Properties

Label 8036.2.a.r
Level 8036
Weight 2
Character orbit 8036.a
Self dual Yes
Analytic conductor 64.168
Analytic rank 0
Dimension 15
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8036.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.1677830643\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{3} \) \( + \beta_{3} q^{5} \) \( + ( 2 + \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{3} \) \( + \beta_{3} q^{5} \) \( + ( 2 + \beta_{2} ) q^{9} \) \( + ( 1 - \beta_{3} + \beta_{5} - \beta_{12} ) q^{11} \) \( -\beta_{14} q^{13} \) \( + ( \beta_{2} + \beta_{5} + \beta_{8} - \beta_{12} + \beta_{14} ) q^{15} \) \( + \beta_{9} q^{17} \) \( + ( -1 + \beta_{2} - \beta_{3} - \beta_{10} + \beta_{11} ) q^{19} \) \( + ( \beta_{3} + \beta_{4} - \beta_{7} + \beta_{8} + \beta_{10} - \beta_{11} ) q^{23} \) \( + ( 2 - \beta_{3} - \beta_{4} + \beta_{5} ) q^{25} \) \( + ( -1 + \beta_{1} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{27} \) \( + ( 1 + \beta_{1} - \beta_{5} - \beta_{6} ) q^{29} \) \( + ( -2 + \beta_{1} - \beta_{3} - \beta_{12} ) q^{31} \) \( + ( 1 + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{10} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{33} \) \( + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{7} + \beta_{10} - \beta_{12} ) q^{37} \) \( + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} ) q^{39} \) \(+ q^{41}\) \( + ( 1 + \beta_{2} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{10} + \beta_{12} + \beta_{13} ) q^{43} \) \( + ( 1 - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{9} + \beta_{10} + \beta_{13} ) q^{45} \) \( + ( 2 + \beta_{3} + \beta_{5} + \beta_{6} - \beta_{12} - \beta_{13} ) q^{47} \) \( + ( 3 + 2 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{12} + \beta_{14} ) q^{51} \) \( + ( 1 - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{13} ) q^{53} \) \( + ( -2 + 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + \beta_{4} + 2 \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{12} - \beta_{13} + \beta_{14} ) q^{55} \) \( + ( 2 + \beta_{1} - 2 \beta_{8} - \beta_{10} + \beta_{11} - \beta_{13} - \beta_{14} ) q^{57} \) \( + ( 1 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{6} + \beta_{10} - \beta_{13} ) q^{59} \) \( + ( -\beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{8} + \beta_{10} - \beta_{11} - \beta_{13} ) q^{61} \) \( + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + 3 \beta_{7} - \beta_{9} + 2 \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{65} \) \( + ( 1 - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{12} - \beta_{14} ) q^{67} \) \( + ( -2 - \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{6} + \beta_{7} - \beta_{9} + 2 \beta_{10} + \beta_{11} - 2 \beta_{12} + \beta_{13} + \beta_{14} ) q^{69} \) \( + ( -\beta_{1} - \beta_{2} + \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{71} \) \( + ( 1 + 2 \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{73} \) \( + ( 4 + 3 \beta_{1} + 2 \beta_{2} + \beta_{3} - 3 \beta_{6} + \beta_{8} - \beta_{10} - 3 \beta_{11} + \beta_{12} + \beta_{14} ) q^{75} \) \( + ( 1 - 3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} - 2 \beta_{8} - \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{13} ) q^{79} \) \( + ( 2 + \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} - 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{13} + \beta_{14} ) q^{81} \) \( + ( -1 - \beta_{2} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{11} - \beta_{12} - \beta_{13} ) q^{83} \) \( + ( 3 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{5} - \beta_{6} + 2 \beta_{8} - \beta_{10} - 3 \beta_{11} + \beta_{12} - \beta_{13} ) q^{85} \) \( + ( 2 - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} + 3 \beta_{11} - \beta_{12} + \beta_{13} ) q^{87} \) \( + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{8} + \beta_{10} + \beta_{13} + \beta_{14} ) q^{89} \) \( + ( 4 - 3 \beta_{1} - \beta_{2} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{11} + \beta_{12} + \beta_{13} - 2 \beta_{14} ) q^{93} \) \( + ( -2 + 3 \beta_{1} - \beta_{2} - \beta_{5} - 3 \beta_{6} + \beta_{8} - \beta_{9} - 2 \beta_{11} + \beta_{12} ) q^{95} \) \( + ( \beta_{1} + \beta_{3} + \beta_{5} - \beta_{7} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{14} ) q^{97} \) \( + ( \beta_{1} + \beta_{5} + 2 \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - 2 \beta_{12} - 3 \beta_{13} + \beta_{14} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(15q \) \(\mathstrut +\mathstrut q^{3} \) \(\mathstrut -\mathstrut 3q^{5} \) \(\mathstrut +\mathstrut 30q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(15q \) \(\mathstrut +\mathstrut q^{3} \) \(\mathstrut -\mathstrut 3q^{5} \) \(\mathstrut +\mathstrut 30q^{9} \) \(\mathstrut +\mathstrut 9q^{11} \) \(\mathstrut -\mathstrut 7q^{13} \) \(\mathstrut +\mathstrut 2q^{15} \) \(\mathstrut -\mathstrut 3q^{17} \) \(\mathstrut -\mathstrut 7q^{19} \) \(\mathstrut -\mathstrut q^{23} \) \(\mathstrut +\mathstrut 32q^{25} \) \(\mathstrut -\mathstrut 11q^{27} \) \(\mathstrut +\mathstrut 18q^{29} \) \(\mathstrut -\mathstrut 30q^{31} \) \(\mathstrut +\mathstrut 16q^{33} \) \(\mathstrut +\mathstrut 23q^{37} \) \(\mathstrut +\mathstrut 5q^{39} \) \(\mathstrut +\mathstrut 15q^{41} \) \(\mathstrut +\mathstrut 12q^{43} \) \(\mathstrut +\mathstrut 13q^{45} \) \(\mathstrut +\mathstrut 16q^{47} \) \(\mathstrut +\mathstrut 29q^{51} \) \(\mathstrut +\mathstrut 33q^{53} \) \(\mathstrut -\mathstrut 37q^{55} \) \(\mathstrut +\mathstrut 16q^{57} \) \(\mathstrut +\mathstrut 10q^{59} \) \(\mathstrut -\mathstrut q^{61} \) \(\mathstrut +\mathstrut 16q^{65} \) \(\mathstrut +\mathstrut 20q^{67} \) \(\mathstrut -\mathstrut 21q^{69} \) \(\mathstrut +\mathstrut 5q^{71} \) \(\mathstrut +\mathstrut 3q^{73} \) \(\mathstrut +\mathstrut 51q^{75} \) \(\mathstrut +\mathstrut 25q^{79} \) \(\mathstrut +\mathstrut 43q^{81} \) \(\mathstrut -\mathstrut 18q^{83} \) \(\mathstrut +\mathstrut 36q^{85} \) \(\mathstrut +\mathstrut 53q^{87} \) \(\mathstrut +\mathstrut 11q^{89} \) \(\mathstrut +\mathstrut 65q^{93} \) \(\mathstrut -\mathstrut 30q^{95} \) \(\mathstrut -\mathstrut 