Properties

Label 8046.2.a.t
Level 8046
Weight 2
Character orbit 8046.a
Self dual yes
Analytic conductor 64.248
Analytic rank 0
Dimension 16
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8046.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.2476334663\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + \beta_{1} q^{5} + \beta_{7} q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + \beta_{1} q^{5} + \beta_{7} q^{7} + q^{8} + \beta_{1} q^{10} -\beta_{4} q^{11} + \beta_{9} q^{13} + \beta_{7} q^{14} + q^{16} -\beta_{2} q^{17} + ( 1 + \beta_{14} ) q^{19} + \beta_{1} q^{20} -\beta_{4} q^{22} + ( 1 + \beta_{15} ) q^{23} + ( 2 + \beta_{1} + \beta_{3} + \beta_{4} - \beta_{6} ) q^{25} + \beta_{9} q^{26} + \beta_{7} q^{28} -\beta_{5} q^{29} + ( 2 + \beta_{6} - \beta_{11} + \beta_{14} ) q^{31} + q^{32} -\beta_{2} q^{34} + ( 1 + \beta_{1} + \beta_{2} + \beta_{7} - \beta_{9} + \beta_{13} ) q^{35} + ( 1 - \beta_{1} + \beta_{2} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{37} + ( 1 + \beta_{14} ) q^{38} + \beta_{1} q^{40} + ( -\beta_{1} - \beta_{2} + \beta_{6} - \beta_{8} - \beta_{10} - \beta_{13} - \beta_{14} ) q^{41} + ( \beta_{4} + \beta_{11} - \beta_{12} - \beta_{14} - \beta_{15} ) q^{43} -\beta_{4} q^{44} + ( 1 + \beta_{15} ) q^{46} + ( 1 - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} + \beta_{10} + \beta_{13} - \beta_{15} ) q^{47} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} - \beta_{9} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{15} ) q^{49} + ( 2 + \beta_{1} + \beta_{3} + \beta_{4} - \beta_{6} ) q^{50} + \beta_{9} q^{52} + ( 2 \beta_{1} - 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{53} + ( -2 \beta_{2} - \beta_{4} + 2 \beta_{9} - \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{55} + \beta_{7} q^{56} -\beta_{5} q^{58} + ( -1 + 2 \beta_{1} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{15} ) q^{59} + ( 2 - \beta_{1} - \beta_{3} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{12} + \beta_{13} + \beta_{14} ) q^{61} + ( 2 + \beta_{6} - \beta_{11} + \beta_{14} ) q^{62} + q^{64} + ( 3 - \beta_{1} - \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{65} + ( 2 + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} - \beta_{12} + \beta_{13} ) q^{67} -\beta_{2} q^{68} + ( 1 + \beta_{1} + \beta_{2} + \beta_{7} - \beta_{9} + \beta_{13} ) q^{70} + ( 3 \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{6} - \beta_{8} - \beta_{9} + 2 \beta_{11} - \beta_{13} ) q^{71} + ( -1 + \beta_{2} - \beta_{4} + \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{14} ) q^{73} + ( 1 - \beta_{1} + \beta_{2} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{74} + ( 1 + \beta_{14} ) q^{76} + ( -\beta_{1} - \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{10} - 2 \beta_{11} + \beta_{13} ) q^{77} + ( 3 - \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{13} - \beta_{14} ) q^{79} + \beta_{1} q^{80} + ( -\beta_{1} - \beta_{2} + \beta_{6} - \beta_{8} - \beta_{10} - \beta_{13} - \beta_{14} ) q^{82} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{11} + \beta_{12} - 3 \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{83} + ( 3 - \beta_{2} + \beta_{5} - \beta_{7} - \beta_{14} ) q^{85} + ( \beta_{4} + \beta_{11} - \beta_{12} - \beta_{14} - \beta_{15} ) q^{86} -\beta_{4} q^{88} + ( -\beta_{1} - \beta_{2} - \beta_{3} + \beta_{6} + \beta_{8} - \beta_{10} - \beta_{11} + \beta_{13} - \beta_{15} ) q^{89} + ( 1 - 2 \beta_{1} - \beta_{3} + \beta_{4} + \beta_{6} + \beta_{8} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{14} - \beta_{15} ) q^{91} + ( 1 + \beta_{15} ) q^{92} + ( 1 - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} + \beta_{10} + \beta_{13} - \beta_{15} ) q^{94} + ( 4 - \beta_{1} + \beta_{2} + \beta_{4} - \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + 2 \beta_{13} + 2 \beta_{14} ) q^{95} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{10} + \beta_{11} + \beta_{14} + \beta_{15} ) q^{97} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} - \beta_{9} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{15} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 16q^{2} + 16q^{4} + 4q^{5} + 6q^{7} + 16q^{8} + O(q^{10}) \) \( 16q + 16q^{2} + 16q^{4} + 4q^{5} + 6q^{7} + 16q^{8} + 4q^{10} + 6q^{11} + 6q^{13} + 6q^{14} + 16q^{16} + q^{17} + 10q^{19} + 4q^{20} + 6q^{22} + 10q^{23} + 28q^{25} + 6q^{26} + 6q^{28} + 6q^{29} + 21q^{31} + 16q^{32} + q^{34} + 16q^{35} + 17q^{37} + 10q^{38} + 4q^{40} - 4q^{41} + 16q^{43} + 6q^{44} + 10q^{46} + 25q^{47} + 36q^{49} + 28q^{50} + 6q^{52} + 14q^{53} + 19q^{55} + 6q^{56} + 6q^{58} + 6q^{59} + 23q^{61} + 21q^{62} + 16q^{64} + 