Properties

Label 4005.2.a.w
Level 4005
Weight 2
Character orbit 4005.a
Self dual Yes
Analytic conductor 31.980
Analytic rank 0
Dimension 17
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4005.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(31.9800860095\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{2} \) \( + ( 1 + \beta_{2} ) q^{4} \) \(- q^{5}\) \( + ( 1 + \beta_{8} ) q^{7} \) \( + ( -1 - \beta_{1} - \beta_{3} ) q^{8} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{2} \) \( + ( 1 + \beta_{2} ) q^{4} \) \(- q^{5}\) \( + ( 1 + \beta_{8} ) q^{7} \) \( + ( -1 - \beta_{1} - \beta_{3} ) q^{8} \) \( + \beta_{1} q^{10} \) \( -\beta_{14} q^{11} \) \( -\beta_{9} q^{13} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{6} - \beta_{7} + \beta_{12} - \beta_{13} + \beta_{15} ) q^{14} \) \( + ( 2 + \beta_{1} + 2 \beta_{2} - \beta_{10} + \beta_{11} ) q^{16} \) \( + ( -1 - \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} + \beta_{10} - \beta_{12} ) q^{17} \) \( + ( 2 - \beta_{16} ) q^{19} \) \( + ( -1 - \beta_{2} ) q^{20} \) \( + \beta_{5} q^{22} \) \( + ( -1 - \beta_{4} ) q^{23} \) \(+ q^{25}\) \( + ( 2 - \beta_{1} + 2 \beta_{2} - \beta_{5} + \beta_{6} + \beta_{9} + \beta_{11} + \beta_{13} ) q^{26} \) \( + ( 2 + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{13} ) q^{28} \) \( + ( \beta_{5} - \beta_{6} - \beta_{9} - \beta_{10} - \beta_{16} ) q^{29} \) \( + ( 1 - \beta_{6} - \beta_{9} ) q^{31} \) \( + ( -1 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{8} - \beta_{11} ) q^{32} \) \( + ( 1 + \beta_{2} + \beta_{4} - \beta_{7} - \beta_{14} + \beta_{16} ) q^{34} \) \( + ( -1 - \beta_{8} ) q^{35} \) \( + ( -\beta_{1} - \beta_{2} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{37} \) \( + ( -1 - 2 \beta_{1} - \beta_{6} - \beta_{9} + \beta_{12} + \beta_{16} ) q^{38} \) \( + ( 1 + \beta_{1} + \beta_{3} ) q^{40} \) \( + ( -\beta_{1} - \beta_{8} - \beta_{15} ) q^{41} \) \( + ( 1 - \beta_{1} + \beta_{6} + \beta_{9} + \beta_{10} - \beta_{15} + \beta_{16} ) q^{43} \) \( + ( 1 - \beta_{1} + \beta_{4} + \beta_{6} + \beta_{9} - 2 \beta_{12} - \beta_{15} + \beta_{16} ) q^{44} \) \( + ( \beta_{2} - 2 \beta_{3} - 2 \beta_{6} - 2 \beta_{7} - \beta_{11} + 2 \beta_{12} - \beta_{13} + \beta_{15} ) q^{46} \) \( + ( -2 - \beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{47} \) \( + ( 4 - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{49} \) \( -\beta_{1} q^{50} \) \( + ( -2 \beta_{1} + \beta_{4} - \beta_{6} + \beta_{7} - 2 \beta_{9} - \beta_{11} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{52} \) \( + ( -\beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} + \beta_{16} ) q^{53} \) \( + \beta_{14} q^{55} \) \( + ( -2 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} + 3 \beta_{12} - 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{56} \) \( + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{7} + 2 \beta_{9} - \beta_{14} - \beta_{15} + \beta_{16} ) q^{58} \) \( + ( 2 - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} + 2 \beta_{7} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{59} \) \( + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{12} - \beta_{14} ) q^{61} \) \( + ( -2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{7} + \beta_{9} + \beta_{14} - \beta_{16} ) q^{62} \) \( + ( 3 + \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{64} \) \( + \beta_{9} q^{65} \) \( + ( 3 - \beta_{1} - \beta_{5} + \beta_{15} ) q^{67} \) \( + ( 1 - 2 \beta_{2} + 2 \beta_{3} + \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} - 2 \beta_{12} + 2 \beta_{13} - \beta_{15} - \beta_{16} ) q^{68} \) \( + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{70} \) \( + ( 1 + 2 \beta_{3} - \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{9} - 2 \beta_{12} + 2 \beta_{13} + \beta_{14} - \beta_{16} ) q^{71} \) \( + ( 3 - \beta_{2} + \beta_{3} + \beta_{7} - \beta_{9} - \beta_{12} + 2 \beta_{13} - \beta_{14} - \beta_{16} ) q^{73} \) \( + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{74} \) \( + ( 4 + 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} + 3 \beta_{9} - \beta_{12} + \beta_{14} ) q^{76} \) \( + ( \beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} - \beta_{9} + \beta_{12} + 2 \beta_{13} - 2 \beta_{14} + \beta_{16} ) q^{77} \) \( + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{8} - 2 \beta_{9} - \beta_{11} - 2 \beta_{12} + \beta_{13} - \beta_{14} ) q^{79} \) \( + ( -2 - \beta_{1} - 2 \beta_{2} + \beta_{10} - \beta_{11} ) q^{80} \) \( + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} - \beta_{16} ) q^{82} \) \( + ( -2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} - \beta_{9} - 2 \beta_{12} + \beta_{14} - \beta_{16} ) q^{83} \) \( + ( 1 + \beta_{2} - \beta_{3} - \beta_{6} - \beta_{7} - \beta_{10} + \beta_{12} ) q^{85} \) \( + ( 1 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{12} - \beta_{16} ) q^{86} \) \( + ( -2 \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{5} + 3 \beta_{7} - 2 \beta_{8} - 3 \beta_{9} + \beta_{11} - 3 \beta_{12} + \beta_{13} - \beta_{15} - \beta_{16} ) q^{88} \) \(+ q^{89}\) \( + ( 2 - 3 \beta_{3} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} - \beta_{11} + 3 \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{91} \) \( + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{11} + \beta_{13} + \beta_{14} - \beta_{16} ) q^{92} \) \( + ( 3 + \beta_{1} + \beta_{2} - \beta_{4} + 2 \beta_{9} + \beta_{11} - \beta_{14} ) q^{94} \) \( + ( -2 + \beta_{16} ) q^{95} \) \( + ( 3 - 3 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} - \beta_{4} - 3 \beta_{7} + 3 \beta_{9} + \beta_{11} + 3 \beta_{12} - \beta_{13} + \beta_{16} ) q^{97} \) \( + ( -\beta_{1} - \beta_{3} - \beta_{6} - \beta_{7} + 3 \beta_{12} - 2 \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(17q \) \(\mathstrut -\mathstrut 5q^{2} \) \(\mathstrut +\mathstrut 21q^{4} \) \(\mathstrut -\mathstrut 17q^{5} \) \(\mathstrut +\mathstrut 12q^{7} \) \(\mathstrut -\mathstrut 15q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(17q \) \(\mathstrut -\mathstrut 5q^{2} \) \(\mathstrut +\mathstrut 21q^{4} \) \(\mathstrut -\mathstrut 17q^{5} \) \(\mathstrut +\mathstrut 12q^{7} \) \(\mathstrut -\mathstrut 15q^{8} \) \(\mathstrut +\mathstrut 5q^{10} \) \(\mathstrut -\mathstrut 2q^{11} \) \(\mathstrut +\mathstrut 8q^{13} \) \(\mathstrut -\mathstrut 4q^{14} \) \(\mathstrut +\mathstrut 33q^{16} \) \(\mathstrut -\mathstrut 10q^{17} \) \(\mathstrut +\mathstrut 32q^{19} \) \(\mathstrut -\mathstrut 21q^{20} \) \(\mathstrut +\mathstrut 8q^{22} \) \(\mathstrut -\mathstrut 15q^{23} \) \(\mathstrut +\mathstrut 17q^{25} \) \(\mathstrut +\mathstrut 15q^{26} \) \(\mathstrut +\mathstrut 24q^{28} \) \(\mathstrut -\mathstrut q^{29} \) \(\mathstrut +\mathstrut 18q^{31} \) \(\mathstrut -\mathstrut 25q^{32} \) \(\mathstrut +\mathstrut 14q^{34} \) \(\mathstrut -\mathstrut 12q^{35} \) \(\mathstrut +\mathstrut 12q^{37} \) \(\mathstrut -\mathstrut 22q^{38} \) \(\mathstrut +\mathstrut 15q^{40} \) \(\mathstrut +\mathstrut 7q^{41} \) \(\mathstrut +\mathstrut 28q^{43} \) \(\mathstrut +\mathstrut 14q^{44} \) \(\mathstrut +\mathstrut 4q^{46} \) \(\mathstrut -\mathstrut 26q^{47} \) \(\mathstrut +\mathstrut 41q^{49} \) \(\mathstrut -\mathstrut 