Properties

Label 6010.2.a.c
Level 6010
Weight 2
Character orbit 6010.a
Self dual Yes
Analytic conductor 47.990
Analytic rank 1
Dimension 16
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6010 = 2 \cdot 5 \cdot 601 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6010.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(47.9900916148\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16}\mathstrut -\mathstrut \) \(8\) \(x^{15}\mathstrut +\mathstrut \) \(9\) \(x^{14}\mathstrut +\mathstrut \) \(75\) \(x^{13}\mathstrut -\mathstrut \) \(178\) \(x^{12}\mathstrut -\mathstrut \) \(232\) \(x^{11}\mathstrut +\mathstrut \) \(872\) \(x^{10}\mathstrut +\mathstrut \) \(228\) \(x^{9}\mathstrut -\mathstrut \) \(1986\) \(x^{8}\mathstrut +\mathstrut \) \(164\) \(x^{7}\mathstrut +\mathstrut \) \(2332\) \(x^{6}\mathstrut -\mathstrut \) \(440\) \(x^{5}\mathstrut -\mathstrut \) \(1344\) \(x^{4}\mathstrut +\mathstrut \) \(244\) \(x^{3}\mathstrut +\mathstrut \) \(295\) \(x^{2}\mathstrut -\mathstrut \) \(41\) \(x\mathstrut -\mathstrut \) \(10\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(\beta_{3}\)\(=\)\((\)\(25986\) \(\nu^{15}\mathstrut -\mathstrut \) \(3039938\) \(\nu^{14}\mathstrut +\mathstrut \) \(22373644\) \(\nu^{13}\mathstrut -\mathstrut \) \(24822547\) \(\nu^{12}\mathstrut -\mathstrut \) \(180887926\) \(\nu^{11}\mathstrut +\mathstrut \) \(419917416\) \(\nu^{10}\mathstrut +\mathstrut \) \(422463844\) \(\nu^{9}\mathstrut -\mathstrut \) \(1628277055\) \(\nu^{8}\mathstrut -\mathstrut \) \(171959962\) \(\nu^{7}\mathstrut +\mathstrut \) \(2649268970\) \(\nu^{6}\mathstrut -\mathstrut \) \(438850745\) \(\nu^{5}\mathstrut -\mathstrut \) \(1879892501\) \(\nu^{4}\mathstrut +\mathstrut \) \(414738405\) \(\nu^{3}\mathstrut +\mathstrut \) \(462320623\) \(\nu^{2}\mathstrut -\mathstrut \) \(89999479\) \(\nu\mathstrut -\mathstrut \) \(17499053\)\()/3115183\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(75052\) \(\nu^{15}\mathstrut -\mathstrut \) \(416320\) \(\nu^{14}\mathstrut +\mathstrut \) \(7114944\) \(\nu^{13}\mathstrut -\mathstrut \) \(14421854\) \(\nu^{12}\mathstrut -\mathstrut \) \(48394711\) \(\nu^{11}\mathstrut +\mathstrut \) \(157537117\) \(\nu^{10}\mathstrut +\mathstrut \) \(79579696\) \(\nu^{9}\mathstrut -\mathstrut \) \(536624577\) \(\nu^{8}\mathstrut +\mathstrut \) \(59905022\) \(\nu^{7}\mathstrut +\mathstrut \) \(793919965\) \(\nu^{6}\mathstrut -\mathstrut \) \(225824922\) \(\nu^{5}\mathstrut -\mathstrut \) \(522240622\) \(\nu^{4}\mathstrut +\mathstrut \) \(145436552\) \(\nu^{3}\mathstrut +\mathstrut \) \(127539985\) \(\nu^{2}\mathstrut -\mathstrut \) \(24830756\) \(\nu\mathstrut -\mathstrut \) \(2133268\)\()/3115183\)