20q^{65} + 22q^{67} + q^{68} + 16q^{70} + 10q^{71} + 16q^{73} + 17q^{74} + 10q^{76} - 2q^{77} + 37q^{79} + 4q^{80} - 4q^{82} + 33q^{83} + 43q^{85} + 16q^{86} + 6q^{88} - 3q^{89} + 28q^{91} + 10q^{92} + 25q^{94} + 14q^{95} - 3q^{97} + 36q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 4 x^{15} - 46 x^{14} + 192 x^{13} + 752 x^{12} - 3378 x^{11} - 5277 x^{10} + 27132 x^{9} + 15173 x^{8} - 102466 x^{7} - 16612 x^{6} + 186120 x^{5} + 117 x^{4} - 156722 x^{3} + 10545 x^{2} + 48580 x - 4260\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(3510940170587622749 \nu^{15} + 54063165980008993776 \nu^{14} - 446649497147977730525 \nu^{13} - 2362131666061751543705 \nu^{12} + 15997483119600244134128 \nu^{11} + 34998650290367146864052 \nu^{10} - 243888108418249000106827 \nu^{9} - 195463528202988737623753 \nu^{8} + 1727300021212221508862313 \nu^{7} + 228054816246547158585145 \nu^{6} - 5437010618291029626046681 \nu^{5} + 554714737192770751190957 \nu^{4} + 7122806020940373557724842 \nu^{3} - 1084994680959541801882910 \nu^{2} - 3153158086558842160081373 \nu + 360018488385231990868518\)\()/ \)\(20\!\cdots\!86\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-4754997191594517542 \nu^{15} - 347344005317852013 \nu^{14} + 306419427900463382768 \nu^{13} - 48008625773567493889 \nu^{12} - 7691277146709412834256 \nu^{11} + 2648542829239608234472 \nu^{10} + 94740118210314987568708 \nu^{9} - 44361757837299721428236 \nu^{8} - 592140964809809036745717 \nu^{7} + 297889852475890994427515 \nu^{6} + 1759946342541780069646072 \nu^{5} - 733393797626012409826817 \nu^{4} - 2185973827396544451124070 \nu^{3} + 633969993190004797338812 \nu^{2} + 751807907384308053725495 \nu - 182279319431412172187472\)\()/ \)\(20\!\cdots\!86\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-13448299103641896776 \nu^{15} + 49442470348036594881 \nu^{14} + 611620029613652072375 \nu^{13} - 2310930074346787417717 \nu^{12} - 9795264030036985408166 \nu^{11} + 38805911132736045482119 \nu^{10} + 65932543156139201891047 \nu^{9} - 284786378584871819975426 \nu^{8} - 171250857889975406280714 \nu^{7} + 882011258020087358471534 \nu^{6} + 148367828957537370265324 \nu^{5} - 1038342224814970865346725 \nu^{4} + 10571082055781807851228 \nu^{3} + 351435353595757394132864 \nu^{2} + 23777040021564809954009 \nu - 24726085520350837194582\)\()/ \)\(20\!\cdots\!86\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-4870778526700468213 \nu^{15} + 29345633697477809283 \nu^{14} + 186177135704038830469 \nu^{13} - 1355836877618922716261 \nu^{12} - 1907520179085753062716 \nu^{11} + 22333151733677036755034 \nu^{10} - 3363097415090753015800 \nu^{9} - 158543878194815233550905 \nu^{8} + 134268425017258732238220 \nu^{7} + 461022158906373649936951 \nu^{6} - 536113354289406958185283 \nu^{5} - 484007898366371827581613 \nu^{4} + 691577050241824423415186 \nu^{3} + 55993119859889147955691 \nu^{2} - 254911501489191216163955 \nu + 67706575180520972434908\)\()/ \)\(68\!\cdots\!62\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-18203296295236414318 \nu^{15} + 49095126342718742868 \nu^{14} + 918039457514115455143 \nu^{13} - 2358938700120354911606 \nu^{12} - 17486541176746398242422 \nu^{11} + 41454453961975653716591 \nu^{10} + 160672661366454189459755 \nu^{9} - 329148136422171541403662 \nu^{8} - 763391822699784443026431 \nu^{7} + 1179901110495978352899049 \nu^{6} + 1908314171499317439911396 \nu^{5} - 1771736022440983275173542 \nu^{4} - 2175402745340762643272842 \nu^{3} + 964856999712590496548590 \nu^{2} + 796133294479044558602590 \nu - 63166975439561144920452\)\()/ \)\(20\!\cdots\!86\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-18680626296136902922 \nu^{15} + 27882442447801133532 \nu^{14} + 1011578402276866346314 \nu^{13} - 1384487095617726597038 \nu^{12} - 21240405260219974891711 \nu^{11} + 25585616947736601569180 \nu^{10} + 221654321118125671835564 \nu^{9} - 218954288667199319945671 \nu^{8} - 1220461390768346749202601 \nu^{7} + 873829480776839372239708 \nu^{6} + 3462517067109938316552731 \nu^{5} - 1464583912044165045976069 \nu^{4} - 4484485896875198791434034 \nu^{3} + 862702954632930867937294 \nu^{2} + 2029463639691858322897504 \nu - 81313365316627588069470\)\()/ \)\(20\!\cdots\!86\)\( \)
