Properties

Label 930.2.n.h
Level $930$
Weight $2$
Character orbit 930.n
Analytic conductor $7.426$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.n (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 4 x^{15} + 5 x^{14} - 60 x^{13} + 480 x^{12} - 1202 x^{11} - 147 x^{10} - 2185 x^{9} + 62400 x^{8} - 289745 x^{7} + 699483 x^{6} - 1042897 x^{5} + 1014415 x^{4} - 652345 x^{3} + 274580 x^{2} - 73519 x + 12851\)
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} + \beta_{3} q^{3} + ( -1 - \beta_{3} + \beta_{6} + \beta_{7} ) q^{4} - q^{5} - q^{6} + ( \beta_{4} - \beta_{6} - \beta_{7} + \beta_{9} - \beta_{11} ) q^{7} + \beta_{6} q^{8} -\beta_{6} q^{9} +O(q^{10})\) \( q + \beta_{7} q^{2} + \beta_{3} q^{3} + ( -1 - \beta_{3} + \beta_{6} + \beta_{7} ) q^{4} - q^{5} - q^{6} + ( \beta_{4} - \beta_{6} - \beta_{7} + \beta_{9} - \beta_{11} ) q^{7} + \beta_{6} q^{8} -\beta_{6} q^{9} -\beta_{7} q^{10} + ( -\beta_{4} - \beta_{9} + \beta_{11} - \beta_{14} ) q^{11} -\beta_{7} q^{12} + ( \beta_{1} - \beta_{2} + \beta_{5} - \beta_{8} ) q^{13} + ( 1 - \beta_{6} - \beta_{7} - \beta_{11} ) q^{14} -\beta_{3} q^{15} + \beta_{3} q^{16} + ( -\beta_{1} - \beta_{2} - \beta_{5} - \beta_{7} + 2 \beta_{8} + \beta_{10} ) q^{17} -\beta_{3} q^{18} + ( -1 + \beta_{1} - \beta_{2} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{12} ) q^{19} + ( 1 + \beta_{3} - \beta_{6} - \beta_{7} ) q^{20} + ( -\beta_{3} + \beta_{6} + \beta_{7} - \beta_{9} + \beta_{12} ) q^{21} + ( \beta_{10} + \beta_{11} ) q^{22} + ( -\beta_{9} + \beta_{10} - \beta_{11} ) q^{23} + ( 1 + \beta_{3} - \beta_{6} - \beta_{7} ) q^{24} + q^{25} + ( 2 \beta_{1} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{26} + ( -1 - \beta_{3} + \beta_{6} + \beta_{7} ) q^{27} + ( 1 - \beta_{4} - \beta_{6} ) q^{28} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{6} + 3 \beta_{7} + \beta_{15} ) q^{29} + q^{30} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{9} + \beta_{14} ) q^{31} - q^{32} + ( \beta_{9} - \beta_{12} + \beta_{15} ) q^{33} + ( 2 - \beta_{1} + 2 \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{10} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{34} + ( -\beta_{4} + \beta_{6} + \beta_{7} - \beta_{9} + \beta_{11} ) q^{35} + q^{36} + ( -1 + 2 \beta_{2} - \beta_{4} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{14} ) q^{37} + ( -1 - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} ) q^{38} + ( \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{39} -\beta_{6} q^{40} + ( \beta_{1} + \beta_{4} + 2 \beta_{5} - 4 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} - \beta_{12} ) q^{41} + ( -\beta_{4} + \beta_{6} + \beta_{7} - \beta_{9} + \beta_{11} ) q^{42} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{11} + 2 \beta_{12} + \beta_{13} - \beta_{14} ) q^{43} + ( 1 + \beta_{3} + \beta_{4} - \beta_{7} + \beta_{10} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{44} + \beta_{6} q^{45} + ( 1 + \beta_{3} - 2 \beta_{4} - \beta_{7} + \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{46} + ( 2 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{47} -\beta_{6} q^{48} + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{10} + 2 \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{49} + \beta_{7} q^{50} + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{6} - 2 \beta_{7} + \beta_{14} ) q^{51} + ( -1 + \beta_{1} - \beta_{2} + \beta_{7} + \beta_{8} ) q^{52} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} + 2 \beta_{8} - 2 \beta_{9} + \beta_{13} + \beta_{15} ) q^{53} + \beta_{6} q^{54} + ( \beta_{4} + \beta_{9} - \beta_{11} + \beta_{14} ) q^{55} + ( -\beta_{3} + \beta_{7} + \beta_{12} ) q^{56} + ( -2 \beta_{1} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{12} ) q^{57} + ( -2 + \beta_{1} - 2 \beta_{3} + \beta_{5} + 2 \beta_{6} + 3 \beta_{7} + \beta_{8} + \beta_{14} ) q^{58} + ( 2 + \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} - \beta_{13} ) q^{59} + \beta_{7} q^{60} + ( 3 + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{6} + 