Properties

Label 1003.2.a.i
Level 1003
Weight 2
Character orbit 1003.a
Self dual Yes
Analytic conductor 8.009
Analytic rank 0
Dimension 18
CM No
Inner twists 1

Related objects

Downloads

Learn more about

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: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18}\mathstrut -\mathstrut \) \(5\) \(x^{17}\mathstrut -\mathstrut \) \(16\) \(x^{16}\mathstrut +\mathstrut \) \(112\) \(x^{15}\mathstrut +\mathstrut \) \(52\) \(x^{14}\mathstrut -\mathstrut \) \(984\) \(x^{13}\mathstrut +\mathstrut \) \(431\) \(x^{12}\mathstrut +\mathstrut \) \(4304\) \(x^{11}\mathstrut -\mathstrut \) \(3825\) \(x^{10}\mathstrut -\mathstrut \) \(9826\) \(x^{9}\mathstrut +\mathstrut \) \(11533\) \(x^{8}\mathstrut +\mathstrut \) \(11182\) \(x^{7}\mathstrut -\mathstrut \) \(15697\) \(x^{6}\mathstrut -\mathstrut \) \(5604\) \(x^{5}\mathstrut +\mathstrut \) \(9201\) \(x^{4}\mathstrut +\mathstrut \) \(1189\) \(x^{3}\mathstrut -\mathstrut \) \(1952\) \(x^{2}\mathstrut -\mathstrut \) \(100\) \(x\mathstrut +\mathstrut \) \(28\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\(1603205\) \(\nu^{17}\mathstrut +\mathstrut \) \(27603530\) \(\nu^{16}\mathstrut -\mathstrut \) \(195692906\) \(\nu^{15}\mathstrut -\mathstrut \) \(437835062\) \(\nu^{14}\mathstrut +\mathstrut \) \(3840968938\) \(\nu^{13}\mathstrut +\mathstrut \) \(1553634958\) \(\nu^{12}\mathstrut -\mathstrut \) \(31655264755\) \(\nu^{11}\mathstrut +\mathstrut \) \(8453508555\) \(\nu^{10}\mathstrut +\mathstrut \) \(130824516472\) \(\nu^{9}\mathstrut -\mathstrut \) \(75963419650\) \(\nu^{8}\mathstrut -\mathstrut \) \(276891915821\) \(\nu^{7}\mathstrut +\mathstrut \) \(202318455723\) \(\nu^{6}\mathstrut +\mathstrut \) \(273476087608\) \(\nu^{5}\mathstrut -\mathstrut \) \(212478348524\) \(\nu^{4}\mathstrut -\mathstrut \) \(93338201887\) \(\nu^{3}\mathstrut +\mathstrut \) \(64993586656\) \(\nu^{2}\mathstrut +\mathstrut \) \(5242626976\) \(\nu\mathstrut -\mathstrut \) \(1040316564\)\()/\)\(568793824\)
\(\beta_{4}\)\(=\)\((\)\(4468293\) \(\nu^{17}\mathstrut -\mathstrut \) \(9046678\) \(\nu^{16}\mathstrut -\mathstrut \) \(104332442\) \(\nu^{15}\mathstrut +\mathstrut \) \(172115546\) \(\nu^{14}\mathstrut +\mathstrut \) \(1061307674\) \(\nu^{13}\mathstrut -\mathstrut \) \(1245763682\) \(\nu^{12}\mathstrut -\mathstrut \) \(6337233987\) \(\nu^{11}\mathstrut +\mathstrut \) \(4626655131\) \(\nu^{10}\mathstrut +\mathstrut \) \(23993616376\) \(\nu^{9}\mathstrut -\mathstrut \) \(11457941330\) \(\nu^{8}\mathstrut -\mathstrut \) \(54362020477\) \(\nu^{7}\mathstrut +\mathstrut \) \(23150614939\) \(\nu^{6}\mathstrut +\mathstrut \) \(63467758040\) \(\nu^{5}\mathstrut -\mathstrut \) \(30360885660\) \(\nu^{4}\mathstrut -\mathstrut \) \(29262232015\) \(\nu^{3}\mathstrut +\mathstrut \) \(16437157520\) \(\nu^{2}\mathstrut +\mathstrut \) \(2591601760\) \(\nu\mathstrut -\mathstrut \) \(1680834164\)\()/\)\(568793824\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(6526647\) \(\nu^{17}\mathstrut +\mathstrut \) \(84688962\) \(\nu^{16}\mathstrut -\mathstrut \) \(83459586\) \(\nu^{15}\mathstrut -\mathstrut \) \(1781428510\) \(\nu^{14}\mathstrut +\mathstrut \) \(3847479106\) \(\nu^{13}\mathstrut +\mathstrut \) \(14100829750\) \(\nu^{12}\mathstrut -\mathstrut \) \(39180669471\) \(\nu^{11}\mathstrut -\mathstrut \) \(50706085177\) \(\nu^{10}\mathstrut +\mathstrut \) \(180240532008\) \(\nu^{9}\mathstrut +\mathstrut \) \(72173081990\) \(\nu^{8}\mathstrut -\mathstrut \) \(411676152609\) \(\nu^{7}\mathstrut +\mathstrut \) \(12535086055\) \(\nu^{6}\mathstrut +\mathstrut \) \(440305730504\) \(\nu^{5}\mathstrut -\mathstrut \) \(99577814860\) \(\nu^{4}\mathstrut -\mathstrut \) \(176724978555\) \(\nu^{3}\mathstrut +\mathstrut \) \(33075248352\) \(\nu^{2}\mathstrut +\mathstrut \) \(19243183584\) \(\nu\mathstrut +\mathstrut \) \(3061678300\)\()/\)\(568793824\)
