Properties

Label 6040.2.a.o
Level 6040
Weight 2
Character orbit 6040.a
Self dual Yes
Analytic conductor 48.230
Analytic rank 0
Dimension 15
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6040.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.2296428209\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{3} \) \(+ q^{5}\) \( -\beta_{4} q^{7} \) \( + ( 1 + \beta_{1} + \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{3} \) \(+ q^{5}\) \( -\beta_{4} q^{7} \) \( + ( 1 + \beta_{1} + \beta_{2} ) q^{9} \) \( -\beta_{9} q^{11} \) \( + \beta_{11} q^{13} \) \( + \beta_{1} q^{15} \) \( -\beta_{10} q^{17} \) \( + ( 1 - \beta_{7} - \beta_{8} + \beta_{9} ) q^{19} \) \( + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{10} + \beta_{12} - \beta_{13} ) q^{21} \) \( + ( 2 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} ) q^{23} \) \(+ q^{25}\) \( + ( 1 + 2 \beta_{3} + \beta_{8} + \beta_{10} - \beta_{13} + \beta_{14} ) q^{27} \) \( + ( 1 + \beta_{1} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} + \beta_{13} - \beta_{14} ) q^{29} \) \( + ( 2 - \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{12} - \beta_{14} ) q^{31} \) \( + ( -2 - \beta_{4} - \beta_{6} + \beta_{8} - 2 \beta_{9} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{33} \) \( -\beta_{4} q^{35} \) \( + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{12} + \beta_{13} ) q^{37} \) \( + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{7} - \beta_{8} - \beta_{10} + \beta_{11} - \beta_{12} ) q^{39} \) \( + ( 1 + \beta_{1} - \beta_{2} - \beta_{8} + \beta_{10} ) q^{41} \) \( + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{7} - \beta_{10} + \beta_{13} ) q^{43} \) \( + ( 1 + \beta_{1} + \beta_{2} ) q^{45} \) \( + ( 2 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} + \beta_{11} - \beta_{12} + \beta_{14} ) q^{47} \) \( + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} ) q^{49} \) \( + ( 2 + \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{7} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{14} ) q^{51} \) \( + ( -1 - \beta_{2} - 2 \beta_{4} + \beta_{14} ) q^{53} \) \( -\beta_{9} q^{55} \) \( + ( -1 + 2 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{9} + \beta_{12} + 2 \beta_{13} - \beta_{14} ) q^{57} \) \( + ( 3 + \beta_{2} - 2 \beta_{3} + \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{13} - \beta_{14} ) q^{59} \) \( + ( 1 - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} ) q^{61} \) \( + ( 1 + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{10} - 2 \beta_{11} + 2 \beta_{12} - \beta_{14} ) q^{63} \) \( + \beta_{11} q^{65} \) \( + ( 2 - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{12} - \beta_{14} ) q^{67} \) \( + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{69} \) \( + ( 2 - \beta_{1} + \beta_{4} + \beta_{8} + \beta_{12} ) q^{71} \) \( + ( 2 \beta_{1} - \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{10} - 2 \beta_{12} + \beta_{13} ) q^{73} \) \( + \beta_{1} q^{75} \) \( + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} - 