# Properties

 Label 1050.3.l.h Level 1050 Weight 3 Character orbit 1050.l Analytic conductor 28.610 Analytic rank 0 Dimension 16 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1050.l (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$28.6104277578$$ 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: $$2^{10}\cdot 5^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{2} ) q^{2} + \beta_{4} q^{3} + 2 \beta_{2} q^{4} + ( \beta_{3} + \beta_{4} ) q^{6} + \beta_{7} q^{7} + ( -2 + 2 \beta_{2} ) q^{8} -3 \beta_{2} q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{2} ) q^{2} + \beta_{4} q^{3} + 2 \beta_{2} q^{4} + ( \beta_{3} + \beta_{4} ) q^{6} + \beta_{7} q^{7} + ( -2 + 2 \beta_{2} ) q^{8} -3 \beta_{2} q^{9} + ( 1 - \beta_{3} + \beta_{7} + 2 \beta_{8} - \beta_{12} ) q^{11} + 2 \beta_{3} q^{12} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{10} + 2 \beta_{13} ) q^{13} + ( \beta_{7} - \beta_{8} ) q^{14} -4 q^{16} + ( -4 - 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{7} + \beta_{9} + \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{17} + ( 3 - 3 \beta_{2} ) q^{18} + ( \beta_{2} - \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{13} + \beta_{15} ) q^{19} + \beta_{1} q^{21} + ( 1 + \beta_{2} - \beta_{3} + 2 \beta_{7} - \beta_{9} - \beta_{12} ) q^{22} + ( -2 + 4 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{9} + \beta_{10} + \beta_{12} + 4 \beta_{13} ) q^{23} + ( 2 \beta_{3} - 2 \beta_{4} ) q^{24} + ( 4 + 4 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{10} + \beta_{13} + \beta_{15} ) q^{26} -3 \beta_{3} q^{27} -2 \beta_{8} q^{28} + ( 4 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 2 \beta_{6} + 4 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} + \beta_{10} - 2 \beta_{11} - \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{29} + ( -7 + 3 \beta_{1} - 3 \beta_{3} - 4 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{10} + \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{31} + ( -4 - 4 \beta_{2} ) q^{32} + ( -1 + \beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{11} + \beta_{13} ) q^{33} + ( -8 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - \beta_{6} + 2 \beta_{7} + 2 \beta_{9} - \beta_{10} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{34} + 6 q^{36} + ( 8 + 2 \beta_{1} + 8 \beta_{2} - 3 \beta_{3} - 2 \beta_{5} + 4 \beta_{7} + \beta_{9} - 2 \beta_{11} + \beta_{12} - 2 \beta_{13} + 2 \beta_{14} ) q^{37} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{7} - \beta_{9} + 2 \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{38} + ( -3 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - \beta_{6} - 4 \beta_{7} + 6 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{14} - \beta_{15} ) q^{39} + ( 1 + 5 \beta_{1} + 2 \beta_{3} + 5 \beta_{4} - \beta_{5} + \beta_{7} + 4 \beta_{8} - 2 \beta_{10} - 3 \beta_{12} + 2 \beta_{13} + 2 \beta_{15} ) q^{41} + ( \beta_{1} - \beta_{13} ) q^{42} + ( 4 \beta_{1} - 6 \beta_{4} - 2 \beta_{6} - 8 \beta_{8} - 2 \beta_{10} + 4 \beta_{13} ) q^{43} + ( 2 \beta_{2} + 2 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} ) q^{44} + ( -4 + 8 \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{7} + \beta_{10} + 2 \beta_{12} - \beta_{13} - \beta_{15} ) q^{46} + ( -4 + 2 \beta_{1} - 4 \beta_{2} + 6 \beta_{3} + 2 \beta_{5} + 2 \beta_{7} + 2 \beta_{11} - 4 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{47} -4 \beta_{4} q^{48} + 7 \beta_{2} q^{49} + ( -6 + \beta_{1} - 2 \beta_{3} - 4 \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} + 2 \beta_{10} + 2 \beta_{12} - 2 \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{51} + ( 4 + 4 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - 2 \beta_{13} + 2 \beta_{15} ) q^{52} + ( -4 + 6 \beta_{1} + 4 \beta_{2} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 4 \beta_{8} - \beta_{10} + 2 \beta_{11} + 6 \beta_{13} ) q^{53} + ( -3 \beta_{3} + 3 \beta_{4} ) q^{54} + ( -2 \beta_{7} - 2 \beta_{8} ) q^{56} + ( -6 \beta_{7} - 3 \beta_{9} - 3 \beta_{12} + 3 \beta_{13} + 3 \beta_{15} ) q^{57} + ( -4 - 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} - 8 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} ) q^{58} + ( -2 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} - 2 \beta_{6} - 4 \beta_{7} - 6 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{14} - 2 \beta_{15} ) q^{59} + ( 6 + 4 \beta_{1} + 5 \beta_{3} + 5 \beta_{4} + 2 \beta_{6} - \beta_{10} + \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{61} + ( -7 + 3 \beta_{1} - 7 \beta_{2} - 7 \beta_{3} + \beta_{5} + 3 \beta_{7} + \beta_{9} + \beta_{11} + \beta_{12} - 5 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{62} + 3 \beta_{8} q^{63} -8 \beta_{2} q^{64} + ( -2 + 2 \beta_{1} + 2 \beta_{5} + \beta_{6} - \beta_{14} ) q^{66} + ( 6 + 6 \beta_{1} + 6 \beta_{2} + 6 \beta_{3} + 2 \beta_{5} - 6 \beta_{7} + 2 \beta_{11} - 4 \beta_{13} + 2 \beta_{15} ) q^{67} + ( 8 - 8 \beta_{2} - 2 \beta_{3} + 6 \beta_{4} - 2 \beta_{6} + 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{12} ) q^{68} + ( \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 12 \beta_{7} + 12 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{13} + \beta_{15} ) q^{69} + ( 33 + 8 \beta_{1} - 8 \beta_{3} - 9 \beta_{4} + 4 \beta_{6} - 5 \beta_{7} - 6 \beta_{8} + \beta_{10} + \beta_{12} - \beta_{13} - 4 \beta_{14} - \beta_{15} ) q^{71} + ( 6 + 6 \beta_{2} ) q^{72} + ( 2 + 6 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 23 \beta_{4} + 4 \beta_{6} + 4 \beta_{9} + \beta_{10} - 4 \beta_{12} + 6 \beta_{13} ) q^{73} + ( 16 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} - 4 \beta_{11} - 4 \beta_{13} + 2 \beta_{14} ) q^{74} + ( -2 + 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{7} + 2 \beta_{10} + 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{15} ) q^{76} + ( 11 - \beta_{1} + 11 \beta_{2} + 4 \beta_{3} + \beta_{5} + \beta_{7} + \beta_{11} + \beta_{13} + \beta_{14} ) q^{77} + ( 3 + \beta_{1} - 3 \beta_{2} - \beta_{3} + 6 \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} + 12 \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{78} + ( 38 \beta_{2} + 12 \beta_{3} - 12 \beta_{4} - 4 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - 4 \beta_{13} - 2 \beta_{15} ) q^{79} -9 q^{81} + ( 1 + 5 \beta_{1} + \beta_{2} + 7 \beta_{3} - \beta_{5} + \beta_{7} - 3 \beta_{9} - \beta_{11} - 3 \beta_{12} - \beta_{13} + 4 \beta_{15} ) q^{82} + ( 18 + 2 \beta_{1} - 18 \beta_{2} - 