Properties

Label 1003.2.a.h
Level 1003
Weight 2
Character orbit 1003.a
Self dual Yes
Analytic conductor 8.009
Analytic rank 1
Dimension 16
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1003 = 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1003.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(8.00899532273\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{2} \) \( -\beta_{10} q^{3} \) \( + ( 1 + \beta_{2} ) q^{4} \) \( + ( -1 - \beta_{4} ) q^{5} \) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{5} ) q^{6} \) \( + ( -1 - \beta_{1} + \beta_{3} + \beta_{5} + \beta_{10} - \beta_{11} ) q^{7} \) \( + ( -2 \beta_{1} + \beta_{6} - \beta_{7} ) q^{8} \) \( + ( 1 + \beta_{4} + \beta_{11} ) q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{2} \) \( -\beta_{10} q^{3} \) \( + ( 1 + \beta_{2} ) q^{4} \) \( + ( -1 - \beta_{4} ) q^{5} \) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{5} ) q^{6} \) \( + ( -1 - \beta_{1} + \beta_{3} + \beta_{5} + \beta_{10} - \beta_{11} ) q^{7} \) \( + ( -2 \beta_{1} + \beta_{6} - \beta_{7} ) q^{8} \) \( + ( 1 + \beta_{4} + \beta_{11} ) q^{9} \) \( + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{14} ) q^{10} \) \( + ( -1 + 2 \beta_{1} - \beta_{3} - \beta_{5} - \beta_{9} + \beta_{11} + \beta_{13} ) q^{11} \) \( + ( -1 + 2 \beta_{1} + \beta_{5} + \beta_{7} + \beta_{11} ) q^{12} \) \( + ( -2 + \beta_{1} + \beta_{5} + \beta_{7} + \beta_{10} - \beta_{11} - \beta_{13} ) q^{13} \) \( + ( 1 + 2 \beta_{1} - \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{11} - \beta_{13} - \beta_{14} ) q^{14} \) \( + ( \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} + \beta_{10} + \beta_{11} - \beta_{14} ) q^{15} \) \( + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{9} - \beta_{12} - \beta_{14} ) q^{16} \) \(+ q^{17}\) \( + ( 1 + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{9} - \beta_{14} ) q^{18} \) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} + \beta_{9} + \beta_{10} - \beta_{11} + 2 \beta_{12} ) q^{19} \) \( + ( -3 + \beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{8} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{20} \) \( + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} + \beta_{9} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{21} \) \( + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{12} + 2 \beta_{14} ) q^{22} \) \( + ( \beta_{1} + \beta_{5} - \beta_{8} - \beta_{10} + \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{23} \) \( + ( -4 + 3 \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{6} + \beta_{7} + 2 \beta_{10} - \beta_{11} + \beta_{12} ) q^{24} \) \( + ( 1 + \beta_{2} + 2 \beta_{4} - \beta_{8} + \beta_{11} ) q^{25} \) \( + ( -3 + 2 \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} + 2 \beta_{11} + \beta_{13} - \beta_{14} ) q^{26} \) \( + ( -2 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} + \beta_{10} - 2 \beta_{11} - \beta_{13} + \beta_{14} ) q^{27} \) \( + ( -1 - 3 \beta_{1} + 2 \beta_{3} + \beta_{5} + 3 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} - \beta_{12} + 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{28} \) \( + ( -2 - \beta_{2} - \beta_{5} - \beta_{6} + \beta_{9} - \beta_{13} + \beta_{15} ) q^{29} \) \( + ( 2 - 4 \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} + 4 \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{11} - 2 \beta_{12} - \beta_{13} + \beta_{15} ) q^{30} \) \( + ( \beta_{2} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} - \beta_{12} - \beta_{13} ) q^{31} \) \( + ( 3 - 5 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{10} - \beta_{13} ) q^{32} \) \( + ( -1 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{6} - 2 \beta_{7} + 3 \beta_{10} - 3 \beta_{11} + 2 \beta_{12} + \beta_{14} - \beta_{15} ) q^{33} \) \( -\beta_{1} q^{34} \) \( + ( -1 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{13} - \beta_{15} ) q^{35} \) \( + ( 1 - 5 \beta_{1} + \beta_{3} + 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{11} - \beta_{13} + \beta_{14} ) q^{36} \) \( + ( -2 + \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{13} - \beta_{14} ) q^{37} \) \( + ( -3 + \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{9} + \beta_{11} + 2 \beta_{13} + \beta_{14} ) q^{38} \) \( + ( -1 - \beta_{1} + \beta_{2} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{13} ) q^{39} \) \( + ( -1 + 8 \beta_{1} - \beta_{3} - 3 \beta_{6} + 4 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{13} + \beta_{15} ) q^{40} \) \( + ( -1 - \beta_{3} + \beta_{6} + 2 \beta_{8} - 2 \beta_{10} - \beta_{12} ) q^{41} \) \( + ( -1 + \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{8} - \beta_{10} + \beta_{12} ) q^{42} \) \( + ( 2 - 3 \beta_{1} - \beta_{3} + \beta_{4} + \beta_{6} + \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - \beta_{13} + \beta_{15} ) q^{43} \) \( + ( -2 + 5 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + \beta_{11} - \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{44} \) \( + ( -4 - 3 \beta_{4} - \beta_{6} - 3 \beta_{11} - \beta_{13} ) q^{45} \) \( + ( -3 + \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{11} - \beta_{14} - \beta_{15} ) q^{46} \) \( + ( -5 - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{47} \) \( + ( -4 - 4 \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{6} + 2 \beta_{9} + 2 \beta_{10} - 3 \beta_{11} + \beta_{12} + \beta_{13} + 3 \beta_{14} ) q^{48} \) \( + ( 1 + 6 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + 4 \beta_{11} + \beta_{12} + \beta_{13} - 2 \beta_{14} ) q^{49} \) \( + ( 2 - 5 \beta_{1} + 2 \beta_{2} + \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{50} \) \( -\beta_{10} q^{51} \) \( + ( -2 + 5 \beta_{1} - 3 \beta_{2} + 3 \beta_{4} - \beta_{5} - 3 \beta_{6} - \beta_{8} + \beta_{9} - \beta_{11} + \beta_{14} ) q^{52} \) \( + ( -2 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{9} + \beta_{15} ) q^{53} \) \( + ( 1 - \beta_{1} - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} + \beta_{12} + 2 \beta_{13} - \beta_{15} ) q^{54} \) \( + ( 4 - 5 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{11} - 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{55} \) \( + ( 1 + 2 \beta_{1} - 3 \beta_{3} + \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{10} - \beta_{11} - 3 \beta_{13} - \beta_{14} + \beta_{15} ) q^{56} \) \( + ( 2 - \beta_{1} + 3 \beta_{2} - \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 3 \beta_{11} + \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{57} \) \( + ( 5 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} + 3 \beta_{7} + 2 \beta_{8} - 2 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{58} \) \(+ q^{59}\) \( + ( 4 - 5 \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + \beta_{6} - 4 \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{60} \) \( + ( -2 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} - 3 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{13} - \beta_{15} ) q^{61} \) \( + ( 2 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{8} - \beta_{9} - 3 \beta_{10} + 2 \beta_{11} - \beta_{12} - \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{62} \) \( + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} + 4 \beta_{5} - 4 \beta_{6} + 3 \beta_{7} - \beta_{8} - 2 \beta_{9} + 4 \beta_{10} - 3 \beta_{11} - 4 \beta_{13} + 2 \beta_{15} ) q^{63} \) \( + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{11} - \beta_{12} - \beta_{13} - 2 \beta_{14} ) q^{64} \) \( + ( -2 \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} - 3 \beta_{10} + 3 \beta_{11} - \beta_{12} + 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{65} \) \( + ( 3 + 3 \beta_{2} + \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} - 3 \beta_{8} - \beta_{9} + 2 \beta_{11} - 2 \beta_{12} + \beta_{13} - \beta_{14} ) q^{66} \) \( + ( 1 + \beta_{1} - 3 \beta_{3} + \beta_{5} + \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} - 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{67} \) \( + ( 1 + \beta_{2} ) q^{68} \) \( + ( 2 - 2 \beta_{1} + \beta_{2} + 3 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} - \beta_{12} - 2 \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{69} \) \( + ( 2 - 5 \beta_{1} + \beta_{2} + 4 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} - \beta_{7} - 3 \beta_{10} + \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{70} \) \( + ( -1 + 2 \beta_{4} - \beta_{5} - 2 \beta_{9} + \beta_{10} + 2 \beta_{11} + 2 \beta_{13} + \beta_{14} ) q^{71} \) \( + ( 1 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} + 2 \beta_{11} - \beta_{12} + 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{72} \) \( + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{10} + 3 \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{73} \) \( + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{7} - \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + \beta_{12} - 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{74} \) \( + ( -1 - 2 \beta_{1} + \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} + 4 \beta_{7} - 2 \beta_{8} - 3 \beta_{11} + \beta_{12} - \beta_{13} + 3 \beta_{14} ) q^{75} \) \( + ( -2 + 9 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 2 \beta_{5} - 4 \beta_{6} + 3 \beta_{7} + \beta_{11} - 2 \beta_{13} + \beta_{14} ) q^{76} \) \( + ( -3 - 3 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{8} - 2 \beta_{10} - \beta_{11} - \beta_{13} ) q^{77} \) \( + ( 3 - 4 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + 3 \beta_{10} - 3 \beta_{11} + 2 \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{78} \) \( + ( -2 - \beta_{1} + 2 \beta_{3} - 2 \beta_{5} - \beta_{6} - 3 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{14} ) q^{79} \) \( + ( -9 - 2 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} + 2 \beta_{12} + \beta_{13} + 3 \beta_{14} - \beta_{15} ) q^{80} \) \( + ( -6 + 6 \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{8} + \beta_{10} + 3 \beta_{11} + 3 \beta_{13} - 2 \beta_{14} ) q^{81} \) \( + ( 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} - 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{15} ) q^{82} \) \( + ( -5 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + 2 \beta_{7} + \beta_{9} + \beta_{10} + \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{83} \) \( + ( \beta_{1} + \beta_{3} + 2 \beta_{4} + 3 \beta_{5} - 4 \beta_{6} + 4 \beta_{7} - \beta_{8} - \beta_{9} + 4 \beta_{10} - 3 \beta_{11} - 2 \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{84} \) \( + ( -1 - \beta_{4} ) q^{85} \) \( + ( 5 + \beta_{1} + 5 \beta_{2} - 3 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 3 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} + \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{86} \) \( + ( 1 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} - 3 \beta_{12} - 2 \beta_{13} - 3 \beta_{14} ) q^{87} \) \( + ( -2 \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{88} \) \( + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + 2 \beta_{12} + \beta_{13} + 3 \beta_{14} - 2 \beta_{15} ) q^{89} \) \( + ( -3 + \beta_{1} - 3 \beta_{2} + \beta_{3} + 3 \beta_{4} + 4 \beta_{5} + \beta_{6} + 3 \beta_{9} - \beta_{12} + \beta_{13} + 3 \beta_{14} ) q^{90} \) \( + ( 4 - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{8} - 2 \beta_{10} + 3 \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{15} ) q^{91} \) \( + ( -3 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 3 \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{12} - \beta_{13} + \beta_{15} ) q^{92} \) \( + ( -3 - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{13} - \beta_{14} ) q^{93} \) \( + ( -1 + 8 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{14} + \beta_{15} ) q^{94} \) \( + ( 2 - 5 \beta_{1} + 3 \beta_{2} + \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} - 2 \beta_{10} + \beta_{11} - 2 \beta_{12} + \beta_{13} ) q^{95} \) \( + ( 9 \beta_{1} - \beta_{2} - 4 \beta_{3} - \beta_{5} - \beta_{6} + 3 \beta_{7} + \beta_{8} + \beta_{9} - 4 \beta_{10} + 4 \beta_{11} - \beta_{12} + \beta_{13} ) q^{96} \) \( + ( -1 - 3 \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} - 4 \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} + 3 \beta_{13} - 2 \beta_{15} ) q^{97} \) \( + ( -6 - 7 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} + \beta_{4} + 4 \beta_{5} + 2 \beta_{7} - \beta_{8} + \beta_{9} + 4 \beta_{10} - 5 \beta_{11} + \beta_{12} - \beta_{13} + 4 \beta_{14} + \beta_{15} ) q^{98} \) \( + ( -4 + 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + 3 \beta_{11} + 3 \beta_{13} - \beta_{14} - \beta_{15} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(16q \) \(\mathstrut -\mathstrut 6q^{2} \) \(\mathstrut -\mathstrut 7q^{3} \) \(\mathstrut +\mathstrut 14q^{4} \) \(\mathstrut -\mathstrut 21q^{5} \) \(\mathstrut -\mathstrut 5q^{6} \) \(\mathstrut -\mathstrut 11q^{7} \) \(\mathstrut -\mathstrut 12q^{8} \) \(\mathstrut +\mathstrut 17q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(16q \) \(\mathstrut -\mathstrut 6q^{2} \) \(\mathstrut -\mathstrut 7q^{3} \) \(\mathstrut +\mathstrut 14q^{4} \) \(\mathstrut -\mathstrut 21q^{5} \) \(\mathstrut -\mathstrut 5q^{6} \) \(\mathstrut -\mathstrut 11q^{7} \) \(\mathstrut -\mathstrut 12q^{8} \) \(\mathstrut +\mathstrut 17q^{9} \) \(\mathstrut +\mathstrut 12q^{10} \) \(\mathstrut -\mathstrut 7q^{11} \) \(\mathstrut -\mathstrut 4q^{12} \) \(\mathstrut -\mathstrut 16q^{13} \) \(\mathstrut +\mathstrut 11q^{14} \) \(\mathstrut +\mathstrut 