# Properties

 Label 630.3.o.f Level 630 Weight 3 Character orbit 630.o Analytic conductor 17.166 Analytic rank 0 Dimension 16 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$630 = 2 \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 630.o (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$17.1662566547$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{14}\cdot 5$$ Twist minimal: no (minimal twist has level 210) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{3} ) q^{2} -2 \beta_{3} q^{4} + ( 1 - \beta_{11} ) q^{5} + \beta_{5} q^{7} + ( -2 - 2 \beta_{3} ) q^{8} +O(q^{10})$$ $$q + ( 1 - \beta_{3} ) q^{2} -2 \beta_{3} q^{4} + ( 1 - \beta_{11} ) q^{5} + \beta_{5} q^{7} + ( -2 - 2 \beta_{3} ) q^{8} + ( 1 - \beta_{3} - \beta_{11} + \beta_{12} ) q^{10} + ( 2 \beta_{4} + \beta_{5} - \beta_{14} + \beta_{15} ) q^{11} + ( -2 - 2 \beta_{3} - \beta_{9} - \beta_{11} - \beta_{13} - \beta_{15} ) q^{13} + ( -\beta_{4} + \beta_{5} ) q^{14} -4 q^{16} + ( -3 + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{8} - \beta_{11} + \beta_{15} ) q^{17} + ( 1 - \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{7} - 2 \beta_{8} - \beta_{11} - \beta_{13} - 2 \beta_{14} ) q^{19} + ( -2 \beta_{3} + 2 \beta_{12} ) q^{20} + ( \beta_{2} + 2 \beta_{5} + \beta_{6} - \beta_{14} + \beta_{15} ) q^{22} + ( -1 - \beta_{2} - \beta_{3} - \beta_{6} + 2 \beta_{9} - \beta_{11} - \beta_{13} - \beta_{14} ) q^{23} + ( 3 + \beta_{1} + \beta_{3} + 7 \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{8} - 2 \beta_{11} - \beta_{12} ) q^{25} + ( -5 - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{26} -2 \beta_{4} q^{28} + ( -1 + 4 \beta_{3} - 3 \beta_{4} + 5 \beta_{5} - \beta_{6} - \beta_{8} - 3 \beta_{11} + 3 \beta_{12} + \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{29} + ( -10 - 3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} + 3 \beta_{12} + \beta_{14} - 2 \beta_{15} ) q^{31} + ( -4 + 4 \beta_{3} ) q^{32} + ( -1 + 3 \beta_{2} + 4 \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{8} - \beta_{11} + \beta_{12} + \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{34} + ( -2 + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{8} + \beta_{13} ) q^{35} + ( -9 + \beta_{1} + \beta_{2} + 7 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} - 4 \beta_{11} + 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{37} + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{11} + \beta_{12} - 3 \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{38} + ( -2 - 2 \beta_{3} + 2 \beta_{11} + 2 \beta_{12} ) q^{40} + ( -1 - 2 \beta_{1} - \beta_{3} + 2 \beta_{4} + 3 \beta_{5} - \beta_{6} + 2 \beta_{8} - \beta_{9} - \beta_{10} - 3 \beta_{11} + 2 \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{41} + ( 2 + 3 \beta_{1} + 2 \beta_{3} - 8 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} - 2 \beta_{9} - 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{43} + ( 2 \beta_{2} - 4 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{44} + ( -3 - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{46} + ( -6 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{7} + 2 \beta_{10} - 4 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} ) q^{47} -7 \beta_{3} q^{49} + ( 4 - \beta_{1} - \beta_{2} - \beta_{3} + 5 \beta_{4} + 9 \beta_{5} + 3 \beta_{7} - \beta_{8} - 3 \beta_{11} + \beta_{12} - \beta_{13} - \beta_{15} ) q^{50} + ( -6 + 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} - 2 \beta_{10} + 2 \beta_{12} + 2 \beta_{14} - 2 \beta_{15} ) q^{52} + ( -4 - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{8} + 2 \beta_{9} + \beta_{11} + 4 \beta_{12} - \beta_{13} + \beta_{15} ) q^{53} + ( -1 - 4 \beta_{1} + \beta_{2} + 8 \beta_{3} + 9 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{55} + ( -2 \beta_{4} - 2 \beta_{5} ) q^{56} + ( 4 - 2 \beta_{2} + 6 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} - 2 \beta_{8} + 6 \beta_{12} + 4 \beta_{15} ) q^{58} + ( 2 + 4 \beta_{2} - 6 \beta_{3} - 6 \beta_{5} + 6 \beta_{6} - 4 \beta_{7} + 2 \beta_{11} - 4 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{59} + ( 11 + 4 \beta_{1} + 2 \beta_{2} + \beta_{4} - 5 \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} - 5 \beta_{12} + \beta_{13} - 3 \beta_{14} + 4 \beta_{15} ) q^{61} + ( -11 - 3 \beta_{1} - \beta_{2} + 8 \beta_{3} - \beta_{4} + \beta_{5} - 3 \beta_{7} + \beta_{8} + 2 \beta_{10} + \beta_{11} + 5 \beta_{12} + \beta_{13} - 3 \beta_{15} ) q^{62} + 8 \beta_{3} q^{64} + ( 1 + 4 \beta_{1} + 2 \beta_{2} + 12 \beta_{3} + 5 \beta_{4} + 3 \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} - 4 \beta_{9} + 2 \beta_{10} + \beta_{12} - 4 \beta_{13} - 3 \beta_{14} + \beta_{15} ) q^{65} + ( -6 + 6 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} + 4 \beta_{8} - 4 \beta_{10} + 2 \beta_{11} - 2 \beta_{13} + 4 \beta_{14} - 2 \beta_{15} ) q^{67} + ( 4 + 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{5} + 4 \beta_{6} + 2 \beta_{12} + 2 \beta_{13} + 4 \beta_{14} ) q^{68} + ( 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{13} + \beta_{15} ) q^{70} + ( -41 - 4 \beta_{2} - 5 \beta_{4} + 2 \beta_{5} + 5 \beta_{6} - \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + \beta_{11} + 7 \beta_{12} + \beta_{13} + 4 \beta_{14} - 3 \beta_{15} ) q^{71} + ( -2 - 8 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + 4 \beta_{5} + 8 \beta_{7} - 3 \beta_{9} - 3 \beta_{11} + 4 \beta_{12} + \beta_{13} + 5 \beta_{15} ) q^{73} + ( 18 \beta_{3} + 2 \beta_{5} + 2 \beta_{7} - 4 \beta_{8} - 4 \beta_{11} + 4 \beta_{12} - 2 \beta_{14} + 2 \beta_{15} ) q^{74} + ( 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{6} + 2 \beta_{8} + 2 \beta_{12} - 4 \beta_{13} - 2 \beta_{15} ) q^{76} + ( 9 - \beta_{2} - 10 \beta_{3} + \beta_{4} + \beta_{6} - \beta_{8} + 2 \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{77} + ( -2 - 2 \beta_{2} - 38 \beta_{3} - 4 \beta_{4} + 6 \beta_{5} - 4 \beta_{6} + 8 \beta_{7} + 4 \beta_{8} + 2 \beta_{11} + 2 \beta_{13} + 4 \beta_{14} ) q^{79} + ( -4 + 4 \beta_{11} ) q^{80} + ( -3 - 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + 5 \beta_{5} - 2 \beta_{7} + 3 \beta_{8} - 2 \beta_{10} - \beta_{11} + 5 \beta_{12} + \beta_{13} - \beta_{15} ) q^{82} + ( 12 - 4 \beta_{1} + 2 \beta_{2} + 10 \beta_{3} - 10 \beta_{4} + 4 \beta_{5} + 6 \beta_{6} + 4 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 6 \beta_{11} + 6 \beta_{14} - 4 \beta_{15} ) q^{83} + ( 7 + 7 \beta_{1} - 12 \beta_{3} + 3 \beta_{4} + 12 \beta_{5} + \beta_{6} - 6 \beta_{7} - \beta_{8} + 4 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - 5 \beta_{12} - 4 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{85} + ( 2 + 6 \beta_{1} + 2 \beta_{2} - 10 \beta_{4} - 10 \beta_{5} - 4 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{15} ) q^{86} + ( 2 \beta_{2} - 8 \beta_{4} + 2 \beta_{6} + 2 \beta_{14} - 2 \beta_{15} ) q^{88} + ( -2 \beta_{2} - 55 \beta_{3} - 5 \beta_{4} + 8 \beta_{5} - 2 \beta_{6} + 8 \beta_{7} - 3 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} + 3 \beta_{12} - 2 \beta_{14} + \beta_{15} ) q^{89} + ( 2 + 7 \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + \beta_{8} - 2 \beta_{11} - 3 \beta_{12} - \beta_{15} ) q^{91} + ( -4 + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{8} + 4 \beta_{10} + 2 \beta_{12} ) q^{92} + ( -2 + 2 \beta_{2} + 8 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 4 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + 6 \beta_{12} + 2 \beta_{13} + 2 \beta_{15} ) q^{94} + ( -24 - 8 \beta_{1} - 2 \beta_{2} - 13 \beta_{3} + 21 \beta_{4} + 10 \beta_{5} + \beta_{6} - 6 \beta_{7} - 3 \beta_{8} - 4 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + \beta_{12} + 2 \beta_{13} + 2 \beta_{14} ) q^{95} + ( 43 - 6 \beta_{1} - 2 \beta_{2} - 40 \beta_{3} + 3 \beta_{4} - 27 \beta_{5} + 5 \beta_{6} - 6 \beta_{7} - 3 \beta_{8} + 3 \beta_{10} + 4 \beta_{11} + \beta_{12} - 5 \beta_{14} + \beta_{15} ) q^{97} + ( -7 - 7 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 16q^{2} + 16q^{5} - 32q^{8} + O(q^{10})$$ $$16q + 16q^{2} + 16q^{5} - 32q^{8} + 24q^{10} - 8q^{11} - 32q^{13} - 64q^{16} - 56q^{17} + 16q^{20} - 8q^{22} - 24q^{23} + 40q^{25} - 64q^{26} - 112q^{31} - 64q^{32} - 28q^{35} - 152q^{37} - 16q^{40} - 48q^{46} - 80q^{47} + 72q^{50} - 64q^{52} - 48q^{53} - 24q^{55} + 96q^{58} + 96q^{61} - 112q^{62} - 16q^{65} - 80q^{67} + 112q^{68} - 536q^{71} + 168q^{77} - 64q^{80} + 256q^{83} + 40q^{85} + 16q^{88} - 48q^{92} - 360q^{95} + 688q^{97} - 112q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 8 x^{15} + 32 x^{14} + 152 x^{13} + 1954 x^{12} - 12664 x^{11} + 50336 x^{10} + 231896 x^{9} + 1093889 x^{8} - 4595248 x^{7} + 18837632 x^{6} + 86081152 x^{5} + 178889856 x^{4} + 70149120 x^{3} + 10035200 x^{2} - 7168000 x + 2560000$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$498284030129 \nu^{15} - 3107145446008 \nu^{14} + 4536459111192 \nu^{13} + 151584816096312 \nu^{12} + 884237923385410 \nu^{11} - 4928139696036504 \nu^{10} + 7134008349081616 \nu^{9} + 244453432126191576 \nu^{8} + 409758392839305393 \nu^{7} - 1824569731375289568 \nu^{6} + 2818992904749593592 \nu^{5} + 95046452581399705312 \nu^{4} + 42570659834773128768 \nu^{3} + 9196525750040289280 \nu^{2} - 5587401352259187200 \nu + 534324510814884608000$$$$)/$$$$10\!\cdots\!00$$ $$\beta_{2}$$ $$=$$ $$($$$$-1507199328622388427929525 \nu^{15} + 17970079645683158103311503 \nu^{14} - 86826702835655389258321412 \nu^{13} - 112235197009448860752592032 \nu^{12} - 1794093536667855446399605602 \nu^{11} + 32136734753681945313258983974 \nu^{10} - 135022761934534323422171192576 \nu^{9} - 173521337611490097457645568136 \nu^{8} + 77180454102197290632334158579 \nu^{7} + 15565333085757911374763920921143 \nu^{6} - 47559316792232250224591704366012 \nu^{5} - 66084760112201739257152625941032 \nu^{4} + 353821788205151003380064012198848 \nu^{3} + 1696749097671225799725375195551680 \nu^{2} + 1590560516521259391081802999148800 \nu + 53477185232134471394118710208000$$$$)/$$$$75\!