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.

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

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

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

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8046))\):

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