5q^{50} \) \(\mathstrut +\mathstrut 10q^{52} \) \(\mathstrut -\mathstrut 12q^{53} \) \(\mathstrut +\mathstrut 2q^{55} \) \(\mathstrut -\mathstrut 13q^{56} \) \(\mathstrut +\mathstrut 16q^{58} \) \(\mathstrut +\mathstrut 23q^{59} \) \(\mathstrut +\mathstrut 26q^{61} \) \(\mathstrut -\mathstrut 10q^{62} \) \(\mathstrut +\mathstrut 59q^{64} \) \(\mathstrut -\mathstrut 8q^{65} \) \(\mathstrut +\mathstrut 31q^{67} \) \(\mathstrut +\mathstrut q^{68} \) \(\mathstrut +\mathstrut 4q^{70} \) \(\mathstrut +\mathstrut 2q^{71} \) \(\mathstrut +\mathstrut 33q^{73} \) \(\mathstrut +\mathstrut 10q^{74} \) \(\mathstrut +\mathstrut 66q^{76} \) \(\mathstrut -\mathstrut 12q^{77} \) \(\mathstrut +\mathstrut 33q^{79} \) \(\mathstrut -\mathstrut 33q^{80} \) \(\mathstrut +\mathstrut 30q^{82} \) \(\mathstrut -\mathstrut 13q^{83} \) \(\mathstrut +\mathstrut 10q^{85} \) \(\mathstrut +\mathstrut 20q^{86} \) \(\mathstrut +\mathstrut 12q^{88} \) \(\mathstrut +\mathstrut 17q^{89} \) \(\mathstrut +\mathstrut 40q^{91} \) \(\mathstrut -\mathstrut 16q^{92} \) \(\mathstrut +\mathstrut 38q^{94} \) \(\mathstrut -\mathstrut 32q^{95} \) \(\mathstrut +\mathstrut 45q^{97} \) \(\mathstrut -\mathstrut 2q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{17}\mathstrut -\mathstrut \) \(5\) \(x^{16}\mathstrut -\mathstrut \) \(15\) \(x^{15}\mathstrut +\mathstrut \) \(105\) \(x^{14}\mathstrut +\mathstrut \) \(45\) \(x^{13}\mathstrut -\mathstrut \) \(849\) \(x^{12}\mathstrut +\mathstrut \) \(320\) \(x^{11}\mathstrut +\mathstrut \) \(3371\) \(x^{10}\mathstrut -\mathstrut \) \(2456\) \(x^{9}\mathstrut -\mathstrut \) \(7002\) \(x^{8}\mathstrut +\mathstrut \) \(6279\) \(x^{7}\mathstrut +\mathstrut \) \(7299\) \(x^{6}\mathstrut -\mathstrut \) \(7119\) \(x^{5}\mathstrut -\mathstrut \) \(3066\) \(x^{4}\mathstrut +\mathstrut \) \(3184\) \(x^{3}\mathstrut +\mathstrut \) \(99\) \(x^{2}\mathstrut -\mathstrut \) \(231\) \(x\mathstrut +\mathstrut \) \(24\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu - 1 \)
\(\beta_{4}\)\(=\)\((\)\(685\) \(\nu^{16}\mathstrut -\mathstrut \) \(1978\) \(\nu^{15}\mathstrut -\mathstrut \) \(23375\) \(\nu^{14}\mathstrut +\mathstrut \) \(80742\) \(\nu^{13}\mathstrut +\mathstrut \) \(247035\) \(\nu^{12}\mathstrut -\mathstrut \) \(1063752\) \(\nu^{11}\mathstrut -\mathstrut \) \(927300\) \(\nu^{10}\mathstrut +\mathstrut \) \(6222061\) \(\nu^{9}\mathstrut +\mathstrut \) \(79227\) \(\nu^{8}\mathstrut -\mathstrut \) \(17370991\) \(\nu^{7}\mathstrut +\mathstrut \) \(6231634\) \(\nu^{6}\mathstrut +\mathstrut \) \(22769593\) \(\nu^{5}\mathstrut -\mathstrut \) \(11572552\) \(\nu^{4}\mathstrut -\mathstrut \) \(11774626\) \(\nu^{3}\mathstrut +\mathstrut \) \(6374744\) \(\nu^{2}\mathstrut +\mathstrut \) \(982243\) \(\nu\mathstrut -\mathstrut \) \(368684\)\()/31022\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(1070\) \(\nu^{16}\mathstrut +\mathstrut \) \(1165\) \(\nu^{15}\mathstrut +\mathstrut \) \(40249\) \(\nu^{14}\mathstrut -\mathstrut \) \(75174\) \(\nu^{13}\mathstrut -\mathstrut \) \(484380\) \(\nu^{12}\mathstrut +\mathstrut \) \(1137763\) \(\nu^{11}\mathstrut +\mathstrut \) \(2444697\) \(\nu^{10}\mathstrut -\mathstrut \) \(7082375\) \(\nu^{9}\mathstrut -\mathstrut \) \(4941450\) \(\nu^{8}\mathstrut +\mathstrut \) \(20342129\) \(\nu^{7}\mathstrut +\mathstrut \) \(1670236\) \(\nu^{6}\mathstrut -\mathstrut \) \(26893622\) \(\nu^{5}\mathstrut +\mathstrut \) \(5596139\) \(\nu^{4}\mathstrut +\mathstrut \) \(13977281\) \(\nu^{3}\mathstrut -\mathstrut \) \(4729743\) \(\nu^{2}\mathstrut -\mathstrut \) \(1384065\) \(\nu\mathstrut +\mathstrut \) \(300552\)\()/31022\)
