Properties

Label 6039.2.a.k
Level 6039
Weight 2
Character orbit 6039.a
Self dual Yes
Analytic conductor 48.222
Analytic rank 0
Dimension 19
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6039.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.2216577807\)
Analytic rank: \(0\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{18}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{2} \) \( + ( 1 + \beta_{2} ) q^{4} \) \( + \beta_{3} q^{5} \) \( -\beta_{16} q^{7} \) \( + ( -1 - \beta_{1} - \beta_{5} + \beta_{6} - \beta_{11} - \beta_{13} ) q^{8} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{2} \) \( + ( 1 + \beta_{2} ) q^{4} \) \( + \beta_{3} q^{5} \) \( -\beta_{16} q^{7} \) \( + ( -1 - \beta_{1} - \beta_{5} + \beta_{6} - \beta_{11} - \beta_{13} ) q^{8} \) \( + ( 1 - \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{8} + \beta_{9} + \beta_{13} ) q^{10} \) \(- q^{11}\) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{15} - \beta_{16} ) q^{13} \) \( + ( -1 - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{14} - \beta_{16} + \beta_{17} - \beta_{18} ) q^{14} \) \( + ( 1 + \beta_{2} - \beta_{3} - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{12} ) q^{16} \) \( + ( -\beta_{2} + \beta_{4} - \beta_{6} - \beta_{11} + \beta_{13} - \beta_{16} + \beta_{17} ) q^{17} \) \( + ( 2 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{9} + \beta_{13} + \beta_{18} ) q^{19} \) \( + ( -1 - \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{20} \) \( + \beta_{1} q^{22} \) \( + ( 1 - \beta_{1} - \beta_{4} + \beta_{9} + \beta_{10} + \beta_{13} - \beta_{14} + \beta_{15} + \beta_{17} ) q^{23} \) \( + ( 2 + \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{9} + \beta_{10} - 2 \beta_{11} - 2 \beta_{12} + \beta_{13} - \beta_{14} - \beta_{16} + \beta_{17} ) q^{25} \) \( + ( \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} - 2 \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} ) q^{26} \) \( + ( 2 - \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{5} - \beta_{13} - \beta_{17} ) q^{28} \) \( + ( -1 + \beta_{3} + \beta_{5} - \beta_{8} + \beta_{11} + \beta_{15} - \beta_{17} ) q^{29} \) \( + ( -\beta_{3} + \beta_{8} + \beta_{9} + \beta_{11} + \beta_{13} + \beta_{15} - \beta_{16} - \beta_{17} - \beta_{18} ) q^{31} \) \( + ( -2 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{9} - 2 \beta_{10} + \beta_{11} + \beta_{12} - 3 \beta_{13} + \beta_{14} + \beta_{16} - 2 \beta_{17} - \beta_{18} ) q^{32} \) \( + ( -1 + 2 \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} + \beta_{11} + 2 \beta_{12} - \beta_{14} ) q^{34} \) \( + ( -1 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{9} - \beta_{10} + 2 \beta_{12} - \beta_{13} - 2 \beta_{15} + \beta_{16} + \beta_{18} ) q^{35} \) \( + ( -\beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{8} - \beta_{11} - \beta_{15} - \beta_{16} + \beta_{17} ) q^{37} \) \( + ( 3 + 2 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{8} + \beta_{11} + \beta_{16} + \beta_{17} ) q^{38} \) \( + ( 2 - 3 \beta_{3} + 2 \beta_{4} + \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} - 3 \beta_{11} - \beta_{12} ) q^{40} \) \( + ( -3 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{14} - 2 \beta_{16} ) q^{41} \) \( + ( 2 - \beta_{1} + \beta_{4} - \beta_{5} - 2 \beta_{9} - \beta_{10} + \beta_{14} - \beta_{15} + \beta_{16} + \beta_{17} + \beta_{18} ) q^{43} \) \( + ( -1 - \beta_{2} ) q^{44} \) \( + ( 5 - 2 \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} - 2 \beta_{7} + 2 \beta_{14} - \beta_{15} + 2 \beta_{18} ) q^{46} \) \( + ( 3 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} + \beta_{16} + 2 \beta_{17} + \beta_{18} ) q^{47} \) \( + ( 2 - \beta_{1} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{12} + \beta_{13} + 2 \beta_{15} - \beta_{16} - \beta_{17} - 2 \beta_{18} ) q^{49} \) \( + ( -\beta_{1} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} + \beta_{9} + 2 \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} - \beta_{17} + \beta_{18} ) q^{50} \) \( + ( -3 - 2 \beta_{3} - 2 \beta_{5} + 3 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} - \beta_{18} ) q^{52} \) \( + ( -1 + \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} + \beta_{11} + \beta_{12} + \beta_{15} + \beta_{16} - \beta_{18} ) q^{53} \) \( -\beta_{3} q^{55} \) \( + ( 1 - 4 \beta_{1} + \beta_{2} - 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{11} + \beta_{12} - 2 \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} - \beta_{17} ) q^{56} \) \( + ( -1 + 4 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{14} + \beta_{15} + \beta_{17} - 2 \beta_{18} ) q^{58} \) \( + ( 4 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{14} - \beta_{16} - \beta_{17} - 2 \beta_{18} ) q^{59} \) \(- q^{61}\) \( + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{5} + 2 \beta_{7} - 2 \beta_{9} - \beta_{10} + \beta_{14} - \beta_{15} + \beta_{17} - \beta_{18} ) q^{62} \) \( + ( 2 - \beta_{1} + 4 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} + 2 \beta_{16} - \beta_{17} ) q^{64} \) \( + ( 3 - 3 \beta_{1} - \beta_{2} + \beta_{3} + 4 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 3 \beta_{12} + 2 \beta_{16} + \beta_{17} + 3 \beta_{18} ) q^{65} \) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{12} - 