7q^{15} \) \(\mathstrut +\mathstrut 22q^{16} \) \(\mathstrut +\mathstrut 16q^{17} \) \(\mathstrut +\mathstrut 11q^{18} \) \(\mathstrut +\mathstrut q^{19} \) \(\mathstrut -\mathstrut 43q^{20} \) \(\mathstrut -\mathstrut 8q^{21} \) \(\mathstrut -\mathstrut 18q^{22} \) \(\mathstrut -\mathstrut 8q^{23} \) \(\mathstrut -\mathstrut 39q^{24} \) \(\mathstrut +\mathstrut 23q^{25} \) \(\mathstrut -\mathstrut 49q^{26} \) \(\mathstrut -\mathstrut 7q^{27} \) \(\mathstrut -\mathstrut 15q^{28} \) \(\mathstrut -\mathstrut 39q^{29} \) \(\mathstrut +\mathstrut q^{30} \) \(\mathstrut -\mathstrut 3q^{31} \) \(\mathstrut -\mathstrut 15q^{33} \) \(\mathstrut -\mathstrut 6q^{34} \) \(\mathstrut +\mathstrut 9q^{35} \) \(\mathstrut -\mathstrut 17q^{36} \) \(\mathstrut -\mathstrut 28q^{37} \) \(\mathstrut -\mathstrut 27q^{38} \) \(\mathstrut -\mathstrut 4q^{39} \) \(\mathstrut +\mathstrut 26q^{40} \) \(\mathstrut -\mathstrut 31q^{41} \) \(\mathstrut -\mathstrut 45q^{42} \) \(\mathstrut +\mathstrut 5q^{43} \) \(\mathstrut -\mathstrut 19q^{44} \) \(\mathstrut -\mathstrut 79q^{45} \) \(\mathstrut -\mathstrut 39q^{46} \) \(\mathstrut -\mathstrut 47q^{47} \) \(\mathstrut -\mathstrut 31q^{48} \) \(\mathstrut +\mathstrut 35q^{49} \) \(\mathstrut -\mathstrut 13q^{50} \) \(\mathstrut -\mathstrut 7q^{51} \) \(\mathstrut +\mathstrut 9q^{52} \) \(\mathstrut -\mathstrut 36q^{53} \) \(\mathstrut +\mathstrut 9q^{54} \) \(\mathstrut +\mathstrut 32q^{55} \) \(\mathstrut +\mathstrut 3q^{56} \) \(\mathstrut +\mathstrut 6q^{57} \) \(\mathstrut +\mathstrut 22q^{58} \) \(\mathstrut +\mathstrut 16q^{59} \) \(\mathstrut +\mathstrut 6q^{60} \) \(\mathstrut -\mathstrut 22q^{61} \) \(\mathstrut +\mathstrut q^{62} \) \(\mathstrut -\mathstrut 19q^{63} \) \(\mathstrut +\mathstrut 32q^{64} \) \(\mathstrut -\mathstrut 19q^{65} \) \(\mathstrut +\mathstrut 48q^{66} \) \(\mathstrut +\mathstrut 4q^{67} \) \(\mathstrut +\mathstrut 14q^{68} \) \(\mathstrut +\mathstrut 14q^{69} \) \(\mathstrut -\mathstrut 11q^{70} \) \(\mathstrut -\mathstrut 5q^{71} \) \(\mathstrut +\mathstrut 40q^{72} \) \(\mathstrut -\mathstrut 35q^{73} \) \(\mathstrut +\mathstrut 31q^{74} \) \(\mathstrut -\mathstrut 12q^{75} \) \(\mathstrut +\mathstrut q^{76} \) \(\mathstrut -\mathstrut 79q^{77} \) \(\mathstrut +\mathstrut 31q^{78} \) \(\mathstrut -\mathstrut 48q^{79} \) \(\mathstrut -\mathstrut 127q^{80} \) \(\mathstrut -\mathstrut 28q^{81} \) \(\mathstrut -\mathstrut 7q^{82} \) \(\mathstrut -\mathstrut 42q^{83} \) \(\mathstrut +\mathstrut 28q^{84} \) \(\mathstrut -\mathstrut 21q^{85} \) \(\mathstrut +\mathstrut 58q^{86} \) \(\mathstrut -\mathstrut 16q^{87} \) \(\mathstrut -\mathstrut 2q^{88} \) \(\mathstrut +\mathstrut 20q^{89} \) \(\mathstrut -\mathstrut 14q^{90} \) \(\mathstrut +\mathstrut 49q^{91} \) \(\mathstrut -\mathstrut 21q^{92} \) \(\mathstrut -\mathstrut 55q^{93} \) \(\mathstrut -\mathstrut 22q^{95} \) \(\mathstrut +\mathstrut 24q^{96} \) \(\mathstrut -\mathstrut 2q^{97} \) \(\mathstrut -\mathstrut 78q^{98} \) \(\mathstrut -\mathstrut 35q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16}\mathstrut -\mathstrut \) \(6\) \(x^{15}\mathstrut -\mathstrut \) \(5\) \(x^{14}\mathstrut +\mathstrut \) \(90\) \(x^{13}\mathstrut -\mathstrut \) \(82\) \(x^{12}\mathstrut -\mathstrut \) \(456\) \(x^{11}\mathstrut +\mathstrut \) \(723\) \(x^{10}\mathstrut +\mathstrut \) \(951\) \(x^{9}\mathstrut -\mathstrut \) \(2105\) \(x^{8}\mathstrut -\mathstrut \) \(695\) \(x^{7}\mathstrut +\mathstrut \) \(2641\) \(x^{6}\mathstrut -\mathstrut \) \(151\) \(x^{5}\mathstrut -\mathstrut \) \(1323\) \(x^{4}\mathstrut +\mathstrut \) \(301\) \(x^{3}\mathstrut +\mathstrut \) \(179\) \(x^{2}\mathstrut -\mathstrut \) \(50\) \(x\mathstrut -\mathstrut \) \(1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(21607\) \(\nu^{15}\mathstrut +\mathstrut \) \(206853\) \(\nu^{14}\mathstrut -\mathstrut \) \(210704\) \(\nu^{13}\mathstrut -\mathstrut \) \(2846302\) \(\nu^{12}\mathstrut +\mathstrut \) \(6450296\) \(\nu^{11}\mathstrut +\mathstrut \) \(12562022\) \(\nu^{10}\mathstrut -\mathstrut \) \(38927099\) \(\nu^{9}\mathstrut -\mathstrut \) \(22572604\) \(\nu^{8}\mathstrut +\mathstrut \) \(96459971\) \(\nu^{7}\mathstrut +\mathstrut \) \(21890310\) \(\nu^{6}\mathstrut -\mathstrut \) \(111632131\) \(\nu^{5}\mathstrut -\mathstrut \) \(20768470\) \(\nu^{4}\mathstrut +\mathstrut \) \(58636299\) \(\nu^{3}\mathstrut +\mathstrut \) \(12947770\) \(\nu^{2}\mathstrut -\mathstrut \) \(12584729\) \(\nu\mathstrut -\mathstrut \) \(1629531\)\()/796522\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(46479\) \(\nu^{15}\mathstrut +\mathstrut \) \(448281\) \(\nu^{14}\mathstrut -\mathstrut \) \(494240\) \(\nu^{13}\mathstrut -\mathstrut \) \(6086430\) \(\nu^{12}\mathstrut +\mathstrut \) \(14620862\) \(\nu^{11}\mathstrut +\mathstrut \) \(25713044\) \(\nu^{10}\mathstrut -\mathstrut \) \(88715715\) \(\nu^{9}\mathstrut -\mathstrut \) \(38866840\) \(\nu^{8}\mathstrut +\mathstrut \) \(222583253\) \(\nu^{7}\mathstrut +\mathstrut \) \(17845342\) \(\nu^{6}\mathstrut -\mathstrut \) \(259608919\) \(\nu^{5}\mathstrut -\mathstrut \) \(6096132\) \(\nu^{4}\mathstrut +\mathstrut \) \(129950349\) \(\nu^{3}\mathstrut +\mathstrut \) \(6021412\) \(\nu^{2}\mathstrut -\mathstrut \) \(18751633\) \(\nu\mathstrut -\mathstrut \) \(252191\)\()/796522\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(73855\) \(\nu^{15}\mathstrut +\mathstrut \) \(331382\) \(\nu^{14}\mathstrut +\mathstrut \) \(929938\) \(\nu^{13}\mathstrut -\mathstrut \) \(5685137\) \(\nu^{12}\mathstrut -\mathstrut \) \(2256821\) \(\nu^{11}\mathstrut +\mathstrut \) \(35569812\) \(\nu^{10}\mathstrut -\mathstrut \) \(11490419\) \(\nu^{9}\mathstrut -\mathstrut \) \(102911265\) \(\nu^{8}\mathstrut +\mathstrut \) \(63824178\) \(\nu^{7}\mathstrut +\mathstrut \) \(140772231\) \(\nu^{6}\mathstrut -\mathstrut \) \(102838342\) \(\nu^{5}\mathstrut -\mathstrut \) \(81535535\) \(\nu^{4}\mathstrut +\mathstrut \) \(58146626\) \(\nu^{3}\mathstrut +\mathstrut \) \(13601600\) \(\nu^{2}\mathstrut -\mathstrut \) \(8034099\) \(\nu\mathstrut +\mathstrut \) \(621523\)\()/398261\)
\(\beta_{6}\)\(=\)\((\)\(169225\) \(\nu^{15}\mathstrut -\mathstrut \) \(821071\) \(\nu^{14}\mathstrut -\mathstrut \) \(1814188\) \(\nu^{13}\mathstrut +\mathstrut \) \(13610466\) \(\nu^{12}\mathstrut +\mathstrut \) \(173262\) \(\nu^{11}\mathstrut -\mathstrut \) \(80818546\) \(\nu^{10}\mathstrut +\mathstrut \) \(54087255\) \(\nu^{9}\mathstrut +\mathstrut \) \(216056280\) \(\nu^{8}\mathstrut -\mathstrut \) \(215179431\) \(\nu^{7}\mathstrut -\mathstrut \) \(258192220\) \(\nu^{6}\mathstrut +\mathstrut \) \(314110313\) \(\nu^{5}\mathstrut +\mathstrut \) \(109309352\) \(\nu^{4}\mathstrut -\mathstrut \) \(170098381\) \(\nu^{3}\mathstrut -\mathstrut \) \(1162256\) \(\nu^{2}\mathstrut +\mathstrut \) \(26785741\) \(\nu\mathstrut -\mathstrut \) \(1684399\)\()/796522\)
\(\beta_{7}\)\(=\)\((\)\(169225\) \(\nu^{15}\mathstrut -\mathstrut \) \(821071\) \(\nu^{14}\mathstrut -\mathstrut \) \(1814188\) \(\nu^{13}\mathstrut +\mathstrut \) \(13610466\) \(\nu^{12}\mathstrut +\mathstrut \) \(173262\) \(\nu^{11}\mathstrut -\mathstrut \) \(80818546\) \(\nu^{10}\mathstrut +\mathstrut \) \(54087255\) \(\nu^{9}\mathstrut +\mathstrut \) \(216056280\) \(\nu^{8}\mathstrut -\mathstrut \) \(215179431\) \(\nu^{7}\mathstrut -\mathstrut \) \(258192220\) \(\nu^{6}\mathstrut +\mathstrut \) \(314110313\) \(\nu^{5}\mathstrut +\mathstrut \) \(109309352\) \(\nu^{4}\mathstrut -\mathstrut \) \(169301859\) \(\nu^{3}\mathstrut -\mathstrut \) \(1162256\) \(\nu^{2}\mathstrut +\mathstrut \) \(22006609\) \(\nu\mathstrut -\mathstrut \) \(1684399\)\()/796522\)
\(\beta_{8}\)\(=\)\((\)\(171663\) \(\nu^{15}\mathstrut -\mathstrut \) \(912757\) \(\nu^{14}\mathstrut -\mathstrut \) \(1423874\) \(\nu^{13}\mathstrut +\mathstrut \) \(14140202\) \(\nu^{12}\mathstrut -\mathstrut \) \(4762104\) \(\nu^{11}\mathstrut -\mathstrut \) \(76771974\) \(\nu^{10}\mathstrut +\mathstrut \) \(68975265\) \(\nu^{9}\mathstrut +\mathstrut \) \(188185360\) \(\nu^{8}\mathstrut -\mathstrut \) \(212042053\) \(\nu^{7}\mathstrut -\mathstrut \) \(220121182\) \(\nu^{6}\mathstrut +\mathstrut \) \(263860137\) \(\nu^{5}\mathstrut +\mathstrut \) \(112170862\) \(\nu^{4}\mathstrut -\mathstrut \) \(126473895\) \(\nu^{3}\mathstrut -\mathstrut \) \(17602844\) \(\nu^{2}\mathstrut +\mathstrut \) \(12942671\) \(\nu\mathstrut +\mathstrut \) \(623575\)\()/796522\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(252191\) \(\nu^{15}\mathstrut +\mathstrut \) \(1466667\) \(\nu^{14}\mathstrut +\mathstrut \) \(1709236\) \(\nu^{13}\mathstrut -\mathstrut \) \(23191430\) \(\nu^{12}\mathstrut +\mathstrut \) \(14593232\) \(\nu^{11}\mathstrut +\mathstrut \) \(129619958\) \(\nu^{10}\mathstrut -\mathstrut \) \(156621049\) \(\nu^{9}\mathstrut -\mathstrut \) \(328549356\) \(\nu^{8}\mathstrut +\mathstrut \) \(491995215\) \(\nu^{7}\mathstrut +\mathstrut \) \(397855998\) \(\nu^{6}\mathstrut -\mathstrut \) \(648191089\) \(\nu^{5}\mathstrut -\mathstrut \) \(221528078\) \(\nu^{4}\mathstrut +\mathstrut \) \(327552561\) \(\nu^{3}\mathstrut +\mathstrut \) \(54040858\) \(\nu^{2}\mathstrut -\mathstrut \) \(39120777\) \(\nu\mathstrut -\mathstrut \) \(5345561\)\()/796522\)