\(\beta_{8}\)\(=\)\((\)\(20692202309464622107 \nu^{15} - 106602788567369842998 \nu^{14} - 836188745848262084971 \nu^{13} + 4943127853569302990186 \nu^{12} + 10258360725409116825403 \nu^{11} - 82033368154198672876421 \nu^{10} - 23050035687896949522359 \nu^{9} + 592814066698747032935005 \nu^{8} - 280418240210990345848149 \nu^{7} - 1822311195341477401884556 \nu^{6} + 1264055711338459244767600 \nu^{5} + 2375202383079278267942287 \nu^{4} - 1373061456327431726391002 \nu^{3} - 1262886324364991917297063 \nu^{2} + 195720124585722623829971 \nu + 210691747629644245224078\)\()/ \)\(20\!\cdots\!86\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-17659463270000847856 \nu^{15} + 112674849465302620863 \nu^{14} + 625250951955638454235 \nu^{13} - 5157765380438916256226 \nu^{12} - 4637604499564731844909 \nu^{11} + 83614199760564082451732 \nu^{10} - 49199888289279349400050 \nu^{9} - 576194342513877050219725 \nu^{8} + 744268046981926614860247 \nu^{7} + 1577320504832949407724766 \nu^{6} - 2679422985981498134591914 \nu^{5} - 1534852291258759158021619 \nu^{4} + 3375588418562574107529374 \nu^{3} + 221895584554016563562452 \nu^{2} - 1291392256003253928436280 \nu + 160376672951279446133040\)\()/ \)\(10\!\cdots\!43\)\( \)
\(\beta_{10}\)\(=\)\((\)\(41446131301225921876 \nu^{15} - 221309560807863248148 \nu^{14} - 1679523494623171608055 \nu^{13} + 10310828298257771834060 \nu^{12} + 20675319127343125022560 \nu^{11} - 172410940151103852345224 \nu^{10} - 45686082550627449865763 \nu^{9} + 1261675924020898281944968 \nu^{8} - 602590802585193783419064 \nu^{7} - 3953209669599953752126132 \nu^{6} + 2934260670560762903037718 \nu^{5} + 5092709888365352169739327 \nu^{4} - 4123282599896150901155483 \nu^{3} - 1973874545005638490612996 \nu^{2} + 1748967802382404519286900 \nu - 222702689094607408460388\)\()/ \)\(20\!\cdots\!86\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-4866438783562677883 \nu^{15} + 22246285566973702038 \nu^{14} + 206677531834818634582 \nu^{13} - 1034291517650156260196 \nu^{12} - 2873478397933583498167 \nu^{11} + 17237303226850460052347 \nu^{10} + 12944042514816404321966 \nu^{9} - 125396907799812356302846 \nu^{8} + 13855279387412214435291 \nu^{7} + 388911767651873565294670 \nu^{6} - 159291675933773913033145 \nu^{5} - 500357424058104567886375 \nu^{4} + 237638633584666013966447 \nu^{3} + 220427606220122390496928 \nu^{2} - 97912592885920732283882 \nu + 5016583647462582055500\)\()/ \)\(22\!\cdots\!54\)\( \)
\(\beta_{12}\)\(=\)\((\)\(54161807572233779206 \nu^{15} - 211323136030971909396 \nu^{14} - 2440234131471514014661 \nu^{13} + 9831076057812548631803 \nu^{12} + 38559865071831350975326 \nu^{11} - 163947574382780456645540 \nu^{10} - 255546723242544104710352 \nu^{9} + 1191706119094707520793047 \nu^{8} + 677961313552995903872778 \nu^{7} - 3658570638895919211881203 \nu^{6} - 902767169679213294116693 \nu^{5} + 4443753843270833181423163 \nu^{4} + 789033001420441322697109 \nu^{3} - 1689131766885666756411682 \nu^{2} - 360276986372978627883406 \nu - 51037260949345635209202\)\()/ \)\(20\!\cdots\!86\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-66989303151198893695 \nu^{15} + 295673683157363926320 \nu^{14} + 2887766152022947056667 \nu^{13} - 13761486284723378266076 \nu^{12} - 41549825539123589433571 \nu^{11} + 229720788436435651586402 \nu^{10} + 211722474721151759605196 \nu^{9} - 1674991116134627564037121 \nu^{8} - 58602109763146052505162 \nu^{7} + 5204951215721024369837146 \nu^{6} - 1372098166589069633823307 \nu^{5} - 6642135671264674795050286 \nu^{4} + 2047894553309736616963550 \nu^{3} + 2892533739419266025280514 \nu^{2} - 653446952062992558730487 \nu - 58079592260760411903774\)\()/ \)\(20\!\cdots\!86\)\( \)
\(\beta_{14}\)\(=\)\((\)\(68664277207807360306 \nu^{15} - 279054956636323783530 \nu^{14} - 3042629676831347235895 \nu^{13} + 13036364240141061818840 \nu^{12} + 46378967789974608079609 \nu^{11} - 218945689175130753028484 \nu^{10} - 278577861856521048825758 \nu^{9} + 1612814708289637865808445 \nu^{8} + 483773456373879948239127 \nu^{7} - 5097504700481493617954293 \nu^{6} + 267412040955872840085178 \nu^{5} + 6580240720692693617903482 \nu^{4} - 1098342088884409885459382 \nu^{3} - 2799909864751410264156889 \nu^{2} + 548416957297495466414102 \nu + 1016248819176872185674\)\()/ \)\(20\!\cdots\!86\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-26736485233082331256 \nu^{15} + 123323809602502233672 \nu^{14} + 1125670957927427137738 \nu^{13} - 5714336214752494315817 \nu^{12} - 15316729258858068025048 \nu^{11} + 94653873404939315892413 \nu^{10} + 62914020543105248601287 \nu^{9} - 680024491140876610871542 \nu^{8} + 140458911714210100740333 \nu^{7} + 2045371468023769292966830 \nu^{6} - 1103359887788440174201318 \nu^{5} - 2438720840884623489589888 \nu^{4} + 1656608352161068462963655 \nu^{3} + 891878012826428491401091 \nu^{2} - 705427248925333274689337 \nu + 48063680489732423252586\)\()/ \)\(68\!\cdots\!62\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{6} + \beta_{4} + \beta_{3} + \beta_{1} + 7\)
\(\nu^{3}\)\(=\)\(-\beta_{15} + 2 \beta_{11} + \beta_{10} + \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} + 13 \beta_{1} - 2\)