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} - \beta_{17} + \beta_{18} ) q^{67} \) \( + ( -1 + \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{9} - 2 \beta_{11} + \beta_{12} + 2 \beta_{14} - 2 \beta_{15} + \beta_{16} ) q^{68} \) \( + ( 3 - 3 \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} + 3 \beta_{12} + \beta_{14} + \beta_{16} + \beta_{17} + \beta_{18} ) q^{70} \) \( + ( 3 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} + \beta_{11} - \beta_{12} + \beta_{13} + 2 \beta_{15} - \beta_{16} - 2 \beta_{18} ) q^{71} \) \( + ( 2 - 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 3 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} - \beta_{15} + 2 \beta_{16} - \beta_{17} + \beta_{18} ) q^{73} \) \( + ( 6 - \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 5 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + 4 \beta_{9} + 3 \beta_{10} + \beta_{11} + 4 \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} + \beta_{18} ) q^{74} \) \( + ( -4 - 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} - 2 \beta_{16} - 2 \beta_{17} - \beta_{18} ) q^{76} \) \( + \beta_{16} q^{77} \) \( + ( 3 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} + 2 \beta_{11} + \beta_{12} - 2 \beta_{14} + \beta_{15} - \beta_{16} - 2 \beta_{17} - \beta_{18} ) q^{79} \) \( + ( -1 - \beta_{1} + \beta_{2} + 6 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} - 3 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} + \beta_{15} + \beta_{16} ) q^{80} \) \( + ( -3 + 3 \beta_{1} + \beta_{3} - \beta_{4} + 3 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} + 2 \beta_{10} - \beta_{11} - \beta_{12} - 2 \beta_{14} + \beta_{15} - \beta_{16} + 2 \beta_{17} - \beta_{18} ) q^{82} \) \( + ( 1 - 3 \beta_{3} + 2 \beta_{4} + \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} + 2 \beta_{13} + \beta_{14} - \beta_{15} - \beta_{16} - \beta_{17} + 2 \beta_{18} ) q^{83} \) \( + ( 3 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{7} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - \beta_{13} - 3 \beta_{14} + 2 \beta_{17} + \beta_{18} ) q^{85} \) \( + ( 5 - 3 \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + 3 \beta_{9} + 4 \beta_{11} + 2 \beta_{12} - \beta_{14} + 3 \beta_{15} + 2 \beta_{16} - 2 \beta_{17} + \beta_{18} ) q^{86} \) \( + ( 1 + \beta_{1} + \beta_{5} - \beta_{6} + \beta_{11} + \beta_{13} ) q^{88} \) \( + ( \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{7} + \beta_{10} - 2 \beta_{12} + \beta_{14} - 2 \beta_{15} - 2 \beta_{16} - \beta_{17} + \beta_{18} ) q^{89} \) \( + ( 3 - \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{11} + \beta_{12} - \beta_{13} + 2 \beta_{15} + 2 \beta_{16} - \beta_{17} ) q^{91} \) \( + ( 2 - 3 \beta_{1} + \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{11} - \beta_{12} - \beta_{13} - 2 \beta_{14} + \beta_{16} ) q^{92} \) \( + ( 3 - 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - \beta_{12} + \beta_{13} - \beta_{14} - 2 \beta_{16} - \beta_{17} + \beta_{18} ) q^{94} \) \( + ( 3 - 3 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{10} + 4 \beta_{11} - \beta_{14} - \beta_{15} - \beta_{17} + \beta_{18} ) q^{95} \) \( + ( 3 + \beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{5} - \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{11} + \beta_{12} + 2 \beta_{15} + \beta_{16} ) q^{97} \) \( + ( 3 - \beta_{1} + 3 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 3 \beta_{9} - \beta_{10} + 3 \beta_{12} - \beta_{13} - \beta_{14} + \beta_{16} + \beta_{17} - 2 \beta_{18} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(19q \) \(\mathstrut -\mathstrut 5q^{2} \) \(\mathstrut +\mathstrut 23q^{4} \) \(\mathstrut +\mathstrut 9q^{7} \) \(\mathstrut -\mathstrut 9q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(19q \) \(\mathstrut -\mathstrut 5q^{2} \) \(\mathstrut +\mathstrut 23q^{4} \) \(\mathstrut +\mathstrut 9q^{7} \) \(\mathstrut -\mathstrut 9q^{8} \) \(\mathstrut +\mathstrut 7q^{10} \) \(\mathstrut -\mathstrut 19q^{11} \) \(\mathstrut +\mathstrut 8q^{13} \) \(\mathstrut +\mathstrut 11q^{14} \) \(\mathstrut +\mathstrut 31q^{16} \) \(\mathstrut -\mathstrut 9q^{17} \) \(\mathstrut +\mathstrut 17q^{19} \) \(\mathstrut +\mathstrut 6q^{20} \) \(\mathstrut +\mathstrut 5q^{22} \) \(\mathstrut +\mathstrut 10q^{23} \) \(\mathstrut +\mathstrut 45q^{25} \) \(\mathstrut -\mathstrut 5q^{26} \) \(\mathstrut +\mathstrut 36q^{28} \) \(\mathstrut -\mathstrut 27q^{29} \) \(\mathstrut +\mathstrut 7q^{31} \) \(\mathstrut -\mathstrut 8q^{32} \) \(\mathstrut -\mathstrut 5q^{34} \) \(\mathstrut -\mathstrut 17q^{35} \) \(\mathstrut +\mathstrut 20q^{37} \) \(\mathstrut +\mathstrut 37q^{38} \) \(\mathstrut +\mathstrut 10q^{40} \) \(\mathstrut -\mathstrut 19q^{41} \) \(\mathstrut +\mathstrut 20q^{43} \) \(\mathstrut -\mathstrut 23q^{44} \) \(\mathstrut +\mathstrut 41q^{46} \) \(\mathstrut +\mathstrut 19q^{47} \) \(\mathstrut +\mathstrut 42q^{49} \) \(\mathstrut -\mathstrut 36q^{50} \) \(\mathstrut -\mathstrut 28q^{52} \) \(\mathstrut -\mathstrut 3q^{53} \) \(\mathstrut +\mathstrut 44q^{56} \) \(\mathstrut +\mathstrut 23q^{58} \) \(\mathstrut +\mathstrut 28q^{59} \) \(\mathstrut -\mathstrut 19q^{61} \) \(\mathstrut +\mathstrut 11q^{62} \) \(\mathstrut +\mathstrut 47q^{64} \) \(\mathstrut -\mathstrut 