\(\beta_{5}\)\(=\)\((\)\(180376\) \(\nu^{15}\mathstrut -\mathstrut \) \(2419781\) \(\nu^{14}\mathstrut +\mathstrut \) \(9497505\) \(\nu^{13}\mathstrut +\mathstrut \) \(2661981\) \(\nu^{12}\mathstrut -\mathstrut \) \(91910225\) \(\nu^{11}\mathstrut +\mathstrut \) \(118706510\) \(\nu^{10}\mathstrut +\mathstrut \) \(274101206\) \(\nu^{9}\mathstrut -\mathstrut \) \(560118954\) \(\nu^{8}\mathstrut -\mathstrut \) \(327354943\) \(\nu^{7}\mathstrut +\mathstrut \) \(988201962\) \(\nu^{6}\mathstrut +\mathstrut \) \(157843278\) \(\nu^{5}\mathstrut -\mathstrut \) \(763974869\) \(\nu^{4}\mathstrut -\mathstrut \) \(43939251\) \(\nu^{3}\mathstrut +\mathstrut \) \(233958171\) \(\nu^{2}\mathstrut +\mathstrut \) \(10010483\) \(\nu\mathstrut -\mathstrut \) \(20383548\)\()/3115183\)
\(\beta_{6}\)\(=\)\((\)\(284786\) \(\nu^{15}\mathstrut -\mathstrut \) \(1819192\) \(\nu^{14}\mathstrut -\mathstrut \) \(441924\) \(\nu^{13}\mathstrut +\mathstrut \) \(20718600\) \(\nu^{12}\mathstrut -\mathstrut \) \(13794772\) \(\nu^{11}\mathstrut -\mathstrut \) \(100540960\) \(\nu^{10}\mathstrut +\mathstrut \) \(74146068\) \(\nu^{9}\mathstrut +\mathstrut \) \(279407436\) \(\nu^{8}\mathstrut -\mathstrut \) \(157995531\) \(\nu^{7}\mathstrut -\mathstrut \) \(442765003\) \(\nu^{6}\mathstrut +\mathstrut \) \(150259400\) \(\nu^{5}\mathstrut +\mathstrut \) \(331067185\) \(\nu^{4}\mathstrut -\mathstrut \) \(39287261\) \(\nu^{3}\mathstrut -\mathstrut \) \(67060795\) \(\nu^{2}\mathstrut -\mathstrut \) \(9102667\) \(\nu\mathstrut +\mathstrut \) \(706474\)\()/3115183\)
\(\beta_{7}\)\(=\)\((\)\(362678\) \(\nu^{15}\mathstrut -\mathstrut \) \(2892660\) \(\nu^{14}\mathstrut +\mathstrut \) \(4353048\) \(\nu^{13}\mathstrut +\mathstrut \) \(18928481\) \(\nu^{12}\mathstrut -\mathstrut \) \(57805768\) \(\nu^{11}\mathstrut -\mathstrut \) \(10924214\) \(\nu^{10}\mathstrut +\mathstrut \) \(182827079\) \(\nu^{9}\mathstrut -\mathstrut \) \(139100342\) \(\nu^{8}\mathstrut -\mathstrut \) \(176603303\) \(\nu^{7}\mathstrut +\mathstrut \) \(347372878\) \(\nu^{6}\mathstrut -\mathstrut \) \(81890064\) \(\nu^{5}\mathstrut -\mathstrut \) \(319570964\) \(\nu^{4}\mathstrut +\mathstrut \) \(201791930\) \(\nu^{3}\mathstrut +\mathstrut \) \(116648156\) \(\nu^{2}\mathstrut -\mathstrut \) \(61280117\) \(\nu\mathstrut -\mathstrut \) \(6358506\)\()/3115183\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(384044\) \(\nu^{15}\mathstrut +\mathstrut \) \(3421318\) \(\nu^{14}\mathstrut -\mathstrut \) \(6474594\) \(\nu^{13}\mathstrut -\mathstrut \) \(22475282\) \(\nu^{12}\mathstrut +\mathstrut \) \(82865319\) \(\nu^{11}\mathstrut +\mathstrut \) \(16650207\) \(\nu^{10}\mathstrut -\mathstrut \) \(294055924\) \(\nu^{9}\mathstrut +\mathstrut \) \(129035112\) \(\nu^{8}\mathstrut +\mathstrut \) \(429564885\) \(\nu^{7}\mathstrut -\mathstrut \) \(280058413\) \(\nu^{6}\mathstrut -\mathstrut \) \(230548103\) \(\nu^{5}\mathstrut +\mathstrut \) \(178775499\) \(\nu^{4}\mathstrut -\mathstrut \) \(8887973\) \(\nu^{3}\mathstrut -\mathstrut \) \(34425448\) \(\nu^{2}\mathstrut +\mathstrut \) \(12448056\) \(\nu\mathstrut -\mathstrut \) \(714592\)\()/3115183\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(39704\) \(\nu^{15}\mathstrut +\mathstrut \) \(366746\) \(\nu^{14}\mathstrut -\mathstrut \) \(785482\) \(\nu^{13}\mathstrut -\mathstrut \) \(2133030\) \(\nu^{12}\mathstrut +\mathstrut \) \(9493309\) \(\nu^{11}\mathstrut -\mathstrut \) \(971276\) \(\nu^{10}\mathstrut -\mathstrut \) \(32730627\) \(\nu^{9}\mathstrut +\mathstrut \) \(23795599\) \(\nu^{8}\mathstrut +\mathstrut \) \(47347456\) \(\nu^{7}\mathstrut -\mathstrut \) \(46199287\) \(\nu^{6}\mathstrut -\mathstrut \) \(29093811\) \(\nu^{5}\mathstrut +\mathstrut \) \(29632797\) \(\nu^{4}\mathstrut +\mathstrut \) \(6564984\) \(\nu^{3}\mathstrut -\mathstrut \) \(4266261\) \(\nu^{2}\mathstrut -\mathstrut \) \(556740\) \(\nu\mathstrut -\mathstrut \) \(43793\)\()/163957\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(800612\) \(\nu^{15}\mathstrut +\mathstrut \) \(5318060\) \(\nu^{14}\mathstrut -\mathstrut \) \(244087\) \(\nu^{13}\mathstrut -\mathstrut \) \(58445943\) \(\nu^{12}\mathstrut +\mathstrut \) \(60827688\) \(\nu^{11}\mathstrut +\mathstrut \) \(255794130\) \(\nu^{10}\mathstrut -\mathstrut \) \(323031488\) \(\nu^{9}\mathstrut -\mathstrut \) \(600603130\) \(\nu^{8}\mathstrut +\mathstrut \) \(709158898\) \(\nu^{7}\mathstrut +\mathstrut \) \(816274892\) \(\nu^{6}\mathstrut -\mathstrut \) \(731389267\) \(\nu^{5}\mathstrut -\mathstrut \) \(601061929\) \(\nu^{4}\mathstrut +\mathstrut \) \(323396742\) \(\nu^{3}\mathstrut +\mathstrut \) \(181790025\) \(\nu^{2}\mathstrut -\mathstrut \) \(48865179\) \(\nu\mathstrut -\mathstrut \) \(7297457\)\()/3115183\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(972224\) \(\nu^{15}\mathstrut +\mathstrut \) \(8826787\) \(\nu^{14}\mathstrut -\mathstrut \) \(18013961\) \(\nu^{13}\mathstrut -\mathstrut \) \(54357008\) \(\nu^{12}\mathstrut +\mathstrut \) \(225954995\) \(\nu^{11}\mathstrut +\mathstrut \) \(3659483\) \(\nu^{10}\mathstrut -\mathstrut \) \(818923620\) \(\nu^{9}\mathstrut +\mathstrut \) \(503191029\) \(\nu^{8}\mathstrut +\mathstrut \) \(1324407527\) \(\nu^{7}\mathstrut -\mathstrut \) \(1099582230\) \(\nu^{6}\mathstrut -\mathstrut \) \(1049225189\) \(\nu^{5}\mathstrut +\mathstrut \) \(852144102\) \(\nu^{4}\mathstrut +\mathstrut \) \(392375534\) \(\nu^{3}\mathstrut -\mathstrut \) \(211092724\) \(\nu^{2}\mathstrut -\mathstrut \) \(44134004\) \(\nu\mathstrut +\mathstrut \) \(4252139\)\()/3115183\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(1048336\) \(\nu^{15}\mathstrut +\mathstrut \) \(8878763\) \(\nu^{14}\mathstrut -\mathstrut \) \(14466751\) \(\nu^{13}\mathstrut -\mathstrut \) \(64724264\) \(\nu^{12}\mathstrut +\mathstrut \) \(206543323\) \(\nu^{11}\mathstrut +\mathstrut \) \(92532283\) \(\nu^{10}\mathstrut -\mathstrut \) \(804630339\) \(\nu^{9}\mathstrut +\mathstrut \) \(238488877\) \(\nu^{8}\mathstrut +\mathstrut \) \(1382719669\) \(\nu^{7}\mathstrut -\mathstrut \) \(757194522\) \(\nu^{6}\mathstrut -\mathstrut \) \(1110390251\) \(\nu^{5}\mathstrut +\mathstrut \) \(658088239\) \(\nu^{4}\mathstrut +\mathstrut \) \(363077413\) \(\nu^{3}\mathstrut -\mathstrut \) \(173968012\) \(\nu^{2}\mathstrut -\mathstrut \) \(25743241\) \(\nu\mathstrut +\mathstrut \) \(5288765\)\()/3115183\)
\(\beta_{13}\)\(=\)\((\)\(1124047\) \(\nu^{15}\mathstrut -\mathstrut \) \(8847625\) \(\nu^{14}\mathstrut +\mathstrut \) \(11444848\) \(\nu^{13}\mathstrut +\mathstrut \) \(67320977\) \(\nu^{12}\mathstrut -\mathstrut \) \(173501774\) \(\nu^{11}\mathstrut -\mathstrut \) \(128681409\) \(\nu^{10}\mathstrut +\mathstrut \) \(645278405\) \(\nu^{9}\mathstrut -\mathstrut \) \(69804760\) \(\nu^{8}\mathstrut -\mathstrut \) \(1000042516\) \(\nu^{7}\mathstrut +\mathstrut \) \(424081214\) \(\nu^{6}\mathstrut +\mathstrut \) \(650190464\) \(\nu^{5}\mathstrut -\mathstrut \) \(392052900\) \(\nu^{4}\mathstrut -\mathstrut \) \(119306620\) \(\nu^{3}\mathstrut +\mathstrut \) \(112016908\) \(\nu^{2}\mathstrut -\mathstrut \) \(10778289\) \(\nu\mathstrut -\mathstrut \) \(1358606\)\()/3115183\)
\(\beta_{14}\)\(=\)\((\)\(-\)\(1618763\) \(\nu^{15}\mathstrut +\mathstrut \) \(13952863\) \(\nu^{14}\mathstrut -\mathstrut \) \(25482926\) \(\nu^{13}\mathstrut -\mathstrut \) \(88167362\) \(\nu^{12}\mathstrut +\mathstrut \) \(323120564\) \(\nu^{11}\mathstrut +\mathstrut \) \(34111348\) \(\nu^{10}\mathstrut -\mathstrut \) \(1108457254\) \(\nu^{9}\mathstrut +\mathstrut \) \(654630562\) \(\nu^{8}\mathstrut +\mathstrut \) \(1555714470\) \(\nu^{7}\mathstrut -\mathstrut \) \(1396057392\) \(\nu^{6}\mathstrut -\mathstrut \) \(862892792\) \(\nu^{5}\mathstrut +\mathstrut \) \(958014404\) \(\nu^{4}\mathstrut +\mathstrut \) \(97238878\) \(\nu^{3}\mathstrut -\mathstrut \) \(170186194\) \(\nu^{2}\mathstrut +\mathstrut \) \(27985975\) \(\nu\mathstrut -\mathstrut \) \(7526881\)\()/3115183\)