\(\nu^{4}\)\(=\)\(-4 \beta_{15} + 5 \beta_{14} + 9 \beta_{13} - 4 \beta_{12} + 2 \beta_{11} - 5 \beta_{9} + 3 \beta_{8} - \beta_{7} - 18 \beta_{6} + 2 \beta_{5} + 19 \beta_{4} + 15 \beta_{3} + 4 \beta_{2} + 13 \beta_{1} + 93\)
\(\nu^{5}\)\(=\)\(-24 \beta_{15} + 3 \beta_{14} - 8 \beta_{13} + 57 \beta_{11} + 21 \beta_{10} + \beta_{9} - 2 \beta_{8} + 2 \beta_{7} + 19 \beta_{6} + 23 \beta_{5} + 16 \beta_{4} + 21 \beta_{3} + 20 \beta_{2} + 200 \beta_{1} - 53\)
\(\nu^{6}\)\(=\)\(-122 \beta_{15} + 159 \beta_{14} + 281 \beta_{13} - 131 \beta_{12} + 45 \beta_{11} - 4 \beta_{10} - 157 \beta_{9} + 78 \beta_{8} - 37 \beta_{7} - 310 \beta_{6} + 47 \beta_{5} + 357 \beta_{4} + 234 \beta_{3} + 131 \beta_{2} + 166 \beta_{1} + 1472\)
\(\nu^{7}\)\(=\)\(-489 \beta_{15} + 120 \beta_{14} - 245 \beta_{13} - 10 \beta_{12} + 1261 \beta_{11} + 382 \beta_{10} + 27 \beta_{9} - 63 \beta_{8} + 73 \beta_{7} + 341 \beta_{6} + 432 \beta_{5} + 532 \beta_{4} + 422 \beta_{3} + 389 \beta_{2} + 3319 \beta_{1} - 1063\)
\(\nu^{8}\)\(=\)\(-2798 \beta_{15} + 3772 \beta_{14} + 6527 \beta_{13} - 3144 \beta_{12} + 847 \beta_{11} - 169 \beta_{10} - 3700 \beta_{9} + 1621 \beta_{8} - 861 \beta_{7} - 5452 \beta_{6} + 929 \beta_{5} + 6726 \beta_{4} + 3887 \beta_{3} + 3178 \beta_{2} + 2317 \beta_{1} + 25073\)
\(\nu^{9}\)\(=\)\(-9469 \beta_{15} + 3260 \beta_{14} - 5513 \beta_{13} - 323 \beta_{12} + 25574 \beta_{11} + 6832 \beta_{10} + 550 \beta_{9} - 1446 \beta_{8} + 1901 \beta_{7} + 6147 \beta_{6} + 7835 \beta_{5} + 12914 \beta_{4} + 8377 \beta_{3} + 7550 \beta_{2} + 57730 \beta_{1} - 19466\)
\(\nu^{10}\)\(=\)\(-57976 \beta_{15} + 80254 \beta_{14} + 136295 \beta_{13} - 67183 \beta_{12} + 15744 \beta_{11} - 4571 \beta_{10} - 78485 \beta_{9} + 31418 \beta_{8} - 17499 \beta_{7} - 98040 \beta_{6} + 17808 \beta_{5} + 126778 \beta_{4} + 67744 \beta_{3} + 68611 \beta_{2} + 36089 \beta_{1} + 444808\)
\(\nu^{11}\)\(=\)\(-179989 \beta_{15} + 76046 \beta_{14} - 110725 \beta_{13} - 7806 \beta_{12} + 499669 \beta_{11} + 123118 \beta_{10} + 9595 \beta_{9} - 29317 \beta_{8} + 42671 \beta_{7} + 111124 \beta_{6} + 142422 \beta_{5} + 279211 \beta_{4} + 164467 \beta_{3} + 145881 \beta_{2} + 1035287 \beta_{1} - 344176\)
\(\nu^{12}\)\(=\)\(-1148453 \beta_{15} + 1620472 \beta_{14} + 2709979 \beta_{13} - 1358174 \beta_{12} + 298719 \beta_{11} - 102964 \beta_{10} - 1582246 \beta_{9} + 593377 \beta_{8} - 338031 \beta_{7} - 1791108 \beta_{6} + 340646 \beta_{5} + 2390587 \beta_{4} + 1219457 \beta_{3} + 1398897 \beta_{2} + 619749 \beta_{1} + 8080758\)
\(\nu^{13}\)\(=\)\(-3403966 \beta_{15} + 1646147 \beta_{14} - 2103514 \beta_{13} - 177729 \beta_{12} + 9591388 \beta_{11} + 2243295 \beta_{10} + 144851 \beta_{9} - 559047 \beta_{8} + 885346 \beta_{7} + 2008284 \beta_{6} + 2614828 \beta_{5} + 5717809 \beta_{4} + 3201687 \beta_{3} + 2810165 \beta_{2} + 18928514 \beta_{1} - 5997125\)
\(\nu^{14}\)\(=\)\(-22261980 \beta_{15} + 31813644 \beta_{14} + 52544664 \beta_{13} - 26656843 \beta_{12} + 5795859 \beta_{11} - 2115034 \beta_{10} - 31024473 \beta_{9} + 11102379 \beta_{8} - 6395524 \beta_{7} - 33057569 \beta_{6} + 6531211 \beta_{5} + 45111529 \beta_{4} + 22406504 \beta_{3} + 27649957 \beta_{2} + 11417176 \beta_{1} + 148849998\)
\(\nu^{15}\)\(=\)\(-64406875 \beta_{15} + 34210426 \beta_{14} - 38709000 \beta_{13} - 4036107 \beta_{12} + 182490569 \beta_{11} + 41255743 \beta_{10} + 1760267 \beta_{9} - 10312043 \beta_{8} + 17559892 \beta_{7} + 36231695 \beta_{6} + 48468680 \beta_{5} + 113813010 \beta_{4} + 61957020 \beta_{3} + 54069494 \beta_{2} + 350235205 \beta_{1} - 103774387\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−4.31154
−3.47967
−3.23178
−1.68422
−1.23039
−1.16198
−0.859599
0.0881948
0.939377
1.06521
1.31779
2.46314
3.09186
3.23488
3.39614
4.36260
1.00000 0 1.00000 −4.31154 0 −0.737290 1.00000 0 −4.31154
1.2 1.00000 0 1.00000 −3.47967 0 2.01581 1.00000 0 −3.47967
1.3 1.00000 0 1.00000 −3.23178 0 3.10807 1.00000 0 −3.23178
1.4 1.00000 0 1.00000 −1.68422 0 −3.55865 1.00000 0 −1.68422
1.5 1.00000 0 1.00000 −1.23039 0 0.0132875 1.00000 0 −1.23039
1.6 1.00000 0 1.00000 −1.16198 0 1.20200 1.00000 0 −1.16198
1.7 1.00000 0 1.00000 −0.859599 0 −4.94866 1.00000 0 −0.859599
1.8 1.00000 0 1.00000 0.0881948 0 4.92686 1.00000 0 0.0881948
1.9 1.00000 0 1.00000 0.939377 0 3.42185 1.00000 0 0.939377
1.10 1.00000 0 1.00000 1.06521 0 −4.24065 1.00000 0 1.06521
1.11 1.00000 0 1.00000 1.31779 0 −2.35088 1.00000 0 1.31779
1.12 1.00000 0 1.00000 2.46314 0 0.600028 1.00000 0 2.46314
1.13 1.00000 0 1.00000 3.09186 0 4.92856 1.00000 0 3.09186
1.14 1.00000 0 1.00000 3.23488 0 −1.87212 1.00000 0 3.23488
