Properties

Label 4012.2.a.h
Level 4012
Weight 2
Character orbit 4012.a
Self dual Yes
Analytic conductor 32.036
Analytic rank 1
Dimension 15
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4012.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0359812909\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{5} q^{3} \) \( - \beta_{8} q^{5} \) \( + ( -1 - \beta_{9} ) q^{7} \) \( + ( 1 + \beta_{10} - \beta_{12} + \beta_{14} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{5} q^{3} \) \( - \beta_{8} q^{5} \) \( + ( -1 - \beta_{9} ) q^{7} \) \( + ( 1 + \beta_{10} - \beta_{12} + \beta_{14} ) q^{9} \) \( + ( -1 + \beta_{1} - \beta_{5} + \beta_{7} + \beta_{12} - \beta_{14} ) q^{11} \) \( + ( -1 + \beta_{1} + \beta_{2} - \beta_{5} - \beta_{10} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{13} \) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{7} + \beta_{9} - \beta_{12} ) q^{15} \) \(+ q^{17}\) \( + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{12} ) q^{19} \) \( + ( -2 \beta_{1} + \beta_{4} - \beta_{6} + \beta_{8} - \beta_{13} ) q^{21} \) \( + ( -1 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + 3 \beta_{8} + \beta_{9} - \beta_{13} + \beta_{14} ) q^{23} \) \( + ( - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{8} + \beta_{9} - \beta_{11} - \beta_{12} + \beta_{14} ) q^{25} \) \( + ( -2 - 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{14} ) q^{27} \) \( + ( 3 - \beta_{1} + \beta_{6} + 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + \beta_{11} - 2 \beta_{12} - \beta_{13} + \beta_{14} ) q^{29} \) \( + ( -1 - 2 \beta_{1} + \beta_{3} + \beta_{5} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{31} \) \( + ( -1 - \beta_{3} - \beta_{5} + 2 \beta_{6} + \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{13} ) q^{33} \) \( + ( -2 - \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{13} ) q^{35} \) \( + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} ) q^{37} \) \( + ( \beta_{1} + \beta_{2} + \beta_{3} - 3 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{12} - \beta_{13} ) q^{39} \) \( + ( -3 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{6} + 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{41} \) \( + ( -1 + \beta_{1} + \beta_{3} - \beta_{5} - 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - 2 \beta_{12} - \beta_{13} + \beta_{14} ) q^{43} \) \( + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} - \beta_{8} + \beta_{11} ) q^{45} \) \( + ( -4 - 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{9} - 2 \beta_{10} - \beta_{11} + 2 \beta_{13} - \beta_{14} ) q^{47} \) \( + ( 2 + \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{6} - 2 \beta_{8} + \beta_{9} + \beta_{11} - \beta_{13} - \beta_{14} ) q^{49} \) \( + \beta_{5} q^{51} \) \( + ( -1 + 5 \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - \beta_{8} + 3 \beta_{10} - \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{53} \) \( + ( -1 - 3 \beta_{3} - 2 \beta_{4} - \beta_{6} - \beta_{9} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{55} \) \( + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + \beta_{7} - \beta_{8} - 4 \beta_{9} - 3 \beta_{10} + 2 \beta_{11} + 4 \beta_{12} + \beta_{13} - 2 \beta_{14} ) q^{57} \) \(- q^{59}\) \( + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + \beta_{12} + 2 \beta_{13} ) q^{61} \) \( + ( 5 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{11} + 4 \beta_{12} ) q^{63} \) \( + ( -2 + 3 