25q^{65} \) \(\mathstrut +\mathstrut 3q^{67} \) \(\mathstrut -\mathstrut 38q^{68} \) \(\mathstrut +\mathstrut 3q^{70} \) \(\mathstrut +\mathstrut 19q^{71} \) \(\mathstrut +\mathstrut 20q^{73} \) \(\mathstrut +\mathstrut 22q^{74} \) \(\mathstrut -\mathstrut 25q^{76} \) \(\mathstrut -\mathstrut 9q^{77} \) \(\mathstrut +\mathstrut 69q^{79} \) \(\mathstrut +\mathstrut 36q^{80} \) \(\mathstrut -\mathstrut 61q^{82} \) \(\mathstrut -\mathstrut q^{83} \) \(\mathstrut +\mathstrut 24q^{85} \) \(\mathstrut +\mathstrut 27q^{86} \) \(\mathstrut +\mathstrut 9q^{88} \) \(\mathstrut +\mathstrut 24q^{91} \) \(\mathstrut +\mathstrut 67q^{92} \) \(\mathstrut +\mathstrut 64q^{94} \) \(\mathstrut +\mathstrut 3q^{95} \) \(\mathstrut +\mathstrut 21q^{97} \) \(\mathstrut +\mathstrut 87q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{19}\mathstrut -\mathstrut \) \(5\) \(x^{18}\mathstrut -\mathstrut \) \(18\) \(x^{17}\mathstrut +\mathstrut \) \(122\) \(x^{16}\mathstrut +\mathstrut \) \(78\) \(x^{15}\mathstrut -\mathstrut \) \(1177\) \(x^{14}\mathstrut +\mathstrut \) \(387\) \(x^{13}\mathstrut +\mathstrut \) \(5755\) \(x^{12}\mathstrut -\mathstrut \) \(4673\) \(x^{11}\mathstrut -\mathstrut \) \(15053\) \(x^{10}\mathstrut +\mathstrut \) \(16875\) \(x^{9}\mathstrut +\mathstrut \) \(20141\) \(x^{8}\mathstrut -\mathstrut \) \(28019\) \(x^{7}\mathstrut -\mathstrut \) \(11589\) \(x^{6}\mathstrut +\mathstrut \) \(21077\) \(x^{5}\mathstrut +\mathstrut \) \(1674\) \(x^{4}\mathstrut -\mathstrut \) \(6404\) \(x^{3}\mathstrut +\mathstrut \) \(241\) \(x^{2}\mathstrut +\mathstrut \) \(643\) \(x\mathstrut -\mathstrut \) \(43\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\(8867201804\) \(\nu^{18}\mathstrut -\mathstrut \) \(28397403169\) \(\nu^{17}\mathstrut -\mathstrut \) \(153963852608\) \(\nu^{16}\mathstrut +\mathstrut \) \(585160018840\) \(\nu^{15}\mathstrut +\mathstrut \) \(420015084215\) \(\nu^{14}\mathstrut -\mathstrut \) \(3896315485489\) \(\nu^{13}\mathstrut +\mathstrut \) \(7816695786848\) \(\nu^{12}\mathstrut +\mathstrut \) \(4170318228153\) \(\nu^{11}\mathstrut -\mathstrut \) \(74752830871939\) \(\nu^{10}\mathstrut +\mathstrut \) \(61181608853165\) \(\nu^{9}\mathstrut +\mathstrut \) \(280962923414937\) \(\nu^{8}\mathstrut -\mathstrut \) \(283316198675975\) \(\nu^{7}\mathstrut -\mathstrut \) \(514007340828783\) \(\nu^{6}\mathstrut +\mathstrut \) \(474088918297633\) \(\nu^{5}\mathstrut +\mathstrut \) \(431277028203763\) \(\nu^{4}\mathstrut -\mathstrut \) \(300469578965604\) \(\nu^{3}\mathstrut -\mathstrut \) \(132758950019250\) \(\nu^{2}\mathstrut +\mathstrut \) \(50083912844543\) \(\nu\mathstrut +\mathstrut \) \(7406641729547\)\()/\)\(2584793613763\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(9925308116\) \(\nu^{18}\mathstrut +\mathstrut \) \(211585433401\) \(\nu^{17}\mathstrut -\mathstrut \) \(157290256641\) \(\nu^{16}\mathstrut -\mathstrut \) \(5438380518759\) \(\nu^{15}\mathstrut +\mathstrut \) \(7280102119263\) \(\nu^{14}\mathstrut +\mathstrut \) \(56715822193005\) \(\nu^{13}\mathstrut -\mathstrut \) \(79490940840104\) \(\nu^{12}\mathstrut -\mathstrut \) \(311772296964757\) \(\nu^{11}\mathstrut +\mathstrut \) \(401624582318481\) \(\nu^{10}\mathstrut +\mathstrut \) \(976030566478054\) \(\nu^{9}\mathstrut -\mathstrut \) \(1042517331908497\) \(\nu^{8}\mathstrut -\mathstrut \) \(1744313483302286\) \(\nu^{7}\mathstrut +\mathstrut \) \(1352625850873237\) \(\nu^{6}\mathstrut +\mathstrut \) \(1686914951359080\) \(\nu^{5}\mathstrut -\mathstrut \) \(761921796525769\) \(\nu^{4}\mathstrut -\mathstrut \) \(794326099908997\) \(\nu^{3}\mathstrut +\mathstrut \) \(121134977398858\) \(\nu^{2}\mathstrut +\mathstrut \) \(133177244880778\) \(\nu\mathstrut +\mathstrut \) \(4643140412416\)\()/\)\(2584793613763\)
\(\beta_{5}\)\(=\)\((\)\(16270525372\) \(\nu^{18}\mathstrut -\mathstrut \) \(35753207242\) \(\nu^{17}\mathstrut -\mathstrut \) \(519467802678\) \(\nu^{16}\mathstrut +\mathstrut \) \(1065729788540\) \(\nu^{15}\mathstrut +\mathstrut \) \(6910095072255\) \(\nu^{14}\mathstrut -\mathstrut \) \(13113396066991\) \(\nu^{13}\mathstrut -\mathstrut \) \(49478643130019\) \(\nu^{12}\mathstrut +\mathstrut \) \(86061802478689\) \(\nu^{11}\mathstrut +\mathstrut \) \(204922845192456\) \(\nu^{10}\mathstrut -\mathstrut \) \(323580558431117\) \(\nu^{9}\mathstrut -\mathstrut \) \(488697894791415\) \(\nu^{8}\mathstrut +\mathstrut \) \(692134937669964\) \(\nu^{7}\mathstrut +\mathstrut \) \(625158766804376\) \(\nu^{6}\mathstrut -\mathstrut \) \(777549565519077\) \(\nu^{5}\mathstrut -\mathstrut \) \(360901712332219\) \(\nu^{4}\mathstrut +\mathstrut \) \(372817093364514\) \(\nu^{3}\mathstrut +\mathstrut \) \(64006649549778\) \(\nu^{2}\mathstrut -\mathstrut \) \(47424793625471\) \(\nu\mathstrut +\mathstrut \) \(1737797285981\)\()/\)\(2584793613763\)