2 \beta_{3} - 10 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + 8 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} + 2 \beta_{13} ) q^{83} -2 \beta_{13} q^{84} + ( 8 \beta_{1} - 6 \beta_{3} - 6 \beta_{4} - 2 \beta_{6} - 8 \beta_{7} - 8 \beta_{8} - 2 \beta_{10} + 2 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{86} + ( 7 + 7 \beta_{1} + 7 \beta_{2} + 7 \beta_{3} - 3 \beta_{5} - \beta_{7} + \beta_{9} - 3 \beta_{11} + \beta_{12} - 3 \beta_{13} + 2 \beta_{14} + 4 \beta_{15} ) q^{87} + ( -2 + 2 \beta_{2} + 2 \beta_{3} - 8 \beta_{8} - 2 \beta_{9} + 2 \beta_{12} ) q^{88} + ( -59 \beta_{2} - 17 \beta_{3} + 17 \beta_{4} + \beta_{6} + 8 \beta_{7} - 8 \beta_{8} + 3 \beta_{9} - 3 \beta_{11} - 11 \beta_{13} + \beta_{14} ) q^{89} + ( 1 - \beta_{1} + 9 \beta_{3} + 10 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{8} + \beta_{10} - \beta_{12} - \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{91} + ( -4 + 8 \beta_{1} - 4 \beta_{2} + 4 \beta_{7} + 2 \beta_{9} + 2 \beta_{12} - 10 \beta_{13} - 2 \beta_{15} ) q^{92} + ( -11 + 11 \beta_{2} - 2 \beta_{3} - 7 \beta_{4} + 4 \beta_{6} + 12 \beta_{8} + 2 \beta_{9} - \beta_{10} - 2 \beta_{12} ) q^{93} + ( -8 \beta_{2} + 6 \beta_{3} - 6 \beta_{4} - 2 \beta_{6} + 4 \beta_{7} - 2 \beta_{10} + 4 \beta_{11} - 6 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{94} + ( -4 \beta_{3} - 4 \beta_{4} ) q^{96} + ( -42 - 6 \beta_{1} - 42 \beta_{2} + 11 \beta_{3} - 28 \beta_{7} - 2 \beta_{9} - 2 \beta_{12} + 3 \beta_{13} - 4 \beta_{14} - 3 \beta_{15} ) q^{97} + ( -7 + 7 \beta_{2} ) q^{98} + ( -3 \beta_{2} - 3 \beta_{7} + 6 \beta_{8} + 3 \beta_{9} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 16q^{2} - 32q^{8} + O(q^{10})$$ $$16q + 16q^{2} - 32q^{8} + 8q^{11} + 32q^{13} - 64q^{16} - 56q^{17} + 48q^{18} + 8q^{22} - 24q^{23} + 64q^{26} - 112q^{31} - 64q^{32} - 24q^{33} + 96q^{36} + 152q^{37} - 48q^{46} - 80q^{47} - 72q^{51} + 64q^{52} - 48q^{53} - 24q^{57} - 96q^{58} + 96q^{61} - 112q^{62} - 48q^{66} + 80q^{67} + 112q^{68} + 536q^{71} + 96q^{72} + 168q^{77} + 48q^{78} - 144q^{81} + 256q^{83} + 144q^{87} - 16q^{88} - 48q^{92} - 192q^{93} - 688q^{97} - 112q^{98} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-124150896048631 \nu^{15} + 759860784649304 \nu^{14} - 962633185803656 \nu^{13} - 37961089316763336 \nu^{12} - 220441925294230158 \nu^{11} + 1204837597044011592 \nu^{10} - 1461099351530560688 \nu^{9} - 59153440044822509928 \nu^{8} - 100458761515785846391 \nu^{7} + 446250282498400102704 \nu^{6} - 588890574148529033256 \nu^{5} - 22232452750211023274336 \nu^{4} - 9966351473395582185920 \nu^{3} - 2157494988382877363200 \nu^{2} + 1309863686202312128000 \nu - 125597356615831969536000$$$$)/$$$$27\!\cdots\!00$$ $$\beta_{2}$$ $$=$$ $$($$$$-50521933035234406935387 \nu^{15} + 355849584339626868196590 \nu^{14} - 1220781585234851432658840 \nu^{13} - 9304805374903308743399240 \nu^{12} - 105717930910748956540229574 \nu^{11} + 546511703953657264363821140 \nu^{10} - 1912893592429770663178189120 \nu^{9} - 14274226782583577122771384520 \nu^{8} - 65924256015119669450932994891 \nu^{7} + 180976850729426810190271470950 \nu^{6} - 719449018130531781232088723240 \nu^{5} - 5305696558618628641218886482240 \nu^{4} - 12993474700473951273694265843648 \nu^{3} - 11620918919670154992391536887680 \nu^{2} - 2225350312372339004103034892800 \nu + 232431648484874621115141632000$$$$)/$$$$11\!