Properties

Label 6045.2.a.bg
Level 6045
Weight 2
Character orbit 6045.a
Self dual Yes
Analytic conductor 48.270
Analytic rank 0
Dimension 16
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6045.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.2695680219\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{2} \) \(- q^{3}\) \( + ( 2 + \beta_{2} ) q^{4} \) \(- q^{5}\) \( + \beta_{1} q^{6} \) \( -\beta_{14} q^{7} \) \( + ( -2 \beta_{1} - \beta_{3} ) q^{8} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{2} \) \(- q^{3}\) \( + ( 2 + \beta_{2} ) q^{4} \) \(- q^{5}\) \( + \beta_{1} q^{6} \) \( -\beta_{14} q^{7} \) \( + ( -2 \beta_{1} - \beta_{3} ) q^{8} \) \(+ q^{9}\) \( + \beta_{1} q^{10} \) \( + ( \beta_{4} + \beta_{10} ) q^{11} \) \( + ( -2 - \beta_{2} ) q^{12} \) \(- q^{13}\) \( + ( -\beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{14} \) \(+ q^{15}\) \( + ( 4 + 2 \beta_{2} - \beta_{6} + \beta_{7} - \beta_{10} + \beta_{11} - \beta_{13} ) q^{16} \) \( + ( -1 - \beta_{1} + \beta_{8} - \beta_{13} ) q^{17} \) \( -\beta_{1} q^{18} \) \( + ( -\beta_{1} + \beta_{2} + \beta_{4} + \beta_{9} + \beta_{10} + \beta_{15} ) q^{19} \) \( + ( -2 - \beta_{2} ) q^{20} \) \( + \beta_{14} q^{21} \) \( + ( 1 + \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{10} + \beta_{11} - 2 \beta_{12} - \beta_{13} - \beta_{14} ) q^{22} \) \( + ( -1 - \beta_{1} + \beta_{7} + \beta_{9} + \beta_{10} - \beta_{13} + \beta_{14} ) q^{23} \) \( + ( 2 \beta_{1} + \beta_{3} ) q^{24} \) \(+ q^{25}\) \( + \beta_{1} q^{26} \) \(- q^{27}\) \( + ( 1 - \beta_{1} - \beta_{2} - \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{12} - 2 \beta_{14} ) q^{28} \) \( + ( -\beta_{1} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{9} - \beta_{11} + \beta_{14} + \beta_{15} ) q^{29} \) \( -\beta_{1} q^{30} \) \(+ q^{31}\) \( + ( -1 - 4 \beta_{1} - 2 \beta_{3} - \beta_{5} + \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{12} + \beta_{13} ) q^{32} \) \( + ( -\beta_{4} - \beta_{10} ) q^{33} \) \( + ( 2 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} - \beta_{8} + 2 \beta_{11} + \beta_{12} ) q^{34} \) \( + \beta_{14} q^{35} \) \( + ( 2 + \beta_{2} ) q^{36} \) \( + ( 1 - \beta_{3} - 2 \beta_{4} - \beta_{6} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{15} ) q^{37} \) \( + ( 1 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} - 2 \beta_{11} - 2 \beta_{12} - \beta_{13} + \beta_{15} ) q^{38} \) \(+ q^{39}\) \( + ( 2 \beta_{1} + \beta_{3} ) q^{40} \) \( + ( -2 \beta_{6} + \beta_{11} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{41} \) \( + ( \beta_{1} - \beta_{2} - \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{42} \) \( + ( -1 + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} + 2 \beta_{10} - \beta_{12} - \beta_{14} ) q^{43} \) \( + ( -2 + \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - 3 \beta_{7} + 3 \beta_{10} - \beta_{11} - 2 \beta_{12} + \beta_{13} - \beta_{14} ) q^{44} \) \(- q^{45}\) \( + ( 1 + 3 \beta_{1} - \beta_{4} + \beta_{5} - \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{14} - 2 \beta_{15} ) q^{46} \) \( + ( 1 - \beta_{1} - \beta_{7} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{15} ) q^{47} \) \( + ( -4 - 2 \beta_{2} + \beta_{6} - \beta_{7} + \beta_{10} - \beta_{11} + \beta_{13} ) q^{48} \) \( + ( 1 + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{8} + \beta_{10} - \beta_{11} - \beta_{15} ) q^{49} \) \( -\beta_{1} q^{50} \) \( + ( 1 + \beta_{1} - \beta_{8} + \beta_{13} ) q^{51} \) \( + ( -2 - \beta_{2} ) q^{52} \) \( + ( -3 + \beta_{2} + 2 \beta_{4} + 2 \beta_{6} - \beta_{7} + \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{53} \) \( + \beta_{1} q^{54} \) \( + ( -\beta_{4} - \beta_{10} ) q^{55} \) \( + ( 4 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} + 2 \beta_{14} ) q^{56} \) \( + ( \beta_{1} - \beta_{2} - \beta_{4} - \beta_{9} - \beta_{10} - \beta_{15} ) q^{57} \) \( + ( -3 \beta_{5} - \beta_{7} - \beta_{8} + \beta_{10} - \beta_{12} + \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{58} \) \( + ( 3 - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} + \beta_{13} - \beta_{14} ) q^{59} \) \( + ( 2 + \beta_{2} ) q^{60} \) \( + ( 2 - \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{11} + \beta_{14} - \beta_{15} ) q^{61} \) \( -\beta_{1} q^{62} \) \( -\beta_{14} q^{63} \) \( + ( 8 - \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} - \beta_{6} + 3 \beta_{7} - \beta_{9} - 4 \beta_{10} + 3 \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{64} \) \(+ q^{65}\) \( + ( -1 - \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{10} - \beta_{11} + 2 \beta_{12} + \beta_{13} + \beta_{14} ) q^{66} \) \( + ( \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{67} \) \( + ( -1 - 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + \beta_{13} - 3 \beta_{14} - \beta_{15} ) q^{68} \) \( + ( 1 + \beta_{1} - \beta_{7} - \beta_{9} - \beta_{10} + \beta_{13} - \beta_{14} ) q^{69} \) \( + ( \beta_{1} - \beta_{2} - \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{70} \) \( + ( -2 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{8} - \beta_{10} + 2 \beta_{12} ) q^{71} \) \( + ( -2 \beta_{1} - \beta_{3} ) q^{72} \) \( + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{11} + \beta_{12} + \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{73} \) \( + ( 1 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - 3 \beta_{4} - \beta_{6} + 3 \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} + 2 \beta_{12} - \beta_{13} + 3 \beta_{14} - \beta_{15} ) q^{74} \) \(- q^{75}\) \( + ( 7 - 2 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} + 5 \beta_{11} + \beta_{13} - 3 \beta_{14} + 2 \beta_{15} ) q^{76} \) \( + ( -3 - \beta_{1} - \beta_{4} + \beta_{8} + \beta_{9} - \beta_{11} + \beta_{12} + \beta_{13} ) q^{77} \) \( -\beta_{1} q^{78} \) \( + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{7} + 2 \beta_{8} - 3 \beta_{11} + \beta_{13} + 2 \beta_{14} ) q^{79} \) \( + ( -4 - 2 \beta_{2} + \beta_{6} - \beta_{7} + \beta_{10} - \beta_{11} + \beta_{13} ) q^{80} \) \(+ q^{81}\) \( + ( 3 + \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{10} - 3 \beta_{11} - 3 \beta_{12} - \beta_{13} + 4 \beta_{14} - \beta_{15} ) q^{82} \) \( + ( -1 - \beta_{1} - 3 \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{7} - 2 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{14} - 2 \beta_{15} ) q^{83} \) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{12} + 2 \beta_{14} ) q^{84} \) \( + ( 1 + \beta_{1} - \beta_{8} + \beta_{13} ) q^{85} \) \( + ( 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} + 3 \beta_{6} - 4 \beta_{7} + \beta_{9} + 5 \beta_{10} - 4 \beta_{11} - 4 \beta_{12} - \beta_{13} ) q^{86} \) \( + ( \beta_{1} - \beta_{4} - \beta_{5} - \beta_{7} - \beta_{9} + \beta_{11} - \beta_{14} - \beta_{15} ) q^{87} \) \( + ( 1 + 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 2 \beta_{7} + \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{12} - \beta_{13} + \beta_{15} ) q^{88} \) \( + ( \beta_{2} + \beta_{4} + 2 \beta_{5} + \beta_{9} + 2 \beta_{10} - \beta_{11} - \beta_{13} - \beta_{15} ) q^{89} \) \( + \beta_{1} q^{90} \) \( + \beta_{14} q^{91} \) \( + ( -5 - 3 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} - 2 \beta_{8} + \beta_{10} + 3 \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{92} \) \(- q^{93}\) \( + ( 5 - 2 \beta_{1} + 3 \beta_{2} + \beta_{4} - \beta_{7} + \beta_{9} + \beta_{10} - 2 \beta_{11} - 2 \beta_{12} + \beta_{15} ) q^{94} \) \( + ( \beta_{1} - \beta_{2} - \beta_{4} - \beta_{9} - \beta_{10} - \beta_{15} ) q^{95} \) \( + ( 1 + 4 \beta_{1} + 2 \beta_{3} + \beta_{5} - \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{96} \) \( + ( 1 - \beta_{1} + 2 \beta_{4} + \beta_{5} + 2 \beta_{10} - \beta_{11} - \beta_{12} - 2 \beta_{14} + \beta_{15} ) q^{97} \) \( + ( 3 - \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - \beta_{6} + 3 \beta_{7} + \beta_{9} - \beta_{11} + \beta_{12} - \beta_{13} - 2 \beta_{15} ) q^{98} \) \( + ( \beta_{4} + \beta_{10} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(16q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 16q^{3} \) \(\mathstrut +\mathstrut 26q^{4} \) \(\mathstrut -\mathstrut 16q^{5} \) \(\mathstrut +\mathstrut 2q^{6} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut -\mathstrut 9q^{8} \) \(\mathstrut +\mathstrut 16q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(16q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 16q^{3} \) \(\mathstrut +\mathstrut 26q^{4} \) \(\mathstrut -\mathstrut 16q^{5} \) \(\mathstrut +\mathstrut 2q^{6} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut -\mathstrut 9q^{8} \) \(\mathstrut +\mathstrut 16q^{9} \) \(\mathstrut +\mathstrut 2q^{10} \) \(\mathstrut +\mathstrut 3q^{11} \) \(\mathstrut -\mathstrut 26q^{12} \) \(\mathstrut -\mathstrut 16q^{13} \) \(\mathstrut -\mathstrut 5q^{14} \) \(\mathstrut +\mathstrut 16q^{15} \) \(\mathstrut +\mathstrut 38q^{16} \) \(\mathstrut -\mathstrut 13q^{17} \) \(\mathstrut -\mathstrut 2q^{18} \) \(\mathstrut -\mathstrut 26q^{20} \) \(\mathstrut +\mathstrut 2q^{21} \) \(\mathstrut +\mathstrut q^{22} \) \(\mathstrut -\mathstrut 15q^{23} \) \(\mathstrut +\mathstrut 9q^{24} \) \(\mathstrut +\mathstrut 16q^{25} \) \(\mathstrut +\mathstrut 2q^{26} \) \(\mathstrut -\mathstrut 16q^{27} \) \(\mathstrut +\mathstrut 8q^{28} \) \(\mathstrut -\mathstrut 4q^{29} \) \(\mathstrut -\mathstrut 2q^{30} \) \(\mathstrut +\mathstrut 16q^{31} \) \(\mathstrut -\mathstrut 30q^{32} \) \(\mathstrut -\mathstrut 3q^{33} \) \(\mathstrut +\mathstrut 29q^{34} \) \(\mathstrut +\mathstrut 2q^{35} \) \(\mathstrut +\mathstrut 26q^{36} \) \(\mathstrut +\mathstrut 12q^{37} \) \(\mathstrut +\mathstrut 16q^{39} \) \(\mathstrut +\mathstrut 9q^{40} \) \(\mathstrut -\mathstrut 12q^{41} \) \(\mathstrut +\mathstrut 5q^{42} \) \(\mathstrut -\mathstrut 7q^{43} \) \(\mathstrut -\mathstrut 13q^{44} \) \(\mathstrut -\mathstrut 16q^{45} \) \(\mathstrut +\mathstrut 14q^{46} \) \(\mathstrut +\mathstrut 17q^{47} \) \(\mathstrut -\mathstrut 38q^{48} \) \(\mathstrut +\mathstrut 16q^{49} \) \(\mathstrut -\mathstrut 2q^{50} \) \(\mathstrut +\mathstrut 13q^{51} \) \(\mathstrut -\mathstrut 26q^{52} \) \(\mathstrut -\mathstrut 36q^{53} \) \(\mathstrut +\mathstrut 2q^{54} \) \(\mathstrut -\mathstrut 3q^{55} \) \(\mathstrut +\mathstrut 41q^{56} \) \(\mathstrut +\mathstrut 16q^{58} \) \(\mathstrut +\mathstrut 53q^{59} \) \(\mathstrut +\mathstrut 26q^{60} \) \(\mathstrut +\mathstrut 34q^{61} \) \(\mathstrut -\mathstrut 2q^{62} \) \(\mathstrut -\mathstrut 2q^{63} \) \(\mathstrut +\mathstrut 79q^{64} \) \(\mathstrut +\mathstrut 16q^{65} \) \(\mathstrut -\mathstrut q^{66} \) \(\mathstrut -\mathstrut 13q^{67} \) \(\mathstrut -\mathstrut 39q^{68} \) \(\mathstrut +\mathstrut 15q^{69} \) \(\mathstrut +\mathstrut 5q^{70} \) \(\mathstrut -\mathstrut 11q^{71} \) \(\mathstrut -\mathstrut 9q^{72} \) \(\mathstrut +\mathstrut 34q^{73} \) \(\mathstrut -\mathstrut 12q^{74} \) \(\mathstrut -\mathstrut 16q^{75} \) \(\mathstrut +\mathstrut 86q^{76} \) \(\mathstrut -\mathstrut 32q^{77} \) \(\mathstrut -\mathstrut 2q^{78} \) \(\mathstrut -\mathstrut 7q^{79} \) \(\mathstrut -\mathstrut 38q^{80} \) \(\mathstrut +\mathstrut 16q^{81} \) \(\mathstrut +\mathstrut 27q^{82} \) \(\mathstrut -\mathstrut 28q^{83} \) \(\mathstrut -\mathstrut 8q^{84} \) \(\mathstrut +\mathstrut 13q^{85} \) \(\mathstrut +\mathstrut 38q^{86} \) \(\mathstrut +\mathstrut 4q^{87} \) \(\mathstrut +\mathstrut 23q^{88} \) \(\mathstrut -\mathstrut 8q^{89} \) \(\mathstrut +\mathstrut 2q^{90} \) \(\mathstrut +\mathstrut 2q^{91} \) \(\mathstrut -\mathstrut 71q^{92} \) \(\mathstrut -\mathstrut 