\(\beta_{10}\)\(=\)\((\)\(174999\) \(\nu^{15}\mathstrut -\mathstrut \) \(976139\) \(\nu^{14}\mathstrut -\mathstrut \) \(1206377\) \(\nu^{13}\mathstrut +\mathstrut \) \(14819972\) \(\nu^{12}\mathstrut -\mathstrut \) \(8664781\) \(\nu^{11}\mathstrut -\mathstrut \) \(77542723\) \(\nu^{10}\mathstrut +\mathstrut \) \(90954465\) \(\nu^{9}\mathstrut +\mathstrut \) \(177914468\) \(\nu^{8}\mathstrut -\mathstrut \) \(265461630\) \(\nu^{7}\mathstrut -\mathstrut \) \(185448483\) \(\nu^{6}\mathstrut +\mathstrut \) \(321400128\) \(\nu^{5}\mathstrut +\mathstrut \) \(76413493\) \(\nu^{4}\mathstrut -\mathstrut \) \(149988142\) \(\nu^{3}\mathstrut -\mathstrut \) \(5471927\) \(\nu^{2}\mathstrut +\mathstrut \) \(17324960\) \(\nu\mathstrut -\mathstrut \) \(317590\)\()/398261\)
\(\beta_{11}\)\(=\)\((\)\(454985\) \(\nu^{15}\mathstrut -\mathstrut \) \(2624923\) \(\nu^{14}\mathstrut -\mathstrut \) \(2947570\) \(\nu^{13}\mathstrut +\mathstrut \) \(40413834\) \(\nu^{12}\mathstrut -\mathstrut \) \(26534880\) \(\nu^{11}\mathstrut -\mathstrut \) \(216678478\) \(\nu^{10}\mathstrut +\mathstrut \) \(267361123\) \(\nu^{9}\mathstrut +\mathstrut \) \(518142928\) \(\nu^{8}\mathstrut -\mathstrut \) \(795429433\) \(\nu^{7}\mathstrut -\mathstrut \) \(580720748\) \(\nu^{6}\mathstrut +\mathstrut \) \(991791603\) \(\nu^{5}\mathstrut +\mathstrut \) \(280288612\) \(\nu^{4}\mathstrut -\mathstrut \) \(474483529\) \(\nu^{3}\mathstrut -\mathstrut \) \(38353282\) \(\nu^{2}\mathstrut +\mathstrut \) \(54032887\) \(\nu\mathstrut -\mathstrut \) \(180195\)\()/796522\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(493455\) \(\nu^{15}\mathstrut +\mathstrut \) \(2751753\) \(\nu^{14}\mathstrut +\mathstrut \) \(3520236\) \(\nu^{13}\mathstrut -\mathstrut \) \(42280306\) \(\nu^{12}\mathstrut +\mathstrut \) \(22964484\) \(\nu^{11}\mathstrut +\mathstrut \) \(226708120\) \(\nu^{10}\mathstrut -\mathstrut \) \(252272051\) \(\nu^{9}\mathstrut -\mathstrut \) \(548042556\) \(\nu^{8}\mathstrut +\mathstrut \) \(755318627\) \(\nu^{7}\mathstrut +\mathstrut \) \(642026498\) \(\nu^{6}\mathstrut -\mathstrut \) \(949139277\) \(\nu^{5}\mathstrut -\mathstrut \) \(351211644\) \(\nu^{4}\mathstrut +\mathstrut \) \(472849827\) \(\nu^{3}\mathstrut +\mathstrut \) \(68918706\) \(\nu^{2}\mathstrut -\mathstrut \) \(60979649\) \(\nu\mathstrut -\mathstrut \) \(1226399\)\()/796522\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(710147\) \(\nu^{15}\mathstrut +\mathstrut \) \(3849819\) \(\nu^{14}\mathstrut +\mathstrut \) \(5665410\) \(\nu^{13}\mathstrut -\mathstrut \) \(60313288\) \(\nu^{12}\mathstrut +\mathstrut \) \(25543250\) \(\nu^{11}\mathstrut +\mathstrut \) \(332167660\) \(\nu^{10}\mathstrut -\mathstrut \) \(336831221\) \(\nu^{9}\mathstrut -\mathstrut \) \(822852954\) \(\nu^{8}\mathstrut +\mathstrut \) \(1066670989\) \(\nu^{7}\mathstrut +\mathstrut \) \(954414268\) \(\nu^{6}\mathstrut -\mathstrut \) \(1385054993\) \(\nu^{5}\mathstrut -\mathstrut \) \(466744348\) \(\nu^{4}\mathstrut +\mathstrut \) \(695675295\) \(\nu^{3}\mathstrut +\mathstrut \) \(66001372\) \(\nu^{2}\mathstrut -\mathstrut \) \(87539399\) \(\nu\mathstrut +\mathstrut \) \(303563\)\()/796522\)
\(\beta_{14}\)\(=\)\((\)\(360387\) \(\nu^{15}\mathstrut -\mathstrut \) \(1988496\) \(\nu^{14}\mathstrut -\mathstrut \) \(2756504\) \(\nu^{13}\mathstrut +\mathstrut \) \(31115804\) \(\nu^{12}\mathstrut -\mathstrut \) \(14693575\) \(\nu^{11}\mathstrut -\mathstrut \) \(171588528\) \(\nu^{10}\mathstrut +\mathstrut \) \(179552242\) \(\nu^{9}\mathstrut +\mathstrut \) \(430148838\) \(\nu^{8}\mathstrut -\mathstrut \) \(560595280\) \(\nu^{7}\mathstrut -\mathstrut \) \(521963732\) \(\nu^{6}\mathstrut +\mathstrut \) \(724676789\) \(\nu^{5}\mathstrut +\mathstrut \) \(293307769\) \(\nu^{4}\mathstrut -\mathstrut \) \(364544169\) \(\nu^{3}\mathstrut -\mathstrut \) \(62155134\) \(\nu^{2}\mathstrut +\mathstrut \) \(47365022\) \(\nu\mathstrut +\mathstrut \) \(1585084\)\()/398261\)
