Properties

Label 4029.2.a.e
Level 4029
Weight 2
Character orbit 4029.a
Self dual Yes
Analytic conductor 32.172
Analytic rank 1
Dimension 18
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4029 = 3 \cdot 17 \cdot 79 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4029.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.1717269744\)
Analytic rank: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18}\mathstrut -\mathstrut \) \(6\) \(x^{17}\mathstrut -\mathstrut \) \(10\) \(x^{16}\mathstrut +\mathstrut \) \(120\) \(x^{15}\mathstrut -\mathstrut \) \(56\) \(x^{14}\mathstrut -\mathstrut \) \(921\) \(x^{13}\mathstrut +\mathstrut \) \(1181\) \(x^{12}\mathstrut +\mathstrut \) \(3316\) \(x^{11}\mathstrut -\mathstrut \) \(6280\) \(x^{10}\mathstrut -\mathstrut \) \(5249\) \(x^{9}\mathstrut +\mathstrut \) \(15005\) \(x^{8}\mathstrut +\mathstrut \) \(1809\) \(x^{7}\mathstrut -\mathstrut \) \(16711\) \(x^{6}\mathstrut +\mathstrut \) \(2434\) \(x^{5}\mathstrut +\mathstrut \) \(8758\) \(x^{4}\mathstrut -\mathstrut \) \(1858\) \(x^{3}\mathstrut -\mathstrut \) \(1942\) \(x^{2}\mathstrut +\mathstrut \) \(318\) \(x\mathstrut +\mathstrut \) \(138\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( 14 \nu^{17} - 113 \nu^{16} + 69 \nu^{15} + 1603 \nu^{14} - 3539 \nu^{13} - 6970 \nu^{12} + 25822 \nu^{11} + 4682 \nu^{10} - 73431 \nu^{9} + 30548 \nu^{8} + 84496 \nu^{7} - 51518 \nu^{6} - 39620 \nu^{5} + 26524 \nu^{4} + 3936 \nu^{3} - 5801 \nu^{2} + 550 \nu + 579 \)\()/65\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(43\) \(\nu^{17}\mathstrut +\mathstrut \) \(309\) \(\nu^{16}\mathstrut +\mathstrut \) \(49\) \(\nu^{15}\mathstrut -\mathstrut \) \(4878\) \(\nu^{14}\mathstrut +\mathstrut \) \(7019\) \(\nu^{13}\mathstrut +\mathstrut \) \(27126\) \(\nu^{12}\mathstrut -\mathstrut \) \(60752\) \(\nu^{11}\mathstrut -\mathstrut \) \(60627\) \(\nu^{10}\mathstrut +\mathstrut \) \(200020\) \(\nu^{9}\mathstrut +\mathstrut \) \(40802\) \(\nu^{8}\mathstrut -\mathstrut \) \(292315\) \(\nu^{7}\mathstrut +\mathstrut \) \(2453\) \(\nu^{6}\mathstrut +\mathstrut \) \(202368\) \(\nu^{5}\mathstrut -\mathstrut \) \(263\) \(\nu^{4}\mathstrut -\mathstrut \) \(58891\) \(\nu^{3}\mathstrut -\mathstrut \) \(3355\) \(\nu^{2}\mathstrut +\mathstrut \) \(5717\) \(\nu\mathstrut +\mathstrut \) \(752\)\()/65\)
\(\beta_{5}\)\(=\)\((\)\(2\) \(\nu^{17}\mathstrut -\mathstrut \) \(44\) \(\nu^{16}\mathstrut +\mathstrut \) \(268\) \(\nu^{15}\mathstrut -\mathstrut \) \(57\) \(\nu^{14}\mathstrut -\mathstrut \) \(4166\) \(\nu^{13}\mathstrut +\mathstrut \) \(8732\) \(\nu^{12}\mathstrut +\mathstrut \) \(19508\) \(\nu^{11}\mathstrut -\mathstrut \) \(69325\) \(\nu^{10}\mathstrut -\mathstrut \) \(19774\) \(\nu^{9}\mathstrut +\mathstrut \) \(212910\) \(\nu^{8}\mathstrut -\mathstrut \) \(65151\) \(\nu^{7}\mathstrut -\mathstrut \) \(282150\) \(\nu^{6}\mathstrut +\mathstrut \) \(127434\) \(\nu^{5}\mathstrut +\mathstrut \) \(175573\) \(\nu^{4}\mathstrut -\mathstrut \) \(67372\) \(\nu^{3}\mathstrut -\mathstrut \) \(46221\) \(\nu^{2}\mathstrut +\mathstrut \) \(10356\) \(\nu\mathstrut +\mathstrut \) \(3901\)\()/65\)