\(\beta_{6}\)\(=\)\((\)\(7745729\) \(\nu^{17}\mathstrut -\mathstrut \) \(11556334\) \(\nu^{16}\mathstrut -\mathstrut \) \(262206418\) \(\nu^{15}\mathstrut +\mathstrut \) \(431026258\) \(\nu^{14}\mathstrut +\mathstrut \) \(3455427122\) \(\nu^{13}\mathstrut -\mathstrut \) \(6013468890\) \(\nu^{12}\mathstrut -\mathstrut \) \(22957637767\) \(\nu^{11}\mathstrut +\mathstrut \) \(41448412623\) \(\nu^{10}\mathstrut +\mathstrut \) \(82189416168\) \(\nu^{9}\mathstrut -\mathstrut \) \(152435041482\) \(\nu^{8}\mathstrut -\mathstrut \) \(154717191481\) \(\nu^{7}\mathstrut +\mathstrut \) \(295042593455\) \(\nu^{6}\mathstrut +\mathstrut \) \(132442588328\) \(\nu^{5}\mathstrut -\mathstrut \) \(267468538284\) \(\nu^{4}\mathstrut -\mathstrut \) \(28420424579\) \(\nu^{3}\mathstrut +\mathstrut \) \(78337756256\) \(\nu^{2}\mathstrut -\mathstrut \) \(4274427680\) \(\nu\mathstrut -\mathstrut \) \(599729124\)\()/\)\(568793824\)
\(\beta_{7}\)\(=\)\((\)\(6002425\) \(\nu^{17}\mathstrut -\mathstrut \) \(5938506\) \(\nu^{16}\mathstrut -\mathstrut \) \(196373670\) \(\nu^{15}\mathstrut +\mathstrut \) \(220312190\) \(\nu^{14}\mathstrut +\mathstrut \) \(2543960726\) \(\nu^{13}\mathstrut -\mathstrut \) \(3055625518\) \(\nu^{12}\mathstrut -\mathstrut \) \(16838568259\) \(\nu^{11}\mathstrut +\mathstrut \) \(20921021015\) \(\nu^{10}\mathstrut +\mathstrut \) \(60917184580\) \(\nu^{9}\mathstrut -\mathstrut \) \(76407838686\) \(\nu^{8}\mathstrut -\mathstrut \) \(118650382749\) \(\nu^{7}\mathstrut +\mathstrut \) \(147120641455\) \(\nu^{6}\mathstrut +\mathstrut \) \(112629948452\) \(\nu^{5}\mathstrut -\mathstrut \) \(133664775808\) \(\nu^{4}\mathstrut -\mathstrut \) \(40703624959\) \(\nu^{3}\mathstrut +\mathstrut \) \(40558943600\) \(\nu^{2}\mathstrut +\mathstrut \) \(4552577048\) \(\nu\mathstrut -\mathstrut \) \(736631868\)\()/\)\(142198456\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(30052531\) \(\nu^{17}\mathstrut +\mathstrut \) \(84099226\) \(\nu^{16}\mathstrut +\mathstrut \) \(722209126\) \(\nu^{15}\mathstrut -\mathstrut \) \(2030406822\) \(\nu^{14}\mathstrut -\mathstrut \) \(6987875910\) \(\nu^{13}\mathstrut +\mathstrut \) \(19790303422\) \(\nu^{12}\mathstrut +\mathstrut \) \(34840412021\) \(\nu^{11}\mathstrut -\mathstrut \) \(100266293789\) \(\nu^{10}\mathstrut -\mathstrut \) \(94156691400\) \(\nu^{9}\mathstrut +\mathstrut \) \(282941145998\) \(\nu^{8}\mathstrut +\mathstrut \) \(127822374123\) \(\nu^{7}\mathstrut -\mathstrut \) \(438843100125\) \(\nu^{6}\mathstrut -\mathstrut \) \(55429121448\) \(\nu^{5}\mathstrut +\mathstrut \) \(336609343060\) \(\nu^{4}\mathstrut -\mathstrut \) \(33137195863\) \(\nu^{3}\mathstrut -\mathstrut \) \(91595317216\) \(\nu^{2}\mathstrut +\mathstrut \) \(19664554208\) \(\nu\mathstrut +\mathstrut \) \(1835590924\)\()/\)\(568793824\)
