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

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

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

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.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1003.2.a.g 10
3.b odd 2 1 9027.2.a.j 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1003.2.a.g 10 1.a even 1 1 trivial
9027.2.a.j 10 3.b odd 2 1

Atkin-Lehner signs

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

Hecke kernels

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

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + 3 T^{2} + 6 T^{3} + 16 T^{4} + 20 T^{5} + 44 T^{6} + 67 T^{7} + 117 T^{8} + 149 T^{9} + 259 T^{10} + 298 T^{11} + 468 T^{12} + 536 T^{13} + 704 T^{14} + 640 T^{15} + 1024 T^{16} + 768 T^{17} + 768 T^{18} + 512 T^{19} + 1024 T^{20} \)
$3$ \( 1 + 7 T + 38 T^{2} + 155 T^{3} + 541 T^{4} + 1621 T^{5} + 4342 T^{6} + 10389 T^{7} + 22635 T^{8} + 44766 T^{9} + 81197 T^{10} + 134298 T^{11} + 203715 T^{12} + 280503 T^{13} + 351702 T^{14} + 393903 T^{15} + 394389 T^{16} + 338985 T^{17} + 249318 T^{18} + 137781 T^{19} + 59049 T^{20} \)
$5$ \( 1 + 12 T + 96 T^{2} + 570 T^{3} + 2783 T^{4} + 11529 T^{5} + 41780 T^{6} + 134057 T^{7} + 385889 T^{8} + 1000919 T^{9} + 2351889 T^{10} + 5004595 T^{11} + 9647225 T^{12} + 16757125 T^{13} + 26112500 T^{14} + 36028125 T^{15} + 43484375 T^{16} + 44531250 T^{17} + 37500000 T^{18} + 23437500 T^{19} + 9765625 T^{20} \)
$7$ \( 1 + 9 T + 80 T^{2} + 468 T^{3} + 2555 T^{4} + 11334 T^{5} + 46888 T^{6} + 168019 T^{7} + 564109 T^{8} + 1680606 T^{9} + 4704893 T^{10} + 11764242 T^{11} + 27641341 T^{12} + 57630517 T^{13} + 112578088 T^{14} + 190490538 T^{15} + 300593195 T^{16} + 385418124 T^{17} + 461184080 T^{18} + 363182463 T^{19} + 282475249 T^{20} \)
$11$ \( 1 - 12 T + 106 T^{2} - 672 T^{3} + 3820 T^{4} - 18679 T^{5} + 86210 T^{6} - 357923 T^{7} + 1411720 T^{8} - 5083546 T^{9} + 17554057 T^{10} - 55919006 T^{11} + 170818120 T^{12} - 476395513 T^{13} + 1262200610 T^{14} - 3008271629 T^{15} + 6767363020 T^{16} - 13095378912 T^{17} + 22722041386 T^{18} - 28295372292 T^{19} + 25937424601 T^{20} \)
$13$ \( 1 + 11 T + 115 T^{2} + 763 T^{3} + 4875 T^{4} + 24307 T^{5} + 120492 T^{6} + 501644 T^{7} + 2136116 T^{8} + 7904440 T^{9} + 30412295 T^{10} + 102757720 T^{11} + 361003604 T^{12} + 1102111868 T^{13} + 3441372012 T^{14} + 9025018951 T^{15} + 23530693875 T^{16} + 47877118471 T^{17} + 93809032915 T^{18} + 116649493103 T^{19} + 137858491849 T^{20} \)
$17$ \( ( 1 + T )^{10} \)
$19$ \( 1 + 5 T + 104 T^{2} + 550 T^{3} + 5320 T^{4} + 27780 T^{5} + 187167 T^{6} + 888731 T^{7} + 5101648 T^{8} + 21055340 T^{9} + 109553353 T^{10} + 400051460 T^{11} + 1841694928 T^{12} + 6095805929 T^{13} + 24391790607 T^{14} + 68786030220 T^{15} + 250284086920 T^{16} + 491629456450 T^{17} + 1766290556264 T^{18} + 1613438488895 T^{19} + 6131066257801 T^{20} \)
$23$ \( 1 + 7 T + 120 T^{2} + 588 T^{3} + 6289 T^{4} + 22668 T^{5} + 201269 T^{6} + 514368 T^{7} + 4737639 T^{8} + 8908707 T^{9} + 104593011 T^{10} + 204900261 T^{11} + 2506211031 T^{12} + 6258315456 T^{13} + 56323318229 T^{14} + 145899023124 T^{15} + 930997705921 T^{16} + 2002037362836 T^{17} + 9397318233720 T^{18} + 12608068630241 T^{19} + 41426511213649 T^{20} \)
