Properties

Label 2009.2.a.s
Level $2009$
Weight $2$
Character orbit 2009.a
Self dual yes
Analytic conductor $16.042$
Analytic rank $0$
Dimension $17$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2009 = 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2009.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.0419457661\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Defining polynomial: \(x^{17} - 3 x^{16} - 25 x^{15} + 77 x^{14} + 247 x^{13} - 790 x^{12} - 1231 x^{11} + 4173 x^{10} + 3251 x^{9} - 12183 x^{8} - 4259 x^{7} + 19567 x^{6} + 2029 x^{5} - 16136 x^{4} + 299 x^{3} + 5775 x^{2} - 312 x - 464\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 287)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} -\beta_{12} q^{3} + ( 1 + \beta_{2} ) q^{4} -\beta_{13} q^{5} + ( -\beta_{4} - \beta_{12} + \beta_{15} - \beta_{16} ) q^{6} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} + \beta_{8} - \beta_{11} + \beta_{13} - \beta_{15} ) q^{8} + ( 2 - \beta_{14} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} -\beta_{12} q^{3} + ( 1 + \beta_{2} ) q^{4} -\beta_{13} q^{5} + ( -\beta_{4} - \beta_{12} + \beta_{15} - \beta_{16} ) q^{6} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} + \beta_{8} - \beta_{11} + \beta_{13} - \beta_{15} ) q^{8} + ( 2 - \beta_{14} ) q^{9} + ( -\beta_{1} - \beta_{4} - \beta_{6} - \beta_{13} + \beta_{15} - \beta_{16} ) q^{10} + ( 1 - \beta_{11} ) q^{11} + ( -\beta_{1} - \beta_{3} + \beta_{4} + \beta_{7} + \beta_{8} - \beta_{11} - 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{15} - \beta_{16} ) q^{12} + ( -1 + \beta_{1} + \beta_{5} - \beta_{12} ) q^{13} + ( 2 - \beta_{1} - \beta_{4} + \beta_{9} - \beta_{10} - \beta_{13} ) q^{15} + ( 1 + 2 \beta_{2} + \beta_{3} - \beta_{9} + \beta_{12} ) q^{16} + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{10} + \beta_{13} ) q^{17} + ( 1 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{8} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{18} + ( \beta_{5} + \beta_{8} ) q^{19} + ( \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} - \beta_{13} - \beta_{14} ) q^{20} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{15} ) q^{22} + ( -\beta_{3} - \beta_{4} + \beta_{7} + \beta_{9} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} - \beta_{16} ) q^{23} + ( 1 - 4 \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{10} - \beta_{11} - \beta_{12} + \beta_{14} - \beta_{16} ) q^{24} + ( 1 + \beta_{3} - \beta_{4} - \beta_{8} - \beta_{13} + \beta_{14} + \beta_{16} ) q^{25} + ( 3 - \beta_{2} - 2 \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{11} - 2 \beta_{13} + 2 \beta_{15} ) q^{26} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} + 2 \beta_{11} - \beta_{12} - 2 \beta_{13} + \beta_{15} + \beta_{16} ) q^{27} + ( \beta_{2} + \beta_{4} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{15} + \beta_{16} ) q^{29} + ( -3 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{8} + \beta_{9} - \beta_{12} + 2 \beta_{13} + \beta_{14} - \beta_{15} - \beta_{16} ) q^{30} + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{12} + \beta_{16} ) q^{31} + ( 1 + \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{11} - \beta_{12} + 2 \beta_{13} - 2 \beta_{15} ) q^{32} + ( \beta_{3} - \beta_{5} - 2 \beta_{8} - \beta_{9} - \beta_{13} + \beta_{14} + 2 \beta_{16} ) q^{33} + ( -2 - \beta_{4} + \beta_{7} - \beta_{11} + \beta_{13} + \beta_{14} + \beta_{16} ) q^{34} + ( 5 - \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{10} + 2 \beta_{11} + \beta_{12} - 3 \beta_{13} - \beta_{14} + 2 \beta_{15} + \beta_{16} ) q^{36} + ( 2 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} + \beta_{9} + 2 \beta_{11} - \beta_{13} + 2 \beta_{15} ) q^{37} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{9} + 2 \beta_{11} + 2 \beta_{12} - 3 \beta_{13} - \beta_{14} + 2 \beta_{15} + \beta_{16} ) q^{38} + ( 3 + \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} + 2 \beta_{15} - \beta_{16} ) q^{39} + ( 3 - \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{5} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + \beta_{11} + 2 \beta_{12} - 3 \beta_{13} - \beta_{14} + 2 \beta_{15} + \beta_{16} ) q^{40} + q^{41} + ( 1 - \beta_{2} - \beta_{6} + \beta_{10} + \beta_{11} + \beta_{12} ) q^{43} + ( \beta_{1} + \beta_{2} - 2 \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} + 2 \beta_{12} - 3 \beta_{13} + \beta_{16} ) q^{44} + ( 1 - \beta_{1} - \beta_{2} - \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} + \beta_{16} ) q^{45} + ( -1 + 2 \beta_{2} + 2 \beta_{5} + 2 \beta_{7} - \beta_{11} - 3 \beta_{12} + 3 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} - \beta_{16} ) q^{46} + ( 1 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{9} + \beta_{13} ) q^{47} + ( -6 + \beta_{1} + \beta_{2} - 3 \beta_{3} + 4 \beta_{4} + \beta_{5} + \beta_{6} + 3 \beta_{7} + 3 \beta_{8} - 3 \beta_{11} - 3 \beta_{12} + 7 \beta_{13} + \beta_{14} - 5 \beta_{15} - 2 \beta_{16} ) q^{48} + ( -3 + 3 \beta_{1} - \beta_{3} + 2 \beta_{5} + \beta_{6} + \beta_{9} + \beta_{10} - \beta_{12} + 3 \beta_{13} + 2 \beta_{14} - \beta_{16} ) q^{50} + ( 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + \beta_{11} + \beta_{12} - \beta_{15} + 2 \beta_{16} ) q^{51} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{7} + \beta_{10} + \beta_{11} + \beta_{13} - \beta_{15} ) q^{52} + ( 1 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{16} ) q^{53} + ( 3 - \beta_{1} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{8} + \beta_{9} - 3 \beta_{12} + 2 \beta_{13} + \beta_{15} - 2 \beta_{16} ) q^{54} + ( -1 + 2 \beta_{2} + \beta_{5} + \beta_{7} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{55} + ( -1 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} - 2 \beta_{12} + 2 \beta_{13} + \beta_{14} - \beta_{15} - 2 \beta_{16} ) q^{57} + ( -1 + 4 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + 3 \beta_{12} - 2 \beta_{13} + 2 \beta_{15} + \beta_{16} ) q^{58} + ( -1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{10} - 2 \beta_{11} + \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{59} + ( 4 - \beta_{1} + \beta_{2} + 3 \beta_{3} - 4 \beta_{4} - \beta_{5} - 2 \beta_{7} - 4 \beta_{8} + 2 \beta_{11} + 2 \beta_{12} - 5 \beta_{13} + \beta_{14} + 3 \beta_{15} + 2 \beta_{16} ) q^{60} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{8} + \beta_{9} - \beta_{11} + \beta_{13} - \beta_{16} ) q^{61} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{12} - \beta_{13} + 2 \beta_{16} ) q^{62} + ( 4 - 2 \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{6} + 2 \beta_{7} - \beta_{10} - \beta_{11} - \beta_{12} + 2 \beta_{15} - 2 \beta_{16} ) q^{64} + ( 2 - 2 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} - 2 \beta_{12} + \beta_{15} - 2 \beta_{16} ) q^{65} + ( -4 + 2 \beta_{1} + \beta_{2} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{11} + 4 \beta_{13} + \beta_{14} - \beta_{15} - \beta_{16} ) q^{66} + ( 5 + \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} - \beta_{8} - \beta_{9} + 3 \beta_{11} + 2 \beta_{12} - 3 \beta_{13} - \beta_{14} + \beta_{15} + 2 \beta_{16} ) q^{67} + ( 2 + \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} + 2 \beta_{11} + 3 \beta_{13} - 2 \beta_{15} + \beta_{16} ) q^{68} + ( 2 + \beta_{1} + \beta_{3} - 2 \beta_{5} - \beta_{6} - \beta_{9} + \beta_{11} + \beta_{12} - 2 \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} ) q^{69} + ( 3 + \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - \beta_{7} - 2 \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} - 2 \beta_{14} - \beta_{15} + \beta_{16} ) q^{71} + ( -2 + 2 \beta_{1} - 2 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + 5 \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} + 5 \beta_{13} - \beta_{14} - 4 \beta_{15} - \beta_{16} ) q^{72} + ( -3 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{73} + ( 2 + 2 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{8} + \beta_{9} + \beta_{16} ) q^{74} + ( 4 - \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{11} + \beta_{12} - \beta_{13} - 2 \beta_{14} - \beta_{15} + \beta_{16} ) q^{75} + ( 2 \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} + \beta_{10} + 2 \beta_{11} + 2 \beta_{12} + \beta_{13} - 2 \beta_{14} + \beta_{16} ) q^{76} + ( 3 + 2 \beta_{1} - 4 \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{11} - 2 \beta_{12} - 4 \beta_{13} - 2 \beta_{14} + \beta_{15} + \beta_{16} ) q^{78} + ( 4 - \beta_{1} + \beta_{4} + \beta_{8} - \beta_{9} - 2 \beta_{12} + \beta_{13} - \beta_{14} - 2 \beta_{15} + \beta_{16} ) q^{79} + ( -6 + 3 \beta_{1} - 2 \beta_{3} + 4 \beta_{4} + \beta_{6} - \beta_{7} - 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + \beta_{12} + 2 \beta_{13} - 3 \beta_{15} + \beta_{16} ) q^{80} + ( 5 - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{12} - \beta_{13} - \beta_{14} + 2 \beta_{15} - 3 \beta_{16} ) q^{81} + \beta_{1} q^{82} + ( -3 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} - \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{83} + ( 3 - 5 \beta_{1} - 3 \beta_{2} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{12} - \beta_{13} ) q^{85} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} + 3 \beta_{12} - 5 \beta_{13} - \beta_{14} + 3 \beta_{15} + 2 \beta_{16} ) q^{86} + ( -1 - \beta_{1} + 2 \beta_{3} - \beta_{5} - \beta_{7} - 3 \beta_{8} - \beta_{9} + \beta_{10} - \beta_{13} + \beta_{14} + \beta_{15} + 2 \beta_{16} ) q^{87} + ( 3 - 2 \beta_{1} + 4 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{11} + \beta_{12} - \beta_{14} - 2 \beta_{15} - \beta_{16} ) q^{88} + ( 1 + 2 \beta_{1} + \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{11} + \beta_{12} + \beta_{16} ) q^{89} + ( -\beta_{2} - 4 \beta_{4} + \beta_{5} - 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + 3 \beta_{11} + \beta_{12} - 2 \beta_{13} - \beta_{14} + 5 \beta_{15} - 2 \beta_{16} ) q^{90} + ( \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} - 3 \beta_{11} - 4 \beta_{12} + 4 \beta_{13} + 3 \beta_{14} - 2 \beta_{15} - \beta_{16} ) q^{92} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{12} + \beta_{15} + \beta_{16} ) q^{93} + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} - \beta_{8} + 2 \beta_{12} + \beta_{14} + \beta_{15} + \beta_{16} ) q^{94} + ( 2 \beta_{1} - \beta_{2} + 2 \beta_{5} + \beta_{7} - \beta_{9} + \beta_{15} ) q^{95} + ( 7 - 6 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - \beta_{5} + 2 \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{11} + \beta_{13} + \beta_{14} + \beta_{15} - \beta_{16} ) q^{96} + ( 1 + 3 \beta_{1} - 3 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{9} - \beta_{12} + 2 \beta_{13} - \beta_{16} ) q^{97} + ( 2 + \beta_{1} + \beta_{2} + 4 \beta_{4} + \beta_{5} + 3 \beta_{6} - \beta_{7} - \beta_{9} + 3 \beta_{10} - \beta_{11} + 3 \beta_{12} + 3 \beta_{13} - 2 \beta_{14} - \beta_{15} + \beta_{16} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 3q^{2} + q^{3} + 25q^{4} - q^{5} + 2q^{6} + 9q^{8} + 26q^{9} + O(q^{10}) \) \( 17q + 3q^{2} + q^{3} + 25q^{4} - q^{5} + 2q^{6} + 9q^{8} + 26q^{9} - 2q^{10} + 15q^{11} + 4q^{12} - 5q^{13} + 24q^{15} + 33q^{16} + 4q^{17} + 10q^{18} + 5q^{19} - 26q^{20} + 16q^{22} + 12q^{23} + 16q^{24} + 24q^{25} + 31q^{26} - 11q^{27} + 14q^{29} - 33q^{30} - 3q^{31} + 16q^{32} + 4q^{33} - 24q^{34} + 57q^{36} + 24q^{37} + 45q^{39} + 36q^{40} + 17q^{41} + 14q^{43} - 9q^{44} - 21q^{45} + 44q^{46} + 19q^{47} - 60q^{48} - 4q^{50} + 2q^{51} - 25q^{52} + 4q^{53} + 68q^{54} + 9q^{55} - 12q^{57} - q^{58} - 27q^{59} + 66q^{60} - q^{61} - 23q^{62} + 75q^{64} + 22q^{65} - 16q^{66} + 49q^{67} + 45q^{68} + 12q^{69} + 40q^{71} - 23q^{72} - 14q^{73} + 33q^{74} + 27q^{75} - 9q^{76} - 12q^{78} + 61q^{79} - 82q^{80} + 53q^{81} + 3q^{82} - 18q^{83} - 13q^{85} - 4q^{86} - 17q^{87} + 74q^{88} + 18q^{89} - 20q^{90} + 28q^{92} - 36q^{93} - 5q^{94} + 20q^{95} + 148q^{96} + 26q^{97} + 38q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{17} - 3 x^{16} - 25 x^{15} + 77 x^{14} + 247 x^{13} - 790 x^{12} - 1231 x^{11} + 4173 x^{10} + 3251 x^{9} - 12183 x^{8} - 4259 x^{7} + 19567 x^{6} + 2029 x^{5} - 16136 x^{4} + 299 x^{3} + 5775 x^{2} - 312 x - 464\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\(1012 \nu^{16} - 2475 \nu^{15} - 26247 \nu^{14} + 61458 \nu^{13} + 275296 \nu^{12} - 601415 \nu^{11} - 1515086 \nu^{10} + 2962988 \nu^{9} + 4740015 \nu^{8} - 7769389 \nu^{7} - 8475370 \nu^{6} + 10431266 \nu^{5} + 8194565 \nu^{4} - 6101901 \nu^{3} - 3684569 \nu^{2} + 936964 \nu + 369456\)\()/10256\)
\(\beta_{4}\)\(=\)\((\)\(-1180 \nu^{16} + 2217 \nu^{15} + 31795 \nu^{14} - 55182 \nu^{13} - 346840 \nu^{12} + 539297 \nu^{11} + 1973136 \nu^{10} - 2633450 \nu^{9} - 6263143 \nu^{8} + 6747799 \nu^{7} + 10960328 \nu^{6} - 8630228 \nu^{5} - 9827649 \nu^{4} + 4543791 \nu^{3} + 3931049 \nu^{2} - 508260 \nu - 376144\)\()/10256\)
\(\beta_{5}\)\(=\)\((\)\(-1754 \nu^{16} + 3579 \nu^{15} + 46691 \nu^{14} - 88224 \nu^{13} - 504106 \nu^{12} + 853001 \nu^{11} + 2854300 \nu^{10} - 4121206 \nu^{9} - 9121531 \nu^{8} + 10481757 \nu^{7} + 16381600 \nu^{6} - 13462460 \nu^{5} - 15462265 \nu^{4} + 7431779 \nu^{3} + 6544913 \nu^{2} - 1066090 \nu - 600696\)\()/10256\)
\(\beta_{6}\)\(=\)\((\)\(1867 \nu^{16} - 3932 \nu^{15} - 49217 \nu^{14} + 95678 \nu^{13} + 526771 \nu^{12} - 909773 \nu^{11} - 2966850 \nu^{10} + 4303909 \nu^{9} + 9491036 \nu^{8} - 10671527 \nu^{7} - 17204720 \nu^{6} + 13322273 \nu^{5} + 16491282 \nu^{4} - 7130644 \nu^{3} - 7025889 \nu^{2} + 900136 \nu + 660992\)\()/10256\)
\(\beta_{7}\)\(=\)\((\)\(-2143 \nu^{16} + 3966 \nu^{15} + 58881 \nu^{14} - 98920 \nu^{13} - 658959 \nu^{12} + 969895 \nu^{11} + 3876772 \nu^{10} - 4764225 \nu^{9} - 12861598 \nu^{8} + 12366051 \nu^{7} + 23858202 \nu^{6} - 16346061 \nu^{5} - 23007752 \nu^{4} + 9489468 \nu^{3} + 9741717 \nu^{2} - 1532230 \nu - 888088\)\()/10256\)
\(\beta_{8}\)\(=\)\((\)\(-1364 \nu^{16} + 2667 \nu^{15} + 36742 \nu^{14} - 65249 \nu^{13} - 402488 \nu^{12} + 623821 \nu^{11} + 2316727 \nu^{10} - 2964258 \nu^{9} - 7522617 \nu^{8} + 7363186 \nu^{7} + 13645839 \nu^{6} - 9177342 \nu^{5} - 12792103 \nu^{4} + 4924294 \nu^{3} + 5184572 \nu^{2} - 703383 \nu - 441220\)\()/5128\)