\(\beta_{15}\)\(=\)\((\)\(1717836\) \(\nu^{15}\mathstrut -\mathstrut \) \(13768724\) \(\nu^{14}\mathstrut +\mathstrut \) \(19404207\) \(\nu^{13}\mathstrut +\mathstrut \) \(100594394\) \(\nu^{12}\mathstrut -\mathstrut \) \(280173236\) \(\nu^{11}\mathstrut -\mathstrut \) \(160180219\) \(\nu^{10}\mathstrut +\mathstrut \) \(1017753578\) \(\nu^{9}\mathstrut -\mathstrut \) \(243563866\) \(\nu^{8}\mathstrut -\mathstrut \) \(1527505049\) \(\nu^{7}\mathstrut +\mathstrut \) \(858348612\) \(\nu^{6}\mathstrut +\mathstrut \) \(939591930\) \(\nu^{5}\mathstrut -\mathstrut \) \(739948253\) \(\nu^{4}\mathstrut -\mathstrut \) \(146521530\) \(\nu^{3}\mathstrut +\mathstrut \) \(214322353\) \(\nu^{2}\mathstrut -\mathstrut \) \(16102888\) \(\nu\mathstrut -\mathstrut \) \(10162035\)\()/3115183\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{15}\mathstrut +\mathstrut \) \(2\) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(-\)\(3\) \(\beta_{15}\mathstrut +\mathstrut \) \(6\) \(\beta_{13}\mathstrut -\mathstrut \) \(2\) \(\beta_{12}\mathstrut +\mathstrut \) \(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(3\) \(\beta_{10}\mathstrut -\mathstrut \) \(3\) \(\beta_{9}\mathstrut +\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(7\) \(\beta_{4}\mathstrut -\mathstrut \) \(4\) \(\beta_{3}\mathstrut +\mathstrut \) \(10\) \(\beta_{2}\mathstrut +\mathstrut \) \(10\) \(\beta_{1}\mathstrut +\mathstrut \) \(5\)
\(\nu^{5}\)\(=\)\(-\)\(13\) \(\beta_{15}\mathstrut +\mathstrut \) \(28\) \(\beta_{13}\mathstrut -\mathstrut \) \(12\) \(\beta_{12}\mathstrut +\mathstrut \) \(13\) \(\beta_{11}\mathstrut +\mathstrut \) \(13\) \(\beta_{10}\mathstrut -\mathstrut \) \(12\) \(\beta_{9}\mathstrut +\mathstrut \) \(9\) \(\beta_{8}\mathstrut -\mathstrut \) \(6\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(31\) \(\beta_{4}\mathstrut -\mathstrut \) \(16\) \(\beta_{3}\mathstrut +\mathstrut \) \(28\) \(\beta_{2}\mathstrut +\mathstrut \) \(34\) \(\beta_{1}\mathstrut -\mathstrut \) \(3\)
\(\nu^{6}\)\(=\)\(-\)\(40\) \(\beta_{15}\mathstrut -\mathstrut \) \(3\) \(\beta_{14}\mathstrut +\mathstrut \) \(87\) \(\beta_{13}\mathstrut -\mathstrut \) \(33\) \(\beta_{12}\mathstrut +\mathstrut \) \(36\) \(\beta_{11}\mathstrut +\mathstrut \) \(42\) \(\beta_{10}\mathstrut -\mathstrut \) \(32\) \(\beta_{9}\mathstrut +\mathstrut \) \(19\) \(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{7}\mathstrut -\mathstrut \) \(31\) \(\beta_{6}\mathstrut +\mathstrut \) \(3\) \(\beta_{5}\mathstrut +\mathstrut \) \(103\) \(\beta_{4}\mathstrut -\mathstrut \) \(56\) \(\beta_{3}\mathstrut +\mathstrut \) \(109\) \(\beta_{2}\mathstrut +\mathstrut \) \(84\) \(\beta_{1}\mathstrut +\mathstrut \) \(6\)