$29$ \( 1 - 10 T + 157 T^{2} - 1430 T^{3} + 14283 T^{4} - 105029 T^{5} + 840303 T^{6} - 5372919 T^{7} + 35889806 T^{8} - 201535046 T^{9} + 1186241053 T^{10} - 5844516334 T^{11} + 30183326846 T^{12} - 131040121491 T^{13} + 594330346143 T^{14} - 2154265468321 T^{15} + 8495861493843 T^{16} - 24667323121870 T^{17} + 78538686834877 T^{18} - 145071459758690 T^{19} + 420707233300201 T^{20} \)
$31$ \( 1 - 13 T + 170 T^{2} - 1670 T^{3} + 14959 T^{4} - 116004 T^{5} + 832653 T^{6} - 5483248 T^{7} + 34521695 T^{8} - 202943201 T^{9} + 1156502631 T^{10} - 6291239231 T^{11} + 33175348895 T^{12} - 163351441168 T^{13} + 768972531213 T^{14} - 3321096032604 T^{15} + 13276167564079 T^{16} - 45946065565370 T^{17} + 144991476364970 T^{18} - 343715088088723 T^{19} + 819628286980801 T^{20} \)
$37$ \( 1 - 12 T + 270 T^{2} - 2244 T^{3} + 30448 T^{4} - 195442 T^{5} + 2093530 T^{6} - 11100210 T^{7} + 105028311 T^{8} - 486723652 T^{9} + 4252341769 T^{10} - 18008775124 T^{11} + 143783757759 T^{12} - 562258937130 T^{13} + 3923612278330 T^{14} - 13552721643994 T^{15} + 78121237701232 T^{16} - 213027132286452 T^{17} + 948369452558670 T^{18} - 1559540877540924 T^{19} + 4808584372417849 T^{20} \)
$41$ \( 1 + 29 T + 588 T^{2} + 9019 T^{3} + 117459 T^{4} + 1312900 T^{5} + 13048499 T^{6} + 115992786 T^{7} + 935973813 T^{8} + 6858355770 T^{9} + 45972909783 T^{10} + 281192586570 T^{11} + 1573371979653 T^{12} + 7994338803906 T^{13} + 36871939582739 T^{14} + 152107606292900 T^{15} + 557942494043619 T^{16} + 1756488796132739 T^{17} + 4695136034723148 T^{18} + 9494076097424869 T^{19} + 13422659310152401 T^{20} \)
$43$ \( 1 + 18 T + 334 T^{2} + 2980 T^{3} + 29029 T^{4} + 136196 T^{5} + 1103813 T^{6} + 3815337 T^{7} + 62800951 T^{8} + 348213533 T^{9} + 3943962421 T^{10} + 14973181919 T^{11} + 116118958399 T^{12} + 303345998859 T^{13} + 3773716988213 T^{14} + 20021961902828 T^{15} + 183502847949421 T^{16} + 810019461098860 T^{17} + 3903858892718734 T^{18} + 9046667014863174 T^{19} + 21611482313284249 T^{20} \)
$47$ \( 1 + 18 T + 494 T^{2} + 6270 T^{3} + 99181 T^{4} + 979232 T^{5} + 11419378 T^{6} + 92557225 T^{7} + 873890510 T^{8} + 6001170605 T^{9} + 47926902307 T^{10} + 282055018435 T^{11} + 1930424136590 T^{12} + 9609568771175 T^{13} + 55722921858418 T^{14} + 224581969894624 T^{15} + 1069093355545549 T^{16} + 3176526965303010 T^{17} + 11762775610909934 T^{18} + 20144348515849806 T^{19} + 52599132235830049 T^{20} \)
$53$ \( 1 + 280 T^{2} - 453 T^{3} + 40547 T^{4} - 95841 T^{5} + 4116700 T^{6} - 10306599 T^{7} + 316060756 T^{8} - 756016153 T^{9} + 18872290141 T^{10} - 40068856109 T^{11} + 887814663604 T^{12} - 1534415539323 T^{13} + 32482743132700 T^{14} - 40080274244613 T^{15} + 898698350697563 T^{16} - 532144146346161 T^{17} + 17432713315181080 T^{18} + 174887470365513049 T^{20} \)
$59$ \( ( 1 + T )^{10} \)
$61$ \( 1 - 8 T + 217 T^{2} - 2336 T^{3} + 31389 T^{4} - 303921 T^{5} + 3563673 T^{6} - 28470976 T^{7} + 295582745 T^{8} - 2240831291 T^{9} + 19394855451 T^{10} - 136690708751 T^{11} + 1099863394145 T^{12} - 6462370603456 T^{13} + 49342049733993 T^{14} - 256690552396221 T^{15} + 1617173030817429 T^{16} - 7341447264945056 T^{17} + 41600486920409977 T^{18} - 93553168742673128 T^{19} + 713342911662882601 T^{20} \)