\(\beta_{9}\)\(=\)\((\)\(33160165\) \(\nu^{17}\mathstrut -\mathstrut \) \(186790710\) \(\nu^{16}\mathstrut -\mathstrut \) \(379778058\) \(\nu^{15}\mathstrut +\mathstrut \) \(3887312362\) \(\nu^{14}\mathstrut -\mathstrut \) \(1510725526\) \(\nu^{13}\mathstrut -\mathstrut \) \(30205793170\) \(\nu^{12}\mathstrut +\mathstrut \) \(40801027885\) \(\nu^{11}\mathstrut +\mathstrut \) \(104573869707\) \(\nu^{10}\mathstrut -\mathstrut \) \(229270906824\) \(\nu^{9}\mathstrut -\mathstrut \) \(131568240834\) \(\nu^{8}\mathstrut +\mathstrut \) \(562653989427\) \(\nu^{7}\mathstrut -\mathstrut \) \(77018484181\) \(\nu^{6}\mathstrut -\mathstrut \) \(608004675912\) \(\nu^{5}\mathstrut +\mathstrut \) \(259080908628\) \(\nu^{4}\mathstrut +\mathstrut \) \(225691904993\) \(\nu^{3}\mathstrut -\mathstrut \) \(93309825920\) \(\nu^{2}\mathstrut -\mathstrut \) \(16222142464\) \(\nu\mathstrut -\mathstrut \) \(1088955028\)\()/\)\(568793824\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(37154163\) \(\nu^{17}\mathstrut +\mathstrut \) \(184167610\) \(\nu^{16}\mathstrut +\mathstrut \) \(566863078\) \(\nu^{15}\mathstrut -\mathstrut \) \(3965573350\) \(\nu^{14}\mathstrut -\mathstrut \) \(1494181414\) \(\nu^{13}\mathstrut +\mathstrut \) \(32718727454\) \(\nu^{12}\mathstrut -\mathstrut \) \(17567079211\) \(\nu^{11}\mathstrut -\mathstrut \) \(128256252797\) \(\nu^{10}\mathstrut +\mathstrut \) \(133661164920\) \(\nu^{9}\mathstrut +\mathstrut \) \(234252289166\) \(\nu^{8}\mathstrut -\mathstrut \) \(352535542229\) \(\nu^{7}\mathstrut -\mathstrut \) \(138565934845\) \(\nu^{6}\mathstrut +\mathstrut \) \(380890440888\) \(\nu^{5}\mathstrut -\mathstrut \) \(65264158156\) \(\nu^{4}\mathstrut -\mathstrut \) \(129377105239\) \(\nu^{3}\mathstrut +\mathstrut \) \(49161902080\) \(\nu^{2}\mathstrut +\mathstrut \) \(7531339520\) \(\nu\mathstrut -\mathstrut \) \(1527210676\)\()/\)\(568793824\)
\(\beta_{11}\)\(=\)\((\)\(6040615\) \(\nu^{17}\mathstrut -\mathstrut \) \(22234052\) \(\nu^{16}\mathstrut -\mathstrut \) \(122475798\) \(\nu^{15}\mathstrut +\mathstrut \) \(499734910\) \(\nu^{14}\mathstrut +\mathstrut \) \(913780746\) \(\nu^{13}\mathstrut -\mathstrut \) \(4407747754\) \(\nu^{12}\mathstrut -\mathstrut \) \(2941782421\) \(\nu^{11}\mathstrut +\mathstrut \) \(19364549555\) \(\nu^{10}\mathstrut +\mathstrut \) \(3047670594\) \(\nu^{9}\mathstrut -\mathstrut \) \(44421256370\) \(\nu^{8}\mathstrut +\mathstrut \) \(2692357581\) \(\nu^{7}\mathstrut +\mathstrut \) \(50814226103\) \(\nu^{6}\mathstrut -\mathstrut \) \(3464216802\) \(\nu^{5}\mathstrut -\mathstrut \) \(25574527844\) \(\nu^{4}\mathstrut -\mathstrut \) \(4369624325\) \(\nu^{3}\mathstrut +\mathstrut \) \(5272197858\) \(\nu^{2}\mathstrut +\mathstrut \) \(2393923808\) \(\nu\mathstrut -\mathstrut \) \(231904632\)\()/71099228\)
\(\beta_{12}\)\(=\)\((\)\(6053175\) \(\nu^{17}\mathstrut -\mathstrut \) \(28282200\) \(\nu^{16}\mathstrut -\mathstrut \) \(97976644\) \(\nu^{15}\mathstrut +\mathstrut \) \(613684104\) \(\nu^{14}\mathstrut +\mathstrut \) \(360506880\) \(\nu^{13}\mathstrut -\mathstrut \) \(5128036560\) \(\nu^{12}\mathstrut +\mathstrut \) \(1941081393\) \(\nu^{11}\mathstrut +\mathstrut \) \(20581292633\) \(\nu^{10}\mathstrut -\mathstrut \) \(18409013782\) \(\nu^{9}\mathstrut -\mathstrut \) \(39689575104\) \(\nu^{8}\mathstrut +\mathstrut \) \(51907281475\) \(\nu^{7}\mathstrut +\mathstrut \) \(29244763413\) \(\nu^{6}\mathstrut -\mathstrut \) \(59382333630\) \(\nu^{5}\mathstrut +\mathstrut \) \(1840425726\) \(\nu^{4}\mathstrut +\mathstrut \) \(22352348333\) \(\nu^{3}\mathstrut -\mathstrut \) \(4229868864\) \(\nu^{2}\mathstrut -\mathstrut \) \(1770614060\) \(\nu\mathstrut -\mathstrut \) \(64456832\)\()/71099228\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(25731019\) \(\nu^{17}\mathstrut +\mathstrut \) \(132425074\) \(\nu^{16}\mathstrut +\mathstrut \) \(392850758\) \(\nu^{15}\mathstrut -\mathstrut \) \(2917574854\) \(\nu^{14}\mathstrut -\mathstrut \) \(986596822\) \(\nu^{13}\mathstrut +\mathstrut \) \(24952854334\) \(\nu^{12}\mathstrut -\mathstrut \) \(13174676275\) \(\nu^{11}\mathstrut -\mathstrut \) \(104117203501\) \(\nu^{10}\mathstrut +\mathstrut \) \(100062459136\) \(\nu^{9}\mathstrut +\mathstrut \) \(216787275502\) \(\nu^{8}\mathstrut -\mathstrut \) \(269465073261\) \(\nu^{7}\mathstrut -\mathstrut \) \(198840292381\) \(\nu^{6}\mathstrut +\mathstrut \) \(303110008560\) \(\nu^{5}\mathstrut +\mathstrut \) \(47579502788\) \(\nu^{4}\mathstrut -\mathstrut \) \(113857594079\) \(\nu^{3}\mathstrut +\mathstrut \) \(4349495704\) \(\nu^{2}\mathstrut +\mathstrut \) \(8570974544\) \(\nu\mathstrut -\mathstrut \) \(301716580\)\()/\)\(284396912\)
\(\beta_{14}\)\(=\)\((\)\(-\)\(27616971\) \(\nu^{17}\mathstrut +\mathstrut \) \(102983154\) \(\nu^{16}\mathstrut +\mathstrut \) \(576714958\) \(\nu^{15}\mathstrut -\mathstrut \) \(2389076942\) \(\nu^{14}\mathstrut -\mathstrut \) \(4512375294\) \(\nu^{13}\mathstrut +\mathstrut \) \(22083367798\) \(\nu^{12}\mathstrut +\mathstrut \) \(15817530869\) \(\nu^{11}\mathstrut -\mathstrut \) \(104218462469\) \(\nu^{10}\mathstrut -\mathstrut \) \(20090353488\) \(\nu^{9}\mathstrut +\mathstrut \) \(267515584966\) \(\nu^{8}\mathstrut -\mathstrut \) \(15371453797\) \(\nu^{7}\mathstrut -\mathstrut \) \(366506974869\) \(\nu^{6}\mathstrut +\mathstrut \) \(56580628080\) \(\nu^{5}\mathstrut +\mathstrut \) \(242495850636\) \(\nu^{4}\mathstrut -\mathstrut \) \(35961353735\) \(\nu^{3}\mathstrut -\mathstrut \) \(59079830624\) \(\nu^{2}\mathstrut +\mathstrut \) \(5349825328\) \(\nu\mathstrut +\mathstrut \) \(656205004\)\()/\)\(284396912\)
\(\beta_{15}\)\(=\)\((\)\(-\)\(55864183\) \(\nu^{17}\mathstrut +\mathstrut \) \(316018786\) \(\nu^{16}\mathstrut +\mathstrut \) \(711941006\) \(\nu^{15}\mathstrut -\mathstrut \) \(6767846030\) \(\nu^{14}\mathstrut +\mathstrut \) \(963704754\) \(\nu^{13}\mathstrut +\mathstrut \) \(55256076454\) \(\nu^{12}\mathstrut -\mathstrut \) \(55249056879\) \(\nu^{11}\mathstrut -\mathstrut \) \(211764044169\) \(\nu^{10}\mathstrut +\mathstrut \) \(329590031528\) \(\nu^{9}\mathstrut +\mathstrut \) \(364158983542\) \(\nu^{8}\mathstrut -\mathstrut \) \(825445861681\) \(\nu^{7}\mathstrut -\mathstrut \) \(153858197321\) \(\nu^{6}\mathstrut +\mathstrut \) \(895533906920\) \(\nu^{5}\mathstrut -\mathstrut \) \(189752917180\) \(\nu^{4}\mathstrut -\mathstrut \) \(324651015307\) \(\nu^{3}\mathstrut +\mathstrut \) \(99470541936\) \(\nu^{2}\mathstrut +\mathstrut \) \(21482869568\) \(\nu\mathstrut -\mathstrut \) \(1512586372\)\()/\)\(568793824\)