1.15 1.00000 0 1.00000 3.39614 0 0.912273 1.00000 0 3.39614
1.16 1.00000 0 1.00000 4.36260 0 2.57952 1.00000 0 4.36260
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8046.2.a.t yes 16
3.b odd 2 1 8046.2.a.s 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8046.2.a.s 16 3.b odd 2 1
8046.2.a.t yes 16 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(149\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8046))\):

\(T_{5}^{16} - \cdots\)
\(T_{11}^{16} - \cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{16} \)
$3$ 1
$5$ \( 1 - 4 T + 34 T^{2} - 108 T^{3} + 532 T^{4} - 1398 T^{5} + 5193 T^{6} - 11758 T^{7} + 36623 T^{8} - 74776 T^{9} + 204933 T^{10} - 390140 T^{11} + 955357 T^{12} - 1737372 T^{13} + 3931885 T^{14} - 7341000 T^{15} + 17363740 T^{16} - 36705000 T^{17} + 98297125 T^{18} - 217171500 T^{19} + 597098125 T^{20} - 1219187500 T^{21} + 3202078125 T^{22} - 5841875000 T^{23} + 14305859375 T^{24} - 22964843750 T^{25} + 50712890625 T^{26} - 68261718750 T^{27} + 129882812500 T^{28} - 131835937500 T^{29} + 207519531250 T^{30} - 122070312500 T^{31} + 152587890625 T^{32} \)
$7$ \( 1 - 6 T + 56 T^{2} - 247 T^{3} + 1352 T^{4} - 4889 T^{5} + 20361 T^{6} - 64630 T^{7} + 228423 T^{8} - 665183 T^{9} + 2128726 T^{10} - 5823317 T^{11} + 17590230 T^{12} - 45668170 T^{13} + 133607858 T^{14} - 333266322 T^{15} + 957088079 T^{16} - 2332864254 T^{17} + 6546785042 T^{18} - 15664182310 T^{19} + 42234142230 T^{20} - 97872488819 T^{21} + 250442485174 T^{22} - 547806803369 T^{23} + 1316813138823 T^{24} - 2608053620410 T^{25} + 5751478544889 T^{26} - 9667150446527 T^{27} + 18713420295752 T^{28} - 23931585570529 T^{29} + 37980492079544 T^{30} - 28485369059658 T^{31} + 33232930569601 T^{32} \)
$11$ \( 1 - 6 T + 57 T^{2} - 196 T^{3} + 986 T^{4} - 1601 T^{5} + 8962 T^{6} - 18443 T^{7} + 194975 T^{8} - 702253 T^{9} + 3225278 T^{10} - 7639011 T^{11} + 20694222 T^{12} - 57979410 T^{13} + 285895823 T^{14} - 1481020044 T^{15} + 5344191760 T^{16} - 16291220484 T^{17} + 34593394583 T^{18} - 77170594710 T^{19} + 302984104302 T^{20} - 1230270360561 T^{21} + 5713776718958 T^{22} - 13684924296263 T^{23} + 41794622822975 T^{24} - 43487629265113 T^{25} + 232451199274162 T^{26} - 456783984648211 T^{27} + 3094490379446906 T^{28} - 6766451580210476 T^{29} + 21645740514244737 T^{30} - 25063489016493906 T^{31} + 45949729863572161 T^{32} \)
$13$ \( 1 - 6 T + 105 T^{2} - 512 T^{3} + 5370 T^{4} - 22684 T^{5} + 184517 T^{6} - 698596 T^{7} + 4833808 T^{8} - 16722072 T^{9} + 103048788 T^{10} - 330007070 T^{11} + 1860613029 T^{12} - 5565962602 T^{13} + 29174544319 T^{14} - 81971188970 T^{15} + 403045310618 T^{16} - 1065625456610 T^{17} + 4930497989911 T^{18} - 12228419836594 T^{19} + 53140968721269 T^{20} - 122529315041510 T^{21} + 497396817357492 T^{22} - 1049285219167224 T^{23} + 3943085685015568 T^{24} - 7408260843980308 T^{25} + 25437235340501933 T^{26} - 40653366378335308 T^{27} + 125110717107722970 T^{28} - 155072054575233536 T^{29} + 413424520498425345 T^{30} - 307115358084544542 T^{31} + 665416609183179841 T^{32} \)
$17$ \( 1 - T + 124 T^{2} - 56 T^{3} + 7233 T^{4} + 829 T^{5} + 265803 T^{6} + 183114 T^{7} + 6946713 T^{8} + 8766926 T^{9} + 138378101 T^{10} + 254560431 T^{11} + 2232877007 T^{12} + 5479687692 T^{13} + 32292139716 T^{14} + 99848317065 T^{15} + 500083148268 T^{16} + 1697421390105 T^{17} + 9332428377924 T^{18} + 26921705630796 T^{19} + 186492120501647 T^{20} + 361439409878367 T^{21} + 3340110960976469 T^{22} + 3597408781129198 T^{23} + 48458584900241433 T^{24} + 21715100416871658 T^{25} + 535857226721045547 T^{26} + 28411402039027757 T^{27} + 4214106641882861313 T^{28} - 554656369842732472 T^{29} + 20878850493365715196 T^{30} - 2862423051509815793 T^{31} + 48661191875666868481 T^{32} \)
$19$ \( 1 - 10 T + 233 T^{2} - 1849 T^{3} + 24703 T^{4} - 164799 T^{5} + 1637491 T^{6} - 9505444 T^{7} + 77594475 T^{8} - 400714476 T^{9} + 2829452813 T^{10} - 13199060119 T^{11} + 83133680565 T^{12} - 354150322561 T^{13} + 2027903567399 T^{14} - 7944825738298 T^{15} + 41795280996480 T^{16} - 150951689027662 T^{17} + 732073187831039 T^{18} - 2429117062445899 T^{19} + 10834064384911365 T^{20} - 32682179561595781 T^{21} + 133114100335513253 T^{22} - 358187345504593764 T^{23} + 1317830657795798475 T^{24} - 3067289840727208876 T^{25} + 10039565817552817291 T^{26} - 19197478176167592981 T^{27} + 54675518445691375183 T^{28} - 77755966421713302091 T^{29} + \)\(18\!\cdots\!93\)\( T^{30} - \)\(15\!\cdots\!90\)\( T^{31} + \)\(28\!\cdots\!81\)\( T^{32} \)