\(\beta_{6}\)\(=\)\((\)\(29\) \(\nu^{17}\mathstrut -\mathstrut \) \(14\) \(\nu^{16}\mathstrut -\mathstrut \) \(1249\) \(\nu^{15}\mathstrut +\mathstrut \) \(2391\) \(\nu^{14}\mathstrut +\mathstrut \) \(14200\) \(\nu^{13}\mathstrut -\mathstrut \) \(37290\) \(\nu^{12}\mathstrut -\mathstrut \) \(61335\) \(\nu^{11}\mathstrut +\mathstrut \) \(222852\) \(\nu^{10}\mathstrut +\mathstrut \) \(75782\) \(\nu^{9}\mathstrut -\mathstrut \) \(603297\) \(\nu^{8}\mathstrut +\mathstrut \) \(117768\) \(\nu^{7}\mathstrut +\mathstrut \) \(733177\) \(\nu^{6}\mathstrut -\mathstrut \) \(264772\) \(\nu^{5}\mathstrut -\mathstrut \) \(417145\) \(\nu^{4}\mathstrut +\mathstrut \) \(142289\) \(\nu^{3}\mathstrut +\mathstrut \) \(100910\) \(\nu^{2}\mathstrut -\mathstrut \) \(22153\) \(\nu\mathstrut -\mathstrut \) \(8052\)\()/65\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(135\) \(\nu^{17}\mathstrut +\mathstrut \) \(747\) \(\nu^{16}\mathstrut +\mathstrut \) \(1306\) \(\nu^{15}\mathstrut -\mathstrut \) \(13046\) \(\nu^{14}\mathstrut +\mathstrut \) \(2862\) \(\nu^{13}\mathstrut +\mathstrut \) \(86395\) \(\nu^{12}\mathstrut -\mathstrut \) \(73626\) \(\nu^{11}\mathstrut -\mathstrut \) \(273530\) \(\nu^{10}\mathstrut +\mathstrut \) \(308629\) \(\nu^{9}\mathstrut +\mathstrut \) \(444545\) \(\nu^{8}\mathstrut -\mathstrut \) \(532954\) \(\nu^{7}\mathstrut -\mathstrut \) \(409050\) \(\nu^{6}\mathstrut +\mathstrut \) \(427849\) \(\nu^{5}\mathstrut +\mathstrut \) \(219953\) \(\nu^{4}\mathstrut -\mathstrut \) \(148926\) \(\nu^{3}\mathstrut -\mathstrut \) \(58277\) \(\nu^{2}\mathstrut +\mathstrut \) \(18011\) \(\nu\mathstrut +\mathstrut \) \(5672\)\()/65\)
\(\beta_{8}\)\(=\)\((\)\(112\) \(\nu^{17}\mathstrut -\mathstrut \) \(553\) \(\nu^{16}\mathstrut -\mathstrut \) \(1554\) \(\nu^{15}\mathstrut +\mathstrut \) \(10770\) \(\nu^{14}\mathstrut +\mathstrut \) \(4760\) \(\nu^{13}\mathstrut -\mathstrut \) \(82553\) \(\nu^{12}\mathstrut +\mathstrut \) \(24680\) \(\nu^{11}\mathstrut +\mathstrut \) \(317060\) \(\nu^{10}\mathstrut -\mathstrut \) \(193847\) \(\nu^{9}\mathstrut -\mathstrut \) \(647065\) \(\nu^{8}\mathstrut +\mathstrut \) \(463002\) \(\nu^{7}\mathstrut +\mathstrut \) \(701215\) \(\nu^{6}\mathstrut -\mathstrut \) \(461754\) \(\nu^{5}\mathstrut -\mathstrut \) \(396078\) \(\nu^{4}\mathstrut +\mathstrut \) \(185785\) \(\nu^{3}\mathstrut +\mathstrut \) \(99153\) \(\nu^{2}\mathstrut -\mathstrut \) \(24551\) \(\nu\mathstrut -\mathstrut \) \(8173\)\()/65\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(122\) \(\nu^{17}\mathstrut +\mathstrut \) \(604\) \(\nu^{16}\mathstrut +\mathstrut \) \(1683\) \(\nu^{15}\mathstrut -\mathstrut \) \(11759\) \(\nu^{14}\mathstrut -\mathstrut \) \(4938\) \(\nu^{13}\mathstrut +\mathstrut \) \(89970\) \(\nu^{12}\mathstrut -\mathstrut \) \(29231\) \(\nu^{11}\mathstrut -\mathstrut \) \(343886\) \(\nu^{10}\mathstrut +\mathstrut \) \(221633\) \(\nu^{9}\mathstrut +\mathstrut \) \(694431\) \(\nu^{8}\mathstrut -\mathstrut \) \(526493\) \(\nu^{7}\mathstrut -\mathstrut \) \(738496\) \(\nu^{6}\mathstrut +\mathstrut \) \(522775\) \(\nu^{5}\mathstrut +\mathstrut \) \(410208\) \(\nu^{4}\mathstrut -\mathstrut \) \(210078\) \(\nu^{3}\mathstrut -\mathstrut \) \(103387\) \(\nu^{2}\mathstrut +\mathstrut \) \(27605\) \(\nu\mathstrut +\mathstrut \) \(9013\)\()/65\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(155\) \(\nu^{17}\mathstrut +\mathstrut \) \(862\) \(\nu^{16}\mathstrut +\mathstrut \) \(1603\) \(\nu^{15}\mathstrut -\mathstrut \) \(15648\) \(\nu^{14}\mathstrut +\mathstrut \) \(2259\) \(\nu^{13}\mathstrut +\mathstrut \) \(109679\) \(\nu^{12}\mathstrut -\mathstrut \) \(85432\) \(\nu^{11}\mathstrut -\mathstrut \) \(377687\) \(\nu^{10}\mathstrut +\mathstrut \) \(393867\) \(\nu^{9}\mathstrut +\mathstrut \) \(687867\) \(\nu^{8}\mathstrut -\mathstrut \) \(755317\) \(\nu^{7}\mathstrut -\mathstrut \) \(698762\) \(\nu^{6}\mathstrut +\mathstrut \) \(664122\) \(\nu^{5}\mathstrut +\mathstrut \) \(395880\) \(\nu^{4}\mathstrut -\mathstrut \) \(244741\) \(\nu^{3}\mathstrut -\mathstrut \) \(102963\) \(\nu^{2}\mathstrut +\mathstrut \) \(30528\) \(\nu\mathstrut +\mathstrut \) \(9380\)\()/65\)
\(\beta_{11}\)\(=\)\((\)\(144\) \(\nu^{17}\mathstrut -\mathstrut \) \(620\) \(\nu^{16}\mathstrut -\mathstrut \) \(2609\) \(\nu^{15}\mathstrut +\mathstrut \) \(13641\) \(\nu^{14}\mathstrut +\mathstrut \) \(15402\) \(\nu^{13}\mathstrut -\mathstrut \) \(118549\) \(\nu^{12}\mathstrut -\mathstrut \) \(15921\) \(\nu^{11}\mathstrut +\mathstrut \) \(513814\) \(\nu^{10}\mathstrut -\mathstrut \) \(165393\) \(\nu^{9}\mathstrut -\mathstrut \) \(1156469\) \(\nu^{8}\mathstrut +\mathstrut \) \(611373\) \(\nu^{7}\mathstrut +\mathstrut \) \(1307034\) \(\nu^{6}\mathstrut -\mathstrut \) \(715802\) \(\nu^{5}\mathstrut -\mathstrut \) \(734691\) \(\nu^{4}\mathstrut +\mathstrut \) \(312192\) \(\nu^{3}\mathstrut +\mathstrut \) \(178552\) \(\nu^{2}\mathstrut -\mathstrut \) \(43208\) \(\nu\mathstrut -\mathstrut \) \(14371\)\()/65\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(216\) \(\nu^{17}\mathstrut +\mathstrut \) \(982\) \(\nu^{16}\mathstrut +\mathstrut \) \(3465\) \(\nu^{15}\mathstrut -\mathstrut \) \(20078\) \(\nu^{14}\mathstrut -\mathstrut \) \(16733\) \(\nu^{13}\mathstrut +\mathstrut \) \(162347\) \(\nu^{12}\mathstrut -\mathstrut \) \(3776\) \(\nu^{11}\mathstrut -\mathstrut \) \(658414\) \(\nu^{10}\mathstrut +\mathstrut \) \(265503\) \(\nu^{9}\mathstrut +\mathstrut \) \(1406109\) \(\nu^{8}\mathstrut -\mathstrut \) \(788223\) \(\nu^{7}\mathstrut -\mathstrut \) \(1549049\) \(\nu^{6}\mathstrut +\mathstrut \) \(853444\) \(\nu^{5}\mathstrut +\mathstrut \) \(864650\) \(\nu^{4}\mathstrut -\mathstrut \) \(357294\) \(\nu^{3}\mathstrut -\mathstrut \) \(210979\) \(\nu^{2}\mathstrut +\mathstrut \) \(48341\) \(\nu\mathstrut +\mathstrut \) \(16948\)\()/65\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(196\) \(\nu^{17}\mathstrut +\mathstrut \) \(880\) \(\nu^{16}\mathstrut +\mathstrut \) \(3311\) \(\nu^{15}\mathstrut -\mathstrut \) \(18659\) \(\nu^{14}\mathstrut -\mathstrut \) \(17313\) \(\nu^{13}\mathstrut +\mathstrut \) \(156626\) \(\nu^{12}\mathstrut +\mathstrut \) \(2869\) \(\nu^{11}\mathstrut -\mathstrut \) \(658621\) \(\nu^{10}\mathstrut +\mathstrut \) \(258772\) \(\nu^{9}\mathstrut +\mathstrut \) \(1450321\) \(\nu^{8}\mathstrut -\mathstrut \) \(829032\) \(\nu^{7}\mathstrut -\mathstrut \) \(1627926\) \(\nu^{6}\mathstrut +\mathstrut \) \(933448\) \(\nu^{5}\mathstrut +\mathstrut \) \(921384\) \(\nu^{4}\mathstrut -\mathstrut \) \(403218\) \(\nu^{3}\mathstrut -\mathstrut \) \(228693\) \(\nu^{2}\mathstrut +\mathstrut \) \(56247\) \(\nu\mathstrut +\mathstrut \) \(18934\)\()/65\)