\cdots\!00$$ $$\beta_{3}$$ $$=$$ $$($$$$-50521933035234406935387 \nu^{15} + 355849584339626868196590 \nu^{14} - 1220781585234851432658840 \nu^{13} - 9304805374903308743399240 \nu^{12} - 105717930910748956540229574 \nu^{11} + 546511703953657264363821140 \nu^{10} - 1912893592429770663178189120 \nu^{9} - 14274226782583577122771384520 \nu^{8} - 65924256015119669450932994891 \nu^{7} + 180976850729426810190271470950 \nu^{6} - 719449018130531781232088723240 \nu^{5} - 5305696558618628641218886482240 \nu^{4} - 12993474700473951273694265843648 \nu^{3} - 11620918919670154992391536887680 \nu^{2} - 2225350312372339004103034892800 \nu + 232431648484874621115141632000$$$$)/$$$$11\!\cdots\!00$$ $$\beta_{4}$$ $$=$$ $$($$$$-12085404159758350350699247 \nu^{15} + 85224520217749198675066170 \nu^{14} - 292450754027165513539470160 \nu^{13} - 2226878906398720288454917560 \nu^{12} - 25263266575828111291938782414 \nu^{11} + 130955780952902562464699541580 \nu^{10} - 458255913561199101206385102480 \nu^{9} - 3418446210754313761500284959880 \nu^{8} - 15752394612797340867688025357631 \nu^{7} + 43372335081959687277592960733330 \nu^{6} - 172352038909657772404504958906560 \nu^{5} - 1270855705906394283493640257820560 \nu^{4} - 3111633157131511050847624678326208 \nu^{3} - 2782839408275349352230347459422080 \nu^{2} - 532880132548126797451977997516800 \nu + 338043110964360494658210332032000$$$$)/$$$$15\!\cdots\!00$$ $$\beta_{5}$$ $$=$$ $$($$$$6056238176411623354901114 \nu^{15} - 42688660290736661449235625 \nu^{14} + 146396752377212370392985060 \nu^{13} + 1114365932830071889909269160 \nu^{12} + 12669223252126716516697758108 \nu^{11} - 65598741434497573017700291490 \nu^{10} + 228974356306765774837195962480 \nu^{9} + 1708385418054973711004999115680 \nu^{8} + 7897245961507910001212281287842 \nu^{7} - 21731443531600977047188683957145 \nu^{6} + 85927199122479636045696950022860 \nu^{5} + 634888487783576647750492215095160 \nu^{4} + 1555603783724061364253191658007936 \nu^{3} + 1391388873664253031217367420877760 \nu^{2} + 266461998719592764984523512185600 \nu + 113359430152712771622478808256000$$$$)/$$$$75\!\cdots\!00$$ $$\beta_{6}$$ $$=$$ $$($$$$-8777584802917452100592915 \nu^{15} + 74241770522206402655967728 \nu^{14} - 325081325120604388154675112 \nu^{13} - 1102626571635748756666051432 \nu^{12} - 16954729091769539577109963782 \nu^{11} + 117319933591638618645769694424 \nu^{10} - 515313230685024011226516720176 \nu^{9} - 1663833911130719816240276739336 \nu^{8} - 9288302061651309344113847415891 \nu^{7} + 42150479723803937902196888918568 \nu^{6} - 195902824516510917785544306872712 \nu^{5} - 614728432832429659646949837550432 \nu^{4} - 1442268565245647741634224725650112 \nu^{3} - 866292116559071007080999323409920 \nu^{2} - 1750255215698528982004688900723200 \nu + 101744118017880288715678636288000$$$$)/$$$$75\!\cdots\!00$$ $$\beta_{7}$$ $$=$$ $$($$$$2256059372367 \nu^{15} - 15491795991004 \nu^{14} + 51193155741176 \nu^{13} + 429277520452136 \nu^{12} + 4778238010708190 \nu^{11} - 23664986684234512 \nu^{10} + 80163851117945648 \nu^{9} + 659096250117860328 \nu^{8} + 3029964043268094319 \nu^{7} - 7728422423215531764 \nu^{6} + 30190303107868824376 \nu^{5} + 245029826542538987136 \nu^{4} + 611701073097399244224 \nu^{3} + 548941248000905818880 \nu^{2} + 105301060960663846400 \nu - 11051711894885376000$$$$)/ 11373131612962816000$$ $$\beta_{8}$$ $$=$$ $$($$$$-53453000446683789795632691 \nu^{15} + 413468129475094484678871402 \nu^{14} - 1584755917723635844287496048 \nu^{13} - 8697885978704018417996591128 \nu^{12} - 105961003111341288722252426070 \nu^{11} + 650291046644204334171770899276 \nu^{10} - 2489257670269199927462816733904 \nu^{9} - 13290507949248193011690515661544 \nu^{8} - 60660729641285327316685378685187 \nu^{7} + 232001360301403406124337624556642 \nu^{6} - 930645725391078941535034491879648 \nu^{5} - 4930577980578256305585405063338128 \nu^{4} - 10338182569908802403245236397660352 \nu^{3} - 5452919232974584984693876198069120 \nu^{2} + 2360523507935563196223389644800 \nu + 750749150510170223960423077248000$$$$)/$$$$15\!\cdots\!00$$ $$\beta_{9}$$ $$=$$ $$($$$$3412338678579729269398729 \nu^{15} - 23503972919102308071304538 \nu^{14} + 78640463768519071837864112 \nu^{13} + 639713175327196616809510632 \nu^{12} + 7257109192570910665843547330 \nu^{11} - 35921176398745244766382383644 \nu^{10} + 123140326784007072026295781776 \nu^{9} + 981603878157766671660618418936 \nu^{8} + 4630744326327588694200050265753 \nu^{7} - 11715006307895645409371321205698 \nu^{6} + 46354481699860174480729913468512 \nu^{5} + 364761822101351710054709910455632 \nu^{4} + 942331764442715348852192670029888 \nu^{3} + 850374348624210207641634207838080 \nu^{2} + 164319503969898351179625507276800 \nu - 103889220948746126317350906752000$$$$)/$$$$94\!\cdots\!00$$ $$\beta_{10}$$ $$=$$ $$($$$$-1791907420770118236739192 \nu^{15} + 12276743089921280580133529 \nu^{14} - 39985022273591084051503196 \nu^{13} - 346072354706028195294085656 \nu^{12} - 3780790853566572992793977560 \nu^{11} + 18792643926343523344187154802 \nu^{10} - 62579192398151045598224168608 \nu^{9} - 531653053519601170802975090288 \nu^{8} - 2384748557168006514317267187104 \nu^{7} + 6165681703593142903658606767609 \nu^{6} - 23583926179124347694854481579396 \nu^{5} - 197734548487628531179290039085656 \nu^{4} - 478048602216806780235957352319744 \nu^{3} - 426677131187122355771571423727040 \nu^{2} - 81255165686903865978536717318400 \nu - 34792390529748269256595542464000$$$$)/$$$$47\!