Properties

Label 8001.2.a.u
Level 8001
Weight 2
Character orbit 8001.a
Self dual Yes
Analytic conductor 63.888
Analytic rank 0
Dimension 18
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8001.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 5 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( 1 + \beta_{2} ) q^{4} \) \( + ( 1 - \beta_{12} ) q^{5} \) \(+ q^{7}\) \( + ( 1 + \beta_{2} + \beta_{7} - \beta_{10} + \beta_{14} + \beta_{16} ) q^{8} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( + ( 1 + \beta_{2} ) q^{4} \) \( + ( 1 - \beta_{12} ) q^{5} \) \(+ q^{7}\) \( + ( 1 + \beta_{2} + \beta_{7} - \beta_{10} + \beta_{14} + \beta_{16} ) q^{8} \) \( + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{12} - \beta_{13} ) q^{10} \) \( + ( 1 - \beta_{4} ) q^{11} \) \( + ( -1 - \beta_{6} ) q^{13} \) \( + \beta_{1} q^{14} \) \( + ( 1 + 2 \beta_{2} + \beta_{6} + \beta_{7} - \beta_{10} + \beta_{13} + \beta_{14} + \beta_{15} + \beta_{16} ) q^{16} \) \( + ( 1 - \beta_{1} - \beta_{3} - \beta_{5} - \beta_{13} - \beta_{17} ) q^{17} \) \( + ( -1 - \beta_{5} - \beta_{9} - \beta_{15} ) q^{19} \) \( + ( 3 - \beta_{3} - \beta_{4} - \beta_{7} + \beta_{10} - 2 \beta_{12} - \beta_{13} + \beta_{15} - \beta_{16} + \beta_{17} ) q^{20} \) \( + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{9} + \beta_{10} + \beta_{11} - 2 \beta_{12} - \beta_{13} + \beta_{14} ) q^{22} \) \( + ( \beta_{1} - \beta_{2} - \beta_{5} - \beta_{7} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{14} - \beta_{15} ) q^{23} \) \( + ( 1 - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} - \beta_{12} + \beta_{14} + \beta_{16} - \beta_{17} ) q^{25} \) \( + ( 1 - \beta_{1} + \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{10} + \beta_{11} - \beta_{14} - 2 \beta_{16} ) q^{26} \) \( + ( 1 + \beta_{2} ) q^{28} \) \( + ( -\beta_{1} + \beta_{2} + \beta_{7} - \beta_{11} + \beta_{12} + \beta_{13} + \beta_{15} ) q^{29} \) \( + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{7} + \beta_{11} - \beta_{15} ) q^{31} \) \( + ( 2 + \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} - \beta_{10} - \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} ) q^{32} \) \( + ( -2 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{16} ) q^{34} \) \( + ( 1 - \beta_{12} ) q^{35} \) \( + ( -\beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{14} + 2 \beta_{15} + \beta_{16} - \beta_{17} ) q^{37} \) \( + ( 1 + 2 \beta_{2} + \beta_{5} + \beta_{8} + \beta_{10} + \beta_{11} + \beta_{13} + \beta_{15} ) q^{38} \) \( + ( 3 - \beta_{3} + 3 \beta_{5} + 2 \beta_{7} - 2 \beta_{8} - 3 \beta_{10} - \beta_{11} - 3 \beta_{12} + 2 \beta_{14} + \beta_{15} + \beta_{16} + 2 \beta_{17} ) q^{40} \) \( + ( 1 - \beta_{2} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{10} - \beta_{13} - \beta_{15} ) q^{41} \) \( + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{9} + \beta_{10} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{43} \) \( + ( 4 - \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + 3 \beta_{5} + \beta_{6} + 3 \beta_{7} - 2 \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} + 3 \beta_{15} + 2 \beta_{16} - \beta_{17} ) q^{44} \) \( + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{12} + 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{46} \) \( + ( 1 - \beta_{1} + 2 \beta_{4} - \beta_{5} - \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{13} - 2 \beta_{15} ) q^{47} \) \(+ q^{49}\) \( + ( 1 + 2 \beta_{2} - \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} + \beta_{16} - \beta_{17} ) q^{50} \) \( + ( -4 + 3 \beta_{1} - 4 \beta_{2} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - \beta_{9} + 3 \beta_{10} + \beta_{11} - \beta_{14} - \beta_{15} - 2 \beta_{16} ) q^{52} \) \( + ( 2 + 2 \beta_{5} + \beta_{6} + 3 \beta_{7} - 2 \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{12} + \beta_{13} + \beta_{16} + \beta_{17} ) q^{53} \) \( + ( 1 + \beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{5} + \beta_{7} - \beta_{10} - \beta_{11} + 2 \beta_{15} ) q^{55} \) \( + ( 1 + \beta_{2} + \beta_{7} - \beta_{10} + \beta_{14} + \beta_{16} ) q^{56} \) \( + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{9} + \beta_{11} + 2 \beta_{12} - 2 \beta_{15} ) q^{58} \) \( + ( 2 - \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{13} - \beta_{15} - \beta_{16} + \beta_{17} ) q^{59} \) \( + ( 1 - \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} - 2 \beta_{13} - 3 \beta_{15} - 2 \beta_{16} + 2 \beta_{17} ) q^{61} \) \( + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} + \beta_{12} + \beta_{15} - \beta_{17} ) q^{62} \) \( + ( 