\(\beta_{14}\)\(=\)\((\)\(22\) \(\nu^{17}\mathstrut -\mathstrut \) \(100\) \(\nu^{16}\mathstrut -\mathstrut \) \(359\) \(\nu^{15}\mathstrut +\mathstrut \) \(2075\) \(\nu^{14}\mathstrut +\mathstrut \) \(1769\) \(\nu^{13}\mathstrut -\mathstrut \) \(17036\) \(\nu^{12}\mathstrut +\mathstrut \) \(363\) \(\nu^{11}\mathstrut +\mathstrut \) \(70104\) \(\nu^{10}\mathstrut -\mathstrut \) \(28892\) \(\nu^{9}\mathstrut -\mathstrut \) \(151444\) \(\nu^{8}\mathstrut +\mathstrut \) \(87357\) \(\nu^{7}\mathstrut +\mathstrut \) \(167684\) \(\nu^{6}\mathstrut -\mathstrut \) \(95454\) \(\nu^{5}\mathstrut -\mathstrut \) \(93920\) \(\nu^{4}\mathstrut +\mathstrut \) \(40201\) \(\nu^{3}\mathstrut +\mathstrut \) \(23037\) \(\nu^{2}\mathstrut -\mathstrut \) \(5501\) \(\nu\mathstrut -\mathstrut \) \(1881\)\()/5\)
\(\beta_{15}\)\(=\)\((\)\(-\)\(277\) \(\nu^{17}\mathstrut +\mathstrut \) \(1206\) \(\nu^{16}\mathstrut +\mathstrut \) \(4872\) \(\nu^{15}\mathstrut -\mathstrut \) \(26003\) \(\nu^{14}\mathstrut -\mathstrut \) \(27548\) \(\nu^{13}\mathstrut +\mathstrut \) \(221606\) \(\nu^{12}\mathstrut +\mathstrut \) \(21694\) \(\nu^{11}\mathstrut -\mathstrut \) \(942989\) \(\nu^{10}\mathstrut +\mathstrut \) \(322714\) \(\nu^{9}\mathstrut +\mathstrut \) \(2087964\) \(\nu^{8}\mathstrut -\mathstrut \) \(1129034\) \(\nu^{7}\mathstrut -\mathstrut \) \(2330124\) \(\nu^{6}\mathstrut +\mathstrut \) \(1296721\) \(\nu^{5}\mathstrut +\mathstrut \) \(1301219\) \(\nu^{4}\mathstrut -\mathstrut \) \(562199\) \(\nu^{3}\mathstrut -\mathstrut \) \(319398\) \(\nu^{2}\mathstrut +\mathstrut \) \(77984\) \(\nu\mathstrut +\mathstrut \) \(26492\)\()/65\)
\(\beta_{16}\)\(=\)\((\)\(-\)\(432\) \(\nu^{17}\mathstrut +\mathstrut \) \(2146\) \(\nu^{16}\mathstrut +\mathstrut \) \(5890\) \(\nu^{15}\mathstrut -\mathstrut \) \(41521\) \(\nu^{14}\mathstrut -\mathstrut \) \(16670\) \(\nu^{13}\mathstrut +\mathstrut \) \(315542\) \(\nu^{12}\mathstrut -\mathstrut \) \(105000\) \(\nu^{11}\mathstrut -\mathstrut \) \(1198437\) \(\nu^{10}\mathstrut +\mathstrut \) \(771181\) \(\nu^{9}\mathstrut +\mathstrut \) \(2411792\) \(\nu^{8}\mathstrut -\mathstrut \) \(1803881\) \(\nu^{7}\mathstrut -\mathstrut \) \(2573717\) \(\nu^{6}\mathstrut +\mathstrut \) \(1778193\) \(\nu^{5}\mathstrut +\mathstrut \) \(1435357\) \(\nu^{4}\mathstrut -\mathstrut \) \(712924\) \(\nu^{3}\mathstrut -\mathstrut \) \(358232\) \(\nu^{2}\mathstrut +\mathstrut \) \(95122\) \(\nu\mathstrut +\mathstrut \) \(30464\)\()/65\)