\(\beta_{6}\)\(=\)\((\)\(37762226457\) \(\nu^{18}\mathstrut -\mathstrut \) \(165903350608\) \(\nu^{17}\mathstrut -\mathstrut \) \(747153495518\) \(\nu^{16}\mathstrut +\mathstrut \) \(3973232345087\) \(\nu^{15}\mathstrut +\mathstrut \) \(4711323004499\) \(\nu^{14}\mathstrut -\mathstrut \) \(37022753494585\) \(\nu^{13}\mathstrut -\mathstrut \) \(4098672552276\) \(\nu^{12}\mathstrut +\mathstrut \) \(168673178705194\) \(\nu^{11}\mathstrut -\mathstrut \) \(72597790484259\) \(\nu^{10}\mathstrut -\mathstrut \) \(372792814215164\) \(\nu^{9}\mathstrut +\mathstrut \) \(311955896020022\) \(\nu^{8}\mathstrut +\mathstrut \) \(273746173403669\) \(\nu^{7}\mathstrut -\mathstrut \) \(483092267921848\) \(\nu^{6}\mathstrut +\mathstrut \) \(262742744285573\) \(\nu^{5}\mathstrut +\mathstrut \) \(256243772589666\) \(\nu^{4}\mathstrut -\mathstrut \) \(424688104742192\) \(\nu^{3}\mathstrut -\mathstrut \) \(12717902390185\) \(\nu^{2}\mathstrut +\mathstrut \) \(114678199755666\) \(\nu\mathstrut -\mathstrut \) \(2002160866471\)\()/\)\(2584793613763\)
\(\beta_{7}\)\(=\)\((\)\(41025764486\) \(\nu^{18}\mathstrut +\mathstrut \) \(68688107954\) \(\nu^{17}\mathstrut -\mathstrut \) \(1546829355593\) \(\nu^{16}\mathstrut -\mathstrut \) \(1494342049450\) \(\nu^{15}\mathstrut +\mathstrut \) \(22839785265251\) \(\nu^{14}\mathstrut +\mathstrut \) \(12033133930515\) \(\nu^{13}\mathstrut -\mathstrut \) \(172638105605592\) \(\nu^{12}\mathstrut -\mathstrut \) \(42177686787597\) \(\nu^{11}\mathstrut +\mathstrut \) \(725805637025464\) \(\nu^{10}\mathstrut +\mathstrut \) \(41070927048001\) \(\nu^{9}\mathstrut -\mathstrut \) \(1708211052099464\) \(\nu^{8}\mathstrut +\mathstrut \) \(119332311754315\) \(\nu^{7}\mathstrut +\mathstrut \) \(2128704632011736\) \(\nu^{6}\mathstrut -\mathstrut \) \(311958245977134\) \(\nu^{5}\mathstrut -\mathstrut \) \(1222257229443266\) \(\nu^{4}\mathstrut +\mathstrut \) \(179493755356720\) \(\nu^{3}\mathstrut +\mathstrut \) \(237228681118952\) \(\nu^{2}\mathstrut -\mathstrut \) \(19813780702\) \(\nu\mathstrut -\mathstrut \) \(165641941058\)\()/\)\(2584793613763\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(44058942396\) \(\nu^{18}\mathstrut +\mathstrut \) \(100366607650\) \(\nu^{17}\mathstrut +\mathstrut \) \(1163734380318\) \(\nu^{16}\mathstrut -\mathstrut \) \(2544906190927\) \(\nu^{15}\mathstrut -\mathstrut \) \(12599484691005\) \(\nu^{14}\mathstrut +\mathstrut \) \(26056387605249\) \(\nu^{13}\mathstrut +\mathstrut \) \(72759363883607\) \(\nu^{12}\mathstrut -\mathstrut \) \(139981328381100\) \(\nu^{11}\mathstrut -\mathstrut \) \(242182034413731\) \(\nu^{10}\mathstrut +\mathstrut \) \(427053290996220\) \(\nu^{9}\mathstrut +\mathstrut \) \(464054899866586\) \(\nu^{8}\mathstrut -\mathstrut \) \(743045495285101\) \(\nu^{7}\mathstrut -\mathstrut \) \(477531549805097\) \(\nu^{6}\mathstrut +\mathstrut \) \(689021693044756\) \(\nu^{5}\mathstrut +\mathstrut \) \(222408147075718\) \(\nu^{4}\mathstrut -\mathstrut \) \(272000230119563\) \(\nu^{3}\mathstrut -\mathstrut \) \(27572301495516\) \(\nu^{2}\mathstrut +\mathstrut \) \(23059854361824\) \(\nu\mathstrut -\mathstrut \) \(246530814813\)\()/\)\(2584793613763\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(57136613064\) \(\nu^{18}\mathstrut +\mathstrut \) \(46120188886\) \(\nu^{17}\mathstrut +\mathstrut \) \(1573939504037\) \(\nu^{16}\mathstrut -\mathstrut \) \(711542003465\) \(\nu^{15}\mathstrut -\mathstrut \) \(18089060782894\) \(\nu^{14}\mathstrut +\mathstrut \) \(888172948827\) \(\nu^{13}\mathstrut +\mathstrut \) \(113473718291916\) \(\nu^{12}\mathstrut +\mathstrut \) \(41171189048558\) \(\nu^{11}\mathstrut -\mathstrut \) \(423076214974525\) \(\nu^{10}\mathstrut -\mathstrut \) \(308367411917029\) \(\nu^{9}\mathstrut +\mathstrut \) \(949735366352185\) \(\nu^{8}\mathstrut +\mathstrut \) \(935542592921969\) \(\nu^{7}\mathstrut -\mathstrut \) \(1236732975882466\) \(\nu^{6}\mathstrut -\mathstrut \) \(1317806710310478\) \(\nu^{5}\mathstrut +\mathstrut \) \(850923535863497\) \(\nu^{4}\mathstrut +\mathstrut \) \(778848775823786\) \(\nu^{3}\mathstrut -\mathstrut \) \(236302917332312\) \(\nu^{2}\mathstrut -\mathstrut \) \(127878466080227\) \(\nu\mathstrut +\mathstrut \) \(8443491451865\)\()/\)\(2584793613763\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(81254538506\) \(\nu^{18}\mathstrut +\mathstrut \) \(654428683551\) \(\nu^{17}\mathstrut +\mathstrut \) \(970372146851\) \(\nu^{16}\mathstrut -\mathstrut \) \(16384821425649\) \(\nu^{15}\mathstrut +\mathstrut \) \(5490595341552\) \(\nu^{14}\mathstrut +\mathstrut \) \(164287162414465\) \(\nu^{13}\mathstrut -\mathstrut \) \(143000610338671\) \(\nu^{12}\mathstrut -\mathstrut \) \(851967355105863\) \(\nu^{11}\mathstrut +\mathstrut \) \(907306072440698\) \(\nu^{10}\mathstrut +\mathstrut \) \(2447482008098315\) \(\nu^{9}\mathstrut -\mathstrut \) \(2686226489462764\) \(\nu^{8}\mathstrut -\mathstrut \) \(3851875830463548\) \(\nu^{7}\mathstrut +\mathstrut \) \(3918184841562988\) \(\nu^{6}\mathstrut +\mathstrut \) \(3081313386395895\) \(\nu^{5}\mathstrut -\mathstrut \) \(2559452836437059\) \(\nu^{4}\mathstrut -\mathstrut \) \(1114097846357755\) \(\nu^{3}\mathstrut +\mathstrut \) \(587551380088491\) \(\nu^{2}\mathstrut +\mathstrut \) \(144837697384176\) \(\nu\mathstrut -\mathstrut \) \(25082161671395\)\()/\)\(2584793613763\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(91424960248\) \(\nu^{18}\mathstrut +\mathstrut \) \(290362672773\) \(\nu^{17}\mathstrut +\mathstrut \) \(2163987038842\) \(\nu^{16}\mathstrut -\mathstrut \) \(7229272850781\) \(\nu^{15}\mathstrut -\mathstrut \) \(19906864553003\) \(\nu^{14}\mathstrut +\mathstrut \) \(71928744979165\) \(\nu^{13}\mathstrut +\mathstrut \) \(89628292185599\) \(\nu^{12}\mathstrut -\mathstrut \) \(369628093368947\) \(\nu^{11}\mathstrut -\mathstrut \) \(196989043393835\) \(\nu^{10}\mathstrut +\mathstrut \) \(1054469865039305\) \(\nu^{9}\mathstrut +\mathstrut \) \(148090368081498\) \(\nu^{8}\mathstrut -\mathstrut \) \(1669129966062270\) \(\nu^{7}\mathstrut +\mathstrut \) \(134975892121205\) \(\nu^{6}\mathstrut +\mathstrut \) \(1394627423083597\) \(\nu^{5}\mathstrut -\mathstrut \) \(249311959857695\) \(\nu^{4}\mathstrut -\mathstrut \) \(557432190432943\) \(\nu^{3}\mathstrut +\mathstrut \) \(94000023879346\) \(\nu^{2}\mathstrut +\mathstrut \) \(76490070141423\) \(\nu\mathstrut -\mathstrut \) \(5331850884506\)\()/\)\(2584793613763\)