2 \beta_{9} - \beta_{12} ) q^{77} \) \( + ( 4 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{8} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{79} \) \( + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{12} ) q^{81} \) \( + ( 3 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{11} - \beta_{13} ) q^{83} \) \( -\beta_{10} q^{85} \) \( + ( 1 + 2 \beta_{1} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{11} - \beta_{12} + \beta_{14} ) q^{87} \) \( + ( 1 - \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{9} - 2 \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{89} \) \( + ( 2 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{91} \) \( + ( -\beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{10} - \beta_{11} + \beta_{12} ) q^{93} \) \( + ( 1 - \beta_{7} - \beta_{8} + \beta_{9} ) q^{95} \) \( + ( -2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + 2 \beta_{7} + \beta_{8} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{97} \) \( + ( -3 \beta_{1} + 2 \beta_{3} - 3 \beta_{5} + \beta_{7} + 3 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} - 2 \beta_{13} + \beta_{14} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(15q \) \(\mathstrut +\mathstrut 5q^{3} \) \(\mathstrut +\mathstrut 15q^{5} \) \(\mathstrut +\mathstrut 7q^{7} \) \(\mathstrut +\mathstrut 18q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(15q \) \(\mathstrut +\mathstrut 5q^{3} \) \(\mathstrut +\mathstrut 15q^{5} \) \(\mathstrut +\mathstrut 7q^{7} \) \(\mathstrut +\mathstrut 18q^{9} \) \(\mathstrut +\mathstrut 7q^{11} \) \(\mathstrut +\mathstrut 2q^{13} \) \(\mathstrut +\mathstrut 5q^{15} \) \(\mathstrut -\mathstrut 3q^{17} \) \(\mathstrut +\mathstrut 8q^{19} \) \(\mathstrut +\mathstrut 7q^{21} \) \(\mathstrut +\mathstrut 15q^{23} \) \(\mathstrut +\mathstrut 15q^{25} \) \(\mathstrut +\mathstrut 23q^{27} \) \(\mathstrut +\mathstrut 5q^{29} \) \(\mathstrut +\mathstrut 27q^{31} \) \(\mathstrut -\mathstrut 5q^{33} \) \(\mathstrut +\mathstrut 7q^{35} \) \(\mathstrut -\mathstrut 4q^{37} \) \(\mathstrut +\mathstrut 11q^{39} \) \(\mathstrut +\mathstrut 20q^{41} \) \(\mathstrut +\mathstrut 25q^{43} \) \(\mathstrut +\mathstrut 18q^{45} \) \(\mathstrut +\mathstrut 35q^{47} \) \(\mathstrut -\mathstrut 14q^{49} \) \(\mathstrut +\mathstrut 25q^{51} \) \(\mathstrut -\mathstrut 2q^{53} \) \(\mathstrut +\mathstrut 7q^{55} \) \(\mathstrut -\mathstrut 24q^{57} \) \(\mathstrut +\mathstrut 39q^{59} \) \(\mathstrut +\mathstrut 23q^{61} \) \(\mathstrut +\mathstrut 39q^{63} \) \(\mathstrut +\mathstrut 2q^{65} \) \(\mathstrut +\mathstrut 32q^{67} \) \(\mathstrut +\mathstrut 13q^{69} \) \(\mathstrut +\mathstrut 30q^{71} \) \(\mathstrut +\mathstrut 7q^{73} \) \(\mathstrut +\mathstrut 5q^{75} \) \(\mathstrut -\mathstrut 4q^{77} \) \(\mathstrut +\mathstrut 38q^{79} \) \(\mathstrut +\mathstrut 11q^{81} \) \(\mathstrut +\mathstrut 29q^{83} \) \(\mathstrut -\mathstrut 3q^{85} \) \(\mathstrut +\mathstrut 4q^{87} \) \(\mathstrut +\mathstrut 19q^{89} \) \(\mathstrut +\mathstrut 