$23$ \( 1 - 10 T + 231 T^{2} - 1965 T^{3} + 25974 T^{4} - 193341 T^{5} + 1907808 T^{6} - 12717628 T^{7} + 103446656 T^{8} - 627139540 T^{9} + 4419557421 T^{10} - 24602786647 T^{11} + 154610899217 T^{12} - 794751432473 T^{13} + 4534379053794 T^{14} - 21563767490965 T^{15} + 112936238951801 T^{16} - 495966652292195 T^{17} + 2398686519457026 T^{18} - 9669740678898991 T^{19} + 43266468647784497 T^{20} - 158351973615911921 T^{21} + 654253111804282269 T^{22} - 2135300664611874380 T^{23} + 8101009555384670336 T^{24} - 22906389519696369764 T^{25} + 79033829505489271392 T^{26} - \)\(18\!\cdots\!07\)\( T^{27} + \)\(56\!\cdots\!54\)\( T^{28} - \)\(99\!\cdots\!95\)\( T^{29} + \)\(26\!\cdots\!79\)\( T^{30} - \)\(26\!\cdots\!70\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \)
$29$ \( 1 - 6 T + 200 T^{2} - 802 T^{3} + 18548 T^{4} - 54884 T^{5} + 1178934 T^{6} - 2930047 T^{7} + 59710763 T^{8} - 138920047 T^{9} + 2550922831 T^{10} - 5911371606 T^{11} + 95302405018 T^{12} - 224903846136 T^{13} + 3197686961249 T^{14} - 7542310153573 T^{15} + 97336321600103 T^{16} - 218726994453617 T^{17} + 2689254734410409 T^{18} - 5485179903410904 T^{19} + 67405580323536058 T^{20} - 121249023805035294 T^{21} + 1517348389950141751 T^{22} - 2396353627590466523 T^{23} + 29870095005914399243 T^{24} - 42506619545157035843 T^{25} + \)\(49\!\cdots\!34\)\( T^{26} - \)\(66\!\cdots\!36\)\( T^{27} + \)\(65\!\cdots\!68\)\( T^{28} - \)\(82\!\cdots\!78\)\( T^{29} + \)\(59\!\cdots\!00\)\( T^{30} - \)\(51\!\cdots\!94\)\( T^{31} + \)\(25\!\cdots\!21\)\( T^{32} \)
$31$ \( 1 - 21 T + 406 T^{2} - 5418 T^{3} + 66194 T^{4} - 684138 T^{5} + 6616575 T^{6} - 57766014 T^{7} + 479436430 T^{8} - 3695295727 T^{9} + 27323117273 T^{10} - 190405288761 T^{11} + 1279391847883 T^{12} - 8167450042813 T^{13} + 50413604197098 T^{14} - 296760287069368 T^{15} + 1691074010419373 T^{16} - 9199568899150408 T^{17} + 48447473633411178 T^{18} - 243316504225442083 T^{19} + 1181545238748756043 T^{20} - 5451141763137271911 T^{21} + 24249367156182181913 T^{22} - \)\(10\!\cdots\!97\)\( T^{23} + \)\(40\!\cdots\!30\)\( T^{24} - \)\(15\!\cdots\!94\)\( T^{25} + \)\(54\!\cdots\!75\)\( T^{26} - \)\(17\!\cdots\!78\)\( T^{27} + \)\(52\!\cdots\!34\)\( T^{28} - \)\(13\!\cdots\!38\)\( T^{29} + \)\(30\!\cdots\!26\)\( T^{30} - \)\(49\!\cdots\!71\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \)
$37$ \( 1 - 17 T + 483 T^{2} - 6261 T^{3} + 105177 T^{4} - 1126916 T^{5} + 14321748 T^{6} - 132506150 T^{7} + 1395476111 T^{8} - 11441164930 T^{9} + 104430953608 T^{10} - 770597570200 T^{11} + 6249885305799 T^{12} - 41874523865947 T^{13} + 306138212842601 T^{14} - 1869964669554675 T^{15} + 12422051133229152 T^{16} - 69188692773522975 T^{17} + 419103213381520769 T^{18} - 2121070257381813391 T^{19} + 11713291294601559639 T^{20} - 53436284772253281400 T^{21} + \)\(26\!\cdots\!72\)\( T^{22} - \)\(10\!\cdots\!90\)\( T^{23} + \)\(49\!\cdots\!31\)\( T^{24} - \)\(17\!\cdots\!50\)\( T^{25} + \)\(68\!\cdots\!52\)\( T^{26} - \)\(20\!\cdots\!08\)\( T^{27} + \)\(69\!\cdots\!37\)\( T^{28} - \)\(15\!\cdots\!17\)\( T^{29} + \)\(43\!\cdots\!87\)\( T^{30} - \)\(56\!\cdots\!81\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \)
$41$ \( 1 + 4 T + 321 T^{2} + 653 T^{3} + 49145 T^{4} + 23684 T^{5} + 4950641 T^{6} - 4328241 T^{7} + 371007733 T^{8} - 804399872 T^{9} + 22106598940 T^{10} - 75786353047 T^{11} + 1107775617097 T^{12} - 5037504154475 T^{13} + 49394198868106 T^{14} - 258965630567270 T^{15} + 2065808864035479 T^{16} - 10617590853258070 T^{17} + 83031648297286186 T^{18} - 347189823830571475 T^{19} + 3130309135543635817 T^{20} - 8780318951670194447 T^{21} + \)\(10\!\cdots\!40\)\( T^{22} - \)\(15\!\cdots\!32\)\( T^{23} + \)\(29\!\cdots\!93\)\( T^{24} - \)\(14\!\cdots\!01\)\( T^{25} + \)\(66\!\cdots\!41\)\( T^{26} + \)\(13\!\cdots\!44\)\( T^{27} + \)\(11\!\cdots\!45\)\( T^{28} + \)\(60\!\cdots\!13\)\( T^{29} + \)\(12\!\cdots\!81\)\( T^{30} + \)\(62\!\cdots\!04\)\( T^{31} + \)\(63\!\cdots\!41\)\( T^{32} \)
$43$ \( 1 - 16 T + 487 T^{2} - 6403 T^{3} + 112261 T^{4} - 1259217 T^{5} + 16530621 T^{6} - 162556277 T^{7} + 1763434449 T^{8} - 15506840106 T^{9} + 146014228119 T^{10} - 1164224623425 T^{11} + 9783076145087 T^{12} - 71359400900301 T^{13} + 543949869004930 T^{14} - 3646005619767259 T^{15} + 25444894236690094 T^{16} - 156778241649992137 T^{17} + 1005763307790115570 T^{18} - 5673571887380231607 T^{19} + 33446390507899580687 T^{20} - \)\(17\!\cdots\!75\)\( T^{21} + \)\(92\!\cdots\!31\)\( T^{22} - \)\(42\!\cdots\!42\)\( T^{23} + \)\(20\!\cdots\!49\)\( T^{24} - \)\(81\!\cdots\!11\)\( T^{25} + \)\(35\!\cdots\!29\)\( T^{26} - \)\(11\!\cdots\!19\)\( T^{27} + \)\(44\!\cdots\!61\)\( T^{28} - \)\(11\!\cdots\!29\)\( T^{29} + \)\(35\!\cdots\!63\)\( T^{30} - \)\(50\!\cdots\!12\)\( T^{31} + \)\(13\!\cdots\!01\)\( T^{32} \)