\(\beta_{12}\)\(=\)\((\)\(110220246484\) \(\nu^{18}\mathstrut -\mathstrut \) \(467343546422\) \(\nu^{17}\mathstrut -\mathstrut \) \(2253974970877\) \(\nu^{16}\mathstrut +\mathstrut \) \(11406225381357\) \(\nu^{15}\mathstrut +\mathstrut \) \(15508655859118\) \(\nu^{14}\mathstrut -\mathstrut \) \(110076102275048\) \(\nu^{13}\mathstrut -\mathstrut \) \(28689902332059\) \(\nu^{12}\mathstrut +\mathstrut \) \(538405030181766\) \(\nu^{11}\mathstrut -\mathstrut \) \(130776269174957\) \(\nu^{10}\mathstrut -\mathstrut \) \(1408591428559113\) \(\nu^{9}\mathstrut +\mathstrut \) \(700069942254179\) \(\nu^{8}\mathstrut +\mathstrut \) \(1883801315016804\) \(\nu^{7}\mathstrut -\mathstrut \) \(1141430209449476\) \(\nu^{6}\mathstrut -\mathstrut \) \(1082982581796351\) \(\nu^{5}\mathstrut +\mathstrut \) \(635720108720707\) \(\nu^{4}\mathstrut +\mathstrut \) \(170395081372216\) \(\nu^{3}\mathstrut -\mathstrut \) \(54746227366930\) \(\nu^{2}\mathstrut +\mathstrut \) \(3421690058101\) \(\nu\mathstrut -\mathstrut \) \(8426703234447\)\()/\)\(2584793613763\)
\(\beta_{13}\)\(=\)\((\)\(112916661333\) \(\nu^{18}\mathstrut -\mathstrut \) \(420512816139\) \(\nu^{17}\mathstrut -\mathstrut \) \(2391672731682\) \(\nu^{16}\mathstrut +\mathstrut \) \(10136775407328\) \(\nu^{15}\mathstrut +\mathstrut \) \(17708092485247\) \(\nu^{14}\mathstrut -\mathstrut \) \(95838102406759\) \(\nu^{13}\mathstrut -\mathstrut \) \(44248321607856\) \(\nu^{12}\mathstrut +\mathstrut \) \(452239469595452\) \(\nu^{11}\mathstrut -\mathstrut \) \(80531592282880\) \(\nu^{10}\mathstrut -\mathstrut \) \(1103682120823352\) \(\nu^{9}\mathstrut +\mathstrut \) \(652563422729939\) \(\nu^{8}\mathstrut +\mathstrut \) \(1250741201795975\) \(\nu^{7}\mathstrut -\mathstrut \) \(1243226926847429\) \(\nu^{6}\mathstrut -\mathstrut \) \(354335113278947\) \(\nu^{5}\mathstrut +\mathstrut \) \(866457444779580\) \(\nu^{4}\mathstrut -\mathstrut \) \(237488214060000\) \(\nu^{3}\mathstrut -\mathstrut \) \(170724575819309\) \(\nu^{2}\mathstrut +\mathstrut \) \(72688955170899\) \(\nu\mathstrut -\mathstrut \) \(992900881709\)\()/\)\(2584793613763\)
\(\beta_{14}\)\(=\)\((\)\(134524452259\) \(\nu^{18}\mathstrut -\mathstrut \) \(595936463557\) \(\nu^{17}\mathstrut -\mathstrut \) \(2794615480726\) \(\nu^{16}\mathstrut +\mathstrut \) \(14870979403509\) \(\nu^{15}\mathstrut +\mathstrut \) \(19764923687112\) \(\nu^{14}\mathstrut -\mathstrut \) \(148126283460976\) \(\nu^{13}\mathstrut -\mathstrut \) \(39968403213030\) \(\nu^{12}\mathstrut +\mathstrut \) \(759359884142781\) \(\nu^{11}\mathstrut -\mathstrut \) \(157363017924653\) \(\nu^{10}\mathstrut -\mathstrut \) \(2143032129859066\) \(\nu^{9}\mathstrut +\mathstrut \) \(934401836140697\) \(\nu^{8}\mathstrut +\mathstrut \) \(3296195916484018\) \(\nu^{7}\mathstrut -\mathstrut \) \(1683719333735166\) \(\nu^{6}\mathstrut -\mathstrut \) \(2591704634942325\) \(\nu^{5}\mathstrut +\mathstrut \) \(1136718336996770\) \(\nu^{4}\mathstrut +\mathstrut \) \(957988530198262\) \(\nu^{3}\mathstrut -\mathstrut \) \(195065901784280\) \(\nu^{2}\mathstrut -\mathstrut \) \(131669286419725\) \(\nu\mathstrut -\mathstrut \) \(6942619491638\)\()/\)\(2584793613763\)
\(\beta_{15}\)\(=\)\((\)\(268063867548\) \(\nu^{18}\mathstrut -\mathstrut \) \(975838766429\) \(\nu^{17}\mathstrut -\mathstrut \) \(5844382212784\) \(\nu^{16}\mathstrut +\mathstrut \) \(23718172730414\) \(\nu^{15}\mathstrut +\mathstrut \) \(46258916319239\) \(\nu^{14}\mathstrut -\mathstrut \) \(227625432361619\) \(\nu^{13}\mathstrut -\mathstrut \) \(147385979907443\) \(\nu^{12}\mathstrut +\mathstrut \) \(1105726169117787\) \(\nu^{11}\mathstrut +\mathstrut \) \(23829240400472\) \(\nu^{10}\mathstrut -\mathstrut \) \(2873186468193146\) \(\nu^{9}\mathstrut +\mathstrut \) \(965965174706451\) \(\nu^{8}\mathstrut +\mathstrut \) \(3844359687237313\) \(\nu^{7}\mathstrut -\mathstrut \) \(2136542032244486\) \(\nu^{6}\mathstrut -\mathstrut \) \(2308864017778112\) \(\nu^{5}\mathstrut +\mathstrut \) \(1541694664772399\) \(\nu^{4}\mathstrut +\mathstrut \) \(490128717085397\) \(\nu^{3}\mathstrut -\mathstrut \) \(296741969124174\) \(\nu^{2}\mathstrut -\mathstrut \) \(29439287899993\) \(\nu\mathstrut +\mathstrut \) \(2625408551097\)\()/\)\(2584793613763\)