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} + \beta_{13} ) q^{65} \) \( + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{5} - 3 \beta_{6} - \beta_{9} - \beta_{10} + 2 \beta_{11} + \beta_{12} - \beta_{13} - 2 \beta_{14} ) q^{67} \) \( + ( -2 - \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{6} - 2 \beta_{9} - \beta_{10} - \beta_{11} + 2 \beta_{12} ) q^{69} \) \( + ( 1 + 3 \beta_{2} - 5 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{10} + 3 \beta_{11} + \beta_{12} - 2 \beta_{14} ) q^{71} \) \( + ( -2 - 2 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} + \beta_{4} - 2 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} + \beta_{10} - 5 \beta_{12} + \beta_{14} ) q^{73} \) \( + ( -3 + 4 \beta_{1} + 3 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{11} + 2 \beta_{12} + \beta_{13} ) q^{75} \) \( + ( 1 - 3 \beta_{1} - 3 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} - \beta_{7} + 3 \beta_{8} + 4 \beta_{9} - \beta_{10} - \beta_{11} - 6 \beta_{12} + 2 \beta_{14} ) q^{77} \) \( + ( -2 + 3 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - 3 \beta_{8} + \beta_{10} + 2 \beta_{12} + \beta_{13} - 3 \beta_{14} ) q^{79} \) \( + ( -1 + 5 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} - 3 \beta_{8} - \beta_{9} + \beta_{10} + 3 \beta_{12} - \beta_{14} ) q^{81} \) \( + ( -3 + 2 \beta_{1} + \beta_{2} - 2 \beta_{4} - 3 \beta_{5} - \beta_{6} + 2 \beta_{7} - 4 \beta_{8} - \beta_{9} + 3 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} - \beta_{13} + \beta_{14} ) q^{83} \) \( - \beta_{8} q^{85} \) \( + ( -3 - 4 \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{4} + 6 \beta_{5} + 3 \beta_{6} + 2 \beta_{8} + 2 \beta_{9} + \beta_{11} - 3 \beta_{12} + 2 \beta_{13} - \beta_{14} ) q^{87} \) \( + ( -3 + 2 \beta_{1} - 4 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} - 2 \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} + \beta_{12} ) q^{89} \) \( + ( -4 \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{5} + 3 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} - 3 \beta_{12} + 2 \beta_{14} ) q^{91} \) \( + ( 2 - \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} + 2 \beta_{6} - 3 \beta_{7} - \beta_{9} - 2 \beta_{10} - \beta_{12} + 3 \beta_{13} + \beta_{14} ) q^{93} \) \( + ( -1 + \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{5} + 3 \beta_{7} - \beta_{8} + \beta_{9} + 5 \beta_{10} - 2 \beta_{11} - 3 \beta_{13} ) q^{95} \) \( + ( - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} ) q^{97} \) \( + ( -6 - \beta_{1} - 3 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} + 3 \beta_{8} + \beta_{9} - 2 \beta_{12} - \beta_{13} + \beta_{14} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(15q \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut -\mathstrut 11q^{7} \) \(\mathstrut +\mathstrut 16q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(15q \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut -\mathstrut 11q^{7} \) \(\mathstrut +\mathstrut 16q^{9} \) \(\mathstrut -\mathstrut 12q^{11} \) \(\mathstrut -\mathstrut 10q^{13} \) \(\mathstrut -\mathstrut 13q^{15} \) \(\mathstrut +\mathstrut 15q^{17} \) \(\mathstrut -\mathstrut 4q^{21} \) \(\mathstrut -\mathstrut 21q^{23} \) \(\mathstrut +\mathstrut 4q^{25} \) \(\mathstrut -\mathstrut 37q^{27} \) \(\mathstrut +\mathstrut 23q^{29} \) \(\mathstrut -\mathstrut 31q^{31} \) \(\mathstrut -\mathstrut 11q^{33} \) \(\mathstrut -\mathstrut 35q^{35} \) \(\mathstrut -\mathstrut 