Properties

Label 4017.2.a.f
Level 4017
Weight 2
Character orbit 4017.a
Self dual Yes
Analytic conductor 32.076
Analytic rank 1
Dimension 19
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4017 = 3 \cdot 13 \cdot 103 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4017.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0759064919\)
Analytic rank: \(1\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{19}\mathstrut -\mathstrut \) \(4\) \(x^{18}\mathstrut -\mathstrut \) \(16\) \(x^{17}\mathstrut +\mathstrut \) \(77\) \(x^{16}\mathstrut +\mathstrut \) \(88\) \(x^{15}\mathstrut -\mathstrut \) \(594\) \(x^{14}\mathstrut -\mathstrut \) \(154\) \(x^{13}\mathstrut +\mathstrut \) \(2388\) \(x^{12}\mathstrut -\mathstrut \) \(278\) \(x^{11}\mathstrut -\mathstrut \) \(5460\) \(x^{10}\mathstrut +\mathstrut \) \(1491\) \(x^{9}\mathstrut +\mathstrut \) \(7285\) \(x^{8}\mathstrut -\mathstrut \) \(2223\) \(x^{7}\mathstrut -\mathstrut \) \(5579\) \(x^{6}\mathstrut +\mathstrut \) \(1430\) \(x^{5}\mathstrut +\mathstrut \) \(2261\) \(x^{4}\mathstrut -\mathstrut \) \(352\) \(x^{3}\mathstrut -\mathstrut \) \(378\) \(x^{2}\mathstrut +\mathstrut \) \(11\) \(x\mathstrut +\mathstrut \) \(4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(94088\) \(\nu^{18}\mathstrut -\mathstrut \) \(626428\) \(\nu^{17}\mathstrut +\mathstrut \) \(3886444\) \(\nu^{16}\mathstrut +\mathstrut \) \(9779883\) \(\nu^{15}\mathstrut -\mathstrut \) \(46409280\) \(\nu^{14}\mathstrut -\mathstrut \) \(53503587\) \(\nu^{13}\mathstrut +\mathstrut \) \(230717285\) \(\nu^{12}\mathstrut +\mathstrut \) \(115056639\) \(\nu^{11}\mathstrut -\mathstrut \) \(445048447\) \(\nu^{10}\mathstrut -\mathstrut \) \(44364492\) \(\nu^{9}\mathstrut -\mathstrut \) \(136140833\) \(\nu^{8}\mathstrut -\mathstrut \) \(125435006\) \(\nu^{7}\mathstrut +\mathstrut \) \(1628375777\) \(\nu^{6}\mathstrut +\mathstrut \) \(70345533\) \(\nu^{5}\mathstrut -\mathstrut \) \(1910156217\) \(\nu^{4}\mathstrut +\mathstrut \) \(67933382\) \(\nu^{3}\mathstrut +\mathstrut \) \(670807958\) \(\nu^{2}\mathstrut -\mathstrut \) \(65076335\) \(\nu\mathstrut -\mathstrut \) \(30277713\)\()/8327001\)
\(\beta_{4}\)\(=\)\((\)\(193859\) \(\nu^{18}\mathstrut -\mathstrut \) \(2142369\) \(\nu^{17}\mathstrut -\mathstrut \) \(4314\) \(\nu^{16}\mathstrut +\mathstrut \) \(40884782\) \(\nu^{15}\mathstrut -\mathstrut \) \(39656466\) \(\nu^{14}\mathstrut -\mathstrut \) \(310348080\) \(\nu^{13}\mathstrut +\mathstrut \) \(377207518\) \(\nu^{12}\mathstrut +\mathstrut \) \(1213875850\) \(\nu^{11}\mathstrut -\mathstrut \) \(1520636534\) \(\nu^{10}\mathstrut -\mathstrut \) \(2649906620\) \(\nu^{9}\mathstrut +\mathstrut \) \(3112759541\) \(\nu^{8}\mathstrut +\mathstrut \) \(3263812942\) \(\nu^{7}\mathstrut -\mathstrut \) \(3299395776\) \(\nu^{6}\mathstrut -\mathstrut \) \(2166951074\) \(\nu^{5}\mathstrut +\mathstrut \) \(1726428678\) \(\nu^{4}\mathstrut +\mathstrut \) \(685986403\) \(\nu^{3}\mathstrut -\mathstrut \) \(385111717\) \(\nu^{2}\mathstrut -\mathstrut \) \(92926590\) \(\nu\mathstrut +\mathstrut \) \(8487263\)\()/8327001\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(301002\) \(\nu^{18}\mathstrut +\mathstrut \) \(2086417\) \(\nu^{17}\mathstrut +\mathstrut \) \(830096\) \(\nu^{16}\mathstrut -\mathstrut \) \(36097151\) \(\nu^{15}\mathstrut +\mathstrut \) \(49187691\) \(\nu^{14}\mathstrut +\mathstrut \) \(235624707\) \(\nu^{13}\mathstrut -\mathstrut \) \(523591869\) \(\nu^{12}\mathstrut -\mathstrut \) \(720016087\) \(\nu^{11}\mathstrut +\mathstrut \) \(2280606051\) \(\nu^{10}\mathstrut +\mathstrut \) \(1009320407\) \(\nu^{9}\mathstrut -\mathstrut \) \(5111196996\) \(\nu^{8}\mathstrut -\mathstrut \) \(464378927\) \(\nu^{7}\mathstrut +\mathstrut \) \(6113602981\) \(\nu^{6}\mathstrut -\mathstrut \) \(87045064\) \(\nu^{5}\mathstrut -\mathstrut \) \(3694969428\) \(\nu^{4}\mathstrut -\mathstrut \) \(38526093\) \(\nu^{3}\mathstrut +\mathstrut \) \(900642533\) \(\nu^{2}\mathstrut +\mathstrut \) \(74082323\) \(\nu\mathstrut -\mathstrut \) \(18473720\)\()/8327001\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(505639\) \(\nu^{18}\mathstrut +\mathstrut \) \(943097\) \(\nu^{17}\mathstrut +\mathstrut \) \(11786836\) \(\nu^{16}\mathstrut -\mathstrut \) \(20247569\) \(\nu^{15}\mathstrut -\mathstrut \) \(112671033\) \(\nu^{14}\mathstrut +\mathstrut \) \(173915817\) \(\nu^{13}\mathstrut +\mathstrut \) \(571733266\) \(\nu^{12}\mathstrut -\mathstrut \) \(761598784\) \(\nu^{11}\mathstrut -\mathstrut \) \(1668257465\) \(\nu^{10}\mathstrut +\mathstrut \) \(1786828862\) \(\nu^{9}\mathstrut +\mathstrut \) \(2842754969\) \(\nu^{8}\mathstrut -\mathstrut \) \(2137689600\) \(\nu^{7}\mathstrut -\mathstrut \) \(2757965584\) \(\nu^{6}\mathstrut +\mathstrut \) \(1068552050\) \(\nu^{5}\mathstrut +\mathstrut \) \(1414007301\) \(\nu^{4}\mathstrut -\mathstrut \) \(61613093\) \(\nu^{3}\mathstrut -\mathstrut \) \(298043445\) \(\nu^{2}\mathstrut -\mathstrut \) \(50846708\) \(\nu\mathstrut -\mathstrut \) \(7102352\)\()/8327001\)
\(\beta_{7}\)\(=\)\((\)\(522452\) \(\nu^{18}\mathstrut -\mathstrut \) \(173763\) \(\nu^{17}\mathstrut -\mathstrut \) \(14520768\) \(\nu^{16}\mathstrut +\mathstrut \) \(6574721\) \(\nu^{15}\mathstrut +\mathstrut \) \(162869580\) \(\nu^{14}\mathstrut -\mathstrut \) \(84744672\) \(\nu^{13}\mathstrut -\mathstrut \) \(963497294\) \(\nu^{12}\mathstrut +\mathstrut \) \(519460555\) \(\nu^{11}\mathstrut +\mathstrut \) \(3292783051\) \(\nu^{10}\mathstrut -\mathstrut \) \(1680536573\) \(\nu^{9}\mathstrut -\mathstrut \) \(6682423192\) \(\nu^{8}\mathstrut +\mathstrut \) \(2902932841\) \(\nu^{7}\mathstrut +\mathstrut \) \(7923527727\) \(\nu^{6}\mathstrut -\mathstrut \) \(2506162403\) \(\nu^{5}\mathstrut -\mathstrut \) \(5081746752\) \(\nu^{4}\mathstrut +\mathstrut \) \(883967110\) \(\nu^{3}\mathstrut +\mathstrut \) \(1412281769\) \(\nu^{2}\mathstrut -\mathstrut \) \(49334292\) \(\nu\mathstrut -\mathstrut \) \(73754683\)\()/8327001\)
\(\beta_{8}\)\(=\)\((\)\(683216\) \(\nu^{18}\mathstrut +\mathstrut \) \(1334850\) \(\nu^{17}\mathstrut -\mathstrut \) \(24023172\) \(\nu^{16}\mathstrut -\mathstrut \) \(16582591\) \(\nu^{15}\mathstrut +\mathstrut \) \(300102768\) \(\nu^{14}\mathstrut +\mathstrut \) \(37272603\) \(\nu^{13}\mathstrut -\mathstrut \) \(1828942592\) \(\nu^{12}\mathstrut +\mathstrut \) \(288782608\) \(\nu^{11}\mathstrut +\mathstrut \) \(6037093933\) \(\nu^{10}\mathstrut -\mathstrut \) \(1718391077\) \(\nu^{9}\mathstrut -\mathstrut \) \(11061827209\) \(\nu^{8}\mathstrut +\mathstrut \) \(3367642315\) \(\nu^{7}\mathstrut +\mathstrut \) \(10924134189\) \(\nu^{6}\mathstrut -\mathstrut \) \(2586060788\) \(\nu^{5}\mathstrut -\mathstrut \) \(5271494415\) \(\nu^{4}\mathstrut +\mathstrut \) \(508159444\) \(\nu^{3}\mathstrut +\mathstrut \) \(919589882\) \(\nu^{2}\mathstrut +\mathstrut \) \(90613854\) \(\nu\mathstrut +\mathstrut \) \(8797055\)\()/8327001\)
\(\beta_{9}\)\(=\)\((\)\(878636\) \(\nu^{18}\mathstrut -\mathstrut \) \(2477135\) \(\nu^{17}\mathstrut -\mathstrut \) \(17742577\) \(\nu^{16}\mathstrut +\mathstrut \) \(51743466\) \(\nu^{15}\mathstrut +\mathstrut \) \(142470306\) \(\nu^{14}\mathstrut -\mathstrut \) \(438788274\) \(\nu^{13}\mathstrut -\mathstrut \) \(579561356\) \(\nu^{12}\mathstrut +\mathstrut \) \(1949589303\) \(\nu^{11}\mathstrut +\mathstrut \) \(1244174596\) \(\nu^{10}\mathstrut -\mathstrut \) \(4869475809\) \(\nu^{9}\mathstrut -\mathstrut \) \(1281877039\) \(\nu^{8}\mathstrut +\mathstrut \) \(6779610320\) \(\nu^{7}\mathstrut +\mathstrut \) \(352120375\) \(\nu^{6}\mathstrut -\mathstrut \) \(4845433314\) \(\nu^{5}\mathstrut +\mathstrut \) \(273029682\) \(\nu^{4}\mathstrut +\mathstrut \) \(1416269452\) \(\nu^{3}\mathstrut -\mathstrut \) \(137691452\) \(\nu^{2}\mathstrut -\mathstrut \) \(46280392\) \(\nu\mathstrut +\mathstrut \) \(9751737\)\()/8327001\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(898500\) \(\nu^{18}\mathstrut +\mathstrut \) \(445742\) \(\nu^{17}\mathstrut +\mathstrut \) \(25408444\) \(\nu^{16}\mathstrut -\mathstrut \) \(16785145\) \(\nu^{15}\mathstrut -\mathstrut \) \(284915496\) \(\nu^{14}\mathstrut +\mathstrut \) \(214341540\) \(\nu^{13}\mathstrut +\mathstrut \) \(1654523163\) \(\nu^{12}\mathstrut -\mathstrut \) \(1301382665\) \(\nu^{11}\mathstrut -\mathstrut \) \(5429734446\) \(\nu^{10}\mathstrut +\mathstrut \) \(4181996197\) \(\nu^{9}\mathstrut +\mathstrut \) \(10277505300\) \(\nu^{8}\mathstrut -\mathstrut \) \(7250315182\) \(\nu^{7}\mathstrut -\mathstrut \) \(10915609351\) \(\nu^{6}\mathstrut +\mathstrut \) \(6490474048\) \(\nu^{5}\mathstrut +\mathstrut \) \(5904177015\) \(\nu^{4}\mathstrut -\mathstrut \) \(2646783813\) \(\nu^{3}\mathstrut -\mathstrut \) \(1196579108\) \(\nu^{2}\mathstrut +\mathstrut \) \(361699234\) \(\nu\mathstrut -\mathstrut \) \(11793478\)\()/8327001\)
\(\beta_{11}\)\(=\)\((\)\(977686\) \(\nu^{18}\mathstrut -\mathstrut \) \(2109846\) \(\nu^{17}\mathstrut -\mathstrut \) \(19576350\) \(\nu^{16}\mathstrut +\mathstrut \) \(40045288\) \(\nu^{15}\mathstrut +\mathstrut \) \(159886860\) \(\nu^{14}\mathstrut -\mathstrut \) \(304111437\) \(\nu^{13}\mathstrut -\mathstrut \) \(701472610\) \(\nu^{12}\mathstrut +\mathstrut \) \(1199138972\) \(\nu^{11}\mathstrut +\mathstrut \) \(1835184089\) \(\nu^{10}\mathstrut -\mathstrut \) \(2664298648\) \(\nu^{9}\mathstrut -\mathstrut \) \(2982909800\) \(\nu^{8}\mathstrut +\mathstrut \) \(3357982253\) \(\nu^{7}\mathstrut +\mathstrut \) \(2953197273\) \(\nu^{6}\mathstrut -\mathstrut \) \(2203868323\) \(\nu^{5}\mathstrut -\mathstrut \) \(1578545622\) \(\nu^{4}\mathstrut +\mathstrut \) \(499205258\) \(\nu^{3}\mathstrut +\mathstrut \) \(333663343\) \(\nu^{2}\mathstrut +\mathstrut \) \(79078185\) \(\nu\mathstrut -\mathstrut \) \(3170015\)\()/8327001\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(1006596\) \(\nu^{18}\mathstrut +\mathstrut \) \(4839121\) \(\nu^{17}\mathstrut +\mathstrut \) \(13408784\) \(\nu^{16}\mathstrut -\mathstrut \) \(90912362\) \(\nu^{15}\mathstrut -\mathstrut \) \(38552868\) \(\nu^{14}\mathstrut +\mathstrut \) \(676127358\) \(\nu^{13}\mathstrut -\mathstrut \) \(210763494\) \(\nu^{12}\mathstrut -\mathstrut \) \(2570195272\) \(\nu^{11}\mathstrut +\mathstrut \) \(1636153110\) \(\nu^{10}\mathstrut +\mathstrut \) \(5380497584\) \(\nu^{9}\mathstrut -\mathstrut \) \(4231425600\) \(\nu^{8}\mathstrut -\mathstrut \) \(6214385801\) \(\nu^{7}\mathstrut +\mathstrut \) \(5188235767\) \(\nu^{6}\mathstrut +\mathstrut \) \(3726959093\) \(\nu^{5}\mathstrut -\mathstrut \) \(2999440299\) \(\nu^{4}\mathstrut -\mathstrut \) \(1002901935\) \(\nu^{3}\mathstrut +\mathstrut \) \(671237324\) \(\nu^{2}\mathstrut +\mathstrut \) \(94367681\) \(\nu\mathstrut -\mathstrut \) \(20840648\)\()/8327001\)
\(\beta_{13}\)\(=\)\((\)\(340719\) \(\nu^{18}\mathstrut -\mathstrut \) \(2041117\) \(\nu^{17}\mathstrut -\mathstrut \) \(2297555\) \(\nu^{16}\mathstrut +\mathstrut \) \(35421487\) \(\nu^{15}\mathstrut -\mathstrut \) \(28351011\) \(\nu^{14}\mathstrut -\mathstrut \) \(233045118\) \(\nu^{13}\mathstrut +\mathstrut \) \(371781427\) \(\nu^{12}\mathstrut +\mathstrut \) \(725834826\) \(\nu^{11}\mathstrut -\mathstrut \) \(1656310546\) \(\nu^{10}\mathstrut -\mathstrut \) \(1066464572\) \(\nu^{9}\mathstrut +\mathstrut \) \(3628535304\) \(\nu^{8}\mathstrut +\mathstrut \) \(562746675\) \(\nu^{7}\mathstrut -\mathstrut \) \(4127234417\) \(\nu^{6}\mathstrut +\mathstrut \) \(120589701\) \(\nu^{5}\mathstrut +\mathstrut \) \(2302393775\) \(\nu^{4}\mathstrut -\mathstrut \) \(140104997\) \(\nu^{3}\mathstrut -\mathstrut \) \(480758306\) \(\nu^{2}\mathstrut +\mathstrut \) \(7664027\) \(\nu\mathstrut -\mathstrut \) \(1711033\)\()/2775667\)