\(\beta_{16}\)\(=\)\((\)\(-\)\(55930331\) \(\nu^{17}\mathstrut +\mathstrut \) \(273896826\) \(\nu^{16}\mathstrut +\mathstrut \) \(890033958\) \(\nu^{15}\mathstrut -\mathstrut \) \(6007265254\) \(\nu^{14}\mathstrut -\mathstrut \) \(3034501318\) \(\nu^{13}\mathstrut +\mathstrut \) \(51151472350\) \(\nu^{12}\mathstrut -\mathstrut \) \(20012118563\) \(\nu^{11}\mathstrut -\mathstrut \) \(212861062133\) \(\nu^{10}\mathstrut +\mathstrut \) \(176131301896\) \(\nu^{9}\mathstrut +\mathstrut \) \(445032492334\) \(\nu^{8}\mathstrut -\mathstrut \) \(484568126045\) \(\nu^{7}\mathstrut -\mathstrut \) \(420831199701\) \(\nu^{6}\mathstrut +\mathstrut \) \(542752259080\) \(\nu^{5}\mathstrut +\mathstrut \) \(125519891236\) \(\nu^{4}\mathstrut -\mathstrut \) \(197884162319\) \(\nu^{3}\mathstrut -\mathstrut \) \(7738166112\) \(\nu^{2}\mathstrut +\mathstrut \) \(11706378208\) \(\nu\mathstrut +\mathstrut \) \(1077401004\)\()/\)\(568793824\)
\(\beta_{17}\)\(=\)\((\)\(49926317\) \(\nu^{17}\mathstrut -\mathstrut \) \(203107614\) \(\nu^{16}\mathstrut -\mathstrut \) \(934344946\) \(\nu^{15}\mathstrut +\mathstrut \) \(4510183266\) \(\nu^{14}\mathstrut +\mathstrut \) \(5763599234\) \(\nu^{13}\mathstrut -\mathstrut \) \(39016219162\) \(\nu^{12}\mathstrut -\mathstrut \) \(8103174819\) \(\nu^{11}\mathstrut +\mathstrut \) \(165748033427\) \(\nu^{10}\mathstrut -\mathstrut \) \(48552214560\) \(\nu^{9}\mathstrut -\mathstrut \) \(356503271578\) \(\nu^{8}\mathstrut +\mathstrut \) \(199306839139\) \(\nu^{7}\mathstrut +\mathstrut \) \(352769386931\) \(\nu^{6}\mathstrut -\mathstrut \) \(244328770576\) \(\nu^{5}\mathstrut -\mathstrut \) \(115388668836\) \(\nu^{4}\mathstrut +\mathstrut \) \(81114719985\) \(\nu^{3}\mathstrut +\mathstrut \) \(3188992512\) \(\nu^{2}\mathstrut -\mathstrut \) \(1751857152\) \(\nu\mathstrut +\mathstrut \) \(555600236\)\()/\)\(284396912\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{16}\mathstrut -\mathstrut \) \(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{16}\mathstrut -\mathstrut \) \(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(6\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(14\)
\(\nu^{5}\)\(=\)\(\beta_{17}\mathstrut +\mathstrut \) \(8\) \(\beta_{16}\mathstrut -\mathstrut \) \(\beta_{15}\mathstrut -\mathstrut \) \(\beta_{14}\mathstrut -\mathstrut \) \(7\) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(9\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(29\) \(\beta_{1}\)
\(\nu^{6}\)\(=\)\(\beta_{17}\mathstrut +\mathstrut \) \(10\) \(\beta_{16}\mathstrut -\mathstrut \) \(\beta_{15}\mathstrut -\mathstrut \) \(11\) \(\beta_{14}\mathstrut -\mathstrut \) \(8\) \(\beta_{13}\mathstrut +\mathstrut \) \(12\) \(\beta_{10}\mathstrut +\mathstrut \) \(10\) \(\beta_{9}\mathstrut +\mathstrut \) \(11\) \(\beta_{8}\mathstrut -\mathstrut \) \(11\) \(\beta_{7}\mathstrut +\mathstrut \) \(11\) \(\beta_{6}\mathstrut +\mathstrut \) \(9\) \(\beta_{5}\mathstrut +\mathstrut \) \(11\) \(\beta_{4}\mathstrut +\mathstrut \) \(11\) \(\beta_{3}\mathstrut +\mathstrut \) \(36\) \(\beta_{2}\mathstrut +\mathstrut \) \(11\) \(\beta_{1}\mathstrut +\mathstrut \) \(76\)
\(\nu^{7}\)\(=\)\(13\) \(\beta_{17}\mathstrut +\mathstrut \) \(55\) \(\beta_{16}\mathstrut -\mathstrut \) \(13\) \(\beta_{15}\mathstrut -\mathstrut \) \(12\) \(\beta_{14}\mathstrut -\mathstrut \) \(42\) \(\beta_{13}\mathstrut -\mathstrut \) \(12\) \(\beta_{12}\mathstrut -\mathstrut \) \(11\) \(\beta_{11}\mathstrut +\mathstrut \) \(15\) \(\beta_{10}\mathstrut +\mathstrut \) \(12\) \(\beta_{8}\mathstrut -\mathstrut \) \(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(3\) \(\beta_{6}\mathstrut +\mathstrut \) \(68\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(12\) \(\beta_{2}\mathstrut +\mathstrut \) \(179\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{8}\)\(=\)\(15\) \(\beta_{17}\mathstrut +\mathstrut \) \(78\) \(\beta_{16}\mathstrut -\mathstrut \) \(17\) \(\beta_{15}\mathstrut -\mathstrut \) \(93\) \(\beta_{14}\mathstrut -\mathstrut \) \(48\) \(\beta_{13}\mathstrut -\mathstrut \) \(3\) \(\beta_{12}\mathstrut +\mathstrut \) \(106\) \(\beta_{10}\mathstrut +\mathstrut \) \(76\) \(\beta_{9}\mathstrut +\mathstrut \) \(91\) \(\beta_{8}\mathstrut -\mathstrut \) \(93\) \(\beta_{7}\mathstrut +\mathstrut \) \(92\) \(\beta_{6}\mathstrut +\mathstrut \) \(65\) \(\beta_{5}\mathstrut +\mathstrut \) \(91\) \(\beta_{4}\mathstrut +\mathstrut \) \(90\) \(\beta_{3}\mathstrut +\mathstrut \) \(224\) \(\beta_{2}\mathstrut +\mathstrut \) \(94\) \(\beta_{1}\mathstrut +\mathstrut \) \(446\)
\(\nu^{9}\)\(=\)\(122\) \(\beta_{17}\mathstrut +\mathstrut \) \(364\) \(\beta_{16}\mathstrut -\mathstrut \) \(127\) \(\beta_{15}\mathstrut -\mathstrut \) \(109\) \(\beta_{14}\mathstrut -\mathstrut \) \(241\) \(\beta_{13}\mathstrut -\mathstrut \) \(109\) \(\beta_{12}\mathstrut -\mathstrut \) \(88\) \(\beta_{11}\mathstrut +\mathstrut \) \(160\) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(106\) \(\beta_{8}\mathstrut -\mathstrut \) \(50\) \(\beta_{7}\mathstrut +\mathstrut \) \(44\) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(486\) \(\beta_{4}\mathstrut +\mathstrut \) \(34\) \(\beta_{3}\mathstrut +\mathstrut \) \(109\) \(\beta_{2}\mathstrut +\mathstrut \) \(1143\) \(\beta_{1}\mathstrut +\mathstrut \) \(32\)
\(\nu^{10}\)\(=\)\(155\) \(\beta_{17}\mathstrut +\mathstrut \) \(564\) \(\beta_{16}\mathstrut -\mathstrut \) \(193\) \(\beta_{15}\mathstrut -\mathstrut \) \(717\) \(\beta_{14}\mathstrut -\mathstrut \) \(254\) \(\beta_{13}\mathstrut -\mathstrut \) \(56\) \(\beta_{12}\mathstrut +\mathstrut \) \(3\) \(\beta_{11}\mathstrut +\mathstrut \) \(844\) \(\beta_{10}\mathstrut +\mathstrut \) \(530\) \(\beta_{9}\mathstrut +\mathstrut \) \(681\) \(\beta_{8}\mathstrut -\mathstrut \) \(719\) \(\beta_{7}\mathstrut +\mathstrut \) \(700\) \(\beta_{6}\mathstrut +\mathstrut \) \(441\) \(\beta_{5}\mathstrut +\mathstrut \) \(688\) \(\beta_{4}\mathstrut +\mathstrut \) \(669\) \(\beta_{3}\mathstrut +\mathstrut \) \(1432\) \(\beta_{2}\mathstrut +\mathstrut \) \(739\) \(\beta_{1}\mathstrut +\mathstrut \) \(2735\)
\(\nu^{11}\)\(=\)\(1011\) \(\beta_{17}\mathstrut +\mathstrut \) \(2384\) \(\beta_{16}\mathstrut -\mathstrut \) \(1107\) \(\beta_{15}\mathstrut -\mathstrut \) \(899\) \(\beta_{14}\mathstrut -\mathstrut \) \(1352\) \(\beta_{13}\mathstrut -\mathstrut \) \(899\) \(\beta_{12}\mathstrut -\mathstrut \) \(625\) \(\beta_{11}\mathstrut +\mathstrut \) \(1474\) \(\beta_{10}\mathstrut -\mathstrut \) \(15\) \(\beta_{9}\mathstrut +\mathstrut \) \(836\) \(\beta_{8}\mathstrut -\mathstrut \) \(564\) \(\beta_{7}\mathstrut +\mathstrut \) \(443\) \(\beta_{6}\mathstrut +\mathstrut \) \(38\) \(\beta_{5}\mathstrut +\mathstrut \) \(3394\) \(\beta_{4}\mathstrut +\mathstrut \) \(393\) \(\beta_{3}\mathstrut +\mathstrut \) \(894\) \(\beta_{2}\mathstrut +\mathstrut \) \(7452\) \(\beta_{1}\mathstrut +\mathstrut \) \(354\)