16q^{97} \) \(\mathstrut -\mathstrut 18q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{15}\mathstrut -\mathstrut \) \(x^{14}\mathstrut -\mathstrut \) \(37\) \(x^{13}\mathstrut +\mathstrut \) \(39\) \(x^{12}\mathstrut +\mathstrut \) \(537\) \(x^{11}\mathstrut -\mathstrut \) \(616\) \(x^{10}\mathstrut -\mathstrut \) \(3853\) \(x^{9}\mathstrut +\mathstrut \) \(4929\) \(x^{8}\mathstrut +\mathstrut \) \(13971\) \(x^{7}\mathstrut -\mathstrut \) \(20311\) \(x^{6}\mathstrut -\mathstrut \) \(22309\) \(x^{5}\mathstrut +\mathstrut \) \(38415\) \(x^{4}\mathstrut +\mathstrut \) \(8429\) \(x^{3}\mathstrut -\mathstrut \) \(22584\) \(x^{2}\mathstrut -\mathstrut \) \(399\) \(x\mathstrut +\mathstrut \) \(3381\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 5 \)
\(\beta_{3}\)\(=\)\((\)\(21568415171\) \(\nu^{14}\mathstrut +\mathstrut \) \(33366538546\) \(\nu^{13}\mathstrut -\mathstrut \) \(780431594905\) \(\nu^{12}\mathstrut -\mathstrut \) \(1076896328346\) \(\nu^{11}\mathstrut +\mathstrut \) \(11058300703621\) \(\nu^{10}\mathstrut +\mathstrut \) \(12550722332099\) \(\nu^{9}\mathstrut -\mathstrut \) \(78350006830010\) \(\nu^{8}\mathstrut -\mathstrut \) \(63368265036347\) \(\nu^{7}\mathstrut +\mathstrut \) \(290964255861732\) \(\nu^{6}\mathstrut +\mathstrut \) \(125086901622927\) \(\nu^{5}\mathstrut -\mathstrut \) \(518478043052718\) \(\nu^{4}\mathstrut -\mathstrut \) \(34154729402677\) \(\nu^{3}\mathstrut +\mathstrut \) \(315331689321588\) \(\nu^{2}\mathstrut -\mathstrut \) \(33588397215636\) \(\nu\mathstrut -\mathstrut \) \(53909708702913\)\()/\)\(8954190428232\)
\(\beta_{4}\)\(=\)\((\)\(207367915591\) \(\nu^{14}\mathstrut +\mathstrut \) \(75537732154\) \(\nu^{13}\mathstrut -\mathstrut \) \(6864511642829\) \(\nu^{12}\mathstrut -\mathstrut \) \(2264494596922\) \(\nu^{11}\mathstrut +\mathstrut \) \(84630564241705\) \(\nu^{10}\mathstrut +\mathstrut \) \(19997951254183\) \(\nu^{9}\mathstrut -\mathstrut \) \(471109920542042\) \(\nu^{8}\mathstrut -\mathstrut \) \(26134945978879\) \(\nu^{7}\mathstrut +\mathstrut \) \(1061905945918156\) \(\nu^{6}\mathstrut -\mathstrut \) \(337765061235149\) \(\nu^{5}\mathstrut -\mathstrut \) \(14256388065710\) \(\nu^{4}\mathstrut +\mathstrut \) \(1086025882031975\) \(\nu^{3}\mathstrut -\mathstrut \) \(2504020310672772\) \(\nu^{2}\mathstrut -\mathstrut \) \(382307010614964\) \(\nu\mathstrut +\mathstrut \) \(964719687046731\)\()/\)\(53725142569392\)
\(\beta_{5}\)\(=\)\((\)\(375675670889\) \(\nu^{14}\mathstrut +\mathstrut \) \(197307815606\) \(\nu^{13}\mathstrut -\mathstrut \) \(13458627125155\) \(\nu^{12}\mathstrut -\mathstrut \) \(5416632772358\) \(\nu^{11}\mathstrut +\mathstrut \) \(187920186390263\) \(\nu^{10}\mathstrut +\mathstrut \) \(41971019614937\) \(\nu^{9}\mathstrut -\mathstrut \) \(1297715569983766\) \(\nu^{8}\mathstrut +\mathstrut \) \(1604986959871\) \(\nu^{7}\mathstrut +\mathstrut \) \(4597551204850484\) \(\nu^{6}\mathstrut -\mathstrut \) \(1062369256372771\) \(\nu^{5}\mathstrut -\mathstrut \) \(7504847710757362\) \(\nu^{4}\mathstrut +\mathstrut \) \(2957473201007065\) \(\nu^{3}\mathstrut +\mathstrut \) \(3676894843258596\) \(\nu^{2}\mathstrut -\mathstrut \) \(925571855654028\) \(\nu\mathstrut -\mathstrut \) \(394220916659787\)\()/\)\(53725142569392\)
