Properties

Label 667.2.a.d
Level $667$
Weight $2$
Character orbit 667.a
Self dual yes
Analytic conductor $5.326$
Analytic rank $0$
Dimension $16$
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: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 3 x^{15} - 22 x^{14} + 68 x^{13} + 187 x^{12} - 597 x^{11} - 795 x^{10} + 2592 x^{9} + 1860 x^{8} - 5877 x^{7} - 2496 x^{6} + 6612 x^{5} + 1842 x^{4} - 3011 x^{3} - 505 x^{2} + 336 x + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 3 x^{15} - 22 x^{14} + 68 x^{13} + 187 x^{12} - 597 x^{11} - 795 x^{10} + 2592 x^{9} + 1860 x^{8} - 5877 x^{7} - 2496 x^{6} + 6612 x^{5} + 1842 x^{4} - 3011 x^{3} - 505 x^{2} + 336 x + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\(-124169 \nu^{15} + 194810 \nu^{14} + 2801696 \nu^{13} - 4124788 \nu^{12} - 24534615 \nu^{11} + 32910414 \nu^{10} + 106985521 \nu^{9} - 125237015 \nu^{8} - 247933907 \nu^{7} + 238939362 \nu^{6} + 295392450 \nu^{5} - 218624434 \nu^{4} - 150942644 \nu^{3} + 82287639 \nu^{2} + 14985488 \nu - 5285248\)\()/1247936\)
\(\beta_{4}\)\(=\)\((\)\(-320937 \nu^{15} + 55514 \nu^{14} + 7810176 \nu^{13} - 279828 \nu^{12} - 74718487 \nu^{11} - 9187314 \nu^{10} + 357397585 \nu^{9} + 105825449 \nu^{8} - 891751731 \nu^{7} - 417755646 \nu^{6} + 1076238434 \nu^{5} + 659418286 \nu^{4} - 486230356 \nu^{3} - 310735913 \nu^{2} + 70510224 \nu + 26486912\)\()/1247936\)
\(\beta_{5}\)\(=\)\((\)\(-399265 \nu^{15} + 390970 \nu^{14} + 9549392 \nu^{13} - 8005092 \nu^{12} - 89853839 \nu^{11} + 59863870 \nu^{10} + 424971913 \nu^{9} - 199268623 \nu^{8} - 1062555835 \nu^{7} + 281768162 \nu^{6} + 1319983010 \nu^{5} - 121294226 \nu^{4} - 647488676 \nu^{3} + 18055599 \nu^{2} + 82151376 \nu + 9101888\)\()/1247936\)
\(\beta_{6}\)\(=\)\((\)\(412871 \nu^{15} - 286326 \nu^{14} - 9954704 \nu^{13} + 5268764 \nu^{12} + 94577321 \nu^{11} - 31443954 \nu^{10} - 451780703 \nu^{9} + 50528745 \nu^{8} + 1135959741 \nu^{7} + 125345330 \nu^{6} - 1394129102 \nu^{5} - 412374178 \nu^{4} + 636921244 \nu^{3} + 221078775 \nu^{2} - 83614256 \nu - 22861184\)\()/1247936\)
\(\beta_{7}\)\(=\)\((\)\(524711 \nu^{15} - 423750 \nu^{14} - 12681760 \nu^{13} + 8084172 \nu^{12} + 121020601 \nu^{11} - 52379826 \nu^{10} - 582618367 \nu^{9} + 118277337 \nu^{8} + 1484699485 \nu^{7} + 41030562 \nu^{6} - 1866965854 \nu^{5} - 405092882 \nu^{4} + 902289932 \nu^{3} + 240369543 \nu^{2} - 128155856 \nu - 29967232\)\()/1247936\)
\(\beta_{8}\)\(=\)\((\)\(17613 \nu^{15} - 15970 \nu^{14} - 416256 \nu^{13} + 315428 \nu^{12} + 3856563 \nu^{11} - 2217254 \nu^{10} - 17880021 \nu^{9} + 6505683 \nu^{8} + 43543519 \nu^{7} - 6272874 \nu^{6} - 51974890 \nu^{5} - 2466278 \nu^{4} + 23503844 \nu^{3} + 2851149 \nu^{2} - 2582832 \nu - 482240\)\()/40256\)
