Properties

Label 8046.2.a.s
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: \( 1 \)
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
4.36260
3.39614
3.23488
3.09186
2.46314
1.31779
1.06521
0.939377
0.0881948
−0.859599
−1.16198
−1.23039
−1.68422
−3.23178
−3.47967
−4.31154
−1.00000 0 1.00000 −4.36260 0 2.57952 −1.00000 0 4.36260
1.2 −1.00000 0 1.00000 −3.39614 0 0.912273 −1.00000 0 3.39614
1.3 −1.00000 0 1.00000 −3.23488 0 −1.87212 −1.00000 0 3.23488
1.4 −1.00000 0 1.00000 −3.09186 0 4.92856 −1.00000 0 3.09186
1.5 −1.00000 0 1.00000 −2.46314 0 0.600028 −1.00000 0 2.46314
1.6 −1.00000 0 1.00000 −1.31779 0 −2.35088 −1.00000 0 1.31779
1.7 −1.00000 0 1.00000 −1.06521 0 −4.24065 −1.00000 0 1.06521
1.8 −1.00000 0 1.00000 −0.939377 0 3.42185 −1.00000 0 0.939377
1.9 −1.00000 0 1.00000 −0.0881948 0 4.92686 −1.00000 0 0.0881948
1.10 −1.00000 0 1.00000 0.859599 0 −4.94866 −1.00000 0 −0.859599
1.11 −1.00000 0 1.00000 1.16198 0 1.20200 −1.00000 0 −1.16198
1.12 −1.00000 0 1.00000 1.23039 0 0.0132875 −1.00000 0 −1.23039
1.13 −1.00000 0 1.00000 1.68422 0 −3.55865 −1.00000 0 −1.68422
1.14 −1.00000 0 1.00000 3.23178 0 3.10807 −1.00000 0 −3.23178
1.15 −1.00000 0 1.00000 3.47967 0 2.01581 −1.00000 0 −3.47967
1.16 −1.00000 0 1.00000 4.31154 0 −0.737290 −1.00000 0 −4.31154
\(n\): 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.

Atkin-Lehner signs

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

Hecke kernels

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

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