$47$ \( 1 - 25 T + 546 T^{2} - 8571 T^{3} + 120717 T^{4} - 1460380 T^{5} + 16368393 T^{6} - 168330661 T^{7} + 1638532375 T^{8} - 15075498317 T^{9} + 132745613029 T^{10} - 1118441523566 T^{11} + 9048628100263 T^{12} - 70411098454645 T^{13} + 527332944636048 T^{14} - 3811977322200313 T^{15} + 26576590641469672 T^{16} - 179162934143414711 T^{17} + 1164878474701030032 T^{18} - 7310291474856607835 T^{19} + 44154418616919456103 T^{20} - \)\(25\!\cdots\!62\)\( T^{21} + \)\(14\!\cdots\!41\)\( T^{22} - \)\(76\!\cdots\!71\)\( T^{23} + \)\(39\!\cdots\!75\)\( T^{24} - \)\(18\!\cdots\!87\)\( T^{25} + \)\(86\!\cdots\!57\)\( T^{26} - \)\(36\!\cdots\!40\)\( T^{27} + \)\(14\!\cdots\!97\)\( T^{28} - \)\(46\!\cdots\!17\)\( T^{29} + \)\(14\!\cdots\!74\)\( T^{30} - \)\(30\!\cdots\!75\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \)
$53$ \( 1 - 14 T + 355 T^{2} - 4539 T^{3} + 75770 T^{4} - 860747 T^{5} + 11214548 T^{6} - 115534366 T^{7} + 1287431724 T^{8} - 12096832200 T^{9} + 119428141805 T^{10} - 1034460940195 T^{11} + 9274173886709 T^{12} - 74390586646347 T^{13} + 613141761069538 T^{14} - 4574216784025259 T^{15} + 34956397362447065 T^{16} - 242433489553338727 T^{17} + 1722315206844332242 T^{18} - 11075047368148202319 T^{19} + 73177692843773517029 T^{20} - \)\(43\!\cdots\!35\)\( T^{21} + \)\(26\!\cdots\!45\)\( T^{22} - \)\(14\!\cdots\!00\)\( T^{23} + \)\(80\!\cdots\!64\)\( T^{24} - \)\(38\!\cdots\!78\)\( T^{25} + \)\(19\!\cdots\!52\)\( T^{26} - \)\(79\!\cdots\!59\)\( T^{27} + \)\(37\!\cdots\!70\)\( T^{28} - \)\(11\!\cdots\!47\)\( T^{29} + \)\(48\!\cdots\!95\)\( T^{30} - \)\(10\!\cdots\!98\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \)
$59$ \( 1 - 6 T + 421 T^{2} - 3377 T^{3} + 93309 T^{4} - 858993 T^{5} + 14636145 T^{6} - 138417394 T^{7} + 1799435159 T^{8} - 16359859240 T^{9} + 180822933617 T^{10} - 1540190906555 T^{11} + 15224681939107 T^{12} - 121397973697163 T^{13} + 1097841079463113 T^{14} - 8232953788291556 T^{15} + 69074281764740816 T^{16} - 485744273509201804 T^{17} + 3821584797611096353 T^{18} - 24932594439949639777 T^{19} + \)\(18\!\cdots\!27\)\( T^{20} - \)\(11\!\cdots\!45\)\( T^{21} + \)\(76\!\cdots\!97\)\( T^{22} - \)\(40\!\cdots\!60\)\( T^{23} + \)\(26\!\cdots\!39\)\( T^{24} - \)\(11\!\cdots\!66\)\( T^{25} + \)\(74\!\cdots\!45\)\( T^{26} - \)\(25\!\cdots\!87\)\( T^{27} + \)\(16\!\cdots\!29\)\( T^{28} - \)\(35\!\cdots\!83\)\( T^{29} + \)\(26\!\cdots\!81\)\( T^{30} - \)\(21\!\cdots\!94\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \)
$61$ \( 1 - 23 T + 623 T^{2} - 9731 T^{3} + 164697 T^{4} - 2062213 T^{5} + 27459612 T^{6} - 297150828 T^{7} + 3400086161 T^{8} - 33134645582 T^{9} + 340076853532 T^{10} - 3053216330899 T^{11} + 28777635020423 T^{12} - 241106926237677 T^{13} + 2115995875564849 T^{14} - 16655372283408603 T^{15} + 137163094019680348 T^{16} - 1015977709287924783 T^{17} + 7873620652976803129 T^{18} - 54726691224354163137 T^{19} + \)\(39\!\cdots\!43\)\( T^{20} - \)\(25\!\cdots\!99\)\( T^{21} + \)\(17\!\cdots\!52\)\( T^{22} - \)\(10\!\cdots\!22\)\( T^{23} + \)\(65\!\cdots\!41\)\( T^{24} - \)\(34\!\cdots\!48\)\( T^{25} + \)\(19\!\cdots\!12\)\( T^{26} - \)\(89\!\cdots\!93\)\( T^{27} + \)\(43\!\cdots\!37\)\( T^{28} - \)\(15\!\cdots\!11\)\( T^{29} + \)\(61\!\cdots\!43\)\( T^{30} - \)\(13\!\cdots\!23\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \)
$67$ \( 1 - 22 T + 501 T^{2} - 6447 T^{3} + 90397 T^{4} - 897211 T^{5} + 11103847 T^{6} - 105422090 T^{7} + 1265907613 T^{8} - 11337838128 T^{9} + 123146992079 T^{10} - 997535874381 T^{11} + 10143521293443 T^{12} - 78621571092433 T^{13} + 783630161714741 T^{14} - 5897580593587028 T^{15} + 55883270158319620 T^{16} - 395137899770330876 T^{17} + 3517715795937472349 T^{18} - 23646459586473426379 T^{19} + \)\(20\!\cdots\!03\)\( T^{20} - \)\(13\!\cdots\!67\)\( T^{21} + \)\(11\!\cdots\!51\)\( T^{22} - \)\(68\!\cdots\!44\)\( T^{23} + \)\(51\!\cdots\!33\)\( T^{24} - \)\(28\!\cdots\!30\)\( T^{25} + \)\(20\!\cdots\!03\)\( T^{26} - \)\(10\!\cdots\!13\)\( T^{27} + \)\(73\!\cdots\!17\)\( T^{28} - \)\(35\!\cdots\!89\)\( T^{29} + \)\(18\!\cdots\!29\)\( T^{30} - \)\(54\!\cdots\!46\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \)
$71$ \( 1 - 10 T + 348 T^{2} - 3306 T^{3} + 79475 T^{4} - 704716 T^{5} + 13269974 T^{6} - 112792963 T^{7} + 1810263908 T^{8} - 14555945407 T^{9} + 207139838222 T^{10} - 1584359881707 T^{11} + 20430446607892 T^{12} - 147793105273060 T^{13} + 1759580852292030 T^{14} - 12008007115543542 T^{15} + 133043632605308137 T^{16} - 852568505203591482 T^{17} + 8870047076404123230 T^{18} - 52896778101386177660 T^{19} + \)\(51\!\cdots\!52\)\( T^{20} - \)\(28\!\cdots\!57\)\( T^{21} + \)\(26\!\cdots\!62\)\( T^{22} - \)\(13\!\cdots\!37\)\( T^{23} + \)\(11\!\cdots\!88\)\( T^{24} - \)\(51\!\cdots\!53\)\( T^{25} + \)\(43\!\cdots\!74\)\( T^{26} - \)\(16\!\cdots\!36\)\( T^{27} + \)\(13\!\cdots\!75\)\( T^{28} - \)\(38\!\cdots\!66\)\( T^{29} + \)\(28\!\cdots\!88\)\( T^{30} - \)\(58\!\cdots\!10\)\( T^{31} + \)\(41\!\cdots\!21\)\( T^{32} \)