\(\beta_{17}\)\(=\)\((\)\(428\) \(\nu^{17}\mathstrut -\mathstrut \) \(1759\) \(\nu^{16}\mathstrut -\mathstrut \) \(8233\) \(\nu^{15}\mathstrut +\mathstrut \) \(39932\) \(\nu^{14}\mathstrut +\mathstrut \) \(53407\) \(\nu^{13}\mathstrut -\mathstrut \) \(356679\) \(\nu^{12}\mathstrut -\mathstrut \) \(90601\) \(\nu^{11}\mathstrut +\mathstrut \) \(1580811\) \(\nu^{10}\mathstrut -\mathstrut \) \(389681\) \(\nu^{9}\mathstrut -\mathstrut \) \(3612191\) \(\nu^{8}\mathstrut +\mathstrut \) \(1731721\) \(\nu^{7}\mathstrut +\mathstrut \) \(4100966\) \(\nu^{6}\mathstrut -\mathstrut \) \(2109774\) \(\nu^{5}\mathstrut -\mathstrut \) \(2302226\) \(\nu^{4}\mathstrut +\mathstrut \) \(935821\) \(\nu^{3}\mathstrut +\mathstrut \) \(560667\) \(\nu^{2}\mathstrut -\mathstrut \) \(131226\) \(\nu\mathstrut -\mathstrut \) \(45468\)\()/65\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{17}\mathstrut -\mathstrut \) \(\beta_{15}\mathstrut +\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{17}\mathstrut -\mathstrut \) \(\beta_{15}\mathstrut +\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(8\) \(\beta_{2}\mathstrut +\mathstrut \) \(14\)
\(\nu^{5}\)\(=\)\(-\)\(9\) \(\beta_{17}\mathstrut +\mathstrut \) \(\beta_{16}\mathstrut -\mathstrut \) \(8\) \(\beta_{15}\mathstrut +\mathstrut \) \(9\) \(\beta_{14}\mathstrut +\mathstrut \) \(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(7\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(11\) \(\beta_{2}\mathstrut +\mathstrut \) \(18\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{6}\)\(=\)\(-\)\(11\) \(\beta_{17}\mathstrut -\mathstrut \) \(\beta_{16}\mathstrut -\mathstrut \) \(11\) \(\beta_{15}\mathstrut +\mathstrut \) \(13\) \(\beta_{14}\mathstrut +\mathstrut \) \(2\) \(\beta_{13}\mathstrut +\mathstrut \) \(2\) \(\beta_{12}\mathstrut +\mathstrut \) \(11\) \(\beta_{10}\mathstrut -\mathstrut \) \(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(10\) \(\beta_{8}\mathstrut +\mathstrut \) \(12\) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(8\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(58\) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(74\)
\(\nu^{7}\)\(=\)\(-\)\(68\) \(\beta_{17}\mathstrut +\mathstrut \) \(9\) \(\beta_{16}\mathstrut -\mathstrut \) \(55\) \(\beta_{15}\mathstrut +\mathstrut \) \(72\) \(\beta_{14}\mathstrut +\mathstrut \) \(4\) \(\beta_{13}\mathstrut +\mathstrut \) \(4\) \(\beta_{12}\mathstrut +\mathstrut \) \(24\) \(\beta_{11}\mathstrut +\mathstrut \) \(14\) \(\beta_{10}\mathstrut -\mathstrut \) \(14\) \(\beta_{9}\mathstrut +\mathstrut \) \(12\) \(\beta_{8}\mathstrut +\mathstrut \) \(46\) \(\beta_{7}\mathstrut +\mathstrut \) \(10\) \(\beta_{6}\mathstrut -\mathstrut \) \(3\) \(\beta_{5}\mathstrut -\mathstrut \) \(9\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(95\) \(\beta_{2}\mathstrut +\mathstrut \) \(91\) \(\beta_{1}\mathstrut +\mathstrut \) \(31\)
\(\nu^{8}\)\(=\)\(-\)\(97\) \(\beta_{17}\mathstrut -\mathstrut \) \(14\) \(\beta_{16}\mathstrut -\mathstrut \) \(91\) \(\beta_{15}\mathstrut +\mathstrut \) \(128\) \(\beta_{14}\mathstrut +\mathstrut \) \(33\) \(\beta_{13}\mathstrut +\mathstrut \) \(31\) \(\beta_{12}\mathstrut +\mathstrut \) \(8\) \(\beta_{11}\mathstrut +\mathstrut \) \(95\) \(\beta_{10}\mathstrut -\mathstrut \) \(31\) \(\beta_{9}\mathstrut +\mathstrut \) \(83\) \(\beta_{8}\mathstrut +\mathstrut \) \(106\) \(\beta_{7}\mathstrut -\mathstrut \) \(23\) \(\beta_{6}\mathstrut -\mathstrut \) \(29\) \(\beta_{5}\mathstrut -\mathstrut \) \(51\) \(\beta_{4}\mathstrut +\mathstrut \) \(11\) \(\beta_{3}\mathstrut +\mathstrut \) \(413\) \(\beta_{2}\mathstrut +\mathstrut \) \(68\) \(\beta_{1}\mathstrut +\mathstrut \) \(423\)