3 + \beta_{1} + 3 \beta_{2} + \beta_{6} + 2 \beta_{7} - 3 \beta_{10} - 2 \beta_{11} + \beta_{14} + \beta_{15} + 2 \beta_{16} ) q^{64} \) \( + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + \beta_{10} + \beta_{11} + 2 \beta_{12} - 2 \beta_{15} - \beta_{17} ) q^{65} \) \( + ( -\beta_{2} - \beta_{3} + \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{10} - \beta_{11} + \beta_{13} ) q^{67} \) \( + ( 2 + \beta_{4} - \beta_{5} - \beta_{6} - 3 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{14} - 3 \beta_{15} - 2 \beta_{16} + \beta_{17} ) q^{68} \) \( + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{12} - \beta_{13} ) q^{70} \) \( + ( 1 - 3 \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} + 2 \beta_{14} + \beta_{15} + 2 \beta_{16} ) q^{71} \) \( + ( -2 + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} + \beta_{12} - 2 \beta_{14} - \beta_{15} - \beta_{16} ) q^{73} \) \( + ( 1 + 2 \beta_{1} - 4 \beta_{2} - \beta_{3} - \beta_{5} - 2 \beta_{7} + \beta_{8} + 2 \beta_{10} + \beta_{11} - 3 \beta_{12} - 2 \beta_{13} - \beta_{14} - 2 \beta_{15} + \beta_{17} ) q^{74} \) \( + ( 1 + 3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} - \beta_{13} + \beta_{14} - \beta_{15} + 2 \beta_{16} - \beta_{17} ) q^{76} \) \( + ( 1 - \beta_{4} ) q^{77} \) \( + ( 3 + \beta_{1} + \beta_{2} + 3 \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{12} + \beta_{13} + \beta_{15} + 2 \beta_{16} - \beta_{17} ) q^{79} \) \( + ( 7 - 4 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 6 \beta_{5} + \beta_{6} + 4 \beta_{7} - 4 \beta_{8} - 5 \beta_{10} - 2 \beta_{11} - 4 \beta_{12} + \beta_{13} + 2 \beta_{14} + 3 \beta_{15} + \beta_{16} + 3 \beta_{17} ) q^{80} \) \( + ( -2 + \beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{8} + \beta_{10} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} - \beta_{17} ) q^{82} \) \( + ( 2 - 2 \beta_{1} - 2 \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{12} - \beta_{17} ) q^{83} \) \( + ( -2 + 2 \beta_{1} - \beta_{2} - 3 \beta_{5} - 3 \beta_{7} + 2 \beta_{8} - \beta_{9} + 3 \beta_{10} - 2 \beta_{12} - \beta_{13} - \beta_{15} - 3 \beta_{16} - \beta_{17} ) q^{85} \) \( + ( 1 + 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{7} + \beta_{8} + \beta_{10} - \beta_{11} - 2 \beta_{12} - \beta_{14} + 2 \beta_{15} - \beta_{16} + \beta_{17} ) q^{86} \) \( + ( 4 + \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{7} - 3 \beta_{8} + \beta_{9} - 3 \beta_{10} - 2 \beta_{12} - 2 \beta_{13} + 3 \beta_{14} + 2 \beta_{16} + \beta_{17} ) q^{88} \) \( + ( 3 - \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{12} + \beta_{13} + 2 \beta_{15} ) q^{89} \) \( + ( -1 - \beta_{6} ) q^{91} \) \( + ( -2 + \beta_{1} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} + 3 \beta_{10} + \beta_{11} + 2 \beta_{12} + \beta_{13} - \beta_{14} - 3 \beta_{15} - 2 \beta_{17} ) q^{92} \) \( + ( -1 - \beta_{1} + 4 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} - 3 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} - \beta_{14} + 3 \beta_{15} - \beta_{16} + \beta_{17} ) q^{94} \) \( + ( -1 - \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{6} + 3 \beta_{7} - \beta_{8} + \beta_{10} - 2 \beta_{11} - \beta_{12} + \beta_{14} + 3 \beta_{15} - \beta_{17} ) q^{95} \) \( + ( -3 + 3 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{7} + \beta_{9} + 2 \beta_{10} - 2 \beta_{12} + \beta_{16} - \beta_{17} ) q^{97} \) \( + \beta_{1} q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(18q \) \(\mathstrut +\mathstrut 6q^{2} \) \(\mathstrut +\mathstrut 22q^{4} \) \(\mathstrut +\mathstrut 10q^{5} \) \(\mathstrut +\mathstrut 18q^{7} \) \(\mathstrut +\mathstrut 21q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(18q \) \(\mathstrut +\mathstrut 6q^{2} \) \(\mathstrut +\mathstrut 22q^{4} \) \(\mathstrut +\mathstrut 10q^{5} \) \(\mathstrut +\mathstrut 18q^{7} \) \(\mathstrut +\mathstrut 21q^{8} \) \(\mathstrut -\mathstrut 4q^{10} \) \(\mathstrut +\mathstrut 9q^{11} \) \(\mathstrut -\mathstrut 25q^{13} \) \(\mathstrut +\mathstrut 6q^{14} \) \(\mathstrut +\mathstrut 34q^{16} \) \(\mathstrut +\mathstrut 17q^{17} \) \(\mathstrut -\mathstrut 5q^{19} \) \(\mathstrut +\mathstrut 21q^{20} \) \(\mathstrut +\mathstrut 5q^{22} \) \(\mathstrut +\mathstrut 14q^{23} \) \(\mathstrut +\mathstrut 28q^{25} \) \(\mathstrut +\mathstrut 8q^{26} \) \(\mathstrut +\mathstrut 22q^{28} \) \(\mathstrut +\mathstrut 17q^{29} \) \(\mathstrut +\mathstrut 5q^{31} \) \(\mathstrut +\mathstrut 53q^{32} \) \(\mathstrut -\mathstrut 19q^{34} \) \(\mathstrut +\mathstrut 10q^{35} \) \(\mathstrut -\mathstrut 15q^{37} \) \(\mathstrut +\mathstrut 22q^{38} \) \(\mathstrut -\mathstrut q^{40} \) \(\mathstrut +\mathstrut 17q^{41} \) \(\mathstrut +\mathstrut q^{43} \) \(\mathstrut +\mathstrut 33q^{44} \) \(\mathstrut +\mathstrut 10q^{46} \) \(\mathstrut +\mathstrut 31q^{47} \) \(\mathstrut +\mathstrut 18q^{49} \) \(\mathstrut +\mathstrut 35q^{50} \) \(\mathstrut -\mathstrut 70q^{52} \) \(\mathstrut +\mathstrut 35q^{53} \) \(\mathstrut +\mathstrut 4q^{55} \) \(\mathstrut +\mathstrut 21q^{56} \) \(\mathstrut +\mathstrut 3q^{58} \) \(\mathstrut +\mathstrut 46q^{59} \) \(\mathstrut -\mathstrut 5q^{61} \) \(\mathstrut +\mathstrut 10q^{62} \) \(\mathstrut +\mathstrut 63q^{64} \) \(\mathstrut +\mathstrut 12q^{65} \) \(\mathstrut +\mathstrut 6q^{67} \) \(\mathstrut +\mathstrut 56q^{68} \) \(\mathstrut -\mathstrut 4q^{70} \) \(\mathstrut +\mathstrut 22q^{71} \) \(\mathstrut -\mathstrut 16q^{73} \) \(\mathstrut -\mathstrut 18q^{74} \) \(\mathstrut +\mathstrut 32q^{76} \) \(\mathstrut +\mathstrut 9q^{77} \) \(\mathstrut +\mathstrut 46q^{79} \) \(\mathstrut +\mathstrut 30q^{80} \) \(\mathstrut -\mathstrut 12q^{82} \) \(\mathstrut +\mathstrut 46q^{83} \) \(\mathstrut +\mathstrut 4q^{85} \) \(\mathstrut -\mathstrut 18q^{86} \) \(\mathstrut +\mathstrut 30q^{88} \) \(\mathstrut +\mathstrut 42q^{89} \) \(\mathstrut -\mathstrut 25q^{91} \) \(\mathstrut +\mathstrut 48q^{92} \) \(\mathstrut +\mathstrut 3q^{94} \) \(\mathstrut +\mathstrut 2q^{95} \) \(\mathstrut -\mathstrut 35q^{97} \) \(\mathstrut +\mathstrut 6q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18}\mathstrut -\mathstrut \) \(6\) \(x^{17}\mathstrut -\mathstrut \) \(11\) \(x^{16}\mathstrut +\mathstrut \) \(123\) \(x^{15}\mathstrut -\mathstrut \) \(35\) \(x^{14}\mathstrut -\mathstrut \) \(982\) \(x^{13}\mathstrut +\mathstrut \) \(988\) \(x^{12}\mathstrut +\mathstrut \) \(3872\) \(x^{11}\mathstrut -\mathstrut \) \(5421\) \(x^{10}\mathstrut -\mathstrut \) \(7882\) \(x^{9}\mathstrut +\mathstrut \) \(13376\) \(x^{8}\mathstrut +\mathstrut \) \(7948\) \(x^{7}\mathstrut -\mathstrut \) \(15795\) \(x^{6}\mathstrut -\mathstrut \) \(3858\) \(x^{5}\mathstrut +\mathstrut \) \(8199\) \(x^{4}\mathstrut +\mathstrut \) \(1453\) \(x^{3}\mathstrut -\mathstrut \) \(1610\) \(x^{2}\mathstrut -\mathstrut \) \(380\) \(x\mathstrut +\mathstrut \) \(16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(9056232\) \(\nu^{17}\mathstrut +\mathstrut \) \(56952713\) \(\nu^{16}\mathstrut +\mathstrut \) \(63697503\) \(\nu^{15}\mathstrut -\mathstrut \) \(1083863790\) \(\nu^{14}\mathstrut +\mathstrut \) \(1061485425\) \(\nu^{13}\mathstrut +\mathstrut \) \(7576909534\) \(\nu^{12}\mathstrut -\mathstrut \) \(14966699558\) \(\nu^{11}\mathstrut -\mathstrut \) \(22572883200\) \(\nu^{10}\mathstrut +\mathstrut \) \(73124433062\) \(\nu^{9}\mathstrut +\mathstrut \) \(17899620183\) \(\nu^{8}\mathstrut -\mathstrut \) \(170430820251\) \(\nu^{7}\mathstrut +\mathstrut \) \(41863070067\) \(\nu^{6}\mathstrut +\mathstrut \) \(192141347449\) \(\nu^{5}\mathstrut -\mathstrut \) \(79413530246\) \(\nu^{4}\mathstrut -\mathstrut \) \(95483657380\) \(\nu^{3}\mathstrut +\mathstrut \) \(30790742789\) \(\nu^{2}\mathstrut +\mathstrut \) \(19199072948\) \(\nu\mathstrut -\mathstrut \) \(228715354\)\()/\)\(365415070\)
\(\beta_{4}\)\(=\)\((\)\(12516449\) \(\nu^{17}\mathstrut -\mathstrut \) \(35576866\) \(\nu^{16}\mathstrut -\mathstrut \) \(322120881\) \(\nu^{15}\mathstrut +\mathstrut \) \(878461675\) \(\nu^{14}\mathstrut +\mathstrut \) \(3459985365\) \(\nu^{13}\mathstrut -\mathstrut \) \(8859247938\) \(\nu^{12}\mathstrut -\mathstrut \) \(20043398304\) \(\nu^{11}\mathstrut +\mathstrut \) \(46965576220\) \(\nu^{10}\mathstrut +\mathstrut \) \(67169495191\) \(\nu^{9}\mathstrut -\mathstrut \) \(139958996266\) \(\nu^{8}\mathstrut -\mathstrut \) \(129295866318\) \(\nu^{7}\mathstrut +\mathstrut \) \(231201547396\) \(\nu^{6}\mathstrut +\mathstrut \) \(134129812547\) \(\nu^{5}\mathstrut -\mathstrut \) \(191805387798\) \(\nu^{4}\mathstrut -\mathstrut \) \(67208803935\) \(\nu^{3}\mathstrut +\mathstrut \) \(60307885677\) \(\nu^{2}\mathstrut +\mathstrut \) \(15314260994\) \(\nu\mathstrut -\mathstrut \) \(1868332582\)\()/\)\(365415070\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(25066381\) \(\nu^{17}\mathstrut +\mathstrut \) \(165457944\) \(\nu^{16}\mathstrut +\mathstrut \) \(197605069\) \(\nu^{15}\mathstrut -\mathstrut \) \(3294331535\) \(\nu^{14}\mathstrut +\mathstrut \) \(2503016525\) \(\nu^{13}\mathstrut +\mathstrut \) \(24999769422\) \(\nu^{12}\mathstrut -\mathstrut \) \(37913753504\) \(\nu^{11}\mathstrut -\mathstrut \) \(89306242820\) \(\nu^{10}\mathstrut +\mathstrut \) \(187798500441\) \(\nu^{9}\mathstrut +\mathstrut \) \(143893003544\) \(\nu^{8}\mathstrut -\mathstrut \) \(436801227938\) \(\nu^{7}\mathstrut -\mathstrut \) \(57150890634\) \(\nu^{6}\mathstrut +\mathstrut \) \(479794667297\) \(\nu^{5}\mathstrut -\mathstrut \) \(71979148918\) \(\nu^{4}\mathstrut -\mathstrut \) \(217408254535\) \(\nu^{3}\mathstrut +\mathstrut \) \(44636251777\) \(\nu^{2}\mathstrut +\mathstrut \) \(35680796674\) \(\nu\mathstrut -\mathstrut \) \(2904833932\)\()/\)\(730830140\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(32625911\) \(\nu^{17}\mathstrut +\mathstrut \) \(257402314\) \(\nu^{16}\mathstrut +\mathstrut \) \(38720489\) \(\nu^{15}\mathstrut -\mathstrut \) \(4899416105\) \(\nu^{14}\mathstrut +\mathstrut \) \(7765546185\) \(\nu^{13}\mathstrut +\mathstrut \) \(34400341342\) \(\nu^{12}\mathstrut -\mathstrut \) \(85621658304\) \(\nu^{11}\mathstrut -\mathstrut \) \(104551063080\) \(\nu^{10}\mathstrut +\mathstrut \) \(388747945111\) \(\nu^{9}\mathstrut +\mathstrut \) \(96698100714\) \(\nu^{8}\mathstrut -\mathstrut \) \(864214618628\) \(\nu^{7}\mathstrut +\mathstrut \) \(143202903956\) \(\nu^{6}\mathstrut +\mathstrut \) \(917584185897\) \(\nu^{5}\mathstrut -\mathstrut \) \(295691214778\) \(\nu^{4}\mathstrut -\mathstrut \) \(402286545165\) \(\nu^{3}\mathstrut +\mathstrut \) \(104139037497\) \(\nu^{2}\mathstrut +\mathstrut \) \(68310107754\) \(\nu\mathstrut -\mathstrut \) \(523550232\)\()/\)\(730830140\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(40185441\) \(\nu^{17}\mathstrut +\mathstrut \) \(349346684\) \(\nu^{16}\mathstrut -\mathstrut \) \(120164091\) \(\nu^{15}\mathstrut -\mathstrut \) \(6504500675\) \(\nu^{14}\mathstrut +\mathstrut \) \(13028075845\) \(\nu^{13}\mathstrut +\mathstrut \) \(43800913262\) \(\nu^{12}\mathstrut -\mathstrut \) \(133329563104\) \(\nu^{11}\mathstrut -\mathstrut \) \(119795883340\) \(\nu^{10}\mathstrut +\mathstrut \) \(589697389781\) \(\nu^{9}\mathstrut +\mathstrut \) \(49503197884\) \(\nu^{8}\mathstrut -\mathstrut \) \(1291628009318\) \(\nu^{7}\mathstrut +\mathstrut \) \(343556698546\) \(\nu^{6}\mathstrut +\mathstrut \) \(1354642874357\) \(\nu^{5}\mathstrut -\mathstrut \) \(520134110778\) \(\nu^{4}\mathstrut -\mathstrut \) \(579856534395\) \(\nu^{3}\mathstrut +\mathstrut \) \(168757634197\) \(\nu^{2}\mathstrut +\mathstrut \) \(87053646174\) \(\nu\mathstrut -\mathstrut \) \(2527247372\)\()/\)\(730830140\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(20878693\) \(\nu^{17}\mathstrut +\mathstrut \) \(77409897\) \(\nu^{16}\mathstrut +\mathstrut \) \(451307397\) \(\nu^{15}\mathstrut -\mathstrut \) \(1771179645\) \(\nu^{14}\mathstrut -\mathstrut \) \(3907179050\) \(\nu^{13}\mathstrut +\mathstrut \) \(16305558241\) \(\nu^{12}\mathstrut +\mathstrut \) \(17519717313\) \(\nu^{11}\mathstrut -\mathstrut \) \(77527845145\) \(\nu^{10}\mathstrut -\mathstrut \) \(44085517162\) \(\nu^{9}\mathstrut +\mathstrut \) \(202940243242\) \(\nu^{8}\mathstrut +\mathstrut \) \(63666071141\) \(\nu^{7}\mathstrut -\mathstrut \) \(287556922462\) \(\nu^{6}\mathstrut -\mathstrut \) \(53922064354\) \(\nu^{5}\mathstrut +\mathstrut \) \(199447839821\) \(\nu^{4}\mathstrut +\mathstrut \) \(29523656040\) \(\nu^{3}\mathstrut -\mathstrut \) \(50247230349\) \(\nu^{2}\mathstrut -\mathstrut \) \(11117097738\) \(\nu\mathstrut +\mathstrut \) \(723097034\)\()/\)\(365415070\)
\(\beta_{9}\)\(=\)\((\)\(8395741\) \(\nu^{17}\mathstrut -\mathstrut \) \(64482942\) \(\nu^{16}\mathstrut -\mathstrut \) \(11290113\) \(\nu^{15}\mathstrut +\mathstrut \) \(1199594231\) \(\nu^{14}\mathstrut -\mathstrut \) \(1945179675\) \(\nu^{13}\mathstrut -\mathstrut \) \(8038582192\) \(\nu^{12}\mathstrut +\mathstrut \) \(21352465070\) \(\nu^{11}\mathstrut +\mathstrut \) \(21465866534\) \(\nu^{10}\mathstrut -\mathstrut \) \(96183962599\) \(\nu^{9}\mathstrut -\mathstrut \) \(5111788540\) \(\nu^{8}\mathstrut +\mathstrut \) \(212368884076\) \(\nu^{7}\mathstrut -\mathstrut \) \(75583142318\) \(\nu^{6}\mathstrut -\mathstrut \) \(226469839587\) \(\nu^{5}\mathstrut +\mathstrut \) \(112899717876\) \(\nu^{4}\mathstrut +\mathstrut \) \(104809072213\) \(\nu^{3}\mathstrut -\mathstrut \) \(38853986253\) \(\nu^{2}\mathstrut -\mathstrut \) \(20711179258\) \(\nu\mathstrut -\mathstrut \) \(76030588\)\()/\)\(146166028\)