\(\beta_{6}\)\(=\)\((\)\(6613\) \(\nu^{16}\mathstrut -\mathstrut \) \(17420\) \(\nu^{15}\mathstrut -\mathstrut \) \(180375\) \(\nu^{14}\mathstrut +\mathstrut \) \(520530\) \(\nu^{13}\mathstrut +\mathstrut \) \(1766251\) \(\nu^{12}\mathstrut -\mathstrut \) \(5814854\) \(\nu^{11}\mathstrut -\mathstrut \) \(7314342\) \(\nu^{10}\mathstrut +\mathstrut \) \(30806921\) \(\nu^{9}\mathstrut +\mathstrut \) \(9574065\) \(\nu^{8}\mathstrut -\mathstrut \) \(81028001\) \(\nu^{7}\mathstrut +\mathstrut \) \(12506146\) \(\nu^{6}\mathstrut +\mathstrut \) \(103349077\) \(\nu^{5}\mathstrut -\mathstrut \) \(39644890\) \(\nu^{4}\mathstrut -\mathstrut \) \(55867828\) \(\nu^{3}\mathstrut +\mathstrut \) \(25227292\) \(\nu^{2}\mathstrut +\mathstrut \) \(7528791\) \(\nu\mathstrut -\mathstrut \) \(1639268\)\()/62044\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(7257\) \(\nu^{16}\mathstrut +\mathstrut \) \(30443\) \(\nu^{15}\mathstrut +\mathstrut \) \(137250\) \(\nu^{14}\mathstrut -\mathstrut \) \(693400\) \(\nu^{13}\mathstrut -\mathstrut \) \(837779\) \(\nu^{12}\mathstrut +\mathstrut \) \(6167935\) \(\nu^{11}\mathstrut +\mathstrut \) \(943861\) \(\nu^{10}\mathstrut -\mathstrut \) \(27194200\) \(\nu^{9}\mathstrut +\mathstrut \) \(8863387\) \(\nu^{8}\mathstrut +\mathstrut \) \(62567552\) \(\nu^{7}\mathstrut -\mathstrut \) \(34862246\) \(\nu^{6}\mathstrut -\mathstrut \) \(72621151\) \(\nu^{5}\mathstrut +\mathstrut \) \(46524231\) \(\nu^{4}\mathstrut +\mathstrut \) \(36780711\) \(\nu^{3}\mathstrut -\mathstrut \) \(21561069\) \(\nu^{2}\mathstrut -\mathstrut \) \(4918520\) \(\nu\mathstrut +\mathstrut \) \(1063920\)\()/62044\)
\(\beta_{8}\)\(=\)\((\)\(6850\) \(\nu^{16}\mathstrut -\mathstrut \) \(35291\) \(\nu^{15}\mathstrut -\mathstrut \) \(94151\) \(\nu^{14}\mathstrut +\mathstrut \) \(714354\) \(\nu^{13}\mathstrut +\mathstrut \) \(174722\) \(\nu^{12}\mathstrut -\mathstrut \) \(5503379\) \(\nu^{11}\mathstrut +\mathstrut \) \(2779047\) \(\nu^{10}\mathstrut +\mathstrut \) \(20604597\) \(\nu^{9}\mathstrut -\mathstrut \) \(16766182\) \(\nu^{8}\mathstrut -\mathstrut \) \(40485931\) \(\nu^{7}\mathstrut +\mathstrut \) \(37622828\) \(\nu^{6}\mathstrut +\mathstrut \) \(41160644\) \(\nu^{5}\mathstrut -\mathstrut \) \(37720701\) \(\nu^{4}\mathstrut -\mathstrut \) \(18925679\) \(\nu^{3}\mathstrut +\mathstrut \) \(14593081\) \(\nu^{2}\mathstrut +\mathstrut \) \(2423683\) \(\nu\mathstrut -\mathstrut \) \(739750\)\()/31022\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(14426\) \(\nu^{16}\mathstrut +\mathstrut \) \(61515\) \(\nu^{15}\mathstrut +\mathstrut \) \(278177\) \(\nu^{14}\mathstrut -\mathstrut \) \(1417322\) \(\nu^{13}\mathstrut -\mathstrut \) \(1771986\) \(\nu^{12}\mathstrut +\mathstrut \) \(12781181\) \(\nu^{11}\mathstrut +\mathstrut \) \(2666987\) \(\nu^{10}\mathstrut -\mathstrut \) \(57266037\) \(\nu^{9}\mathstrut +\mathstrut \) \(14936458\) \(\nu^{8}\mathstrut +\mathstrut \) \(134174293\) \(\nu^{7}\mathstrut -\mathstrut \) \(65290780\) \(\nu^{6}\mathstrut -\mathstrut \) \(158712488\) \(\nu^{5}\mathstrut +\mathstrut \) \(91148769\) \(\nu^{4}\mathstrut +\mathstrut \) \(81480431\) \(\nu^{3}\mathstrut -\mathstrut \) \(44547125\) \(\nu^{2}\mathstrut -\mathstrut \) \(10787091\) \(\nu\mathstrut +\mathstrut \) \(2544040\)\()/62044\)