\(\beta_{16}\)\(=\)\((\)\(-\)\(386309035457\) \(\nu^{18}\mathstrut +\mathstrut \) \(1365268186173\) \(\nu^{17}\mathstrut +\mathstrut \) \(8692512067202\) \(\nu^{16}\mathstrut -\mathstrut \) \(33566135421448\) \(\nu^{15}\mathstrut -\mathstrut \) \(73247230474246\) \(\nu^{14}\mathstrut +\mathstrut \) \(327363575480166\) \(\nu^{13}\mathstrut +\mathstrut \) \(276164257045428\) \(\nu^{12}\mathstrut -\mathstrut \) \(1627427645433792\) \(\nu^{11}\mathstrut -\mathstrut \) \(350590014826373\) \(\nu^{10}\mathstrut +\mathstrut \) \(4379858371730078\) \(\nu^{9}\mathstrut -\mathstrut \) \(537601793521600\) \(\nu^{8}\mathstrut -\mathstrut \) \(6212724005068246\) \(\nu^{7}\mathstrut +\mathstrut \) \(1859899256299250\) \(\nu^{6}\mathstrut +\mathstrut \) \(4173706077317173\) \(\nu^{5}\mathstrut -\mathstrut \) \(1430231827654368\) \(\nu^{4}\mathstrut -\mathstrut \) \(1122823610741571\) \(\nu^{3}\mathstrut +\mathstrut \) \(279954024772659\) \(\nu^{2}\mathstrut +\mathstrut \) \(99943970595050\) \(\nu\mathstrut -\mathstrut \) \(8519539396534\)\()/\)\(2584793613763\)
\(\beta_{17}\)\(=\)\((\)\(-\)\(407596389827\) \(\nu^{18}\mathstrut +\mathstrut \) \(996406897672\) \(\nu^{17}\mathstrut +\mathstrut \) \(10303884149056\) \(\nu^{16}\mathstrut -\mathstrut \) \(24506168703039\) \(\nu^{15}\mathstrut -\mathstrut \) \(104725686151822\) \(\nu^{14}\mathstrut +\mathstrut \) \(239031289075534\) \(\nu^{13}\mathstrut +\mathstrut \) \(553369806506455\) \(\nu^{12}\mathstrut -\mathstrut \) \(1188615976491545\) \(\nu^{11}\mathstrut -\mathstrut \) \(1630922731137518\) \(\nu^{10}\mathstrut +\mathstrut \) \(3205416930871435\) \(\nu^{9}\mathstrut +\mathstrut \) \(2656196462753142\) \(\nu^{8}\mathstrut -\mathstrut \) \(4586218801226622\) \(\nu^{7}\mathstrut -\mathstrut \) \(2216000497099625\) \(\nu^{6}\mathstrut +\mathstrut \) \(3162740167704228\) \(\nu^{5}\mathstrut +\mathstrut \) \(818360171228219\) \(\nu^{4}\mathstrut -\mathstrut \) \(877910452999009\) \(\nu^{3}\mathstrut -\mathstrut \) \(76416261707646\) \(\nu^{2}\mathstrut +\mathstrut \) \(75002246004675\) \(\nu\mathstrut -\mathstrut \) \(7512666382563\)\()/\)\(2584793613763\)
\(\beta_{18}\)\(=\)\((\)\(433257329190\) \(\nu^{18}\mathstrut -\mathstrut \) \(1312562946260\) \(\nu^{17}\mathstrut -\mathstrut \) \(10331601737881\) \(\nu^{16}\mathstrut +\mathstrut \) \(32317494993362\) \(\nu^{15}\mathstrut +\mathstrut \) \(96395574289456\) \(\nu^{14}\mathstrut -\mathstrut \) \(315991261600359\) \(\nu^{13}\mathstrut -\mathstrut \) \(447303281907149\) \(\nu^{12}\mathstrut +\mathstrut \) \(1578531396957927\) \(\nu^{11}\mathstrut +\mathstrut \) \(1065197051025780\) \(\nu^{10}\mathstrut -\mathstrut \) \(4290881606498135\) \(\nu^{9}\mathstrut -\mathstrut \) \(1147835903097151\) \(\nu^{8}\mathstrut +\mathstrut \) \(6222910401208218\) \(\nu^{7}\mathstrut +\mathstrut \) \(239835296100884\) \(\nu^{6}\mathstrut -\mathstrut \) \(4402846506872819\) \(\nu^{5}\mathstrut +\mathstrut \) \(270534203277648\) \(\nu^{4}\mathstrut +\mathstrut \) \(1308086020776475\) \(\nu^{3}\mathstrut -\mathstrut \) \(97910802478304\) \(\nu^{2}\mathstrut -\mathstrut \) \(115787993029379\) \(\nu\mathstrut +\mathstrut \) \(7601132183693\)\()/\)\(2584793613763\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(15\)
\(\nu^{5}\)\(=\)\(\beta_{18}\mathstrut +\mathstrut \) \(2\) \(\beta_{17}\mathstrut -\mathstrut \) \(\beta_{16}\mathstrut -\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(11\) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(7\) \(\beta_{11}\mathstrut +\mathstrut \) \(2\) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(9\) \(\beta_{6}\mathstrut +\mathstrut \) \(11\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(30\) \(\beta_{1}\mathstrut +\mathstrut \) \(10\)
\(\nu^{6}\)\(=\)\(-\)\(\beta_{17}\mathstrut +\mathstrut \) \(2\) \(\beta_{16}\mathstrut +\mathstrut \) \(\beta_{15}\mathstrut +\mathstrut \) \(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{13}\mathstrut -\mathstrut \) \(11\) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(12\) \(\beta_{10}\mathstrut -\mathstrut \) \(12\) \(\beta_{9}\mathstrut +\mathstrut \) \(12\) \(\beta_{8}\mathstrut +\mathstrut \) \(12\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(12\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(11\) \(\beta_{3}\mathstrut +\mathstrut \) \(50\) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(88\)
\(\nu^{7}\)\(=\)\(10\) \(\beta_{18}\mathstrut +\mathstrut \) \(24\) \(\beta_{17}\mathstrut -\mathstrut \) \(15\) \(\beta_{16}\mathstrut -\mathstrut \) \(14\) \(\beta_{14}\mathstrut +\mathstrut \) \(94\) \(\beta_{13}\mathstrut -\mathstrut \) \(12\) \(\beta_{12}\mathstrut +\mathstrut \) \(46\) \(\beta_{11}\mathstrut +\mathstrut \) \(26\) \(\beta_{10}\mathstrut +\mathstrut \) \(12\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(71\) \(\beta_{6}\mathstrut +\mathstrut \) \(97\) \(\beta_{5}\mathstrut -\mathstrut \) \(16\) \(\beta_{4}\mathstrut +\mathstrut \) \(17\) \(\beta_{3}\mathstrut -\mathstrut \) \(14\) \(\beta_{2}\mathstrut +\mathstrut \) \(200\) \(\beta_{1}\mathstrut +\mathstrut \) \(76\)
\(\nu^{8}\)\(=\)\(3\) \(\beta_{18}\mathstrut -\mathstrut \) \(15\) \(\beta_{17}\mathstrut +\mathstrut \) \(31\) \(\beta_{16}\mathstrut +\mathstrut \) \(14\) \(\beta_{15}\mathstrut +\mathstrut \) \(14\) \(\beta_{14}\mathstrut -\mathstrut \) \(18\) \(\beta_{13}\mathstrut -\mathstrut \) \(93\) \(\beta_{12}\mathstrut +\mathstrut \) \(19\) \(\beta_{11}\mathstrut -\mathstrut \) \(112\) \(\beta_{10}\mathstrut -\mathstrut \) \(109\) \(\beta_{9}\mathstrut +\mathstrut \) \(110\) \(\beta_{8}\mathstrut +\mathstrut \) \(109\) \(\beta_{7}\mathstrut +\mathstrut \) \(14\) \(\beta_{6}\mathstrut -\mathstrut \) \(112\) \(\beta_{5}\mathstrut -\mathstrut \) \(12\) \(\beta_{4}\mathstrut -\mathstrut \) \(94\) \(\beta_{3}\mathstrut +\mathstrut \) \(367\) \(\beta_{2}\mathstrut -\mathstrut \) \(23\) \(\beta_{1}\mathstrut +\mathstrut \) \(567\)