\(\nu^{7}\)\(=\)\(-\)\(140\) \(\beta_{15}\mathstrut -\mathstrut \) \(13\) \(\beta_{14}\mathstrut +\mathstrut \) \(319\) \(\beta_{13}\mathstrut -\mathstrut \) \(132\) \(\beta_{12}\mathstrut +\mathstrut \) \(149\) \(\beta_{11}\mathstrut +\mathstrut \) \(148\) \(\beta_{10}\mathstrut -\mathstrut \) \(100\) \(\beta_{9}\mathstrut +\mathstrut \) \(60\) \(\beta_{8}\mathstrut -\mathstrut \) \(21\) \(\beta_{7}\mathstrut -\mathstrut \) \(102\) \(\beta_{6}\mathstrut +\mathstrut \) \(15\) \(\beta_{5}\mathstrut +\mathstrut \) \(368\) \(\beta_{4}\mathstrut -\mathstrut \) \(191\) \(\beta_{3}\mathstrut +\mathstrut \) \(344\) \(\beta_{2}\mathstrut +\mathstrut \) \(259\) \(\beta_{1}\mathstrut -\mathstrut \) \(49\)
\(\nu^{8}\)\(=\)\(-\)\(443\) \(\beta_{15}\mathstrut -\mathstrut \) \(71\) \(\beta_{14}\mathstrut +\mathstrut \) \(1025\) \(\beta_{13}\mathstrut -\mathstrut \) \(403\) \(\beta_{12}\mathstrut +\mathstrut \) \(461\) \(\beta_{11}\mathstrut +\mathstrut \) \(487\) \(\beta_{10}\mathstrut -\mathstrut \) \(262\) \(\beta_{9}\mathstrut +\mathstrut \) \(121\) \(\beta_{8}\mathstrut -\mathstrut \) \(84\) \(\beta_{7}\mathstrut -\mathstrut \) \(396\) \(\beta_{6}\mathstrut +\mathstrut \) \(50\) \(\beta_{5}\mathstrut +\mathstrut \) \(1209\) \(\beta_{4}\mathstrut -\mathstrut \) \(642\) \(\beta_{3}\mathstrut +\mathstrut \) \(1221\) \(\beta_{2}\mathstrut +\mathstrut \) \(701\) \(\beta_{1}\mathstrut -\mathstrut \) \(107\)
\(\nu^{9}\)\(=\)\(-\)\(1476\) \(\beta_{15}\mathstrut -\mathstrut \) \(276\) \(\beta_{14}\mathstrut +\mathstrut \) \(3512\) \(\beta_{13}\mathstrut -\mathstrut \) \(1416\) \(\beta_{12}\mathstrut +\mathstrut \) \(1652\) \(\beta_{11}\mathstrut +\mathstrut \) \(1638\) \(\beta_{10}\mathstrut -\mathstrut \) \(748\) \(\beta_{9}\mathstrut +\mathstrut \) \(301\) \(\beta_{8}\mathstrut -\mathstrut \) \(467\) \(\beta_{7}\mathstrut -\mathstrut \) \(1330\) \(\beta_{6}\mathstrut +\mathstrut \) \(192\) \(\beta_{5}\mathstrut +\mathstrut \) \(4075\) \(\beta_{4}\mathstrut -\mathstrut \) \(2123\) \(\beta_{3}\mathstrut +\mathstrut \) \(3998\) \(\beta_{2}\mathstrut +\mathstrut \) \(2110\) \(\beta_{1}\mathstrut -\mathstrut \) \(629\)
\(\nu^{10}\)\(=\)\(-\)\(4758\) \(\beta_{15}\mathstrut -\mathstrut \) \(1126\) \(\beta_{14}\mathstrut +\mathstrut \) \(11468\) \(\beta_{13}\mathstrut -\mathstrut \) \(4492\) \(\beta_{12}\mathstrut +\mathstrut \) \(5313\) \(\beta_{11}\mathstrut +\mathstrut \) \(5404\) \(\beta_{10}\mathstrut -\mathstrut \) \(1938\) \(\beta_{9}\mathstrut +\mathstrut \) \(438\) \(\beta_{8}\mathstrut -\mathstrut \) \(1772\) \(\beta_{7}\mathstrut -\mathstrut \) \(4716\) \(\beta_{6}\mathstrut +\mathstrut \) \(659\) \(\beta_{5}\mathstrut +\mathstrut \) \(13342\) \(\beta_{4}\mathstrut -\mathstrut \) \(7027\) \(\beta_{3}\mathstrut +\mathstrut \) \(13650\) \(\beta_{2}\mathstrut +\mathstrut \) \(6014\) \(\beta_{1}\mathstrut -\mathstrut \) \(1896\)