\(\beta_{10}\)\(=\)\((\)\(7535\) \(\nu^{16}\mathstrut -\mathstrut \) \(37269\) \(\nu^{15}\mathstrut -\mathstrut \) \(117526\) \(\nu^{14}\mathstrut +\mathstrut \) \(795096\) \(\nu^{13}\mathstrut +\mathstrut \) \(421757\) \(\nu^{12}\mathstrut -\mathstrut \) \(6567131\) \(\nu^{11}\mathstrut +\mathstrut \) \(1851747\) \(\nu^{10}\mathstrut +\mathstrut \) \(26826658\) \(\nu^{9}\mathstrut -\mathstrut \) \(16686955\) \(\nu^{8}\mathstrut -\mathstrut \) \(57856922\) \(\nu^{7}\mathstrut +\mathstrut \) \(43854462\) \(\nu^{6}\mathstrut +\mathstrut \) \(63961259\) \(\nu^{5}\mathstrut -\mathstrut \) \(49324275\) \(\nu^{4}\mathstrut -\mathstrut \) \(30979503\) \(\nu^{3}\mathstrut +\mathstrut \) \(21184979\) \(\nu^{2}\mathstrut +\mathstrut \) \(3902278\) \(\nu\mathstrut -\mathstrut \) \(1263544\)\()/31022\)
\(\beta_{11}\)\(=\)\((\)\(7535\) \(\nu^{16}\mathstrut -\mathstrut \) \(37269\) \(\nu^{15}\mathstrut -\mathstrut \) \(117526\) \(\nu^{14}\mathstrut +\mathstrut \) \(795096\) \(\nu^{13}\mathstrut +\mathstrut \) \(421757\) \(\nu^{12}\mathstrut -\mathstrut \) \(6567131\) \(\nu^{11}\mathstrut +\mathstrut \) \(1851747\) \(\nu^{10}\mathstrut +\mathstrut \) \(26826658\) \(\nu^{9}\mathstrut -\mathstrut \) \(16686955\) \(\nu^{8}\mathstrut -\mathstrut \) \(57856922\) \(\nu^{7}\mathstrut +\mathstrut \) \(43854462\) \(\nu^{6}\mathstrut +\mathstrut \) \(63961259\) \(\nu^{5}\mathstrut -\mathstrut \) \(49293253\) \(\nu^{4}\mathstrut -\mathstrut \) \(30979503\) \(\nu^{3}\mathstrut +\mathstrut \) \(20936803\) \(\nu^{2}\mathstrut +\mathstrut \) \(3871256\) \(\nu\mathstrut -\mathstrut \) \(1015368\)\()/31022\)
\(\beta_{12}\)\(=\)\((\)\(15238\) \(\nu^{16}\mathstrut -\mathstrut \) \(70517\) \(\nu^{15}\mathstrut -\mathstrut \) \(270335\) \(\nu^{14}\mathstrut +\mathstrut \) \(1582686\) \(\nu^{13}\mathstrut +\mathstrut \) \(1432154\) \(\nu^{12}\mathstrut -\mathstrut \) \(13900087\) \(\nu^{11}\mathstrut +\mathstrut \) \(185359\) \(\nu^{10}\mathstrut +\mathstrut \) \(60904047\) \(\nu^{9}\mathstrut -\mathstrut \) \(25130162\) \(\nu^{8}\mathstrut -\mathstrut \) \(141089899\) \(\nu^{7}\mathstrut +\mathstrut \) \(83217368\) \(\nu^{6}\mathstrut +\mathstrut \) \(167347268\) \(\nu^{5}\mathstrut -\mathstrut \) \(106903155\) \(\nu^{4}\mathstrut -\mathstrut \) \(87031309\) \(\nu^{3}\mathstrut +\mathstrut \) \(50859751\) \(\nu^{2}\mathstrut +\mathstrut \) \(11368909\) \(\nu\mathstrut -\mathstrut \) \(3029808\)\()/62044\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(5003\) \(\nu^{16}\mathstrut +\mathstrut \) \(23640\) \(\nu^{15}\mathstrut +\mathstrut \) \(78789\) \(\nu^{14}\mathstrut -\mathstrut \) \(491641\) \(\nu^{13}\mathstrut -\mathstrut \) \(318258\) \(\nu^{12}\mathstrut +\mathstrut \) \(3931692\) \(\nu^{11}\mathstrut -\mathstrut \) \(700684\) \(\nu^{10}\mathstrut -\mathstrut \) \(15463187\) \(\nu^{9}\mathstrut +\mathstrut \) \(8035982\) \(\nu^{8}\mathstrut +\mathstrut \) \(32135471\) \(\nu^{7}\mathstrut -\mathstrut \) \(21250233\) \(\nu^{6}\mathstrut -\mathstrut \) \(34474069\) \(\nu^{5}\mathstrut +\mathstrut \) \(23881305\) \(\nu^{4}\mathstrut +\mathstrut \) \(16421556\) \(\nu^{3}\mathstrut -\mathstrut \) \(10374066\) \(\nu^{2}\mathstrut -\mathstrut \) \(1994666\) \(\nu\mathstrut +\mathstrut \) \(580367\)\()/15511\)
\(\beta_{14}\)\(=\)\((\)\(-\)\(12523\) \(\nu^{16}\mathstrut +\mathstrut \) \(61545\) \(\nu^{15}\mathstrut +\mathstrut \) \(189010\) \(\nu^{14}\mathstrut -\mathstrut \) \(1274666\) \(\nu^{13}\mathstrut -\mathstrut \) \(638709\) \(\nu^{12}\mathstrut +\mathstrut \) \(10147647\) \(\nu^{11}\mathstrut -\mathstrut \) \(2869597\) \(\nu^{10}\mathstrut -\mathstrut \) \(39770336\) \(\nu^{9}\mathstrut +\mathstrut \) \(23674113\) \(\nu^{8}\mathstrut +\mathstrut \) \(82744596\) \(\nu^{7}\mathstrut -\mathstrut \) \(58289788\) \(\nu^{6}\mathstrut -\mathstrut \) \(89735141\) \(\nu^{5}\mathstrut +\mathstrut \) \(62257615\) \(\nu^{4}\mathstrut +\mathstrut \) \(43991657\) \(\nu^{3}\mathstrut -\mathstrut \) \(25895951\) \(\nu^{2}\mathstrut -\mathstrut \) \(5969520\) \(\nu\mathstrut +\mathstrut \) \(1508748\)\()/31022\)