10q^{37} \) \(\mathstrut -\mathstrut 2q^{39} \) \(\mathstrut -\mathstrut 15q^{41} \) \(\mathstrut -\mathstrut 3q^{43} \) \(\mathstrut +\mathstrut 15q^{45} \) \(\mathstrut -\mathstrut 47q^{47} \) \(\mathstrut +\mathstrut 18q^{49} \) \(\mathstrut -\mathstrut q^{51} \) \(\mathstrut -\mathstrut 7q^{53} \) \(\mathstrut -\mathstrut 20q^{55} \) \(\mathstrut -\mathstrut 30q^{57} \) \(\mathstrut -\mathstrut 15q^{59} \) \(\mathstrut +\mathstrut q^{61} \) \(\mathstrut +\mathstrut 19q^{63} \) \(\mathstrut +\mathstrut 11q^{65} \) \(\mathstrut -\mathstrut 20q^{67} \) \(\mathstrut -\mathstrut 24q^{69} \) \(\mathstrut -\mathstrut 13q^{71} \) \(\mathstrut -\mathstrut 24q^{73} \) \(\mathstrut -\mathstrut 10q^{75} \) \(\mathstrut +\mathstrut 21q^{77} \) \(\mathstrut -\mathstrut 34q^{79} \) \(\mathstrut -\mathstrut 21q^{81} \) \(\mathstrut -\mathstrut 40q^{83} \) \(\mathstrut +\mathstrut q^{85} \) \(\mathstrut -\mathstrut 60q^{87} \) \(\mathstrut -\mathstrut 46q^{89} \) \(\mathstrut -\mathstrut 19q^{91} \) \(\mathstrut +\mathstrut 33q^{93} \) \(\mathstrut -\mathstrut 4q^{95} \) \(\mathstrut -\mathstrut 5q^{97} \) \(\mathstrut -\mathstrut 74q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{15}\mathstrut -\mathstrut \) \(5\) \(x^{14}\mathstrut -\mathstrut \) \(16\) \(x^{13}\mathstrut +\mathstrut \) \(123\) \(x^{12}\mathstrut -\mathstrut \) \(76\) \(x^{11}\mathstrut -\mathstrut \) \(630\) \(x^{10}\mathstrut +\mathstrut \) \(937\) \(x^{9}\mathstrut +\mathstrut \) \(1083\) \(x^{8}\mathstrut -\mathstrut \) \(2602\) \(x^{7}\mathstrut -\mathstrut \) \(249\) \(x^{6}\mathstrut +\mathstrut \) \(2736\) \(x^{5}\mathstrut -\mathstrut \) \(801\) \(x^{4}\mathstrut -\mathstrut \) \(900\) \(x^{3}\mathstrut +\mathstrut \) \(429\) \(x^{2}\mathstrut -\mathstrut \) \(36\) \(x\mathstrut -\mathstrut \) \(2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(1642110\) \(\nu^{14}\mathstrut +\mathstrut \) \(5493041\) \(\nu^{13}\mathstrut +\mathstrut \) \(44856541\) \(\nu^{12}\mathstrut -\mathstrut \) \(160648719\) \(\nu^{11}\mathstrut -\mathstrut \) \(341454762\) \(\nu^{10}\mathstrut +\mathstrut \) \(1316309295\) \(\nu^{9}\mathstrut +\mathstrut \) \(1206761264\) \(\nu^{8}\mathstrut -\mathstrut \) \(4604137302\) \(\nu^{7}\mathstrut -\mathstrut \) \(2481282245\) \(\nu^{6}\mathstrut +\mathstrut \) \(7168124400\) \(\nu^{5}\mathstrut +\mathstrut \) \(3230047412\) \(\nu^{4}\mathstrut -\mathstrut \) \(3872105642\) \(\nu^{3}\mathstrut -\mathstrut \) \(2078341617\) \(\nu^{2}\mathstrut -\mathstrut \) \(113490130\) \(\nu\mathstrut +\mathstrut \) \(189641702\)\()/95351182\)
\(\beta_{3}\)\(=\)\((\)\(2499271\) \(\nu^{14}\mathstrut +\mathstrut \) \(5394424\) \(\nu^{13}\mathstrut -\mathstrut \) \(96824570\) \(\nu^{12}\mathstrut -\mathstrut \) \(88043849\) \(\nu^{11}\mathstrut +\mathstrut \) \(1318014655\) \(\nu^{10}\mathstrut -\mathstrut \) \(95090619\) \(\nu^{9}\mathstrut -\mathstrut \) \(6916869158\) \(\nu^{8}\mathstrut +\mathstrut \) \(2968463153\) \(\nu^{7}\mathstrut +\mathstrut \) \(16654248791\) \(\nu^{6}\mathstrut -\mathstrut \) \(9186606266\) \(\nu^{5}\mathstrut -\mathstrut \) \(18331801638\) \(\nu^{4}\mathstrut +\mathstrut \) \(10082491657\) \(\nu^{3}\mathstrut +\mathstrut \) \(7142637423\) \(\nu^{2}\mathstrut -\mathstrut \) \(3199335814\) \(\nu\mathstrut +\mathstrut \) \(131157754\)\()/95351182\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(6589235\) \(\nu^{14}\mathstrut +\mathstrut \) \(21534662\) \(\nu^{13}\mathstrut +\mathstrut \) \(149175107\) \(\nu^{12}\mathstrut -\mathstrut \) \(577550108\) \(\nu^{11}\mathstrut -\mathstrut \) \(627835958\) \(\nu^{10}\mathstrut +\mathstrut \) \(3703715580\) \(\nu^{9}\mathstrut +\mathstrut \) \(438222586\) \(\nu^{8}\mathstrut -\mathstrut \) \(9886195276\) \(\nu^{7}\mathstrut +\mathstrut \) \(1359291740\) \(\nu^{6}\mathstrut +\mathstrut \) \(12036309335\) \(\nu^{5}\mathstrut -\mathstrut \) \(1339292667\) \(\nu^{4}\mathstrut -\mathstrut \) \(5866163863\) \(\nu^{3}\mathstrut -\mathstrut \) \(410465505\) \(\nu^{2}\mathstrut +\mathstrut \) \(485967518\) \(\nu\mathstrut +\mathstrut \) \(11524587\)\()/47675591\)
\(\beta_{5}\)\(=\)\((\)\(29448859\) \(\nu^{14}\mathstrut -\mathstrut \) \(118865162\) \(\nu^{13}\mathstrut -\mathstrut \) \(583179274\) \(\nu^{12}\mathstrut +\mathstrut \) \(3043456985\) \(\nu^{11}\mathstrut +\mathstrut \) \(653319903\) \(\nu^{10}\mathstrut -\mathstrut \) \(17499504791\) \(\nu^{9}\mathstrut +\mathstrut \) \(10520388562\) \(\nu^{8}\mathstrut +\mathstrut \) \(39503795341\) \(\nu^{7}\mathstrut -\mathstrut \) \(36007368575\) \(\nu^{6}\mathstrut -\mathstrut \) \(36114958670\) \(\nu^{5}\mathstrut +\mathstrut \) \(39200943698\) \(\nu^{4}\mathstrut +\mathstrut \) \(8415289279\) \(\nu^{3}\mathstrut -\mathstrut \) \(12848394501\) \(\nu^{2}\mathstrut +\mathstrut \) \(2319111788\) \(\nu\mathstrut +\mathstrut \) \(129215570\)\()/95351182\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(36873004\) \(\nu^{14}\mathstrut +\mathstrut \) \(144992745\) \(\nu^{13}\mathstrut +\mathstrut \) \(729566385\) \(\nu^{12}\mathstrut -\mathstrut \) \(3708988537\) \(\nu^{11}\mathstrut -\mathstrut \) \(818596384\) \(\nu^{10}\mathstrut +\mathstrut \) \(21093381481\) \(\nu^{9}\mathstrut -\mathstrut \) \(13361532648\) \(\nu^{8}\mathstrut -\mathstrut \) \(46378126822\) \(\nu^{7}\mathstrut +\mathstrut \) \(46596966433\) \(\nu^{6}\mathstrut +\mathstrut \) \(39124095638\) \(\nu^{5}\mathstrut -\mathstrut \) \(52573837040\) \(\nu^{4}\mathstrut -\mathstrut \) \(4706759198\) \(\nu^{3}\mathstrut +\mathstrut \) \(18396452745\) \(\nu^{2}\mathstrut -\mathstrut \) \(4564703394\) \(\nu\mathstrut -\mathstrut \) \(133290618\)\()/95351182\)
\(\beta_{7}\)\(=\)\((\)\(40746102\) \(\nu^{14}\mathstrut -\mathstrut \) \(141104307\) \(\nu^{13}\mathstrut -\mathstrut \) \(875338947\) \(\nu^{12}\mathstrut +\mathstrut \) \(3685347851\) \(\nu^{11}\mathstrut +\mathstrut \) \(2730849730\) \(\nu^{10}\mathstrut -\mathstrut \) \(22011930647\) \(\nu^{9}\mathstrut +\mathstrut \) \(3355203376\) \(\nu^{8}\mathstrut +\mathstrut \) \(53261736674\) \(\nu^{7}\mathstrut -\mathstrut \) \(21911047283\) \(\nu^{6}\mathstrut -\mathstrut \) \(56230036218\) \(\nu^{5}\mathstrut +\mathstrut \) \(23460592692\) \(\nu^{4}\mathstrut +\mathstrut \) \(21085698418\) \(\nu^{3}\mathstrut -\mathstrut \) \(4854984431\) \(\nu^{2}\mathstrut +\mathstrut \) \(398646228\) \(\nu\mathstrut -\mathstrut \) \(338210566\)\()/95351182\)
\(\beta_{8}\)\(=\)\((\)\(95633725\) \(\nu^{14}\mathstrut -\mathstrut \) \(359609000\) \(\nu^{13}\mathstrut -\mathstrut \) \(1951460100\) \(\nu^{12}\mathstrut +\mathstrut \) \(9276232291\) \(\nu^{11}\mathstrut +\mathstrut \) \(3654449555\) \(\nu^{10}\mathstrut -\mathstrut \) \(53869328967\) \(\nu^{9}\mathstrut +\mathstrut \) \(25740619298\) \(\nu^{8}\mathstrut +\mathstrut \) \(123383929169\) \(\nu^{7}\mathstrut -\mathstrut \) \(100008537973\) \(\nu^{6}\mathstrut -\mathstrut \) \(115872881722\) \(\nu^{5}\mathstrut +\mathstrut \) \(115167642186\) \(\nu^{4}\mathstrut +\mathstrut \) \(30532653209\) \(\nu^{3}\mathstrut -\mathstrut \) \(39836754771\) \(\nu^{2}\mathstrut +\mathstrut \) \(5001928278\) \(\nu\mathstrut +\mathstrut \) \(443495554\)\()/\)\(190702364\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(97204111\) \(\nu^{14}\mathstrut +\mathstrut \) \(408284470\) \(\nu^{13}\mathstrut +\mathstrut \) \(1824797366\) \(\nu^{12}\mathstrut -\mathstrut \) \(10322410111\) \(\nu^{11}\mathstrut +\mathstrut \) \(386823059\) \(\nu^{10}\mathstrut +\mathstrut \) \(56972906955\) \(\nu^{9}\mathstrut -\mathstrut \) \(50219018070\) \(\nu^{8}\mathstrut -\mathstrut \) \(118538250583\) \(\nu^{7}\mathstrut +\mathstrut \) \(158702692833\) \(\nu^{6}\mathstrut +\mathstrut \) \(87957399718\) \(\nu^{5}\mathstrut -\mathstrut \) \(176258359614\) \(\nu^{4}\mathstrut +\mathstrut \) \(646244605\) \(\nu^{3}\mathstrut +\mathstrut \) \(62525219875\) \(\nu^{2}\mathstrut -\mathstrut \) \(14572316814\) \(\nu\mathstrut -\mathstrut \) \(167261742\)\()/\)\(190702364\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(25842475\) \(\nu^{14}\mathstrut +\mathstrut \) \(101169385\) \(\nu^{13}\mathstrut +\mathstrut \) \(521083651\) \(\nu^{12}\mathstrut -\mathstrut \) \(2604048863\) \(\nu^{11}\mathstrut -\mathstrut \) \(816126103\) \(\nu^{10}\mathstrut +\mathstrut \) \(15153025831\) \(\nu^{9}\mathstrut -\mathstrut \) \(7864783050\) \(\nu^{8}\mathstrut -\mathstrut \) \(34957337633\) \(\nu^{7}\mathstrut +\mathstrut \) \(28573976493\) \(\nu^{6}\mathstrut +\mathstrut \) \(33373023966\) \(\nu^{5}\mathstrut -\mathstrut \) \(31339622646\) \(\nu^{4}\mathstrut -\mathstrut \) \(8963008460\) \(\nu^{3}\mathstrut +\mathstrut \) \(9722679174\) \(\nu^{2}\mathstrut -\mathstrut \) \(1839898369\) \(\nu\mathstrut +\mathstrut \) \(106478707\)\()/47675591\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(58215653\) \(\nu^{14}\mathstrut +\mathstrut \) \(202536060\) \(\nu^{13}\mathstrut +\mathstrut \) \(1239397782\) \(\nu^{12}\mathstrut -\mathstrut \) \(5282011523\) \(\nu^{11}\mathstrut -\mathstrut \) \(3579431491\) \(\nu^{10}\mathstrut +\mathstrut \) \(31339492215\) \(\nu^{9}\mathstrut -\mathstrut \) \(7527204066\) \(\nu^{8}\mathstrut -\mathstrut \) \(74223474393\) \(\nu^{7}\mathstrut +\mathstrut \) \(40946834245\) \(\nu^{6}\mathstrut +\mathstrut \) \(73844820286\) \(\nu^{5}\mathstrut -\mathstrut \) \(48459183198\) \(\nu^{4}\mathstrut -\mathstrut \) \(22486916493\) \(\nu^{3}\mathstrut +\mathstrut \) \(15280826425\) \(\nu^{2}\mathstrut -\mathstrut \) \(2796392008\) \(\nu\mathstrut +\mathstrut \) \(74376318\)\()/95351182\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(121421599\) \(\nu^{14}\mathstrut +\mathstrut \) \(480687854\) \(\nu^{13}\mathstrut +\mathstrut \) \(2412644590\) \(\nu^{12}\mathstrut -\mathstrut \) \(12328562947\) \(\nu^{11}\mathstrut -\mathstrut \) \(2924433725\) \(\nu^{10}\mathstrut +\mathstrut \) \(70935144335\) \(\nu^{9}\mathstrut -\mathstrut \) \(42646712690\) \(\nu^{8}\mathstrut -\mathstrut \) \(160312566091\) \(\nu^{7}\mathstrut +\mathstrut \) \(149689508201\) \(\nu^{6}\mathstrut +\mathstrut \) \(146614988770\) \(\nu^{5}\mathstrut -\mathstrut \) \(167221293562\) \(\nu^{4}\mathstrut -\mathstrut \) \(33298989487\) \(\nu^{3}\mathstrut +\mathstrut \) \(55815466299\) \(\nu^{2}\mathstrut -\mathstrut \) \(10559674518\) \(\nu\mathstrut +\mathstrut \) \(210174674\)\()/\)\(190702364\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(166794261\) \(\nu^{14}\mathstrut +\mathstrut \) \(629928074\) \(\nu^{13}\mathstrut +\mathstrut \) \(3417073022\) \(\nu^{12}\mathstrut -\mathstrut \) \(16278312837\) \(\nu^{11}\mathstrut -\mathstrut \) \(6694066567\) \(\nu^{10}\mathstrut +\mathstrut \) \(95271285377\) \(\nu^{9}\mathstrut -\mathstrut \) \(42808440098\) \(\nu^{8}\mathstrut -\mathstrut \) \(221681072765\) \(\nu^{7}\mathstrut +\mathstrut \) \(167971062563\) \(\nu^{6}\mathstrut +\mathstrut \) \(215663490182\) \(\nu^{5}\mathstrut -\mathstrut \) \(190929742802\) \(\nu^{4}\mathstrut -\mathstrut \) \(63378890185\) \(\nu^{3}\mathstrut +\mathstrut \) \(63563825561\) \(\nu^{2}\mathstrut -\mathstrut \) \(8886145330\) \(\nu\mathstrut -\mathstrut \) \(204973042\)\()/\)\(190702364\)
\(\beta_{14}\)\(=\)\((\)\(-\)\(169653351\) \(\nu^{14}\mathstrut +\mathstrut \) \(605403962\) \(\nu^{13}\mathstrut +\mathstrut \) \(3601868958\) \(\nu^{12}\mathstrut -\mathstrut \) \(15798177543\) \(\nu^{11}\mathstrut -\mathstrut \) \(10130008829\) \(\nu^{10}\mathstrut +\mathstrut \) \(94608841675\) \(\nu^{9}\mathstrut -\mathstrut \) \(23155286746\) \(\nu^{8}\mathstrut -\mathstrut \) \(229847533475\) \(\nu^{7}\mathstrut +\mathstrut \) \(120212213389\) \(\nu^{6}\mathstrut +\mathstrut \) \(244549054426\) \(\nu^{5}\mathstrut -\mathstrut \) \(139871902806\) \(\nu^{4}\mathstrut -\mathstrut \) \(94024452139\) \(\nu^{3}\mathstrut +\mathstrut \) \(44401361523\) \(\nu^{2}\mathstrut -\mathstrut \) \(227999114\) \(\nu\mathstrut -\mathstrut \) \(898759938\)\()/\)\(190702364\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(3\) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(3\) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(2\) \(\beta_{8}\mathstrut -\mathstrut \) \(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(4\) \(\beta_{3}\mathstrut -\mathstrut \) \(3\) \(\beta_{2}\mathstrut +\mathstrut \) \(9\) \(\beta_{1}\mathstrut -\mathstrut \) \(5\)
\(\nu^{4}\)\(=\)\(11\) \(\beta_{14}\mathstrut -\mathstrut \) \(17\) \(\beta_{13}\mathstrut -\mathstrut \) \(9\) \(\beta_{12}\mathstrut +\mathstrut \) \(2\) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(15\) \(\beta_{9}\mathstrut -\mathstrut \) \(6\) \(\beta_{8}\mathstrut +\mathstrut \) \(3\) \(\beta_{7}\mathstrut -\mathstrut \) \(4\) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(12\) \(\beta_{4}\mathstrut +\mathstrut \) \(6\) \(\beta_{3}\mathstrut +\mathstrut \) \(11\) \(\beta_{2}\mathstrut -\mathstrut \) \(17\) \(\beta_{1}\mathstrut +\mathstrut \) \(46\)
\(\nu^{5}\)\(=\)\(-\)\(9\) \(\beta_{14}\mathstrut +\mathstrut \) \(64\) \(\beta_{13}\mathstrut +\mathstrut \) \(5\) \(\beta_{12}\mathstrut -\mathstrut \) \(21\) \(\beta_{11}\mathstrut -\mathstrut \) \(52\) \(\beta_{10}\mathstrut -\mathstrut \) \(28\) \(\beta_{9}\mathstrut +\mathstrut \) \(32\) \(\beta_{8}\mathstrut -\mathstrut \) \(56\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(9\) \(\beta_{4}\mathstrut +\mathstrut \) \(65\) \(\beta_{3}\mathstrut -\mathstrut \) \(56\) \(\beta_{2}\mathstrut +\mathstrut \) \(134\) \(\beta_{1}\mathstrut -\mathstrut \) \(120\)
\(\nu^{6}\)\(=\)\(159\) \(\beta_{14}\mathstrut -\mathstrut \) \(293\) \(\beta_{13}\mathstrut -\mathstrut \) \(127\) \(\beta_{12}\mathstrut +\mathstrut \) \(44\) \(\beta_{11}\mathstrut +\mathstrut \) \(7\) \(\beta_{10}\mathstrut +\mathstrut \) \(244\) \(\beta_{9}\mathstrut -\mathstrut \) \(129\) \(\beta_{8}\mathstrut +\mathstrut \) \(82\) \(\beta_{7}\mathstrut -\mathstrut \) \(72\) \(\beta_{6}\mathstrut -\mathstrut \) \(36\) \(\beta_{5}\mathstrut +\mathstrut \) \(177\) \(\beta_{4}\mathstrut +\mathstrut \) \(30\) \(\beta_{3}\mathstrut +\mathstrut \) \(235\) \(\beta_{2}\mathstrut -\mathstrut \) \(338\) \(\beta_{1}\mathstrut +\mathstrut \) \(726\)