16q^{93} \) \(\mathstrut +\mathstrut 66q^{94} \) \(\mathstrut +\mathstrut 30q^{96} \) \(\mathstrut +\mathstrut 4q^{97} \) \(\mathstrut +\mathstrut 22q^{98} \) \(\mathstrut +\mathstrut 3q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16}\mathstrut -\mathstrut \) \(2\) \(x^{15}\mathstrut -\mathstrut \) \(27\) \(x^{14}\mathstrut +\mathstrut \) \(51\) \(x^{13}\mathstrut +\mathstrut \) \(294\) \(x^{12}\mathstrut -\mathstrut \) \(517\) \(x^{11}\mathstrut -\mathstrut \) \(1657\) \(x^{10}\mathstrut +\mathstrut \) \(2678\) \(x^{9}\mathstrut +\mathstrut \) \(5124\) \(x^{8}\mathstrut -\mathstrut \) \(7583\) \(x^{7}\mathstrut -\mathstrut \) \(8358\) \(x^{6}\mathstrut +\mathstrut \) \(11579\) \(x^{5}\mathstrut +\mathstrut \) \(6027\) \(x^{4}\mathstrut -\mathstrut \) \(8572\) \(x^{3}\mathstrut -\mathstrut \) \(780\) \(x^{2}\mathstrut +\mathstrut \) \(2178\) \(x\mathstrut -\mathstrut \) \(428\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 6 \nu \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(401375\) \(\nu^{15}\mathstrut +\mathstrut \) \(5831475\) \(\nu^{14}\mathstrut -\mathstrut \) \(6105068\) \(\nu^{13}\mathstrut -\mathstrut \) \(127878505\) \(\nu^{12}\mathstrut +\mathstrut \) \(266535327\) \(\nu^{11}\mathstrut +\mathstrut \) \(1041027406\) \(\nu^{10}\mathstrut -\mathstrut \) \(2652072475\) \(\nu^{9}\mathstrut -\mathstrut \) \(3840982829\) \(\nu^{8}\mathstrut +\mathstrut \) \(11616670989\) \(\nu^{7}\mathstrut +\mathstrut \) \(5927851406\) \(\nu^{6}\mathstrut -\mathstrut \) \(24399073216\) \(\nu^{5}\mathstrut -\mathstrut \) \(891629561\) \(\nu^{4}\mathstrut +\mathstrut \) \(22883291190\) \(\nu^{3}\mathstrut -\mathstrut \) \(5347435986\) \(\nu^{2}\mathstrut -\mathstrut \) \(7260966308\) \(\nu\mathstrut +\mathstrut \) \(2656763704\)\()/66641158\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(675635\) \(\nu^{15}\mathstrut -\mathstrut \) \(166555\) \(\nu^{14}\mathstrut +\mathstrut \) \(24375904\) \(\nu^{13}\mathstrut -\mathstrut \) \(4276819\) \(\nu^{12}\mathstrut -\mathstrut \) \(340606193\) \(\nu^{11}\mathstrut +\mathstrut \) \(144806888\) \(\nu^{10}\mathstrut +\mathstrut \) \(2381186265\) \(\nu^{9}\mathstrut -\mathstrut \) \(1333309911\) \(\nu^{8}\mathstrut -\mathstrut \) \(8903486735\) \(\nu^{7}\mathstrut +\mathstrut \) \(5448761876\) \(\nu^{6}\mathstrut +\mathstrut \) \(17484344834\) \(\nu^{5}\mathstrut -\mathstrut \) \(10588021223\) \(\nu^{4}\mathstrut -\mathstrut \) \(16197805786\) \(\nu^{3}\mathstrut +\mathstrut \) \(9022573940\) \(\nu^{2}\mathstrut +\mathstrut \) \(5350743020\) \(\nu\mathstrut -\mathstrut \) \(2433705296\)\()/66641158\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(2488534\) \(\nu^{15}\mathstrut +\mathstrut \) \(15164660\) \(\nu^{14}\mathstrut +\mathstrut \) \(28420427\) \(\nu^{13}\mathstrut -\mathstrut \) \(329766812\) \(\nu^{12}\mathstrut +\mathstrut \) \(146821522\) \(\nu^{11}\mathstrut +\mathstrut \) \(2652036733\) \(\nu^{10}\mathstrut -\mathstrut \) \(3443080522\) \(\nu^{9}\mathstrut -\mathstrut \) \(9616165022\) \(\nu^{8}\mathstrut +\mathstrut \) \(18411647961\) \(\nu^{7}\mathstrut +\mathstrut \) \(14581580542\) \(\nu^{6}\mathstrut -\mathstrut \) \(42361156993\) \(\nu^{5}\mathstrut -\mathstrut \) \(3092968523\) \(\nu^{4}\mathstrut +\mathstrut \) \(41723176503\) \(\nu^{3}\mathstrut -\mathstrut \) \(9364329458\) \(\nu^{2}\mathstrut -\mathstrut \) \(13136412272\) \(\nu\mathstrut +\mathstrut \) \(3930275208\)\()/66641158\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(1368167\) \(\nu^{15}\mathstrut +\mathstrut \) \(5740111\) \(\nu^{14}\mathstrut +\mathstrut \) \(22545964\) \(\nu^{13}\mathstrut -\mathstrut \) \(124455102\) \(\nu^{12}\mathstrut -\mathstrut \) \(74325399\) \(\nu^{11}\mathstrut +\mathstrut \) \(997145395\) \(\nu^{10}\mathstrut -\mathstrut \) \(579282729\) \(\nu^{9}\mathstrut -\mathstrut \) \(3598060725\) \(\nu^{8}\mathstrut +\mathstrut \) \(4843358589\) \(\nu^{7}\mathstrut +\mathstrut \) \(5408123199\) \(\nu^{6}\mathstrut -\mathstrut \) \(12986541279\) \(\nu^{5}\mathstrut -\mathstrut \) \(998374179\) \(\nu^{4}\mathstrut +\mathstrut \) \(14233568984\) \(\nu^{3}\mathstrut -\mathstrut \) \(3814744471\) \(\nu^{2}\mathstrut -\mathstrut \) \(5053913356\) \(\nu\mathstrut +\mathstrut \) \(1615863693\)\()/33320579\)