\(\beta_{15}\)\(=\)\((\)\(-\)\(16906\) \(\nu^{16}\mathstrut +\mathstrut \) \(80451\) \(\nu^{15}\mathstrut +\mathstrut \) \(276079\) \(\nu^{14}\mathstrut -\mathstrut \) \(1727532\) \(\nu^{13}\mathstrut -\mathstrut \) \(1200628\) \(\nu^{12}\mathstrut +\mathstrut \) \(14402921\) \(\nu^{11}\mathstrut -\mathstrut \) \(2192535\) \(\nu^{10}\mathstrut -\mathstrut \) \(59629455\) \(\nu^{9}\mathstrut +\mathstrut \) \(30378002\) \(\nu^{8}\mathstrut +\mathstrut \) \(130973989\) \(\nu^{7}\mathstrut -\mathstrut \) \(86449694\) \(\nu^{6}\mathstrut -\mathstrut \) \(148302258\) \(\nu^{5}\mathstrut +\mathstrut \) \(102577437\) \(\nu^{4}\mathstrut +\mathstrut \) \(74063549\) \(\nu^{3}\mathstrut -\mathstrut \) \(46121451\) \(\nu^{2}\mathstrut -\mathstrut \) \(9366361\) \(\nu\mathstrut +\mathstrut \) \(2614408\)\()/31022\)
\(\beta_{16}\)\(=\)\((\)\(-\)\(38615\) \(\nu^{16}\mathstrut +\mathstrut \) \(177511\) \(\nu^{15}\mathstrut +\mathstrut \) \(642124\) \(\nu^{14}\mathstrut -\mathstrut \) \(3780588\) \(\nu^{13}\mathstrut -\mathstrut \) \(3038785\) \(\nu^{12}\mathstrut +\mathstrut \) \(31183239\) \(\nu^{11}\mathstrut -\mathstrut \) \(2039975\) \(\nu^{10}\mathstrut -\mathstrut \) \(127378426\) \(\nu^{9}\mathstrut +\mathstrut \) \(54823177\) \(\nu^{8}\mathstrut +\mathstrut \) \(275564722\) \(\nu^{7}\mathstrut -\mathstrut \) \(161135054\) \(\nu^{6}\mathstrut -\mathstrut \) \(306983937\) \(\nu^{5}\mathstrut +\mathstrut \) \(190626927\) \(\nu^{4}\mathstrut +\mathstrut \) \(150613483\) \(\nu^{3}\mathstrut -\mathstrut \) \(84980589\) \(\nu^{2}\mathstrut -\mathstrut \) \(18623890\) \(\nu\mathstrut +\mathstrut \) \(4923384\)\()/62044\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(8\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(16\)
\(\nu^{5}\)\(=\)\(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(9\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(30\) \(\beta_{1}\mathstrut +\mathstrut \) \(9\)
\(\nu^{6}\)\(=\)\(11\) \(\beta_{11}\mathstrut -\mathstrut \) \(9\) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(57\) \(\beta_{2}\mathstrut +\mathstrut \) \(11\) \(\beta_{1}\mathstrut +\mathstrut \) \(99\)
\(\nu^{7}\)\(=\)\(-\)\(\beta_{16}\mathstrut +\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{13}\mathstrut +\mathstrut \) \(14\) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(11\) \(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(12\) \(\beta_{4}\mathstrut +\mathstrut \) \(69\) \(\beta_{3}\mathstrut +\mathstrut \) \(15\) \(\beta_{2}\mathstrut +\mathstrut \) \(195\) \(\beta_{1}\mathstrut +\mathstrut \) \(71\)
\(\nu^{8}\)\(=\)\(-\)\(2\) \(\beta_{16}\mathstrut -\mathstrut \) \(\beta_{15}\mathstrut +\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(4\) \(\beta_{13}\mathstrut -\mathstrut \) \(2\) \(\beta_{12}\mathstrut +\mathstrut \) \(95\) \(\beta_{11}\mathstrut -\mathstrut \) \(67\) \(\beta_{10}\mathstrut +\mathstrut \) \(12\) \(\beta_{9}\mathstrut -\mathstrut \) \(13\) \(\beta_{8}\mathstrut +\mathstrut \) \(16\) \(\beta_{7}\mathstrut +\mathstrut \) \(14\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(15\) \(\beta_{4}\mathstrut +\mathstrut \) \(33\) \(\beta_{3}\mathstrut +\mathstrut \) \(396\) \(\beta_{2}\mathstrut +\mathstrut \) \(100\) \(\beta_{1}\mathstrut +\mathstrut \) \(654\)
