Properties

 Label 1470.3.f.d Level 1470 Weight 3 Character orbit 1470.f Analytic conductor 40.055 Analytic rank 0 Dimension 16 CM no Inner twists 2

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1470.f (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$40.0545988610$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{16}\cdot 7^{2}$$ Twist minimal: no (minimal twist has level 210) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{6} q^{3} + 2 q^{4} -\beta_{7} q^{5} -\beta_{8} q^{6} + 2 \beta_{1} q^{8} -3 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{6} q^{3} + 2 q^{4} -\beta_{7} q^{5} -\beta_{8} q^{6} + 2 \beta_{1} q^{8} -3 q^{9} + \beta_{10} q^{10} + ( 1 - \beta_{1} + \beta_{2} - \beta_{12} ) q^{11} + 2 \beta_{6} q^{12} + ( 2 \beta_{6} - \beta_{9} ) q^{13} -\beta_{2} q^{15} + 4 q^{16} + ( \beta_{4} + \beta_{7} + 5 \beta_{10} ) q^{17} -3 \beta_{1} q^{18} + ( -3 \beta_{6} - 5 \beta_{7} - 2 \beta_{8} + \beta_{14} ) q^{19} -2 \beta_{7} q^{20} + ( -3 + \beta_{1} + \beta_{3} + \beta_{13} ) q^{22} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{23} -2 \beta_{8} q^{24} -5 q^{25} + ( -\beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{26} -3 \beta_{6} q^{27} + ( 4 + 5 \beta_{1} - 6 \beta_{2} + \beta_{5} - \beta_{11} + \beta_{12} ) q^{29} -\beta_{3} q^{30} + ( -\beta_{4} - 6 \beta_{6} - \beta_{7} + \beta_{9} + 7 \beta_{10} ) q^{31} + 4 \beta_{1} q^{32} + ( \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{14} ) q^{33} + ( \beta_{4} - \beta_{6} - 11 \beta_{7} + \beta_{9} - \beta_{10} ) q^{34} -6 q^{36} + ( -6 + 6 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} - \beta_{5} - \beta_{11} + \beta_{12} + \beta_{13} ) q^{37} + ( 3 \beta_{6} - \beta_{7} + 3 \beta_{8} + 5 \beta_{10} - \beta_{15} ) q^{38} + ( -5 - \beta_{2} + \beta_{11} + \beta_{12} ) q^{39} + 2 \beta_{10} q^{40} + ( -2 \beta_{4} - 4 \beta_{6} - 2 \beta_{7} - 4 \beta_{8} + \beta_{9} + 3 \beta_{10} - \beta_{14} + \beta_{15} ) q^{41} + ( -3 + 5 \beta_{1} - 2 \beta_{3} - \beta_{5} + \beta_{11} - \beta_{12} - 2 \beta_{13} ) q^{43} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{12} ) q^{44} + 3 \beta_{7} q^{45} + ( -2 - \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{5} - \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{46} + ( -2 \beta_{6} - 6 \beta_{7} + 2 \beta_{9} + 12 \beta_{10} ) q^{47} + 4 \beta_{6} q^{48} -5 \beta_{1} q^{50} + ( 2 + 3 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} + \beta_{5} - \beta_{12} + \beta_{13} ) q^{51} + ( 4 \beta_{6} - 2 \beta_{9} ) q^{52} + ( -4 - 14 \beta_{1} + 2 \beta_{3} - 2 \beta_{11} + 2 \beta_{13} ) q^{53} + 3 \beta_{8} q^{54} + ( 2 \beta_{6} - \beta_{7} + \beta_{9} - \beta_{10} - \beta_{15} ) q^{55} + ( 10 - 3 \beta_{1} - 4 \beta_{2} + \beta_{11} - 2 \beta_{12} ) q^{57} + ( 11 + 5 \beta_{1} + 2 \beta_{2} - 7 \beta_{3} + 2 \beta_{5} - \beta_{13} ) q^{58} + ( -5 \beta_{6} + 17 \beta_{7} - 6 \beta_{8} - \beta_{9} + 14 \beta_{10} - 3 \beta_{14} + 2 \beta_{15} ) q^{59} -2 \beta_{2} q^{60} + ( 3 \beta_{4} + 7 \beta_{6} - 4 \beta_{7} - 4 \beta_{8} - 3 \beta_{9} + 9 \beta_{10} - \beta_{14} ) q^{61} + ( 2 \beta_{6} - 12 \beta_{7} + 5 \beta_{8} - 2 \beta_{9} ) q^{62} + 8 q^{64} + ( -2 - 3 \beta_{1} - \beta_{2} - \beta_{5} - \beta_{11} - \beta_{12} ) q^{65} + ( -\beta_{4} - 4 \beta_{6} + \beta_{9} + 2 \beta_{10} + \beta_{15} ) q^{66} + ( 20 + 29 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{67} + ( 2 \beta_{4} + 2 \beta_{7} + 10 \beta_{10} ) q^{68} + ( \beta_{4} + \beta_{6} + 2 \beta_{9} - 2 \beta_{10} - \beta_{15} ) q^{69} + ( -9 - 3 \beta_{1} - 9 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + \beta_{12} + 2 \beta_{13} ) q^{71} -6 \beta_{1} q^{72} + ( -\beta_{4} + 13 \beta_{6} + 2 \beta_{7} + 12 \beta_{8} + 2 \beta_{9} + 7 \beta_{10} + 