\cdots\!00$$ $$\beta_{3}$$ $$=$$ $$($$$$-7312607874029475001394987 \nu^{15} + 50322989299182724918321074 \nu^{14} - 168283183559866103821772176 \nu^{13} - 1371720702266455093133696536 \nu^{12} - 15547180873839092314908209030 \nu^{11} + 76809414399688904270132589212 \nu^{10} - 263505718309346300935072397648 \nu^{9} - 2102021021189220528568128879528 \nu^{8} - 9911131083816128297130191166779 \nu^{7} + 25009453348432333318581627650154 \nu^{6} - 99192306063654602485356687469376 \nu^{5} - 780014500024526499493845235763536 \nu^{4} - 2014834486401151120674936909338304 \nu^{3} - 1818182355719495270833534626247040 \nu^{2} - 351325748136599941624346108518400 \nu + 222122905273351925296690427776000$$$$)/$$$$15\!\cdots\!00$$ $$\beta_{4}$$ $$=$$ $$($$$$3829248573711961810228154 \nu^{15} - 26239924034908349561047733 \nu^{14} + 85716107970375280109650292 \nu^{13} + 738472474668082305176147912 \nu^{12} + 8080492117822719953566872060 \nu^{11} - 40115168057085959901727168954 \nu^{10} + 134228933858105791931167618416 \nu^{9} + 1135855513549067340147092753376 \nu^{8} + 5097784662125274247991681038018 \nu^{7} - 13137999125753012940783686579493 \nu^{6} + 50574570308491286816955297317692 \nu^{5} + 422996013884134839010228418101912 \nu^{4} + 1022192686228881977575596028483968 \nu^{3} + 912283111994794320793370813810880 \nu^{2} + 173723596390269875933568240172800 \nu + 74392129685396065048852419008000$$$$)/$$$$75\!\cdots\!00$$ $$\beta_{5}$$ $$=$$ $$($$$$8434758074635845174981811 \nu^{15} - 72884793664688762327427496 \nu^{14} + 329286172060042198589089224 \nu^{13} + 970844817532966381882679464 \nu^{12} + 16406783357784092286256768006 \nu^{11} - 116078749946814566963122790728 \nu^{10} + 524082693362606559108631080752 \nu^{9} + 1488190510116819840272411790472 \nu^{8} + 9281398469286679055313488293555 \nu^{7} - 42746338127717678855584877772256 \nu^{6} + 197242012453008650162361763243624 \nu^{5} + 560505715214402813371124692098464 \nu^{4} + 1579073718550839377173770069070528 \nu^{3} + 335304029931824859067057245818880 \nu^{2} - 204933875925357469386506273011200 \nu - 791690269400822440126720933632000$$$$)/$$$$15\!\cdots\!00$$ $$\beta_{6}$$ $$=$$ $$($$$$2703350908857430681504226 \nu^{15} - 21121139726752262958470857 \nu^{14} + 85615020415050855998432068 \nu^{13} + 400180941001107979947439848 \nu^{12} + 5470121097635201079614833260 \nu^{11} - 32772269636178963252888012066 \nu^{10} + 135335521255521909352386882864 \nu^{9} + 609896432357399170382037323104 \nu^{8} + 3249015252369852414260909288202 \nu^{7} - 11155203340322902555583466472697 \nu^{6} + 51697072134057534901962276570668 \nu^{5} + 226905386235990131470393653849848 \nu^{4} + 592392329795398244339777298698112 \nu^{3} + 534989378035413890928655655721920 \nu^{2} + 700480090955541156514822825491200 \nu + 43416482023106197952486779072000$$$$)/$$$$37\!