3 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{13} ) q^{61} + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{7} - \beta_{10} + \beta_{11} ) q^{62} + ( \beta_{3} - \beta_{7} - \beta_{12} ) q^{63} -\beta_{7} q^{64} + ( -\beta_{1} + \beta_{2} - \beta_{5} + \beta_{8} ) q^{65} + ( \beta_{4} + \beta_{9} - \beta_{11} + \beta_{14} ) q^{66} + ( -6 - 2 \beta_{2} + \beta_{4} - \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{12} - \beta_{14} ) q^{67} + ( 2 \beta_{2} + \beta_{6} - \beta_{8} - \beta_{13} ) q^{68} + ( -2 \beta_{4} - \beta_{9} + \beta_{11} - \beta_{12} + \beta_{14} ) q^{69} + ( -1 + \beta_{6} + \beta_{7} + \beta_{11} ) q^{70} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - 6 \beta_{6} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{11} + \beta_{13} + \beta_{15} ) q^{71} + \beta_{7} q^{72} + ( 2 - \beta_{1} + 2 \beta_{3} - \beta_{5} - 3 \beta_{6} - 4 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{14} - \beta_{15} ) q^{73} + ( -1 - \beta_{1} - \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{74} + \beta_{3} q^{75} + ( \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{11} ) q^{76} + ( \beta_{1} - \beta_{2} - 6 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} - 3 \beta_{11} + 3 \beta_{12} + \beta_{13} ) q^{77} + ( -\beta_{1} + \beta_{2} - \beta_{5} + \beta_{8} ) q^{78} + ( 1 - \beta_{1} + \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{10} + 2 \beta_{11} - \beta_{13} - \beta_{15} ) q^{79} -\beta_{3} q^{80} -\beta_{7} q^{81} + ( 4 + \beta_{1} + \beta_{2} + 4 \beta_{3} - 4 \beta_{6} - 4 \beta_{7} + \beta_{9} - \beta_{11} - \beta_{12} ) q^{82} + ( -2 + \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{83} + ( -1 + \beta_{6} + \beta_{7} + \beta_{11} ) q^{84} + ( \beta_{1} + \beta_{2} + \beta_{5} + \beta_{7} - 2 \beta_{8} - \beta_{10} ) q^{85} + ( \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} - 2 \beta_{9} + \beta_{10} + 2 \beta_{11} + \beta_{12} - \beta_{15} ) q^{86} + ( -1 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{13} ) q^{87} + ( 1 + \beta_{3} - \beta_{7} - \beta_{12} - \beta_{13} ) q^{88} + ( \beta_{4} - 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + 3 \beta_{12} + \beta_{14} - 2 \beta_{15} ) q^{89} + \beta_{3} q^{90} + ( 2 - 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} - \beta_{8} + 3 \beta_{9} - 3 \beta_{12} ) q^{91} + ( 1 + \beta_{3} + \beta_{4} - \beta_{7} + \beta_{9} + \beta_{12} - \beta_{13} ) q^{92} + ( -\beta_{1} - \beta_{4} - \beta_{6} + \beta_{8} + \beta_{11} - \beta_{12} - \beta_{15} ) q^{93} + ( 2 \beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{13} + \beta_{14} ) q^{94} + ( 1 - \beta_{1} + \beta_{2} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{12} ) q^{95} -\beta_{3} q^{96} + ( 1 + \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} + 6 \beta_{6} + 5 \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{10} - 2 \beta_{11} + \beta_{12} - \beta_{14} + \beta_{15} ) q^{97} + ( 2 - 2 \beta_{1} + 3 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} - 4 \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{13} + \beta_{14} ) q^{98} + ( -1 - \beta_{3} + \beta_{7} + \beta_{12} + \beta_{13} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 4q^{2} - 4q^{3} - 4q^{4} - 16q^{5} - 16q^{6} - 7q^{7} + 4q^{8} - 4q^{9} + O(q^{10}) \) \( 16q + 4q^{2} - 4q^{3} - 4q^{4} - 16q^{5} - 16q^{6} - 7q^{7} + 4q^{8} - 4q^{9} - 4q^{10} + 4q^{11} - 4q^{12} + 2q^{13} + 7q^{14} + 4q^{15} - 4q^{16} + 5q^{17} + 4q^{18} - 2q^{19} + 4q^{20} + 8q^{21} - 4q^{22} - 4q^{23} + 4q^{24} + 16q^{25} + 8q^{26} - 4q^{27} + 8q^{28} + 5q^{29} + 16q^{30} - q^{31} - 16q^{32} - q^{33} - 5q^{34} + 7q^{35} + 16q^{36} - 20q^{37} - 3q^{38} + 2q^{39} - 4q^{40} - 17q^{41} + 7q^{42} - 4q^{43} - q^{44} + 4q^{45} - 6q^{46} + 2q^{47} - 4q^{48} - 21q^{49} + 4q^{50} + 5q^{51} + 2q^{52} + 9q^{53} + 4q^{54} - 4q^{55} + 2q^{56} - 2q^{57} - 5q^{58} - 3q^{59} + 4q^{60} + 70q^{61} - 4q^{62} - 2q^{63} - 4q^{64} - 2q^{65} - 4q^{66} - 66q^{67} - 