Properties

Label 667.2.a.b
Level $667$
Weight $2$
Character orbit 667.a
Self dual yes
Analytic conductor $5.326$
Analytic rank $1$
Dimension $12$
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 3 x^{11} - 13 x^{10} + 41 x^{9} + 54 x^{8} - 188 x^{7} - 77 x^{6} + 342 x^{5} + 13 x^{4} - 215 x^{3} + 9 x^{2} + 37 x - 5\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 3 x^{11} - 13 x^{10} + 41 x^{9} + 54 x^{8} - 188 x^{7} - 77 x^{6} + 342 x^{5} + 13 x^{4} - 215 x^{3} + 9 x^{2} + 37 x - 5\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 69 \nu^{11} - 2268 \nu^{10} + 4126 \nu^{9} + 30343 \nu^{8} - 62480 \nu^{7} - 131017 \nu^{6} + 277030 \nu^{5} + 194815 \nu^{4} - 433403 \nu^{3} - 31183 \nu^{2} + 158295 \nu - 20718 \)\()/6469\)
\(\beta_{3}\)\(=\)\((\)\( -127 \nu^{11} - 607 \nu^{10} + 4875 \nu^{9} + 7810 \nu^{8} - 51132 \nu^{7} - 30082 \nu^{6} + 209382 \nu^{5} + 24036 \nu^{4} - 328362 \nu^{3} + 49707 \nu^{2} + 117880 \nu - 29932 \)\()/6469\)
\(\beta_{4}\)\(=\)\((\)\( -353 \nu^{11} - 210 \nu^{10} + 8049 \nu^{9} + 773 \nu^{8} - 63527 \nu^{7} + 13625 \nu^{6} + 210856 \nu^{5} - 84627 \nu^{4} - 276488 \nu^{3} + 135717 \nu^{2} + 84019 \nu - 25638 \)\()/6469\)
\(\beta_{5}\)\(=\)\((\)\( 549 \nu^{11} - 1451 \nu^{10} - 8798 \nu^{9} + 21760 \nu^{8} + 52179 \nu^{7} - 114560 \nu^{6} - 143208 \nu^{5} + 255406 \nu^{4} + 177916 \nu^{3} - 216607 \nu^{2} - 75949 \nu + 34852 \)\()/6469\)
\(\beta_{6}\)\(=\)\((\)\( -874 \nu^{11} + 2852 \nu^{10} + 10271 \nu^{9} - 37175 \nu^{8} - 32306 \nu^{7} + 154428 \nu^{6} + 1464 \nu^{5} - 218601 \nu^{4} + 86000 \nu^{3} + 34877 \nu^{2} - 19087 \nu + 23075 \)\()/6469\)
\(\beta_{7}\)\(=\)\((\)\( 1153 \nu^{11} - 2741 \nu^{10} - 16651 \nu^{9} + 36674 \nu^{8} + 84275 \nu^{7} - 163016 \nu^{6} - 182412 \nu^{5} + 282647 \nu^{4} + 156809 \nu^{3} - 153652 \nu^{2} - 43159 \nu + 10438 \)\()/6469\)
\(\beta_{8}\)\(=\)\((\)\( -1227 \nu^{11} + 2642 \nu^{10} + 18320 \nu^{9} - 36402 \nu^{8} - 95833 \nu^{7} + 168053 \nu^{6} + 212320 \nu^{5} - 303228 \nu^{4} - 190488 \nu^{3} + 177063 \nu^{2} + 64932 \nu - 21970 \)\()/6469\)
\(\beta_{9}\)\(=\)\((\)\( 1231 \nu^{11} - 3336 \nu^{10} - 16487 \nu^{9} + 44255 \nu^{8} + 73554 \nu^{7} - 190180 \nu^{6} - 130539 \nu^{5} + 291927 \nu^{4} + 89329 \nu^{3} - 89336 \nu^{2} - 43380 \nu - 10451 \)\()/6469\)
\(\beta_{10}\)\(=\)\((\)\( 2257 \nu^{11} - 6684 \nu^{10} - 28263 \nu^{9} + 88739 \nu^{8} + 106697 \nu^{7} - 383278 \nu^{6} - 103569 \nu^{5} + 605079 \nu^{4} - 88693 \nu^{3} - 238564 \nu^{2} + 63686 \nu + 8869 \)\()/6469\)