\(\beta_{6}\)\(=\)\((\)\(563847518531\) \(\nu^{14}\mathstrut +\mathstrut \) \(271104011042\) \(\nu^{13}\mathstrut -\mathstrut \) \(20576643380593\) \(\nu^{12}\mathstrut -\mathstrut \) \(7746814774274\) \(\nu^{11}\mathstrut +\mathstrut \) \(294326513071613\) \(\nu^{10}\mathstrut +\mathstrut \) \(65601128365283\) \(\nu^{9}\mathstrut -\mathstrut \) \(2097712145040178\) \(\nu^{8}\mathstrut -\mathstrut \) \(64560226435499\) \(\nu^{7}\mathstrut +\mathstrut \) \(7768694051421020\) \(\nu^{6}\mathstrut -\mathstrut \) \(1280840055975121\) \(\nu^{5}\mathstrut -\mathstrut \) \(13752243630567190\) \(\nu^{4}\mathstrut +\mathstrut \) \(4090409019404467\) \(\nu^{3}\mathstrut +\mathstrut \) \(8604345326302092\) \(\nu^{2}\mathstrut -\mathstrut \) \(1583632566957636\) \(\nu\mathstrut -\mathstrut \) \(1413258738046953\)\()/\)\(53725142569392\)
\(\beta_{7}\)\(=\)\((\)\(82318075711\) \(\nu^{14}\mathstrut +\mathstrut \) \(110890419958\) \(\nu^{13}\mathstrut -\mathstrut \) \(2903659812299\) \(\nu^{12}\mathstrut -\mathstrut \) \(3508288802422\) \(\nu^{11}\mathstrut +\mathstrut \) \(39739641971875\) \(\nu^{10}\mathstrut +\mathstrut \) \(39661374923485\) \(\nu^{9}\mathstrut -\mathstrut \) \(268865569213064\) \(\nu^{8}\mathstrut -\mathstrut \) \(191585391007741\) \(\nu^{7}\mathstrut +\mathstrut \) \(943989351011626\) \(\nu^{6}\mathstrut +\mathstrut \) \(359635444491391\) \(\nu^{5}\mathstrut -\mathstrut \) \(1587383470520204\) \(\nu^{4}\mathstrut -\mathstrut \) \(134303162335843\) \(\nu^{3}\mathstrut +\mathstrut \) \(916971510040800\) \(\nu^{2}\mathstrut +\mathstrut \) \(51742333507248\) \(\nu\mathstrut -\mathstrut \) \(123529112322861\)\()/\)\(6715642821174\)
\(\beta_{8}\)\(=\)\((\)\(57933555161\) \(\nu^{14}\mathstrut +\mathstrut \) \(19902931966\) \(\nu^{13}\mathstrut -\mathstrut \) \(2038445185159\) \(\nu^{12}\mathstrut -\mathstrut \) \(636604082118\) \(\nu^{11}\mathstrut +\mathstrut \) \(27895029041707\) \(\nu^{10}\mathstrut +\mathstrut \) \(6543610028837\) \(\nu^{9}\mathstrut -\mathstrut \) \(188732383570406\) \(\nu^{8}\mathstrut -\mathstrut \) \(21783985515293\) \(\nu^{7}\mathstrut +\mathstrut \) \(661794329691888\) \(\nu^{6}\mathstrut -\mathstrut \) \(12151856350959\) \(\nu^{5}\mathstrut -\mathstrut \) \(1127435686751094\) \(\nu^{4}\mathstrut +\mathstrut \) \(128904705950189\) \(\nu^{3}\mathstrut +\mathstrut \) \(743133409360908\) \(\nu^{2}\mathstrut -\mathstrut \) \(36292597607964\) \(\nu\mathstrut -\mathstrut \) \(132293240921499\)\()/\)\(4477095214116\)