3 \beta_{14} ) q^{73} + ( 15 - 4 \beta_{1} - 10 \beta_{2} - \beta_{3} + 2 \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{74} -5 \beta_{6} q^{75} + ( -6 \beta_{6} - 10 \beta_{7} - 4 \beta_{8} + 2 \beta_{14} ) q^{76} + ( -6 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{11} - \beta_{13} ) q^{78} + ( -34 + \beta_{1} - 4 \beta_{2} - 10 \beta_{3} - \beta_{11} - 2 \beta_{12} + 2 \beta_{13} ) q^{79} -4 \beta_{7} q^{80} + 9 q^{81} + ( -\beta_{4} + 12 \beta_{6} - 2 \beta_{7} + 4 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} - 2 \beta_{14} + \beta_{15} ) q^{82} + ( 3 \beta_{4} + 13 \beta_{6} + 4 \beta_{7} - 4 \beta_{8} - 4 \beta_{9} - 4 \beta_{10} + 2 \beta_{14} - \beta_{15} ) q^{83} + ( 7 - 26 \beta_{1} - \beta_{3} + \beta_{5} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{85} + ( 9 - 6 \beta_{1} - 6 \beta_{2} + \beta_{3} - 2 \beta_{5} + 4 \beta_{12} + \beta_{13} ) q^{86} + ( -2 \beta_{4} + 4 \beta_{6} + 18 \beta_{7} - 6 \beta_{8} - \beta_{9} + \beta_{10} + 3 \beta_{14} - \beta_{15} ) q^{87} + ( -6 + 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{13} ) q^{88} + ( -2 \beta_{4} - 12 \beta_{6} + 12 \beta_{8} - 3 \beta_{9} + 25 \beta_{10} + 3 \beta_{14} - \beta_{15} ) q^{89} -3 \beta_{10} q^{90} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} ) q^{92} + ( 15 - 3 \beta_{1} - \beta_{2} - 7 \beta_{3} - \beta_{5} - \beta_{11} - \beta_{13} ) q^{93} + ( 2 \beta_{4} + 2 \beta_{6} - 22 \beta_{7} - 2 \beta_{9} + 4 \beta_{10} ) q^{94} + ( -25 - 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + \beta_{11} - 2 \beta_{13} ) q^{95} -4 \beta_{8} q^{96} + ( -\beta_{4} - 13 \beta_{6} + 28 \beta_{8} + 3 \beta_{9} - 17 \beta_{10} + \beta_{14} ) q^{97} + ( -3 + 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{12} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 32q^{4} - 48q^{9} + O(q^{10})$$ $$16q + 32q^{4} - 48q^{9} + 8q^{11} + 64q^{16} - 48q^{22} + 24q^{23} - 80q^{25} + 72q^{29} - 96q^{36} - 88q^{37} - 72q^{39} - 56q^{43} + 16q^{44} - 16q^{46} + 24q^{51} - 64q^{53} + 144q^{57} + 176q^{58} + 128q^{64} - 40q^{65} + 328q^{67} - 136q^{71} + 224q^{74} - 560q^{79} + 144q^{81} + 120q^{85} + 176q^{86} - 96q^{88} + 48q^{92} + 240q^{93} - 400q^{95} - 24q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 92 x^{14} - 112 x^{13} + 5846 x^{12} - 7728 x^{11} + 197216 x^{10} - 298200 x^{9} + 4836403 x^{8} - 6808704 x^{7} + 64376800 x^{6} - 91953512 x^{5} + 595763862 x^{4} - 630430976 x^{3} + 1087013404 x^{2} + 294123256 x + 101626561$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$57754008957283349 \nu^{15} + 676562680999836294 \nu^{14} + 8239103585403158159 \nu^{13} + 54078590776251682475 \nu^{12} + 455975368108340251668 \nu^{11} + 2793391292439344623383 \nu^{10} + 16545566601055515429559 \nu^{9} + 76261143494901216401580 \nu^{8} + 306572240997688184788012 \nu^{7} + 1467974756910134628134465 \nu^{6} + 3497231496721962844392765 \nu^{5} + 10591214719874231223793348 \nu^{4} - 4685579583089306764764583 \nu^{3} + 35874681437029057265441973 \nu^{2} + 9955887688266404429946106 \nu - 1202728311133779179554909109$$$$)/$$$$85\!\cdots\!24$$ $$\beta_{2}$$ $$=$$ $$($$$$6718070224756 \nu^{15} - 35399436274870 \nu^{14} + 824042900206056 \nu^{13} - 3468795000369922 \nu^{12} + 55447463307528888 \nu^{11} - 237157576567025547 \nu^{10} + 2160971883833844512 \nu^{9} - 7103633821675122284 \nu^{8} + 46638150914308904520 \nu^{7} - 152233236071607452553 \nu^{6} + 595135449699402838792 \nu^{5} - 1018214320678792586196 \nu^{4} + 1279768421429813362272 \nu^{3} - 459537667512276799289 \nu^{2} - 103279705437214175760 \nu + 206657596524136518657064$$$$)/$$$$53\!