\(\beta_{14}\)\(=\)\((\)\(-\)\(1173795\) \(\nu^{18}\mathstrut +\mathstrut \) \(4091147\) \(\nu^{17}\mathstrut +\mathstrut \) \(20237422\) \(\nu^{16}\mathstrut -\mathstrut \) \(78194119\) \(\nu^{15}\mathstrut -\mathstrut \) \(131370000\) \(\nu^{14}\mathstrut +\mathstrut \) \(595242228\) \(\nu^{13}\mathstrut +\mathstrut \) \(398312019\) \(\nu^{12}\mathstrut -\mathstrut \) \(2332250702\) \(\nu^{11}\mathstrut -\mathstrut \) \(555378540\) \(\nu^{10}\mathstrut +\mathstrut \) \(5073257425\) \(\nu^{9}\mathstrut +\mathstrut \) \(286658199\) \(\nu^{8}\mathstrut -\mathstrut \) \(6164640280\) \(\nu^{7}\mathstrut -\mathstrut \) \(112710592\) \(\nu^{6}\mathstrut +\mathstrut \) \(4020429766\) \(\nu^{5}\mathstrut +\mathstrut \) \(306252117\) \(\nu^{4}\mathstrut -\mathstrut \) \(1293543924\) \(\nu^{3}\mathstrut -\mathstrut \) \(227428052\) \(\nu^{2}\mathstrut +\mathstrut \) \(163481722\) \(\nu\mathstrut +\mathstrut \) \(33234527\)\()/8327001\)
\(\beta_{15}\)\(=\)\((\)\(-\)\(1656344\) \(\nu^{18}\mathstrut +\mathstrut \) \(2850571\) \(\nu^{17}\mathstrut +\mathstrut \) \(37818140\) \(\nu^{16}\mathstrut -\mathstrut \) \(60186352\) \(\nu^{15}\mathstrut -\mathstrut \) \(356551428\) \(\nu^{14}\mathstrut +\mathstrut \) \(518525517\) \(\nu^{13}\mathstrut +\mathstrut \) \(1803133061\) \(\nu^{12}\mathstrut -\mathstrut \) \(2357316008\) \(\nu^{11}\mathstrut -\mathstrut \) \(5303138830\) \(\nu^{10}\mathstrut +\mathstrut \) \(6102042916\) \(\nu^{9}\mathstrut +\mathstrut \) \(9166163902\) \(\nu^{8}\mathstrut -\mathstrut \) \(9045153015\) \(\nu^{7}\mathstrut -\mathstrut \) \(8918685770\) \(\nu^{6}\mathstrut +\mathstrut \) \(7316823022\) \(\nu^{5}\mathstrut +\mathstrut \) \(4354535292\) \(\nu^{4}\mathstrut -\mathstrut \) \(2833548478\) \(\nu^{3}\mathstrut -\mathstrut \) \(808126617\) \(\nu^{2}\mathstrut +\mathstrut \) \(355144607\) \(\nu\mathstrut +\mathstrut \) \(21257525\)\()/8327001\)
\(\beta_{16}\)\(=\)\((\)\(2105177\) \(\nu^{18}\mathstrut -\mathstrut \) \(11038000\) \(\nu^{17}\mathstrut -\mathstrut \) \(22037027\) \(\nu^{16}\mathstrut +\mathstrut \) \(196981321\) \(\nu^{15}\mathstrut -\mathstrut \) \(25058403\) \(\nu^{14}\mathstrut -\mathstrut \) \(1357359756\) \(\nu^{13}\mathstrut +\mathstrut \) \(1150153099\) \(\nu^{12}\mathstrut +\mathstrut \) \(4592467277\) \(\nu^{11}\mathstrut -\mathstrut \) \(5703282992\) \(\nu^{10}\mathstrut -\mathstrut \) \(7991010202\) \(\nu^{9}\mathstrut +\mathstrut \) \(12419678693\) \(\nu^{8}\mathstrut +\mathstrut \) \(6739037490\) \(\nu^{7}\mathstrut -\mathstrut \) \(13149766900\) \(\nu^{6}\mathstrut -\mathstrut \) \(2090427664\) \(\nu^{5}\mathstrut +\mathstrut \) \(6362522892\) \(\nu^{4}\mathstrut -\mathstrut \) \(218917406\) \(\nu^{3}\mathstrut -\mathstrut \) \(1069751136\) \(\nu^{2}\mathstrut +\mathstrut \) \(159048874\) \(\nu\mathstrut +\mathstrut \) \(6865267\)\()/8327001\)
\(\beta_{17}\)\(=\)\((\)\(-\)\(855796\) \(\nu^{18}\mathstrut +\mathstrut \) \(3359640\) \(\nu^{17}\mathstrut +\mathstrut \) \(13123579\) \(\nu^{16}\mathstrut -\mathstrut \) \(62837734\) \(\nu^{15}\mathstrut -\mathstrut \) \(64697412\) \(\nu^{14}\mathstrut +\mathstrut \) \(464512414\) \(\nu^{13}\mathstrut +\mathstrut \) \(54323182\) \(\nu^{12}\mathstrut -\mathstrut \) \(1750271618\) \(\nu^{11}\mathstrut +\mathstrut \) \(521784388\) \(\nu^{10}\mathstrut +\mathstrut \) \(3618039992\) \(\nu^{9}\mathstrut -\mathstrut \) \(1830878338\) \(\nu^{8}\mathstrut -\mathstrut \) \(4115548654\) \(\nu^{7}\mathstrut +\mathstrut \) \(2467406610\) \(\nu^{6}\mathstrut +\mathstrut \) \(2447706825\) \(\nu^{5}\mathstrut -\mathstrut \) \(1492732591\) \(\nu^{4}\mathstrut -\mathstrut \) \(668587229\) \(\nu^{3}\mathstrut +\mathstrut \) \(345567051\) \(\nu^{2}\mathstrut +\mathstrut \) \(64549695\) \(\nu\mathstrut -\mathstrut \) \(10034449\)\()/2775667\)