\cdots\!00$$ $$\beta_{11}$$ $$=$$ $$($$$$29369391430993776549242657 \nu^{15} - 234752825343122954764952696 \nu^{14} + 931403725433463611273107864 \nu^{13} + 4528659966012185459267448504 \nu^{12} + 57152262588460708403280624418 \nu^{11} - 372339390852850080288812888088 \nu^{10} + 1462824424856672409223525337872 \nu^{9} + 6910952861315796051643861922392 \nu^{8} + 31739225467634626177448390357793 \nu^{7} - 135945802433372310116560565795936 \nu^{6} + 545738373979820199663499762745464 \nu^{5} + 2564623033441214030440116202017504 \nu^{4} + 5101702365002084263095418587284032 \nu^{3} + 1652595275103126127268440486789120 \nu^{2} - 488077145624801586094222748684800 \nu - 154424235478291493398759673088000$$$$)/$$$$75\!\cdots\!00$$ $$\beta_{12}$$ $$=$$ $$($$$$33748983840104420496354041 \nu^{15} - 276588239709545424544104452 \nu^{14} + 1139686289961773367211583048 \nu^{13} + 4864429024102505288540376728 \nu^{12} + 65148253732557025727405518770 \nu^{11} - 439191029038663453727895892976 \nu^{10} + 1795939749414398495782778605904 \nu^{9} + 7407789726927897658768044912344 \nu^{8} + 35698063581887795467163947659737 \nu^{7} - 160707111571222275251171900418092 \nu^{6} + 673359068251502610668644337808648 \nu^{5} + 2749003015767474911914702680761728 \nu^{4} + 5581662438726636442298377681906752 \nu^{3} + 1748046373385579006534464146392320 \nu^{2} + 827381353100050330290685450867200 \nu - 13556691495783048346065701888000$$$$)/$$$$75\!\cdots\!00$$ $$\beta_{13}$$ $$=$$ $$($$$$4955856064949048488893960 \nu^{15} - 40730009400841889037483081 \nu^{14} + 168403974109603378470151324 \nu^{13} + 709834574265647494182988664 \nu^{12} + 9557535311357750667568232824 \nu^{11} - 64720079266248378638438018898 \nu^{10} + 265533250379439632170611253152 \nu^{9} + 1081697621767113158489583473072 \nu^{8} + 5233599884399459371581852989872 \nu^{7} - 23719114928058737333969647109161 \nu^{6} + 99602828269490304490450790739524 \nu^{5} + 401805382218616894700708223300664 \nu^{4} + 818028966462865206534708372063744 \nu^{3} + 243994044675541917193691356527040 \nu^{2} + 126466386156125953953863727878400 \nu - 3658088019850964697623236416000$$$$)/$$$$94\!\cdots\!00$$ $$\beta_{14}$$ $$=$$ $$($$$$43136369546463387987857517 \nu^{15} - 365620184134954700262732931 \nu^{14} + 1546473273046397165975051604 \nu^{13} + 5884114166706164314372148544 \nu^{12} + 81220154548791279920124616178 \nu^{11} - 586013418928779212849080250318 \nu^{10} + 2435027751677853805931316147392 \nu^{9} + 8940333135123891365164420498312 \nu^{8} + 42513426596138906143189654014693 \nu^{7} - 220006139215002140207901233137771 \nu^{6} + 909864607443146925496142029564204 \nu^{5} + 3313466430217182716408137894865544 \nu^{4} + 5984389005390320008999784103464512 \nu^{3} - 359154337198876821720458290470080 \nu^{2} + 4735112833339007106336508947200 \nu - 302620871803708696152200273088000$$$$)/$$$$75\!\cdots\!00$$ $$\beta_{15}$$ $$=$$ $$($$$$65196613732168144527114713 \nu^{15} - 538987862975039906091868764 \nu^{14} + 2225981274042425398223395176 \nu^{13} + 9347599508626451088434718936 \nu^{12} + 124777143491149335127304736562 \nu^{11} - 859603350184844994130705189792 \nu^{10} + 3502549277391697915664705524048 \nu^{9} + 14235180667441472243487072048728 \nu^{8} + 67340663352449631842871029974137 \nu^{7} - 318401580604409937167107256737924 \nu^{6} + 1308784224636813158773593035161576 \nu^{5} + 5282853462056585681244785460447936 \nu^{4} + 10193963705550330718278853441434688 \nu^{3} + 1596955145704081586052408181518080 \nu^{2} - 73703741187705017000901847283200 \nu - 230645924071229951444989696512000$$$$)/$$$$75\!\cdots\!00$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$3 \beta_{15} + \beta_{13} - 5 \beta_{12} - 5 \beta_{11} - 3 \beta_{8} - \beta_{5} + 3 \beta_{4} - 8 \beta_{3} + \beta_{2} + 9$$$$)/10$$ $$\nu^{2}$$ $$=$$ $$($$$$6 \beta_{15} + 5 \beta_{14} - 3 \beta_{13} - 5 \beta_{12} - 15 \beta_{11} - \beta_{8} - 10 \beta_{7} + 5 \beta_{6} - 37 \beta_{5} + 41 \beta_{4} - 236 \beta_{3} + 2 \beta_{2} + 3$$$$)/10$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{15} + 40 \beta_{14} - 123 \beta_{13} + 205 \beta_{12} - 205 \beta_{11} + 20 \beta_{9} - 41 \beta_{8} - 60 \beta_{7} + 40 \beta_{6} + 123 \beta_{5} + 121 \beta_{4} - 506 \beta_{3} - 83 \beta_{2} + 60 \beta_{1} - 547$$$$)/10$$ $$\nu^{4}$$ $$=$$ $$($$$$-164 \beta_{15} + 121 \beta_{14} - 45 \beta_{13} + 247 \beta_{12} - 41 \beta_{11} + 24 \beta_{10} + 24 \beta_{9} + 43 \beta_{8} + 57 \beta_{6} + 583 \beta_{5} + 317 \beta_{4} + 2 \beta_{3} - 102 \beta_{2} + 130 \beta_{1} - 2075$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-9647 \beta_{15} + 3000 \beta_{14} - 1749 \beta_{13} + 10445 \beta_{12} + 10145 \beta_{11} + 2100 \beta_{10} + 5547 \beta_{8} + 4600 \beta_{7} - 3000 \beta_{6} + 18049 \beta_{5} - 5547 \beta_{4} + 37592 \beta_{3} - 5849 \beta_{2} + 4600 \beta_{1} - 