\cdots\!38$$ $$\beta_{3}$$ $$=$$ $$($$$$3871767397361 \nu^{15} + 79093222616710 \nu^{14} + 341679513333957 \nu^{13} + 5895790087314937 \nu^{12} + 14035722749452236 \nu^{11} + 371621017202953791 \nu^{10} + 356604274247547817 \nu^{9} + 10456974871241558252 \nu^{8} + 4802805224198506740 \nu^{7} + 233091302320357629453 \nu^{6} + 23538017273856016799 \nu^{5} + 1467948791527369196388 \nu^{4} - 940733060256762782559 \nu^{3} + 3491995997388483650717 \nu^{2} + 957085277114840834430 \nu - 148193606010062737313275$$$$)/$$$$27\!\cdots\!92$$ $$\beta_{4}$$ $$=$$ $$($$$$-$$$$24\!\cdots\!58$$$$\nu^{15} +$$$$21\!\cdots\!04$$$$\nu^{14} -$$$$23\!\cdots\!64$$$$\nu^{13} +$$$$33\!\cdots\!95$$$$\nu^{12} -$$$$15\!\cdots\!72$$$$\nu^{11} +$$$$22\!\cdots\!77$$$$\nu^{10} -$$$$56\!\cdots\!72$$$$\nu^{9} +$$$$65\!\cdots\!74$$$$\nu^{8} -$$$$14\!\cdots\!24$$$$\nu^{7} +$$$$15\!\cdots\!19$$$$\nu^{6} -$$$$21\!\cdots\!44$$$$\nu^{5} +$$$$16\!\cdots\!38$$$$\nu^{4} -$$$$19\!\cdots\!74$$$$\nu^{3} +$$$$23\!\cdots\!01$$$$\nu^{2} -$$$$45\!\cdots\!04$$$$\nu -$$$$49\!\cdots\!51$$$$)/$$$$95\!\cdots\!44$$ $$\beta_{5}$$ $$=$$ $$($$$$-$$$$15\!\cdots\!15$$$$\nu^{15} +$$$$21\!\cdots\!74$$$$\nu^{14} -$$$$13\!\cdots\!09$$$$\nu^{13} +$$$$34\!\cdots\!07$$$$\nu^{12} -$$$$88\!\cdots\!16$$$$\nu^{11} +$$$$20\!\cdots\!79$$$$\nu^{10} -$$$$29\!\cdots\!65$$$$\nu^{9} +$$$$64\!\cdots\!12$$$$\nu^{8} -$$$$71\!\cdots\!88$$$$\nu^{7} +$$$$11\!\cdots\!81$$$$\nu^{6} -$$$$89\!\cdots\!51$$$$\nu^{5} +$$$$93\!\cdots\!96$$$$\nu^{4} -$$$$86\!\cdots\!83$$$$\nu^{3} -$$$$71\!\cdots\!15$$$$\nu^{2} -$$$$22\!\cdots\!34$$$$\nu -$$$$23\!\cdots\!33$$$$)/$$$$31\!\cdots\!08$$ $$\beta_{6}$$ $$=$$ $$($$$$34\!\cdots\!04$$$$\nu^{15} -$$$$10\!\cdots\!36$$$$\nu^{14} +$$$$31\!\cdots\!40$$$$\nu^{13} -$$$$48\!\cdots\!52$$$$\nu^{12} +$$$$20\!\cdots\!68$$$$\nu^{11} -$$$$32\!\cdots\!16$$$$\nu^{10} +$$$$68\!\cdots\!40$$$$\nu^{9} -$$$$12\!\cdots\!18$$$$\nu^{8} +$$$$16\!\cdots\!16$$$$\nu^{7} -$$$$28\!\cdots\!40$$$$\nu^{6} +$$$$22\!\cdots\!48$$$$\nu^{5} -$$$$37\!\cdots\!60$$$$\nu^{4} +$$$$20\!\cdots\!48$$$$\nu^{3} -$$$$27\!\cdots\!84$$$$\nu^{2} +$$$$36\!\cdots\!44$$$$\nu +$$$$35\!\cdots\!31$$$$)/$$$$62\!\cdots\!01$$ $$\beta_{7}$$ $$=$$ $$($$$$-536662839109972 \nu^{15} + 184355843936000 \nu^{14} - 49316294058364856 \nu^{13} + 76985389887345138 \nu^{12} - 3154550976828630456 \nu^{11} + 5211291917675260305 \nu^{10} - 106964341125832723328 \nu^{9} + 195705915902593012444 \nu^{8} - 2639716566132467353984 \nu^{7} + 4517022991968883377451 \nu^{6} - 35333929028214918917480 \nu^{5} + 60517159199633559261440 \nu^{4} - 328552596355156909606256 \nu^{3} + 436496862559525516173889 \nu^{2} - 567954327233848070808304 \nu - 55329661987849902130608$$$$)/$$$$75\!\cdots\!34$$ $$\beta_{8}$$ $$=$$ $$($$$$2130173036201 \nu^{15} - 540413699312 \nu^{14} + 193661918213181 \nu^{13} - 293621077625441 \nu^{12} + 12285060311199138 \nu^{11} - 19810557668746935 \nu^{10} + 410965646117486839 \nu^{9} - 746713194852137098 \nu^{8} + 10036111646725656126 \nu^{7} - 17105922536607134721 \nu^{6} + 131897626701278295101 \nu^{5} - 228882972939831674094 \nu^{4} + 1220130262910161491531 \nu^{3} - 1620423665127954679375 \nu^{2} + 2091806790152256573456 \nu + 203021156853115775993$$$$)/$$$$25\!\cdots\!08$$ $$\beta_{9}$$ $$=$$ $$($$$$27\!\cdots\!01$$$$\nu^{15} -$$$$90\!\cdots\!76$$$$\nu^{14} +$$$$24\!\cdots\!57$$$$\nu^{13} -$$$$35\!\cdots\!28$$$$\nu^{12} +$$$$15\!\cdots\!94$$$$\nu^{11} -$$$$23\!\cdots\!02$$$$\nu^{10} +$$$$52\!\cdots\!75$$$$\nu^{9} -$$$$85\!\cdots\!28$$$$\nu^{8} +$$$$12\!\cdots\!46$$$$\nu^{7} -$$$$19\!\cdots\!78$$$$\nu^{6} +$$$$16\!\cdots\!77$$$$\nu^{5} -$$$$25\!\cdots\!96$$$$\nu^{4} +$$$$15\!\cdots\!23$$$$\nu^{3} -$$$$18\!\cdots\!22$$$$\nu^{2} +$$$$29\!\cdots\!00$$$$\nu +$$$$31\!\cdots\!96$$$$)/$$$$31\!