\(\beta_{10}\)\(=\)\((\)\(50383667\) \(\nu^{17}\mathstrut -\mathstrut \) \(161191148\) \(\nu^{16}\mathstrut -\mathstrut \) \(1204101853\) \(\nu^{15}\mathstrut +\mathstrut \) \(3787887405\) \(\nu^{14}\mathstrut +\mathstrut \) \(11956797505\) \(\nu^{13}\mathstrut -\mathstrut \) \(35940666424\) \(\nu^{12}\mathstrut -\mathstrut \) \(64188753322\) \(\nu^{11}\mathstrut +\mathstrut \) \(176493100810\) \(\nu^{10}\mathstrut +\mathstrut \) \(201153933563\) \(\nu^{9}\mathstrut -\mathstrut \) \(476359522038\) \(\nu^{8}\mathstrut -\mathstrut \) \(365447236374\) \(\nu^{7}\mathstrut +\mathstrut \) \(688564895308\) \(\nu^{6}\mathstrut +\mathstrut \) \(355164023581\) \(\nu^{5}\mathstrut -\mathstrut \) \(472380557164\) \(\nu^{4}\mathstrut -\mathstrut \) \(157197227985\) \(\nu^{3}\mathstrut +\mathstrut \) \(110627356231\) \(\nu^{2}\mathstrut +\mathstrut \) \(31151563462\) \(\nu\mathstrut -\mathstrut \) \(2009629216\)\()/\)\(730830140\)
\(\beta_{11}\)\(=\)\((\)\(80549203\) \(\nu^{17}\mathstrut -\mathstrut \) \(492648012\) \(\nu^{16}\mathstrut -\mathstrut \) \(821028837\) \(\nu^{15}\mathstrut +\mathstrut \) \(9925282045\) \(\nu^{14}\mathstrut -\mathstrut \) \(3917777155\) \(\nu^{13}\mathstrut -\mathstrut \) \(77138965156\) \(\nu^{12}\mathstrut +\mathstrut \) \(85375182182\) \(\nu^{11}\mathstrut +\mathstrut \) \(290745846970\) \(\nu^{10}\mathstrut -\mathstrut \) \(439435499053\) \(\nu^{9}\mathstrut -\mathstrut \) \(542716950162\) \(\nu^{8}\mathstrut +\mathstrut \) \(1006286981314\) \(\nu^{7}\mathstrut +\mathstrut \) \(442575263692\) \(\nu^{6}\mathstrut -\mathstrut \) \(1037561853751\) \(\nu^{5}\mathstrut -\mathstrut \) \(95993813136\) \(\nu^{4}\mathstrut +\mathstrut \) \(391615540795\) \(\nu^{3}\mathstrut -\mathstrut \) \(4720823941\) \(\nu^{2}\mathstrut -\mathstrut \) \(38822492922\) \(\nu\mathstrut +\mathstrut \) \(3906261256\)\()/\)\(730830140\)
\(\beta_{12}\)\(=\)\((\)\(100575831\) \(\nu^{17}\mathstrut -\mathstrut \) \(597870654\) \(\nu^{16}\mathstrut -\mathstrut \) \(1070376159\) \(\nu^{15}\mathstrut +\mathstrut \) \(11961026425\) \(\nu^{14}\mathstrut -\mathstrut \) \(3958878145\) \(\nu^{13}\mathstrut -\mathstrut \) \(91593180032\) \(\nu^{12}\mathstrut +\mathstrut \) \(99303778874\) \(\nu^{11}\mathstrut +\mathstrut \) \(333681577930\) \(\nu^{10}\mathstrut -\mathstrut \) \(522661745161\) \(\nu^{9}\mathstrut -\mathstrut \) \(568242407784\) \(\nu^{8}\mathstrut +\mathstrut \) \(1220905295328\) \(\nu^{7}\mathstrut +\mathstrut \) \(318287907994\) \(\nu^{6}\mathstrut -\mathstrut \) \(1310588271257\) \(\nu^{5}\mathstrut +\mathstrut \) \(131621699248\) \(\nu^{4}\mathstrut +\mathstrut \) \(559009857575\) \(\nu^{3}\mathstrut -\mathstrut \) \(98821283147\) \(\nu^{2}\mathstrut -\mathstrut \) \(79152820674\) \(\nu\mathstrut +\mathstrut \) \(4483050272\)\()/\)\(730830140\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(101911933\) \(\nu^{17}\mathstrut +\mathstrut \) \(591076942\) \(\nu^{16}\mathstrut +\mathstrut \) \(1177422127\) \(\nu^{15}\mathstrut -\mathstrut \) \(11988946255\) \(\nu^{14}\mathstrut +\mathstrut \) \(2088222575\) \(\nu^{13}\mathstrut +\mathstrut \) \(94092714686\) \(\nu^{12}\mathstrut -\mathstrut \) \(85031622852\) \(\nu^{11}\mathstrut -\mathstrut \) \(359907129280\) \(\nu^{10}\mathstrut +\mathstrut \) \(466570180813\) \(\nu^{9}\mathstrut +\mathstrut \) \(687964139822\) \(\nu^{8}\mathstrut -\mathstrut \) \(1102540718984\) \(\nu^{7}\mathstrut -\mathstrut \) \(586410163532\) \(\nu^{6}\mathstrut +\mathstrut \) \(1177689206511\) \(\nu^{5}\mathstrut +\mathstrut \) \(145204756046\) \(\nu^{4}\mathstrut -\mathstrut \) \(478582900535\) \(\nu^{3}\mathstrut -\mathstrut \) \(1332536689\) \(\nu^{2}\mathstrut +\mathstrut \) \(52828018842\) \(\nu\mathstrut +\mathstrut \) \(1025152864\)\()/\)\(730830140\)
\(\beta_{14}\)\(=\)\((\)\(-\)\(10957014\) \(\nu^{17}\mathstrut +\mathstrut \) \(55541023\) \(\nu^{16}\mathstrut +\mathstrut \) \(166873740\) \(\nu^{15}\mathstrut -\mathstrut \) \(1164950524\) \(\nu^{14}\mathstrut -\mathstrut \) \(620946577\) \(\nu^{13}\mathstrut +\mathstrut \) \(9612588205\) \(\nu^{12}\mathstrut -\mathstrut \) \(2152998819\) \(\nu^{11}\mathstrut -\mathstrut \) \(39807766547\) \(\nu^{10}\mathstrut +\mathstrut \) \(21114688373\) \(\nu^{9}\mathstrut +\mathstrut \) \(87382928254\) \(\nu^{8}\mathstrut -\mathstrut \) \(54458600727\) \(\nu^{7}\mathstrut -\mathstrut \) \(98803608930\) \(\nu^{6}\mathstrut +\mathstrut \) \(54227047741\) \(\nu^{5}\mathstrut +\mathstrut \) \(52720255893\) \(\nu^{4}\mathstrut -\mathstrut \) \(14672012293\) \(\nu^{3}\mathstrut -\mathstrut \) \(12177807586\) \(\nu^{2}\mathstrut -\mathstrut \) \(852708688\) \(\nu\mathstrut +\mathstrut \) \(267274566\)\()/73083014\)