\(\beta_{8}\)\(=\)\((\)\(2803945\) \(\nu^{15}\mathstrut -\mathstrut \) \(19153621\) \(\nu^{14}\mathstrut -\mathstrut \) \(38546041\) \(\nu^{13}\mathstrut +\mathstrut \) \(439852093\) \(\nu^{12}\mathstrut -\mathstrut \) \(11139427\) \(\nu^{11}\mathstrut -\mathstrut \) \(3861029905\) \(\nu^{10}\mathstrut +\mathstrut \) \(2468886049\) \(\nu^{9}\mathstrut +\mathstrut \) \(16331960235\) \(\nu^{8}\mathstrut -\mathstrut \) \(14429646670\) \(\nu^{7}\mathstrut -\mathstrut \) \(34346738054\) \(\nu^{6}\mathstrut +\mathstrut \) \(33292767751\) \(\nu^{5}\mathstrut +\mathstrut \) \(32323253840\) \(\nu^{4}\mathstrut -\mathstrut \) \(31297948725\) \(\nu^{3}\mathstrut -\mathstrut \) \(8620241002\) \(\nu^{2}\mathstrut +\mathstrut \) \(8548552388\) \(\nu\mathstrut -\mathstrut \) \(1177545982\)\()/66641158\)
\(\beta_{9}\)\(=\)\((\)\(2831059\) \(\nu^{15}\mathstrut -\mathstrut \) \(2064367\) \(\nu^{14}\mathstrut -\mathstrut \) \(89938815\) \(\nu^{13}\mathstrut +\mathstrut \) \(67211901\) \(\nu^{12}\mathstrut +\mathstrut \) \(1147216629\) \(\nu^{11}\mathstrut -\mathstrut \) \(852753033\) \(\nu^{10}\mathstrut -\mathstrut \) \(7510769565\) \(\nu^{9}\mathstrut +\mathstrut \) \(5399648587\) \(\nu^{8}\mathstrut +\mathstrut \) \(26756629056\) \(\nu^{7}\mathstrut -\mathstrut \) \(18150614914\) \(\nu^{6}\mathstrut -\mathstrut \) \(50523613135\) \(\nu^{5}\mathstrut +\mathstrut \) \(31714926198\) \(\nu^{4}\mathstrut +\mathstrut \) \(44928219971\) \(\nu^{3}\mathstrut -\mathstrut \) \(25301350906\) \(\nu^{2}\mathstrut -\mathstrut \) \(13542412620\) \(\nu\mathstrut +\mathstrut \) \(6127161078\)\()/66641158\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(3037054\) \(\nu^{15}\mathstrut +\mathstrut \) \(3168600\) \(\nu^{14}\mathstrut +\mathstrut \) \(89382371\) \(\nu^{13}\mathstrut -\mathstrut \) \(82563440\) \(\nu^{12}\mathstrut -\mathstrut \) \(1067461518\) \(\nu^{11}\mathstrut +\mathstrut \) \(859595697\) \(\nu^{10}\mathstrut +\mathstrut \) \(6623436958\) \(\nu^{9}\mathstrut -\mathstrut \) \(4600819186\) \(\nu^{8}\mathstrut -\mathstrut \) \(22628667487\) \(\nu^{7}\mathstrut +\mathstrut \) \(13556760324\) \(\nu^{6}\mathstrut +\mathstrut \) \(41339037949\) \(\nu^{5}\mathstrut -\mathstrut \) \(21686057951\) \(\nu^{4}\mathstrut -\mathstrut \) \(35772605869\) \(\nu^{3}\mathstrut +\mathstrut \) \(16776685232\) \(\nu^{2}\mathstrut +\mathstrut \) \(10687542066\) \(\nu\mathstrut -\mathstrut \) \(4318069210\)\()/66641158\)
\(\beta_{11}\)\(=\)\((\)\(6782934\) \(\nu^{15}\mathstrut -\mathstrut \) \(25215264\) \(\nu^{14}\mathstrut -\mathstrut \) \(135487993\) \(\nu^{13}\mathstrut +\mathstrut \) \(572705046\) \(\nu^{12}\mathstrut +\mathstrut \) \(914873352\) \(\nu^{11}\mathstrut -\mathstrut \) \(4945595009\) \(\nu^{10}\mathstrut -\mathstrut \) \(1961321472\) \(\nu^{9}\mathstrut +\mathstrut \) \(20406762218\) \(\nu^{8}\mathstrut -\mathstrut \) \(3214863019\) \(\nu^{7}\mathstrut -\mathstrut \) \(41288117178\) \(\nu^{6}\mathstrut +\mathstrut \) \(18923186773\) \(\nu^{5}\mathstrut +\mathstrut \) \(36425878673\) \(\nu^{4}\mathstrut -\mathstrut \) \(23543309537\) \(\nu^{3}\mathstrut -\mathstrut \) \(8447540368\) \(\nu^{2}\mathstrut +\mathstrut \) \(7707270874\) \(\nu\mathstrut -\mathstrut \) \(1296448684\)\()/66641158\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(7250209\) \(\nu^{15}\mathstrut +\mathstrut \) \(23370511\) \(\nu^{14}\mathstrut +\mathstrut \) \(161207194\) \(\nu^{13}\mathstrut -\mathstrut \) \(539774147\) \(\nu^{12}\mathstrut -\mathstrut \) \(1356968093\) \(\nu^{11}\mathstrut +\mathstrut \) \(4782945210\) \(\nu^{10}\mathstrut +\mathstrut \) \(5456991977\) \(\nu^{9}\mathstrut -\mathstrut \) \(20586373127\) \(\nu^{8}\mathstrut -\mathstrut \) \(10968126505\) \(\nu^{7}\mathstrut +\mathstrut \) \(44968494856\) \(\nu^{6}\mathstrut +\mathstrut \) \(10655400508\) \(\nu^{5}\mathstrut -\mathstrut \) \(47064899461\) \(\nu^{4}\mathstrut -\mathstrut \) \(5136433166\) \(\nu^{3}\mathstrut +\mathstrut \) \(19765596698\) \(\nu^{2}\mathstrut +\mathstrut \) \(1757645388\) \(\nu\mathstrut -\mathstrut \) \(2141597206\)\()/66641158\)
