\!48$$$$\beta_{3}) q^{93} +(-$$$$14\!\cdots\!48$$$$+$$$$90\!\cdots\!68$$$$\beta_{1} -$$$$58\!\cdots\!20$$$$\beta_{2} +$$$$50\!\cdots\!16$$$$\beta_{3}) q^{94} +(-$$$$22\!\cdots\!40$$$$-$$$$11\!\cdots\!76$$$$\beta_{1} -$$$$45\!\cdots\!36$$$$\beta_{2} +$$$$20\!\cdots\!08$$$$\beta_{3}) q^{95} +($$$$35\!\cdots\!44$$$$-$$$$55\!\cdots\!44$$$$\beta_{1} +$$$$12\!\cdots\!60$$$$\beta_{2} +$$$$22\!\cdots\!96$$$$\beta_{3}) q^{96} +($$$$11\!\cdots\!34$$$$+$$$$93\!\cdots\!72$$$$\beta_{1} +$$$$73\!\cdots\!80$$$$\beta_{2} -$$$$65\!\cdots\!64$$$$\beta_{3}) q^{97} +(-$$$$10\!\cdots\!21$$$$-$$$$11\!\cdots\!25$$$$\beta_{1} -$$$$26\!\cdots\!00$$$$\beta_{2} -$$$$47\!\cdots\!28$$$$\beta_{3}) q^{98} +($$$$78\!\cdots\!74$$$$-$$$$38\!\cdots\!18$$$$\beta_{1} -$$$$20\!\cdots\!70$$$$\beta_{2} -$$$$40\!\cdots\!48$$$$\beta_{3}) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 437562q^{2} + 1549681956q^{3} + 346098955492q^{4} - 4099829756904q^{5} + 169520484007818q^{6} + 6605809948153184q^{7} + 140484113342159976q^{8} + 600378541187996484q^{9} + O(q^{10})$$ $$4q + 437562q^{2} + 1549681956q^{3} + 346098955492q^{4} - 4099829756904q^{5} + 169520484007818q^{6} + 6605809948153184q^{7} + 140484113342159976q^{8} + 600378541187996484q^{9} - 21511023001649316q^{10} + 20953708852195292976q^{11} +$$$$13\!\cdots\!88$$$$q^{12} + 51830892788989874168q^{13} -$$$$18\!\cdots\!12$$$$q^{14} -$$$$15\!\cdots\!56$$$$q^{15} +$$$$55\!\cdots\!40$$$$q^{16} +$$$$81\!\cdots\!28$$$$q^{17} +$$$$65\!\cdots\!02$$$$q^{18} -$$$$54\!\cdots\!32$$$$q^{19} -$$$$35\!\cdots\!76$$$$q^{20} +$$$$25\!\cdots\!76$$$$q^{21} +$$$$23\!\cdots\!76$$$$q^{22} -$$$$61\!\cdots\!88$$$$q^{23} +$$$$54\!\cdots\!64$$$$q^{24} +$$$$95\!\cdots\!16$$$$q^{25} +$$$$42\!\cdots\!88$$$$q^{26} +$$$$23\!\cdots\!76$$$$q^{27} +$$$$24\!\cdots\!48$$$$q^{28} +$$$$41\!\cdots\!36$$$$q^{29} -$$$$83\!\cdots\!24$$$$q^{30} +$$$$89\!\cdots\!64$$$$q^{31} +$$$$36\!\cdots\!84$$$$q^{32} +$$$$81\!\cdots\!64$$$$q^{33} +$$$$31\!\cdots\!24$$$$q^{34} +$$$$42\!\cdots\!40$$$$q^{35} +$$$$51\!\cdots\!32$$$$q^{36} +$$$$55\!\cdots\!24$$$$q^{37} -$$$$73\!\cdots\!68$$$$q^{38} +$$$$20\!\cdots\!52$$$$q^{39} -$$$$26\!\cdots\!52$$$$q^{40} -$$$$86\!\cdots\!76$$$$q^{41} -$$$$70\!\cdots\!68$$$$q^{42} -$$$$50\!\cdots\!80$$$$q^{43} +$$$$28\!\cdots\!36$$$$q^{44} -$$$$61\!\cdots\!84$$$$q^{45} -$$$$14\!