\(\beta_{15}\)\(=\)\((\)\(33634461\) \(\nu^{17}\mathstrut -\mathstrut \) \(212119814\) \(\nu^{16}\mathstrut -\mathstrut \) \(304035654\) \(\nu^{15}\mathstrut +\mathstrut \) \(4222090590\) \(\nu^{14}\mathstrut -\mathstrut \) \(2463442190\) \(\nu^{13}\mathstrut -\mathstrut \) \(32123264007\) \(\nu^{12}\mathstrut +\mathstrut \) \(42663320289\) \(\nu^{11}\mathstrut +\mathstrut \) \(116114548090\) \(\nu^{10}\mathstrut -\mathstrut \) \(213829531481\) \(\nu^{9}\mathstrut -\mathstrut \) \(196165560134\) \(\nu^{8}\mathstrut +\mathstrut \) \(491688834403\) \(\nu^{7}\mathstrut +\mathstrut \) \(110801814894\) \(\nu^{6}\mathstrut -\mathstrut \) \(523818348102\) \(\nu^{5}\mathstrut +\mathstrut \) \(37804322218\) \(\nu^{4}\mathstrut +\mathstrut \) \(220034653890\) \(\nu^{3}\mathstrut -\mathstrut \) \(26980577947\) \(\nu^{2}\mathstrut -\mathstrut \) \(29553701509\) \(\nu\mathstrut +\mathstrut \) \(1153552087\)\()/\)\(182707535\)
\(\beta_{16}\)\(=\)\((\)\(100069624\) \(\nu^{17}\mathstrut -\mathstrut \) \(532974031\) \(\nu^{16}\mathstrut -\mathstrut \) \(1376337581\) \(\nu^{15}\mathstrut +\mathstrut \) \(10970946660\) \(\nu^{14}\mathstrut +\mathstrut \) \(2569093715\) \(\nu^{13}\mathstrut -\mathstrut \) \(87933730868\) \(\nu^{12}\mathstrut +\mathstrut \) \(45335398986\) \(\nu^{11}\mathstrut +\mathstrut \) \(347183324810\) \(\nu^{10}\mathstrut -\mathstrut \) \(299845169974\) \(\nu^{9}\mathstrut -\mathstrut \) \(699846001231\) \(\nu^{8}\mathstrut +\mathstrut \) \(735383390107\) \(\nu^{7}\mathstrut +\mathstrut \) \(666522143031\) \(\nu^{6}\mathstrut -\mathstrut \) \(770874664093\) \(\nu^{5}\mathstrut -\mathstrut \) \(239724502658\) \(\nu^{4}\mathstrut +\mathstrut \) \(285055129740\) \(\nu^{3}\mathstrut +\mathstrut \) \(31458483877\) \(\nu^{2}\mathstrut -\mathstrut \) \(25149158196\) \(\nu\mathstrut -\mathstrut \) \(346733612\)\()/\)\(365415070\)
\(\beta_{17}\)\(=\)\((\)\(212880147\) \(\nu^{17}\mathstrut -\mathstrut \) \(1116211818\) \(\nu^{16}\mathstrut -\mathstrut \) \(2947927153\) \(\nu^{15}\mathstrut +\mathstrut \) \(22837760465\) \(\nu^{14}\mathstrut +\mathstrut \) \(5940214235\) \(\nu^{13}\mathstrut -\mathstrut \) \(181212148554\) \(\nu^{12}\mathstrut +\mathstrut \) \(92102702768\) \(\nu^{11}\mathstrut +\mathstrut \) \(702455085660\) \(\nu^{10}\mathstrut -\mathstrut \) \(618936385827\) \(\nu^{9}\mathstrut -\mathstrut \) \(1362506463858\) \(\nu^{8}\mathstrut +\mathstrut \) \(1525835050196\) \(\nu^{7}\mathstrut +\mathstrut \) \(1169851753908\) \(\nu^{6}\mathstrut -\mathstrut \) \(1615051615589\) \(\nu^{5}\mathstrut -\mathstrut \) \(258422227194\) \(\nu^{4}\mathstrut +\mathstrut \) \(624723711805\) \(\nu^{3}\mathstrut -\mathstrut \) \(36412665269\) \(\nu^{2}\mathstrut -\mathstrut \) \(71450856478\) \(\nu\mathstrut +\mathstrut \) \(5373162964\)\()/\)\(730830140\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{16}\mathstrut +\mathstrut \) \(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(\beta_{16}\mathstrut +\mathstrut \) \(\beta_{15}\mathstrut +\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(8\) \(\beta_{2}\mathstrut +\mathstrut \) \(15\)
\(\nu^{5}\)\(=\)\(9\) \(\beta_{16}\mathstrut -\mathstrut \) \(\beta_{15}\mathstrut +\mathstrut \) \(9\) \(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{13}\mathstrut -\mathstrut \) \(9\) \(\beta_{10}\mathstrut +\mathstrut \) \(8\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(9\) \(\beta_{2}\mathstrut +\mathstrut \) \(21\) \(\beta_{1}\mathstrut +\mathstrut \) \(10\)
\(\nu^{6}\)\(=\)\(12\) \(\beta_{16}\mathstrut +\mathstrut \) \(11\) \(\beta_{15}\mathstrut +\mathstrut \) \(11\) \(\beta_{14}\mathstrut +\mathstrut \) \(10\) \(\beta_{13}\mathstrut -\mathstrut \) \(2\) \(\beta_{11}\mathstrut -\mathstrut \) \(13\) \(\beta_{10}\mathstrut +\mathstrut \) \(12\) \(\beta_{7}\mathstrut +\mathstrut \) \(11\) \(\beta_{6}\mathstrut +\mathstrut \) \(59\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(89\)
\(\nu^{7}\)\(=\)\(70\) \(\beta_{16}\mathstrut -\mathstrut \) \(11\) \(\beta_{15}\mathstrut +\mathstrut \) \(69\) \(\beta_{14}\mathstrut -\mathstrut \) \(12\) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(72\) \(\beta_{10}\mathstrut +\mathstrut \) \(2\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(59\) \(\beta_{7}\mathstrut +\mathstrut \) \(15\) \(\beta_{6}\mathstrut -\mathstrut \) \(14\) \(\beta_{5}\mathstrut +\mathstrut \) \(70\) \(\beta_{2}\mathstrut +\mathstrut \) \(125\) \(\beta_{1}\mathstrut +\mathstrut \) \(86\)