20q^{68} - 4q^{69} - 7q^{70} - 25q^{71} + 4q^{72} - 10q^{73} - 15q^{74} - 4q^{75} + 3q^{76} + 26q^{77} - 2q^{78} + 14q^{79} + 4q^{80} - 4q^{81} + 17q^{82} + 20q^{83} - 7q^{84} - 5q^{85} - 6q^{86} - 20q^{87} + 6q^{88} - 13q^{89} - 4q^{90} - 4q^{91} - 4q^{92} + 4q^{93} - 12q^{94} + 2q^{95} + 4q^{96} + 55q^{97} - 14q^{98} - 6q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 4 x^{15} + 5 x^{14} - 60 x^{13} + 480 x^{12} - 1202 x^{11} - 147 x^{10} - 2185 x^{9} + 62400 x^{8} - 289745 x^{7} + 699483 x^{6} - 1042897 x^{5} + 1014415 x^{4} - 652345 x^{3} + 274580 x^{2} - 73519 x + 12851\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(100088349171015463663880816 \nu^{15} - 504646929512976890497221763 \nu^{14} + 454642135257016153777677250 \nu^{13} - 5847932363375733433252536267 \nu^{12} + 54074430053964935169047259599 \nu^{11} - 143449490434185843031757914498 \nu^{10} - 43719634977046924108962551686 \nu^{9} - 60390001173110533704205244515 \nu^{8} + 7011061192510602284544734103246 \nu^{7} - 33158490879816571942413260174007 \nu^{6} + 76807429790805254368589160739546 \nu^{5} - 104931746814290847736868416041548 \nu^{4} + 87227013597503290618417020629940 \nu^{3} - 43400069378555293292301566317112 \nu^{2} + 11456330628240068357637938427999 \nu - 3265763473393338711863191411345\)\()/ \)\(21\!\cdots\!40\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-115503386275919477965347643 \nu^{15} + 226222368817637316589270116 \nu^{14} - 75170297328050902412616558 \nu^{13} + 6702361361979112055382235630 \nu^{12} - 41750337174754291989349315977 \nu^{11} + 51346841598413227880090960875 \nu^{10} + 135832888339413745073980002557 \nu^{9} + 513205486148578354014107077599 \nu^{8} - 6217748222086829339666660957995 \nu^{7} + 20627455348358108371990451602931 \nu^{6} - 36551277121293757643195285806656 \nu^{5} + 38666701433427233655510616524551 \nu^{4} - 27090209423483029764701745141093 \nu^{3} + 15419305728016429960557834157345 \nu^{2} - 9639099017232392800687731037822 \nu + 2432054689959008736271955604856\)\()/ \)\(21\!\cdots\!40\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-182306882795156956529944138 \nu^{15} + 1206689587068232865481313397 \nu^{14} - 1399083663172131995284811573 \nu^{13} + 10218524284159446343073124390 \nu^{12} - 116118567510895265871703004548 \nu^{11} + 364462689261061743860368382693 \nu^{10} - 15543625802022652588377384016 \nu^{9} - 331588296670156476849489703246 \nu^{8} - 14263644062230145642642778705360 \nu^{7} + 76328246556930830542379479752654 \nu^{6} - 186568536869247490647807826188050 \nu^{5} + 264478533471293026541042531043397 \nu^{4} - 228845247118184021493610268948343 \nu^{3} + 126918923437692669143338134916660 \nu^{2} - 39653728578724565795390752563108 \nu + 9766064023617170590224092808989\)\()/ \)\(21\!\cdots\!40\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-267872744379589568741842489 \nu^{15} + 499851217189271908425729415 \nu^{14} - 54753381720615092665945564 \nu^{13} + 15309311579452963923643222418 \nu^{12} - 95728483213498980599729338907 \nu^{11} + 105298205995885655479447816164 \nu^{10} + 355650734130836024649468337700 \nu^{9} + 1190642370840898739739411649342 \nu^{8} - 14454443936656341920986335022544 \nu^{7} + 46115674763981167670419858083106 \nu^{6} - 75742042806446422186923284178235 \nu^{5} + 69243020313937186563261420590115 \nu^{4} - 34869099510987746139326388104544 \nu^{3} + 8911382321366949719358888482908 \nu^{2} - 3228901293194488933352712275657 \nu + 574824402489950639176648206282\)\()/ \)\(21\!\cdots\!40\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-303136675452287903298322056 \nu^{15} + 747990876795248058889024325 \nu^{14} - 291133853146961018527918331 \nu^{13} + 17300678585491174499833895837 \nu^{12} - 118580669734855555864339572253 \nu^{11} + 178197010178434026884837785186 \nu^{10} + 362418380979748745566623022785 \nu^{9} + 1090044670431285135647326039808 \nu^{8} - 17277728006105657375126805122591 \nu^{7} + 61408588995994646032989610028824 \nu^{6} - 112206139941989709882640354513475 \nu^{5} + 115687146179786325569310928705295 \nu^{4} - 64368938012934330668982904574676 \nu^{3} + 13818261754071428769608772973467 \nu^{2} + 48858195008461400178894845422 \nu + 479955009670246896417948626818\)\()/ \)\(21\!