$67$ \( 1 + 6 T + 329 T^{2} + 2024 T^{3} + 62805 T^{4} + 348054 T^{5} + 8142587 T^{6} + 41127531 T^{7} + 790515460 T^{8} + 3576798755 T^{9} + 59745142595 T^{10} + 239645516585 T^{11} + 3548623899940 T^{12} + 12369639606153 T^{13} + 164082255890027 T^{14} + 469916443991778 T^{15} + 5681238692124045 T^{16} + 12266880289173752 T^{17} + 133596265916134889 T^{18} + 163239206377769682 T^{19} + 1822837804551761449 T^{20} \)
$71$ \( 1 - 8 T + 422 T^{2} - 1784 T^{3} + 78397 T^{4} - 137528 T^{5} + 9660861 T^{6} - 1206535 T^{7} + 911777254 T^{8} + 707944132 T^{9} + 70230028469 T^{10} + 50264033372 T^{11} + 4596269137414 T^{12} - 431832148385 T^{13} + 245498717917341 T^{14} - 248132054184328 T^{15} + 10042677958554637 T^{16} - 16225694362569544 T^{17} + 272507990185711142 T^{18} - 366788005747592248 T^{19} + 3255243551009881201 T^{20} \)
$73$ \( 1 + 41 T + 1176 T^{2} + 25011 T^{3} + 446720 T^{4} + 6818286 T^{5} + 92020459 T^{6} + 1105799804 T^{7} + 12010240509 T^{8} + 118061925228 T^{9} + 1058028065433 T^{10} + 8618520541644 T^{11} + 64002571672461 T^{12} + 430174922352668 T^{13} + 2613219171612619 T^{14} + 14134795019549598 T^{15} + 67604025567822080 T^{16} + 276306484361135067 T^{17} + 948397068067439256 T^{18} + 2413735055038984433 T^{19} + 4297625829703557649 T^{20} \)
$79$ \( 1 - 3 T + 518 T^{2} - 1587 T^{3} + 128745 T^{4} - 390976 T^{5} + 20517279 T^{6} - 60195058 T^{7} + 2367861681 T^{8} - 6490084302 T^{9} + 210779814221 T^{10} - 512716659858 T^{11} + 14777824751121 T^{12} - 29678511201262 T^{13} + 799149678949599 T^{14} - 1203055202655424 T^{15} + 31296294461051145 T^{16} - 30476603561034333 T^{17} + 785862363531598598 T^{18} - 359554787947854957 T^{19} + 9468276082626847201 T^{20} \)
$83$ \( 1 + 369 T^{2} + 611 T^{3} + 68049 T^{4} + 305165 T^{5} + 8173289 T^{6} + 66211883 T^{7} + 744395127 T^{8} + 8505557036 T^{9} + 61617763203 T^{10} + 705961233988 T^{11} + 5128138029903 T^{12} + 37859093944921 T^{13} + 387890572987769 T^{14} + 1202057337821095 T^{15} + 22247965467387081 T^{16} + 16580127154662097 T^{17} + 831095833659306129 T^{18} + 15516041187205853449 T^{20} \)
$89$ \( 1 + 45 T + 1573 T^{2} + 37721 T^{3} + 776454 T^{4} + 13065786 T^{5} + 196990215 T^{6} + 2578456481 T^{7} + 30964392375 T^{8} + 331742873437 T^{9} + 3299461837305 T^{10} + 29525115735893 T^{11} + 245268952002375 T^{12} + 1817731886954089 T^{13} + 12359607544171815 T^{14} + 72960125771911914 T^{15} + 385883111291832294 T^{16} + 1668450183594249409 T^{17} + 6192254191369373413 T^{18} + 15766038166836834405 T^{19} + 31181719929966183601 T^{20} \)
$97$ \( 1 + 5 T + 559 T^{2} + 3358 T^{3} + 163611 T^{4} + 1065617 T^{5} + 31994112 T^{6} + 210736090 T^{7} + 4604742021 T^{8} + 28659825956 T^{9} + 507252820407 T^{10} + 2780003117732 T^{11} + 43326017675589 T^{12} + 192333139468570 T^{13} + 2832415731593472 T^{14} + 9150815762643569 T^{15} + 136283382698438619 T^{16} + 271320639277503454 T^{17} + 4381125379256721199 T^{18} + 3801155293272826085 T^{19} + 73742412689492826049 T^{20} \)
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