\(\beta_{9}\)\(=\)\((\)\(263299916363\) \(\nu^{14}\mathstrut +\mathstrut \) \(284951510738\) \(\nu^{13}\mathstrut -\mathstrut \) \(9037992446569\) \(\nu^{12}\mathstrut -\mathstrut \) \(9319581543506\) \(\nu^{11}\mathstrut +\mathstrut \) \(119443657067957\) \(\nu^{10}\mathstrut +\mathstrut \) \(110071063789547\) \(\nu^{9}\mathstrut -\mathstrut \) \(772667003836402\) \(\nu^{8}\mathstrut -\mathstrut \) \(576510242603315\) \(\nu^{7}\mathstrut +\mathstrut \) \(2574917570991548\) \(\nu^{6}\mathstrut +\mathstrut \) \(1337367111898247\) \(\nu^{5}\mathstrut -\mathstrut \) \(4158684124869190\) \(\nu^{4}\mathstrut -\mathstrut \) \(1188084234706517\) \(\nu^{3}\mathstrut +\mathstrut \) \(2549675888785068\) \(\nu^{2}\mathstrut +\mathstrut \) \(395328021911436\) \(\nu\mathstrut -\mathstrut \) \(369589303008177\)\()/\)\(17908380856464\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(803771605457\) \(\nu^{14}\mathstrut +\mathstrut \) \(55554951370\) \(\nu^{13}\mathstrut +\mathstrut \) \(29873873602171\) \(\nu^{12}\mathstrut -\mathstrut \) \(2773231651450\) \(\nu^{11}\mathstrut -\mathstrut \) \(437414324713775\) \(\nu^{10}\mathstrut +\mathstrut \) \(64143993774511\) \(\nu^{9}\mathstrut +\mathstrut \) \(3202662100683478\) \(\nu^{8}\mathstrut -\mathstrut \) \(708242318071255\) \(\nu^{7}\mathstrut -\mathstrut \) \(12199681840975508\) \(\nu^{6}\mathstrut +\mathstrut \) \(3575923666457563\) \(\nu^{5}\mathstrut +\mathstrut \) \(22280485010412706\) \(\nu^{4}\mathstrut -\mathstrut \) \(7066331684287153\) \(\nu^{3}\mathstrut -\mathstrut \) \(14659545336879396\) \(\nu^{2}\mathstrut +\mathstrut \) \(2212465125784428\) \(\nu\mathstrut +\mathstrut \) \(2285567947233123\)\()/\)\(53725142569392\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(527518007087\) \(\nu^{14}\mathstrut -\mathstrut \) \(247737754298\) \(\nu^{13}\mathstrut +\mathstrut \) \(19237736393941\) \(\nu^{12}\mathstrut +\mathstrut \) \(7202431283834\) \(\nu^{11}\mathstrut -\mathstrut \) \(274960302342017\) \(\nu^{10}\mathstrut -\mathstrut \) \(63910333187519\) \(\nu^{9}\mathstrut +\mathstrut \) \(1958752009116682\) \(\nu^{8}\mathstrut +\mathstrut \) \(105751315240871\) \(\nu^{7}\mathstrut -\mathstrut \) \(7259458623838364\) \(\nu^{6}\mathstrut +\mathstrut \) \(925907072139781\) \(\nu^{5}\mathstrut +\mathstrut \) \(12894254808423982\) \(\nu^{4}\mathstrut -\mathstrut \) \(3195696128916367\) \(\nu^{3}\mathstrut -\mathstrut \) \(8114892589426428\) \(\nu^{2}\mathstrut +\mathstrut \) \(1033679062742580\) \(\nu\mathstrut +\mathstrut \) \(1300628388320541\)\()/\)\(17908380856464\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(1991383769819\) \(\nu^{14}\mathstrut -\mathstrut \) \(970794976610\) \(\nu^{13}\mathstrut +\mathstrut \) \(72904083529129\) \(\nu^{12}\mathstrut +\mathstrut \) \(29524283624978\) \(\nu^{11}\mathstrut -\mathstrut \) \(1045949435856725\) \(\nu^{10}\mathstrut -\mathstrut \) \(291365982179867\) \(\nu^{9}\mathstrut +\mathstrut \) \(7472488513327378\) \(\nu^{8}\mathstrut +\mathstrut \) \(870143948269859\) \(\nu^{7}\mathstrut -\mathstrut \) \(27720511397395436\) \(\nu^{6}\mathstrut +\mathstrut \) \(1482781049982937\) \(\nu^{5}\mathstrut +\mathstrut \) \(49218218809824022\) \(\nu^{4}\mathstrut -\mathstrut \) \(8617048669258075\) \(\nu^{3}\mathstrut -\mathstrut \) \(31166858944757532\) \(\nu^{2}\mathstrut +\mathstrut \) \(2619386748959364\) \(\nu\mathstrut +\mathstrut \) \(5020217462879745\)\()/\)\(53725142569392\)