\(\beta_{13}\)\(=\)\((\)\(9572188\) \(\nu^{15}\mathstrut -\mathstrut \) \(32068302\) \(\nu^{14}\mathstrut -\mathstrut \) \(208198863\) \(\nu^{13}\mathstrut +\mathstrut \) \(736125094\) \(\nu^{12}\mathstrut +\mathstrut \) \(1686862550\) \(\nu^{11}\mathstrut -\mathstrut \) \(6462936649\) \(\nu^{10}\mathstrut -\mathstrut \) \(6300243366\) \(\nu^{9}\mathstrut +\mathstrut \) \(27427624976\) \(\nu^{8}\mathstrut +\mathstrut \) \(10688873685\) \(\nu^{7}\mathstrut -\mathstrut \) \(58610211646\) \(\nu^{6}\mathstrut -\mathstrut \) \(6027776741\) \(\nu^{5}\mathstrut +\mathstrut \) \(59141515631\) \(\nu^{4}\mathstrut -\mathstrut \) \(1026742203\) \(\nu^{3}\mathstrut -\mathstrut \) \(22956255820\) \(\nu^{2}\mathstrut +\mathstrut \) \(48314368\) \(\nu\mathstrut +\mathstrut \) \(1789943440\)\()/66641158\)
\(\beta_{14}\)\(=\)\((\)\(9720317\) \(\nu^{15}\mathstrut -\mathstrut \) \(40614744\) \(\nu^{14}\mathstrut -\mathstrut \) \(177132377\) \(\nu^{13}\mathstrut +\mathstrut \) \(911381417\) \(\nu^{12}\mathstrut +\mathstrut \) \(923369914\) \(\nu^{11}\mathstrut -\mathstrut \) \(7729231159\) \(\nu^{10}\mathstrut +\mathstrut \) \(543799535\) \(\nu^{9}\mathstrut +\mathstrut \) \(30983285268\) \(\nu^{8}\mathstrut -\mathstrut \) \(18471962304\) \(\nu^{7}\mathstrut -\mathstrut \) \(59479946487\) \(\nu^{6}\mathstrut +\mathstrut \) \(55300353944\) \(\nu^{5}\mathstrut +\mathstrut \) \(46082550133\) \(\nu^{4}\mathstrut -\mathstrut \) \(59309144451\) \(\nu^{3}\mathstrut -\mathstrut \) \(3826053620\) \(\nu^{2}\mathstrut +\mathstrut \) \(18415371574\) \(\nu\mathstrut -\mathstrut \) \(4303214968\)\()/66641158\)
\(\beta_{15}\)\(=\)\((\)\(-\)\(12686056\) \(\nu^{15}\mathstrut +\mathstrut \) \(41128413\) \(\nu^{14}\mathstrut +\mathstrut \) \(279181880\) \(\nu^{13}\mathstrut -\mathstrut \) \(947900454\) \(\nu^{12}\mathstrut -\mathstrut \) \(2295935223\) \(\nu^{11}\mathstrut +\mathstrut \) \(8361431816\) \(\nu^{10}\mathstrut +\mathstrut \) \(8718170552\) \(\nu^{9}\mathstrut -\mathstrut \) \(35648126769\) \(\nu^{8}\mathstrut -\mathstrut \) \(14882591054\) \(\nu^{7}\mathstrut +\mathstrut \) \(76333936461\) \(\nu^{6}\mathstrut +\mathstrut \) \(7413524609\) \(\nu^{5}\mathstrut -\mathstrut \) \(76558797477\) \(\nu^{4}\mathstrut +\mathstrut \) \(4251737464\) \(\nu^{3}\mathstrut +\mathstrut \) \(29046054534\) \(\nu^{2}\mathstrut -\mathstrut \) \(1865853208\) \(\nu\mathstrut -\mathstrut \) \(2285385432\)\()/66641158\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(6\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(8\) \(\beta_{2}\mathstrut +\mathstrut \) \(24\)
\(\nu^{5}\)\(=\)\(-\)\(\beta_{13}\mathstrut -\mathstrut \) \(2\) \(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(10\) \(\beta_{3}\mathstrut +\mathstrut \) \(40\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{6}\)\(=\)\(-\)\(11\) \(\beta_{13}\mathstrut +\mathstrut \) \(2\) \(\beta_{12}\mathstrut +\mathstrut \) \(13\) \(\beta_{11}\mathstrut -\mathstrut \) \(14\) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(13\) \(\beta_{7}\mathstrut -\mathstrut \) \(11\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(57\) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(160\)