\(\nu^{12}\)\(=\)\(1380\) \(\beta_{17}\mathstrut +\mathstrut \) \(3965\) \(\beta_{16}\mathstrut -\mathstrut \) \(1845\) \(\beta_{15}\mathstrut -\mathstrut \) \(5304\) \(\beta_{14}\mathstrut -\mathstrut \) \(1209\) \(\beta_{13}\mathstrut -\mathstrut \) \(688\) \(\beta_{12}\mathstrut +\mathstrut \) \(63\) \(\beta_{11}\mathstrut +\mathstrut \) \(6414\) \(\beta_{10}\mathstrut +\mathstrut \) \(3576\) \(\beta_{9}\mathstrut +\mathstrut \) \(4870\) \(\beta_{8}\mathstrut -\mathstrut \) \(5346\) \(\beta_{7}\mathstrut +\mathstrut \) \(5093\) \(\beta_{6}\mathstrut +\mathstrut \) \(2927\) \(\beta_{5}\mathstrut +\mathstrut \) \(5013\) \(\beta_{4}\mathstrut +\mathstrut \) \(4793\) \(\beta_{3}\mathstrut +\mathstrut \) \(9313\) \(\beta_{2}\mathstrut +\mathstrut \) \(5616\) \(\beta_{1}\mathstrut +\mathstrut \) \(17216\)
\(\nu^{13}\)\(=\)\(7879\) \(\beta_{17}\mathstrut +\mathstrut \) \(15598\) \(\beta_{16}\mathstrut -\mathstrut \) \(9065\) \(\beta_{15}\mathstrut -\mathstrut \) \(7081\) \(\beta_{14}\mathstrut -\mathstrut \) \(7445\) \(\beta_{13}\mathstrut -\mathstrut \) \(7094\) \(\beta_{12}\mathstrut -\mathstrut \) \(4182\) \(\beta_{11}\mathstrut +\mathstrut \) \(12518\) \(\beta_{10}\mathstrut -\mathstrut \) \(143\) \(\beta_{9}\mathstrut +\mathstrut \) \(6244\) \(\beta_{8}\mathstrut -\mathstrut \) \(5391\) \(\beta_{7}\mathstrut +\mathstrut \) \(3825\) \(\beta_{6}\mathstrut +\mathstrut \) \(464\) \(\beta_{5}\mathstrut +\mathstrut \) \(23452\) \(\beta_{4}\mathstrut +\mathstrut \) \(3845\) \(\beta_{3}\mathstrut +\mathstrut \) \(6972\) \(\beta_{2}\mathstrut +\mathstrut \) \(49257\) \(\beta_{1}\mathstrut +\mathstrut \) \(3370\)
\(\nu^{14}\)\(=\)\(11387\) \(\beta_{17}\mathstrut +\mathstrut \) \(27571\) \(\beta_{16}\mathstrut -\mathstrut \) \(16076\) \(\beta_{15}\mathstrut -\mathstrut \) \(38425\) \(\beta_{14}\mathstrut -\mathstrut \) \(4911\) \(\beta_{13}\mathstrut -\mathstrut \) \(7041\) \(\beta_{12}\mathstrut +\mathstrut \) \(850\) \(\beta_{11}\mathstrut +\mathstrut \) \(47581\) \(\beta_{10}\mathstrut +\mathstrut \) \(23809\) \(\beta_{9}\mathstrut +\mathstrut \) \(34021\) \(\beta_{8}\mathstrut -\mathstrut \) \(38996\) \(\beta_{7}\mathstrut +\mathstrut \) \(36135\) \(\beta_{6}\mathstrut +\mathstrut \) \(19272\) \(\beta_{5}\mathstrut +\mathstrut \) \(35895\) \(\beta_{4}\mathstrut +\mathstrut \) \(33854\) \(\beta_{3}\mathstrut +\mathstrut \) \(61221\) \(\beta_{2}\mathstrut +\mathstrut \) \(42021\) \(\beta_{1}\mathstrut +\mathstrut \) \(110171\)
\(\nu^{15}\)\(=\)\(59325\) \(\beta_{17}\mathstrut +\mathstrut \) \(102323\) \(\beta_{16}\mathstrut -\mathstrut \) \(71375\) \(\beta_{15}\mathstrut -\mathstrut \) \(54323\) \(\beta_{14}\mathstrut -\mathstrut \) \(40109\) \(\beta_{13}\mathstrut -\mathstrut \) \(54653\) \(\beta_{12}\mathstrut -\mathstrut \) \(26980\) \(\beta_{11}\mathstrut +\mathstrut \) \(101089\) \(\beta_{10}\mathstrut -\mathstrut \) \(1090\) \(\beta_{9}\mathstrut +\mathstrut \) \(45257\) \(\beta_{8}\mathstrut -\mathstrut \) \(47120\) \(\beta_{7}\mathstrut +\mathstrut \) \(30483\) \(\beta_{6}\mathstrut +\mathstrut \) \(4651\) \(\beta_{5}\mathstrut +\mathstrut \) \(161238\) \(\beta_{4}\mathstrut +\mathstrut \) \(34282\) \(\beta_{3}\mathstrut +\mathstrut \) \(52807\) \(\beta_{2}\mathstrut +\mathstrut \) \(328761\) \(\beta_{1}\mathstrut +\mathstrut \) \(29609\)