\(\nu^{9}\)\(=\)\(-\)\(16\) \(\beta_{16}\mathstrut -\mathstrut \) \(\beta_{15}\mathstrut +\mathstrut \) \(15\) \(\beta_{14}\mathstrut +\mathstrut \) \(20\) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(141\) \(\beta_{11}\mathstrut -\mathstrut \) \(17\) \(\beta_{10}\mathstrut +\mathstrut \) \(16\) \(\beta_{9}\mathstrut -\mathstrut \) \(89\) \(\beta_{8}\mathstrut +\mathstrut \) \(35\) \(\beta_{7}\mathstrut +\mathstrut \) \(19\) \(\beta_{6}\mathstrut -\mathstrut \) \(110\) \(\beta_{4}\mathstrut +\mathstrut \) \(510\) \(\beta_{3}\mathstrut +\mathstrut \) \(160\) \(\beta_{2}\mathstrut +\mathstrut \) \(1317\) \(\beta_{1}\mathstrut +\mathstrut \) \(556\)
\(\nu^{10}\)\(=\)\(-\)\(34\) \(\beta_{16}\mathstrut -\mathstrut \) \(21\) \(\beta_{15}\mathstrut +\mathstrut \) \(18\) \(\beta_{14}\mathstrut +\mathstrut \) \(73\) \(\beta_{13}\mathstrut -\mathstrut \) \(38\) \(\beta_{12}\mathstrut +\mathstrut \) \(758\) \(\beta_{11}\mathstrut -\mathstrut \) \(475\) \(\beta_{10}\mathstrut +\mathstrut \) \(108\) \(\beta_{9}\mathstrut -\mathstrut \) \(119\) \(\beta_{8}\mathstrut +\mathstrut \) \(182\) \(\beta_{7}\mathstrut +\mathstrut \) \(146\) \(\beta_{6}\mathstrut +\mathstrut \) \(16\) \(\beta_{5}\mathstrut -\mathstrut \) \(160\) \(\beta_{4}\mathstrut +\mathstrut \) \(374\) \(\beta_{3}\mathstrut +\mathstrut \) \(2735\) \(\beta_{2}\mathstrut +\mathstrut \) \(863\) \(\beta_{1}\mathstrut +\mathstrut \) \(4461\)
\(\nu^{11}\)\(=\)\(-\)\(175\) \(\beta_{16}\mathstrut -\mathstrut \) \(25\) \(\beta_{15}\mathstrut +\mathstrut \) \(157\) \(\beta_{14}\mathstrut +\mathstrut \) \(258\) \(\beta_{13}\mathstrut -\mathstrut \) \(29\) \(\beta_{12}\mathstrut +\mathstrut \) \(1254\) \(\beta_{11}\mathstrut -\mathstrut \) \(192\) \(\beta_{10}\mathstrut +\mathstrut \) \(177\) \(\beta_{9}\mathstrut -\mathstrut \) \(643\) \(\beta_{8}\mathstrut +\mathstrut \) \(419\) \(\beta_{7}\mathstrut +\mathstrut \) \(240\) \(\beta_{6}\mathstrut -\mathstrut \) \(3\) \(\beta_{5}\mathstrut -\mathstrut \) \(920\) \(\beta_{4}\mathstrut +\mathstrut \) \(3738\) \(\beta_{3}\mathstrut +\mathstrut \) \(1489\) \(\beta_{2}\mathstrut +\mathstrut \) \(9078\) \(\beta_{1}\mathstrut +\mathstrut \) \(4378\)
\(\nu^{12}\)\(=\)\(-\)\(387\) \(\beta_{16}\mathstrut -\mathstrut \) \(280\) \(\beta_{15}\mathstrut +\mathstrut \) \(209\) \(\beta_{14}\mathstrut +\mathstrut \) \(891\) \(\beta_{13}\mathstrut -\mathstrut \) \(477\) \(\beta_{12}\mathstrut +\mathstrut \) \(5849\) \(\beta_{11}\mathstrut -\mathstrut \) \(3319\) \(\beta_{10}\mathstrut +\mathstrut \) \(891\) \(\beta_{9}\mathstrut -\mathstrut \) \(946\) \(\beta_{8}\mathstrut +\mathstrut \) \(1795\) \(\beta_{7}\mathstrut +\mathstrut \) \(1366\) \(\beta_{6}\mathstrut +\mathstrut \) \(166\) \(\beta_{5}\mathstrut -\mathstrut \) \(1491\) \(\beta_{4}\mathstrut +\mathstrut \) \(3637\) \(\beta_{3}\mathstrut +\mathstrut \) \(18897\) \(\beta_{2}\mathstrut +\mathstrut \) \(7255\) \(\beta_{1}\mathstrut +\mathstrut \) \(30985\)
\(\nu^{13}\)\(=\)\(-\)\(1639\) \(\beta_{16}\mathstrut -\mathstrut \) \(379\) \(\beta_{15}\mathstrut +\mathstrut \) \(1418\) \(\beta_{14}\mathstrut +\mathstrut \) \(2756\) \(\beta_{13}\mathstrut -\mathstrut \) \(465\) \(\beta_{12}\mathstrut +\mathstrut \) \(10502\) \(\beta_{11}\mathstrut -\mathstrut \) \(1842\) \(\beta_{10}\mathstrut +\mathstrut \) \(1700\) \(\beta_{9}\mathstrut -\mathstrut \) \(4398\) \(\beta_{8}\mathstrut +\mathstrut \) \(4260\) \(\beta_{7}\mathstrut +\mathstrut \) \(2539\) \(\beta_{6}\mathstrut -\mathstrut \) \(73\) \(\beta_{5}\mathstrut -\mathstrut \) \(7381\) \(\beta_{4}\mathstrut +\mathstrut \) \(27380\) \(\beta_{3}\mathstrut +\mathstrut \) \(12932\) \(\beta_{2}\mathstrut +\mathstrut \) \(63356\) \(\beta_{1}\mathstrut +\mathstrut \) \(34535\)