\cdots\!96$$$$q^{46} +$$$$42\!\cdots\!20$$$$q^{47} +$$$$21\!\cdots\!60$$$$q^{48} +$$$$40\!\cdots\!20$$$$q^{49} +$$$$10\!\cdots\!14$$$$q^{50} +$$$$31\!\cdots\!92$$$$q^{51} +$$$$18\!\cdots\!68$$$$q^{52} -$$$$12\!\cdots\!88$$$$q^{53} +$$$$25\!\cdots\!78$$$$q^{54} -$$$$32\!\cdots\!24$$$$q^{55} +$$$$49\!\cdots\!80$$$$q^{56} -$$$$21\!\cdots\!48$$$$q^{57} -$$$$75\!\cdots\!56$$$$q^{58} -$$$$13\!\cdots\!88$$$$q^{59} -$$$$13\!\cdots\!64$$$$q^{60} -$$$$12\!\cdots\!60$$$$q^{61} -$$$$37\!\cdots\!92$$$$q^{62} +$$$$99\!\cdots\!64$$$$q^{63} +$$$$85\!\cdots\!28$$$$q^{64} +$$$$15\!\cdots\!68$$$$q^{65} +$$$$92\!\cdots\!64$$$$q^{66} +$$$$16\!\cdots\!48$$$$q^{67} +$$$$63\!\cdots\!84$$$$q^{68} -$$$$23\!\cdots\!32$$$$q^{69} +$$$$82\!\cdots\!60$$$$q^{70} +$$$$10\!\cdots\!88$$$$q^{71} +$$$$21\!\cdots\!96$$$$q^{72} -$$$$19\!\cdots\!48$$$$q^{73} -$$$$89\!\cdots\!12$$$$q^{74} +$$$$36\!\cdots\!24$$$$q^{75} -$$$$95\!\cdots\!68$$$$q^{76} -$$$$25\!\cdots\!92$$$$q^{77} +$$$$16\!\cdots\!32$$$$q^{78} +$$$$42\!\cdots\!20$$$$q^{79} -$$$$95\!\cdots\!36$$$$q^{80} +$$$$90\!\cdots\!64$$$$q^{81} +$$$$33\!\cdots\!48$$$$q^{82} -$$$$46\!\cdots\!24$$$$q^{83} +$$$$95\!\cdots\!72$$$$q^{84} +$$$$18\!\cdots\!12$$$$q^{85} -$$$$36\!\cdots\!04$$$$q^{86} +$$$$16\!\cdots\!04$$$$q^{87} +$$$$42\!\cdots\!56$$$$q^{88} -$$$$31\!\cdots\!52$$$$q^{89} -$$$$32\!\cdots\!36$$$$q^{90} +$$$$26\!\cdots\!24$$$$q^{91} -$$$$16\!\cdots\!16$$$$q^{92} +$$$$34\!\cdots\!96$$$$q^{93} -$$$$57\!\cdots\!48$$$$q^{94} -$$$$89\!\cdots\!56$$$$q^{95} +$$$$14\!\cdots\!76$$$$q^{96} +$$$$44\!\cdots\!48$$$$q^{97} -$$$$43\!\cdots\!78$$$$q^{98} +$$$$31\!\cdots\!96$$$$q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 11777633936 x^{2} - 35120319927360 x + 11967042111800832000$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$6 \nu - 1$$ $$\beta_{2}$$ $$=$$ $$($$$$27 \nu^{3} - 128691 \nu^{2} - 262552030440 \nu + 46478406852080$$$$)/680$$ $$\beta_{3}$$ $$=$$ $$36 \nu^{2} - 161070 \nu - 211997370590$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1} + 1$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 26845 \beta_{1} + 211997397435$$$$)/36$$ $$\nu^{3}$$ $$=$$ $$($$$$14299 \beta_{3} + 2720 \beta_{2} + 175418543615 \beta_{1} + 2845612193201705$$$$)/108$$

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
 105009. 31743.2 −35434.6 −101317.
