Properties

Label 667.2.a.a
Level $667$
Weight $2$
Character orbit 667.a
Self dual yes
Analytic conductor $5.326$
Analytic rank $1$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 667 = 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 667.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.32602181482\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 3 x^{9} - 10 x^{8} + 32 x^{7} + 32 x^{6} - 118 x^{5} - 29 x^{4} + 182 x^{3} - 28 x^{2} - 101 x + 43\)
Coefficient ring: \(\Z[a_1, a_2]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 3 x^{9} - 10 x^{8} + 32 x^{7} + 32 x^{6} - 118 x^{5} - 29 x^{4} + 182 x^{3} - 28 x^{2} - 101 x + 43\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{8} - 3 \nu^{7} - 8 \nu^{6} + 26 \nu^{5} + 16 \nu^{4} - 66 \nu^{3} + 3 \nu^{2} + 50 \nu - 22 \)
\(\beta_{4}\)\(=\)\( \nu^{8} - 4 \nu^{7} - 6 \nu^{6} + 36 \nu^{5} - \nu^{4} - 98 \nu^{3} + 45 \nu^{2} + 83 \nu - 53 \)
\(\beta_{5}\)\(=\)\( -\nu^{9} + 4 \nu^{8} + 5 \nu^{7} - 34 \nu^{6} + 10 \nu^{5} + 82 \nu^{4} - 69 \nu^{3} - 47 \nu^{2} + 72 \nu - 22 \)
\(\beta_{6}\)\(=\)\( 2 \nu^{8} - 7 \nu^{7} - 14 \nu^{6} + 63 \nu^{5} + 14 \nu^{4} - 172 \nu^{3} + 54 \nu^{2} + 146 \nu - 83 \)
\(\beta_{7}\)\(=\)\( 3 \nu^{9} - 10 \nu^{8} - 23 \nu^{7} + 91 \nu^{6} + 39 \nu^{5} - 253 \nu^{4} + 32 \nu^{3} + 226 \nu^{2} - 84 \nu - 17 \)
\(\beta_{8}\)\(=\)\( \nu^{9} - \nu^{8} - 16 \nu^{7} + 14 \nu^{6} + 89 \nu^{5} - 69 \nu^{4} - 200 \nu^{3} + 144 \nu^{2} + 153 \nu - 109 \)
\(\beta_{9}\)\(=\)\( 2 \nu^{9} - 9 \nu^{8} - 7 \nu^{7} + 77 \nu^{6} - 49 \nu^{5} - 186 \nu^{4} + 225 \nu^{3} + 94 \nu^{2} - 226 \nu + 78 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{8} - \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{2} + 4 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-\beta_{9} + \beta_{7} + \beta_{5} - 2 \beta_{4} - \beta_{3} + 8 \beta_{2} + 2 \beta_{1} + 13\)
\(\nu^{5}\)\(=\)\(-\beta_{9} + 8 \beta_{8} + \beta_{7} - 7 \beta_{6} + 9 \beta_{5} - 11 \beta_{4} - 2 \beta_{3} + 18 \beta_{2} + 21 \beta_{1} + 11\)
\(\nu^{6}\)\(=\)\(-8 \beta_{9} + 3 \beta_{8} + 9 \beta_{7} + 14 \beta_{5} - 24 \beta_{4} - 11 \beta_{3} + 57 \beta_{2} + 23 \beta_{1} + 69\)
\(\nu^{7}\)\(=\)\(-9 \beta_{9} + 54 \beta_{8} + 11 \beta_{7} - 38 \beta_{6} + 69 \beta_{5} - 93 \beta_{4} - 24 \beta_{3} + 136 \beta_{2} + 127 \beta_{1} + 90\)
\(\nu^{8}\)\(=\)\(-49 \beta_{9} + 44 \beta_{8} + 63 \beta_{7} + 2 \beta_{6} + 135 \beta_{5} - 219 \beta_{4} - 91 \beta_{3} + 397 \beta_{2} + 201 \beta_{1} + 407\)