\cdots\!48$$ $$\beta_{10}$$ $$=$$ $$($$$$-4448842632361015 \nu^{15} + 1527746136142088 \nu^{14} - 406573176070056149 \nu^{13} + 638852366768395737 \nu^{12} - 25956155317369110414 \nu^{11} + 43002282249586977093 \nu^{10} - 874597310627211904019 \nu^{9} + 1616020902079177673194 \nu^{8} - 21496564169263011322522 \nu^{7} + 37198980049890495166591 \nu^{6} - 284874104240500582055717 \nu^{5} + 500982390617084205830630 \nu^{4} - 2644373246682812474960417 \nu^{3} + 3517427264905839032095693 \nu^{2} - 4518691295624378534050984 \nu - 437660350242222703221165$$$$)/$$$$42\!\cdots\!36$$ $$\beta_{11}$$ $$=$$ $$($$$$-$$$$13\!\cdots\!03$$$$\nu^{15} -$$$$20\!\cdots\!62$$$$\nu^{14} -$$$$12\!\cdots\!73$$$$\nu^{13} -$$$$27\!\cdots\!91$$$$\nu^{12} -$$$$79\!\cdots\!12$$$$\nu^{11} -$$$$10\!\cdots\!97$$$$\nu^{10} -$$$$25\!\cdots\!49$$$$\nu^{9} +$$$$24\!\cdots\!04$$$$\nu^{8} -$$$$60\!\cdots\!56$$$$\nu^{7} -$$$$17\!\cdots\!11$$$$\nu^{6} -$$$$74\!\cdots\!67$$$$\nu^{5} +$$$$59\!\cdots\!92$$$$\nu^{4} -$$$$58\!\cdots\!31$$$$\nu^{3} -$$$$22\!\cdots\!95$$$$\nu^{2} -$$$$64\!\cdots\!58$$$$\nu -$$$$91\!\cdots\!03$$$$)/$$$$95\!\cdots\!24$$ $$\beta_{12}$$ $$=$$ $$($$$$50\!\cdots\!76$$$$\nu^{15} +$$$$88\!\cdots\!98$$$$\nu^{14} +$$$$46\!\cdots\!52$$$$\nu^{13} +$$$$23\!\cdots\!41$$$$\nu^{12} +$$$$28\!\cdots\!28$$$$\nu^{11} +$$$$10\!\cdots\!43$$$$\nu^{10} +$$$$93\!\cdots\!48$$$$\nu^{9} +$$$$98\!\cdots\!44$$$$\nu^{8} +$$$$22\!\cdots\!24$$$$\nu^{7} +$$$$33\!\cdots\!21$$$$\nu^{6} +$$$$26\!\cdots\!88$$$$\nu^{5} +$$$$45\!\cdots\!72$$$$\nu^{4} +$$$$22\!\cdots\!04$$$$\nu^{3} +$$$$87\!\cdots\!65$$$$\nu^{2} +$$$$24\!\cdots\!32$$$$\nu +$$$$54\!\cdots\!09$$$$)/$$$$31\!\cdots\!08$$ $$\beta_{13}$$ $$=$$ $$($$$$35\!\cdots\!52$$$$\nu^{15} +$$$$57\!\cdots\!36$$$$\nu^{14} +$$$$32\!\cdots\!91$$$$\nu^{13} +$$$$11\!\cdots\!82$$$$\nu^{12} +$$$$20\!\cdots\!52$$$$\nu^{11} +$$$$50\!\cdots\!62$$$$\nu^{10} +$$$$65\!\cdots\!39$$$$\nu^{9} +$$$$94\!\cdots\!40$$$$\nu^{8} +$$$$15\!\cdots\!28$$$$\nu^{7} +$$$$13\!\cdots\!36$$$$\nu^{6} +$$$$18\!\cdots\!25$$$$\nu^{5} -$$$$51\!\cdots\!08$$$$\nu^{4} +$$$$16\!\cdots\!40$$$$\nu^{3} +$$$$59\!\cdots\!82$$$$\nu^{2} +$$$$16\!\cdots\!14$$$$\nu +$$$$40\!\cdots\!54$$$$)/$$$$15\!\cdots\!54$$ $$\beta_{14}$$ $$=$$ $$($$$$15\!\cdots\!08$$$$\nu^{15} -$$$$45\!\cdots\!56$$$$\nu^{14} +$$$$14\!\cdots\!19$$$$\nu^{13} -$$$$17\!\cdots\!22$$$$\nu^{12} +$$$$90\!\cdots\!98$$$$\nu^{11} -$$$$11\!\cdots\!64$$$$\nu^{10} +$$$$30\!\cdots\!90$$$$\nu^{9} -$$$$46\!\cdots\!82$$$$\nu^{8} +$$$$76\!\cdots\!56$$$$\nu^{7} -$$$$10\!\cdots\!86$$$$\nu^{6} +$$$$10\!\cdots\!75$$$$\nu^{5} -$$$$14\!\cdots\!36$$$$\nu^{4} +$$$$97\!\cdots\!41$$$$\nu^{3} -$$$$10\!\cdots\!00$$$$\nu^{2} +$$$$21\!\cdots\!10$$$$\nu +$$$$24\!\cdots\!97$$$$)/$$$$23\!\cdots\!61$$ $$\beta_{15}$$ $$=$$ $$($$$$96\!\cdots\!17$$$$\nu^{15} -$$$$19\!\cdots\!56$$$$\nu^{14} +$$$$89\!\cdots\!11$$$$\nu^{13} -$$$$10\!\cdots\!49$$$$\nu^{12} +$$$$57\!\cdots\!70$$$$\nu^{11} -$$$$75\!\cdots\!51$$$$\nu^{10} +$$$$19\!\cdots\!13$$$$\nu^{9} -$$$$29\!\cdots\!14$$$$\nu^{8} +$$$$47\!\cdots\!18$$$$\nu^{7} -$$$$67\!\cdots\!65$$$$\nu^{6} +$$$$64\!\cdots\!95$$$$\nu^{5} -$$$$92\!\cdots\!50$$$$\nu^{4} +$$$$60\!\cdots\!39$$$$\nu^{3} -$$$$64\!\cdots\!11$$$$\nu^{2} +$$$$13\!\cdots\!12$$$$\nu +$$$$14\!\cdots\!41$$$$)/$$$$10\!\cdots\!52$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$4 \beta_{15} - 5 \beta_{14} - 3 \beta_{13} + 2 \beta_{12} - 2 \beta_{11} + 5 \beta_{10} + 5 \beta_{9} - 4 \beta_{8} - 2 \beta_{7} - 3 \beta_{6} + \beta_{5} - \beta_{4} - 3 \beta_{3} - \beta_{2} - 10 \beta_{1} - 1$$$$)/28$$ $$\nu^{2}$$ $$=$$ $$($$$$-5 \beta_{15} + 8 \beta_{14} - 2 \beta_{13} + 6 \beta_{12} + \beta_{11} - \beta_{10} + 6 \beta_{9} + 96 \beta_{8} - 127 \beta_{7} - 327 \beta_{6} + 3 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} + 39 \beta_{2} - 100 \beta_{1} - 325$$$$)/28$$ $$\nu^{3}$$ $$=$$ $$($$$$17 \beta_{13} - 18 \beta_{12} + 8 \beta_{11} - 3 \beta_{5} + 5 \beta_{3} + 9 \beta_{2} + 88 \beta_{1} + 