\(\nu^{11}\)\(=\)\(-\)\(15673\) \(\beta_{15}\mathstrut -\mathstrut \) \(4126\) \(\beta_{14}\mathstrut +\mathstrut \) \(38386\) \(\beta_{13}\mathstrut -\mathstrut \) \(15058\) \(\beta_{12}\mathstrut +\mathstrut \) \(18058\) \(\beta_{11}\mathstrut +\mathstrut \) \(17929\) \(\beta_{10}\mathstrut -\mathstrut \) \(5242\) \(\beta_{9}\mathstrut +\mathstrut \) \(396\) \(\beta_{8}\mathstrut -\mathstrut \) \(7120\) \(\beta_{7}\mathstrut -\mathstrut \) \(15845\) \(\beta_{6}\mathstrut +\mathstrut \) \(2354\) \(\beta_{5}\mathstrut +\mathstrut \) \(44113\) \(\beta_{4}\mathstrut -\mathstrut \) \(23081\) \(\beta_{3}\mathstrut +\mathstrut \) \(45129\) \(\beta_{2}\mathstrut +\mathstrut \) \(18116\) \(\beta_{1}\mathstrut -\mathstrut \) \(7473\)
\(\nu^{12}\)\(=\)\(-\)\(51058\) \(\beta_{15}\mathstrut -\mathstrut \) \(15215\) \(\beta_{14}\mathstrut +\mathstrut \) \(126139\) \(\beta_{13}\mathstrut -\mathstrut \) \(48520\) \(\beta_{12}\mathstrut +\mathstrut \) \(58871\) \(\beta_{11}\mathstrut +\mathstrut \) \(59106\) \(\beta_{10}\mathstrut -\mathstrut \) \(13264\) \(\beta_{9}\mathstrut -\mathstrut \) \(2769\) \(\beta_{8}\mathstrut -\mathstrut \) \(25673\) \(\beta_{7}\mathstrut -\mathstrut \) \(54176\) \(\beta_{6}\mathstrut +\mathstrut \) \(8095\) \(\beta_{5}\mathstrut +\mathstrut \) \(144315\) \(\beta_{4}\mathstrut -\mathstrut \) \(75954\) \(\beta_{3}\mathstrut +\mathstrut \) \(151396\) \(\beta_{2}\mathstrut +\mathstrut \) \(53466\) \(\beta_{1}\mathstrut -\mathstrut \) \(24198\)
\(\nu^{13}\)\(=\)\(-\)\(167860\) \(\beta_{15}\mathstrut -\mathstrut \) \(53769\) \(\beta_{14}\mathstrut +\mathstrut \) \(418395\) \(\beta_{13}\mathstrut -\mathstrut \) \(159874\) \(\beta_{12}\mathstrut +\mathstrut \) \(196018\) \(\beta_{11}\mathstrut +\mathstrut \) \(195119\) \(\beta_{10}\mathstrut -\mathstrut \) \(33961\) \(\beta_{9}\mathstrut -\mathstrut \) \(18334\) \(\beta_{8}\mathstrut -\mathstrut \) \(93348\) \(\beta_{7}\mathstrut -\mathstrut \) \(181236\) \(\beta_{6}\mathstrut +\mathstrut \) \(28096\) \(\beta_{5}\mathstrut +\mathstrut \) \(474039\) \(\beta_{4}\mathstrut -\mathstrut \) \(249156\) \(\beta_{3}\mathstrut +\mathstrut \) \(501159\) \(\beta_{2}\mathstrut +\mathstrut \) \(162718\) \(\beta_{1}\mathstrut -\mathstrut \) \(85607\)
\(\nu^{14}\)\(=\)\(-\)\(549791\) \(\beta_{15}\mathstrut -\mathstrut \) \(189315\) \(\beta_{14}\mathstrut +\mathstrut \) \(1377611\) \(\beta_{13}\mathstrut -\mathstrut \) \(518871\) \(\beta_{12}\mathstrut +\mathstrut \) \(642097\) \(\beta_{11}\mathstrut +\mathstrut \) \(642570\) \(\beta_{10}\mathstrut -\mathstrut \) \(81445\) \(\beta_{9}\mathstrut -\mathstrut \) \(90804\) \(\beta_{8}\mathstrut -\mathstrut \) \(325797\) \(\beta_{7}\mathstrut -\mathstrut \) \(609317\) \(\beta_{6}\mathstrut +\mathstrut \) \(95967\) \(\beta_{5}\mathstrut +\mathstrut \) \(1551184\) \(\beta_{4}\mathstrut -\mathstrut \) \(818281\) \(\beta_{3}\mathstrut +\mathstrut \) \(1666889\) \(\beta_{2}\mathstrut +\mathstrut \) \(492790\) \(\beta_{1}\mathstrut -\mathstrut \) \(282370\)