\cdots\!40\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-377010885008550592635170302 \nu^{15} + 1207427945649844646161159503 \nu^{14} - 1126200521084205735727803000 \nu^{13} + 22174549622106308707655890687 \nu^{12} - 163417371209079704173802329449 \nu^{11} + 334456611542427467808780770740 \nu^{10} + 244825176904747990556811878906 \nu^{9} + 1121551359007511542759477454115 \nu^{8} - 22374828441420286757267304025546 \nu^{7} + 92210168445583023861593953931687 \nu^{6} - 201605750209487674045943340563664 \nu^{5} + 271048754683930058578452567723368 \nu^{4} - 233879717801903313221670989844210 \nu^{3} + 128285239893197968371082792482892 \nu^{2} - 42950638073387430024675954919489 \nu + 9113472569919452397003935131039\)\()/ \)\(21\!\cdots\!40\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-503663920665630164013962945 \nu^{15} + 1002197851240818395839851072 \nu^{14} - 282595196296671715204568514 \nu^{13} + 29000599817140021357660775081 \nu^{12} - 183216711918740965233986244918 \nu^{11} + 224152096552739237292521715200 \nu^{10} + 616481321266530319309290815344 \nu^{9} + 2180305452371444825110443614547 \nu^{8} - 27293517244814522692065536240648 \nu^{7} + 90357052785126394232859837645019 \nu^{6} - 157533476410016784343039828495455 \nu^{5} + 159321178479541014038777804722867 \nu^{4} - 94797850303136008032073262120034 \nu^{3} + 30987203107776527178396313260026 \nu^{2} - 7098801861775648865124317028278 \nu + 2059158419521791850509330766475\)\()/ \)\(21\!\cdots\!40\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-518549229960251998528431137 \nu^{15} + 989157524666281919369366434 \nu^{14} + 219163726885988166627892367 \nu^{13} + 30346526668960121806764604674 \nu^{12} - 185519241148340628443533109927 \nu^{11} + 191202705848398583062940683291 \nu^{10} + 730390620006498078382050859498 \nu^{9} + 2424218001665229620936329302004 \nu^{8} - 28233627730416117829065491385762 \nu^{7} + 87407427703535750466598416090276 \nu^{6} - 141105452588202400804814409297217 \nu^{5} + 127418347560501672873393333651854 \nu^{4} - 62886220089114974500497879332673 \nu^{3} + 10305409031734646902691458361954 \nu^{2} + 874226421602522561923634868153 \nu - 205900280071002643813471879295\)\()/ \)\(21\!\cdots\!40\)\( \)
\(\beta_{10}\)\(=\)\((\)\(788700379896405494890752555 \nu^{15} - 4621452763321439701998160646 \nu^{14} + 6254053842500698028538683198 \nu^{13} - 45934281425549416118110566107 \nu^{12} + 463230412206541667024545714378 \nu^{11} - 1446810089832881397383268506926 \nu^{10} + 257223609794673786313726186338 \nu^{9} + 560993925611314061206736757241 \nu^{8} + 56548087063458456479711253666386 \nu^{7} - 306529111562104477893346154821639 \nu^{6} + 774215921924026353917596617145943 \nu^{5} - 1134291525768723825577383217116641 \nu^{4} + 1006702723937138817262060110916128 \nu^{3} - 548941249359476044134657373812252 \nu^{2} + 180199231884638643953488654443548 \nu - 31198773127888251811745150524201\)\()/ \)\(21\!\cdots\!40\)\( \)
\(\beta_{11}\)\(=\)\((\)\(869353466431330371903191501 \nu^{15} - 2108004905987356028977897649 \nu^{14} + 759860611709209652450015066 \nu^{13} - 50729752878012917167408265149 \nu^{12} + 337303569558653406429887667358 \nu^{11} - 498508858208141055811311413385 \nu^{10} - 993364797184466314387748529133 \nu^{9} - 3423835277263753672098696739928 \nu^{8} + 49187287715602256801757642459367 \nu^{7} - 172918481352990911241888575002064 \nu^{6} + 323085877307013050685314664106622 \nu^{5} - 362603490548528797217608476150159 \nu^{4} + 259822062461139103116628253069541 \nu^{3} - 115622013530275985840736139791419 \nu^{2} + 34177983342865802383241560846813 \nu - 5270416253687425795164482118507\)\()/ \)\(21\!\cdots\!40\)\( \)
