Properties

Label 1003.2.a.g
Level 1003
Weight 2
Character orbit 1003.a
Self dual Yes
Analytic conductor 8.009
Analytic rank 1
Dimension 10
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10}\mathstrut -\mathstrut \) \(3\) \(x^{9}\mathstrut -\mathstrut \) \(10\) \(x^{8}\mathstrut +\mathstrut \) \(34\) \(x^{7}\mathstrut +\mathstrut \) \(28\) \(x^{6}\mathstrut -\mathstrut \) \(129\) \(x^{5}\mathstrut -\mathstrut \) \(3\) \(x^{4}\mathstrut +\mathstrut \) \(178\) \(x^{3}\mathstrut -\mathstrut \) \(56\) \(x^{2}\mathstrut -\mathstrut \) \(56\) \(x\mathstrut +\mathstrut \) \(15\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( -4 \nu^{9} + 2 \nu^{8} + 45 \nu^{7} - 13 \nu^{6} - 162 \nu^{5} + 20 \nu^{4} + 188 \nu^{3} - 11 \nu^{2} - 31 \nu + 3 \)\()/7\)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{9} + \nu^{8} + 26 \nu^{7} - 10 \nu^{6} - 116 \nu^{5} + 31 \nu^{4} + 199 \nu^{3} - 30 \nu^{2} - 96 \nu + 5 \)\()/7\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{9} - 4 \nu^{8} - 13 \nu^{7} + 47 \nu^{6} + 58 \nu^{5} - 173 \nu^{4} - 96 \nu^{3} + 197 \nu^{2} + 34 \nu - 20 \)\()/7\)
\(\beta_{6}\)\(=\)\((\)\( 3 \nu^{9} - 5 \nu^{8} - 32 \nu^{7} + 50 \nu^{6} + 111 \nu^{5} - 162 \nu^{4} - 134 \nu^{3} + 178 \nu^{2} + 39 \nu - 25 \)\()/7\)
\(\beta_{7}\)\(=\)\((\)\( 6 \nu^{9} - 3 \nu^{8} - 71 \nu^{7} + 30 \nu^{6} + 271 \nu^{5} - 107 \nu^{4} - 352 \nu^{3} + 160 \nu^{2} + 99 \nu - 43 \)\()/7\)
\(\beta_{8}\)\(=\)\((\)\( 5 \nu^{9} - 6 \nu^{8} - 58 \nu^{7} + 60 \nu^{6} + 227 \nu^{5} - 200 \nu^{4} - 326 \nu^{3} + 243 \nu^{2} + 107 \nu - 51 \)\()/7\)
\(\beta_{9}\)\(=\)\((\)\( 5 \nu^{9} - 6 \nu^{8} - 58 \nu^{7} + 60 \nu^{6} + 227 \nu^{5} - 200 \nu^{4} - 333 \nu^{3} + 243 \nu^{2} + 135 \nu - 51 \)\()/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(5\) \(\beta_{2}\mathstrut +\mathstrut \) \(12\)
\(\nu^{5}\)\(=\)\(-\)\(7\) \(\beta_{9}\mathstrut +\mathstrut \) \(7\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(18\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{6}\)\(=\)\(-\)\(10\) \(\beta_{9}\mathstrut +\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(9\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(8\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(23\) \(\beta_{2}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(51\)
\(\nu^{7}\)\(=\)\(-\)\(44\) \(\beta_{9}\mathstrut +\mathstrut \) \(42\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(13\) \(\beta_{6}\mathstrut -\mathstrut \) \(11\) \(\beta_{5}\mathstrut -\mathstrut \) \(10\) \(\beta_{4}\mathstrut +\mathstrut \) \(11\) \(\beta_{3}\mathstrut +\mathstrut \) \(85\) \(\beta_{1}\mathstrut +\mathstrut \) \(9\)
\(\nu^{8}\)\(=\)\(-\)\(76\) \(\beta_{9}\mathstrut +\mathstrut \) \(25\) \(\beta_{8}\mathstrut +\mathstrut \) \(12\) \(\beta_{7}\mathstrut +\mathstrut \) \(63\) \(\beta_{6}\mathstrut -\mathstrut \) \(14\) \(\beta_{5}\mathstrut -\mathstrut \) \(52\) \(\beta_{4}\mathstrut +\mathstrut \) \(24\) \(\beta_{3}\mathstrut +\mathstrut \) \(103\) \(\beta_{2}\mathstrut +\mathstrut \) \(24\) \(\beta_{1}\mathstrut +\mathstrut \) \(223\)
\(\nu^{9}\)\(=\)\(-\)\(269\) \(\beta_{9}\mathstrut +\mathstrut \) \(242\) \(\beta_{8}\mathstrut +\mathstrut \) \(14\) \(\beta_{7}\mathstrut +\mathstrut \) \(113\) \(\beta_{6}\mathstrut -\mathstrut \) \(87\) \(\beta_{5}\mathstrut -\mathstrut \) \(77\) \(\beta_{4}\mathstrut +\mathstrut \) \(87\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(413\) \(\beta_{1}\mathstrut +\mathstrut \) \(59\)

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
−1.99318
−1.54725
2.08521
2.40612
−0.598829
1.06592
2.16584
−2.13534
0.252877
1.29864
−2.65122 −2.99318 5.02897 −3.73588 7.93559 2.50921 −8.03048 5.95914 9.90463
1.2 −2.40787 −2.54725 3.79784 0.977935 6.13345 −2.76854 −4.32896 3.48849 −2.35474
1.3 −1.86330 1.08521 1.47190 0.335552 −2.02208 −1.78229 0.984011 −1.82232 −0.625236
1.4 −1.85341 1.40612 1.43512 −4.08879 −2.60611 1.71195 1.04695 −1.02283 7.57819
1.5 −0.441455 −1.59883 −1.80512 −1.49805 0.705811 −0.804687 1.67979 −0.443745 0.661322
1.6 0.755134 0.0659171 −1.42977 1.80730 0.0497762 −1.15845 −2.58994 −2.99565 1.36475
1.7 1.26414 1.16584 −0.401948 −0.0160405 1.47379 −4.37637 −3.03640 −1.64081 −0.0202774
1.8 1.86166 −3.13534 1.46576 −1.05085 −5.83692 1.46101 −0.994564 6.83035 −1.95633
1.9 2.03516 −0.747123 2.14188 −1.25900 −1.52052 0.391654 0.288756 −2.44181 −2.56227
1.10 2.30116 0.298636 3.29536 −3.47217 0.687210 −4.18349 2.98084 −2.91082 −7.99004
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1003))\):

\(T_{2}^{10} + \cdots\)
\(T_{3}^{10} + \cdots\)