\(\nu^{16}\)\(=\)\(89935\) \(\beta_{17}\mathstrut +\mathstrut \) \(190938\) \(\beta_{16}\mathstrut -\mathstrut \) \(132420\) \(\beta_{15}\mathstrut -\mathstrut \) \(275216\) \(\beta_{14}\mathstrut -\mathstrut \) \(13036\) \(\beta_{13}\mathstrut -\mathstrut \) \(65140\) \(\beta_{12}\mathstrut +\mathstrut \) \(9396\) \(\beta_{11}\mathstrut +\mathstrut \) \(348233\) \(\beta_{10}\mathstrut +\mathstrut \) \(157735\) \(\beta_{9}\mathstrut +\mathstrut \) \(234565\) \(\beta_{8}\mathstrut -\mathstrut \) \(281620\) \(\beta_{7}\mathstrut +\mathstrut \) \(252377\) \(\beta_{6}\mathstrut +\mathstrut \) \(126578\) \(\beta_{5}\mathstrut +\mathstrut \) \(254710\) \(\beta_{4}\mathstrut +\mathstrut \) \(237978\) \(\beta_{3}\mathstrut +\mathstrut \) \(405279\) \(\beta_{2}\mathstrut +\mathstrut \) \(311854\) \(\beta_{1}\mathstrut +\mathstrut \) \(712952\)
\(\nu^{17}\)\(=\)\(437508\) \(\beta_{17}\mathstrut +\mathstrut \) \(673994\) \(\beta_{16}\mathstrut -\mathstrut \) \(547300\) \(\beta_{15}\mathstrut -\mathstrut \) \(409785\) \(\beta_{14}\mathstrut -\mathstrut \) \(209403\) \(\beta_{13}\mathstrut -\mathstrut \) \(414859\) \(\beta_{12}\mathstrut -\mathstrut \) \(169425\) \(\beta_{11}\mathstrut +\mathstrut \) \(789771\) \(\beta_{10}\mathstrut -\mathstrut \) \(7063\) \(\beta_{9}\mathstrut +\mathstrut \) \(322144\) \(\beta_{8}\mathstrut -\mathstrut \) \(390021\) \(\beta_{7}\mathstrut +\mathstrut \) \(231689\) \(\beta_{6}\mathstrut +\mathstrut \) \(41738\) \(\beta_{5}\mathstrut +\mathstrut \) \(1105998\) \(\beta_{4}\mathstrut +\mathstrut \) \(288301\) \(\beta_{3}\mathstrut +\mathstrut \) \(392490\) \(\beta_{2}\mathstrut +\mathstrut \) \(2210366\) \(\beta_{1}\mathstrut +\mathstrut \) \(247515\)

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
−2.55956
−2.46111
−1.89397
−1.84714
−1.38222
−0.714378
−0.690432
−0.149899
0.101226
0.715815
1.14480
1.32090
1.64867
1.79692
2.26170
2.45136
2.58784
2.66948
−2.55956 −0.639116 4.55133 2.79521 1.63585 −2.94129 −6.53027 −2.59153 −7.15450
1.2 −2.46111 2.84715 4.05705 −0.414665 −7.00714 −0.439244 −5.06262 5.10626 1.02054
1.3 −1.89397 −0.0387693 1.58712 3.00387 0.0734279 0.643981 0.781977 −2.99850 −5.68925
1.4 −1.84714 −2.22957 1.41194 −0.984600 4.11833 2.90768 1.08623 1.97096 1.81870
1.5 −1.38222 1.10475 −0.0894560 −3.03767 −1.52702 −3.07306 2.88810 −1.77952 4.19874
1.6 −0.714378 −2.62504 −1.48966 0.882657 1.87527 −0.838797 2.49294 3.89086 −0.630551
1.7 −0.690432 1.58357 −1.52330 3.69903 −1.09335 2.15550 2.43260 −0.492301 −2.55393
1.8 −0.149899 −2.07180 −1.97753 4.08629 0.310561 5.08021 0.596228 1.29235 −0.612531
1.9 0.101226 −0.700214 −1.98975 0.497951 −0.0708800 −5.07659 −0.403867 −2.50970 0.0504057
1.10 0.715815 3.10981 −1.48761 0.799569 2.22605 −0.891306 −2.49648 6.67092 0.572344
1.11 1.14480 −1.66658 −0.689428 −2.63259 −1.90791 −1.31875 −3.07886 −0.222506 −3.01379
1.12 1.32090 −3.03201 −0.255233 3.39315 −4.00497 −0.854531 −2.97893 6.19308 4.48200
1.13 1.64867 2.75148 0.718114 4.35847 4.53629 −3.72637 −2.11341 4.57066 7.18568
1.14 1.79692 1.63689 1.22893 2.71894 2.94136 4.07184 −1.38555 −0.320597 4.88573
1.15 2.26170 −1.91993 3.11529 −0.621807 −4.34231 3.74378 2.52246 0.686140 −1.40634
1.16 2.45136 2.33620 4.00917 −2.57056 5.72687 2.10143 4.92520 2.45783 −6.30136
1.17 2.58784 −0.402595 4.69689 3.31536 −1.04185 −2.14234 6.97912 −2.83792 8.57961
1.18 2.66948 0.955769 5.12613 1.71138 2.55141 −0.402139 8.34515 −2.08651 4.56850
\(n\): 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
\(17\) \(-1\)
\(59\) \(1\)

Hecke kernels

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

\(T_{2}^{18} - \cdots\)
\(T_{3}^{18} - \cdots\)