\(\beta_{11}\)\(=\)\((\)\( -2687 \nu^{11} + 10130 \nu^{10} + 28520 \nu^{9} - 134015 \nu^{8} - 63186 \nu^{7} + 578643 \nu^{6} - 126167 \nu^{5} - 926609 \nu^{4} + 441457 \nu^{3} + 391735 \nu^{2} - 161659 \nu - 20106 \)\()/6469\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{8} - \beta_{6} - \beta_{4} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{10} - \beta_{9} + 7 \beta_{8} - \beta_{7} - 7 \beta_{6} + \beta_{5} - 6 \beta_{4} - \beta_{2} + 15\)
\(\nu^{5}\)\(=\)\(\beta_{11} + 2 \beta_{10} + 2 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + 10 \beta_{5} + 10 \beta_{4} + 10 \beta_{3} - 9 \beta_{2} + 30 \beta_{1}\)
\(\nu^{6}\)\(=\)\(\beta_{11} + 11 \beta_{10} - 8 \beta_{9} + 47 \beta_{8} - 10 \beta_{7} - 45 \beta_{6} + 11 \beta_{5} - 35 \beta_{4} + \beta_{3} - 10 \beta_{2} + 2 \beta_{1} + 86\)
\(\nu^{7}\)\(=\)\(10 \beta_{11} + 21 \beta_{10} + 3 \beta_{9} + 24 \beta_{8} + 20 \beta_{7} + 22 \beta_{6} + 78 \beta_{5} + 76 \beta_{4} + 78 \beta_{3} - 68 \beta_{2} + 192 \beta_{1} + 2\)
\(\nu^{8}\)\(=\)\(12 \beta_{11} + 90 \beta_{10} - 50 \beta_{9} + 314 \beta_{8} - 77 \beta_{7} - 283 \beta_{6} + 94 \beta_{5} - 209 \beta_{4} + 19 \beta_{3} - 81 \beta_{2} + 32 \beta_{1} + 527\)
\(\nu^{9}\)\(=\)\(74 \beta_{11} + 167 \beta_{10} + 43 \beta_{9} + 214 \beta_{8} + 148 \beta_{7} + 178 \beta_{6} + 559 \beta_{5} + 524 \beta_{4} + 565 \beta_{3} - 488 \beta_{2} + 1261 \beta_{1} + 40\)
\(\nu^{10}\)\(=\)\(99 \beta_{11} + 661 \beta_{10} - 284 \beta_{9} + 2098 \beta_{8} - 546 \beta_{7} - 1773 \beta_{6} + 724 \beta_{5} - 1281 \beta_{4} + 227 \beta_{3} - 622 \beta_{2} + 351 \beta_{1} + 3354\)
\(\nu^{11}\)\(=\)\(491 \beta_{11} + 1212 \beta_{10} + 431 \beta_{9} + 1715 \beta_{8} + 980 \beta_{7} + 1286 \beta_{6} + 3859 \beta_{5} + 3450 \beta_{4} + 3981 \beta_{3} - 3436 \beta_{2} + 8380 \beta_{1} + 515\)

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
2.66955
2.58646
1.99588
1.49364
1.08419
0.431373
0.147789
−0.613795
−0.724122
−1.65670
−1.89304
−2.52122
−2.66955 0.744453 5.12647 −2.89555 −1.98735 −0.268248 −8.34627 −2.44579 7.72982
1.2 −2.58646 −2.87567 4.68978 −3.69848 7.43782 −0.298143 −6.95702 5.26950 9.56597
1.3 −1.99588 0.764948 1.98352 2.38535 −1.52674 −0.101071 0.0328940 −2.41485 −4.76086
1.4 −1.49364 −1.48170 0.230961 −1.29771 2.21312 1.16738 2.64231 −0.804576 1.93831
1.5 −1.08419 1.50990 −0.824542 0.786014 −1.63701 −4.63453 3.06233 −0.720207 −0.852185
1.6 −0.431373 2.75201 −1.81392 −3.65141 −1.18714 2.43060 1.64522 4.57354 1.57512
1.7 −0.147789 −3.02697 −1.97816 0.429801 0.447353 0.404424 0.587929 6.16253 −0.0635200