\(\nu^{9}\)\(=\)\(-\)\(496\) \(\beta_{17}\mathstrut +\mathstrut \) \(59\) \(\beta_{16}\mathstrut -\mathstrut \) \(369\) \(\beta_{15}\mathstrut +\mathstrut \) \(564\) \(\beta_{14}\mathstrut +\mathstrut \) \(74\) \(\beta_{13}\mathstrut +\mathstrut \) \(68\) \(\beta_{12}\mathstrut +\mathstrut \) \(220\) \(\beta_{11}\mathstrut +\mathstrut \) \(142\) \(\beta_{10}\mathstrut -\mathstrut \) \(145\) \(\beta_{9}\mathstrut +\mathstrut \) \(121\) \(\beta_{8}\mathstrut +\mathstrut \) \(313\) \(\beta_{7}\mathstrut +\mathstrut \) \(77\) \(\beta_{6}\mathstrut -\mathstrut \) \(51\) \(\beta_{5}\mathstrut -\mathstrut \) \(59\) \(\beta_{4}\mathstrut +\mathstrut \) \(9\) \(\beta_{3}\mathstrut +\mathstrut \) \(762\) \(\beta_{2}\mathstrut +\mathstrut \) \(503\) \(\beta_{1}\mathstrut +\mathstrut \) \(320\)
\(\nu^{10}\)\(=\)\(-\)\(808\) \(\beta_{17}\mathstrut -\mathstrut \) \(135\) \(\beta_{16}\mathstrut -\mathstrut \) \(691\) \(\beta_{15}\mathstrut +\mathstrut \) \(1146\) \(\beta_{14}\mathstrut +\mathstrut \) \(380\) \(\beta_{13}\mathstrut +\mathstrut \) \(339\) \(\beta_{12}\mathstrut +\mathstrut \) \(158\) \(\beta_{11}\mathstrut +\mathstrut \) \(755\) \(\beta_{10}\mathstrut -\mathstrut \) \(341\) \(\beta_{9}\mathstrut +\mathstrut \) \(667\) \(\beta_{8}\mathstrut +\mathstrut \) \(845\) \(\beta_{7}\mathstrut -\mathstrut \) \(179\) \(\beta_{6}\mathstrut -\mathstrut \) \(300\) \(\beta_{5}\mathstrut -\mathstrut \) \(298\) \(\beta_{4}\mathstrut +\mathstrut \) \(76\) \(\beta_{3}\mathstrut +\mathstrut \) \(2941\) \(\beta_{2}\mathstrut +\mathstrut \) \(635\) \(\beta_{1}\mathstrut +\mathstrut \) \(2563\)
\(\nu^{11}\)\(=\)\(-\)\(3610\) \(\beta_{17}\mathstrut +\mathstrut \) \(329\) \(\beta_{16}\mathstrut -\mathstrut \) \(2486\) \(\beta_{15}\mathstrut +\mathstrut \) \(4402\) \(\beta_{14}\mathstrut +\mathstrut \) \(917\) \(\beta_{13}\mathstrut +\mathstrut \) \(797\) \(\beta_{12}\mathstrut +\mathstrut \) \(1853\) \(\beta_{11}\mathstrut +\mathstrut \) \(1278\) \(\beta_{10}\mathstrut -\mathstrut \) \(1331\) \(\beta_{9}\mathstrut +\mathstrut \) \(1156\) \(\beta_{8}\mathstrut +\mathstrut \) \(2212\) \(\beta_{7}\mathstrut +\mathstrut \) \(556\) \(\beta_{6}\mathstrut -\mathstrut \) \(592\) \(\beta_{5}\mathstrut -\mathstrut \) \(330\) \(\beta_{4}\mathstrut +\mathstrut \) \(28\) \(\beta_{3}\mathstrut +\mathstrut \) \(5927\) \(\beta_{2}\mathstrut +\mathstrut \) \(2965\) \(\beta_{1}\mathstrut +\mathstrut \) \(2811\)
\(\nu^{12}\)\(=\)\(-\)\(6584\) \(\beta_{17}\mathstrut -\mathstrut \) \(1131\) \(\beta_{16}\mathstrut -\mathstrut \) \(5089\) \(\beta_{15}\mathstrut +\mathstrut \) \(9796\) \(\beta_{14}\mathstrut +\mathstrut \) \(3786\) \(\beta_{13}\mathstrut +\mathstrut \) \(3242\) \(\beta_{12}\mathstrut +\mathstrut \) \(2049\) \(\beta_{11}\mathstrut +\mathstrut \) \(5792\) \(\beta_{10}\mathstrut -\mathstrut \) \(3264\) \(\beta_{9}\mathstrut +\mathstrut \) \(5352\) \(\beta_{8}\mathstrut +\mathstrut \) \(6450\) \(\beta_{7}\mathstrut -\mathstrut \) \(1147\) \(\beta_{6}\mathstrut -\mathstrut \) \(2732\) \(\beta_{5}\mathstrut -\mathstrut \) \(1624\) \(\beta_{4}\mathstrut +\mathstrut \) \(364\) \(\beta_{3}\mathstrut +\mathstrut \) \(21058\) \(\beta_{2}\mathstrut +\mathstrut \) \(5097\) \(\beta_{1}\mathstrut +\mathstrut \) \(16233\)