\(\beta_{15}\)\(=\)\((\)\(-\)\(488289\) \(\nu^{15}\mathstrut +\mathstrut \) \(2675091\) \(\nu^{14}\mathstrut +\mathstrut \) \(3747450\) \(\nu^{13}\mathstrut -\mathstrut \) \(41578626\) \(\nu^{12}\mathstrut +\mathstrut \) \(19291176\) \(\nu^{11}\mathstrut +\mathstrut \) \(226475122\) \(\nu^{10}\mathstrut -\mathstrut \) \(236146200\) \(\nu^{9}\mathstrut -\mathstrut \) \(554950973\) \(\nu^{8}\mathstrut +\mathstrut \) \(728495661\) \(\nu^{7}\mathstrut +\mathstrut \) \(644160968\) \(\nu^{6}\mathstrut -\mathstrut \) \(927386318\) \(\nu^{5}\mathstrut -\mathstrut \) \(329011326\) \(\nu^{4}\mathstrut +\mathstrut \) \(461460405\) \(\nu^{3}\mathstrut +\mathstrut \) \(57412699\) \(\nu^{2}\mathstrut -\mathstrut \) \(59282695\) \(\nu\mathstrut -\mathstrut \) \(1545655\)\()/398261\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(6\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(15\)
\(\nu^{5}\)\(=\)\(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(10\) \(\beta_{7}\mathstrut -\mathstrut \) \(9\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(41\) \(\beta_{1}\mathstrut -\mathstrut \) \(3\)
\(\nu^{6}\)\(=\)\(-\)\(12\) \(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{13}\mathstrut -\mathstrut \) \(11\) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(10\) \(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(3\) \(\beta_{5}\mathstrut +\mathstrut \) \(10\) \(\beta_{4}\mathstrut -\mathstrut \) \(11\) \(\beta_{3}\mathstrut +\mathstrut \) \(47\) \(\beta_{2}\mathstrut +\mathstrut \) \(11\) \(\beta_{1}\mathstrut +\mathstrut \) \(88\)
\(\nu^{7}\)\(=\)\(2\) \(\beta_{15}\mathstrut -\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(9\) \(\beta_{13}\mathstrut +\mathstrut \) \(14\) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(81\) \(\beta_{7}\mathstrut -\mathstrut \) \(73\) \(\beta_{6}\mathstrut -\mathstrut \) \(15\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(26\) \(\beta_{3}\mathstrut -\mathstrut \) \(13\) \(\beta_{2}\mathstrut +\mathstrut \) \(291\) \(\beta_{1}\mathstrut -\mathstrut \) \(38\)
\(\nu^{8}\)\(=\)\(\beta_{15}\mathstrut -\mathstrut \) \(107\) \(\beta_{14}\mathstrut -\mathstrut \) \(17\) \(\beta_{13}\mathstrut -\mathstrut \) \(91\) \(\beta_{12}\mathstrut +\mathstrut \) \(11\) \(\beta_{11}\mathstrut +\mathstrut \) \(4\) \(\beta_{10}\mathstrut -\mathstrut \) \(81\) \(\beta_{9}\mathstrut -\mathstrut \) \(30\) \(\beta_{8}\mathstrut -\mathstrut \) \(14\) \(\beta_{7}\mathstrut -\mathstrut \) \(19\) \(\beta_{6}\mathstrut -\mathstrut \) \(45\) \(\beta_{5}\mathstrut +\mathstrut \) \(80\) \(\beta_{4}\mathstrut -\mathstrut \) \(97\) \(\beta_{3}\mathstrut +\mathstrut \) \(317\) \(\beta_{2}\mathstrut +\mathstrut \) \(97\) \(\beta_{1}\mathstrut +\mathstrut \) \(560\)
\(\nu^{9}\)\(=\)\(30\) \(\beta_{15}\mathstrut -\mathstrut \) \(19\) \(\beta_{14}\mathstrut +\mathstrut \) \(54\) \(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(137\) \(\beta_{10}\mathstrut -\mathstrut \) \(13\) \(\beta_{9}\mathstrut -\mathstrut \) \(22\) \(\beta_{8}\mathstrut +\mathstrut \) \(612\) \(\beta_{7}\mathstrut -\mathstrut \) \(574\) \(\beta_{6}\mathstrut -\mathstrut \) \(160\) \(\beta_{5}\mathstrut +\mathstrut \) \(14\) \(\beta_{4}\mathstrut -\mathstrut \) \(251\) \(\beta_{3}\mathstrut -\mathstrut \) \(127\) \(\beta_{2}\mathstrut +\mathstrut \) \(2094\) \(\beta_{1}\mathstrut -\mathstrut \) \(348\)
\(\nu^{10}\)\(=\)\(22\) \(\beta_{15}\mathstrut -\mathstrut \) \(865\) \(\beta_{14}\mathstrut -\mathstrut \) \(201\) \(\beta_{13}\mathstrut -\mathstrut \) \(688\) \(\beta_{12}\mathstrut +\mathstrut \) \(92\) \(\beta_{11}\mathstrut +\mathstrut \) \(67\) \(\beta_{10}\mathstrut -\mathstrut \) \(613\) \(\beta_{9}\mathstrut -\mathstrut \) \(314\) \(\beta_{8}\mathstrut -\mathstrut \) \(132\) \(\beta_{7}\mathstrut -\mathstrut \) \(237\) \(\beta_{6}\mathstrut -\mathstrut \) \(473\) \(\beta_{5}\mathstrut +\mathstrut \) \(605\) \(\beta_{4}\mathstrut -\mathstrut \) \(806\) \(\beta_{3}\mathstrut +\mathstrut \) \(2160\) \(\beta_{2}\mathstrut +\mathstrut \) \(810\) \(\beta_{1}\mathstrut +\mathstrut \) \(3732\)
\(\nu^{11}\)\(=\)\(314\) \(\beta_{15}\mathstrut -\mathstrut \) \(237\) \(\beta_{14}\mathstrut +\mathstrut \) \(232\) \(\beta_{13}\mathstrut +\mathstrut \) \(19\) \(\beta_{12}\mathstrut -\mathstrut \) \(12\) \(\beta_{11}\mathstrut +\mathstrut \) \(1173\) \(\beta_{10}\mathstrut -\mathstrut \) \(119\) \(\beta_{9}\mathstrut -\mathstrut \) \(299\) \(\beta_{8}\mathstrut +\mathstrut \) \(4496\) \(\beta_{7}\mathstrut -\mathstrut \) \(4445\) \(\beta_{6}\mathstrut -\mathstrut \) \(1484\) \(\beta_{5}\mathstrut +\mathstrut \) \(152\) \(\beta_{4}\mathstrut -\mathstrut \) \(2178\) \(\beta_{3}\mathstrut -\mathstrut \) \(1105\) \(\beta_{2}\mathstrut +\mathstrut \) \(15170\) \(\beta_{1}\mathstrut -\mathstrut \) \(2828\)