\(\beta_{9}\)\(=\)\((\)\(-3287 \nu^{16} + 5231 \nu^{15} + 91227 \nu^{14} - 129349 \nu^{13} - 1033873 \nu^{12} + 1253924 \nu^{11} + 6178111 \nu^{10} - 6069903 \nu^{9} - 20897723 \nu^{8} + 15475041 \nu^{7} + 39708899 \nu^{6} - 20066009 \nu^{5} - 39356425 \nu^{4} + 11481724 \nu^{3} + 16962947 \nu^{2} - 1798211 \nu - 1559124\)\()/10256\)
\(\beta_{10}\)\(=\)\((\)\( 3 \nu^{16} - 6 \nu^{15} - 80 \nu^{14} + 149 \nu^{13} + 865 \nu^{12} - 1455 \nu^{11} - 4903 \nu^{10} + 7123 \nu^{9} + 15686 \nu^{8} - 18420 \nu^{7} - 28247 \nu^{6} + 24079 \nu^{5} + 26846 \nu^{4} - 13355 \nu^{3} - 11494 \nu^{2} + 1741 \nu + 1060 \)\()/8\)
\(\beta_{11}\)\(=\)\((\)\(-3916 \nu^{16} + 7264 \nu^{15} + 106581 \nu^{14} - 180711 \nu^{13} - 1181046 \nu^{12} + 1767956 \nu^{11} + 6885565 \nu^{10} - 8673558 \nu^{9} - 22700848 \nu^{8} + 22512843 \nu^{7} + 42065275 \nu^{6} - 29744822 \nu^{5} - 40791454 \nu^{4} + 17112577 \nu^{3} + 17391111 \nu^{2} - 2597117 \nu - 1550588\)\()/10256\)
\(\beta_{12}\)\(=\)\((\)\(-4299 \nu^{16} + 7706 \nu^{15} + 117474 \nu^{14} - 190807 \nu^{13} - 1309169 \nu^{12} + 1855339 \nu^{11} + 7693197 \nu^{10} - 9032891 \nu^{9} - 25637738 \nu^{8} + 23244430 \nu^{7} + 48184269 \nu^{6} - 30497275 \nu^{5} - 47540734 \nu^{4} + 17583625 \nu^{3} + 20565468 \nu^{2} - 2735175 \nu - 1836276\)\()/10256\)
\(\beta_{13}\)\(=\)\((\)\(5598 \nu^{16} - 10221 \nu^{15} - 151810 \nu^{14} + 252761 \nu^{13} + 1674636 \nu^{12} - 2452715 \nu^{11} - 9707837 \nu^{10} + 11899406 \nu^{9} + 31773615 \nu^{8} - 30428332 \nu^{7} - 58311343 \nu^{6} + 39456992 \nu^{5} + 55816819 \nu^{4} - 22265532 \nu^{3} - 23455130 \nu^{2} + 3373513 \nu + 2094940\)\()/10256\)
\(\beta_{14}\)\(=\)\((\)\(-6218 \nu^{16} + 11375 \nu^{15} + 167462 \nu^{14} - 280147 \nu^{13} - 1831712 \nu^{12} + 2704861 \nu^{11} + 10510963 \nu^{10} - 13042958 \nu^{9} - 33991461 \nu^{8} + 33100320 \nu^{7} + 61499141 \nu^{6} - 42465504 \nu^{5} - 57875093 \nu^{4} + 23498876 \nu^{3} + 23881994 \nu^{2} - 3371975 \nu - 2091540\)\()/10256\)
\(\beta_{15}\)\(=\)\((\)\(6461 \nu^{16} - 11391 \nu^{15} - 176082 \nu^{14} + 282012 \nu^{13} + 1955341 \nu^{12} - 2742090 \nu^{11} - 11438576 \nu^{10} + 13351919 \nu^{9} + 37917107 \nu^{8} - 34369142 \nu^{7} - 70853962 \nu^{6} + 45107909 \nu^{5} + 69493135 \nu^{4} - 26024729 \nu^{3} - 29877112 \nu^{2} + 4070056 \nu + 2647712\)\()/10256\)
\(\beta_{16}\)\(=\)\((\)\(6749 \nu^{16} - 11315 \nu^{15} - 185135 \nu^{14} + 280685 \nu^{13} + 2070479 \nu^{12} - 2735598 \nu^{11} - 12198073 \nu^{10} + 13356571 \nu^{9} + 40687701 \nu^{8} - 34486543 \nu^{7} - 76377301 \nu^{6} + 45417349 \nu^{5} + 75076479 \nu^{4} - 26301276 \nu^{3} - 32282079 \nu^{2} + 4135927 \nu + 2865396\)\()/10256\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(-\beta_{15} + \beta_{13} - \beta_{11} + \beta_{8} + \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{12} - \beta_{9} + \beta_{3} + 8 \beta_{2} + 15\)
\(\nu^{5}\)\(=\)\(-10 \beta_{15} + 10 \beta_{13} - \beta_{12} - 9 \beta_{11} + 9 \beta_{8} + \beta_{7} + 8 \beta_{6} - \beta_{5} + 10 \beta_{4} - 9 \beta_{3} + 8 \beta_{2} + 29 \beta_{1} + 1\)
\(\nu^{6}\)\(=\)\(-2 \beta_{16} + 2 \beta_{15} + 9 \beta_{12} - \beta_{11} - \beta_{10} - 10 \beta_{9} + 2 \beta_{7} - \beta_{6} - 2 \beta_{4} + 10 \beta_{3} + 57 \beta_{2} - 2 \beta_{1} + 90\)
\(\nu^{7}\)\(=\)\(-2 \beta_{16} - 85 \beta_{15} - \beta_{14} + 85 \beta_{13} - 17 \beta_{12} - 70 \beta_{11} + \beta_{10} - \beta_{9} + 71 \beta_{8} + 15 \beta_{7} + 56 \beta_{6} - 10 \beta_{5} + 84 \beta_{4} - 70 \beta_{3} + 56 \beta_{2} + 181 \beta_{1} + 8\)
\(\nu^{8}\)\(=\)\(-30 \beta_{16} + 33 \beta_{15} - 4 \beta_{13} + 63 \beta_{12} - 12 \beta_{11} - 17 \beta_{10} - 79 \beta_{9} + 29 \beta_{7} - 16 \beta_{6} - 5 \beta_{5} - 36 \beta_{4} + 81 \beta_{3} + 397 \beta_{2} - 38 \beta_{1} + 590\)