51$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$198 \beta_{15} - 272 \beta_{14} - 124 \beta_{13} + 232 \beta_{12} + 34 \beta_{11} - 834 \beta_{10} - 176 \beta_{9} - 2228 \beta_{8} + 3100 \beta_{7} + 5749 \beta_{6} + 74 \beta_{5} + 24 \beta_{4} + 212 \beta_{3} + 892 \beta_{2} - 2476 \beta_{1} - 5765$$$$)/14$$ $$\nu^{5}$$ $$=$$ $$($$$$-5990 \beta_{15} + 8667 \beta_{14} - 5413 \beta_{13} + 6530 \beta_{12} - 1560 \beta_{11} + 9155 \beta_{10} - 2003 \beta_{9} + 20200 \beta_{8} - 9948 \beta_{7} - 19409 \beta_{6} + 577 \beta_{5} + 4259 \beta_{4} + 1055 \beta_{3} - 325 \beta_{2} - 33126 \beta_{1} - 25805$$$$)/28$$ $$\nu^{6}$$ $$=$$ $$($$$$2622 \beta_{13} - 4310 \beta_{12} - 373 \beta_{11} - 811 \beta_{5} - 4770 \beta_{3} - 11579 \beta_{2} + 35956 \beta_{1} + 67649$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$279262 \beta_{15} - 400797 \beta_{14} - 259745 \beta_{13} + 334798 \beta_{12} - 46482 \beta_{11} - 699547 \beta_{10} + 17911 \beta_{9} - 1109420 \beta_{8} + 965760 \beta_{7} + 1345299 \beta_{6} + 19517 \beta_{5} - 220711 \beta_{4} + 140095 \beta_{3} + 120721 \beta_{2} - 1730928 \beta_{1} - 1632009$$$$)/28$$ $$\nu^{8}$$ $$=$$ $$($$$$-691284 \beta_{15} + 964464 \beta_{14} - 585096 \beta_{13} + 900336 \beta_{12} + 42060 \beta_{11} + 3656460 \beta_{10} + 399360 \beta_{9} + 5207880 \beta_{8} - 7456656 \beta_{7} - 10260655 \beta_{6} + 106188 \beta_{5} + 372720 \beta_{4} + 988392 \beta_{3} + 1906704 \beta_{2} - 6545064 \beta_{1} - 10697503$$$$)/14$$ $$\nu^{9}$$ $$=$$ $$($$$$1846509 \beta_{13} - 2471702 \beta_{12} + 211658 \beta_{11} - 112891 \beta_{5} - 1436811 \beta_{3} - 1653581 \beta_{2} + 13011958 \beta_{1} + 13889443$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$77787413 \beta_{15} - 109133120 \beta_{14} - 69460730 \beta_{13} + 103071894 \beta_{12} + 2265457 \beta_{11} - 403264199 \beta_{10} - 38142078 \beta_{9} - 527802720 \beta_{8} + 757264471 \beta_{7} + 971458359 \beta_{6} + 8326683 \beta_{5} - 52807364 \beta_{4} + 108492262 \beta_{3} + 185727303 \beta_{2} - 689743204 \beta_{1} - 1030326949$$$$)/28$$ $$\nu^{11}$$ $$=$$ $$($$$$-696502112 \beta_{15} + 990432227 \beta_{14} - 660025487 \beta_{13} + 903385734 \beta_{12} - 50569868 \beta_{11} + 2558370023 \beta_{10} + 142220511 \beta_{9} + 3268336104 \beta_{8} - 4115827654 \beta_{7} - 4912684271 \beta_{6} + 36476625 \beta_{5} + 587072237 \beta_{4} + 620622637 \beta_{3} + 798700893 \beta_{2} - 4845840568 \beta_{1} - 5586803001$$$$)/28$$ $$\nu^{12}$$ $$=$$ $$283303276 \beta_{13} - 411884296 \beta_{12} - 2844970 \beta_{11} - 25153442 \beta_{5} - 415417460 \beta_{3} - 664916932 \beta_{2} + 2616526588 \beta_{1} + 3695031179$$ $$\nu^{13}$$ $$=$$ $$($$$$36136952786 \beta_{15} - 51274366995 \beta_{14} - 34314451069 \beta_{13} + 47593238282 \beta_{12} - 1858626996 \beta_{11} - 144106197839 \beta_{10} - 9561533221 \beta_{9} - 177814191664 \beta_{8} + 238342892184 \beta_{7} + 279407517869 \beta_{6} + 1822501717 \beta_{5} - 30669447635 \beta_{4} + 35989748351 \beta_{3} + 48945848111 \beta_{2} - 259758006294 \beta_{1} - 313758094217$$$$)/28$$ $$\nu^{14}$$ $$=$$ $$($$$$-237558015007 \beta_{15} + 335044352924 \beta_{14} - 221473349274 \beta_{13} + 318262664810 \beta_{12} - 697022381 \beta_{11} + 1167123223825 \beta_{10} + 95395270774 \beta_{9} + 1430841773632 \beta_{8} - 2050689201961 \beta_{7} - 2461353139957 \beta_{6} + 16084665733 \beta_{5} + 189304017808 \beta_{4} + 309855069606 \beta_{3} + 476981853485 \beta_{2} - 1955190144364 \beta_{1} - 2667438845879$$$$)/28$$ $$\nu^{15}$$ $$=$$ $$($$$$257998257311 \beta_{13} - 360676661674 \beta_{12} + 10562294418 \beta_{11} - 13560958727 \beta_{5} - 289138472617 \beta_{3} - 405091585843 \beta_{2} + 1997234018400 \beta_{1} + 2480567013939$$$$)/2$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times$$.