\(\nu^{7}\)\(=\)\(-\)\(251\) \(\beta_{14}\mathstrut +\mathstrut \) \(1195\) \(\beta_{13}\mathstrut +\mathstrut \) \(149\) \(\beta_{12}\mathstrut -\mathstrut \) \(368\) \(\beta_{11}\mathstrut -\mathstrut \) \(819\) \(\beta_{10}\mathstrut -\mathstrut \) \(593\) \(\beta_{9}\mathstrut +\mathstrut \) \(552\) \(\beta_{8}\mathstrut -\mathstrut \) \(937\) \(\beta_{7}\mathstrut +\mathstrut \) \(75\) \(\beta_{6}\mathstrut +\mathstrut \) \(62\) \(\beta_{5}\mathstrut -\mathstrut \) \(261\) \(\beta_{4}\mathstrut +\mathstrut \) \(993\) \(\beta_{3}\mathstrut -\mathstrut \) \(1000\) \(\beta_{2}\mathstrut +\mathstrut \) \(2231\) \(\beta_{1}\mathstrut -\mathstrut \) \(2367\)
\(\nu^{8}\)\(=\)\(2546\) \(\beta_{14}\mathstrut -\mathstrut \) \(5203\) \(\beta_{13}\mathstrut -\mathstrut \) \(2027\) \(\beta_{12}\mathstrut +\mathstrut \) \(893\) \(\beta_{11}\mathstrut +\mathstrut \) \(592\) \(\beta_{10}\mathstrut +\mathstrut \) \(4113\) \(\beta_{9}\mathstrut -\mathstrut \) \(2393\) \(\beta_{8}\mathstrut +\mathstrut \) \(1838\) \(\beta_{7}\mathstrut -\mathstrut \) \(1179\) \(\beta_{6}\mathstrut -\mathstrut \) \(590\) \(\beta_{5}\mathstrut +\mathstrut \) \(2837\) \(\beta_{4}\mathstrut -\mathstrut \) \(224\) \(\beta_{3}\mathstrut +\mathstrut \) \(4348\) \(\beta_{2}\mathstrut -\mathstrut \) \(6583\) \(\beta_{1}\mathstrut +\mathstrut \) \(12440\)
\(\nu^{9}\)\(=\)\(-\)\(5510\) \(\beta_{14}\mathstrut +\mathstrut \) \(21747\) \(\beta_{13}\mathstrut +\mathstrut \) \(3494\) \(\beta_{12}\mathstrut -\mathstrut \) \(6292\) \(\beta_{11}\mathstrut -\mathstrut \) \(12997\) \(\beta_{10}\mathstrut -\mathstrut \) \(11707\) \(\beta_{9}\mathstrut +\mathstrut \) \(9878\) \(\beta_{8}\mathstrut -\mathstrut \) \(15645\) \(\beta_{7}\mathstrut +\mathstrut \) \(1882\) \(\beta_{6}\mathstrut +\mathstrut \) \(1381\) \(\beta_{5}\mathstrut -\mathstrut \) \(5848\) \(\beta_{4}\mathstrut +\mathstrut \) \(15431\) \(\beta_{3}\mathstrut -\mathstrut \) \(18007\) \(\beta_{2}\mathstrut +\mathstrut \) \(38305\) \(\beta_{1}\mathstrut -\mathstrut \) \(44481\)
\(\nu^{10}\)\(=\)\(42402\) \(\beta_{14}\mathstrut -\mathstrut \) \(93352\) \(\beta_{13}\mathstrut -\mathstrut \) \(33426\) \(\beta_{12}\mathstrut +\mathstrut \) \(17563\) \(\beta_{11}\mathstrut +\mathstrut \) \(17148\) \(\beta_{10}\mathstrut +\mathstrut \) \(70572\) \(\beta_{9}\mathstrut -\mathstrut \) \(43240\) \(\beta_{8}\mathstrut +\mathstrut \) \(37953\) \(\beta_{7}\mathstrut -\mathstrut \) \(19397\) \(\beta_{6}\mathstrut -\mathstrut \) \(9858\) \(\beta_{5}\mathstrut +\mathstrut \) \(47106\) \(\beta_{4}\mathstrut -\mathstrut \) \(13249\) \(\beta_{3}\mathstrut +\mathstrut \) \(78536\) \(\beta_{2}\mathstrut -\mathstrut \) \(125154\) \(\beta_{1}\mathstrut +\mathstrut \) \(218298\)
\(\nu^{11}\)\(=\)\(-\)\(111435\) \(\beta_{14}\mathstrut +\mathstrut \) \(393291\) \(\beta_{13}\mathstrut +\mathstrut \) \(74135\) \(\beta_{12}\mathstrut -\mathstrut \) \(108079\) \(\beta_{11}\mathstrut -\mathstrut \) \(210704\) \(\beta_{10}\mathstrut -\mathstrut \) \(224050\) \(\beta_{9}\mathstrut +\mathstrut \) \(178411\) \(\beta_{8}\mathstrut -\mathstrut \) \(264708\) \(\beta_{7}\mathstrut +\mathstrut \) \(40958\) \(\beta_{6}\mathstrut +\mathstrut \) \(27717\) \(\beta_{5}\mathstrut -\mathstrut \) \(119611\) \(\beta_{4}\mathstrut +\mathstrut \) \(245731\) \(\beta_{3}\mathstrut -\mathstrut \) \(325439\) \(\beta_{2}\mathstrut +\mathstrut \) \(666991\) \(\beta_{1}\mathstrut -\mathstrut \) \(821757\)