\cdots\!00$$ $$\beta_{7}$$ $$=$$ $$($$$$12085404159758350350699247 \nu^{15} - 85224520217749198675066170 \nu^{14} + 292450754027165513539470160 \nu^{13} + 2226878906398720288454917560 \nu^{12} + 25263266575828111291938782414 \nu^{11} - 130955780952902562464699541580 \nu^{10} + 458255913561199101206385102480 \nu^{9} + 3418446210754313761500284959880 \nu^{8} + 15752394612797340867688025357631 \nu^{7} - 43372335081959687277592960733330 \nu^{6} + 172352038909657772404504958906560 \nu^{5} + 1270855705906394283493640257820560 \nu^{4} + 3111633157131511050847624678326208 \nu^{3} + 2782839408275349352230347459422080 \nu^{2} + 532880132548126797451977997516800 \nu - 338043110964360494658210332032000$$$$)/$$$$15\!\cdots\!00$$ $$\beta_{8}$$ $$=$$ $$($$$$-6056238176411623354901114 \nu^{15} + 42688660290736661449235625 \nu^{14} - 146396752377212370392985060 \nu^{13} - 1114365932830071889909269160 \nu^{12} - 12669223252126716516697758108 \nu^{11} + 65598741434497573017700291490 \nu^{10} - 228974356306765774837195962480 \nu^{9} - 1708385418054973711004999115680 \nu^{8} - 7897245961507910001212281287842 \nu^{7} + 21731443531600977047188683957145 \nu^{6} - 85927199122479636045696950022860 \nu^{5} - 634888487783576647750492215095160 \nu^{4} - 1555603783724061364253191658007936 \nu^{3} - 1391388873664253031217367420877760 \nu^{2} - 266461998719592764984523512185600 \nu - 113359430152712771622478808256000$$$$)/$$$$75\!\cdots\!00$$ $$\beta_{9}$$ $$=$$ $$($$$$14129833027634573328418325 \nu^{15} - 139027753904031006612800022 \nu^{14} + 667898763729717798692161488 \nu^{13} + 1240561408564195207045804968 \nu^{12} + 24040930028724245351326487098 \nu^{11} - 229201540500665581650189292276 \nu^{10} + 1055514404460438472878574440624 \nu^{9} + 1845934094669355616190820667864 \nu^{8} + 10026035755381385551980140386629 \nu^{7} - 92357140706999333666351278578782 \nu^{6} + 394337167632079767709453607041888 \nu^{5} + 681418496046837343739531629625968 \nu^{4} + 513303202736939391134214036605248 \nu^{3} - 3148453244888932714110481083573120 \nu^{2} + 34588423261938521837652363724800 \nu + 284070615775820469186311384448000$$$$)/$$$$15\!\cdots\!00$$ $$\beta_{10}$$ $$=$$ $$($$$$9668013063977981500352306 \nu^{15} - 82269111088638989379317089 \nu^{14} + 371219892585702008246536996 \nu^{13} + 1120905125163472401458438056 \nu^{12} + 18946840984161357852443170508 \nu^{11} - 128967308020168938157932429522 \nu^{10} + 591433866196449550916053714608 \nu^{9} + 1684986168098138022742646619488 \nu^{8} + 10699991996617570136301704778266 \nu^{7} - 45017079953250546758762804361489 \nu^{6} + 227697201764607633474125119937996 \nu^{5} + 622367781092282082439843183168056 \nu^{4} + 1778878067376571140283514896411520 \nu^{3} + 1626861080351121141621570863179200 \nu^{2} + 3363881345957517157644805153760000 \nu + 131357250168685668066320480960000$$$$)/$$$$75\!\cdots\!00$$ $$\beta_{11}$$ $$=$$ $$($$$$13428386924650328789132886 \nu^{15} - 95277250258724264876361625 \nu^{14} + 326014514436759550605148740 \nu^{13} + 2468664225355349332858656840 \nu^{12} + 28018034445128638980749316692 \nu^{11} - 147785160815340224918590602210 \nu^{10} + 508437929524523712541921295920 \nu^{9} + 3785911360249835478377779872320 \nu^{8} + 17455229625444557705980955874558 \nu^{7} - 50347774701495496827378148425705 \nu^{6} + 190370004235474841905042468432940 \nu^{5} + 1407092593745364166072964645630840 \nu^{4} + 3447289313445791967409692127220864 \nu^{3} + 2530298869302846424708257483311040 \nu^{2} + 590464581344660932680019676486400 \nu + 79515996867061186837374272704000$$$$)/$$$$75\!