\(\beta_{12}\)\(=\)\((\)\(382306067038565199188628297 \nu^{15} - 1303380308547599180511016712 \nu^{14} + 986066700325309062561149647 \nu^{13} - 22133266738924779659388042330 \nu^{12} + 170480760036591981179295144711 \nu^{11} - 349793652786747953849602009449 \nu^{10} - 314360188230947969818196581514 \nu^{9} - 980981852631985348913937310556 \nu^{8} + 23490814052570436238089832262690 \nu^{7} - 96084889956705373575105397840508 \nu^{6} + 202659491444639721842802213760201 \nu^{5} - 255691876575853809392122090724392 \nu^{4} + 201483663659922155834322821485027 \nu^{3} - 99703375972654416108123640534970 \nu^{2} + 28641892237880940898368945389351 \nu - 5791193462065655939061142801083\)\()/ \)\(72\!\cdots\!80\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-210275423051595494468756737 \nu^{15} + 500353247209301819223902434 \nu^{14} + 114814928755048929855325446 \nu^{13} + 11801306107096129801194460576 \nu^{12} - 81463163707893036199964952185 \nu^{11} + 100515084411656034427654104685 \nu^{10} + 340179166571771590866760313743 \nu^{9} + 770814188867555948785038486167 \nu^{8} - 12309445378945838296509965959913 \nu^{7} + 39852316526119363026363821191595 \nu^{6} - 61369562242114449624446527250944 \nu^{5} + 42679782373106444789769290154809 \nu^{4} - 1424776518427986213284890956559 \nu^{3} - 13421241367535276593691962916289 \nu^{2} + 5063616394208549728693987156220 \nu + 319795813502296267486975825134\)\()/ \)\(36\!\cdots\!40\)\( \)
\(\beta_{14}\)\(=\)\((\)\(1388435052432364503015752334 \nu^{15} - 2755255288765990758332602177 \nu^{14} - 177887877536057609612869175 \nu^{13} - 80502899186472262199372967070 \nu^{12} + 504705356790352127816102098232 \nu^{11} - 560572951910491767041162742629 \nu^{10} - 1906723682287971562805434154124 \nu^{9} - 6248573537414079529684389348910 \nu^{8} + 76267579565248447408459411465660 \nu^{7} - 241484930949502139829568376793986 \nu^{6} + 397597679334625399319820889835770 \nu^{5} - 374265776337441214595670676115197 \nu^{4} + 212703665017386032521073131376415 \nu^{3} - 68231728438525916585257521335400 \nu^{2} - 1750069753612177430015605423388 \nu + 10842162982775897890267915150423\)\()/ \)\(21\!\cdots\!40\)\( \)
\(\beta_{15}\)\(=\)\((\)\(1951635990038649212520217421 \nu^{15} - 5071832855639565665263879113 \nu^{14} + 1915593218679674504264323524 \nu^{13} - 113164257454784593874520596348 \nu^{12} + 779688858582523868565277238265 \nu^{11} - 1211215611306665219226477114332 \nu^{10} - 2240010788297304500290039503376 \nu^{9} - 7237763174904591845715032321112 \nu^{8} + 113009077133159295483652092455444 \nu^{7} - 403544013108358691896268324031940 \nu^{6} + 758836381690581799714825159708323 \nu^{5} - 861548472871865166027017879891203 \nu^{4} + 643091977077265779540497368089684 \nu^{3} - 342037012825561670131904154106628 \nu^{2} + 124554632889870292325673053255015 \nu - 23157786579480468350279748096404\)\()/ \)\(21\!\cdots\!40\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} - \beta_{9} - 2 \beta_{8} + 6 \beta_{7} - \beta_{6} + \beta_{3} + \beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{15} - 2 \beta_{14} + \beta_{12} - 2 \beta_{11} - 3 \beta_{10} + 15 \beta_{8} - 17 \beta_{7} - 21 \beta_{6} - 3 \beta_{5} + 9 \beta_{3} - 12 \beta_{2} - 6 \beta_{1} + 15\)
\(\nu^{4}\)\(=\)\(-17 \beta_{15} + 21 \beta_{14} - 4 \beta_{13} - 6 \beta_{12} + 9 \beta_{11} + 15 \beta_{10} - 9 \beta_{9} - 14 \beta_{8} + 42 \beta_{7} + 93 \beta_{6} - 24 \beta_{5} - 13 \beta_{4} - 78 \beta_{3} + 7 \beta_{2} + 29 \beta_{1} - 114\)
\(\nu^{5}\)\(=\)\(77 \beta_{15} - 47 \beta_{14} + 66 \beta_{13} + 17 \beta_{12} - 14 \beta_{11} - 10 \beta_{10} - 20 \beta_{9} - 123 \beta_{8} + 313 \beta_{7} - 253 \beta_{6} + 194 \beta_{5} - 21 \beta_{4} - 47 \beta_{3} + 105 \beta_{2} - 55 \beta_{1} + 66\)
\(\nu^{6}\)\(=\)\(-125 \beta_{15} - 106 \beta_{14} - 260 \beta_{13} - 55 \beta_{12} + 60 \beta_{11} - 189 \beta_{10} + 159 \beta_{9} + 744 \beta_{8} - 2198 \beta_{7} - 313 \beta_{6} - 203 \beta_{5} + 47 \beta_{4} + 1503 \beta_{3} - 1154 \beta_{2} - 144 \beta_{1} + 1431\)