\(\beta_{9}\)\(=\)\((\)\(767889 \nu^{15} - 741754 \nu^{14} - 18434160 \nu^{13} + 14936836 \nu^{12} + 174451711 \nu^{11} - 108143006 \nu^{10} - 832081593 \nu^{9} + 334114239 \nu^{8} + 2103849867 \nu^{7} - 371571234 \nu^{6} - 2644257954 \nu^{5} - 37531118 \nu^{4} + 1301242372 \nu^{3} + 105743457 \nu^{2} - 165493584 \nu - 25501376\)\()/1247936\)
\(\beta_{10}\)\(=\)\((\)\(781759 \nu^{15} - 787334 \nu^{14} - 18711408 \nu^{13} + 15967900 \nu^{12} + 176274769 \nu^{11} - 117193986 \nu^{10} - 834663895 \nu^{9} + 373891409 \nu^{8} + 2083463077 \nu^{7} - 467484126 \nu^{6} - 2550207070 \nu^{5} + 95845550 \nu^{4} + 1166520668 \nu^{3} + 4288591 \nu^{2} - 120432048 \nu - 9694400\)\()/1247936\)
\(\beta_{11}\)\(=\)\((\)\(-785211 \nu^{15} + 785182 \nu^{14} + 18568432 \nu^{13} - 15860300 \nu^{12} - 172213973 \nu^{11} + 116053706 \nu^{10} + 799917747 \nu^{9} - 371328085 \nu^{8} - 1954600569 \nu^{7} + 478390486 \nu^{6} + 2346845462 \nu^{5} - 130836902 \nu^{4} - 1069127244 \nu^{3} - 5825419 \nu^{2} + 112498544 \nu + 12619200\)\()/1247936\)
\(\beta_{12}\)\(=\)\((\)\(-25785 \nu^{15} + 22922 \nu^{14} + 619504 \nu^{13} - 451684 \nu^{12} - 5863511 \nu^{11} + 3136078 \nu^{10} + 27912865 \nu^{9} - 8752791 \nu^{8} - 70040835 \nu^{7} + 6098098 \nu^{6} + 86000690 \nu^{5} + 8993566 \nu^{4} - 39174116 \nu^{3} - 6494857 \nu^{2} + 4130640 \nu + 1090240\)\()/40256\)
\(\beta_{13}\)\(=\)\((\)\(-203897 \nu^{15} + 168786 \nu^{14} + 4840280 \nu^{13} - 3291564 \nu^{12} - 45092591 \nu^{11} + 22541886 \nu^{10} + 210501617 \nu^{9} - 61808511 \nu^{8} - 517413267 \nu^{7} + 41832298 \nu^{6} + 627687626 \nu^{5} + 64022150 \nu^{4} - 296442140 \nu^{3} - 51140001 \nu^{2} + 37586128 \nu + 8409984\)\()/311984\)
\(\beta_{14}\)\(=\)\((\)\(-268687 \nu^{15} + 248146 \nu^{14} + 6403548 \nu^{13} - 4997680 \nu^{12} - 59968933 \nu^{11} + 36315186 \nu^{10} + 281841111 \nu^{9} - 114014061 \nu^{8} - 698535541 \nu^{7} + 137042970 \nu^{6} + 855515802 \nu^{5} - 15085386 \nu^{4} - 406520288 \nu^{3} - 17897027 \nu^{2} + 48368920 \nu + 5982064\)\()/311984\)
\(\beta_{15}\)\(=\)\((\)\(743697 \nu^{15} - 561258 \nu^{14} - 17803568 \nu^{13} + 10903780 \nu^{12} + 167505279 \nu^{11} - 74079278 \nu^{10} - 790224857 \nu^{9} + 198902831 \nu^{8} + 1960071611 \nu^{7} - 119133362 \nu^{6} - 2384502434 \nu^{5} - 224585838 \nu^{4} + 1101186116 \nu^{3} + 139655233 \nu^{2} - 120861728 \nu - 23649344\)\()/623968\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{13} - \beta_{11} - \beta_{10} + \beta_{6} - \beta_{5} + 6 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{12} + \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + 8 \beta_{2} - \beta_{1} + 14\)
\(\nu^{5}\)\(=\)\(\beta_{14} + 10 \beta_{13} - \beta_{12} - 10 \beta_{11} - 10 \beta_{10} + \beta_{9} - \beta_{7} + 9 \beta_{6} - 10 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 39 \beta_{1} + 7\)