−520663. 3.87420e8 1.33651e11 −2.63441e12 −2.01716e14 8.34393e15 1.97212e15 1.50095e17 1.37164e18
1.2 −81067.1 3.87420e8 −1.30867e11 −7.16603e12 −3.14071e13 −5.96869e15 2.17508e16 1.50095e17 5.80929e17
1.3 322000. 3.87420e8 −3.37552e10 1.53382e13 1.24749e14 2.48687e15 −5.51245e16 1.50095e17 4.93889e18
1.4 717293. 3.87420e8 3.77070e11 −9.63758e12 2.77894e14 1.74370e15 1.71886e17 1.50095e17 −6.91297e18
 $$n$$: 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 3.38.a.b 4
3.b odd 2 1 9.38.a.c 4

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

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 437562 T_{2}^{3} - 352197132768 T_{2}^{2} +$$$$95\!\cdots\!24$$$$T_{2} +$$$$97\!\cdots\!04$$ acting on $$S_{38}^{\mathrm{new}}(\Gamma_0(3))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 437562 T + 197558681120 T^{2} - 85300907961581568 T^{3} +$$$$26\!\cdots\!16$$$$T^{4} -$$$$11\!\cdots\!96$$$$T^{5} +$$$$37\!\cdots\!80$$$$T^{6} -$$$$11\!\cdots\!76$$$$T^{7} +$$$$35\!\cdots\!56$$$$T^{8}$$
$3$ $$( 1 - 387420489 T )^{4}$$
$5$ $$1 + 4099829756904 T +$$$$10\!\cdots\!00$$$$T^{2} -$$$$66\!\cdots\!00$$$$T^{3} +$$$$20\!\cdots\!50$$$$T^{4} -$$$$48\!\cdots\!00$$$$T^{5} +$$$$56\!\cdots\!00$$$$T^{6} +$$$$15\!\cdots\!00$$$$T^{7} +$$$$28\!\cdots\!25$$$$T^{8}$$
$7$ $$1 - 6605809948153184 T +$$$$38\!\cdots\!32$$$$T^{2} -$$$$16\!\cdots\!44$$$$T^{3} +$$$$53\!\cdots\!50$$$$T^{4} -$$$$31\!\cdots\!08$$$$T^{5} +$$$$13\!\cdots\!68$$$$T^{6} -$$$$42\!\cdots\!12$$$$T^{7} +$$$$11\!\cdots\!01$$$$T^{8}$$
$11$ $$1 - 20953708852195292976 T +$$$$92\!\cdots\!92$$$$T^{2} -$$$$12\!\cdots\!20$$$$T^{3} +$$$$41\!\cdots\!06$$$$T^{4} -$$$$43\!\cdots\!20$$$$T^{5} +$$$$10\!\cdots\!72$$$$T^{6} -$$$$82\!\cdots\!36$$$$T^{7} +$$$$13\!\cdots\!81$$$$T^{8}$$
$13$ $$1 - 51830892788989874168 T +$$$$47\!\cdots\!04$$$$T^{2} -$$$$17\!\cdots\!76$$$$T^{3} +$$$$10\!\cdots\!50$$$$T^{4} -$$$$29\!\cdots\!08$$$$T^{5} +$$$$12\!\cdots\!56$$$$T^{6} -$$$$23\!\cdots\!16$$$$T^{7} +$$$$73\!\cdots\!21$$$$T^{8}$$
$17$ $$1 -$$$$81\!\cdots\!28$$$$T +$$$$10\!\cdots\!52$$$$T^{2} -$$$$71\!\cdots\!40$$$$T^{3} +$$$$44\!\cdots\!26$$$$T^{4} -$$$$24\!\cdots\!80$$$$T^{5} +$$$$11\!\cdots\!08$$$$T^{6} -$$$$30\!\cdots\!24$$$$T^{7} +$$$$12\!\cdots\!41$$$$T^{8}$$