\(\beta_{18}\)\(=\)\((\)\(-\)\(2617292\) \(\nu^{18}\mathstrut +\mathstrut \) \(11645805\) \(\nu^{17}\mathstrut +\mathstrut \) \(34882692\) \(\nu^{16}\mathstrut -\mathstrut \) \(210313979\) \(\nu^{15}\mathstrut -\mathstrut \) \(106884618\) \(\nu^{14}\mathstrut +\mathstrut \) \(1474350357\) \(\nu^{13}\mathstrut -\mathstrut \) \(434695399\) \(\nu^{12}\mathstrut -\mathstrut \) \(5118043786\) \(\nu^{11}\mathstrut +\mathstrut \) \(3503256218\) \(\nu^{10}\mathstrut +\mathstrut \) \(9280859786\) \(\nu^{9}\mathstrut -\mathstrut \) \(8597176955\) \(\nu^{8}\mathstrut -\mathstrut \) \(8469958429\) \(\nu^{7}\mathstrut +\mathstrut \) \(9654354819\) \(\nu^{6}\mathstrut +\mathstrut \) \(3352119782\) \(\nu^{5}\mathstrut -\mathstrut \) \(4978722603\) \(\nu^{4}\mathstrut -\mathstrut \) \(328728832\) \(\nu^{3}\mathstrut +\mathstrut \) \(963132781\) \(\nu^{2}\mathstrut -\mathstrut \) \(24618681\) \(\nu\mathstrut -\mathstrut \) \(25074710\)\()/8327001\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{17}\mathstrut +\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{17}\mathstrut +\mathstrut \) \(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(14\)
\(\nu^{5}\)\(=\)\(-\)\(\beta_{18}\mathstrut -\mathstrut \) \(9\) \(\beta_{17}\mathstrut -\mathstrut \) \(\beta_{16}\mathstrut +\mathstrut \) \(8\) \(\beta_{14}\mathstrut -\mathstrut \) \(2\) \(\beta_{13}\mathstrut +\mathstrut \) \(9\) \(\beta_{12}\mathstrut -\mathstrut \) \(7\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(10\) \(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(10\) \(\beta_{4}\mathstrut +\mathstrut \) \(9\) \(\beta_{3}\mathstrut +\mathstrut \) \(11\) \(\beta_{2}\mathstrut +\mathstrut \) \(19\) \(\beta_{1}\mathstrut +\mathstrut \) \(10\)
\(\nu^{6}\)\(=\)\(-\)\(14\) \(\beta_{17}\mathstrut -\mathstrut \) \(2\) \(\beta_{16}\mathstrut -\mathstrut \) \(\beta_{15}\mathstrut +\mathstrut \) \(12\) \(\beta_{14}\mathstrut -\mathstrut \) \(11\) \(\beta_{13}\mathstrut +\mathstrut \) \(2\) \(\beta_{12}\mathstrut +\mathstrut \) \(3\) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(12\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(12\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(11\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(46\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(77\)
\(\nu^{7}\)\(=\)\(-\)\(12\) \(\beta_{18}\mathstrut -\mathstrut \) \(72\) \(\beta_{17}\mathstrut -\mathstrut \) \(14\) \(\beta_{16}\mathstrut +\mathstrut \) \(58\) \(\beta_{14}\mathstrut -\mathstrut \) \(26\) \(\beta_{13}\mathstrut +\mathstrut \) \(67\) \(\beta_{12}\mathstrut -\mathstrut \) \(44\) \(\beta_{11}\mathstrut +\mathstrut \) \(14\) \(\beta_{10}\mathstrut +\mathstrut \) \(78\) \(\beta_{9}\mathstrut -\mathstrut \) \(25\) \(\beta_{8}\mathstrut -\mathstrut \) \(10\) \(\beta_{7}\mathstrut -\mathstrut \) \(13\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(81\) \(\beta_{4}\mathstrut +\mathstrut \) \(69\) \(\beta_{3}\mathstrut +\mathstrut \) \(93\) \(\beta_{2}\mathstrut +\mathstrut \) \(101\) \(\beta_{1}\mathstrut +\mathstrut \) \(84\)