\(\nu^{6}\)\(=\)\(\beta_{14} - \beta_{13} + 11 \beta_{12} + 12 \beta_{11} + 11 \beta_{10} + 12 \beta_{9} + 12 \beta_{8} - 13 \beta_{7} + 13 \beta_{6} + 10 \beta_{5} + \beta_{4} + \beta_{3} + 59 \beta_{2} - 13 \beta_{1} + 76\)
\(\nu^{7}\)\(=\)\(-\beta_{15} + 13 \beta_{14} + 83 \beta_{13} - 13 \beta_{12} - 80 \beta_{11} - 81 \beta_{10} + 11 \beta_{9} + 2 \beta_{8} - 12 \beta_{7} + 69 \beta_{6} - 87 \beta_{5} - 15 \beta_{4} - 16 \beta_{3} - 25 \beta_{2} + 265 \beta_{1} + 42\)
\(\nu^{8}\)\(=\)\(16 \beta_{14} - 19 \beta_{13} + 96 \beta_{12} + 112 \beta_{11} + 100 \beta_{10} + 108 \beta_{9} + 106 \beta_{8} - 124 \beta_{7} + 123 \beta_{6} + 81 \beta_{5} + 14 \beta_{4} + 12 \beta_{3} + 429 \beta_{2} - 125 \beta_{1} + 452\)
\(\nu^{9}\)\(=\)\(-17 \beta_{15} + 125 \beta_{14} + 651 \beta_{13} - 131 \beta_{12} - 597 \beta_{11} - 618 \beta_{10} + 88 \beta_{9} + 34 \beta_{8} - 110 \beta_{7} + 509 \beta_{6} - 722 \beta_{5} - 159 \beta_{4} - 180 \beta_{3} - 237 \beta_{2} + 1852 \beta_{1} + 250\)
\(\nu^{10}\)\(=\)\(-3 \beta_{15} + 171 \beta_{14} - 237 \beta_{13} + 773 \beta_{12} + 963 \beta_{11} + 854 \beta_{10} + 878 \beta_{9} + 843 \beta_{8} - 1061 \beta_{7} + 1039 \beta_{6} + 622 \beta_{5} + 139 \beta_{4} + 100 \beta_{3} + 3121 \beta_{2} - 1076 \beta_{1} + 2853\)
\(\nu^{11}\)\(=\)\(-196 \beta_{15} + 1071 \beta_{14} + 4994 \beta_{13} - 1199 \beta_{12} - 4347 \beta_{11} - 4624 \beta_{10} + 619 \beta_{9} + 379 \beta_{8} - 912 \beta_{7} + 3716 \beta_{6} - 5836 \beta_{5} - 1472 \beta_{4} - 1758 \beta_{3} - 2040 \beta_{2} + 13193 \beta_{1} + 1529\)
\(\nu^{12}\)\(=\)\(-63 \beta_{15} + 1545 \beta_{14} - 2457 \beta_{13} + 6004 \beta_{12} + 7979 \beta_{11} + 7061 \beta_{10} + 6817 \beta_{9} + 6418 \beta_{8} - 8638 \beta_{7} + 8344 \beta_{6} + 4706 \beta_{5} + 1215 \beta_{4} + 708 \beta_{3} + 22820 \beta_{2} - 8807 \beta_{1} + 18746\)
\(\nu^{13}\)\(=\)\(-1927 \beta_{15} + 8665 \beta_{14} + 37940 \beta_{13} - 10419 \beta_{12} - 31431 \beta_{11} - 34406 \beta_{10} + 4041 \beta_{9} + 3535 \beta_{8} - 7203 \beta_{7} + 27095 \beta_{6} - 46361 \beta_{5} - 12728 \beta_{4} - 15911 \beta_{3} - 16793 \beta_{2} + 95276 \beta_{1} + 9651\)
\(\nu^{14}\)\(=\)\(-838 \beta_{15} + 12813 \beta_{14} - 23034 \beta_{13} + 45821 \beta_{12} + 64739 \beta_{11} + 57186 \beta_{10} + 51735 \beta_{9} + 47943 \beta_{8} - 68494 \beta_{7} + 65322 \beta_{6} + 35510 \beta_{5} + 10007 \beta_{4} + 4501 \beta_{3} + 167840 \beta_{2} - 70290 \beta_{1} + 126701\)
\(\nu^{15}\)\(=\)\(-17420 \beta_{15} + 67905 \beta_{14} + 286953 \beta_{13} - 87569 \beta_{12} - 227366 \beta_{11} - 255923 \beta_{10} + 24932 \beta_{9} + 30047 \beta_{8} - 55434 \beta_{7} + 197977 \beta_{6} - 363846 \beta_{5} - 105776 \beta_{4} - 137308 \beta_{3} - 134974 \beta_{2} + 695043 \beta_{1} + 62508\)