\(\nu^{7}\)\(=\)\(-1461 \beta_{15} + 1747 \beta_{14} - 76 \beta_{13} - 35 \beta_{12} + 497 \beta_{11} + 1004 \beta_{10} - 329 \beta_{9} - 1269 \beta_{8} + 4756 \beta_{7} + 7693 \beta_{6} - 2723 \beta_{5} - 784 \beta_{4} - 10114 \beta_{3} + 4205 \beta_{2} + 947 \beta_{1} - 7413\)
\(\nu^{8}\)\(=\)\(9546 \beta_{15} - 6080 \beta_{14} + 5055 \beta_{13} + 1045 \beta_{12} - 3131 \beta_{11} - 1193 \beta_{10} + 448 \beta_{9} - 13782 \beta_{8} + 20068 \beta_{7} - 28538 \beta_{6} + 24166 \beta_{5} + 3225 \beta_{4} + 21089 \beta_{3} + 3560 \beta_{2} + 5702 \beta_{1} + 7283\)
\(\nu^{9}\)\(=\)\(-15929 \beta_{15} - 13124 \beta_{14} - 22955 \beta_{13} + 5109 \beta_{12} + 5760 \beta_{11} - 18837 \beta_{10} + 5751 \beta_{9} + 85400 \beta_{8} - 194668 \beta_{7} - 6803 \beta_{6} - 71117 \beta_{5} + 5595 \beta_{4} + 118055 \beta_{3} - 95074 \beta_{2} - 56248 \beta_{1} + 155369\)
\(\nu^{10}\)\(=\)\(-116284 \beta_{15} + 193114 \beta_{14} + 7843 \beta_{13} - 34519 \beta_{12} + 9183 \beta_{11} + 108355 \beta_{10} - 23245 \beta_{9} - 83283 \beta_{8} + 486849 \beta_{7} + 560240 \beta_{6} - 164952 \beta_{5} - 67108 \beta_{4} - 1018155 \beta_{3} + 401806 \beta_{2} + 187837 \beta_{1} - 900992\)
\(\nu^{11}\)\(=\)\(955552 \beta_{15} - 705841 \beta_{14} + 476916 \beta_{13} + 104737 \beta_{12} - 181288 \beta_{11} - 91126 \beta_{10} - 83936 \beta_{9} - 1588169 \beta_{8} + 2073500 \beta_{7} - 2432456 \beta_{6} + 2278660 \beta_{5} + 262872 \beta_{4} + 2739368 \beta_{3} + 18327 \beta_{2} + 377605 \beta_{1} + 882110\)
\(\nu^{12}\)\(=\)\(-2111653 \beta_{15} - 781722 \beta_{14} - 2379691 \beta_{13} + 321855 \beta_{12} + 488402 \beta_{11} - 2065085 \beta_{10} + 779820 \beta_{9} + 10703011 \beta_{8} - 21073771 \beta_{7} - 1713515 \beta_{6} - 8108443 \beta_{5} + 36776 \beta_{4} + 7835953 \beta_{3} - 8896330 \beta_{2} - 6986192 \beta_{1} + 16014786\)
\(\nu^{13}\)\(=\)\(-10935492 \beta_{15} + 20327298 \beta_{14} + 854213 \beta_{13} - 5074260 \beta_{12} + 1981295 \beta_{11} + 13082702 \beta_{10} - 2193955 \beta_{9} - 19308202 \beta_{8} + 58774773 \beta_{7} + 64696874 \beta_{6} - 8380445 \beta_{5} - 6489657 \beta_{4} - 97955533 \beta_{3} + 41380226 \beta_{2} + 30106900 \beta_{1} - 105329623\)
\(\nu^{14}\)\(=\)\(101049082 \beta_{15} - 85424041 \beta_{14} + 50277050 \beta_{13} + 19745720 \beta_{12} - 18439137 \beta_{11} - 20162415 \beta_{10} - 8696951 \beta_{9} - 142814260 \beta_{8} + 185199403 \beta_{7} - 282787056 \beta_{6} + 217585654 \beta_{5} + 27409694 \beta_{4} + 300537399 \beta_{3} - 2332073 \beta_{2} - 9683246 \beta_{1} + 180623205\)
\(\nu^{15}\)\(=\)\(-264747964 \beta_{15} - 18099316 \beta_{14} - 278040932 \beta_{13} + 2672106 \beta_{12} + 45283488 \beta_{11} - 193091310 \beta_{10} + 107693910 \beta_{9} + 1161791948 \beta_{8} - 2299972646 \beta_{7} - 55935446 \beta_{6} - 870534016 \beta_{5} + 8990302 \beta_{4} + 696446026 \beta_{3} - 932109758 \beta_{2} - 586009046 \beta_{1} + 1439366547\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/930\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
481.1
2.41373 + 0.0663824i
−3.22434 + 1.89830i
0.602406 + 0.654917i
1.20820 + 0.458083i
1.36070 + 1.34711i
0.109569 0.374923i
−2.52433 4.00017i
2.05406 + 2.30144i
1.36070 1.34711i
0.109569 + 0.374923i
−2.52433 + 4.00017i
2.05406 2.30144i
2.41373 0.0663824i
−3.22434 1.89830i
0.602406 0.654917i
1.20820 0.458083i
0.809017 0.587785i −0.809017 0.587785i 0.309017 0.951057i −1.00000 −1.00000 −1.42081 + 4.37281i −0.309017 0.951057i 0.309017 + 0.951057i −0.809017 + 0.587785i
481.2 0.809017 0.587785i −0.809017 0.587785i 0.309017 0.951057i −1.00000 −1.00000 −0.690690 + 2.12572i −0.309017 0.951057i 0.309017 + 0.951057i −0.809017 + 0.587785i
481.3 0.809017 0.587785i −0.809017 0.587785i 0.309017 0.951057i −1.00000 −1.00000 −0.186222 + 0.573132i −0.309017 0.951057i 0.309017 + 0.951057i −0.809017 + 0.587785i
481.4 0.809017 0.587785i −0.809017 0.587785i 0.309017 0.951057i −1.00000 −1.00000 1.10674 3.40620i −0.309017 0.951057i 0.309017 + 0.951057i −0.809017 + 0.587785i
721.1 −0.309017 + 0.951057i 0.309017 + 0.951057i −0.809017 0.587785i −1.00000 −1.00000 −3.70406 2.69116i 0.809017 0.587785i −0.809017 + 0.587785i 0.309017 0.951057i