16q^{91} \) \(\mathstrut +\mathstrut 8q^{93} \) \(\mathstrut +\mathstrut 8q^{95} \) \(\mathstrut -\mathstrut 8q^{97} \) \(\mathstrut +\mathstrut 16q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{15}\mathstrut -\mathstrut \) \(5\) \(x^{14}\mathstrut -\mathstrut \) \(19\) \(x^{13}\mathstrut +\mathstrut \) \(119\) \(x^{12}\mathstrut +\mathstrut \) \(106\) \(x^{11}\mathstrut -\mathstrut \) \(1063\) \(x^{10}\mathstrut -\mathstrut \) \(48\) \(x^{9}\mathstrut +\mathstrut \) \(4510\) \(x^{8}\mathstrut -\mathstrut \) \(1130\) \(x^{7}\mathstrut -\mathstrut \) \(9512\) \(x^{6}\mathstrut +\mathstrut \) \(2839\) \(x^{5}\mathstrut +\mathstrut \) \(9383\) \(x^{4}\mathstrut -\mathstrut \) \(1781\) \(x^{3}\mathstrut -\mathstrut \) \(3692\) \(x^{2}\mathstrut -\mathstrut \) \(92\) \(x\mathstrut +\mathstrut \) \(272\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{3}\)\(=\)\((\)\(51463483\) \(\nu^{14}\mathstrut +\mathstrut \) \(296143482\) \(\nu^{13}\mathstrut -\mathstrut \) \(3862202797\) \(\nu^{12}\mathstrut -\mathstrut \) \(2330841226\) \(\nu^{11}\mathstrut +\mathstrut \) \(65739514714\) \(\nu^{10}\mathstrut -\mathstrut \) \(30526050827\) \(\nu^{9}\mathstrut -\mathstrut \) \(444606565557\) \(\nu^{8}\mathstrut +\mathstrut \) \(351625975913\) \(\nu^{7}\mathstrut +\mathstrut \) \(1337952074763\) \(\nu^{6}\mathstrut -\mathstrut \) \(992862234469\) \(\nu^{5}\mathstrut -\mathstrut \) \(1827288171804\) \(\nu^{4}\mathstrut +\mathstrut \) \(681038903767\) \(\nu^{3}\mathstrut +\mathstrut \) \(1072103286778\) \(\nu^{2}\mathstrut +\mathstrut \) \(85289558920\) \(\nu\mathstrut -\mathstrut \) \(143292511468\)\()/\)\(12081368188\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(27789758\) \(\nu^{14}\mathstrut +\mathstrut \) \(300288361\) \(\nu^{13}\mathstrut -\mathstrut \) \(435211827\) \(\nu^{12}\mathstrut -\mathstrut \) \(5214821099\) \(\nu^{11}\mathstrut +\mathstrut \) \(16744647679\) \(\nu^{10}\mathstrut +\mathstrut \) \(24819789456\) \(\nu^{9}\mathstrut -\mathstrut \) \(137458096755\) \(\nu^{8}\mathstrut -\mathstrut \) \(2344772800\) \(\nu^{7}\mathstrut +\mathstrut \) \(437211178876\) \(\nu^{6}\mathstrut -\mathstrut \) \(171566778738\) \(\nu^{5}\mathstrut -\mathstrut \) \(575397610136\) \(\nu^{4}\mathstrut +\mathstrut \) \(225268302897\) \(\nu^{3}\mathstrut +\mathstrut \) \(289850381323\) \(\nu^{2}\mathstrut -\mathstrut \) \(53755931065\) \(\nu\mathstrut -\mathstrut \) \(30335382422\)\()/\)\(6040684094\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(188871903\) \(\nu^{14}\mathstrut +\mathstrut \) \(1585090827\) \(\nu^{13}\mathstrut +\mathstrut \) \(256024289\) \(\nu^{12}\mathstrut -\mathstrut \) \(32706135385\) \(\nu^{11}\mathstrut +\mathstrut \) \(52077502738\) \(\nu^{10}\mathstrut +\mathstrut \) \(234372149881\) \(\nu^{9}\mathstrut -\mathstrut \) \(550515010672\) \(\nu^{8}\mathstrut -\mathstrut \) \(709454325414\) \(\nu^{7}\mathstrut +\mathstrut \) \(2150552052470\) \(\nu^{6}\mathstrut +\mathstrut \) \(955609215160\) \(\nu^{5}\mathstrut -\mathstrut \) \(3632423495489\) \(\nu^{4}\mathstrut -\mathstrut \) \(672744784145\) \(\nu^{3}\mathstrut +\mathstrut \) \(2325457038591\) \(\nu^{2}\mathstrut +\mathstrut \) \(285961104688\) \(\nu\mathstrut -\mathstrut \) \(220043410060\)\()/\)\(12081368188\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(242873093\) \(\nu^{14}\mathstrut +\mathstrut \) \(1541595868\) \(\nu^{13}\mathstrut +\mathstrut \) \(2637655539\) \(\nu^{12}\mathstrut -\mathstrut \) \(33195175548\) \(\nu^{11}\mathstrut +\mathstrut \) \(16857698830\) \(\nu^{10}\mathstrut +\mathstrut \) \(255613144249\) \(\nu^{9}\mathstrut -\mathstrut \) \(316603050899\) \(\nu^{8}\mathstrut -\mathstrut \) \(876914364661\) \(\nu^{7}\mathstrut +\mathstrut \) \(1394341411749\) \(\nu^{6}\mathstrut +\mathstrut \) \(1428656202461\) \(\nu^{5}\mathstrut -\mathstrut \) \(2424571091646\) \(\nu^{4}\mathstrut -\mathstrut \) \(1139882821129\) \(\nu^{3}\mathstrut +\mathstrut \) \(1467780879808\) \(\nu^{2}\mathstrut +\mathstrut \) \(413451710076\) \(\nu\mathstrut -\mathstrut \) \(97377907560\)\()/\)\(12081368188\)
\(\beta_{7}\)\(=\)\((\)\(251785062\) \(\nu^{14}\mathstrut -\mathstrut \) \(222305117\) \(\nu^{13}\mathstrut -\mathstrut \) \(9395987162\) \(\nu^{12}\mathstrut +\mathstrut \) \(9538824983\) \(\nu^{11}\mathstrut +\mathstrut \) \(129733377740\) \(\nu^{10}\mathstrut -\mathstrut \) \(129407958028\) \(\nu^{9}\mathstrut -\mathstrut \) \(849831887969\) \(\nu^{8}\mathstrut +\mathstrut \) \(716255588487\) \(\nu^{7}\mathstrut +\mathstrut \) \(2799926769177\) \(\nu^{6}\mathstrut -\mathstrut \) \(1535663954249\) \(\nu^{5}\mathstrut -\mathstrut \) \(4504168015099\) \(\nu^{4}\mathstrut +\mathstrut \) \(731743354380\) \(\nu^{3}\mathstrut +\mathstrut \) \(2853659478227\) \(\nu^{2}\mathstrut +\mathstrut \) \(428370401980\) \(\nu\mathstrut -\mathstrut \) \(227271071436\)\()/\)\(12081368188\)
\(\beta_{8}\)\(=\)\((\)\(415665579\) \(\nu^{14}\mathstrut -\mathstrut \) \(2446939882\) \(\nu^{13}\mathstrut -\mathstrut \) \(5421483309\) \(\nu^{12}\mathstrut +\mathstrut \) \(52091499282\) \(\nu^{11}\mathstrut -\mathstrut \) \(4294609990\) \(\nu^{10}\mathstrut -\mathstrut \) \(394273351227\) \(\nu^{9}\mathstrut +\mathstrut \) \(291814675639\) \(\nu^{8}\mathstrut +\mathstrut \) \(1315802467441\) \(\nu^{7}\mathstrut -\mathstrut \) \(1214599094941\) \(\nu^{6}\mathstrut -\mathstrut \) \(2035676214033\) \(\nu^{5}\mathstrut +\mathstrut \) \(1719381749876\) \(\nu^{4}\mathstrut +\mathstrut \) \(1402677546603\) \(\nu^{3}\mathstrut -\mathstrut \) \(866708269342\) \(\nu^{2}\mathstrut -\mathstrut \) \(356475898440\) \(\nu\mathstrut +\mathstrut \) \(128826450508\)\()/\)\(12081368188\)