\(\nu^{9}\)\(=\)\(-37 \beta_{16} - 687 \beta_{15} - 17 \beta_{14} + 694 \beta_{13} - 199 \beta_{12} - 525 \beta_{11} + 15 \beta_{10} - 14 \beta_{9} + 549 \beta_{8} + 159 \beta_{7} + 386 \beta_{6} - 72 \beta_{5} + 670 \beta_{4} - 529 \beta_{3} + 381 \beta_{2} + 1179 \beta_{1} + 31\)
\(\nu^{10}\)\(=\)\(-317 \beta_{16} + 372 \beta_{15} + \beta_{14} - 71 \beta_{13} + 404 \beta_{12} - 107 \beta_{11} - 199 \beta_{10} - 580 \beta_{9} - \beta_{8} + 304 \beta_{7} - 177 \beta_{6} - 89 \beta_{5} - 433 \beta_{4} + 621 \beta_{3} + 2761 \beta_{2} - 474 \beta_{1} + 4053\)
\(\nu^{11}\)\(=\)\(-449 \beta_{16} - 5431 \beta_{15} - 189 \beta_{14} + 5576 \beta_{13} - 1984 \beta_{12} - 3893 \beta_{11} + 162 \beta_{10} - 135 \beta_{9} + 4226 \beta_{8} + 1478 \beta_{7} + 2679 \beta_{6} - 447 \beta_{5} + 5232 \beta_{4} - 3979 \beta_{3} + 2572 \beta_{2} + 7899 \beta_{1} - 95\)
\(\nu^{12}\)\(=\)\(-2913 \beta_{16} + 3597 \beta_{15} + 30 \beta_{14} - 844 \beta_{13} + 2485 \beta_{12} - 857 \beta_{11} - 1987 \beta_{10} - 4141 \beta_{9} - 35 \beta_{8} + 2816 \beta_{7} - 1690 \beta_{6} - 1050 \beta_{5} - 4399 \beta_{4} + 4694 \beta_{3} + 19295 \beta_{2} - 4926 \beta_{1} + 28587\)
\(\nu^{13}\)\(=\)\(-4545 \beta_{16} - 42440 \beta_{15} - 1756 \beta_{14} + 44376 \beta_{13} - 18137 \beta_{12} - 28787 \beta_{11} + 1541 \beta_{10} - 1105 \beta_{9} + 32458 \beta_{8} + 12868 \beta_{7} + 18811 \beta_{6} - 2466 \beta_{5} + 40437 \beta_{4} - 29965 \beta_{3} + 17321 \beta_{2} + 54013 \beta_{1} - 3431\)
\(\nu^{14}\)\(=\)\(-24926 \beta_{16} + 32170 \beta_{15} + 512 \beta_{14} - 8489 \beta_{13} + 14921 \beta_{12} - 6486 \beta_{11} - 18198 \beta_{10} - 29250 \beta_{9} - 641 \beta_{8} + 24485 \beta_{7} - 14964 \beta_{6} - 10401 \beta_{5} - 40861 \beta_{4} + 35447 \beta_{3} + 135755 \beta_{2} - 46372 \beta_{1} + 204915\)
\(\nu^{15}\)\(=\)\(-41693 \beta_{16} - 329388 \beta_{15} - 14858 \beta_{14} + 350609 \beta_{13} - 157302 \beta_{12} - 212980 \beta_{11} + 13730 \beta_{10} - 8196 \beta_{9} + 248831 \beta_{8} + 107815 \beta_{7} + 133680 \beta_{6} - 11591 \beta_{5} + 310770 \beta_{4} - 226181 \beta_{3} + 116542 \beta_{2} + 375301 \beta_{1} - 46076\)
\(\nu^{16}\)\(=\)\(-204679 \beta_{16} + 275110 \beta_{15} + 6695 \beta_{14} - 78272 \beta_{13} + 88141 \beta_{12} - 47204 \beta_{11} - 158101 \beta_{10} - 205953 \beta_{9} - 8799 \beta_{8} + 204928 \beta_{7} - 126807 \beta_{6} - 93843 \beta_{5} - 359680 \beta_{4} + 268516 \beta_{3} + 961978 \beta_{2} - 411186 \beta_{1} + 1484844\)

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
−2.76801
−2.48154
−1.98553
−1.77896
−1.66890
−0.972044
−0.902479
−0.290774
0.381923
1.11543
1.17557
1.20954
1.75743
2.36949
2.52950
2.57424
2.73510
−2.76801 −2.85049 5.66188 −3.02107 7.89019 0 −10.1361 5.12530 8.36234
1.2 −2.48154 1.40932 4.15802 −1.01447 −3.49728 0 −5.35521 −1.01382 2.51746
1.3 −1.98553 2.88783 1.94233 3.91440 −5.73388 0 0.114499 5.33958 −7.77216
1.4 −1.77896 1.20779 1.16468 −2.53283 −2.14861 0 1.48599 −1.54123 4.50580
1.5 −1.66890 −0.213548 0.785231 1.00334 0.356391 0 2.02733 −2.95440 −1.67447
1.6 −0.972044 1.50665 −1.05513 2.83631 −1.46453 0 2.96972 −0.730003 −2.75702
1.7 −0.902479 −3.32518 −1.18553 −0.938362 3.00091 0 2.87488 8.05685 0.846852
1.8 −0.290774 −1.26632 −1.91545 3.23790 0.368212 0 1.13851 −1.39645 −0.941496
1.9 0.381923 −0.350486 −1.85413 −2.79025 −0.133859 0 −1.47198 −2.87716 −1.06566
1.10 1.11543 −2.75409 −0.755825 −1.34462 −3.07199 0 −3.07392 4.58503 −1.49982
1.11 1.17557 2.72860 −0.618039 −2.82853 3.20766 0 −3.07768 4.44528 −3.32513