\(\beta_{13}\)\(=\)\((\)\(154414062065\) \(\nu^{14}\mathstrut +\mathstrut \) \(69132074966\) \(\nu^{13}\mathstrut -\mathstrut \) \(5550942035752\) \(\nu^{12}\mathstrut -\mathstrut \) \(2103794805215\) \(\nu^{11}\mathstrut +\mathstrut \) \(77954228306660\) \(\nu^{10}\mathstrut +\mathstrut \) \(20374588096121\) \(\nu^{9}\mathstrut -\mathstrut \) \(543496021568218\) \(\nu^{8}\mathstrut -\mathstrut \) \(55206873249419\) \(\nu^{7}\mathstrut +\mathstrut \) \(1963368044552309\) \(\nu^{6}\mathstrut -\mathstrut \) \(139122061304389\) \(\nu^{5}\mathstrut -\mathstrut \) \(3386733041584858\) \(\nu^{4}\mathstrut +\mathstrut \) \(660872437651219\) \(\nu^{3}\mathstrut +\mathstrut \) \(2062137225183060\) \(\nu^{2}\mathstrut -\mathstrut \) \(196898900419467\) \(\nu\mathstrut -\mathstrut \) \(312802565393409\)\()/\)\(3357821410587\)
\(\beta_{14}\)\(=\)\((\)\(-\)\(1366326190169\) \(\nu^{14}\mathstrut -\mathstrut \) \(650669688638\) \(\nu^{13}\mathstrut +\mathstrut \) \(49657832878051\) \(\nu^{12}\mathstrut +\mathstrut \) \(19718267961758\) \(\nu^{11}\mathstrut -\mathstrut \) \(706794387141431\) \(\nu^{10}\mathstrut -\mathstrut \) \(191670870598865\) \(\nu^{9}\mathstrut +\mathstrut \) \(5008459392835390\) \(\nu^{8}\mathstrut +\mathstrut \) \(533869176269825\) \(\nu^{7}\mathstrut -\mathstrut \) \(18440308332790964\) \(\nu^{6}\mathstrut +\mathstrut \) \(1233561484274971\) \(\nu^{5}\mathstrut +\mathstrut \) \(32538031186207330\) \(\nu^{4}\mathstrut -\mathstrut \) \(6160094617298017\) \(\nu^{3}\mathstrut -\mathstrut \) \(20547001848439524\) \(\nu^{2}\mathstrut +\mathstrut \) \(1854323154805428\) \(\nu\mathstrut +\mathstrut \) \(3416523056642787\)\()/\)\(26862571284696\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(5\)
\(\nu^{3}\)\(=\)\(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(7\) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(11\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(38\)
\(\nu^{5}\)\(=\)\(-\)\(3\) \(\beta_{14}\mathstrut +\mathstrut \) \(2\) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(14\) \(\beta_{9}\mathstrut -\mathstrut \) \(16\) \(\beta_{8}\mathstrut -\mathstrut \) \(14\) \(\beta_{7}\mathstrut +\mathstrut \) \(12\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(4\) \(\beta_{3}\mathstrut +\mathstrut \) \(59\) \(\beta_{1}\mathstrut -\mathstrut \) \(5\)
\(\nu^{6}\)\(=\)\(14\) \(\beta_{14}\mathstrut +\mathstrut \) \(14\) \(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(2\) \(\beta_{11}\mathstrut -\mathstrut \) \(17\) \(\beta_{10}\mathstrut +\mathstrut \) \(17\) \(\beta_{9}\mathstrut -\mathstrut \) \(32\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(23\) \(\beta_{6}\mathstrut -\mathstrut \) \(19\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(21\) \(\beta_{3}\mathstrut +\mathstrut \) \(116\) \(\beta_{2}\mathstrut +\mathstrut \) \(27\) \(\beta_{1}\mathstrut +\mathstrut \) \(333\)