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
−2.75586
−2.42429
−1.80038
−1.63671
−1.28888
−1.03000
−0.319955
−0.208883
0.445942
0.745705
1.84222
1.88851
2.13862
2.14844
2.51658
2.73896
−2.75586 2.22549 5.59478 3.97063 −6.13314 1.71912 −9.90674 1.95279 −10.9425
1.2 −2.42429 −1.02036 3.87720 −0.275174 2.47365 −5.16310 −4.55089 −1.95887 0.667103
1.3 −1.80038 −3.34524 1.24138 2.67126 6.02272 1.66993 1.36580 8.19063 −4.80929
1.4 −1.63671 −0.492566 0.678809 −1.90242 0.806186 0.625725 2.16240 −2.75738 3.11371
1.5 −1.28888 3.25447 −0.338793 4.18846 −4.19461 −3.67888 3.01442 7.59155 −5.39841
1.6 −1.03000 −0.920112 −0.939097 3.95819 0.947716 4.48231 3.02727 −2.15339 −4.07695
1.7 −0.319955 2.34279 −1.89763 1.26502 −0.749588 1.17463 1.24707 2.48866 −0.404751
1.8 −0.208883 −2.00378 −1.95637 −2.17278 0.418556 −2.92851 0.826420 1.01513 0.453858
1.9 0.445942 −0.0589774 −1.80114 2.46971 −0.0263005 −2.81741 −1.69509 −2.99652 1.10135
1.10 0.745705 2.46941 −1.44392 −2.81935 1.84145 3.27962 −2.56815 3.09797 −2.10240
1.11 1.84222 1.43653 1.39376 2.07719 2.64640 4.15058 −1.11683 −0.936385 3.82664
1.12 1.88851 0.985425 1.56646 3.37829 1.86098 1.01841 −0.818750 −2.02894 6.37991
1.13 2.13862 3.19578 2.57368 −0.343330 6.83454 −2.49131 1.22687 7.21299 −0.734251
1.14 2.14844 −1.74064 2.61580 −1.42439 −3.73966 3.61731 1.32302 0.0298167 −3.06021
1.15 2.51658 −2.82858 4.33319 2.87619 −7.11834 −3.18638 5.87166 5.00084 7.23817
1.16 2.73896 1.50037 5.50188 −1.91750 4.10945 −0.472052 9.59151 −0.748891 −5.25195
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
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.d 16
3.b odd 2 1 6003.2.a.q 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
667.2.a.d 16 1.a even 1 1 trivial
6003.2.a.q 16 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{16} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(667))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 64 + 336 T - 505 T^{2} - 3011 T^{3} + 1842 T^{4} + 6612 T^{5} - 2496 T^{6} - 5877 T^{7} + 1860 T^{8} + 2592 T^{9} - 795 T^{10} - 597 T^{11} + 187 T^{12} + 68 T^{13} - 22 T^{14} - 3 T^{15} + T^{16} \)
$3$ \( 256 + 4736 T + 5760 T^{2} - 17000 T^{3} - 15553 T^{4} + 25490 T^{5} + 13436 T^{6} - 19520 T^{7} - 4215 T^{8} + 7682 T^{9} + 170 T^{10} - 1533 T^{11} + 142 T^{12} + 144 T^{13} - 23 T^{14} - 5 T^{15} + T^{16} \)
$5$ \( -33344 - 189152 T - 126004 T^{2} + 506820 T^{3} + 121383 T^{4} - 455162 T^{5} + 11026 T^{6} + 186834 T^{7} - 37488 T^{8} - 36247 T^{9} + 13033 T^{10} + 2509 T^{11} - 1738 T^{12} + 112 T^{13} + 73 T^{14} - 16 T^{15} + T^{16} \)
$7$ \( -278528 + 462848 T + 1052416 T^{2} - 2507520 T^{3} + 722656 T^{4} + 1341600 T^{5} - 697249 T^{6} - 303266 T^{7} + 198765 T^{8} + 36349 T^{9} - 27297 T^{10} - 2397 T^{11} + 1967 T^{12} + 80 T^{13} - 71 T^{14} - T^{15} + T^{16} \)