\(\beta_{9}\)\(=\)\((\)\(479001604\) \(\nu^{14}\mathstrut -\mathstrut \) \(2804333197\) \(\nu^{13}\mathstrut -\mathstrut \) \(6084620332\) \(\nu^{12}\mathstrut +\mathstrut \) \(59209221923\) \(\nu^{11}\mathstrut -\mathstrut \) \(9181665128\) \(\nu^{10}\mathstrut -\mathstrut \) \(441413544266\) \(\nu^{9}\mathstrut +\mathstrut \) \(378816583577\) \(\nu^{8}\mathstrut +\mathstrut \) \(1430938992849\) \(\nu^{7}\mathstrut -\mathstrut \) \(1607917534101\) \(\nu^{6}\mathstrut -\mathstrut \) \(2101418850603\) \(\nu^{5}\mathstrut +\mathstrut \) \(2476262659081\) \(\nu^{4}\mathstrut +\mathstrut \) \(1346946165510\) \(\nu^{3}\mathstrut -\mathstrut \) \(1432893306293\) \(\nu^{2}\mathstrut -\mathstrut \) \(314929020980\) \(\nu\mathstrut +\mathstrut \) \(173737611300\)\()/\)\(12081368188\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(487587165\) \(\nu^{14}\mathstrut +\mathstrut \) \(2274354782\) \(\nu^{13}\mathstrut +\mathstrut \) \(9272430227\) \(\nu^{12}\mathstrut -\mathstrut \) \(50346781486\) \(\nu^{11}\mathstrut -\mathstrut \) \(59586397546\) \(\nu^{10}\mathstrut +\mathstrut \) \(404281181793\) \(\nu^{9}\mathstrut +\mathstrut \) \(178170846139\) \(\nu^{8}\mathstrut -\mathstrut \) \(1465135221811\) \(\nu^{7}\mathstrut -\mathstrut \) \(471718289029\) \(\nu^{6}\mathstrut +\mathstrut \) \(2427465610639\) \(\nu^{5}\mathstrut +\mathstrut \) \(1190241002576\) \(\nu^{4}\mathstrut -\mathstrut \) \(1433391997877\) \(\nu^{3}\mathstrut -\mathstrut \) \(1152555848606\) \(\nu^{2}\mathstrut -\mathstrut \) \(32675415816\) \(\nu\mathstrut +\mathstrut \) \(152024446264\)\()/\)\(12081368188\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(506155106\) \(\nu^{14}\mathstrut +\mathstrut \) \(2103881269\) \(\nu^{13}\mathstrut +\mathstrut \) \(10476774266\) \(\nu^{12}\mathstrut -\mathstrut \) \(46165780471\) \(\nu^{11}\mathstrut -\mathstrut \) \(81681713292\) \(\nu^{10}\mathstrut +\mathstrut \) \(365806609900\) \(\nu^{9}\mathstrut +\mathstrut \) \(353505059153\) \(\nu^{8}\mathstrut -\mathstrut \) \(1301878920475\) \(\nu^{7}\mathstrut -\mathstrut \) \(1129281014833\) \(\nu^{6}\mathstrut +\mathstrut \) \(2124704721045\) \(\nu^{5}\mathstrut +\mathstrut \) \(2279730369643\) \(\nu^{4}\mathstrut -\mathstrut \) \(1274103604372\) \(\nu^{3}\mathstrut -\mathstrut \) \(1730832238587\) \(\nu^{2}\mathstrut -\mathstrut \) \(25532616328\) \(\nu\mathstrut +\mathstrut \) \(159337477636\)\()/\)\(12081368188\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(684215577\) \(\nu^{14}\mathstrut +\mathstrut \) \(3616262202\) \(\nu^{13}\mathstrut +\mathstrut \) \(11062015193\) \(\nu^{12}\mathstrut -\mathstrut \) \(79740997314\) \(\nu^{11}\mathstrut -\mathstrut \) \(36111135028\) \(\nu^{10}\mathstrut +\mathstrut \) \(636153945801\) \(\nu^{9}\mathstrut -\mathstrut \) \(184429878961\) \(\nu^{8}\mathstrut -\mathstrut \) \(2285036527033\) \(\nu^{7}\mathstrut +\mathstrut \) \(1197729844849\) \(\nu^{6}\mathstrut +\mathstrut \) \(3791951019845\) \(\nu^{5}\mathstrut -\mathstrut \) \(2060349550034\) \(\nu^{4}\mathstrut -\mathstrut \) \(2500942006723\) \(\nu^{3}\mathstrut +\mathstrut \) \(1280199146170\) \(\nu^{2}\mathstrut +\mathstrut \) \(331332041730\) \(\nu\mathstrut -\mathstrut \) \(221398514376\)\()/\)\(12081368188\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(742080601\) \(\nu^{14}\mathstrut +\mathstrut \) \(3659749286\) \(\nu^{13}\mathstrut +\mathstrut \) \(13526986483\) \(\nu^{12}\mathstrut -\mathstrut \) \(82608253774\) \(\nu^{11}\mathstrut -\mathstrut \) \(72888759814\) \(\nu^{10}\mathstrut +\mathstrut \) \(682685487577\) \(\nu^{9}\mathstrut +\mathstrut \) \(70024666691\) \(\nu^{8}\mathstrut -\mathstrut \) \(2582238722659\) \(\nu^{7}\mathstrut +\mathstrut \) \(317735359163\) \(\nu^{6}\mathstrut +\mathstrut \) \(4576682979143\) \(\nu^{5}\mathstrut -\mathstrut \) \(574562294568\) \(\nu^{4}\mathstrut -\mathstrut \) \(3223118061977\) \(\nu^{3}\mathstrut +\mathstrut \) \(295253018306\) \(\nu^{2}\mathstrut +\mathstrut \) \(437755334836\) \(\nu\mathstrut -\mathstrut \) \(100634081820\)\()/\)\(12081368188\)
\(\beta_{14}\)\(=\)\((\)\(-\)\(773085981\) \(\nu^{14}\mathstrut +\mathstrut \) \(3240047422\) \(\nu^{13}\mathstrut +\mathstrut \) \(17400445159\) \(\nu^{12}\mathstrut -\mathstrut \) \(79691289118\) \(\nu^{11}\mathstrut -\mathstrut \) \(140486781706\) \(\nu^{10}\mathstrut +\mathstrut \) \(733729758665\) \(\nu^{9}\mathstrut +\mathstrut \) \(489252276027\) \(\nu^{8}\mathstrut -\mathstrut \) \(3136157920115\) \(\nu^{7}\mathstrut -\mathstrut \) \(671851406393\) \(\nu^{6}\mathstrut +\mathstrut \) \(6170618051475\) \(\nu^{5}\mathstrut +\mathstrut \) \(170391296588\) \(\nu^{4}\mathstrut -\mathstrut \) \(4542400050049\) \(\nu^{3}\mathstrut +\mathstrut \) \(170310562698\) \(\nu^{2}\mathstrut +\mathstrut \) \(583839322124\) \(\nu\mathstrut -\mathstrut \) \(106981323844\)\()/\)\(12081368188\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(6\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(10\) \(\beta_{2}\mathstrut +\mathstrut \) \(11\) \(\beta_{1}\mathstrut +\mathstrut \) \(26\)
\(\nu^{5}\)\(=\)\(12\) \(\beta_{14}\mathstrut -\mathstrut \) \(12\) \(\beta_{13}\mathstrut -\mathstrut \) \(4\) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(11\) \(\beta_{10}\mathstrut -\mathstrut \) \(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(9\) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(4\) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(22\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(48\) \(\beta_{1}\mathstrut +\mathstrut \) \(11\)
\(\nu^{6}\)\(=\)\(\beta_{14}\mathstrut +\mathstrut \) \(3\) \(\beta_{13}\mathstrut -\mathstrut \) \(20\) \(\beta_{12}\mathstrut +\mathstrut \) \(12\) \(\beta_{10}\mathstrut -\mathstrut \) \(17\) \(\beta_{9}\mathstrut +\mathstrut \) \(14\) \(\beta_{8}\mathstrut -\mathstrut \) \(2\) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(17\) \(\beta_{5}\mathstrut -\mathstrut \) \(12\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(95\) \(\beta_{2}\mathstrut +\mathstrut \) \(112\) \(\beta_{1}\mathstrut +\mathstrut \) \(199\)
\(\nu^{7}\)\(=\)\(124\) \(\beta_{14}\mathstrut -\mathstrut \) \(118\) \(\beta_{13}\mathstrut -\mathstrut \) \(75\) \(\beta_{12}\mathstrut +\mathstrut \) \(20\) \(\beta_{11}\mathstrut +\mathstrut \) \(104\) \(\beta_{10}\mathstrut -\mathstrut \) \(36\) \(\beta_{9}\mathstrut +\mathstrut \) \(75\) \(\beta_{8}\mathstrut -\mathstrut \) \(14\) \(\beta_{7}\mathstrut -\mathstrut \) \(38\) \(\beta_{6}\mathstrut +\mathstrut \) \(72\) \(\beta_{5}\mathstrut -\mathstrut \) \(23\) \(\beta_{4}\mathstrut +\mathstrut \) \(204\) \(\beta_{3}\mathstrut +\mathstrut \) \(46\) \(\beta_{2}\mathstrut +\mathstrut \) \(432\) \(\beta_{1}\mathstrut +\mathstrut \) \(116\)