$19$ $$1 +$$$$54\!\cdots\!32$$$$T +$$$$55\!\cdots\!72$$$$T^{2} +$$$$24\!\cdots\!04$$$$T^{3} +$$$$16\!\cdots\!74$$$$T^{4} +$$$$51\!\cdots\!56$$$$T^{5} +$$$$23\!\cdots\!12$$$$T^{6} +$$$$47\!\cdots\!08$$$$T^{7} +$$$$18\!\cdots\!41$$$$T^{8}$$
$23$ $$1 +$$$$61\!\cdots\!88$$$$T +$$$$98\!\cdots\!68$$$$T^{2} +$$$$39\!\cdots\!72$$$$T^{3} +$$$$20\!\cdots\!90$$$$T^{4} +$$$$95\!\cdots\!16$$$$T^{5} +$$$$57\!\cdots\!12$$$$T^{6} +$$$$86\!\cdots\!76$$$$T^{7} +$$$$34\!\cdots\!81$$$$T^{8}$$
$29$ $$1 -$$$$41\!\cdots\!36$$$$T +$$$$10\!\cdots\!80$$$$T^{2} -$$$$16\!\cdots\!32$$$$T^{3} +$$$$21\!\cdots\!78$$$$T^{4} -$$$$21\!\cdots\!88$$$$T^{5} +$$$$17\!\cdots\!80$$$$T^{6} -$$$$88\!\cdots\!44$$$$T^{7} +$$$$27\!\cdots\!61$$$$T^{8}$$
$31$ $$1 -$$$$89\!\cdots\!64$$$$T +$$$$72\!\cdots\!68$$$$T^{2} -$$$$36\!\cdots\!12$$$$T^{3} +$$$$17\!\cdots\!54$$$$T^{4} -$$$$55\!\cdots\!32$$$$T^{5} +$$$$16\!\cdots\!28$$$$T^{6} -$$$$31\!\cdots\!84$$$$T^{7} +$$$$52\!\cdots\!41$$$$T^{8}$$
$37$ $$1 -$$$$55\!\cdots\!24$$$$T +$$$$22\!\cdots\!72$$$$T^{2} -$$$$20\!\cdots\!44$$$$T^{3} +$$$$28\!\cdots\!70$$$$T^{4} -$$$$21\!\cdots\!48$$$$T^{5} +$$$$25\!\cdots\!08$$$$T^{6} -$$$$65\!\cdots\!12$$$$T^{7} +$$$$12\!\cdots\!21$$$$T^{8}$$
$41$ $$1 +$$$$86\!\cdots\!76$$$$T +$$$$17\!\cdots\!88$$$$T^{2} +$$$$10\!\cdots\!28$$$$T^{3} +$$$$11\!\cdots\!34$$$$T^{4} +$$$$49\!\cdots\!68$$$$T^{5} +$$$$38\!\cdots\!68$$$$T^{6} +$$$$90\!\cdots\!16$$$$T^{7} +$$$$49\!\cdots\!21$$$$T^{8}$$
$43$ $$1 +$$$$50\!\cdots\!80$$$$T +$$$$85\!\cdots\!80$$$$T^{2} +$$$$56\!\cdots\!40$$$$T^{3} +$$$$31\!\cdots\!98$$$$T^{4} +$$$$15\!\cdots\!20$$$$T^{5} +$$$$64\!\cdots\!20$$$$T^{6} +$$$$10\!\cdots\!60$$$$T^{7} +$$$$56\!\cdots\!01$$$$T^{8}$$
$47$ $$1 -$$$$42\!\cdots\!20$$$$T +$$$$24\!\cdots\!20$$$$T^{2} -$$$$10\!\cdots\!40$$$$T^{3} +$$$$24\!\cdots\!38$$$$T^{4} -$$$$75\!\cdots\!80$$$$T^{5} +$$$$13\!\cdots\!80$$$$T^{6} -$$$$17\!\cdots\!60$$$$T^{7} +$$$$29\!\cdots\!61$$$$T^{8}$$
$53$ $$1 +$$$$12\!\cdots\!88$$$$T +$$$$20\!\cdots\!88$$$$T^{2} +$$$$19\!\cdots\!32$$$$T^{3} +$$$$18\!\cdots\!50$$$$T^{4} +$$$$12\!\cdots\!16$$$$T^{5} +$$$$79\!\cdots\!72$$$$T^{6} +$$$$31\!\cdots\!36$$$$T^{7} +$$$$15\!\cdots\!61$$$$T^{8}$$