\cdots\!00$$ $$\beta_{12}$$ $$=$$ $$($$$$3404401966607173877425988 \nu^{15} - 28307063272516326875381915 \nu^{14} + 117756865846291539249127380 \nu^{13} + 480385076756020571623172680 \nu^{12} + 6507430709809207300491768896 \nu^{11} - 45229501636240344791660292950 \nu^{10} + 185293017945287983690134265440 \nu^{9} + 732062379870015137712893598640 \nu^{8} + 3502339757578052353297544394044 \nu^{7} - 16870639247760785733127918275355 \nu^{6} + 69103137912046253068044979963980 \nu^{5} + 272269485278723106271564459864680 \nu^{4} + 526392895267692344984883448867072 \nu^{3} + 20598020740194583964273158630720 \nu^{2} - 108919511236600054251394453228800 \nu + 23011798155364052313163481152000$$$$)/$$$$18\!\cdots\!00$$ $$\beta_{13}$$ $$=$$ $$($$$$-2097730881 \nu^{15} + 14409577452 \nu^{14} - 47557870248 \nu^{13} - 399345069528 \nu^{12} - 4443245195650 \nu^{11} + 22040222469376 \nu^{10} - 74450035079504 \nu^{9} - 613147040366744 \nu^{8} - 2818574049615457 \nu^{7} + 7210077963870292 \nu^{6} - 28041792915591848 \nu^{5} - 227948096200534528 \nu^{4} - 569226084296757312 \nu^{3} - 510847011841073920 \nu^{2} - 97996854480243200 \nu + 10285863897088000$$$$)/ 11314720808960000$$ $$\beta_{14}$$ $$=$$ $$($$$$55340626917352680234536855 \nu^{15} - 448999850966826273318055418 \nu^{14} + 1822484469846211499663774672 \nu^{13} + 8198850832251074129599542392 \nu^{12} + 107243768305371799717320206622 \nu^{11} - 712930028410174509849510994444 \nu^{10} + 2867735110082173279479725159056 \nu^{9} + 12497293042934816331478823353416 \nu^{8} + 59211492079032463231993548028391 \nu^{7} - 260940383955764793658263364589458 \nu^{6} + 1072746719544806691245649613534272 \nu^{5} + 4637585547270709318518120721781392 \nu^{4} + 9424101164824769813013080053776832 \nu^{3} + 2823314684767143272028526453659520 \nu^{2} + 330148562707684344358019661619200 \nu - 259119882623177043296570553728000$$$$)/$$$$75\!\cdots\!00$$ $$\beta_{15}$$ $$=$$ $$($$$$303722929320017955947921539 \nu^{15} - 2441714352744758148157979674 \nu^{14} + 9811958801528103138100681456 \nu^{13} + 45809024378751100125056577816 \nu^{12} + 591601321613599416662896665974 \nu^{11} - 3870242021740782277143203820332 \nu^{10} + 15435979975198357718332290647888 \nu^{9} + 69881089967765443707729606788968 \nu^{8} + 329409472637030401174417018228435 \nu^{7} - 1409395537133264350096185765428114 \nu^{6} + 5775972091841201670688510441625056 \nu^{5} + 25944777282824431327728140088012816 \nu^{4} + 53288345966417140025275244418473152 \nu^{3} + 19036102356136401049313401888462720 \nu^{2} + 2558545494384426331242712330291200 \nu - 1188010392247008601362800236928000$$$$)/$$$$15\!\cdots\!00$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{13} - 3 \beta_{12} - \beta_{11} - 2 \beta_{10} + 3 \beta_{9} - \beta_{7} + 4 \beta_{6} + \beta_{5} - 4 \beta_{4} - 3 \beta_{3} - 7 \beta_{2} - \beta_{1} + 7$$$$)/10$$ $$\nu^{2}$$ $$=$$ $$($$$$-3 \beta_{15} + 6 \beta_{14} - \beta_{13} + 2 \beta_{11} - 3 \beta_{10} + 4 \beta_{9} + 40 \beta_{8} - 34 \beta_{7} + 6 \beta_{6} - 31 \beta_{4} + 31 \beta_{3} - 226 \beta_{2}$$$$)/10$$ $$\nu^{3}$$ $$=$$ $$($$$$-82 \beta_{15} + 164 \beta_{14} - 163 \beta_{13} + 