\(\nu^{8}\)\(=\)\(27\) \(\beta_{14}\mathstrut +\mathstrut \) \(54\) \(\beta_{13}\mathstrut -\mathstrut \) \(283\) \(\beta_{12}\mathstrut +\mathstrut \) \(4\) \(\beta_{11}\mathstrut +\mathstrut \) \(119\) \(\beta_{10}\mathstrut -\mathstrut \) \(207\) \(\beta_{9}\mathstrut +\mathstrut \) \(151\) \(\beta_{8}\mathstrut -\mathstrut \) \(41\) \(\beta_{7}\mathstrut -\mathstrut \) \(35\) \(\beta_{6}\mathstrut +\mathstrut \) \(220\) \(\beta_{5}\mathstrut -\mathstrut \) \(105\) \(\beta_{4}\mathstrut +\mathstrut \) \(25\) \(\beta_{3}\mathstrut +\mathstrut \) \(905\) \(\beta_{2}\mathstrut +\mathstrut \) \(1128\) \(\beta_{1}\mathstrut +\mathstrut \) \(1662\)
\(\nu^{9}\)\(=\)\(1232\) \(\beta_{14}\mathstrut -\mathstrut \) \(1086\) \(\beta_{13}\mathstrut -\mathstrut \) \(1028\) \(\beta_{12}\mathstrut +\mathstrut \) \(290\) \(\beta_{11}\mathstrut +\mathstrut \) \(940\) \(\beta_{10}\mathstrut -\mathstrut \) \(464\) \(\beta_{9}\mathstrut +\mathstrut \) \(629\) \(\beta_{8}\mathstrut -\mathstrut \) \(149\) \(\beta_{7}\mathstrut -\mathstrut \) \(507\) \(\beta_{6}\mathstrut +\mathstrut \) \(951\) \(\beta_{5}\mathstrut -\mathstrut \) \(176\) \(\beta_{4}\mathstrut +\mathstrut \) \(1813\) \(\beta_{3}\mathstrut +\mathstrut \) \(717\) \(\beta_{2}\mathstrut +\mathstrut \) \(4101\) \(\beta_{1}\mathstrut +\mathstrut \) \(1256\)
\(\nu^{10}\)\(=\)\(459\) \(\beta_{14}\mathstrut +\mathstrut \) \(704\) \(\beta_{13}\mathstrut -\mathstrut \) \(3513\) \(\beta_{12}\mathstrut +\mathstrut \) \(142\) \(\beta_{11}\mathstrut +\mathstrut \) \(1099\) \(\beta_{10}\mathstrut -\mathstrut \) \(2240\) \(\beta_{9}\mathstrut +\mathstrut \) \(1483\) \(\beta_{8}\mathstrut -\mathstrut \) \(574\) \(\beta_{7}\mathstrut -\mathstrut \) \(480\) \(\beta_{6}\mathstrut +\mathstrut \) \(2622\) \(\beta_{5}\mathstrut -\mathstrut \) \(767\) \(\beta_{4}\mathstrut +\mathstrut \) \(358\) \(\beta_{3}\mathstrut +\mathstrut \) \(8724\) \(\beta_{2}\mathstrut +\mathstrut \) \(11382\) \(\beta_{1}\mathstrut +\mathstrut \) \(14627\)
\(\nu^{11}\)\(=\)\(12015\) \(\beta_{14}\mathstrut -\mathstrut \) \(9653\) \(\beta_{13}\mathstrut -\mathstrut \) \(12485\) \(\beta_{12}\mathstrut +\mathstrut \) \(3653\) \(\beta_{11}\mathstrut +\mathstrut \) \(8306\) \(\beta_{10}\mathstrut -\mathstrut \) \(5241\) \(\beta_{9}\mathstrut +\mathstrut \) \(5309\) \(\beta_{8}\mathstrut -\mathstrut \) \(1534\) \(\beta_{7}\mathstrut -\mathstrut \) \(5876\) \(\beta_{6}\mathstrut +\mathstrut \) \(11250\) \(\beta_{5}\mathstrut -\mathstrut \) \(927\) \(\beta_{4}\mathstrut +\mathstrut \) \(15892\) \(\beta_{3}\mathstrut +\mathstrut \) \(9504\) \(\beta_{2}\mathstrut +\mathstrut \) \(40012\) \(\beta_{1}\mathstrut +\mathstrut \) \(13786\)