\(\nu^{7}\)\(=\)\(\beta_{15}\mathstrut -\mathstrut \) \(\beta_{14}\mathstrut -\mathstrut \) \(12\) \(\beta_{13}\mathstrut -\mathstrut \) \(28\) \(\beta_{12}\mathstrut -\mathstrut \) \(11\) \(\beta_{11}\mathstrut +\mathstrut \) \(16\) \(\beta_{10}\mathstrut +\mathstrut \) \(13\) \(\beta_{9}\mathstrut -\mathstrut \) \(13\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(16\) \(\beta_{5}\mathstrut +\mathstrut \) \(4\) \(\beta_{4}\mathstrut +\mathstrut \) \(84\) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(280\) \(\beta_{1}\mathstrut +\mathstrut \) \(8\)
\(\nu^{8}\)\(=\)\(-\)\(\beta_{15}\mathstrut +\mathstrut \) \(3\) \(\beta_{14}\mathstrut -\mathstrut \) \(95\) \(\beta_{13}\mathstrut +\mathstrut \) \(36\) \(\beta_{12}\mathstrut +\mathstrut \) \(124\) \(\beta_{11}\mathstrut -\mathstrut \) \(143\) \(\beta_{10}\mathstrut -\mathstrut \) \(12\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(130\) \(\beta_{7}\mathstrut -\mathstrut \) \(98\) \(\beta_{6}\mathstrut +\mathstrut \) \(17\) \(\beta_{5}\mathstrut -\mathstrut \) \(33\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(401\) \(\beta_{2}\mathstrut -\mathstrut \) \(22\) \(\beta_{1}\mathstrut +\mathstrut \) \(1116\)
\(\nu^{9}\)\(=\)\(18\) \(\beta_{15}\mathstrut -\mathstrut \) \(20\) \(\beta_{14}\mathstrut -\mathstrut \) \(109\) \(\beta_{13}\mathstrut -\mathstrut \) \(286\) \(\beta_{12}\mathstrut -\mathstrut \) \(88\) \(\beta_{11}\mathstrut +\mathstrut \) \(179\) \(\beta_{10}\mathstrut +\mathstrut \) \(127\) \(\beta_{9}\mathstrut +\mathstrut \) \(2\) \(\beta_{8}\mathstrut -\mathstrut \) \(131\) \(\beta_{7}\mathstrut +\mathstrut \) \(17\) \(\beta_{6}\mathstrut +\mathstrut \) \(176\) \(\beta_{5}\mathstrut +\mathstrut \) \(73\) \(\beta_{4}\mathstrut +\mathstrut \) \(672\) \(\beta_{3}\mathstrut -\mathstrut \) \(36\) \(\beta_{2}\mathstrut +\mathstrut \) \(2011\) \(\beta_{1}\mathstrut +\mathstrut \) \(31\)
\(\nu^{10}\)\(=\)\(-\)\(20\) \(\beta_{15}\mathstrut +\mathstrut \) \(50\) \(\beta_{14}\mathstrut -\mathstrut \) \(759\) \(\beta_{13}\mathstrut +\mathstrut \) \(433\) \(\beta_{12}\mathstrut +\mathstrut \) \(1068\) \(\beta_{11}\mathstrut -\mathstrut \) \(1297\) \(\beta_{10}\mathstrut -\mathstrut \) \(108\) \(\beta_{9}\mathstrut -\mathstrut \) \(21\) \(\beta_{8}\mathstrut +\mathstrut \) \(1177\) \(\beta_{7}\mathstrut -\mathstrut \) \(821\) \(\beta_{6}\mathstrut +\mathstrut \) \(191\) \(\beta_{5}\mathstrut -\mathstrut \) \(381\) \(\beta_{4}\mathstrut -\mathstrut \) \(20\) \(\beta_{3}\mathstrut +\mathstrut \) \(2833\) \(\beta_{2}\mathstrut -\mathstrut \) \(295\) \(\beta_{1}\mathstrut +\mathstrut \) \(7978\)
\(\nu^{11}\)\(=\)\(222\) \(\beta_{15}\mathstrut -\mathstrut \) \(264\) \(\beta_{14}\mathstrut -\mathstrut \) \(894\) \(\beta_{13}\mathstrut -\mathstrut \) \(2602\) \(\beta_{12}\mathstrut -\mathstrut \) \(626\) \(\beta_{11}\mathstrut +\mathstrut \) \(1741\) \(\beta_{10}\mathstrut +\mathstrut \) \(1118\) \(\beta_{9}\mathstrut +\mathstrut \) \(38\) \(\beta_{8}\mathstrut -\mathstrut \) \(1209\) \(\beta_{7}\mathstrut +\mathstrut \) \(199\) \(\beta_{6}\mathstrut +\mathstrut \) \(1659\) \(\beta_{5}\mathstrut +\mathstrut \) \(885\) \(\beta_{4}\mathstrut +\mathstrut \) \(5274\) \(\beta_{3}\mathstrut -\mathstrut \) \(443\) \(\beta_{2}\mathstrut +\mathstrut \) \(14667\) \(\beta_{1}\mathstrut -\mathstrut \) \(104\)
\(\nu^{12}\)\(=\)\(-\)\(256\) \(\beta_{15}\mathstrut +\mathstrut \) \(558\) \(\beta_{14}\mathstrut -\mathstrut \) \(5871\) \(\beta_{13}\mathstrut +\mathstrut \) \(4401\) \(\beta_{12}\mathstrut +\mathstrut \) \(8814\) \(\beta_{11}\mathstrut -\mathstrut \) \(11088\) \(\beta_{10}\mathstrut -\mathstrut \) \(890\) \(\beta_{9}\mathstrut -\mathstrut \) \(281\) \(\beta_{8}\mathstrut +\mathstrut \) \(10116\) \(\beta_{7}\mathstrut -\mathstrut \) \(6694\) \(\beta_{6}\mathstrut +\mathstrut \) \(1799\) \(\beta_{5}\mathstrut -\mathstrut \) \(3789\) \(\beta_{4}\mathstrut -\mathstrut \) \(272\) \(\beta_{3}\mathstrut +\mathstrut \) \(20171\) \(\beta_{2}\mathstrut -\mathstrut \) \(3228\) \(\beta_{1}\mathstrut +\mathstrut \) \(57910\)