 $$n$$ $$491$$ $$1081$$ $$1177$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
391.1
 −0.141814 − 0.245629i 0.848921 + 1.47037i −2.10711 − 3.64962i 2.81422 + 4.87437i −2.10711 + 3.64962i 2.81422 − 4.87437i −0.141814 + 0.245629i 0.848921 − 1.47037i −2.63284 − 4.56021i 1.92573 + 3.33546i 2.96377 + 5.13339i −3.67087 − 6.35814i 2.96377 − 5.13339i −3.67087 + 6.35814i −2.63284 + 4.56021i 1.92573 − 3.33546i
−1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 −2.82843 −3.00000 3.16228i
391.2 −1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 −2.82843 −3.00000 3.16228i
391.3 −1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 −2.82843 −3.00000 3.16228i
391.4 −1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 −2.82843 −3.00000 3.16228i
391.5 −1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 −2.82843 −3.00000 3.16228i
391.6 −1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 −2.82843 −3.00000 3.16228i
391.7 −1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 −2.82843 −3.00000 3.16228i
391.8 −1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 −2.82843 −3.00000 3.16228i
391.9 1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 2.82843 −3.00000 3.16228i
391.10 1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 2.82843 −3.00000 3.16228i
391.11 1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 2.82843 −3.00000 3.16228i
391.12 1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 2.82843 −3.00000 3.16228i
391.13 1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 2.82843 −3.00000 3.16228i
391.14 1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 2.82843 −3.00000 3.16228i
391.15 1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 2.82843 −3.00000 3.16228i
391.16 1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 2.82843 −3.00000 3.16228i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 391.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1470.3.f.d 16
7.b odd 2 1 inner 1470.3.f.d 16
7.c even 3 1 210.3.o.b 16
7.d odd 6 1 210.3.o.b 16
21.g even 6 1 630.3.v.c 16
21.h odd 6 1 630.3.v.c 16
35.i odd 6 1 1050.3.p.i 16
35.j even 6 1 1050.3.p.i 16
35.k even 12 2 1050.3.q.e 32
35.l odd 12 2 1050.3.q.e 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.o.b 16 7.c even 3 1
210.3.o.b 16 7.d odd 6 1
630.3.v.c 16 21.g even 6 1
630.3.v.c 16 21.h odd 6 1
1050.3.p.i 16 35.i odd 6 1
1050.3.p.i 16 35.j even 6 1
1050.3.q.e 32 35.k even 12 2
1050.3.q.e 32 35.l odd 12 2
1470.3.f.d 16 1.a even 1 1 trivial
1470.3.f.d 16 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{8} - \cdots$$ acting on $$S_{3}^{\mathrm{new}}(1470, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T^{2} )^{8}$$
$3$ $$( 1 + 3 T^{2} )^{8}$$
$5$ $$( 1 + 5 T^{2} )^{8}$$
$7$ 1
$11$ $$( 1 - 4 T + 380 T^{2} - 2036 T^{3} + 82148 T^{4} - 539228 T^{5} + 12902676 T^{6} - 93038220 T^{7} + 1637770150 T^{8} - 11257624620 T^{9} + 188908079316 T^{10} - 955275294908 T^{11} + 17609153356388 T^{12} - 52808596487636 T^{13} + 1192602783153980 T^{14} - 1518999334332964 T^{15} + 45949729863572161 T^{16} )^{2}$$
$13$ $$1 - 1336 T^{2} + 822548 T^{4} - 307870064 T^{6} + 77337463770 T^{8} - 13349067637096 T^{10} + 1484717408181296 T^{12} - 75519988778385192 T^{14} - 843289248891793565 T^{16} -$$$$21\!\cdots\!12$$$$T^{18} +$$$$12\!\cdots\!16$$$$T^{20} -$$$$31\!\cdots\!76$$$$T^{22} +$$$$51\!\cdots\!70$$$$T^{24} -$$$$58\!\cdots\!64$$$$T^{26} +$$$$44\!\cdots\!28$$$$T^{28} -$$$$20\!\cdots\!56$$$$T^{30} +$$$$44\!\cdots\!81$$$$T^{32}$$
$17$ $$1 - 696 T^{2} + 231064 T^{4} - 37149384 T^{6} + 6926154940 T^{8} - 4433534480088 T^{10} + 1832268292772776 T^{12} - 374043336733365672 T^{14} + 58828830910943250694 T^{16} -$$$$31\!\cdots\!12$$$$T^{18} +$$$$12\!\cdots\!16$$$$T^{20} -$$$$25\!\cdots\!68$$$$T^{22} +$$$$33\!\cdots\!40$$$$T^{24} -$$$$15\!\cdots\!84$$$$T^{26} +$$$$78\!\cdots\!44$$$$T^{28} -$$$$19\!\cdots\!36$$$$T^{30} +$$$$23\!\cdots\!61$$$$T^{32}$$