\(\nu^{9}\)\(=\)\(-61 \beta_{9} + 355 \beta_{8} + 93 \beta_{7} - 183 \beta_{6} + 511 \beta_{5} - 730 \beta_{4} - 212 \beta_{3} + 981 \beta_{2} + 827 \beta_{1} + 676\)

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
2.67549
2.37954
1.68137
1.21378
0.788514
0.685481
−1.31926
−1.35371
−1.57663
−2.17460
−2.67549 −1.95508 5.15827 1.93239 5.23080 0.721979 −8.44992 0.822337 −5.17011
1.2 −2.37954 0.653256 3.66223 −2.05928 −1.55445 5.08987 −3.95536 −2.57326 4.90014
1.3 −1.68137 −2.54941 0.827018 −1.60084 4.28651 −4.81497 1.97222 3.49949 2.69161
1.4 −1.21378 −1.04091 −0.526738 2.40259 1.26344 −0.534912 3.06690 −1.91650 −2.91621
1.5 −0.788514 −2.59585 −1.37825 −4.42591 2.04687 4.56405 2.66379 3.73846 3.48989
1.6 −0.685481 1.17341 −1.53012 −0.380940 −0.804349 −0.405330 2.41983 −1.62311 0.261127
1.7 1.31926 1.82466 −0.259562 −4.11119 2.40720 −2.74867 −2.98094 0.329400 −5.42371
1.8 1.35371 −1.32895 −0.167476 1.38626 −1.79900 −1.97610 −2.93413 −1.23390 1.87659
1.9 1.57663 0.266494 0.485749 −1.88241 0.420161 −0.868755 −2.38741 −2.92898 −2.96785
1.10 2.17460 −3.44762 2.72887 −1.26068 −7.49718 1.97284 1.58501 8.88607 −2.74147
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(23\) \(-1\)
\(29\) \(-1\)

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 667.2.a.a 10
3.b odd 2 1 6003.2.a.l 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
667.2.a.a 10 1.a even 1 1 trivial
6003.2.a.l 10 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{10} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(667))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 43 + 101 T - 28 T^{2} - 182 T^{3} - 29 T^{4} + 118 T^{5} + 32 T^{6} - 32 T^{7} - 10 T^{8} + 3 T^{9} + T^{10} \)
$3$ \( 23 - 78 T - 93 T^{2} + 188 T^{3} + 198 T^{4} - 86 T^{5} - 136 T^{6} - 19 T^{7} + 22 T^{8} + 9 T^{9} + T^{10} \)
$5$ \( -349 - 1352 T - 961 T^{2} + 884 T^{3} + 1109 T^{4} + 27 T^{5} - 310 T^{6} - 82 T^{7} + 20 T^{8} + 10 T^{9} + T^{10} \)
$7$ \( 163 + 694 T + 369 T^{2} - 1511 T^{3} - 1549 T^{4} + 311 T^{5} + 523 T^{6} + 12 T^{7} - 43 T^{8} - T^{9} + T^{10} \)
$11$ \( 9599 + 17651 T - 4483 T^{2} - 20603 T^{3} - 7444 T^{4} + 2387 T^{5} + 1266 T^{6} - 69 T^{7} - 64 T^{8} + T^{10} \)
$13$ \( 18783 - 8241 T - 29180 T^{2} - 375 T^{3} + 12186 T^{4} + 2731 T^{5} - 1191 T^{6} - 383 T^{7} + 15 T^{8} + 13 T^{9} + T^{10} \)
$17$ \( 7129 + 158491 T + 608126 T^{2} + 479267 T^{3} + 110112 T^{4} - 14084 T^{5} - 8944 T^{6} - 806 T^{7} + 109 T^{8} + 22 T^{9} + T^{10} \)