\(\nu^{13}\)\(=\)\(2329\) \(\beta_{15}\mathstrut -\mathstrut \) \(2883\) \(\beta_{14}\mathstrut -\mathstrut \) \(6973\) \(\beta_{13}\mathstrut -\mathstrut \) \(22371\) \(\beta_{12}\mathstrut -\mathstrut \) \(4217\) \(\beta_{11}\mathstrut +\mathstrut \) \(15779\) \(\beta_{10}\mathstrut +\mathstrut \) \(9372\) \(\beta_{9}\mathstrut +\mathstrut \) \(463\) \(\beta_{8}\mathstrut -\mathstrut \) \(10660\) \(\beta_{7}\mathstrut +\mathstrut \) \(2001\) \(\beta_{6}\mathstrut +\mathstrut \) \(14430\) \(\beta_{5}\mathstrut +\mathstrut \) \(9029\) \(\beta_{4}\mathstrut +\mathstrut \) \(41011\) \(\beta_{3}\mathstrut -\mathstrut \) \(4649\) \(\beta_{2}\mathstrut +\mathstrut \) \(108061\) \(\beta_{1}\mathstrut -\mathstrut \) \(3689\)
\(\nu^{14}\)\(=\)\(-\)\(2679\) \(\beta_{15}\mathstrut +\mathstrut \) \(5285\) \(\beta_{14}\mathstrut -\mathstrut \) \(44720\) \(\beta_{13}\mathstrut +\mathstrut \) \(40924\) \(\beta_{12}\mathstrut +\mathstrut \) \(71241\) \(\beta_{11}\mathstrut -\mathstrut \) \(91612\) \(\beta_{10}\mathstrut -\mathstrut \) \(7100\) \(\beta_{9}\mathstrut -\mathstrut \) \(3097\) \(\beta_{8}\mathstrut +\mathstrut \) \(84234\) \(\beta_{7}\mathstrut -\mathstrut \) \(53729\) \(\beta_{6}\mathstrut +\mathstrut \) \(15410\) \(\beta_{5}\mathstrut -\mathstrut \) \(34746\) \(\beta_{4}\mathstrut -\mathstrut \) \(3137\) \(\beta_{3}\mathstrut +\mathstrut \) \(144819\) \(\beta_{2}\mathstrut -\mathstrut \) \(31788\) \(\beta_{1}\mathstrut +\mathstrut \) \(424747\)
\(\nu^{15}\)\(=\)\(22339\) \(\beta_{15}\mathstrut -\mathstrut \) \(28273\) \(\beta_{14}\mathstrut -\mathstrut \) \(52836\) \(\beta_{13}\mathstrut -\mathstrut \) \(186173\) \(\beta_{12}\mathstrut -\mathstrut \) \(27613\) \(\beta_{11}\mathstrut +\mathstrut \) \(137244\) \(\beta_{10}\mathstrut +\mathstrut \) \(76526\) \(\beta_{9}\mathstrut +\mathstrut \) \(4611\) \(\beta_{8}\mathstrut -\mathstrut \) \(91276\) \(\beta_{7}\mathstrut +\mathstrut \) \(18619\) \(\beta_{6}\mathstrut +\mathstrut \) \(119753\) \(\beta_{5}\mathstrut +\mathstrut \) \(84052\) \(\beta_{4}\mathstrut +\mathstrut \) \(317266\) \(\beta_{3}\mathstrut -\mathstrut \) \(44788\) \(\beta_{2}\mathstrut +\mathstrut \) \(801922\) \(\beta_{1}\mathstrut -\mathstrut \) \(50618\)

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
2.74907
2.67164
2.46449
2.34208
1.69426
1.17778
0.976688
0.399270
0.323092
−0.733028
−1.26786
−1.73595
−1.89108
−1.90524
−2.48293
−2.78228
−2.74907 −1.00000 5.55740 −1.00000 2.74907 −0.783225 −9.77954 1.00000 2.74907
1.2 −2.67164 −1.00000 5.13767 −1.00000 2.67164 1.81220 −8.38272 1.00000 2.67164
1.3 −2.46449 −1.00000 4.07370 −1.00000 2.46449 −4.85009 −5.11060 1.00000 2.46449
1.4 −2.34208 −1.00000 3.48532 −1.00000 2.34208 3.99026 −3.47875 1.00000 2.34208
1.5 −1.69426 −1.00000 0.870532 −1.00000 1.69426 0.135304 1.91362 1.00000 1.69426
1.6 −1.17778 −1.00000 −0.612829 −1.00000 1.17778 −0.648351 3.07734 1.00000 1.17778
1.7 −0.976688 −1.00000 −1.04608 −1.00000 0.976688 2.96131 2.97507 1.00000 0.976688
1.8 −0.399270 −1.00000 −1.84058 −1.00000 0.399270 −2.18061 1.53343 1.00000 0.399270
1.9 −0.323092 −1.00000 −1.89561 −1.00000 0.323092 1.90658 1.25864 1.00000 0.323092
1.10 0.733028 −1.00000 −1.46267 −1.00000 −0.733028 −2.61356 −2.53823 1.00000 −0.733028
1.11 1.26786 −1.00000 −0.392523 −1.00000 −1.26786 −1.11922 −3.03339 1.00000 −1.26786
1.12 1.73595 −1.00000 1.01354 −1.00000 −1.73595 −1.77896 −1.71245 1.00000 −1.73595
1.13 1.89108 −1.00000 1.57618 −1.00000 −1.89108 −4.66347 −0.801474 1.00000 −1.89108
1.14 1.90524 −1.00000 1.62994 −1.00000 −1.90524 4.91854 −0.705044 1.00000 −1.90524
1.15 2.48293 −1.00000 4.16494 −1.00000 −2.48293 −1.71749 5.37540 1.00000 −2.48293
1.16 2.78228 −1.00000 5.74107 −1.00000 −2.78228 2.63079 10.4087 1.00000 −2.78228
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(13\) \(1\)
\(31\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6045))\):

\(T_{2}^{16} + \cdots\)
\(T_{7}^{16} + \cdots\)
\(T_{11}^{16} - \cdots\)