\(\nu^{8}\)\(=\)\(-\)\(4\) \(\beta_{18}\mathstrut -\mathstrut \) \(138\) \(\beta_{17}\mathstrut -\mathstrut \) \(29\) \(\beta_{16}\mathstrut -\mathstrut \) \(11\) \(\beta_{15}\mathstrut +\mathstrut \) \(109\) \(\beta_{14}\mathstrut -\mathstrut \) \(98\) \(\beta_{13}\mathstrut +\mathstrut \) \(33\) \(\beta_{12}\mathstrut -\mathstrut \) \(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(40\) \(\beta_{10}\mathstrut +\mathstrut \) \(21\) \(\beta_{9}\mathstrut -\mathstrut \) \(108\) \(\beta_{8}\mathstrut +\mathstrut \) \(11\) \(\beta_{7}\mathstrut -\mathstrut \) \(104\) \(\beta_{6}\mathstrut +\mathstrut \) \(11\) \(\beta_{5}\mathstrut -\mathstrut \) \(100\) \(\beta_{4}\mathstrut +\mathstrut \) \(32\) \(\beta_{3}\mathstrut +\mathstrut \) \(315\) \(\beta_{2}\mathstrut +\mathstrut \) \(17\) \(\beta_{1}\mathstrut +\mathstrut \) \(470\)
\(\nu^{9}\)\(=\)\(-\)\(106\) \(\beta_{18}\mathstrut -\mathstrut \) \(557\) \(\beta_{17}\mathstrut -\mathstrut \) \(135\) \(\beta_{16}\mathstrut -\mathstrut \) \(2\) \(\beta_{15}\mathstrut +\mathstrut \) \(424\) \(\beta_{14}\mathstrut -\mathstrut \) \(247\) \(\beta_{13}\mathstrut +\mathstrut \) \(479\) \(\beta_{12}\mathstrut -\mathstrut \) \(280\) \(\beta_{11}\mathstrut +\mathstrut \) \(138\) \(\beta_{10}\mathstrut +\mathstrut \) \(569\) \(\beta_{9}\mathstrut -\mathstrut \) \(233\) \(\beta_{8}\mathstrut -\mathstrut \) \(74\) \(\beta_{7}\mathstrut -\mathstrut \) \(127\) \(\beta_{6}\mathstrut -\mathstrut \) \(13\) \(\beta_{5}\mathstrut -\mathstrut \) \(615\) \(\beta_{4}\mathstrut +\mathstrut \) \(505\) \(\beta_{3}\mathstrut +\mathstrut \) \(732\) \(\beta_{2}\mathstrut +\mathstrut \) \(581\) \(\beta_{1}\mathstrut +\mathstrut \) \(673\)
\(\nu^{10}\)\(=\)\(-\)\(69\) \(\beta_{18}\mathstrut -\mathstrut \) \(1198\) \(\beta_{17}\mathstrut -\mathstrut \) \(294\) \(\beta_{16}\mathstrut -\mathstrut \) \(90\) \(\beta_{15}\mathstrut +\mathstrut \) \(905\) \(\beta_{14}\mathstrut -\mathstrut \) \(809\) \(\beta_{13}\mathstrut +\mathstrut \) \(372\) \(\beta_{12}\mathstrut -\mathstrut \) \(47\) \(\beta_{11}\mathstrut +\mathstrut \) \(385\) \(\beta_{10}\mathstrut +\mathstrut \) \(275\) \(\beta_{9}\mathstrut -\mathstrut \) \(881\) \(\beta_{8}\mathstrut +\mathstrut \) \(88\) \(\beta_{7}\mathstrut -\mathstrut \) \(811\) \(\beta_{6}\mathstrut +\mathstrut \) \(88\) \(\beta_{5}\mathstrut -\mathstrut \) \(859\) \(\beta_{4}\mathstrut +\mathstrut \) \(358\) \(\beta_{3}\mathstrut +\mathstrut \) \(2244\) \(\beta_{2}\mathstrut +\mathstrut \) \(197\) \(\beta_{1}\mathstrut +\mathstrut \) \(3082\)
\(\nu^{11}\)\(=\)\(-\)\(842\) \(\beta_{18}\mathstrut -\mathstrut \) \(4248\) \(\beta_{17}\mathstrut -\mathstrut \) \(1138\) \(\beta_{16}\mathstrut -\mathstrut \) \(39\) \(\beta_{15}\mathstrut +\mathstrut \) \(3141\) \(\beta_{14}\mathstrut -\mathstrut \) \(2099\) \(\beta_{13}\mathstrut +\mathstrut \) \(3406\) \(\beta_{12}\mathstrut -\mathstrut \) \(1834\) \(\beta_{11}\mathstrut +\mathstrut \) \(1194\) \(\beta_{10}\mathstrut +\mathstrut \) \(4067\) \(\beta_{9}\mathstrut -\mathstrut \) \(1964\) \(\beta_{8}\mathstrut -\mathstrut \) \(495\) \(\beta_{7}\mathstrut -\mathstrut \) \(1120\) \(\beta_{6}\mathstrut -\mathstrut \) \(117\) \(\beta_{5}\mathstrut -\mathstrut \) \(4562\) \(\beta_{4}\mathstrut +\mathstrut \) \(3638\) \(\beta_{3}\mathstrut +\mathstrut \) \(5625\) \(\beta_{2}\mathstrut +\mathstrut \) \(3549\) \(\beta_{1}\mathstrut +\mathstrut \) \(5298\)
\(\nu^{12}\)\(=\)\(-\)\(795\) \(\beta_{18}\mathstrut -\mathstrut \) \(9808\) \(\beta_{17}\mathstrut -\mathstrut \) \(2598\) \(\beta_{16}\mathstrut -\mathstrut \) \(675\) \(\beta_{15}\mathstrut +\mathstrut \) \(7229\) \(\beta_{14}\mathstrut -\mathstrut \) \(6434\) \(\beta_{13}\mathstrut +\mathstrut \) \(3585\) \(\beta_{12}\mathstrut -\mathstrut \) \(655\) \(\beta_{11}\mathstrut +\mathstrut \) \(3293\) \(\beta_{10}\mathstrut +\mathstrut \) \(2928\) \(\beta_{9}\mathstrut -\mathstrut \) \(6887\) \(\beta_{8}\mathstrut +\mathstrut \) \(618\) \(\beta_{7}\mathstrut -\mathstrut \) \(6073\) \(\beta_{6}\mathstrut +\mathstrut \) \(627\) \(\beta_{5}\mathstrut -\mathstrut \) \(7156\) \(\beta_{4}\mathstrut +\mathstrut \) \(3462\) \(\beta_{3}\mathstrut +\mathstrut \) \(16405\) \(\beta_{2}\mathstrut +\mathstrut \) \(1961\) \(\beta_{1}\mathstrut +\mathstrut \) \(21211\)