\(\nu^{8}\)\(=\)\(\beta_{17}\mathstrut +\mathstrut \) \(110\) \(\beta_{16}\mathstrut +\mathstrut \) \(92\) \(\beta_{15}\mathstrut +\mathstrut \) \(96\) \(\beta_{14}\mathstrut +\mathstrut \) \(77\) \(\beta_{13}\mathstrut +\mathstrut \) \(2\) \(\beta_{12}\mathstrut -\mathstrut \) \(29\) \(\beta_{11}\mathstrut -\mathstrut \) \(130\) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(113\) \(\beta_{7}\mathstrut +\mathstrut \) \(97\) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(427\) \(\beta_{2}\mathstrut +\mathstrut \) \(15\) \(\beta_{1}\mathstrut +\mathstrut \) \(575\)
\(\nu^{9}\)\(=\)\(3\) \(\beta_{17}\mathstrut +\mathstrut \) \(525\) \(\beta_{16}\mathstrut -\mathstrut \) \(88\) \(\beta_{15}\mathstrut +\mathstrut \) \(510\) \(\beta_{14}\mathstrut -\mathstrut \) \(106\) \(\beta_{13}\mathstrut -\mathstrut \) \(14\) \(\beta_{12}\mathstrut -\mathstrut \) \(21\) \(\beta_{11}\mathstrut -\mathstrut \) \(565\) \(\beta_{10}\mathstrut +\mathstrut \) \(30\) \(\beta_{9}\mathstrut -\mathstrut \) \(18\) \(\beta_{8}\mathstrut +\mathstrut \) \(436\) \(\beta_{7}\mathstrut +\mathstrut \) \(158\) \(\beta_{6}\mathstrut -\mathstrut \) \(139\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(4\) \(\beta_{3}\mathstrut +\mathstrut \) \(537\) \(\beta_{2}\mathstrut +\mathstrut \) \(790\) \(\beta_{1}\mathstrut +\mathstrut \) \(698\)
\(\nu^{10}\)\(=\)\(17\) \(\beta_{17}\mathstrut +\mathstrut \) \(926\) \(\beta_{16}\mathstrut +\mathstrut \) \(703\) \(\beta_{15}\mathstrut +\mathstrut \) \(786\) \(\beta_{14}\mathstrut +\mathstrut \) \(543\) \(\beta_{13}\mathstrut +\mathstrut \) \(32\) \(\beta_{12}\mathstrut -\mathstrut \) \(298\) \(\beta_{11}\mathstrut -\mathstrut \) \(1175\) \(\beta_{10}\mathstrut +\mathstrut \) \(24\) \(\beta_{9}\mathstrut -\mathstrut \) \(43\) \(\beta_{8}\mathstrut +\mathstrut \) \(976\) \(\beta_{7}\mathstrut +\mathstrut \) \(799\) \(\beta_{6}\mathstrut -\mathstrut \) \(36\) \(\beta_{5}\mathstrut -\mathstrut \) \(17\) \(\beta_{4}\mathstrut +\mathstrut \) \(4\) \(\beta_{3}\mathstrut +\mathstrut \) \(3077\) \(\beta_{2}\mathstrut +\mathstrut \) \(159\) \(\beta_{1}\mathstrut +\mathstrut \) \(3893\)
\(\nu^{11}\)\(=\)\(56\) \(\beta_{17}\mathstrut +\mathstrut \) \(3904\) \(\beta_{16}\mathstrut -\mathstrut \) \(606\) \(\beta_{15}\mathstrut +\mathstrut \) \(3746\) \(\beta_{14}\mathstrut -\mathstrut \) \(833\) \(\beta_{13}\mathstrut -\mathstrut \) \(137\) \(\beta_{12}\mathstrut -\mathstrut \) \(280\) \(\beta_{11}\mathstrut -\mathstrut \) \(4416\) \(\beta_{10}\mathstrut +\mathstrut \) \(319\) \(\beta_{9}\mathstrut -\mathstrut \) \(224\) \(\beta_{8}\mathstrut +\mathstrut \) \(3268\) \(\beta_{7}\mathstrut +\mathstrut \) \(1458\) \(\beta_{6}\mathstrut -\mathstrut \) \(1195\) \(\beta_{5}\mathstrut +\mathstrut \) \(11\) \(\beta_{4}\mathstrut +\mathstrut \) \(79\) \(\beta_{3}\mathstrut +\mathstrut \) \(4141\) \(\beta_{2}\mathstrut +\mathstrut \) \(5147\) \(\beta_{1}\mathstrut +\mathstrut \) \(5515\)
\(\nu^{12}\)\(=\)\(201\) \(\beta_{17}\mathstrut +\mathstrut \) \(7513\) \(\beta_{16}\mathstrut +\mathstrut \) \(5177\) \(\beta_{15}\mathstrut +\mathstrut \) \(6290\) \(\beta_{14}\mathstrut +\mathstrut \) \(3679\) \(\beta_{13}\mathstrut +\mathstrut \) \(330\) \(\beta_{12}\mathstrut -\mathstrut \) \(2677\) \(\beta_{11}\mathstrut -\mathstrut \) \(10057\) \(\beta_{10}\mathstrut +\mathstrut \) \(348\) \(\beta_{9}\mathstrut -\mathstrut \) \(586\) \(\beta_{8}\mathstrut +\mathstrut \) \(8073\) \(\beta_{7}\mathstrut +\mathstrut \) \(6386\) \(\beta_{6}\mathstrut -\mathstrut \) \(437\) \(\beta_{5}\mathstrut -\mathstrut \) \(205\) \(\beta_{4}\mathstrut +\mathstrut \) \(99\) \(\beta_{3}\mathstrut +\mathstrut \) \(22189\) \(\beta_{2}\mathstrut +\mathstrut \) \(1480\) \(\beta_{1}\mathstrut +\mathstrut \) \(27107\)
\(\nu^{13}\)\(=\)\(688\) \(\beta_{17}\mathstrut +\mathstrut \) \(29038\) \(\beta_{16}\mathstrut -\mathstrut \) \(3737\) \(\beta_{15}\mathstrut +\mathstrut \) \(27583\) \(\beta_{14}\mathstrut -\mathstrut \) \(6182\) \(\beta_{13}\mathstrut -\mathstrut \) \(1180\) \(\beta_{12}\mathstrut -\mathstrut \) \(3051\) \(\beta_{11}\mathstrut -\mathstrut \) \(34447\) \(\beta_{10}\mathstrut +\mathstrut \) \(2981\) \(\beta_{9}\mathstrut -\mathstrut \) \(2384\) \(\beta_{8}\mathstrut +\mathstrut \) \(24810\) \(\beta_{7}\mathstrut +\mathstrut \) \(12611\) \(\beta_{6}\mathstrut -\mathstrut \) \(9514\) \(\beta_{5}\mathstrut +\mathstrut \) \(26\) \(\beta_{4}\mathstrut +\mathstrut \) \(1019\) \(\beta_{3}\mathstrut +\mathstrut \) \(32081\) \(\beta_{2}\mathstrut +\mathstrut \) \(34132\) \(\beta_{1}\mathstrut +\mathstrut \) \(43005\)