$19$ $$1 - 2440 T^{2} + 3262484 T^{4} - 3093518480 T^{6} + 2289552719514 T^{8} - 1393085088559960 T^{10} + 717656314443664304 T^{12} -$$$$31\!\cdots\!00$$$$T^{14} +$$$$12\!\cdots\!75$$$$T^{16} -$$$$41\!\cdots\!00$$$$T^{18} +$$$$12\!\cdots\!64$$$$T^{20} -$$$$30\!\cdots\!60$$$$T^{22} +$$$$66\!\cdots\!34$$$$T^{24} -$$$$11\!\cdots\!80$$$$T^{26} +$$$$15\!\cdots\!64$$$$T^{28} -$$$$15\!\cdots\!40$$$$T^{30} +$$$$83\!\cdots\!61$$$$T^{32}$$
$23$ $$( 1 - 12 T + 1948 T^{2} - 32316 T^{3} + 2328740 T^{4} - 41047380 T^{5} + 1879055412 T^{6} - 32536827396 T^{7} + 1148877345254 T^{8} - 17211981692484 T^{9} + 525836745549492 T^{10} - 6076485389420820 T^{11} + 182365923863275940 T^{12} - 1338739136380281084 T^{13} + 42689688393575585308 T^{14} -$$$$13\!\cdots\!08$$$$T^{15} +$$$$61\!\cdots\!61$$$$T^{16} )^{2}$$
$29$ $$( 1 - 36 T + 1444 T^{2} + 19116 T^{3} - 1274860 T^{4} + 68800452 T^{5} + 84102156 T^{6} - 30035473452 T^{7} + 2426711187974 T^{8} - 25259833173132 T^{9} + 59483856997836 T^{10} + 40924113344941092 T^{11} - 637744142027460460 T^{12} + 8042239471766642316 T^{13} +$$$$51\!\cdots\!04$$$$T^{14} -$$$$10\!\cdots\!16$$$$T^{15} +$$$$25\!\cdots\!21$$$$T^{16} )^{2}$$
$31$ $$1 - 8776 T^{2} + 39983444 T^{4} - 123311980304 T^{6} + 285704968947930 T^{8} - 524575072927909528 T^{10} +$$$$78\!\cdots\!96$$$$T^{12} -$$$$98\!\cdots\!72$$$$T^{14} +$$$$10\!\cdots\!99$$$$T^{16} -$$$$90\!\cdots\!12$$$$T^{18} +$$$$67\!\cdots\!36$$$$T^{20} -$$$$41\!\cdots\!08$$$$T^{22} +$$$$20\!\cdots\!30$$$$T^{24} -$$$$82\!\cdots\!04$$$$T^{26} +$$$$24\!\cdots\!24$$$$T^{28} -$$$$50\!\cdots\!16$$$$T^{30} +$$$$52\!\cdots\!61$$$$T^{32}$$
$37$ $$( 1 + 44 T + 3892 T^{2} + 79760 T^{3} + 7985418 T^{4} + 149405340 T^{5} + 14878534320 T^{6} + 188674612764 T^{7} + 20151694610467 T^{8} + 258295544873916 T^{9} + 27884768759705520 T^{10} + 383333226483624060 T^{11} + 28048616655970923978 T^{12} +$$$$38\!\cdots\!40$$$$T^{13} +$$$$25\!\cdots\!52$$$$T^{14} +$$$$39\!\cdots\!16$$$$T^{15} +$$$$12\!\cdots\!41$$$$T^{16} )^{2}$$
$41$ $$1 - 11416 T^{2} + 66218264 T^{4} - 265318066664 T^{6} + 835818107018940 T^{8} - 2208514761298495288 T^{10} +$$$$50\!\cdots\!56$$$$T^{12} -$$$$10\!\cdots\!32$$$$T^{14} +$$$$18\!\cdots\!14$$$$T^{16} -$$$$28\!\cdots\!52$$$$T^{18} +$$$$40\!\cdots\!76$$$$T^{20} -$$$$49\!\cdots\!28$$$$T^{22} +$$$$53\!\cdots\!40$$$$T^{24} -$$$$47\!\cdots\!64$$$$T^{26} +$$$$33\!\cdots\!04$$$$T^{28} -$$$$16\!\cdots\!36$$$$T^{30} +$$$$40\!\cdots\!81$$$$T^{32}$$
$43$ $$( 1 + 28 T + 5732 T^{2} + 194288 T^{3} + 21755610 T^{4} + 646615948 T^{5} + 59916131504 T^{6} + 1611056221164 T^{7} + 122468758030675 T^{8} + 2978842952932236 T^{9} + 204841330302006704 T^{10} + 4087494160581305452 T^{11} +$$$$25\!\cdots\!10$$$$T^{12} +$$$$41\!\cdots\!12$$$$T^{13} +$$$$22\!\cdots\!32$$$$T^{14} +$$$$20\!\cdots\!72$$$$T^{15} +$$$$13\!\cdots\!01$$$$T^{16} )^{2}$$
$47$ $$1 - 19504 T^{2} + 194566136 T^{4} - 1312184963216 T^{6} + 6686766048287772 T^{8} - 27242570426384646448 T^{10} +$$$$91\!\cdots\!04$$$$T^{12} -$$$$25\!\cdots\!12$$$$T^{14} +$$$$61\!\cdots\!10$$$$T^{16} -$$$$12\!\cdots\!72$$$$T^{18} +$$$$21\!\cdots\!44$$$$T^{20} -$$$$31\!\cdots\!68$$$$T^{22} +$$$$37\!\cdots\!12$$$$T^{24} -$$$$36\!\cdots\!16$$$$T^{26} +$$$$26\!\cdots\!16$$$$T^{28} -$$$$12\!\cdots\!44$$$$T^{30} +$$$$32\!\cdots\!41$$$$T^{32}$$
$53$ $$( 1 + 32 T + 13096 T^{2} + 382176 T^{3} + 87745820 T^{4} + 2185324448 T^{5} + 391392084888 T^{6} + 8408381790560 T^{7} + 1270026486053126 T^{8} + 23619144449683040 T^{9} + 3088271809359151128 T^{10} + 48436320249504581792 T^{11} +$$$$54\!\cdots\!20$$$$T^{12} +$$$$66\!\cdots\!24$$$$T^{13} +$$$$64\!\cdots\!36$$$$T^{14} +$$$$44\!\cdots\!08$$$$T^{15} +$$$$38\!\cdots\!21$$$$T^{16} )^{2}$$