$73$ \( 1 - 16 T + 657 T^{2} - 8691 T^{3} + 193935 T^{4} - 2148045 T^{5} + 33995389 T^{6} - 313704285 T^{7} + 3864817095 T^{8} - 28416856962 T^{9} + 279048539417 T^{10} - 1324790621337 T^{11} + 9344920298771 T^{12} + 29861440907009 T^{13} - 425662097598134 T^{14} + 10097340072817335 T^{15} - 68988484687694674 T^{16} + 737105825315665455 T^{17} - 2268353318100456086 T^{18} + 11616608157321920153 T^{19} + \)\(26\!\cdots\!11\)\( T^{20} - \)\(27\!\cdots\!41\)\( T^{21} + \)\(42\!\cdots\!13\)\( T^{22} - \)\(31\!\cdots\!14\)\( T^{23} + \)\(31\!\cdots\!95\)\( T^{24} - \)\(18\!\cdots\!05\)\( T^{25} + \)\(14\!\cdots\!61\)\( T^{26} - \)\(67\!\cdots\!65\)\( T^{27} + \)\(44\!\cdots\!35\)\( T^{28} - \)\(14\!\cdots\!03\)\( T^{29} + \)\(80\!\cdots\!13\)\( T^{30} - \)\(14\!\cdots\!12\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \)
$79$ \( 1 - 37 T + 1293 T^{2} - 30205 T^{3} + 659418 T^{4} - 11821632 T^{5} + 200229184 T^{6} - 2987108633 T^{7} + 42498417027 T^{8} - 550360270237 T^{9} + 6842422501064 T^{10} - 78818982243014 T^{11} + 875542846416182 T^{12} - 9101674831788723 T^{13} + 91493489547618347 T^{14} - 865201334798351465 T^{15} + 7922241862830505496 T^{16} - 68350905449069765735 T^{17} + \)\(57\!\cdots\!27\)\( T^{18} - \)\(44\!\cdots\!97\)\( T^{19} + \)\(34\!\cdots\!42\)\( T^{20} - \)\(24\!\cdots\!86\)\( T^{21} + \)\(16\!\cdots\!44\)\( T^{22} - \)\(10\!\cdots\!83\)\( T^{23} + \)\(64\!\cdots\!47\)\( T^{24} - \)\(35\!\cdots\!27\)\( T^{25} + \)\(18\!\cdots\!84\)\( T^{26} - \)\(88\!\cdots\!28\)\( T^{27} + \)\(38\!\cdots\!38\)\( T^{28} - \)\(14\!\cdots\!95\)\( T^{29} + \)\(47\!\cdots\!33\)\( T^{30} - \)\(10\!\cdots\!63\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \)
$83$ \( 1 - 33 T + 1122 T^{2} - 24494 T^{3} + 511386 T^{4} - 8622810 T^{5} + 138526414 T^{6} - 1949218058 T^{7} + 26348423747 T^{8} - 325157069944 T^{9} + 3892530023484 T^{10} - 43563085621070 T^{11} + 475900306593478 T^{12} - 4916499499128146 T^{13} + 49636759814357764 T^{14} - 475513781064524147 T^{15} + 4446561469434356760 T^{16} - 39467643828355504201 T^{17} + \)\(34\!\cdots\!96\)\( T^{18} - \)\(28\!\cdots\!02\)\( T^{19} + \)\(22\!\cdots\!38\)\( T^{20} - \)\(17\!\cdots\!10\)\( T^{21} + \)\(12\!\cdots\!96\)\( T^{22} - \)\(88\!\cdots\!88\)\( T^{23} + \)\(59\!\cdots\!27\)\( T^{24} - \)\(36\!\cdots\!74\)\( T^{25} + \)\(21\!\cdots\!86\)\( T^{26} - \)\(11\!\cdots\!70\)\( T^{27} + \)\(54\!\cdots\!46\)\( T^{28} - \)\(21\!\cdots\!22\)\( T^{29} + \)\(82\!\cdots\!38\)\( T^{30} - \)\(20\!\cdots\!31\)\( T^{31} + \)\(50\!\cdots\!81\)\( T^{32} \)
$89$ \( 1 + 3 T + 912 T^{2} + 3691 T^{3} + 403650 T^{4} + 2005170 T^{5} + 116248542 T^{6} + 662095412 T^{7} + 24600752616 T^{8} + 152253821721 T^{9} + 4085470877920 T^{10} + 26287156415963 T^{11} + 553939919786416 T^{12} + 3563810505243158 T^{13} + 62861306538071442 T^{14} + 389212626420763073 T^{15} + 6056913619349440659 T^{16} + 34639923751447913497 T^{17} + \)\(49\!\cdots\!82\)\( T^{18} + \)\(25\!\cdots\!02\)\( T^{19} + \)\(34\!\cdots\!56\)\( T^{20} + \)\(14\!\cdots\!87\)\( T^{21} + \)\(20\!\cdots\!20\)\( T^{22} + \)\(67\!\cdots\!09\)\( T^{23} + \)\(96\!\cdots\!96\)\( T^{24} + \)\(23\!\cdots\!08\)\( T^{25} + \)\(36\!\cdots\!42\)\( T^{26} + \)\(55\!\cdots\!30\)\( T^{27} + \)\(99\!\cdots\!50\)\( T^{28} + \)\(81\!\cdots\!79\)\( T^{29} + \)\(17\!\cdots\!92\)\( T^{30} + \)\(52\!\cdots\!47\)\( T^{31} + \)\(15\!\cdots\!61\)\( T^{32} \)
$97$ \( 1 + 3 T + 627 T^{2} + 2464 T^{3} + 205235 T^{4} + 966481 T^{5} + 46606109 T^{6} + 250233298 T^{7} + 8260841717 T^{8} + 48204496350 T^{9} + 1220378751821 T^{10} + 7404607701747 T^{11} + 156319166890829 T^{12} + 949668455511340 T^{13} + 17772397707910755 T^{14} + 105004232277145469 T^{15} + 1815235835119368212 T^{16} + 10185410530883110493 T^{17} + \)\(16\!\cdots\!95\)\( T^{18} + \)\(86\!\cdots\!20\)\( T^{19} + \)\(13\!\cdots\!49\)\( T^{20} + \)\(63\!\cdots\!79\)\( T^{21} + \)\(10\!\cdots\!09\)\( T^{22} + \)\(38\!\cdots\!50\)\( T^{23} + \)\(64\!\cdots\!37\)\( T^{24} + \)\(19\!\cdots\!66\)\( T^{25} + \)\(34\!\cdots\!41\)\( T^{26} + \)\(69\!\cdots\!93\)\( T^{27} + \)\(14\!\cdots\!35\)\( T^{28} + \)\(16\!\cdots\!28\)\( T^{29} + \)\(40\!\cdots\!63\)\( T^{30} + \)\(18\!\cdots\!79\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \)
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