\(\nu^{9}\)\(=\)\(78\) \(\beta_{18}\mathstrut +\mathstrut \) \(218\) \(\beta_{17}\mathstrut -\mathstrut \) \(153\) \(\beta_{16}\mathstrut +\mathstrut \) \(2\) \(\beta_{15}\mathstrut -\mathstrut \) \(142\) \(\beta_{14}\mathstrut +\mathstrut \) \(746\) \(\beta_{13}\mathstrut -\mathstrut \) \(99\) \(\beta_{12}\mathstrut +\mathstrut \) \(317\) \(\beta_{11}\mathstrut +\mathstrut \) \(249\) \(\beta_{10}\mathstrut +\mathstrut \) \(111\) \(\beta_{9}\mathstrut -\mathstrut \) \(22\) \(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{7}\mathstrut -\mathstrut \) \(549\) \(\beta_{6}\mathstrut +\mathstrut \) \(799\) \(\beta_{5}\mathstrut -\mathstrut \) \(171\) \(\beta_{4}\mathstrut +\mathstrut \) \(196\) \(\beta_{3}\mathstrut -\mathstrut \) \(145\) \(\beta_{2}\mathstrut +\mathstrut \) \(1410\) \(\beta_{1}\mathstrut +\mathstrut \) \(534\)
\(\nu^{10}\)\(=\)\(47\) \(\beta_{18}\mathstrut -\mathstrut \) \(163\) \(\beta_{17}\mathstrut +\mathstrut \) \(334\) \(\beta_{16}\mathstrut +\mathstrut \) \(143\) \(\beta_{15}\mathstrut +\mathstrut \) \(143\) \(\beta_{14}\mathstrut -\mathstrut \) \(224\) \(\beta_{13}\mathstrut -\mathstrut \) \(724\) \(\beta_{12}\mathstrut +\mathstrut \) \(229\) \(\beta_{11}\mathstrut -\mathstrut \) \(958\) \(\beta_{10}\mathstrut -\mathstrut \) \(905\) \(\beta_{9}\mathstrut +\mathstrut \) \(915\) \(\beta_{8}\mathstrut +\mathstrut \) \(905\) \(\beta_{7}\mathstrut +\mathstrut \) \(148\) \(\beta_{6}\mathstrut -\mathstrut \) \(967\) \(\beta_{5}\mathstrut -\mathstrut \) \(107\) \(\beta_{4}\mathstrut -\mathstrut \) \(744\) \(\beta_{3}\mathstrut +\mathstrut \) \(2742\) \(\beta_{2}\mathstrut -\mathstrut \) \(313\) \(\beta_{1}\mathstrut +\mathstrut \) \(3874\)
\(\nu^{11}\)\(=\)\(570\) \(\beta_{18}\mathstrut +\mathstrut \) \(1806\) \(\beta_{17}\mathstrut -\mathstrut \) \(1356\) \(\beta_{16}\mathstrut +\mathstrut \) \(33\) \(\beta_{15}\mathstrut -\mathstrut \) \(1270\) \(\beta_{14}\mathstrut +\mathstrut \) \(5763\) \(\beta_{13}\mathstrut -\mathstrut \) \(699\) \(\beta_{12}\mathstrut +\mathstrut \) \(2290\) \(\beta_{11}\mathstrut +\mathstrut \) \(2145\) \(\beta_{10}\mathstrut +\mathstrut \) \(945\) \(\beta_{9}\mathstrut -\mathstrut \) \(306\) \(\beta_{8}\mathstrut +\mathstrut \) \(29\) \(\beta_{7}\mathstrut -\mathstrut \) \(4239\) \(\beta_{6}\mathstrut +\mathstrut \) \(6393\) \(\beta_{5}\mathstrut -\mathstrut \) \(1566\) \(\beta_{4}\mathstrut +\mathstrut \) \(1931\) \(\beta_{3}\mathstrut -\mathstrut \) \(1357\) \(\beta_{2}\mathstrut +\mathstrut \) \(10260\) \(\beta_{1}\mathstrut +\mathstrut \) \(3645\)
\(\nu^{12}\)\(=\)\(492\) \(\beta_{18}\mathstrut -\mathstrut \) \(1566\) \(\beta_{17}\mathstrut +\mathstrut \) \(3118\) \(\beta_{16}\mathstrut +\mathstrut \) \(1289\) \(\beta_{15}\mathstrut +\mathstrut \) \(1305\) \(\beta_{14}\mathstrut -\mathstrut \) \(2393\) \(\beta_{13}\mathstrut -\mathstrut \) \(5465\) \(\beta_{12}\mathstrut +\mathstrut \) \(2259\) \(\beta_{11}\mathstrut -\mathstrut \) \(7879\) \(\beta_{10}\mathstrut -\mathstrut \) \(7253\) \(\beta_{9}\mathstrut +\mathstrut \) \(7277\) \(\beta_{8}\mathstrut +\mathstrut \) \(7237\) \(\beta_{7}\mathstrut +\mathstrut \) \(1434\) \(\beta_{6}\mathstrut -\mathstrut \) \(8102\) \(\beta_{5}\mathstrut -\mathstrut \) \(862\) \(\beta_{4}\mathstrut -\mathstrut \) \(5721\) \(\beta_{3}\mathstrut +\mathstrut \) \(20716\) \(\beta_{2}\mathstrut -\mathstrut \) \(3451\) \(\beta_{1}\mathstrut +\mathstrut \) \(27441\)
\(\nu^{13}\)\(=\)\(4103\) \(\beta_{18}\mathstrut +\mathstrut \) \(14404\) \(\beta_{17}\mathstrut -\mathstrut \) \(11290\) \(\beta_{16}\mathstrut +\mathstrut \) \(341\) \(\beta_{15}\mathstrut -\mathstrut \) \(10672\) \(\beta_{14}\mathstrut +\mathstrut \) \(44078\) \(\beta_{13}\mathstrut -\mathstrut \) \(4515\) \(\beta_{12}\mathstrut +\mathstrut \) \(17038\) \(\beta_{11}\mathstrut +\mathstrut \) \(17659\) \(\beta_{10}\mathstrut +\mathstrut \) \(7783\) \(\beta_{9}\mathstrut -\mathstrut \) \(3478\) \(\beta_{8}\mathstrut +\mathstrut \) \(235\) \(\beta_{7}\mathstrut -\mathstrut \) \(32754\) \(\beta_{6}\mathstrut +\mathstrut \) \(50414\) \(\beta_{5}\mathstrut -\mathstrut \) \(13319\) \(\beta_{4}\mathstrut +\mathstrut \) \(17533\) \(\beta_{3}\mathstrut -\mathstrut \) \(12149\) \(\beta_{2}\mathstrut +\mathstrut \) \(76103\) \(\beta_{1}\mathstrut +\mathstrut \) \(24525\)
\(\nu^{14}\)\(=\)\(4351\) \(\beta_{18}\mathstrut -\mathstrut \) \(14133\) \(\beta_{17}\mathstrut +\mathstrut \) \(27117\) \(\beta_{16}\mathstrut +\mathstrut \) \(10875\) \(\beta_{15}\mathstrut +\mathstrut \) \(11326\) \(\beta_{14}\mathstrut -\mathstrut \) \(23509\) \(\beta_{13}\mathstrut -\mathstrut \) \(40789\) \(\beta_{12}\mathstrut +\mathstrut \) \(19984\) \(\beta_{11}\mathstrut -\mathstrut \) \(63529\) \(\beta_{10}\mathstrut -\mathstrut \) \(57279\) \(\beta_{9}\mathstrut +\mathstrut \) \(56578\) \(\beta_{8}\mathstrut +\mathstrut \) \(56857\) \(\beta_{7}\mathstrut +\mathstrut \) \(13359\) \(\beta_{6}\mathstrut -\mathstrut \) \(67001\) \(\beta_{5}\mathstrut -\mathstrut \) \(6621\) \(\beta_{4}\mathstrut -\mathstrut \) \(43553\) \(\beta_{3}\mathstrut +\mathstrut \) \(157618\) \(\beta_{2}\mathstrut -\mathstrut \) \(34216\) \(\beta_{1}\mathstrut +\mathstrut \) \(198716\)