\(\nu^{14}\)\(=\)\(2053\) \(\beta_{17}\mathstrut +\mathstrut \) \(59783\) \(\beta_{16}\mathstrut +\mathstrut \) \(37549\) \(\beta_{15}\mathstrut +\mathstrut \) \(49823\) \(\beta_{14}\mathstrut +\mathstrut \) \(24374\) \(\beta_{13}\mathstrut +\mathstrut \) \(2749\) \(\beta_{12}\mathstrut -\mathstrut \) \(22488\) \(\beta_{11}\mathstrut -\mathstrut \) \(83230\) \(\beta_{10}\mathstrut +\mathstrut \) \(4044\) \(\beta_{9}\mathstrut -\mathstrut \) \(6536\) \(\beta_{8}\mathstrut +\mathstrut \) \(65123\) \(\beta_{7}\mathstrut +\mathstrut \) \(50205\) \(\beta_{6}\mathstrut -\mathstrut \) \(4458\) \(\beta_{5}\mathstrut -\mathstrut \) \(2173\) \(\beta_{4}\mathstrut +\mathstrut \) \(1509\) \(\beta_{3}\mathstrut +\mathstrut \) \(160445\) \(\beta_{2}\mathstrut +\mathstrut \) \(12948\) \(\beta_{1}\mathstrut +\mathstrut \) \(192283\)
\(\nu^{15}\)\(=\)\(7080\) \(\beta_{17}\mathstrut +\mathstrut \) \(216596\) \(\beta_{16}\mathstrut -\mathstrut \) \(20422\) \(\beta_{15}\mathstrut +\mathstrut \) \(204059\) \(\beta_{14}\mathstrut -\mathstrut \) \(44489\) \(\beta_{13}\mathstrut -\mathstrut \) \(9693\) \(\beta_{12}\mathstrut -\mathstrut \) \(29755\) \(\beta_{11}\mathstrut -\mathstrut \) \(268136\) \(\beta_{10}\mathstrut +\mathstrut \) \(26170\) \(\beta_{9}\mathstrut -\mathstrut \) \(23273\) \(\beta_{8}\mathstrut +\mathstrut \) \(190018\) \(\beta_{7}\mathstrut +\mathstrut \) \(105128\) \(\beta_{6}\mathstrut -\mathstrut \) \(72351\) \(\beta_{5}\mathstrut -\mathstrut \) \(933\) \(\beta_{4}\mathstrut +\mathstrut \) \(10914\) \(\beta_{3}\mathstrut +\mathstrut \) \(248954\) \(\beta_{2}\mathstrut +\mathstrut \) \(229108\) \(\beta_{1}\mathstrut +\mathstrut \) \(333140\)
\(\nu^{16}\)\(=\)\(19439\) \(\beta_{17}\mathstrut +\mathstrut \) \(470153\) \(\beta_{16}\mathstrut +\mathstrut \) \(271056\) \(\beta_{15}\mathstrut +\mathstrut \) \(392329\) \(\beta_{14}\mathstrut +\mathstrut \) \(159039\) \(\beta_{13}\mathstrut +\mathstrut \) \(19758\) \(\beta_{12}\mathstrut -\mathstrut \) \(181934\) \(\beta_{11}\mathstrut -\mathstrut \) \(673701\) \(\beta_{10}\mathstrut +\mathstrut \) \(41602\) \(\beta_{9}\mathstrut -\mathstrut \) \(65334\) \(\beta_{8}\mathstrut +\mathstrut \) \(517172\) \(\beta_{7}\mathstrut +\mathstrut \) \(390689\) \(\beta_{6}\mathstrut -\mathstrut \) \(41142\) \(\beta_{5}\mathstrut -\mathstrut \) \(21540\) \(\beta_{4}\mathstrut +\mathstrut \) \(18347\) \(\beta_{3}\mathstrut +\mathstrut \) \(1164251\) \(\beta_{2}\mathstrut +\mathstrut \) \(109440\) \(\beta_{1}\mathstrut +\mathstrut \) \(1382442\)
\(\nu^{17}\)\(=\)\(66426\) \(\beta_{17}\mathstrut +\mathstrut \) \(1620828\) \(\beta_{16}\mathstrut -\mathstrut \) \(91207\) \(\beta_{15}\mathstrut +\mathstrut \) \(1516971\) \(\beta_{14}\mathstrut -\mathstrut \) \(314852\) \(\beta_{13}\mathstrut -\mathstrut \) \(78760\) \(\beta_{12}\mathstrut -\mathstrut \) \(271151\) \(\beta_{11}\mathstrut -\mathstrut \) \(2082370\) \(\beta_{10}\mathstrut +\mathstrut \) \(221821\) \(\beta_{9}\mathstrut -\mathstrut \) \(215137\) \(\beta_{8}\mathstrut +\mathstrut \) \(1462990\) \(\beta_{7}\mathstrut +\mathstrut \) \(856189\) \(\beta_{6}\mathstrut -\mathstrut \) \(534253\) \(\beta_{5}\mathstrut -\mathstrut \) \(20129\) \(\beta_{4}\mathstrut +\mathstrut \) \(105540\) \(\beta_{3}\mathstrut +\mathstrut \) \(1931064\) \(\beta_{2}\mathstrut +\mathstrut \) \(1552681\) \(\beta_{1}\mathstrut +\mathstrut \) \(2572050\)

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
−2.55461
−2.24824
−1.90615
−1.70725
−1.33766
−0.665348
−0.432278
−0.375443
0.0366427
0.803961
0.981962
1.52505
1.60620
1.77085
2.40246
2.59597
2.73355
2.77035
−2.55461 0 4.52602 2.12871 0 1.00000 −6.45298 0 −5.43801
1.2 −2.24824 0 3.05458 3.31434 0 1.00000 −2.37095 0 −7.45143
1.3 −1.90615 0 1.63341 −1.97979 0 1.00000 0.698768 0 3.77379
1.4 −1.70725 0 0.914706 −1.08728 0 1.00000 1.85287 0 1.85625
1.5 −1.33766 0 −0.210663 0.660635 0 1.00000 2.95712 0 −0.883705
1.6 −0.665348 0 −1.55731 3.70415 0 1.00000 2.36685 0 −2.46455
1.7 −0.432278 0 −1.81314 4.10000 0 1.00000 1.64833 0 −1.77234
1.8 −0.375443 0 −1.85904 −2.49678 0 1.00000 1.44885 0 0.937397
1.9 0.0366427 0 −1.99866 −1.02187 0 1.00000 −0.146522 0 −0.0374440
1.10 0.803961 0 −1.35365 −0.343335 0 1.00000 −2.69620 0 −0.276028
1.11 0.981962 0 −1.03575 −2.59441 0 1.00000 −2.98099 0 −2.54761
1.12 1.52505 0 0.325779 1.71268 0 1.00000 −2.55327 0 2.61192
1.13 1.60620 0 0.579871 3.54808 0 1.00000 −2.28101 0 5.69892
1.14 1.77085 0 1.13590 −1.38376 0 1.00000 −1.53019 0 −2.45042
1.15 2.40246 0 3.77179 0.666645 0 1.00000 4.25666 0 1.60159
1.16 2.59597 0 4.73905 1.63234 0 1.00000 7.11048 0 4.23750
1.17 2.73355 0 5.47227 −4.25416 0 1.00000 9.49162 0 −11.6289
1.18 2.77035 0 5.67483 3.69380 0 1.00000 10.1806 0 10.2331
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)
\(127\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\(T_{2}^{18} - \cdots\)
\(T_{5}^{18} - \cdots\)