83 \beta_{12} - 41 \beta_{11} + 83 \beta_{9} + 45 \beta_{7} - 41 \beta_{5} + 487 \beta_{3} - 507 \beta_{2} + 81 \beta_{1} - 507$$$$)/10$$ $$\nu^{4}$$ $$=$$ $$($$$$-43 \beta_{15} + 102 \beta_{14} - 43 \beta_{13} + 24 \beta_{12} + 43 \beta_{10} - 360 \beta_{8} - 336 \beta_{7} - 102 \beta_{6} - 2 \beta_{5} + 419 \beta_{4} + 443 \beta_{3} + 98 \beta_{1} - 1854$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$5949 \beta_{13} + 2547 \beta_{12} + 1749 \beta_{11} + 3798 \beta_{10} - 2547 \beta_{9} - 9200 \beta_{8} + 1749 \beta_{7} - 8396 \beta_{6} - 1749 \beta_{5} + 27896 \beta_{4} + 2547 \beta_{3} + 31743 \beta_{2} + 5949 \beta_{1} - 31743$$$$)/10$$ $$\nu^{6}$$ $$=$$ $$($$$$14507 \beta_{15} - 36214 \beta_{14} + 64769 \beta_{13} + 5462 \beta_{11} + 14507 \beta_{10} - 4076 \beta_{9} - 91560 \beta_{8} + 92946 \beta_{7} - 36214 \beta_{6} + 146239 \beta_{4} - 146239 \beta_{3} + 461594 \beta_{2}$$$$)/10$$ $$\nu^{7}$$ $$=$$ $$($$$$205098 \beta_{15} - 485396 \beta_{14} + 601307 \beta_{13} - 84587 \beta_{12} + 83849 \beta_{11} - 84587 \beta_{9} + 630595 \beta_{7} + 83849 \beta_{5} - 1922343 \beta_{3} + 2033723 \beta_{2} - 396209 \beta_{1} + 2033723$$$$)/10$$ $$\nu^{8}$$ $$=$$ $$($$$$192619 \beta_{15} - 488582 \beta_{14} + 192619 \beta_{13} - 39232 \beta_{12} - 192619 \beta_{10} + 1044200 \beta_{8} + 1004968 \beta_{7} + 488582 \beta_{6} + 100650 \beta_{5} - 1989915 \beta_{4} - 2029147 \beta_{3} - 771274 \beta_{1} + 5233270$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-25728301 \beta_{13} - 3142203 \beta_{12} - 4516101 \beta_{11} - 12183502 \beta_{10} + 3142203 \beta_{9} + 49978800 \beta_{8} - 4516101 \beta_{7} + 29953404 \beta_{6} + 4516101 \beta_{5} - 119908704 \beta_{4} - 3142203 \beta_{3} - 132013007 \beta_{2} - 25728301 \beta_{1} + 132013007$$$$)/10$$ $$\nu^{10}$$ $$=$$ $$($$$$-63463483 \beta_{15} + 161696566 \beta_{14} - 333391961 \beta_{13} - 37338078 \beta_{11} - 63463483 \beta_{10} + 12045844 \beta_{9} + 318812840 \beta_{8} - 344105074 \beta_{7} + 161696566 \beta_{6} - 664393991 \beta_{4} + 664393991 \beta_{3} - 1596905586 \beta_{2}$$$$)/10$$ $$\nu^{11}$$ $$=$$ $$($$$$-761028322 \beta_{15} + 1906866244 \beta_{14} - 2426251123 \beta_{13} + 135164643 \beta_{12} - 264747561 \beta_{11} + 135164643 \beta_{9} - 3350066355 \beta_{7} - 264747561 \beta_{5} + 7946493927 \beta_{3} - 8610221147 \beta_{2} + 1665222801 \beta_{1} - 8610221147$$$$)/10$$ $$\nu^{12}$$ $$=$$ $$($$$$-832972299 \beta_{15} + 2124161894 \beta_{14} - 832972299 \beta_{13} + 160514152 \beta_{12} + 832972299 \beta_{10} - 4033147240 \beta_{8} - 3872633088 \beta_{7} - 2124161894 \beta_{6} - 515340178 \beta_{5} + 8778369011 \beta_{4} + 8938883163 \beta_{3} + 3638955634 \beta_{1} - 20219529966$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$107997866749 \beta_{13} + 6752694947 \beta_{12} + 16340771349 \beta_{11} + 48694529798 \beta_{10} - 6752694947 \beta_{9} - 221797625200 \beta_{8} + 16340771349 \beta_{7} - 123110989996 \beta_{6} - 16340771349 \beta_{5} + 509042269896 \beta_{4} + 6752694947 \beta_{3} + 562349185343 \beta_{2} + 107997866749 \beta_{1} - 562349185343$$$$)/10$$ $$\nu^{14}$$ $$=$$ $$($$$$272692226027 \beta_{15} - 695324153654 \beta_{14} + 1478556161809 \beta_{13} + 172593321382 \beta_{11} + 272692226027 \beta_{10} - 54013602236 \beta_{9} - 1297300654760 \beta_{8} + 1415880373906 \beta_{7} - 695324153654 \beta_{6} + 2882927847279 \beta_{4} - 2882927847279 \beta_{3} + 6509919300234 \beta_{2}$$$$)/10$$ $$\nu^{15}$$ $$=$$ $$($$$$3150306189658 \beta_{15} - 7997847164916 \beta_{14} + 10171665397547 \beta_{13} - 379132207227 \beta_{12} + 1037265330729 \beta_{11} - 379132207227 \beta_{9} + 14841031824195 \beta_{7} + 1037265330729 \beta_{5} - 33572794085703 \beta_{3} + 36735471689883 \beta_{2} - 7021359207889 \beta_{1} + 36735471689883$$$$)/10$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$701$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1
 −1.37832 − 1.37832i 3.60306 + 3.60306i 5.71348 + 5.71348i −3.48873 − 3.48873i −0.394902 − 0.394902i 0.170157 + 0.170157i −3.99135 − 3.99135i 3.76660 + 3.76660i −1.37832 + 1.37832i 3.60306 − 3.60306i 5.71348 − 5.71348i −3.48873 + 3.48873i −0.394902 + 0.394902i 0.170157 − 0.170157i −3.99135 + 3.99135i 3.76660 − 3.76660i
1.00000 1.00000i −1.22474 1.22474i 2.00000i 0 −2.44949 −1.87083 + 1.87083i −2.00000 2.00000i 3.00000i 0
43.2 1.00000 1.00000i −1.22474 1.22474i 2.00000i 0 −2.44949 −1.87083 + 1.87083i −2.00000 2.00000i 3.00000i 0
43.3 1.00000 1.00000i −1.22474 1.22474i 2.00000i 0 −2.44949 1.87083 1.87083i −2.00000 2.00000i 3.00000i 0
43.4 1.00000 1.00000i −1.22474 1.22474i 2.00000i 0 −2.44949 1.87083 1.87083i −2.00000 2.00000i 3.00000i 0
43.5 1.00000 1.00000i 1.22474 + 1.22474i 2.00000i 0 2.44949 −1.87083 + 1.87083i −2.00000 2.00000i 3.00000i 0
43.6 1.00000 1.00000i 1.22474 + 1.22474i 2.00000i 0 2.44949 −1.87083 + 1.87083i −2.00000 2.00000i 3.00000i 0
43.7 1.00000 1.00000i 1.22474 + 1.22474i 2.00000i 0 2.44949 1.87083 1.87083i −2.00000 2.00000i 3.00000i 0
43.8 1.00000 1.00000i 1.22474 + 1.22474i 2.00000i 0 2.44949 1.87083 1.87083i −2.00000 2.00000i 3.00000i 0
757.1 1.00000 + 1.00000i −1.22474 + 1.22474i 2.00000i 0 −2.44949 −1.87083 1.87083i −2.00000 + 2.00000i 3.00000i 0
757.2 1.00000 + 1.00000i −1.22474 + 1.22474i 2.00000i 0 −2.44949 −1.87083 1.87083i −2.00000 + 2.00000i 3.00000i 0
757.3 1.00000 + 1.00000i −1.22474 + 1.22474i 2.00000i 0 −2.44949 1.87083 + 1.87083i −2.00000 + 2.00000i 3.00000i 0
757.4 1.00000 + 1.00000i −1.22474 + 1.22474i 2.00000i 0 −2.44949 1.87083 + 1.87083i −2.00000 + 2.00000i 3.00000i 0
757.5 1.00000 + 1.00000i 1.22474 1.22474i 2.00000i 0 2.44949 −1.87083 1.87083i −2.00000 + 2.00000i 3.00000i 0
757.6 1.00000 + 1.00000i 1.22474 1.22474i 2.00000i 0 2.44949 −1.87083 1.87083i −2.00000 + 2.00000i 3.00000i 0
757.7 1.00000 + 1.00000i 1.22474 1.22474i 2.00000i 0 2.44949 1.87083 + 1.87083i −2.00000 + 2.00000i 3.00000i 0
757.8 1.00000 + 1.00000i 1.22474 1.22474i 2.00000i 0 2.44949 1.87083 + 1.87083i −2.00000 + 2.00000i 3.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 757.8 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 1050.3.l.h 16
5.b even 2 1 210.3.l.b 16
5.c odd 4 1 210.3.l.b 16
5.c odd 4 1 inner 1050.3.l.h 16
15.d odd 2 1 630.3.o.f 16
15.e even 4 1 630.3.o.f 16

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1050, [\chi])$$:

 $$T_{11}^{8} - \cdots$$ $$T_{13}^{16} - \cdots$$

## Hecke characteristic polynomials

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