$59$ $$1 - 1896 T^{2} + 10373176 T^{4} - 69423767064 T^{6} + 392354089819132 T^{8} - 1032139977647109192 T^{10} +$$$$50\!\cdots\!84$$$$T^{12} -$$$$17\!\cdots\!48$$$$T^{14} +$$$$83\!\cdots\!30$$$$T^{16} -$$$$21\!\cdots\!28$$$$T^{18} +$$$$73\!\cdots\!64$$$$T^{20} -$$$$18\!\cdots\!52$$$$T^{22} +$$$$84\!\cdots\!12$$$$T^{24} -$$$$18\!\cdots\!64$$$$T^{26} +$$$$32\!\cdots\!36$$$$T^{28} -$$$$72\!\cdots\!16$$$$T^{30} +$$$$46\!\cdots\!81$$$$T^{32}$$
$61$ $$1 - 26032 T^{2} + 316572152 T^{4} - 2428041481232 T^{6} + 13620146914246812 T^{8} - 63272070928644615856 T^{10} +$$$$27\!\cdots\!52$$$$T^{12} -$$$$11\!\cdots\!52$$$$T^{14} +$$$$44\!\cdots\!86$$$$T^{16} -$$$$15\!\cdots\!32$$$$T^{18} +$$$$52\!\cdots\!12$$$$T^{20} -$$$$16\!\cdots\!76$$$$T^{22} +$$$$50\!\cdots\!32$$$$T^{24} -$$$$12\!\cdots\!32$$$$T^{26} +$$$$22\!\cdots\!32$$$$T^{28} -$$$$25\!\cdots\!92$$$$T^{30} +$$$$13\!\cdots\!21$$$$T^{32}$$
$67$ $$( 1 - 164 T + 33236 T^{2} - 4024336 T^{3} + 476556074 T^{4} - 45254634868 T^{5} + 4016466262608 T^{6} - 308250966055572 T^{7} + 22055966830747363 T^{8} - 1383738586623462708 T^{9} + 80936297650231583568 T^{10} -$$$$40\!\cdots\!92$$$$T^{11} +$$$$19\!\cdots\!34$$$$T^{12} -$$$$73\!\cdots\!64$$$$T^{13} +$$$$27\!\cdots\!96$$$$T^{14} -$$$$60\!\cdots\!56$$$$T^{15} +$$$$16\!\cdots\!81$$$$T^{16} )^{2}$$
$71$ $$( 1 + 68 T + 26092 T^{2} + 2195028 T^{3} + 351820772 T^{4} + 28773766364 T^{5} + 3188424927972 T^{6} + 217589962697708 T^{7} + 19655929349959526 T^{8} + 1096871001959146028 T^{9} + 81023237162072440932 T^{10} +$$$$36\!\cdots\!44$$$$T^{11} +$$$$22\!\cdots\!92$$$$T^{12} +$$$$71\!\cdots\!28$$$$T^{13} +$$$$42\!\cdots\!72$$$$T^{14} +$$$$56\!\cdots\!08$$$$T^{15} +$$$$41\!\cdots\!21$$$$T^{16} )^{2}$$
$73$ $$1 - 44312 T^{2} + 949379444 T^{4} - 13136848413808 T^{6} + 133216272007717754 T^{8} -$$$$10\!\cdots\!88$$$$T^{10} +$$$$73\!\cdots\!20$$$$T^{12} -$$$$44\!\cdots\!60$$$$T^{14} +$$$$24\!\cdots\!47$$$$T^{16} -$$$$12\!\cdots\!60$$$$T^{18} +$$$$58\!\cdots\!20$$$$T^{20} -$$$$24\!\cdots\!48$$$$T^{22} +$$$$86\!\cdots\!94$$$$T^{24} -$$$$24\!\cdots\!08$$$$T^{26} +$$$$49\!\cdots\!04$$$$T^{28} -$$$$66\!\cdots\!72$$$$T^{30} +$$$$42\!\cdots\!21$$$$T^{32}$$
$79$ $$( 1 + 280 T + 63868 T^{2} + 9752992 T^{3} + 1312957026 T^{4} + 143007366360 T^{5} + 14416274240064 T^{6} + 1263875597787768 T^{7} + 105782687301458731 T^{8} + 7887847605793460088 T^{9} +$$$$56\!\cdots\!84$$$$T^{10} +$$$$34\!\cdots\!60$$$$T^{11} +$$$$19\!\cdots\!86$$$$T^{12} +$$$$92\!\cdots\!92$$$$T^{13} +$$$$37\!\cdots\!88$$$$T^{14} +$$$$10\!\cdots\!80$$$$T^{15} +$$$$23\!\cdots\!21$$$$T^{16} )^{2}$$
$83$ $$1 - 66184 T^{2} + 2113671608 T^{4} - 43986302935160 T^{6} + 678322851950602236 T^{8} -$$$$83\!\cdots\!72$$$$T^{10} +$$$$84\!\cdots\!60$$$$T^{12} -$$$$72\!\cdots\!04$$$$T^{14} +$$$$53\!\cdots\!06$$$$T^{16} -$$$$34\!\cdots\!84$$$$T^{18} +$$$$19\!\cdots\!60$$$$T^{20} -$$$$88\!\cdots\!92$$$$T^{22} +$$$$34\!\cdots\!16$$$$T^{24} -$$$$10\!\cdots\!60$$$$T^{26} +$$$$24\!\cdots\!68$$$$T^{28} -$$$$35\!\cdots\!44$$$$T^{30} +$$$$25\!\cdots\!61$$$$T^{32}$$
$89$ $$1 - 19336 T^{2} + 310289528 T^{4} - 4184900827256 T^{6} + 49417825883213052 T^{8} -$$$$50\!\cdots\!60$$$$T^{10} +$$$$49\!\cdots\!52$$$$T^{12} -$$$$43\!\cdots\!88$$$$T^{14} +$$$$35\!\cdots\!50$$$$T^{16} -$$$$26\!\cdots\!08$$$$T^{18} +$$$$19\!\cdots\!12$$$$T^{20} -$$$$12\!\cdots\!60$$$$T^{22} +$$$$76\!\cdots\!72$$$$T^{24} -$$$$40\!\cdots\!56$$$$T^{26} +$$$$18\!\cdots\!48$$$$T^{28} -$$$$74\!\cdots\!16$$$$T^{30} +$$$$24\!\cdots\!21$$$$T^{32}$$
$97$ $$1 - 70448 T^{2} + 2604307832 T^{4} - 66719407643536 T^{6} + 1323480529449253148 T^{8} -$$$$21\!\cdots\!72$$$$T^{10} +$$$$29\!\cdots\!68$$$$T^{12} -$$$$34\!\cdots\!84$$$$T^{14} +$$$$34\!\cdots\!38$$$$T^{16} -$$$$30\!\cdots\!04$$$$T^{18} +$$$$23\!\cdots\!48$$$$T^{20} -$$$$14\!\cdots\!52$$$$T^{22} +$$$$81\!\cdots\!08$$$$T^{24} -$$$$36\!\cdots\!36$$$$T^{26} +$$$$12\!\cdots\!92$$$$T^{28} -$$$$30\!\cdots\!28$$$$T^{30} +$$$$37\!\cdots\!41$$$$T^{32}$$