\(\nu^{15}\)\(=\)\(-\)\(1807643\) \(\beta_{15}\mathstrut -\mathstrut \) \(653596\) \(\beta_{14}\mathstrut +\mathstrut \) \(4551335\) \(\beta_{13}\mathstrut -\mathstrut \) \(1699915\) \(\beta_{12}\mathstrut +\mathstrut \) \(2120493\) \(\beta_{11}\mathstrut +\mathstrut \) \(2116648\) \(\beta_{10}\mathstrut -\mathstrut \) \(189849\) \(\beta_{9}\mathstrut -\mathstrut \) \(374428\) \(\beta_{8}\mathstrut -\mathstrut \) \(1134451\) \(\beta_{7}\mathstrut -\mathstrut \) \(2029069\) \(\beta_{6}\mathstrut +\mathstrut \) \(327799\) \(\beta_{5}\mathstrut +\mathstrut \) \(5085340\) \(\beta_{4}\mathstrut -\mathstrut \) \(2684287\) \(\beta_{3}\mathstrut +\mathstrut \) \(5511955\) \(\beta_{2}\mathstrut +\mathstrut \) \(1519951\) \(\beta_{1}\mathstrut -\mathstrut \) \(960385\)

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
−2.06059
−1.45194
−1.28136
−1.23485
−0.981279
−0.624714
−0.136067
0.293003
0.630852
1.27770
1.52769
1.74152
1.84416
2.41044
2.75851
3.28691
1.00000 −3.06059 1.00000 1.00000 −3.06059 0.302669 1.00000 6.36721 1.00000
1.2 1.00000 −2.45194 1.00000 1.00000 −2.45194 1.53125 1.00000 3.01199 1.00000
1.3 1.00000 −2.28136 1.00000 1.00000 −2.28136 1.99437 1.00000 2.20462 1.00000
1.4 1.00000 −2.23485 1.00000 1.00000 −2.23485 −0.971978 1.00000 1.99454 1.00000
1.5 1.00000 −1.98128 1.00000 1.00000 −1.98128 −4.22080 1.00000 0.925466 1.00000
1.6 1.00000 −1.62471 1.00000 1.00000 −1.62471 −0.333284 1.00000 −0.360304 1.00000
1.7 1.00000 −1.13607 1.00000 1.00000 −1.13607 −1.31630 1.00000 −1.70935 1.00000
1.8 1.00000 −0.706997 1.00000 1.00000 −0.706997 −2.05365 1.00000 −2.50015 1.00000
1.9 1.00000 −0.369148 1.00000 1.00000 −0.369148 4.16284 1.00000 −2.86373 1.00000
1.10 1.00000 0.277700 1.00000 1.00000 0.277700 2.55233 1.00000 −2.92288 1.00000
1.11 1.00000 0.527686 1.00000 1.00000 0.527686 −1.45681 1.00000 −2.72155 1.00000
1.12 1.00000 0.741521 1.00000 1.00000 0.741521 −2.92320 1.00000 −2.45015 1.00000
1.13 1.00000 0.844163 1.00000 1.00000 0.844163 −0.532701 1.00000 −2.28739 1.00000
1.14 1.00000 1.41044 1.00000 1.00000 1.41044 −3.15104 1.00000 −1.01066 1.00000
1.15 1.00000 1.75851 1.00000 1.00000 1.75851 −0.115571 1.00000 0.0923708 1.00000
1.16 1.00000 2.28691 1.00000 1.00000 2.28691 −3.46813 1.00000 2.22998 1.00000
\(n\): 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
\(2\) \(-1\)
\(5\) \(-1\)
\(601\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{16} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6010))\).