\(\nu^{13}\)\(=\)\(-\)\(6394\) \(\beta_{18}\mathstrut -\mathstrut \) \(32191\) \(\beta_{17}\mathstrut -\mathstrut \) \(9047\) \(\beta_{16}\mathstrut -\mathstrut \) \(487\) \(\beta_{15}\mathstrut +\mathstrut \) \(23457\) \(\beta_{14}\mathstrut -\mathstrut \) \(16956\) \(\beta_{13}\mathstrut +\mathstrut \) \(24330\) \(\beta_{12}\mathstrut -\mathstrut \) \(12333\) \(\beta_{11}\mathstrut +\mathstrut \) \(9727\) \(\beta_{10}\mathstrut +\mathstrut \) \(28948\) \(\beta_{9}\mathstrut -\mathstrut \) \(15834\) \(\beta_{8}\mathstrut -\mathstrut \) \(3182\) \(\beta_{7}\mathstrut -\mathstrut \) \(9401\) \(\beta_{6}\mathstrut -\mathstrut \) \(900\) \(\beta_{5}\mathstrut -\mathstrut \) \(33585\) \(\beta_{4}\mathstrut +\mathstrut \) \(26118\) \(\beta_{3}\mathstrut +\mathstrut \) \(42845\) \(\beta_{2}\mathstrut +\mathstrut \) \(22727\) \(\beta_{1}\mathstrut +\mathstrut \) \(41335\)
\(\nu^{14}\)\(=\)\(-\)\(7734\) \(\beta_{18}\mathstrut -\mathstrut \) \(77881\) \(\beta_{17}\mathstrut -\mathstrut \) \(21490\) \(\beta_{16}\mathstrut -\mathstrut \) \(4932\) \(\beta_{15}\mathstrut +\mathstrut \) \(56619\) \(\beta_{14}\mathstrut -\mathstrut \) \(50140\) \(\beta_{13}\mathstrut +\mathstrut \) \(31827\) \(\beta_{12}\mathstrut -\mathstrut \) \(7254\) \(\beta_{11}\mathstrut +\mathstrut \) \(26683\) \(\beta_{10}\mathstrut +\mathstrut \) \(27859\) \(\beta_{9}\mathstrut -\mathstrut \) \(52759\) \(\beta_{8}\mathstrut +\mathstrut \) \(4032\) \(\beta_{7}\mathstrut -\mathstrut \) \(44793\) \(\beta_{6}\mathstrut +\mathstrut \) \(4228\) \(\beta_{5}\mathstrut -\mathstrut \) \(58332\) \(\beta_{4}\mathstrut +\mathstrut \) \(30980\) \(\beta_{3}\mathstrut +\mathstrut \) \(121774\) \(\beta_{2}\mathstrut +\mathstrut \) \(18008\) \(\beta_{1}\mathstrut +\mathstrut \) \(150690\)
\(\nu^{15}\)\(=\)\(-\)\(47601\) \(\beta_{18}\mathstrut -\mathstrut \) \(243283\) \(\beta_{17}\mathstrut -\mathstrut \) \(69979\) \(\beta_{16}\mathstrut -\mathstrut \) \(5003\) \(\beta_{15}\mathstrut +\mathstrut \) \(175946\) \(\beta_{14}\mathstrut -\mathstrut \) \(133442\) \(\beta_{13}\mathstrut +\mathstrut \) \(175055\) \(\beta_{12}\mathstrut -\mathstrut \) \(84696\) \(\beta_{11}\mathstrut +\mathstrut \) \(76823\) \(\beta_{10}\mathstrut +\mathstrut \) \(206548\) \(\beta_{9}\mathstrut -\mathstrut \) \(124793\) \(\beta_{8}\mathstrut -\mathstrut \) \(20156\) \(\beta_{7}\mathstrut -\mathstrut \) \(76699\) \(\beta_{6}\mathstrut -\mathstrut \) \(6336\) \(\beta_{5}\mathstrut -\mathstrut \) \(247030\) \(\beta_{4}\mathstrut +\mathstrut \) \(187922\) \(\beta_{3}\mathstrut +\mathstrut \) \(325246\) \(\beta_{2}\mathstrut +\mathstrut \) \(151046\) \(\beta_{1}\mathstrut +\mathstrut \) \(320588\)
\(\nu^{16}\)\(=\)\(-\)\(68849\) \(\beta_{18}\mathstrut -\mathstrut \) \(607884\) \(\beta_{17}\mathstrut -\mathstrut \) \(171603\) \(\beta_{16}\mathstrut -\mathstrut \) \(35856\) \(\beta_{15}\mathstrut +\mathstrut \) \(438532\) \(\beta_{14}\mathstrut -\mathstrut \) \(386211\) \(\beta_{13}\mathstrut +\mathstrut \) \(269172\) \(\beta_{12}\mathstrut -\mathstrut \) \(70882\) \(\beta_{11}\mathstrut +\mathstrut \) \(210265\) \(\beta_{10}\mathstrut +\mathstrut \) \(247615\) \(\beta_{9}\mathstrut -\mathstrut \) \(400233\) \(\beta_{8}\mathstrut +\mathstrut \) \(24997\) \(\beta_{7}\mathstrut -\mathstrut \) \(329050\) \(\beta_{6}\mathstrut +\mathstrut \) \(27674\) \(\beta_{5}\mathstrut -\mathstrut \) \(467720\) \(\beta_{4}\mathstrut +\mathstrut \) \(264410\) \(\beta_{3}\mathstrut +\mathstrut \) \(911826\) \(\beta_{2}\mathstrut +\mathstrut \) \(157234\) \(\beta_{1}\mathstrut +\mathstrut \) \(1092865\)