\(\nu^{13}\)\(=\)\(-\)\(26459\) \(\beta_{17}\mathstrut +\mathstrut \) \(1550\) \(\beta_{16}\mathstrut -\mathstrut \) \(16953\) \(\beta_{15}\mathstrut +\mathstrut \) \(34359\) \(\beta_{14}\mathstrut +\mathstrut \) \(9597\) \(\beta_{13}\mathstrut +\mathstrut \) \(8023\) \(\beta_{12}\mathstrut +\mathstrut \) \(15112\) \(\beta_{11}\mathstrut +\mathstrut \) \(10842\) \(\beta_{10}\mathstrut -\mathstrut \) \(11482\) \(\beta_{9}\mathstrut +\mathstrut \) \(10629\) \(\beta_{8}\mathstrut +\mathstrut \) \(16051\) \(\beta_{7}\mathstrut +\mathstrut \) \(3997\) \(\beta_{6}\mathstrut -\mathstrut \) \(5882\) \(\beta_{5}\mathstrut -\mathstrut \) \(1533\) \(\beta_{4}\mathstrut -\mathstrut \) \(334\) \(\beta_{3}\mathstrut +\mathstrut \) \(45407\) \(\beta_{2}\mathstrut +\mathstrut \) \(18285\) \(\beta_{1}\mathstrut +\mathstrut \) \(22809\)
\(\nu^{14}\)\(=\)\(-\)\(52969\) \(\beta_{17}\mathstrut -\mathstrut \) \(8906\) \(\beta_{16}\mathstrut -\mathstrut \) \(37087\) \(\beta_{15}\mathstrut +\mathstrut \) \(81465\) \(\beta_{14}\mathstrut +\mathstrut \) \(34992\) \(\beta_{13}\mathstrut +\mathstrut \) \(29017\) \(\beta_{12}\mathstrut +\mathstrut \) \(22094\) \(\beta_{11}\mathstrut +\mathstrut \) \(43760\) \(\beta_{10}\mathstrut -\mathstrut \) \(29041\) \(\beta_{9}\mathstrut +\mathstrut \) \(43080\) \(\beta_{8}\mathstrut +\mathstrut \) \(48256\) \(\beta_{7}\mathstrut -\mathstrut \) \(6221\) \(\beta_{6}\mathstrut -\mathstrut \) \(23424\) \(\beta_{5}\mathstrut -\mathstrut \) \(8004\) \(\beta_{4}\mathstrut +\mathstrut \) \(593\) \(\beta_{3}\mathstrut +\mathstrut \) \(151811\) \(\beta_{2}\mathstrut +\mathstrut \) \(37924\) \(\beta_{1}\mathstrut +\mathstrut \) \(106330\)
\(\nu^{15}\)\(=\)\(-\)\(195665\) \(\beta_{17}\mathstrut +\mathstrut \) \(5142\) \(\beta_{16}\mathstrut -\mathstrut \) \(117226\) \(\beta_{15}\mathstrut +\mathstrut \) \(268286\) \(\beta_{14}\mathstrut +\mathstrut \) \(91728\) \(\beta_{13}\mathstrut +\mathstrut \) \(74535\) \(\beta_{12}\mathstrut +\mathstrut \) \(121651\) \(\beta_{11}\mathstrut +\mathstrut \) \(88902\) \(\beta_{10}\mathstrut -\mathstrut \) \(95548\) \(\beta_{9}\mathstrut +\mathstrut \) \(94623\) \(\beta_{8}\mathstrut +\mathstrut \) \(118320\) \(\beta_{7}\mathstrut +\mathstrut \) \(29284\) \(\beta_{6}\mathstrut -\mathstrut \) \(53895\) \(\beta_{5}\mathstrut -\mathstrut \) \(4358\) \(\beta_{4}\mathstrut -\mathstrut \) \(7528\) \(\beta_{3}\mathstrut +\mathstrut \) \(345008\) \(\beta_{2}\mathstrut +\mathstrut \) \(116336\) \(\beta_{1}\mathstrut +\mathstrut \) \(177289\)
\(\nu^{16}\)\(=\)\(-\)\(422173\) \(\beta_{17}\mathstrut -\mathstrut \) \(68279\) \(\beta_{16}\mathstrut -\mathstrut \) \(269753\) \(\beta_{15}\mathstrut +\mathstrut \) \(665226\) \(\beta_{14}\mathstrut +\mathstrut \) \(309301\) \(\beta_{13}\mathstrut +\mathstrut \) \(250081\) \(\beta_{12}\mathstrut +\mathstrut \) \(215125\) \(\beta_{11}\mathstrut +\mathstrut \) \(328704\) \(\beta_{10}\mathstrut -\mathstrut \) \(247336\) \(\beta_{9}\mathstrut +\mathstrut \) \(347441\) \(\beta_{8}\mathstrut +\mathstrut \) \(357734\) \(\beta_{7}\mathstrut -\mathstrut \) \(26075\) \(\beta_{6}\mathstrut -\mathstrut \) \(194520\) \(\beta_{5}\mathstrut -\mathstrut \) \(31734\) \(\beta_{4}\mathstrut -\mathstrut \) \(13195\) \(\beta_{3}\mathstrut +\mathstrut \) \(1101869\) \(\beta_{2}\mathstrut +\mathstrut \) \(270608\) \(\beta_{1}\mathstrut +\mathstrut \) \(714456\)