39641$$$$)/10$$ $$\nu^{6}$$ $$=$$ $$($$$$-36214 \beta_{15} - 16245 \beta_{14} + 14507 \beta_{13} + 9045 \beta_{12} + 86935 \beta_{11} + 11200 \beta_{10} - 11200 \beta_{9} + 19969 \beta_{8} + 46890 \beta_{7} - 46645 \beta_{6} + 107453 \beta_{5} - 111529 \beta_{4} + 524484 \beta_{3} - 32138 \beta_{2} - 14507$$$$)/10$$ $$\nu^{7}$$ $$=$$ $$($$$$120511 \beta_{15} - 316960 \beta_{14} + 288947 \beta_{13} - 569245 \beta_{12} + 606645 \beta_{11} - 156180 \beta_{9} + 83849 \beta_{8} + 309940 \beta_{7} - 316960 \beta_{6} - 401547 \beta_{5} - 799769 \beta_{4} + 2266834 \beta_{3} + 84587 \beta_{2} - 309940 \beta_{1} + 2350683$$$$)/10$$ $$\nu^{8}$$ $$=$$ $$($$$$834588 \beta_{15} - 641969 \beta_{14} + 293269 \beta_{13} - 1169783 \beta_{12} + 91969 \beta_{11} - 167656 \beta_{10} - 167656 \beta_{9} - 192619 \beta_{8} - 195313 \beta_{6} - 2174751 \beta_{5} - 851581 \beta_{4} - 100650 \beta_{3} + 488582 \beta_{2} - 653010 \beta_{1} + 6263171$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$38994703 \beta_{15} - 13557400 \beta_{14} + 4516101 \beta_{13} - 37620805 \beta_{12} - 34469505 \beta_{11} - 10606100 \beta_{10} - 16699603 \beta_{8} - 20304400 \beta_{7} + 13557400 \beta_{6} - 90347401 \beta_{5} + 16699603 \beta_{4} - 158824208 \beta_{3} + 26811201 \beta_{2} - 20304400 \beta_{1} + 166491609$$$$)/10$$ $$\nu^{10}$$ $$=$$ $$($$$$161696566 \beta_{15} + 60895005 \beta_{14} - 63463483 \beta_{13} - 26125405 \beta_{12} - 386856615 \beta_{11} - 58147600 \beta_{10} + 58147600 \beta_{9} - 100801561 \beta_{8} - 220901210 \beta_{7} + 213114205 \beta_{6} - 407568557 \beta_{5} + 419614401 \beta_{4} - 1870914796 \beta_{3} + 149650722 \beta_{2} + 63463483$$$$)/10$$ $$\nu^{11}$$ $$=$$ $$($$$$-625863679 \beta_{15} + 1506954040 \beta_{14} - 1025775883 \beta_{13} + 2171613805 \beta_{12} - 2403147005 \beta_{11} + 700237620 \beta_{9} - 264747561 \beta_{8} - 1322030060 \beta_{7} + 1506954040 \beta_{6} + 1642118683 \beta_{5} + 3620395641 \beta_{4} - 9852427626 \beta_{3} - 135164643 \beta_{2} + 1322030060 \beta_{1} - 10117175187$$$$)/10$$ $$\nu^{12}$$ $$=$$ $$($$$$-3629592340 \beta_{15} + 2796620041 \beta_{14} - 1348312477 \beta_{13} + 5081296087 \beta_{12} - 317632121 \beta_{11} + 780903864 \beta_{10} + 780903864 \beta_{9} + 832972299 \beta_{8} + 775849417 \beta_{6} + 8953929175 \beta_{5} + 3200174941 \beta_{4} + 515340178 \beta_{3} - 2124161894 \beta_{2} + 2939178850 \beta_{1} - 24624971723$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$-165052824847 \beta_{15} + 58282606200 \beta_{14} - 16340771349 \beta_{13} + 155464748445 \beta_{12} + 139451761345 \beta_{11} + 45828547700 \beta_{10} + 65035301147 \beta_{8} + 86049090600 \beta_{7} - 58282606200 \beta_{6} + 396438526449 \beta_{5} - 65035301147 \beta_{4} + 678707480392 \beta_{3} - 116358295049 \beta_{2} + 86049090600 \beta_{1} - 711061238841$$$$)/10$$ $$\nu^{14}$$ $$=$$ $$($$$$-695324153654 \beta_{15} - 250038606245 \beta_{14} + 272692226027 \beta_{13} + 100098904645 \beta_{12} + 1663340533335 \beta_{11} + 258317653600 \beta_{10} - 258317653600 \beta_{9} + 445285547409 \beta_{8} + 968782543690 \beta_{7} - 914002777445 \beta_{6} + 1688572599933 \beta_{5} - 1742586202169 \beta_{4} + 7673960683924 \beta_{3} - 641310551418 \beta_{2} - 272692226027$$$$)/10$$ $$\nu^{15}$$ $$=$$ $$($$$$2771173982431 \beta_{15} - 6581449626960 \beta_{14} + 4187571520387 \beta_{13} - 9035112495645 \beta_{12} + 10110888023845 \beta_{11} - 2992046938580 \beta_{9} + 1037265330729 \beta_{8} + 5605701099940 \beta_{7} - 6581449626960 \beta_{6} - 6960581834187 \beta_{5} - 15599296238649 \beta_{4} + 42279655986114 \beta_{3} + 379132207227 \beta_{2} - 5605701099940 \beta_{1} + 43316921316843$$$$)/10$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/630\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$281$$ $$451$$ $$\chi(n)$$ $$-\beta_{3}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 −0.394902 − 0.394902i 3.60306 + 3.60306i −3.99135 − 3.99135i 5.71348 + 5.71348i −3.48873 − 3.48873i 3.76660 + 3.76660i −1.37832 − 1.37832i 0.170157 + 0.170157i −0.394902 + 0.394902i 3.60306 − 3.60306i −3.99135 + 3.99135i 5.71348 − 5.71348i −3.48873 + 3.48873i 3.76660 − 3.76660i −1.37832 + 1.37832i 0.170157 − 0.170157i
1.00000 + 1.00000i 0 2.00000i −4.80148 1.39490i 0 1.87083 + 1.87083i −2.00000 + 2.00000i 0 −3.40658 6.19639i
127.2 1.00000 + 1.00000i 0 2.00000i −4.26896 + 2.60306i 0 1.87083 + 1.87083i −2.00000 + 2.00000i 0 −6.87203 1.66590i
127.3 1.00000 + 1.00000i 0 2.00000i −0.294013 4.99135i 0 −1.87083 1.87083i −2.00000 + 2.00000i 0 4.69734 5.28536i
127.4 1.00000 + 1.00000i 0 2.00000i 1.66827 + 4.71348i 0 −1.87083 1.87083i −2.00000 + 2.00000i 0 −3.04521 + 6.38175i
127.5 1.00000 + 1.00000i 0 2.00000i 2.20256 4.48873i 0 −1.87083 1.87083i −2.00000 + 2.00000i 0 6.69129 2.28617i
127.6 1.00000 + 1.00000i 0 2.00000i 4.16484 + 2.76660i 0 −1.87083 1.87083i −2.00000 + 2.00000i 0 1.39824 + 6.93144i
127.7 1.00000 + 1.00000i 0 2.00000i 4.39814 2.37832i 0 1.87083 + 1.87083i −2.00000 + 2.00000i 0 6.77646 + 2.01982i
127.8 1.00000 + 1.00000i 0 2.00000i 4.93066 0.829843i 0 1.87083 + 1.87083i −2.00000 + 2.00000i 0 5.76050 + 4.10081i