1.8 0.613795 −0.0223201 −1.62326 −0.782514 −0.0137000 3.32687 −2.22394 −2.99950 −0.480303
1.9 0.724122 1.92599 −1.47565 −1.17886 1.39465 −4.09140 −2.51679 0.709431 −0.853636
1.10 1.65670 −1.05120 0.744648 1.18716 −1.74152 −3.31827 −2.07974 −1.89498 1.96676
1.11 1.89304 −0.392615 1.58359 −4.13664 −0.743235 2.17357 −0.788273 −2.84585 −7.83081
1.12 2.52122 −1.84683 4.35654 −3.14716 −4.65625 −3.79117 5.94135 0.410766 −7.93467
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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.b 12
3.b odd 2 1 6003.2.a.n 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
667.2.a.b 12 1.a even 1 1 trivial
6003.2.a.n 12 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{12} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(667))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -5 - 37 T + 9 T^{2} + 215 T^{3} + 13 T^{4} - 342 T^{5} - 77 T^{6} + 188 T^{7} + 54 T^{8} - 41 T^{9} - 13 T^{10} + 3 T^{11} + T^{12} \)
$3$ \( -1 - 46 T - 48 T^{2} + 256 T^{3} + 129 T^{4} - 382 T^{5} - 150 T^{6} + 207 T^{7} + 74 T^{8} - 44 T^{9} - 15 T^{10} + 3 T^{11} + T^{12} \)
$5$ \( 583 - 300 T - 3214 T^{2} - 660 T^{3} + 4808 T^{4} + 2991 T^{5} - 1627 T^{6} - 1943 T^{7} - 396 T^{8} + 162 T^{9} + 91 T^{10} + 16 T^{11} + T^{12} \)
$7$ \( -16 - 216 T - 413 T^{2} + 1988 T^{3} + 3679 T^{4} - 5425 T^{5} - 741 T^{6} + 1631 T^{7} + 125 T^{8} - 182 T^{9} - 19 T^{10} + 7 T^{11} + T^{12} \)
$11$ \( 266075 + 1063507 T + 1146714 T^{2} + 279150 T^{3} - 213703 T^{4} - 115816 T^{5} + 4140 T^{6} + 11450 T^{7} + 1046 T^{8} - 443 T^{9} - 63 T^{10} + 6 T^{11} + T^{12} \)
$13$ \( 3090655 + 6460857 T + 2072413 T^{2} - 1602192 T^{3} - 915400 T^{4} + 42364 T^{5} + 102285 T^{6} + 12082 T^{7} - 3802 T^{8} - 872 T^{9} + 10 T^{10} + 15 T^{11} + T^{12} \)
$17$ \( -9596 + 31240 T + 67893 T^{2} - 103277 T^{3} - 91550 T^{4} + 39627 T^{5} + 31076 T^{6} - 1682 T^{7} - 3484 T^{8} - 436 T^{9} + 71 T^{10} + 18 T^{11} + T^{12} \)
$19$ \( -149440 + 1324256 T + 1241085 T^{2} - 870280 T^{3} - 1308005 T^{4} - 400570 T^{5} + 43638 T^{6} + 38666 T^{7} + 3160 T^{8} - 897 T^{9} - 119 T^{10} + 6 T^{11} + T^{12} \)
$23$ \( ( 1 + T )^{12} \)
$29$ \( ( 1 + T )^{12} \)
$31$ \( 169 - 29220 T - 13018 T^{2} + 97636 T^{3} - 8705 T^{4} - 90550 T^{5} + 52027 T^{6} - 4023 T^{7} - 3333 T^{8} + 703 T^{9} + 25 T^{10} - 16 T^{11} + T^{12} \)
$37$ \( -362875340 + 492689776 T - 132885771 T^{2} - 67960955 T^{3} + 29813443 T^{4} + 2097862 T^{5} - 1538543 T^{6} - 12921 T^{7} + 32317 T^{8} - 139 T^{9} - 300 T^{10} + T^{11} + T^{12} \)