\(\nu^{17}\)\(=\)\(-\)\(1459248\) \(\beta_{17}\mathstrut -\mathstrut \) \(4804\) \(\beta_{16}\mathstrut -\mathstrut \) \(821669\) \(\beta_{15}\mathstrut +\mathstrut \) \(2094864\) \(\beta_{14}\mathstrut +\mathstrut \) \(829624\) \(\beta_{13}\mathstrut +\mathstrut \) \(659726\) \(\beta_{12}\mathstrut +\mathstrut \) \(973948\) \(\beta_{11}\mathstrut +\mathstrut \) \(713477\) \(\beta_{10}\mathstrut -\mathstrut \) \(777349\) \(\beta_{9}\mathstrut +\mathstrut \) \(820353\) \(\beta_{8}\mathstrut +\mathstrut \) \(879716\) \(\beta_{7}\mathstrut +\mathstrut \) \(220389\) \(\beta_{6}\mathstrut -\mathstrut \) \(470656\) \(\beta_{5}\mathstrut +\mathstrut \) \(19846\) \(\beta_{4}\mathstrut -\mathstrut \) \(96717\) \(\beta_{3}\mathstrut +\mathstrut \) \(2609286\) \(\beta_{2}\mathstrut +\mathstrut \) \(756043\) \(\beta_{1}\mathstrut +\mathstrut \) \(1344775\)

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
2.77821
2.62992
2.53461
1.99378
1.64567
1.59625
1.47280
1.44853
0.615823
0.498176
−0.239152
−0.540222
−0.823313
−0.926931
−2.08696
−2.08785
−2.10151
−2.40783
−2.77821 1.00000 5.71847 1.71746 −2.77821 −2.08361 −10.3307 1.00000 −4.77148
1.2 −2.62992 1.00000 4.91649 0.484241 −2.62992 0.944316 −7.67014 1.00000 −1.27352
1.3 −2.53461 1.00000 4.42426 −2.65587 −2.53461 1.73204 −6.14455 1.00000 6.73160
1.4 −1.99378 1.00000 1.97517 1.24008 −1.99378 −4.06333 0.0495043 1.00000 −2.47246
1.5 −1.64567 1.00000 0.708234 −0.419467 −1.64567 1.49188 2.12582 1.00000 0.690305
1.6 −1.59625 1.00000 0.548008 −4.15667 −1.59625 −4.26341 2.31774 1.00000 6.63507
1.7 −1.47280 1.00000 0.169132 4.03939 −1.47280 0.912411 2.69650 1.00000 −5.94920
1.8 −1.44853 1.00000 0.0982249 1.34868 −1.44853 −1.91511 2.75477 1.00000 −1.95360
1.9 −0.615823 1.00000 −1.62076 −3.91500 −0.615823 4.24139 2.22975 1.00000 2.41095
1.10 −0.498176 1.00000 −1.75182 2.85615 −0.498176 −0.756391 1.86907 1.00000 −1.42286
1.11 0.239152 1.00000 −1.94281 −0.863011 0.239152 −3.18045 −0.942930 1.00000 −0.206391
1.12 0.540222 1.00000 −1.70816 3.09244 0.540222 −3.94821 −2.00323 1.00000 1.67061
1.13 0.823313 1.00000 −1.32216 0.963284 0.823313 4.26360 −2.73517 1.00000 0.793084
1.14 0.926931 1.00000 −1.14080 −2.07435 0.926931 1.24239 −2.91130 1.00000 −1.92278
1.15 2.08696 1.00000 2.35542 −2.36854 2.08696 −0.791085 0.741746 1.00000 −4.94306
1.16 2.08785 1.00000 2.35911 −3.16874 2.08785 0.308283 0.749773 1.00000 −6.61585
1.17 2.10151 1.00000 2.41637 −0.968130 2.10151 −2.79477 0.874998 1.00000 −2.03454
1.18 2.40783 1.00000 3.79762 −0.151954 2.40783 −4.33995 4.32836 1.00000 −0.365880
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(17\) \(-1\)
\(79\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4029))\):

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