1.12 1.20954 0.150404 −0.537006 3.32743 0.181921 0 −3.06862 −2.97738 4.02467
1.13 1.75743 3.22128 1.08856 1.92164 5.66118 0 −1.60179 7.37665 3.37714
1.14 2.36949 −1.64632 3.61450 −4.02976 −3.90094 0 3.82554 −0.289637 −9.54849
1.15 2.52950 −2.69617 4.39838 0.644502 −6.81996 0 6.06670 4.26931 1.63027
1.16 2.57424 0.468252 4.62673 2.24224 1.20539 0 6.76185 −2.78074 5.77207
1.17 2.73510 2.52246 5.48079 −1.62786 6.89919 0 9.52032 3.36281 −4.45235
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(1\)
\(41\) \(-1\)

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 2009.2.a.s 17
7.b odd 2 1 2009.2.a.r 17
7.c even 3 2 287.2.e.d 34
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.2.e.d 34 7.c even 3 2
2009.2.a.r 17 7.b odd 2 1
2009.2.a.s 17 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2009))\):

\(T_{2}^{17} - \cdots\)
\(T_{3}^{17} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -464 - 312 T + 5775 T^{2} + 299 T^{3} - 16136 T^{4} + 2029 T^{5} + 19567 T^{6} - 4259 T^{7} - 12183 T^{8} + 3251 T^{9} + 4173 T^{10} - 1231 T^{11} - 790 T^{12} + 247 T^{13} + 77 T^{14} - 25 T^{15} - 3 T^{16} + T^{17} \)
$3$ \( -127 + 265 T + 4959 T^{2} - 3186 T^{3} - 32237 T^{4} + 24596 T^{5} + 40386 T^{6} - 32352 T^{7} - 20338 T^{8} + 16946 T^{9} + 4973 T^{10} - 4316 T^{11} - 631 T^{12} + 572 T^{13} + 40 T^{14} - 38 T^{15} - T^{16} + T^{17} \)
$5$ \( 169001 + 61397 T - 653363 T^{2} - 355972 T^{3} + 811339 T^{4} + 518708 T^{5} - 425932 T^{6} - 297618 T^{7} + 113888 T^{8} + 85362 T^{9} - 16747 T^{10} - 13444 T^{11} + 1367 T^{12} + 1180 T^{13} - 58 T^{14} - 54 T^{15} + T^{16} + T^{17} \)
$7$ \( T^{17} \)
$11$ \( -62868571 - 129243158 T + 3761764 T^{2} + 122935187 T^{3} + 13856243 T^{4} - 53174669 T^{5} - 2699737 T^{6} + 12480155 T^{7} - 481704 T^{8} - 1613435 T^{9} + 188058 T^{10} + 109379 T^{11} - 20511 T^{12} - 3199 T^{13} + 937 T^{14} + 3 T^{15} - 15 T^{16} + T^{17} \)
$13$ \( 1087488 + 18825216 T + 56161280 T^{2} + 44394240 T^{3} - 21097856 T^{4} - 38994240 T^{5} - 6758560 T^{6} + 8982224 T^{7} + 3520268 T^{8} - 650892 T^{9} - 493123 T^{10} - 13003 T^{11} + 28264 T^{12} + 3149 T^{13} - 649 T^{14} - 103 T^{15} + 5 T^{16} + T^{17} \)
$17$ \( 153257984 - 585559040 T + 69142016 T^{2} + 1184486656 T^{3} - 704501824 T^{4} - 478914016 T^{5} + 432553848 T^{6} - 15851352 T^{7} - 50397787 T^{8} + 6958903 T^{9} + 2521015 T^{10} - 460175 T^{11} - 63565 T^{12} + 13419 T^{13} + 799 T^{14} - 186 T^{15} - 4 T^{16} + T^{17} \)
$19$ \( 105370047 - 295748919 T + 37926001 T^{2} + 457139466 T^{3} - 303188182 T^{4} - 107120792 T^{5} + 132532289 T^{6} - 7279278 T^{7} - 18742522 T^{8} + 3036416 T^{9} + 1234905 T^{10} - 257464 T^{11} - 42210 T^{12} + 9530 T^{13} + 733 T^{14} - 160 T^{15} - 5 T^{16} + T^{17} \)
$23$ \( 7445248 + 204398848 T - 310086016 T^{2} - 610729664 T^{3} + 1476834176 T^{4} - 855421616 T^{5} - 68443960 T^{6} + 203985436 T^{7} - 38618675 T^{8} - 12601794 T^{9} + 4163593 T^{10} + 176298 T^{11} - 152477 T^{12} + 5235 T^{13} + 2283 T^{14} - 152 T^{15} - 12 T^{16} + T^{17} \)
$29$ \( 313574400 - 177116160 T - 4828136192 T^{2} + 3189524480 T^{3} + 3058408064 T^{4} - 2419986240 T^{5} - 181843232 T^{6} + 435623168 T^{7} - 53604692 T^{8} - 25099520 T^{9} + 5524823 T^{10} + 480668 T^{11} - 191075 T^{12} + 1747 T^{13} + 2756 T^{14} - 138 T^{15} - 14 T^{16} + T^{17} \)
$31$ \( -430336 + 6605056 T - 32256384 T^{2} + 65064640 T^{3} - 40607264 T^{4} - 35975568 T^{5} + 52036728 T^{6} - 6360800 T^{7} - 12878111 T^{8} + 4090859 T^{9} + 714743 T^{10} - 360962 T^{11} - 7715 T^{12} + 12130 T^{13} - 188 T^{14} - 180 T^{15} + 3 T^{16} + T^{17} \)