$11$ \( 192704 - 2800416 T + 11769660 T^{2} - 10788296 T^{3} - 3648929 T^{4} + 7486347 T^{5} - 602830 T^{6} - 1826592 T^{7} + 349083 T^{8} + 210270 T^{9} - 48606 T^{10} - 12314 T^{11} + 3032 T^{12} + 355 T^{13} - 89 T^{14} - 4 T^{15} + T^{16} \)
$13$ \( 2778588 - 5587680 T - 25609189 T^{2} + 59312151 T^{3} - 25878106 T^{4} - 17375677 T^{5} + 13935237 T^{6} + 487996 T^{7} - 2204535 T^{8} + 262700 T^{9} + 143883 T^{10} - 29994 T^{11} - 3368 T^{12} + 1175 T^{13} - 15 T^{14} - 15 T^{15} + T^{16} \)
$17$ \( 157952 - 177760 T - 2752156 T^{2} + 5853200 T^{3} + 2824521 T^{4} - 10365785 T^{5} + 1442989 T^{6} + 4897926 T^{7} - 1789102 T^{8} - 418537 T^{9} + 264344 T^{10} - 12608 T^{11} - 9977 T^{12} + 1466 T^{13} + 46 T^{14} - 20 T^{15} + T^{16} \)
$19$ \( -10881280 + 28087104 T + 58816980 T^{2} - 181769172 T^{3} + 105569981 T^{4} + 28426102 T^{5} - 33205918 T^{6} + 118384 T^{7} + 3936405 T^{8} - 245618 T^{9} - 240288 T^{10} + 16703 T^{11} + 8087 T^{12} - 437 T^{13} - 142 T^{14} + 4 T^{15} + T^{16} \)
$23$ \( ( -1 + T )^{16} \)
$29$ \( ( 1 + T )^{16} \)
$31$ \( 148582656 + 8717559264 T + 2371223812 T^{2} - 5508249476 T^{3} - 526978023 T^{4} + 1382678132 T^{5} - 73895026 T^{6} - 148682956 T^{7} + 23279091 T^{8} + 5436106 T^{9} - 1223305 T^{10} - 80111 T^{11} + 26471 T^{12} + 411 T^{13} - 263 T^{14} + T^{16} \)
$37$ \( 78877952 + 360276192 T - 825260124 T^{2} - 2577467016 T^{3} + 2111918221 T^{4} + 2388536953 T^{5} - 688859210 T^{6} - 228216883 T^{7} + 56882942 T^{8} + 8696813 T^{9} - 2035868 T^{10} - 160616 T^{11} + 35983 T^{12} + 1436 T^{13} - 307 T^{14} - 5 T^{15} + T^{16} \)
$41$ \( 4709632 + 30296704 T + 46444496 T^{2} + 152752 T^{3} - 48516648 T^{4} - 29183744 T^{5} + 7240949 T^{6} + 11013179 T^{7} + 1775085 T^{8} - 1068376 T^{9} - 368239 T^{10} + 4820 T^{11} + 12878 T^{12} + 568 T^{13} - 177 T^{14} - 7 T^{15} + T^{16} \)
$43$ \( 489266892 + 2796193224 T + 4712736955 T^{2} + 1720403525 T^{3} - 1823254235 T^{4} - 1005582196 T^{5} + 237675257 T^{6} + 157411583 T^{7} - 10979894 T^{8} - 10194352 T^{9} + 42825 T^{10} + 294863 T^{11} + 8158 T^{12} - 3760 T^{13} - 175 T^{14} + 17 T^{15} + T^{16} \)
$47$ \( -3749718976 + 13316209472 T - 11963351428 T^{2} - 929984284 T^{3} + 5625927051 T^{4} - 1987146058 T^{5} - 310961688 T^{6} + 283399393 T^{7} - 28600325 T^{8} - 10819537 T^{9} + 2516889 T^{10} + 14480 T^{11} - 51128 T^{12} + 4341 T^{13} + 114 T^{14} - 29 T^{15} + T^{16} \)
$53$ \( 114419215472 + 649249375200 T - 1675777940616 T^{2} + 1490268381784 T^{3} - 638044950697 T^{4} + 121230037608 T^{5} + 4222244827 T^{6} - 6641141654 T^{7} + 1149683887 T^{8} - 16941187 T^{9} - 19942965 T^{10} + 2690377 T^{11} - 76948 T^{12} - 13227 T^{13} + 1497 T^{14} - 63 T^{15} + T^{16} \)