\(\nu^{15}\)\(=\)\(29563\) \(\beta_{18}\mathstrut +\mathstrut \) \(112951\) \(\beta_{17}\mathstrut -\mathstrut \) \(91244\) \(\beta_{16}\mathstrut +\mathstrut \) \(2777\) \(\beta_{15}\mathstrut -\mathstrut \) \(86679\) \(\beta_{14}\mathstrut +\mathstrut \) \(336132\) \(\beta_{13}\mathstrut -\mathstrut \) \(27266\) \(\beta_{12}\mathstrut +\mathstrut \) \(128648\) \(\beta_{11}\mathstrut +\mathstrut \) \(142361\) \(\beta_{10}\mathstrut +\mathstrut \) \(63190\) \(\beta_{9}\mathstrut -\mathstrut \) \(35398\) \(\beta_{8}\mathstrut +\mathstrut \) \(1034\) \(\beta_{7}\mathstrut -\mathstrut \) \(253130\) \(\beta_{6}\mathstrut +\mathstrut \) \(394442\) \(\beta_{5}\mathstrut -\mathstrut \) \(108916\) \(\beta_{4}\mathstrut +\mathstrut \) \(151745\) \(\beta_{3}\mathstrut -\mathstrut \) \(106241\) \(\beta_{2}\mathstrut +\mathstrut \) \(571467\) \(\beta_{1}\mathstrut +\mathstrut \) \(163128\)
\(\nu^{16}\)\(=\)\(35212\) \(\beta_{18}\mathstrut -\mathstrut \) \(122951\) \(\beta_{17}\mathstrut +\mathstrut \) \(226841\) \(\beta_{16}\mathstrut +\mathstrut \) \(88185\) \(\beta_{15}\mathstrut +\mathstrut \) \(95848\) \(\beta_{14}\mathstrut -\mathstrut \) \(219042\) \(\beta_{13}\mathstrut -\mathstrut \) \(303570\) \(\beta_{12}\mathstrut +\mathstrut \) \(165507\) \(\beta_{11}\mathstrut -\mathstrut \) \(506776\) \(\beta_{10}\mathstrut -\mathstrut \) \(449554\) \(\beta_{9}\mathstrut +\mathstrut \) \(434837\) \(\beta_{8}\mathstrut +\mathstrut \) \(442742\) \(\beta_{7}\mathstrut +\mathstrut \) \(121414\) \(\beta_{6}\mathstrut -\mathstrut \) \(550426\) \(\beta_{5}\mathstrut -\mathstrut \) \(49387\) \(\beta_{4}\mathstrut -\mathstrut \) \(331180\) \(\beta_{3}\mathstrut +\mathstrut \) \(1204731\) \(\beta_{2}\mathstrut -\mathstrut \) \(319245\) \(\beta_{1}\mathstrut +\mathstrut \) \(1458972\)
\(\nu^{17}\)\(=\)\(214159\) \(\beta_{18}\mathstrut +\mathstrut \) \(879024\) \(\beta_{17}\mathstrut -\mathstrut \) \(726921\) \(\beta_{16}\mathstrut +\mathstrut \) \(18877\) \(\beta_{15}\mathstrut -\mathstrut \) \(690523\) \(\beta_{14}\mathstrut +\mathstrut \) \(2563542\) \(\beta_{13}\mathstrut -\mathstrut \) \(153522\) \(\beta_{12}\mathstrut +\mathstrut \) \(977051\) \(\beta_{11}\mathstrut +\mathstrut \) \(1136319\) \(\beta_{10}\mathstrut +\mathstrut \) \(509856\) \(\beta_{9}\mathstrut -\mathstrut \) \(336785\) \(\beta_{8}\mathstrut -\mathstrut \) \(4112\) \(\beta_{7}\mathstrut -\mathstrut \) \(1955646\) \(\beta_{6}\mathstrut +\mathstrut \) \(3072741\) \(\beta_{5}\mathstrut -\mathstrut \) \(871174\) \(\beta_{4}\mathstrut +\mathstrut \) \(1274242\) \(\beta_{3}\mathstrut -\mathstrut \) \(915300\) \(\beta_{2}\mathstrut +\mathstrut \) \(4326695\) \(\beta_{1}\mathstrut +\mathstrut \) \(1070635\)
\(\nu^{18}\)\(=\)\(270373\) \(\beta_{18}\mathstrut -\mathstrut \) \(1044853\) \(\beta_{17}\mathstrut +\mathstrut \) \(1855069\) \(\beta_{16}\mathstrut +\mathstrut \) \(697382\) \(\beta_{15}\mathstrut +\mathstrut \) \(799771\) \(\beta_{14}\mathstrut -\mathstrut \) \(1968542\) \(\beta_{13}\mathstrut -\mathstrut \) \(2261487\) \(\beta_{12}\mathstrut +\mathstrut \) \(1313376\) \(\beta_{11}\mathstrut -\mathstrut \) \(4017999\) \(\beta_{10}\mathstrut -\mathstrut \) \(3519120\) \(\beta_{9}\mathstrut +\mathstrut \) \(3323516\) \(\beta_{8}\mathstrut +\mathstrut \) \(3431715\) \(\beta_{7}\mathstrut +\mathstrut \) \(1082768\) \(\beta_{6}\mathstrut -\mathstrut \) \(4503238\) \(\beta_{5}\mathstrut -\mathstrut \) \(360082\) \(\beta_{4}\mathstrut -\mathstrut \) \(2526317\) \(\beta_{3}\mathstrut +\mathstrut \) \(9236634\) \(\beta_{2}\mathstrut -\mathstrut \) \(2867553\) \(\beta_{1}\mathstrut +\mathstrut \) \(10807509\)

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
2.72847
2.69542
2.34443
2.13634
2.04293
1.98166
1.31423
1.04858
0.686089
0.488177
0.0681863
−0.436741
−0.556474
−0.976920
−1.62941
−1.82302
−2.08441
−2.21976
−2.80778
−2.72847 0 5.44452 3.87909 0 0.707409 −9.39826 0 −10.5840
1.2 −2.69542 0 5.26531 −1.92020 0 2.04756 −8.80138 0 5.17575
1.3 −2.34443 0 3.49635 −3.40234 0 −2.95885 −3.50809 0 7.97654
1.4 −2.13634 0 2.56396 0.997990 0 −0.526710 −1.20482 0 −2.13205
1.5 −2.04293 0 2.17358 −0.258135 0 3.70333 −0.354615 0 0.527352
1.6 −1.98166 0 1.92698 −2.60057 0 −2.18138 0.144707 0 5.15344
1.7 −1.31423 0 −0.272811 1.38757 0 2.03012 2.98699 0 −1.82358
1.8 −1.04858 0 −0.900472 4.37157 0 −3.70774 3.04139 0 −4.58396
1.9 −0.686089 0 −1.52928 −3.77631 0 4.48234 2.42140 0 2.59089
1.10 −0.488177 0 −1.76168 −1.92726 0 0.259128 1.83637 0 0.940847
1.11 −0.0681863 0 −1.99535 3.91488 0 0.303195 0.272428 0 −0.266941
1.12 0.436741 0 −1.80926 −3.47884 0 −1.93516 −1.66366 0 −1.51935
1.13 0.556474 0 −1.69034 −0.874867 0 4.50273 −2.05358 0 −0.486841
1.14 0.976920 0 −1.04563 2.45987 0 −4.72156 −2.97533 0 2.40309
1.15 1.62941 0 0.654988 −2.04510 0 −3.70972 −2.19158 0 −3.33231
1.16 1.82302 0 1.32341 3.15385 0 −0.287436 −1.23344 0 5.74953
1.17 2.08441 0 2.34475 −2.98183 0 3.78550 0.718591 0 −6.21535
1.18 2.21976 0 2.92734 2.17512 0 5.05675 2.05847 0 4.82825
1.19 2.80778 0 5.88364 0.925514 0 2.15050 10.9044 0 2.59864
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.19
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(11\) \(1\)
\(61\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{19} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6039))\).