\(\nu^{12}\)\(=\)\(299\) \(\beta_{15}\mathstrut -\mathstrut \) \(6703\) \(\beta_{14}\mathstrut -\mathstrut \) \(2027\) \(\beta_{13}\mathstrut -\mathstrut \) \(5027\) \(\beta_{12}\mathstrut +\mathstrut \) \(711\) \(\beta_{11}\mathstrut +\mathstrut \) \(754\) \(\beta_{10}\mathstrut -\mathstrut \) \(4506\) \(\beta_{9}\mathstrut -\mathstrut \) \(2868\) \(\beta_{8}\mathstrut -\mathstrut \) \(1052\) \(\beta_{7}\mathstrut -\mathstrut \) \(2474\) \(\beta_{6}\mathstrut -\mathstrut \) \(4314\) \(\beta_{5}\mathstrut +\mathstrut \) \(4514\) \(\beta_{4}\mathstrut -\mathstrut \) \(6531\) \(\beta_{3}\mathstrut +\mathstrut \) \(14860\) \(\beta_{2}\mathstrut +\mathstrut \) \(6644\) \(\beta_{1}\mathstrut +\mathstrut \) \(25562\)
\(\nu^{13}\)\(=\)\(2868\) \(\beta_{15}\mathstrut -\mathstrut \) \(2464\) \(\beta_{14}\mathstrut +\mathstrut \) \(304\) \(\beta_{13}\mathstrut +\mathstrut \) \(211\) \(\beta_{12}\mathstrut -\mathstrut \) \(87\) \(\beta_{11}\mathstrut +\mathstrut \) \(9428\) \(\beta_{10}\mathstrut -\mathstrut \) \(960\) \(\beta_{9}\mathstrut -\mathstrut \) \(3294\) \(\beta_{8}\mathstrut +\mathstrut \) \(32663\) \(\beta_{7}\mathstrut -\mathstrut \) \(34104\) \(\beta_{6}\mathstrut -\mathstrut \) \(12770\) \(\beta_{5}\mathstrut +\mathstrut \) \(1525\) \(\beta_{4}\mathstrut -\mathstrut \) \(17978\) \(\beta_{3}\mathstrut -\mathstrut \) \(9022\) \(\beta_{2}\mathstrut +\mathstrut \) \(110347\) \(\beta_{1}\mathstrut -\mathstrut \) \(21700\)
\(\nu^{14}\)\(=\)\(3294\) \(\beta_{15}\mathstrut -\mathstrut \) \(50861\) \(\beta_{14}\mathstrut -\mathstrut \) \(18715\) \(\beta_{13}\mathstrut -\mathstrut \) \(36261\) \(\beta_{12}\mathstrut +\mathstrut \) \(5341\) \(\beta_{11}\mathstrut +\mathstrut \) \(7218\) \(\beta_{10}\mathstrut -\mathstrut \) \(32672\) \(\beta_{9}\mathstrut -\mathstrut \) \(24544\) \(\beta_{8}\mathstrut -\mathstrut \) \(7599\) \(\beta_{7}\mathstrut -\mathstrut \) \(23460\) \(\beta_{6}\mathstrut -\mathstrut \) \(36587\) \(\beta_{5}\mathstrut +\mathstrut \) \(33654\) \(\beta_{4}\mathstrut -\mathstrut \) \(52104\) \(\beta_{3}\mathstrut +\mathstrut \) \(103054\) \(\beta_{2}\mathstrut +\mathstrut \) \(54037\) \(\beta_{1}\mathstrut +\mathstrut \) \(178066\)
\(\nu^{15}\)\(=\)\(24544\) \(\beta_{15}\mathstrut -\mathstrut \) \(23195\) \(\beta_{14}\mathstrut -\mathstrut \) \(8241\) \(\beta_{13}\mathstrut +\mathstrut \) \(1770\) \(\beta_{12}\mathstrut -\mathstrut \) \(452\) \(\beta_{11}\mathstrut +\mathstrut \) \(73300\) \(\beta_{10}\mathstrut -\mathstrut \) \(7352\) \(\beta_{9}\mathstrut -\mathstrut \) \(32375\) \(\beta_{8}\mathstrut +\mathstrut \) \(236445\) \(\beta_{7}\mathstrut -\mathstrut \) \(260023\) \(\beta_{6}\mathstrut -\mathstrut \) \(105031\) \(\beta_{5}\mathstrut +\mathstrut \) \(14633\) \(\beta_{4}\mathstrut -\mathstrut \) \(144381\) \(\beta_{3}\mathstrut -\mathstrut \) \(70781\) \(\beta_{2}\mathstrut +\mathstrut \) \(804918\) \(\beta_{1}\mathstrut -\mathstrut \) \(161242\)

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
2.74400
2.52760
2.33579
1.98811
1.79724
1.62876
0.886486
0.431919
0.377947
−0.0187805
−0.446420
−0.896165
−1.35393
−1.38820
−1.94258
−2.67178
−2.74400 −0.794276 5.52956 −3.63969 2.17950 −4.59580 −9.68512 −2.36913 9.98732
1.2 −2.52760 −0.131380 4.38874 −1.05573 0.332076 3.96203 −6.03776 −2.98274 2.66845
1.3 −2.33579 2.58736 3.45592 −4.30332 −6.04353 1.79793 −3.40073 3.69442 10.0517
1.4 −1.98811 1.88888 1.95260 1.43913 −3.75530 −4.79587 0.0942457 0.567851 −2.86115
1.5 −1.79724 −2.09283 1.23006 −2.26932 3.76130 −0.408229 1.38377 1.37993 4.07850
1.6 −1.62876 −1.50871 0.652867 1.81397 2.45732 −1.88099 2.19416 −0.723806 −2.95453
1.7 −0.886486 −3.35617 −1.21414 −3.79855 2.97520 −4.56031 2.84929 8.26386 3.36736
1.8 −0.431919 0.385290 −1.81345 −0.726091 −0.166414 1.71462 1.64710 −2.85155 0.313613
1.9 −0.377947 −1.60628 −1.85716 0.658559 0.607088 −1.52100 1.45780 −0.419863 −0.248900
1.10 0.0187805 1.61675 −1.99965 −1.12710 0.0303633 −0.878091 −0.0751153 −0.386118 −0.0211674
1.11 0.446420 −3.01975 −1.80071 −2.85809 −1.34808 4.83764 −1.69671 6.11887 −1.27591
1.12 0.896165 1.99385 −1.19689 −0.516029 1.78681 −1.20609 −2.86494 0.975426 −0.462447
1.13 1.35393 −1.79865 −0.166881 1.77529 −2.43525 2.41510 −2.93380 0.235159 2.40362
1.14 1.38820 2.64276 −0.0729129 −3.88583 3.66866 −3.47551 −2.87761 3.98417 −5.39429
1.15 1.94258 −1.53769 1.77364 1.30751 −2.98709 −3.17249 −0.439731 −0.635509 2.53995
1.16 2.67178 −2.26915 5.13841 −3.81472 −6.06267 0.767070 8.38515 2.14904 −10.1921
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(17\) \(-1\)
\(59\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1003))\):

\(T_{2}^{16} + \cdots\)
\(T_{3}^{16} + \cdots\)