$59$ \( 1514229698560 + 2848352324096 T + 38592008592 T^{2} - 677964912680 T^{3} + 17032174569 T^{4} + 55601933241 T^{5} - 2866027128 T^{6} - 2248556651 T^{7} + 151189086 T^{8} + 50199707 T^{9} - 3870547 T^{10} - 628879 T^{11} + 52568 T^{12} + 4127 T^{13} - 363 T^{14} - 11 T^{15} + T^{16} \)
$61$ \( 1260900834304 - 1721765584768 T - 657416162160 T^{2} + 783388811456 T^{3} + 241028860056 T^{4} - 60227071248 T^{5} - 19693569343 T^{6} + 1835213786 T^{7} + 688769878 T^{8} - 26104720 T^{9} - 12264003 T^{10} + 173556 T^{11} + 115786 T^{12} - 442 T^{13} - 546 T^{14} + T^{16} \)
$67$ \( 10321323188224 - 4187483445248 T - 2540905354304 T^{2} + 822271112608 T^{3} + 248818401452 T^{4} - 64286170948 T^{5} - 12669928353 T^{6} + 2595172170 T^{7} + 371787768 T^{8} - 58539276 T^{9} - 6550150 T^{10} + 739996 T^{11} + 68939 T^{12} - 4877 T^{13} - 402 T^{14} + 13 T^{15} + T^{16} \)
$71$ \( 13531730176 + 10907213312 T - 48434469940 T^{2} - 76460846648 T^{3} - 30989190473 T^{4} + 3753944191 T^{5} + 4715805584 T^{6} + 455793341 T^{7} - 216911388 T^{8} - 37916416 T^{9} + 3434685 T^{10} + 930098 T^{11} - 3931 T^{12} - 8507 T^{13} - 252 T^{14} + 23 T^{15} + T^{16} \)
$73$ \( -194107921412096 + 22440956319232 T + 72897514237616 T^{2} - 3221198995104 T^{3} - 6234212087576 T^{4} - 50073176560 T^{5} + 224095705123 T^{6} + 9366628506 T^{7} - 3960613110 T^{8} - 264827235 T^{9} + 34680545 T^{10} + 3209788 T^{11} - 122910 T^{12} - 17973 T^{13} - 66 T^{14} + 38 T^{15} + T^{16} \)
$79$ \( -238723830651340 - 517010111001372 T - 384299796406363 T^{2} - 108274373950300 T^{3} - 3438594995398 T^{4} + 4141849474220 T^{5} + 712918813246 T^{6} - 5475604861 T^{7} - 11123646624 T^{8} - 713628937 T^{9} + 51912104 T^{10} + 6761840 T^{11} + 6249 T^{12} - 22977 T^{13} - 571 T^{14} + 27 T^{15} + T^{16} \)
$83$ \( -4205662739456 + 4054497565696 T + 3291040884864 T^{2} - 1087698930112 T^{3} - 598619629724 T^{4} + 114475591208 T^{5} + 44481748949 T^{6} - 5920639933 T^{7} - 1508513674 T^{8} + 164764364 T^{9} + 21804711 T^{10} - 2550131 T^{11} - 101415 T^{12} + 17303 T^{13} - 132 T^{14} - 36 T^{15} + T^{16} \)
$89$ \( -16474890357760 - 8883118201984 T + 5111658599664 T^{2} + 2965007549088 T^{3} - 437074318649 T^{4} - 320887789194 T^{5} + 7653282748 T^{6} + 14351472399 T^{7} + 421941931 T^{8} - 270855086 T^{9} - 13167326 T^{10} + 2415843 T^{11} + 137150 T^{12} - 10112 T^{13} - 613 T^{14} + 16 T^{15} + T^{16} \)
$97$ \( -7348432195646208 - 71679253206528 T + 986976636800112 T^{2} + 60450089250208 T^{3} - 45364562106584 T^{4} - 4255390108176 T^{5} + 911941533591 T^{6} + 111646572778 T^{7} - 8235719551 T^{8} - 1378720110 T^{9} + 24813298 T^{10} + 8566709 T^{11} + 78665 T^{12} - 25832 T^{13} - 626 T^{14} + 30 T^{15} + T^{16} \)
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