\(\nu^{12}\)\(=\)\(721157\) \(\beta_{14}\mathstrut -\mathstrut \) \(1680103\) \(\beta_{13}\mathstrut -\mathstrut \) \(561369\) \(\beta_{12}\mathstrut +\mathstrut \) \(336971\) \(\beta_{11}\mathstrut +\mathstrut \) \(394848\) \(\beta_{10}\mathstrut +\mathstrut \) \(1225651\) \(\beta_{9}\mathstrut -\mathstrut \) \(777610\) \(\beta_{8}\mathstrut +\mathstrut \) \(748242\) \(\beta_{7}\mathstrut -\mathstrut \) \(324384\) \(\beta_{6}\mathstrut -\mathstrut \) \(168384\) \(\beta_{5}\mathstrut +\mathstrut \) \(798611\) \(\beta_{4}\mathstrut -\mathstrut \) \(357149\) \(\beta_{3}\mathstrut +\mathstrut \) \(1412830\) \(\beta_{2}\mathstrut -\mathstrut \) \(2340970\) \(\beta_{1}\mathstrut +\mathstrut \) \(3867318\)
\(\nu^{13}\)\(=\)\(-\)\(2164334\) \(\beta_{14}\mathstrut +\mathstrut \) \(7100193\) \(\beta_{13}\mathstrut +\mathstrut \) \(1487194\) \(\beta_{12}\mathstrut -\mathstrut \) \(1874470\) \(\beta_{11}\mathstrut -\mathstrut \) \(3488558\) \(\beta_{10}\mathstrut -\mathstrut \) \(4209635\) \(\beta_{9}\mathstrut +\mathstrut \) \(3226956\) \(\beta_{8}\mathstrut -\mathstrut \) \(4541011\) \(\beta_{7}\mathstrut +\mathstrut \) \(831376\) \(\beta_{6}\mathstrut +\mathstrut \) \(532924\) \(\beta_{5}\mathstrut -\mathstrut \) \(2338680\) \(\beta_{4}\mathstrut +\mathstrut \) \(4004710\) \(\beta_{3}\mathstrut -\mathstrut \) \(5884307\) \(\beta_{2}\mathstrut +\mathstrut \) \(11720405\) \(\beta_{1}\mathstrut -\mathstrut \) \(15058170\)
\(\nu^{14}\)\(=\)\(12443114\) \(\beta_{14}\mathstrut -\mathstrut \) \(30267026\) \(\beta_{13}\mathstrut -\mathstrut \) \(9571250\) \(\beta_{12}\mathstrut +\mathstrut \) \(6348517\) \(\beta_{11}\mathstrut +\mathstrut \) \(8244380\) \(\beta_{10}\mathstrut +\mathstrut \) \(21483762\) \(\beta_{9}\mathstrut -\mathstrut \) \(13979884\) \(\beta_{8}\mathstrut +\mathstrut \) \(14335354\) \(\beta_{7}\mathstrut -\mathstrut \) \(5512014\) \(\beta_{6}\mathstrut -\mathstrut \) \(2919908\) \(\beta_{5}\mathstrut +\mathstrut \) \(13743120\) \(\beta_{4}\mathstrut -\mathstrut \) \(7984739\) \(\beta_{3}\mathstrut +\mathstrut \) \(25411016\) \(\beta_{2}\mathstrut -\mathstrut \) \(43314563\) \(\beta_{1}\mathstrut +\mathstrut \) \(68870090\)

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
1.28051
2.10632
−4.24628
−1.49960
−1.20072
2.23095
1.18786
−0.0375146
0.346366
−1.64466
3.62663
−0.829690
1.66914
0.186961
1.82373
0 −3.40908 0 1.59062 0 2.94577 0 8.62180 0
1.2 0 −2.74294 0 −0.560055 0 1.03063 0 4.52374 0
1.3 0 −2.34833 0 3.93023 0 −4.81217 0 2.51465 0
1.4 0 −2.30933 0 2.18242 0 −0.264202 0 2.33302 0
1.5 0 −2.21367 0 −3.70336 0 −3.08925 0 1.90033 0
1.6 0 −0.598056 0 −0.722695 0 −0.897746 0 −2.64233 0
1.7 0 −0.460784 0 −0.730164 0 3.58746 0 −2.78768 0
1.8 0 0.249278 0 −1.04042 0 −3.44893 0 −2.93786 0
1.9 0 0.915888 0 1.84475 0 −3.15913 0 −2.16115 0
1.10 0 1.48779 0 1.28557 0 0.0101628 0 −0.786472 0
1.11 0 1.57548 0 3.44709 0 −4.23195 0 −0.517858 0
1.12 0 1.84628 0 −2.98774 0 −0.920155 0 0.408737 0
1.13 0 2.17000 0 −3.20913 0 1.70252 0 1.70890 0
1.14 0 2.17219 0 −1.55586 0 3.67456 0 1.71840 0
1.15 0 2.66529 0 1.22875 0 −3.12758 0 4.10376 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
\(2\) \(-1\)
\(17\) \(-1\)
\(59\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4012))\):

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