253.1 1.00000 1.00000i 0 2.00000i −4.80148 + 1.39490i 0 1.87083 1.87083i −2.00000 2.00000i 0 −3.40658 + 6.19639i
253.2 1.00000 1.00000i 0 2.00000i −4.26896 2.60306i 0 1.87083 1.87083i −2.00000 2.00000i 0 −6.87203 + 1.66590i
253.3 1.00000 1.00000i 0 2.00000i −0.294013 + 4.99135i 0 −1.87083 + 1.87083i −2.00000 2.00000i 0 4.69734 + 5.28536i
253.4 1.00000 1.00000i 0 2.00000i 1.66827 4.71348i 0 −1.87083 + 1.87083i −2.00000 2.00000i 0 −3.04521 6.38175i
253.5 1.00000 1.00000i 0 2.00000i 2.20256 + 4.48873i 0 −1.87083 + 1.87083i −2.00000 2.00000i 0 6.69129 + 2.28617i
253.6 1.00000 1.00000i 0 2.00000i 4.16484 2.76660i 0 −1.87083 + 1.87083i −2.00000 2.00000i 0 1.39824 6.93144i
253.7 1.00000 1.00000i 0 2.00000i 4.39814 + 2.37832i 0 1.87083 1.87083i −2.00000 2.00000i 0 6.77646 2.01982i
253.8 1.00000 1.00000i 0 2.00000i 4.93066 + 0.829843i 0 1.87083 1.87083i −2.00000 2.00000i 0 5.76050 4.10081i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 253.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.3.o.f 16
3.b odd 2 1 210.3.l.b 16
5.c odd 4 1 inner 630.3.o.f 16
15.d odd 2 1 1050.3.l.h 16
15.e even 4 1 210.3.l.b 16
15.e even 4 1 1050.3.l.h 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.l.b 16 3.b odd 2 1
210.3.l.b 16 15.e even 4 1
630.3.o.f 16 1.a even 1 1 trivial
630.3.o.f 16 5.c odd 4 1 inner
1050.3.l.h 16 15.d odd 2 1
1050.3.l.h 16 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{8} + \cdots$$ acting on $$S_{3}^{\mathrm{new}}(630, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T + 2 T^{2} )^{8}$$
$3$ 1
$5$ $$1 - 16 T + 108 T^{2} - 240 T^{3} - 1736 T^{4} + 15856 T^{5} - 26268 T^{6} - 353520 T^{7} + 2938350 T^{8} - 8838000 T^{9} - 16417500 T^{10} + 247750000 T^{11} - 678125000 T^{12} - 2343750000 T^{13} + 26367187500 T^{14} - 97656250000 T^{15} + 152587890625 T^{16}$$
$7$ $$( 1 + 49 T^{4} )^{4}$$
$11$ $$( 1 + 4 T + 428 T^{2} + 3020 T^{3} + 112132 T^{4} + 750180 T^{5} + 21255892 T^{6} + 132232716 T^{7} + 2897491254 T^{8} + 16000158636 T^{9} + 311207514772 T^{10} + 1328989630980 T^{11} + 24036490044292 T^{12} + 78331022295020 T^{13} + 1343247345236588 T^{14} + 1518999334332964 T^{15} + 45949729863572161 T^{16} )^{2}$$
$13$ $$1 + 32 T + 512 T^{2} + 12624 T^{3} + 210400 T^{4} + 1218608 T^{5} + 10953344 T^{6} - 12361888 T^{7} - 9595251012 T^{8} - 144909422048 T^{9} - 1442470310272 T^{10} - 30698489798640 T^{11} - 199614992290272 T^{12} + 2586969858854768 T^{13} + 29532641854346496 T^{14} + 799237423444942944 T^{15} + 19007129532539001606 T^{16} +$$$$13\!\cdots\!36$$$$T^{17} +$$$$84\!\cdots\!56$$$$T^{18} +$$$$12\!\cdots\!12$$$$T^{19} -$$$$16\!\cdots\!12$$$$T^{20} -$$$$42\!\cdots\!60$$$$T^{21} -$$$$33\!\cdots\!32$$$$T^{22} -$$$$57\!\cdots\!72$$$$T^{23} -$$$$63\!\cdots\!92$$$$T^{24} -$$$$13\!\cdots\!52$$$$T^{25} +$$$$20\!\cdots\!44$$$$T^{26} +$$$$39\!\cdots\!52$$$$T^{27} +$$$$11\!\cdots\!00$$$$T^{28} +$$$$11\!\cdots\!16$$$$T^{29} +$$$$79\!\cdots\!52$$$$T^{30} +$$$$83\!\cdots\!68$$$$T^{31} +$$$$44\!\cdots\!81$$$$T^{32}$$
$17$ $$1 + 56 T + 1568 T^{2} + 29832 T^{3} + 267808 T^{4} - 3397960 T^{5} - 165234592 T^{6} - 3465012472 T^{7} - 48572233284 T^{8} - 491667950696 T^{9} - 4716183819232 T^{10} - 39445096533336 T^{11} + 280624802792928 T^{12} + 26208629063564312 T^{13} + 761979769352889696 T^{14} + 15987258252632453928 T^{15} +$$$$28\!\cdots\!74$$$$T^{16} +$$$$46\!\cdots\!92$$$$T^{17} +$$$$63\!\cdots\!16$$$$T^{18} +$$$$63\!\cdots\!28$$$$T^{19} +$$$$19\!\cdots\!48$$$$T^{20} -$$$$79\!\cdots\!64$$$$T^{21} -$$$$27\!\cdots\!52$$$$T^{22} -$$$$82\!\cdots\!84$$$$T^{23} -$$$$23\!\cdots\!04$$$$T^{24} -$$$$48\!\cdots\!48$$$$T^{25} -$$$$67\!\cdots\!92$$$$T^{26} -$$$$39\!\cdots\!40$$$$T^{27} +$$$$90\!\cdots\!68$$$$T^{28} +$$$$29\!\cdots\!08$$$$T^{29} +$$$$44\!\cdots\!88$$$$T^{30} +$$$$45\!\cdots\!44$$$$T^{31} +$$$$23\!\cdots\!61$$$$T^{32}$$
$19$ $$1 - 2056 T^{2} + 2230944 T^{4} - 1603740248 T^{6} + 819184531004 T^{8} - 297045269538504 T^{10} + 69987022118810464 T^{12} - 7175141044809308632 T^{14} -$$$$43\!\cdots\!46$$$$T^{16} -$$$$93\!\cdots\!72$$$$T^{18} +$$$$11\!\cdots\!24$$$$T^{20} -$$$$65\!\cdots\!44$$$$T^{22} +$$$$23\!\cdots\!24$$$$T^{24} -$$$$60\!\cdots\!48$$$$T^{26} +$$$$10\!\cdots\!24$$$$T^{28} -$$$$13\!\cdots\!96$$$$T^{30} +$$$$83\!\cdots\!61$$$$T^{32}$$
$23$ $$1 + 24 T + 288 T^{2} + 17192 T^{3} + 221472 T^{4} + 2391768 T^{5} + 141400928 T^{6} + 3172207336 T^{7} + 137674628156 T^{8} + 274022502200 T^{9} - 6993233578720 T^{10} + 103772124341000 T^{11} - 60508914798085408 T^{12} - 933592691352233992 T^{13} - 15427273581773601184 T^{14} -$$$$57\!\cdots\!36$$$$T^{15} -$$$$64\!\cdots\!26$$$$T^{16} -$$$$30\!\cdots\!44$$$$T^{17} -$$$$43\!\cdots\!44$$$$T^{18} -$$$$13\!\cdots\!88$$$$T^{19} -$$$$47\!\cdots\!48$$$$T^{20} +$$$$42\!\cdots\!00$$$$T^{21} -$$$$15\!\cdots\!20$$$$T^{22} +$$$$31\!\cdots\!00$$$$T^{23} +$$$$84\!\cdots\!16$$$$T^{24} +$$$$10\!\cdots\!84$$$$T^{25} +$$$$24\!\cdots\!28$$$$T^{26} +$$$$21\!\cdots\!72$$$$T^{27} +$$$$10\!\cdots\!52$$$$T^{28} +$$$$43\!\cdots\!88$$$$T^{29} +$$$$38\!\cdots\!28$$$$T^{30} +$$$$17\!\cdots\!76$$$$T^{31} +$$$$37\!\cdots\!21$$$$T^{32}$$