721.2 −0.309017 + 0.951057i 0.309017 + 0.951057i −0.809017 0.587785i −1.00000 −1.00000 −0.0930287 0.0675893i 0.809017 0.587785i −0.809017 + 0.587785i 0.309017 0.951057i
721.3 −0.309017 + 0.951057i 0.309017 + 0.951057i −0.809017 0.587785i −1.00000 −1.00000 0.170108 + 0.123590i 0.809017 0.587785i −0.809017 + 0.587785i 0.309017 0.951057i
721.4 −0.309017 + 0.951057i 0.309017 + 0.951057i −0.809017 0.587785i −1.00000 −1.00000 1.31796 + 0.957557i 0.809017 0.587785i −0.809017 + 0.587785i 0.309017 0.951057i
841.1 −0.309017 0.951057i 0.309017 0.951057i −0.809017 + 0.587785i −1.00000 −1.00000 −3.70406 + 2.69116i 0.809017 + 0.587785i −0.809017 0.587785i 0.309017 + 0.951057i
841.2 −0.309017 0.951057i 0.309017 0.951057i −0.809017 + 0.587785i −1.00000 −1.00000 −0.0930287 + 0.0675893i 0.809017 + 0.587785i −0.809017 0.587785i 0.309017 + 0.951057i
841.3 −0.309017 0.951057i 0.309017 0.951057i −0.809017 + 0.587785i −1.00000 −1.00000 0.170108 0.123590i 0.809017 + 0.587785i −0.809017 0.587785i 0.309017 + 0.951057i
841.4 −0.309017 0.951057i 0.309017 0.951057i −0.809017 + 0.587785i −1.00000 −1.00000 1.31796 0.957557i 0.809017 + 0.587785i −0.809017 0.587785i 0.309017 + 0.951057i
901.1 0.809017 + 0.587785i −0.809017 + 0.587785i 0.309017 + 0.951057i −1.00000 −1.00000 −1.42081 4.37281i −0.309017 + 0.951057i 0.309017 0.951057i −0.809017 0.587785i
901.2 0.809017 + 0.587785i −0.809017 + 0.587785i 0.309017 + 0.951057i −1.00000 −1.00000 −0.690690 2.12572i −0.309017 + 0.951057i 0.309017 0.951057i −0.809017 0.587785i
901.3 0.809017 + 0.587785i −0.809017 + 0.587785i 0.309017 + 0.951057i −1.00000 −1.00000 −0.186222 0.573132i −0.309017 + 0.951057i 0.309017 0.951057i −0.809017 0.587785i
901.4 0.809017 + 0.587785i −0.809017 + 0.587785i 0.309017 + 0.951057i −1.00000 −1.00000 1.10674 + 3.40620i −0.309017 + 0.951057i 0.309017 0.951057i −0.809017 0.587785i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 901.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.n.h 16
31.d even 5 1 inner 930.2.n.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.n.h 16 1.a even 1 1 trivial
930.2.n.h 16 31.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{7}^{16} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \)
$3$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
$5$ \( ( 1 + T )^{16} \)
$7$ \( 16 + 112 T - 56 T^{2} - 4074 T^{3} + 24239 T^{4} + 6515 T^{5} + 73817 T^{6} - 28408 T^{7} + 16548 T^{8} - 3063 T^{9} + 5737 T^{10} + 1445 T^{11} + 714 T^{12} + 176 T^{13} + 49 T^{14} + 7 T^{15} + T^{16} \)
$11$ \( 2624400 + 8748000 T + 21222000 T^{2} + 24149250 T^{3} + 17308675 T^{4} - 6832375 T^{5} + 16978940 T^{6} - 7337230 T^{7} + 1927015 T^{8} - 262190 T^{9} + 44145 T^{10} - 5605 T^{11} + 1611 T^{12} - 219 T^{13} + 41 T^{14} - 4 T^{15} + T^{16} \)
$13$ \( 1256641 + 1229737 T + 21649855 T^{2} - 43483105 T^{3} + 39554180 T^{4} - 18830749 T^{5} + 8242642 T^{6} - 3121605 T^{7} + 971955 T^{8} - 184380 T^{9} + 46187 T^{10} - 6226 T^{11} + 995 T^{12} - 45 T^{13} + 20 T^{14} - 2 T^{15} + T^{16} \)
$17$ \( 97846342416 + 19657854576 T + 15459159744 T^{2} + 4803275082 T^{3} + 1930076341 T^{4} + 92753798 T^{5} + 115502484 T^{6} - 2361738 T^{7} + 3660776 T^{8} - 235492 T^{9} + 118641 T^{10} - 1561 T^{11} + 3594 T^{12} - 308 T^{13} + 51 T^{14} - 5 T^{15} + T^{16} \)
$19$ \( 331776 - 2515968 T + 7375104 T^{2} + 1977096 T^{3} + 21231289 T^{4} + 5386455 T^{5} + 11298737 T^{6} + 6359687 T^{7} + 2186988 T^{8} + 390467 T^{9} + 122042 T^{10} + 15305 T^{11} + 1784 T^{12} - 9 T^{13} + 19 T^{14} + 2 T^{15} + T^{16} \)
$23$ \( 1881824400 + 2687391000 T + 7123347900 T^{2} + 3636814350 T^{3} + 4210608775 T^{4} - 590379825 T^{5} + 148665665 T^{6} + 12865555 T^{7} + 6166210 T^{8} + 774285 T^{9} + 319350 T^{10} + 35515 T^{11} + 4746 T^{12} - 91 T^{13} + T^{14} + 4 T^{15} + T^{16} \)
$29$ \( 104976 - 279936 T + 3300264 T^{2} - 19202382 T^{3} + 79994221 T^{4} - 27416788 T^{5} + 573318019 T^{6} - 195839417 T^{7} + 53481071 T^{8} - 7376403 T^{9} + 676831 T^{10} - 44644 T^{11} + 8569 T^{12} - 472 T^{13} + 26 T^{14} - 5 T^{15} + T^{16} \)