\(\nu^{12}\)\(=\)\(6395\) \(\beta_{14}\mathstrut +\mathstrut \) \(8160\) \(\beta_{13}\mathstrut -\mathstrut \) \(40841\) \(\beta_{12}\mathstrut +\mathstrut \) \(2923\) \(\beta_{11}\mathstrut +\mathstrut \) \(9710\) \(\beta_{10}\mathstrut -\mathstrut \) \(22904\) \(\beta_{9}\mathstrut +\mathstrut \) \(13839\) \(\beta_{8}\mathstrut -\mathstrut \) \(6947\) \(\beta_{7}\mathstrut -\mathstrut \) \(6074\) \(\beta_{6}\mathstrut +\mathstrut \) \(30172\) \(\beta_{5}\mathstrut -\mathstrut \) \(4507\) \(\beta_{4}\mathstrut +\mathstrut \) \(4079\) \(\beta_{3}\mathstrut +\mathstrut \) \(85067\) \(\beta_{2}\mathstrut +\mathstrut \) \(115276\) \(\beta_{1}\mathstrut +\mathstrut \) \(133140\)
\(\nu^{13}\)\(=\)\(115881\) \(\beta_{14}\mathstrut -\mathstrut \) \(83676\) \(\beta_{13}\mathstrut -\mathstrut \) \(142654\) \(\beta_{12}\mathstrut +\mathstrut \) \(42692\) \(\beta_{11}\mathstrut +\mathstrut \) \(72263\) \(\beta_{10}\mathstrut -\mathstrut \) \(55264\) \(\beta_{9}\mathstrut +\mathstrut \) \(44717\) \(\beta_{8}\mathstrut -\mathstrut \) \(16198\) \(\beta_{7}\mathstrut -\mathstrut \) \(63515\) \(\beta_{6}\mathstrut +\mathstrut \) \(126484\) \(\beta_{5}\mathstrut -\mathstrut \) \(452\) \(\beta_{4}\mathstrut +\mathstrut \) \(138536\) \(\beta_{3}\mathstrut +\mathstrut \) \(115721\) \(\beta_{2}\mathstrut +\mathstrut \) \(396524\) \(\beta_{1}\mathstrut +\mathstrut \) \(151248\)
\(\nu^{14}\)\(=\)\(80022\) \(\beta_{14}\mathstrut +\mathstrut \) \(89726\) \(\beta_{13}\mathstrut -\mathstrut \) \(457023\) \(\beta_{12}\mathstrut +\mathstrut \) \(46968\) \(\beta_{11}\mathstrut +\mathstrut \) \(82775\) \(\beta_{10}\mathstrut -\mathstrut \) \(226572\) \(\beta_{9}\mathstrut +\mathstrut \) \(124444\) \(\beta_{8}\mathstrut -\mathstrut \) \(78624\) \(\beta_{7}\mathstrut -\mathstrut \) \(73382\) \(\beta_{6}\mathstrut +\mathstrut \) \(340166\) \(\beta_{5}\mathstrut -\mathstrut \) \(14341\) \(\beta_{4}\mathstrut +\mathstrut \) \(41128\) \(\beta_{3}\mathstrut +\mathstrut \) \(837082\) \(\beta_{2}\mathstrut +\mathstrut \) \(1171125\) \(\beta_{1}\mathstrut +\mathstrut \) \(1240185\)

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
−2.95828
−2.53958
−1.77658
−1.76219
−1.07149
−0.532237
−0.422275
0.266097
1.14431
1.18532
2.06490
2.38022
2.79329
3.03078
3.19771
0 −2.95828 0 1.00000 0 2.63281 0 5.75141 0
1.2 0 −2.53958 0 1.00000 0 −0.798381 0 3.44946 0
1.3 0 −1.77658 0 1.00000 0 2.60627 0 0.156227 0
1.4 0 −1.76219 0 1.00000 0 −1.99837 0 0.105317 0
1.5 0 −1.07149 0 1.00000 0 1.66476 0 −1.85192 0
1.6 0 −0.532237 0 1.00000 0 −2.74903 0 −2.71672 0
1.7 0 −0.422275 0 1.00000 0 −2.22380 0 −2.82168 0
1.8 0 0.266097 0 1.00000 0 3.77999 0 −2.92919 0
1.9 0 1.14431 0 1.00000 0 0.357411 0 −1.69055 0
1.10 0 1.18532 0 1.00000 0 1.34515 0 −1.59502 0
1.11 0 2.06490 0 1.00000 0 −3.39301 0 1.26381 0
1.12 0 2.38022 0 1.00000 0 −1.25626 0 2.66546 0
1.13 0 2.79329 0 1.00000 0 4.28460 0 4.80245 0
1.14 0 3.03078 0 1.00000 0 3.09463 0 6.18562 0
1.15 0 3.19771 0 1.00000 0 −0.346742 0 7.22533 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
\(2\) \(1\)
\(5\) \(-1\)
\(151\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6040))\):

\(T_{3}^{15} - \cdots\)
\(T_{7}^{15} - \cdots\)
\(T_{11}^{15} - \cdots\)