$29$ $$1 - 4032 T^{2} + 8105976 T^{4} - 11130637632 T^{6} + 11901856158364 T^{8} - 11208853722734016 T^{10} + 10516466734892486856 T^{12} -$$$$99\!\cdots\!40$$$$T^{14} +$$$$87\!\cdots\!46$$$$T^{16} -$$$$70\!\cdots\!40$$$$T^{18} +$$$$52\!\cdots\!16$$$$T^{20} -$$$$39\!\cdots\!56$$$$T^{22} +$$$$29\!\cdots\!44$$$$T^{24} -$$$$19\!\cdots\!32$$$$T^{26} +$$$$10\!\cdots\!56$$$$T^{28} -$$$$35\!\cdots\!52$$$$T^{30} +$$$$62\!\cdots\!41$$$$T^{32}$$
$31$ $$( 1 + 56 T + 4484 T^{2} + 186776 T^{3} + 8992264 T^{4} + 304202984 T^{5} + 11800345036 T^{6} + 348556649544 T^{7} + 12266333612430 T^{8} + 334962940211784 T^{9} + 10897866447991756 T^{10} + 269981268071184104 T^{11} + 7669421371903356424 T^{12} +$$$$15\!\cdots\!76$$$$T^{13} +$$$$35\!\cdots\!24$$$$T^{14} +$$$$42\!\cdots\!76$$$$T^{15} +$$$$72\!\cdots\!81$$$$T^{16} )^{2}$$
$37$ $$1 + 152 T + 11552 T^{2} + 598344 T^{3} + 25962168 T^{4} + 1167158904 T^{5} + 56500959840 T^{6} + 2639490056104 T^{7} + 111109511071516 T^{8} + 4242418231122872 T^{9} + 154876079427922976 T^{10} + 5762441784032775208 T^{11} +$$$$22\!\cdots\!72$$$$T^{12} +$$$$89\!\cdots\!08$$$$T^{13} +$$$$32\!\cdots\!12$$$$T^{14} +$$$$10\!\cdots\!40$$$$T^{15} +$$$$37\!\cdots\!46$$$$T^{16} +$$$$14\!\cdots\!60$$$$T^{17} +$$$$60\!\cdots\!32$$$$T^{18} +$$$$22\!\cdots\!72$$$$T^{19} +$$$$79\!\cdots\!12$$$$T^{20} +$$$$27\!\cdots\!92$$$$T^{21} +$$$$10\!\cdots\!56$$$$T^{22} +$$$$38\!\cdots\!08$$$$T^{23} +$$$$13\!\cdots\!56$$$$T^{24} +$$$$44\!\cdots\!16$$$$T^{25} +$$$$13\!\cdots\!40$$$$T^{26} +$$$$36\!\cdots\!76$$$$T^{27} +$$$$11\!\cdots\!48$$$$T^{28} +$$$$35\!\cdots\!96$$$$T^{29} +$$$$93\!\cdots\!92$$$$T^{30} +$$$$16\!\cdots\!48$$$$T^{31} +$$$$15\!\cdots\!81$$$$T^{32}$$
$41$ $$( 1 + 8252 T^{2} + 19936 T^{3} + 34068728 T^{4} + 127251488 T^{5} + 92291784532 T^{6} + 367862933760 T^{7} + 180432835211182 T^{8} + 618377591650560 T^{9} + 260794525350928852 T^{10} + 604457832822360608 T^{11} +$$$$27\!\cdots\!88$$$$T^{12} +$$$$26\!\cdots\!36$$$$T^{13} +$$$$18\!\cdots\!12$$$$T^{14} +$$$$63\!\cdots\!41$$$$T^{16} )^{2}$$
$43$ $$1 - 158592 T^{3} + 2725896 T^{4} - 61004160 T^{5} + 12575711232 T^{6} - 618250768128 T^{7} + 18017190018460 T^{8} - 968512496926464 T^{9} + 65630298643144704 T^{10} - 3092793950726971008 T^{11} +$$$$10\!\cdots\!80$$$$T^{12} -$$$$37\!\cdots\!52$$$$T^{13} +$$$$33\!\cdots\!64$$$$T^{14} -$$$$12\!\cdots\!48$$$$T^{15} +$$$$20\!\cdots\!26$$$$T^{16} -$$$$22\!\cdots\!52$$$$T^{17} +$$$$11\!\cdots\!64$$$$T^{18} -$$$$23\!\cdots\!48$$$$T^{19} +$$$$11\!\cdots\!80$$$$T^{20} -$$$$66\!\cdots\!92$$$$T^{21} +$$$$26\!\cdots\!04$$$$T^{22} -$$$$71\!\cdots\!36$$$$T^{23} +$$$$24\!\cdots\!60$$$$T^{24} -$$$$15\!\cdots\!72$$$$T^{25} +$$$$58\!\cdots\!32$$$$T^{26} -$$$$52\!\cdots\!40$$$$T^{27} +$$$$43\!\cdots\!96$$$$T^{28} -$$$$46\!\cdots\!08$$$$T^{29} +$$$$18\!\cdots\!01$$$$T^{32}$$
$47$ $$1 + 80 T + 3200 T^{2} + 185424 T^{3} + 10961416 T^{4} + 416599088 T^{5} + 15442425728 T^{6} + 918151881776 T^{7} + 40326873674268 T^{8} + 452413353633424 T^{9} - 5098985333045632 T^{10} - 1563817770500267376 T^{11} -$$$$31\!\cdots\!76$$$$T^{12} -$$$$17\!\cdots\!60$$$$T^{13} -$$$$66\!\cdots\!16$$$$T^{14} -$$$$36\!\cdots\!52$$$$T^{15} -$$$$19\!\cdots\!38$$$$T^{16} -$$$$79\!\cdots\!68$$$$T^{17} -$$$$32\!\cdots\!96$$$$T^{18} -$$$$19\!\cdots\!40$$$$T^{19} -$$$$75\!\cdots\!36$$$$T^{20} -$$$$82\!\cdots\!24$$$$T^{21} -$$$$59\!\cdots\!12$$$$T^{22} +$$$$11\!\cdots\!56$$$$T^{23} +$$$$22\!\cdots\!28$$$$T^{24} +$$$$11\!\cdots\!64$$$$T^{25} +$$$$42\!\cdots\!28$$$$T^{26} +$$$$25\!\cdots\!92$$$$T^{27} +$$$$14\!\cdots\!96$$$$T^{28} +$$$$55\!\cdots\!96$$$$T^{29} +$$$$21\!\cdots\!00$$$$T^{30} +$$$$11\!\cdots\!20$$$$T^{31} +$$$$32\!\cdots\!41$$$$T^{32}$$
$53$ $$1 + 48 T + 1152 T^{2} + 172480 T^{3} + 14460640 T^{4} + 33746752 T^{5} - 164137984 T^{6} - 7272556080 T^{7} - 144052676687940 T^{8} - 6754246786116880 T^{9} - 156566570729557248 T^{10} - 21067810577807322688 T^{11} -$$$$28\!\cdots\!60$$$$T^{12} +$$$$50\!\cdots\!00$$$$T^{13} +$$$$11\!\cdots\!80$$$$T^{14} +$$$$15\!\cdots\!88$$$$T^{15} +$$$$20\!\cdots\!78$$$$T^{16} +$$$$43\!\cdots\!92$$$$T^{17} +$$$$90\!\cdots\!80$$$$T^{18} +$$$$11\!\cdots\!00$$$$T^{19} -$$$$17\!\cdots\!60$$$$T^{20} -$$$$36\!\cdots\!12$$$$T^{21} -$$$$76\!\cdots\!68$$$$T^{22} -$$$$93\!\cdots\!20$$$$T^{23} -$$$$55\!\cdots\!40$$$$T^{24} -$$$$79\!\cdots\!20$$$$T^{25} -$$$$50\!\cdots\!84$$$$T^{26} +$$$$28\!\cdots\!68$$$$T^{27} +$$$$34\!\cdots\!40$$$$T^{28} +$$$$11\!\cdots\!20$$$$T^{29} +$$$$21\!\cdots\!72$$$$T^{30} +$$$$25\!\cdots\!52$$$$T^{31} +$$$$15\!\cdots\!41$$$$T^{32}$$
$59$ $$1 - 14320 T^{2} + 157767608 T^{4} - 1244890527696 T^{6} + 8207935216533276 T^{8} - 45520264315988706288 T^{10} +$$$$21\!\cdots\!36$$$$T^{12} -$$$$91\!\cdots\!96$$$$T^{14} +$$$$33\!\cdots\!38$$$$T^{16} -$$$$11\!\cdots\!56$$$$T^{18} +$$$$32\!\cdots\!56$$$$T^{20} -$$$$80\!\cdots\!28$$$$T^{22} +$$$$17\!\cdots\!16$$$$T^{24} -$$$$32\!\cdots\!96$$$$T^{26} +$$$$49\!\cdots\!88$$$$T^{28} -$$$$54\!\cdots\!20$$$$T^{30} +$$$$46\!\cdots\!81$$$$T^{32}$$