$59$ $$1 +$$$$13\!\cdots\!88$$$$T +$$$$62\!\cdots\!12$$$$T^{2} -$$$$19\!\cdots\!84$$$$T^{3} -$$$$29\!\cdots\!66$$$$T^{4} -$$$$64\!\cdots\!96$$$$T^{5} +$$$$68\!\cdots\!32$$$$T^{6} +$$$$48\!\cdots\!92$$$$T^{7} +$$$$12\!\cdots\!21$$$$T^{8}$$
$61$ $$1 +$$$$12\!\cdots\!60$$$$T +$$$$43\!\cdots\!16$$$$T^{2} +$$$$36\!\cdots\!40$$$$T^{3} +$$$$72\!\cdots\!46$$$$T^{4} +$$$$41\!\cdots\!40$$$$T^{5} +$$$$56\!\cdots\!56$$$$T^{6} +$$$$17\!\cdots\!60$$$$T^{7} +$$$$16\!\cdots\!81$$$$T^{8}$$
$67$ $$1 -$$$$16\!\cdots\!48$$$$T +$$$$20\!\cdots\!72$$$$T^{2} -$$$$16\!\cdots\!00$$$$T^{3} +$$$$11\!\cdots\!66$$$$T^{4} -$$$$61\!\cdots\!00$$$$T^{5} +$$$$27\!\cdots\!88$$$$T^{6} -$$$$80\!\cdots\!84$$$$T^{7} +$$$$18\!\cdots\!41$$$$T^{8}$$
$71$ $$1 -$$$$10\!\cdots\!88$$$$T +$$$$55\!\cdots\!68$$$$T^{2} -$$$$47\!\cdots\!16$$$$T^{3} +$$$$14\!\cdots\!70$$$$T^{4} -$$$$14\!\cdots\!56$$$$T^{5} +$$$$55\!\cdots\!08$$$$T^{6} -$$$$33\!\cdots\!48$$$$T^{7} +$$$$96\!\cdots\!61$$$$T^{8}$$
$73$ $$1 +$$$$19\!\cdots\!48$$$$T +$$$$28\!\cdots\!68$$$$T^{2} +$$$$45\!\cdots\!52$$$$T^{3} +$$$$35\!\cdots\!90$$$$T^{4} +$$$$39\!\cdots\!56$$$$T^{5} +$$$$22\!\cdots\!12$$$$T^{6} +$$$$13\!\cdots\!96$$$$T^{7} +$$$$59\!\cdots\!81$$$$T^{8}$$
$79$ $$1 -$$$$42\!\cdots\!20$$$$T +$$$$83\!\cdots\!36$$$$T^{2} -$$$$13\!\cdots\!40$$$$T^{3} -$$$$27\!\cdots\!14$$$$T^{4} -$$$$21\!\cdots\!60$$$$T^{5} +$$$$22\!\cdots\!16$$$$T^{6} -$$$$18\!\cdots\!80$$$$T^{7} +$$$$70\!\cdots\!61$$$$T^{8}$$
$83$ $$1 +$$$$46\!\cdots\!24$$$$T +$$$$47\!\cdots\!60$$$$T^{2} +$$$$14\!\cdots\!44$$$$T^{3} +$$$$74\!\cdots\!46$$$$T^{4} +$$$$14\!\cdots\!12$$$$T^{5} +$$$$48\!\cdots\!40$$$$T^{6} +$$$$48\!\cdots\!08$$$$T^{7} +$$$$10\!\cdots\!41$$$$T^{8}$$
$89$ $$1 +$$$$31\!\cdots\!52$$$$T +$$$$35\!\cdots\!52$$$$T^{2} +$$$$89\!\cdots\!84$$$$T^{3} +$$$$60\!\cdots\!34$$$$T^{4} +$$$$12\!\cdots\!36$$$$T^{5} +$$$$64\!\cdots\!32$$$$T^{6} +$$$$75\!\cdots\!28$$$$T^{7} +$$$$32\!\cdots\!81$$$$T^{8}$$
$97$ $$1 -$$$$44\!\cdots\!48$$$$T +$$$$13\!\cdots\!12$$$$T^{2} -$$$$24\!\cdots\!40$$$$T^{3} +$$$$18\!\cdots\!86$$$$T^{4} -$$$$80\!\cdots\!80$$$$T^{5} +$$$$13\!\cdots\!28$$$$T^{6} -$$$$15\!\cdots\!44$$$$T^{7} +$$$$11\!\cdots\!61$$$$T^{8}$$