$31$ \( 852891037441 + 27512614111 T - 48812702455 T^{2} - 1431457550 T^{3} + 1496104020 T^{4} + 152619293 T^{5} - 47850112 T^{6} - 3952190 T^{7} + 1742455 T^{8} - 127490 T^{9} - 49792 T^{10} + 5123 T^{11} + 1620 T^{12} - 50 T^{13} - 55 T^{14} + T^{15} + T^{16} \)
$37$ \( ( -223421 - 137404 T + 32060 T^{2} + 26438 T^{3} + 325 T^{4} - 1322 T^{5} - 118 T^{6} + 10 T^{7} + T^{8} )^{2} \)
$41$ \( 3088876550400 + 918901756800 T + 326938602240 T^{2} + 91658991840 T^{3} + 33154340176 T^{4} + 3243619432 T^{5} + 1369899176 T^{6} + 251595910 T^{7} + 85663685 T^{8} + 15672912 T^{9} + 2504949 T^{10} + 305327 T^{11} + 40220 T^{12} + 3345 T^{13} + 261 T^{14} + 17 T^{15} + T^{16} \)
$43$ \( 210996910336 - 204483412416 T + 273098326800 T^{2} - 119025473820 T^{3} + 64478771405 T^{4} - 11053139193 T^{5} + 3006014983 T^{6} + 183954050 T^{7} + 8556840 T^{8} - 6488735 T^{9} + 951503 T^{10} + 59082 T^{11} + 11800 T^{12} - 620 T^{13} - 20 T^{14} + 4 T^{15} + T^{16} \)
$47$ \( 153836528400 + 25545288600 T + 123746788800 T^{2} - 29318354400 T^{3} + 26190010225 T^{4} - 1992767275 T^{5} + 64782035 T^{6} + 55978380 T^{7} + 46773750 T^{8} + 5067865 T^{9} + 694625 T^{10} - 4030 T^{11} + 5466 T^{12} + 462 T^{13} + 74 T^{14} - 2 T^{15} + T^{16} \)
$53$ \( 17398665216 - 126988729344 T + 362981905920 T^{2} - 91483303680 T^{3} + 71300346640 T^{4} - 15476050752 T^{5} + 6203264688 T^{6} - 1142577610 T^{7} + 225665715 T^{8} - 32683995 T^{9} + 4263423 T^{10} - 363117 T^{11} + 21870 T^{12} - 5 T^{13} + 15 T^{14} - 9 T^{15} + T^{16} \)
$59$ \( 42855614816400 + 6007846026600 T + 5745582517500 T^{2} - 157422410550 T^{3} + 178408244925 T^{4} - 8967697650 T^{5} + 4504740435 T^{6} - 303381045 T^{7} + 104202115 T^{8} - 10237955 T^{9} + 2633545 T^{10} - 222290 T^{11} + 23131 T^{12} - 88 T^{13} + 14 T^{14} + 3 T^{15} + T^{16} \)
$61$ \( ( 356400 - 467100 T - 222255 T^{2} + 206925 T^{3} - 40090 T^{4} + 1405 T^{5} + 330 T^{6} - 35 T^{7} + T^{8} )^{2} \)
$67$ \( ( 29553984 + 16069320 T + 2470977 T^{2} - 114846 T^{3} - 64070 T^{4} - 4449 T^{5} + 195 T^{6} + 33 T^{7} + T^{8} )^{2} \)
$71$ \( 23444964000000 + 25382732400000 T + 18109625040000 T^{2} + 6929191644000 T^{3} + 1491666840400 T^{4} - 17788230200 T^{5} + 22681953400 T^{6} + 5329877450 T^{7} + 1251186375 T^{8} + 163642725 T^{9} + 19898885 T^{10} + 1769435 T^{11} + 158800 T^{12} + 9685 T^{13} + 555 T^{14} + 25 T^{15} + T^{16} \)
$73$ \( 233420394496 + 725848066048 T + 927480123776 T^{2} + 301348671624 T^{3} + 89071952001 T^{4} + 20477207159 T^{5} + 4408033141 T^{6} + 635668019 T^{7} + 95073056 T^{8} + 9519319 T^{9} + 1306214 T^{10} + 112153 T^{11} + 16584 T^{12} + 991 T^{13} + 259 T^{14} + 10 T^{15} + T^{16} \)
$79$ \( 8008281121 - 87468696336 T + 365698267121 T^{2} + 4975271218 T^{3} + 150687600039 T^{4} - 46660662450 T^{5} + 8966017598 T^{6} - 771141274 T^{7} + 99403903 T^{8} - 15704676 T^{9} + 5204998 T^{10} - 786040 T^{11} + 95164 T^{12} - 6778 T^{13} + 396 T^{14} - 14 T^{15} + T^{16} \)
$83$ \( 129413376 - 2922175872 T + 26457627456 T^{2} - 45812049216 T^{3} + 31431785776 T^{4} + 2391581984 T^{5} + 5288373976 T^{6} + 330151904 T^{7} + 169809521 T^{8} + 15638204 T^{9} + 3219804 T^{10} + 31628 T^{11} + 42534 T^{12} - 1944 T^{13} + 279 T^{14} - 20 T^{15} + T^{16} \)
$89$ \( 1992678175051776 - 1016182561397760 T + 267257691281664 T^{2} - 13948482999744 T^{3} + 6148830526704 T^{4} + 1392871188168 T^{5} + 359836830144 T^{6} + 33096601644 T^{7} + 3422920021 T^{8} + 113750689 T^{9} + 15368016 T^{10} + 882739 T^{11} + 102231 T^{12} + 4659 T^{13} + 456 T^{14} + 13 T^{15} + T^{16} \)
$97$ \( 15953601686416 + 24916345847048 T + 64298624972136 T^{2} + 23357843892524 T^{3} + 2587259823781 T^{4} - 513703069761 T^{5} + 72193698241 T^{6} + 1828816974 T^{7} + 232010206 T^{8} - 168794431 T^{9} + 47629399 T^{10} - 6183397 T^{11} + 584654 T^{12} - 37634 T^{13} + 1789 T^{14} - 55 T^{15} + T^{16} \)
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