$61$ $$( 1 - 48 T + 22640 T^{2} - 673936 T^{3} + 215932124 T^{4} - 3175993520 T^{5} + 1217381291920 T^{6} - 6110146782096 T^{7} + 5029652273881990 T^{8} - 22735856176179216 T^{9} + 16855667804298904720 T^{10} -$$$$16\!\cdots\!20$$$$T^{11} +$$$$41\!\cdots\!44$$$$T^{12} -$$$$48\!\cdots\!36$$$$T^{13} +$$$$60\!\cdots\!40$$$$T^{14} -$$$$47\!\cdots\!68$$$$T^{15} +$$$$36\!\cdots\!61$$$$T^{16} )^{2}$$
$67$ $$1 + 80 T + 3200 T^{2} - 609968 T^{3} - 25803832 T^{4} + 2674348464 T^{5} + 482550620032 T^{6} + 12521071726512 T^{7} - 1529742891382500 T^{8} - 116256625005179504 T^{9} + 3984919157383710336 T^{10} +$$$$89\!\cdots\!92$$$$T^{11} +$$$$25\!\cdots\!88$$$$T^{12} -$$$$21\!\cdots\!76$$$$T^{13} -$$$$18\!\cdots\!68$$$$T^{14} +$$$$71\!\cdots\!08$$$$T^{15} +$$$$10\!\cdots\!46$$$$T^{16} +$$$$31\!\cdots\!12$$$$T^{17} -$$$$38\!\cdots\!28$$$$T^{18} -$$$$19\!\cdots\!44$$$$T^{19} +$$$$10\!\cdots\!08$$$$T^{20} +$$$$16\!\cdots\!08$$$$T^{21} +$$$$32\!\cdots\!96$$$$T^{22} -$$$$42\!\cdots\!16$$$$T^{23} -$$$$25\!\cdots\!00$$$$T^{24} +$$$$92\!\cdots\!08$$$$T^{25} +$$$$16\!\cdots\!32$$$$T^{26} +$$$$39\!\cdots\!96$$$$T^{27} -$$$$17\!\cdots\!72$$$$T^{28} -$$$$18\!\cdots\!92$$$$T^{29} +$$$$43\!\cdots\!00$$$$T^{30} +$$$$48\!\cdots\!20$$$$T^{31} +$$$$27\!\cdots\!61$$$$T^{32}$$
$71$ $$( 1 + 268 T + 44204 T^{2} + 5071044 T^{3} + 471192708 T^{4} + 37316092140 T^{5} + 2755016710036 T^{6} + 195026045020868 T^{7} + 13901563235868662 T^{8} + 983126292950195588 T^{9} + 70009605785104330516 T^{10} +$$$$47\!\cdots\!40$$$$T^{11} +$$$$30\!\cdots\!88$$$$T^{12} +$$$$16\!\cdots\!44$$$$T^{13} +$$$$72\!\cdots\!64$$$$T^{14} +$$$$22\!\cdots\!08$$$$T^{15} +$$$$41\!\cdots\!21$$$$T^{16} )^{2}$$
$73$ $$1 - 638960 T^{3} - 86919200 T^{4} - 369113360 T^{5} + 204134940800 T^{6} + 33110468899200 T^{7} + 5206311620706876 T^{8} + 26720496193596800 T^{9} - 3344897125184227200 T^{10} -$$$$13\!\cdots\!20$$$$T^{11} -$$$$22\!\cdots\!00$$$$T^{12} -$$$$53\!\cdots\!20$$$$T^{13} +$$$$75\!\cdots\!00$$$$T^{14} +$$$$36\!\cdots\!00$$$$T^{15} +$$$$78\!\cdots\!66$$$$T^{16} +$$$$19\!\cdots\!00$$$$T^{17} +$$$$21\!\cdots\!00$$$$T^{18} -$$$$80\!\cdots\!80$$$$T^{19} -$$$$17\!\cdots\!00$$$$T^{20} -$$$$59\!\cdots\!80$$$$T^{21} -$$$$76\!\cdots\!00$$$$T^{22} +$$$$32\!\cdots\!00$$$$T^{23} +$$$$33\!\cdots\!36$$$$T^{24} +$$$$11\!\cdots\!00$$$$T^{25} +$$$$37\!\cdots\!00$$$$T^{26} -$$$$36\!\cdots\!40$$$$T^{27} -$$$$45\!\cdots\!00$$$$T^{28} -$$$$17\!\cdots\!40$$$$T^{29} +$$$$42\!\cdots\!21$$$$T^{32}$$
$79$ $$1 - 52208 T^{2} + 1391680568 T^{4} - 25119513250000 T^{6} + 343967562165562012 T^{8} -$$$$37\!\cdots\!72$$$$T^{10} +$$$$34\!\cdots\!24$$$$T^{12} -$$$$27\!\cdots\!40$$$$T^{14} +$$$$18\!\cdots\!30$$$$T^{16} -$$$$10\!\cdots\!40$$$$T^{18} +$$$$52\!\cdots\!64$$$$T^{20} -$$$$22\!\cdots\!52$$$$T^{22} +$$$$79\!\cdots\!52$$$$T^{24} -$$$$22\!\cdots\!00$$$$T^{26} +$$$$48\!\cdots\!08$$$$T^{28} -$$$$71\!\cdots\!88$$$$T^{30} +$$$$52\!\cdots\!41$$$$T^{32}$$
$83$ $$1 - 256 T + 32768 T^{2} - 2489472 T^{3} + 78882184 T^{4} - 642592896 T^{5} + 678427795456 T^{6} - 150782876556800 T^{7} + 17212084280490396 T^{8} - 1275723698299503616 T^{9} + 84640034952293916672 T^{10} -$$$$62\!\cdots\!76$$$$T^{11} +$$$$45\!\cdots\!48$$$$T^{12} -$$$$40\!\cdots\!04$$$$T^{13} +$$$$47\!\cdots\!44$$$$T^{14} -$$$$59\!\cdots\!24$$$$T^{15} +$$$$59\!\cdots\!02$$$$T^{16} -$$$$41\!\cdots\!36$$$$T^{17} +$$$$22\!\cdots\!24$$$$T^{18} -$$$$13\!\cdots\!76$$$$T^{19} +$$$$10\!\cdots\!68$$$$T^{20} -$$$$96\!\cdots\!24$$$$T^{21} +$$$$90\!\cdots\!92$$$$T^{22} -$$$$93\!\cdots\!64$$$$T^{23} +$$$$87\!\cdots\!76$$$$T^{24} -$$$$52\!\cdots\!00$$$$T^{25} +$$$$16\!\cdots\!56$$$$T^{26} -$$$$10\!\cdots\!44$$$$T^{27} +$$$$90\!\cdots\!64$$$$T^{28} -$$$$19\!\cdots\!68$$$$T^{29} +$$$$17\!\cdots\!88$$$$T^{30} -$$$$95\!\cdots\!44$$$$T^{31} +$$$$25\!\cdots\!61$$$$T^{32}$$
$89$ $$1 - 60568 T^{2} + 1632136448 T^{4} - 25689569431240 T^{6} + 260052896713761532 T^{8} -$$$$17\!\cdots\!32$$$$T^{10} +$$$$70\!\cdots\!24$$$$T^{12} -$$$$68\!\cdots\!20$$$$T^{14} -$$$$92\!\cdots\!90$$$$T^{16} -$$$$43\!\cdots\!20$$$$T^{18} +$$$$27\!\cdots\!44$$$$T^{20} -$$$$42\!\cdots\!72$$$$T^{22} +$$$$40\!\cdots\!52$$$$T^{24} -$$$$24\!\cdots\!40$$$$T^{26} +$$$$99\!\cdots\!68$$$$T^{28} -$$$$23\!\cdots\!08$$$$T^{30} +$$$$24\!\cdots\!21$$$$T^{32}$$
$97$ $$1 - 688 T + 236672 T^{2} - 56991072 T^{3} + 10854023776 T^{4} - 1693042723360 T^{5} + 219961022412800 T^{6} - 24020408470599760 T^{7} + 2158083026478171708 T^{8} -$$$$14\!\cdots\!44$$$$T^{9} +$$$$52\!\cdots\!16$$$$T^{10} +$$$$44\!\cdots\!04$$$$T^{11} -$$$$12\!\cdots\!36$$$$T^{12} +$$$$17\!\cdots\!52$$$$T^{13} -$$$$20\!\cdots\!48$$$$T^{14} +$$$$21\!\cdots\!08$$$$T^{15} -$$$$20\!\cdots\!18$$$$T^{16} +$$$$19\!\cdots\!72$$$$T^{17} -$$$$18\!\cdots\!88$$$$T^{18} +$$$$14\!\cdots\!08$$$$T^{19} -$$$$97\!\cdots\!96$$$$T^{20} +$$$$32\!\cdots\!96$$$$T^{21} +$$$$36\!\cdots\!56$$$$T^{22} -$$$$96\!\cdots\!36$$$$T^{23} +$$$$13\!\cdots\!68$$$$T^{24} -$$$$13\!\cdots\!40$$$$T^{25} +$$$$11\!\cdots\!00$$$$T^{26} -$$$$86\!\cdots\!40$$$$T^{27} +$$$$52\!\cdots\!56$$$$T^{28} -$$$$25\!\cdots\!88$$$$T^{29} +$$$$10\!\cdots\!92$$$$T^{30} -$$$$27\!\cdots\!12$$$$T^{31} +$$$$37\!\cdots\!41$$$$T^{32}$$