\(\nu^{7}\)\(=\)\(-\)\(62\) \(\beta_{14}\mathstrut -\mathstrut \) \(10\) \(\beta_{13}\mathstrut +\mathstrut \) \(44\) \(\beta_{12}\mathstrut -\mathstrut \) \(7\) \(\beta_{11}\mathstrut +\mathstrut \) \(12\) \(\beta_{10}\mathstrut +\mathstrut \) \(170\) \(\beta_{9}\mathstrut -\mathstrut \) \(202\) \(\beta_{8}\mathstrut -\mathstrut \) \(166\) \(\beta_{7}\mathstrut +\mathstrut \) \(132\) \(\beta_{6}\mathstrut -\mathstrut \) \(29\) \(\beta_{5}\mathstrut +\mathstrut \) \(10\) \(\beta_{4}\mathstrut -\mathstrut \) \(67\) \(\beta_{3}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(558\) \(\beta_{1}\mathstrut +\mathstrut \) \(38\)
\(\nu^{8}\)\(=\)\(147\) \(\beta_{14}\mathstrut +\mathstrut \) \(152\) \(\beta_{13}\mathstrut +\mathstrut \) \(30\) \(\beta_{12}\mathstrut +\mathstrut \) \(60\) \(\beta_{11}\mathstrut -\mathstrut \) \(243\) \(\beta_{10}\mathstrut +\mathstrut \) \(234\) \(\beta_{9}\mathstrut -\mathstrut \) \(419\) \(\beta_{8}\mathstrut +\mathstrut \) \(27\) \(\beta_{7}\mathstrut +\mathstrut \) \(370\) \(\beta_{6}\mathstrut -\mathstrut \) \(263\) \(\beta_{5}\mathstrut +\mathstrut \) \(28\) \(\beta_{4}\mathstrut -\mathstrut \) \(320\) \(\beta_{3}\mathstrut +\mathstrut \) \(1240\) \(\beta_{2}\mathstrut +\mathstrut \) \(451\) \(\beta_{1}\mathstrut +\mathstrut \) \(3159\)
\(\nu^{9}\)\(=\)\(-\)\(927\) \(\beta_{14}\mathstrut -\mathstrut \) \(256\) \(\beta_{13}\mathstrut +\mathstrut \) \(685\) \(\beta_{12}\mathstrut -\mathstrut \) \(156\) \(\beta_{11}\mathstrut +\mathstrut \) \(52\) \(\beta_{10}\mathstrut +\mathstrut \) \(2001\) \(\beta_{9}\mathstrut -\mathstrut \) \(2390\) \(\beta_{8}\mathstrut -\mathstrut \) \(1858\) \(\beta_{7}\mathstrut +\mathstrut \) \(1452\) \(\beta_{6}\mathstrut -\mathstrut \) \(508\) \(\beta_{5}\mathstrut +\mathstrut \) \(250\) \(\beta_{4}\mathstrut -\mathstrut \) \(903\) \(\beta_{3}\mathstrut +\mathstrut \) \(228\) \(\beta_{2}\mathstrut +\mathstrut \) \(5654\) \(\beta_{1}\mathstrut +\mathstrut \) \(1342\)
\(\nu^{10}\)\(=\)\(1327\) \(\beta_{14}\mathstrut +\mathstrut \) \(1492\) \(\beta_{13}\mathstrut +\mathstrut \) \(627\) \(\beta_{12}\mathstrut +\mathstrut \) \(1096\) \(\beta_{11}\mathstrut -\mathstrut \) \(3269\) \(\beta_{10}\mathstrut +\mathstrut \) \(3039\) \(\beta_{9}\mathstrut -\mathstrut \) \(5210\) \(\beta_{8}\mathstrut +\mathstrut \) \(459\) \(\beta_{7}\mathstrut +\mathstrut \) \(5154\) \(\beta_{6}\mathstrut -\mathstrut \) \(3302\) \(\beta_{5}\mathstrut +\mathstrut \) \(521\) \(\beta_{4}\mathstrut -\mathstrut \) \(4318\) \(\beta_{3}\mathstrut +\mathstrut \) \(13447\) \(\beta_{2}\mathstrut +\mathstrut \) \(6366\) \(\beta_{1}\mathstrut +\mathstrut \) \(31539\)
\(\nu^{11}\)\(=\)\(-\)\(12261\) \(\beta_{14}\mathstrut -\mathstrut \) \(4464\) \(\beta_{13}\mathstrut +\mathstrut \) \(9349\) \(\beta_{12}\mathstrut -\mathstrut \) \(2401\) \(\beta_{11}\mathstrut -\mathstrut \) \(866\) \(\beta_{10}\mathstrut +\mathstrut \) \(23339\) \(\beta_{9}\mathstrut -\mathstrut \) \(27666\) \(\beta_{8}\mathstrut -\mathstrut \) \(20283\) \(\beta_{7}\mathstrut +\mathstrut \) \(16290\) \(\beta_{6}\mathstrut -\mathstrut \) \(7403\) \(\beta_{5}\mathstrut +\mathstrut \) \(4286\) \(\beta_{4}\mathstrut -\mathstrut \) \(11529\) \(\beta_{3}\mathstrut +\mathstrut \) \(4656\) \(\beta_{2}\mathstrut +\mathstrut \) \(59734\) \(\beta_{1}\mathstrut +\mathstrut \) \(23366\)