\(\nu^{14}\)\(=\)\(-\)\(3726\) \(\beta_{16}\mathstrut -\mathstrut \) \(3056\) \(\beta_{15}\mathstrut +\mathstrut \) \(2014\) \(\beta_{14}\mathstrut +\mathstrut \) \(9160\) \(\beta_{13}\mathstrut -\mathstrut \) \(4993\) \(\beta_{12}\mathstrut +\mathstrut \) \(44424\) \(\beta_{11}\mathstrut -\mathstrut \) \(23120\) \(\beta_{10}\mathstrut +\mathstrut \) \(7152\) \(\beta_{9}\mathstrut -\mathstrut \) \(6988\) \(\beta_{8}\mathstrut +\mathstrut \) \(16345\) \(\beta_{7}\mathstrut +\mathstrut \) \(12081\) \(\beta_{6}\mathstrut +\mathstrut \) \(1413\) \(\beta_{5}\mathstrut -\mathstrut \) \(12968\) \(\beta_{4}\mathstrut +\mathstrut \) \(32623\) \(\beta_{3}\mathstrut +\mathstrut \) \(130973\) \(\beta_{2}\mathstrut +\mathstrut \) \(59806\) \(\beta_{1}\mathstrut +\mathstrut \) \(217789\)
\(\nu^{15}\)\(=\)\(-\)\(14164\) \(\beta_{16}\mathstrut -\mathstrut \) \(4567\) \(\beta_{15}\mathstrut +\mathstrut \) \(11851\) \(\beta_{14}\mathstrut +\mathstrut \) \(26556\) \(\beta_{13}\mathstrut -\mathstrut \) \(5729\) \(\beta_{12}\mathstrut +\mathstrut \) \(85086\) \(\beta_{11}\mathstrut -\mathstrut \) \(16278\) \(\beta_{10}\mathstrut +\mathstrut \) \(15288\) \(\beta_{9}\mathstrut -\mathstrut \) \(29152\) \(\beta_{8}\mathstrut +\mathstrut \) \(39557\) \(\beta_{7}\mathstrut +\mathstrut \) \(24381\) \(\beta_{6}\mathstrut -\mathstrut \) \(1106\) \(\beta_{5}\mathstrut -\mathstrut \) \(57918\) \(\beta_{4}\mathstrut +\mathstrut \) \(200835\) \(\beta_{3}\mathstrut +\mathstrut \) \(107939\) \(\beta_{2}\mathstrut +\mathstrut \) \(446006\) \(\beta_{1}\mathstrut +\mathstrut \) \(271845\)
\(\nu^{16}\)\(=\)\(-\)\(32872\) \(\beta_{16}\mathstrut -\mathstrut \) \(29872\) \(\beta_{15}\mathstrut +\mathstrut \) \(17602\) \(\beta_{14}\mathstrut +\mathstrut \) \(85786\) \(\beta_{13}\mathstrut -\mathstrut \) \(47253\) \(\beta_{12}\mathstrut +\mathstrut \) \(334746\) \(\beta_{11}\mathstrut -\mathstrut \) \(161278\) \(\beta_{10}\mathstrut +\mathstrut \) \(57104\) \(\beta_{9}\mathstrut -\mathstrut \) \(49396\) \(\beta_{8}\mathstrut +\mathstrut \) \(141417\) \(\beta_{7}\mathstrut +\mathstrut \) \(103157\) \(\beta_{6}\mathstrut +\mathstrut \) \(10648\) \(\beta_{5}\mathstrut -\mathstrut \) \(108361\) \(\beta_{4}\mathstrut +\mathstrut \) \(278635\) \(\beta_{3}\mathstrut +\mathstrut \) \(911836\) \(\beta_{2}\mathstrut +\mathstrut \) \(485008\) \(\beta_{1}\mathstrut +\mathstrut \) \(1544374\)

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
2.74886
2.71108
2.50383
2.12059
1.89951
1.69575
1.08305
0.829580
0.184653
0.159847
−0.320785
−1.02928
−1.30924
−1.58492
−1.59605
−2.53341
−2.56306
−2.74886 0 5.55624 −1.00000 0 4.26511 −9.77563 0 2.74886
1.2 −2.71108 0 5.34994 −1.00000 0 −0.946340 −9.08194 0 2.71108
1.3 −2.50383 0 4.26915 −1.00000 0 −2.88177 −5.68155 0 2.50383
1.4 −2.12059 0 2.49688 −1.00000 0 4.22149 −1.05368 0 2.12059
1.5 −1.89951 0 1.60815 −1.00000 0 2.75204 0.744322 0 1.89951
1.6 −1.69575 0 0.875568 −1.00000 0 −1.30460 1.90676 0 1.69575
1.7 −1.08305 0 −0.826999 −1.00000 0 −2.85554 3.06179 0 1.08305
1.8 −0.829580 0 −1.31180 −1.00000 0 1.06171 2.74740 0 0.829580
1.9 −0.184653 0 −1.96590 −1.00000 0 2.68806 0.732314 0 0.184653
1.10 −0.159847 0 −1.97445 −1.00000 0 −1.46854 0.635302 0 0.159847
1.11 0.320785 0 −1.89710 −1.00000 0 4.94917 −1.25013 0 −0.320785
1.12 1.02928 0 −0.940578 −1.00000 0 −4.61830 −3.02668 0 −1.02928
1.13 1.30924 0 −0.285886 −1.00000 0 4.28141 −2.99278 0 −1.30924
1.14 1.58492 0 0.511971 −1.00000 0 −2.52227 −2.35841 0 −1.58492
1.15 1.59605 0 0.547372 −1.00000 0 1.65189 −2.31847 0 −1.59605
1.16 2.53341 0 4.41816 −1.00000 0 3.58669 6.12619 0 −2.53341
1.17 2.56306 0 4.56927 −1.00000 0 −0.860224 6.58519 0 −2.56306
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(89\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4005))\):

\(T_{2}^{17} + \cdots\)
\(T_{7}^{17} - \cdots\)
\(T_{11}^{17} + \cdots\)