$41$ \( -27754352 + 39955432 T + 95038325 T^{2} + 51050139 T^{3} + 5620365 T^{4} - 2820556 T^{5} - 688351 T^{6} + 36992 T^{7} + 19334 T^{8} + 100 T^{9} - 225 T^{10} - 3 T^{11} + T^{12} \)
$43$ \( -229352509 - 456409123 T - 332516092 T^{2} - 98139927 T^{3} + 120819 T^{4} + 7019312 T^{5} + 1432697 T^{6} - 4348 T^{7} - 34372 T^{8} - 3715 T^{9} + 18 T^{10} + 23 T^{11} + T^{12} \)
$47$ \( -40804117 - 3930358 T + 195233100 T^{2} - 139288371 T^{3} - 32908165 T^{4} + 14545371 T^{5} + 4020829 T^{6} + 22788 T^{7} - 68952 T^{8} - 5007 T^{9} + 222 T^{10} + 35 T^{11} + T^{12} \)
$53$ \( -142956229 - 192812894 T + 103125523 T^{2} + 155347564 T^{3} + 43362431 T^{4} - 4447789 T^{5} - 4586185 T^{6} - 947515 T^{7} - 74630 T^{8} + 1491 T^{9} + 683 T^{10} + 45 T^{11} + T^{12} \)
$59$ \( 8891929600 - 5050998336 T - 1293585763 T^{2} + 690028123 T^{3} + 110629045 T^{4} - 27279008 T^{5} - 3722991 T^{6} + 472329 T^{7} + 56094 T^{8} - 3752 T^{9} - 386 T^{10} + 11 T^{11} + T^{12} \)
$61$ \( -18119916 + 27947196 T + 5427069 T^{2} - 22849850 T^{3} + 6151210 T^{4} + 2446190 T^{5} - 674995 T^{6} - 90746 T^{7} + 23370 T^{8} + 1218 T^{9} - 288 T^{10} - 4 T^{11} + T^{12} \)
$67$ \( -409376228 + 2093598060 T + 1996739575 T^{2} + 354520568 T^{3} - 86438366 T^{4} - 25317130 T^{5} + 841992 T^{6} + 575866 T^{7} + 10401 T^{8} - 5505 T^{9} - 222 T^{10} + 19 T^{11} + T^{12} \)
$71$ \( -226642180 + 276404288 T + 107691351 T^{2} - 184276655 T^{3} + 12049787 T^{4} + 17462558 T^{5} - 1037483 T^{6} - 560972 T^{7} + 24172 T^{8} + 6170 T^{9} - 295 T^{10} - 19 T^{11} + T^{12} \)
$73$ \( 2131153580 - 2044166852 T - 522062685 T^{2} + 468676890 T^{3} + 138848386 T^{4} - 8381771 T^{5} - 5196543 T^{6} - 113916 T^{7} + 68014 T^{8} + 2423 T^{9} - 406 T^{10} - 10 T^{11} + T^{12} \)
$79$ \( -36136523 + 19569650 T + 115383457 T^{2} + 8905014 T^{3} - 42217641 T^{4} + 6564553 T^{5} + 1741621 T^{6} - 410928 T^{7} - 2683 T^{8} + 5302 T^{9} - 222 T^{10} - 17 T^{11} + T^{12} \)
$83$ \( 14844146512 - 25209439576 T + 16164213229 T^{2} - 4686902763 T^{3} + 446283618 T^{4} + 68049836 T^{5} - 16696731 T^{6} + 220621 T^{7} + 153191 T^{8} - 4819 T^{9} - 638 T^{10} + 12 T^{11} + T^{12} \)
$89$ \( -12272416784 - 453908952 T + 5874747827 T^{2} + 1422097864 T^{3} - 227836659 T^{4} - 76569835 T^{5} + 1332308 T^{6} + 1362447 T^{7} + 35172 T^{8} - 9078 T^{9} - 384 T^{10} + 20 T^{11} + T^{12} \)
$97$ \( -6155578420 - 3227158204 T + 1049161983 T^{2} + 559443202 T^{3} - 42211711 T^{4} - 31011322 T^{5} - 422404 T^{6} + 648971 T^{7} + 33165 T^{8} - 5224 T^{9} - 368 T^{10} + 12 T^{11} + T^{12} \)
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