\(\nu^{12}\)\(=\)\(10196\) \(\beta_{14}\mathstrut +\mathstrut \) \(13647\) \(\beta_{13}\mathstrut +\mathstrut \) \(10931\) \(\beta_{12}\mathstrut +\mathstrut \) \(16261\) \(\beta_{11}\mathstrut -\mathstrut \) \(42318\) \(\beta_{10}\mathstrut +\mathstrut \) \(38527\) \(\beta_{9}\mathstrut -\mathstrut \) \(63562\) \(\beta_{8}\mathstrut +\mathstrut \) \(6361\) \(\beta_{7}\mathstrut +\mathstrut \) \(66741\) \(\beta_{6}\mathstrut -\mathstrut \) \(40030\) \(\beta_{5}\mathstrut +\mathstrut \) \(8268\) \(\beta_{4}\mathstrut -\mathstrut \) \(54924\) \(\beta_{3}\mathstrut +\mathstrut \) \(147413\) \(\beta_{2}\mathstrut +\mathstrut \) \(83308\) \(\beta_{1}\mathstrut +\mathstrut \) \(326176\)
\(\nu^{13}\)\(=\)\(-\)\(152923\) \(\beta_{14}\mathstrut -\mathstrut \) \(66240\) \(\beta_{13}\mathstrut +\mathstrut \) \(119702\) \(\beta_{12}\mathstrut -\mathstrut \) \(31483\) \(\beta_{11}\mathstrut -\mathstrut \) \(28360\) \(\beta_{10}\mathstrut +\mathstrut \) \(271267\) \(\beta_{9}\mathstrut -\mathstrut \) \(318032\) \(\beta_{8}\mathstrut -\mathstrut \) \(218879\) \(\beta_{7}\mathstrut +\mathstrut \) \(186813\) \(\beta_{6}\mathstrut -\mathstrut \) \(99003\) \(\beta_{5}\mathstrut +\mathstrut \) \(62906\) \(\beta_{4}\mathstrut -\mathstrut \) \(144488\) \(\beta_{3}\mathstrut +\mathstrut \) \(77660\) \(\beta_{2}\mathstrut +\mathstrut \) \(647792\) \(\beta_{1}\mathstrut +\mathstrut \) \(337884\)
\(\nu^{14}\)\(=\)\(56795\) \(\beta_{14}\mathstrut +\mathstrut \) \(115353\) \(\beta_{13}\mathstrut +\mathstrut \) \(170227\) \(\beta_{12}\mathstrut +\mathstrut \) \(216472\) \(\beta_{11}\mathstrut -\mathstrut \) \(532792\) \(\beta_{10}\mathstrut +\mathstrut \) \(481792\) \(\beta_{9}\mathstrut -\mathstrut \) \(768045\) \(\beta_{8}\mathstrut +\mathstrut \) \(78391\) \(\beta_{7}\mathstrut +\mathstrut \) \(829524\) \(\beta_{6}\mathstrut -\mathstrut \) \(478852\) \(\beta_{5}\mathstrut +\mathstrut \) \(120857\) \(\beta_{4}\mathstrut -\mathstrut \) \(676566\) \(\beta_{3}\mathstrut +\mathstrut \) \(1629219\) \(\beta_{2}\mathstrut +\mathstrut \) \(1048772\) \(\beta_{1}\mathstrut +\mathstrut \) \(3460210\)

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
−3.24213
−2.90634
−2.73742
−2.47735
−2.11769
−0.741382
−0.449590
0.515082
1.27444
1.40418
1.72996
1.93225
2.29181
3.10047
3.42372
0 −3.24213 0 −1.49679 0 0 0 7.51143 0
1.2 0 −2.90634 0 −0.506103 0 0 0 5.44682 0
1.3 0 −2.73742 0 0.863054 0 0 0 4.49344 0
1.4 0 −2.47735 0 3.49844 0 0 0 3.13725 0
1.5 0 −2.11769 0 −3.93656 0 0 0 1.48461 0
1.6 0 −0.741382 0 2.48663 0 0 0 −2.45035 0
1.7 0 −0.449590 0 0.787803 0 0 0 −2.79787 0
1.8 0 0.515082 0 −2.19964 0 0 0 −2.73469 0
1.9 0 1.27444 0 −4.37429 0 0 0 −1.37580 0
1.10 0 1.40418 0 −3.61567 0 0 0 −1.02828 0
1.11 0 1.72996 0 0.138351 0 0 0 −0.00725198 0
1.12 0 1.93225 0 0.475465 0 0 0 0.733587 0
1.13 0 2.29181 0 3.92303 0 0 0 2.25239 0
1.14 0 3.10047 0 3.31369 0 0 0 6.61290 0
1.15 0 3.42372 0 −2.35740 0 0 0 8.72183 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(41\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8036))\):

\(T_{3}^{15} - \cdots\)
\(T_{5}^{15} + \cdots\)
\(T_{11}^{15} - \cdots\)