$19$ \( 5125 - 14850 T + 3405 T^{2} + 19484 T^{3} - 12688 T^{4} - 2894 T^{5} + 2532 T^{6} - 31 T^{7} - 95 T^{8} + 2 T^{9} + T^{10} \)
$23$ \( ( -1 + T )^{10} \)
$29$ \( ( -1 + T )^{10} \)
$31$ \( -416511 - 182910 T + 538645 T^{2} + 434016 T^{3} + 84336 T^{4} - 16370 T^{5} - 7879 T^{6} - 587 T^{7} + 120 T^{8} + 22 T^{9} + T^{10} \)
$37$ \( 113329 - 619141 T + 583829 T^{2} - 75968 T^{3} - 80807 T^{4} + 17815 T^{5} + 3909 T^{6} - 753 T^{7} - 98 T^{8} + 9 T^{9} + T^{10} \)
$41$ \( 4666817 - 6083027 T - 1740461 T^{2} + 2126640 T^{3} + 788149 T^{4} + 20974 T^{5} - 22070 T^{6} - 2502 T^{7} + 85 T^{8} + 25 T^{9} + T^{10} \)
$43$ \( -66033 - 206415 T + 1537075 T^{2} + 286892 T^{3} - 223064 T^{4} - 24114 T^{5} + 10801 T^{6} + 496 T^{7} - 183 T^{8} - 3 T^{9} + T^{10} \)
$47$ \( -176953097 - 154758116 T - 39553665 T^{2} + 1482969 T^{3} + 2151750 T^{4} + 257710 T^{5} - 15319 T^{6} - 4634 T^{7} - 157 T^{8} + 17 T^{9} + T^{10} \)
$53$ \( 17317103 - 32296042 T + 9485180 T^{2} + 5729168 T^{3} - 690421 T^{4} - 516073 T^{5} - 64602 T^{6} + 46 T^{7} + 578 T^{8} + 43 T^{9} + T^{10} \)
$59$ \( -603431 + 542507 T + 891773 T^{2} - 408708 T^{3} - 288995 T^{4} + 47573 T^{5} + 14402 T^{6} - 1124 T^{7} - 222 T^{8} + 7 T^{9} + T^{10} \)
$61$ \( 315433 - 1110274 T + 1144042 T^{2} - 207658 T^{3} - 170109 T^{4} + 30010 T^{5} + 8292 T^{6} - 886 T^{7} - 168 T^{8} + 6 T^{9} + T^{10} \)
$67$ \( -5078341 + 7583074 T - 1950028 T^{2} - 1956292 T^{3} + 1283318 T^{4} - 242980 T^{5} + 3647 T^{6} + 3375 T^{7} - 230 T^{8} - 11 T^{9} + T^{10} \)
$71$ \( -249705089 - 245890995 T - 77899753 T^{2} - 3646430 T^{3} + 2706285 T^{4} + 454660 T^{5} - 4640 T^{6} - 5434 T^{7} - 223 T^{8} + 17 T^{9} + T^{10} \)
$73$ \( 874591 + 2048712 T + 2594 T^{2} - 833467 T^{3} - 277325 T^{4} + 23680 T^{5} + 30036 T^{6} + 6991 T^{7} + 782 T^{8} + 44 T^{9} + T^{10} \)
$79$ \( -6631087 - 11903942 T + 4861482 T^{2} + 1986480 T^{3} - 581955 T^{4} - 106703 T^{5} + 23752 T^{6} + 1901 T^{7} - 337 T^{8} - 5 T^{9} + T^{10} \)
$83$ \( 3705241 + 16906559 T + 19007530 T^{2} + 8478304 T^{3} + 1401287 T^{4} - 66855 T^{5} - 46267 T^{6} - 3681 T^{7} + 176 T^{8} + 32 T^{9} + T^{10} \)
$89$ \( 12231211 + 35137538 T + 17448573 T^{2} - 2690649 T^{3} - 1662364 T^{4} + 139489 T^{5} + 38432 T^{6} - 2168 T^{7} - 336 T^{8} + 10 T^{9} + T^{10} \)
$97$ \( 86444003 - 145056960 T + 62427033 T^{2} + 2092222 T^{3} - 4275256 T^{4} - 21765 T^{5} + 83827 T^{6} + 1294 T^{7} - 514 T^{8} - 6 T^{9} + T^{10} \)
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