\(\nu^{17}\)\(=\)\(-\)\(351573\) \(\beta_{18}\mathstrut -\mathstrut \) \(1836887\) \(\beta_{17}\mathstrut -\mathstrut \) \(534255\) \(\beta_{16}\mathstrut -\mathstrut \) \(46235\) \(\beta_{15}\mathstrut +\mathstrut \) \(1323034\) \(\beta_{14}\mathstrut -\mathstrut \) \(1035008\) \(\beta_{13}\mathstrut +\mathstrut \) \(1268980\) \(\beta_{12}\mathstrut -\mathstrut \) \(591374\) \(\beta_{11}\mathstrut +\mathstrut \) \(596476\) \(\beta_{10}\mathstrut +\mathstrut \) \(1481566\) \(\beta_{9}\mathstrut -\mathstrut \) \(971076\) \(\beta_{8}\mathstrut -\mathstrut \) \(127320\) \(\beta_{7}\mathstrut -\mathstrut \) \(614363\) \(\beta_{6}\mathstrut -\mathstrut \) \(42098\) \(\beta_{5}\mathstrut -\mathstrut \) \(1820672\) \(\beta_{4}\mathstrut +\mathstrut \) \(1358569\) \(\beta_{3}\mathstrut +\mathstrut \) \(2465860\) \(\beta_{2}\mathstrut +\mathstrut \) \(1033344\) \(\beta_{1}\mathstrut +\mathstrut \) \(2475244\)
\(\nu^{18}\)\(=\)\(-\)\(581913\) \(\beta_{18}\mathstrut -\mathstrut \) \(4696666\) \(\beta_{17}\mathstrut -\mathstrut \) \(1343260\) \(\beta_{16}\mathstrut -\mathstrut \) \(261250\) \(\beta_{15}\mathstrut +\mathstrut \) \(3373548\) \(\beta_{14}\mathstrut -\mathstrut \) \(2954518\) \(\beta_{13}\mathstrut +\mathstrut \) \(2207094\) \(\beta_{12}\mathstrut -\mathstrut \) \(641881\) \(\beta_{11}\mathstrut +\mathstrut \) \(1631484\) \(\beta_{10}\mathstrut +\mathstrut \) \(2105746\) \(\beta_{9}\mathstrut -\mathstrut \) \(3022494\) \(\beta_{8}\mathstrut +\mathstrut \) \(148509\) \(\beta_{7}\mathstrut -\mathstrut \) \(2419455\) \(\beta_{6}\mathstrut +\mathstrut \) \(178143\) \(\beta_{5}\mathstrut -\mathstrut \) \(3703210\) \(\beta_{4}\mathstrut +\mathstrut \) \(2187142\) \(\beta_{3}\mathstrut +\mathstrut \) \(6861491\) \(\beta_{2}\mathstrut +\mathstrut \) \(1326141\) \(\beta_{1}\mathstrut +\mathstrut \) \(8033073\)

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
2.75283
2.17652
2.10013
2.02307
1.82121
1.44458
1.21153
0.950625
0.725598
0.116858
−0.0952190
−0.569180
−0.810100
−0.929067
−1.09685
−1.49588
−1.78092
−2.08649
−2.45923
−2.75283 1.00000 5.57808 1.28742 −2.75283 −1.78893 −9.84984 1.00000 −3.54405
1.2 −2.17652 1.00000 2.73724 −0.898923 −2.17652 0.611535 −1.60461 1.00000 1.95652
1.3 −2.10013 1.00000 2.41053 −2.56145 −2.10013 −4.38780 −0.862160 1.00000 5.37937
1.4 −2.02307 1.00000 2.09279 −0.298497 −2.02307 0.750849 −0.187724 1.00000 0.603878
1.5 −1.82121 1.00000 1.31679 3.98129 −1.82121 −0.917766 1.24426 1.00000 −7.25075
1.6 −1.44458 1.00000 0.0867978 −1.93878 −1.44458 1.10404 2.76376 1.00000 2.80071
1.7 −1.21153 1.00000 −0.532186 2.05730 −1.21153 −4.52276 3.06783 1.00000 −2.49249
1.8 −0.950625 1.00000 −1.09631 −0.0164301 −0.950625 3.21492 2.94343 1.00000 0.0156189
1.9 −0.725598 1.00000 −1.47351 −3.68550 −0.725598 −0.281235 2.52037 1.00000 2.67419
1.10 −0.116858 1.00000 −1.98634 −1.39349 −0.116858 −4.81215 0.465837 1.00000 0.162841
1.11 0.0952190 1.00000 −1.99093 2.07279 0.0952190 −1.92020 −0.380013 1.00000 0.197369
1.12 0.569180 1.00000 −1.67603 1.84267 0.569180 1.78912 −2.09233 1.00000 1.04881
1.13 0.810100 1.00000 −1.34374 −3.50747 0.810100 −0.645139 −2.70876 1.00000 −2.84140
1.14 0.929067 1.00000 −1.13683 3.14855 0.929067 −4.69509 −2.91433 1.00000 2.92521
1.15 1.09685 1.00000 −0.796918 0.842996 1.09685 0.628687 −3.06780 1.00000 0.924641
1.16 1.49588 1.00000 0.237660 −0.502630 1.49588 −1.16688 −2.63625 1.00000 −0.751875
1.17 1.78092 1.00000 1.17168 −1.00354 1.78092 −0.242679 −1.47518 1.00000 −1.78722
1.18 2.08649 1.00000 2.35342 0.146235 2.08649 −2.80287 0.737414 1.00000 0.305116
1.19 2.45923 1.00000 4.04782 −2.57255 2.45923 −2.91567 5.03608 1.00000 −6.32649
\(n\): 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
\(3\) \(-1\)
\(13\) \(-1\)
\(103\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4017))\):

\(T_{2}^{19} + \cdots\)
\(T_{23}^{19} + \cdots\)