$37$ \( 8594641728 - 16457484096 T - 39739879152 T^{2} + 44556574176 T^{3} + 4653251712 T^{4} - 10762224048 T^{5} + 326718648 T^{6} + 1076009208 T^{7} - 87777660 T^{8} - 54827572 T^{9} + 6310529 T^{10} + 1452632 T^{11} - 217177 T^{12} - 16957 T^{13} + 3658 T^{14} + 6 T^{15} - 24 T^{16} + T^{17} \)
$41$ \( ( -1 + T )^{17} \)
$43$ \( -9456975872 + 28875489280 T - 22154190848 T^{2} - 13888847872 T^{3} + 29182514176 T^{4} - 14107059712 T^{5} + 452041344 T^{6} + 1509440960 T^{7} - 301799136 T^{8} - 50151124 T^{9} + 17075893 T^{10} + 342580 T^{11} - 394543 T^{12} + 12429 T^{13} + 4012 T^{14} - 230 T^{15} - 14 T^{16} + T^{17} \)
$47$ \( 134296840532 - 1165150192960 T + 2912473268312 T^{2} - 2569279921756 T^{3} + 943840049972 T^{4} - 67014446792 T^{5} - 53191560244 T^{6} + 14043574080 T^{7} + 14046301 T^{8} - 430969204 T^{9} + 41248791 T^{10} + 4542195 T^{11} - 863440 T^{12} - 2001 T^{13} + 6835 T^{14} - 235 T^{15} - 19 T^{16} + T^{17} \)
$53$ \( -351924508928 - 515409680896 T + 504418232320 T^{2} + 464659844224 T^{3} - 410319909408 T^{4} + 642259104 T^{5} + 51142146464 T^{6} - 5075060264 T^{7} - 2431504637 T^{8} + 322258400 T^{9} + 53305484 T^{10} - 8117814 T^{11} - 550119 T^{12} + 94652 T^{13} + 2514 T^{14} - 505 T^{15} - 4 T^{16} + T^{17} \)
$59$ \( 12653127175424 - 6790657948160 T - 5566010822528 T^{2} + 2762922784448 T^{3} + 862175276896 T^{4} - 380892733520 T^{5} - 66310846664 T^{6} + 22579453016 T^{7} + 3166398183 T^{8} - 646218164 T^{9} - 92686071 T^{10} + 8487813 T^{11} + 1493948 T^{12} - 30597 T^{13} - 11215 T^{14} - 213 T^{15} + 27 T^{16} + T^{17} \)
$61$ \( -44785569988 + 194299954900 T + 712685731397 T^{2} + 599537928223 T^{3} + 106355690804 T^{4} - 76406745109 T^{5} - 32993841706 T^{6} + 152549989 T^{7} + 1942958785 T^{8} + 198986759 T^{9} - 41653239 T^{10} - 6610551 T^{11} + 372574 T^{12} + 84393 T^{13} - 1300 T^{14} - 477 T^{15} + T^{16} + T^{17} \)
$67$ \( -40677894128 + 112434940808 T - 22620573996 T^{2} - 142007298440 T^{3} + 79085168620 T^{4} + 31738286372 T^{5} - 21343937800 T^{6} - 1889083204 T^{7} + 1952816196 T^{8} - 55531050 T^{9} - 72168291 T^{10} + 7304460 T^{11} + 823860 T^{12} - 159549 T^{13} + 3876 T^{14} + 662 T^{15} - 49 T^{16} + T^{17} \)
$71$ \( 22487137032128 - 4388316927488 T - 13907696636611 T^{2} + 3715058105628 T^{3} + 2203885385704 T^{4} - 623451604792 T^{5} - 138612687761 T^{6} + 42849944915 T^{7} + 3264278786 T^{8} - 1418426731 T^{9} + 359597 T^{10} + 22513400 T^{11} - 1033558 T^{12} - 147456 T^{13} + 12263 T^{14} + 167 T^{15} - 40 T^{16} + T^{17} \)
$73$ \( 144806948824176 + 55591128604032 T - 45580614999712 T^{2} - 13020445493104 T^{3} + 4869319675096 T^{4} + 1086268015976 T^{5} - 257679285968 T^{6} - 44949652648 T^{7} + 7638131987 T^{8} + 1043323138 T^{9} - 131423299 T^{10} - 14297187 T^{11} + 1290259 T^{12} + 116057 T^{13} - 6665 T^{14} - 521 T^{15} + 14 T^{16} + T^{17} \)
$79$ \( -15425861826749 + 59989479692318 T + 11604471640103 T^{2} - 20600063035317 T^{3} + 216001799000 T^{4} + 2286923271563 T^{5} - 284370358265 T^{6} - 81908151343 T^{7} + 17477709162 T^{8} + 594976537 T^{9} - 370599189 T^{10} + 17418957 T^{11} + 2759692 T^{12} - 300979 T^{13} + 1139 T^{14} + 1159 T^{15} - 61 T^{16} + T^{17} \)
$83$ \( -558878703263744 - 1973165870845952 T - 133915783893504 T^{2} + 292869823596032 T^{3} + 40466442683776 T^{4} - 13934229429568 T^{5} - 2628314999232 T^{6} + 243235758560 T^{7} + 67728580232 T^{8} - 931775672 T^{9} - 835788337 T^{10} - 17147096 T^{11} + 5202305 T^{12} + 199233 T^{13} - 15638 T^{14} - 762 T^{15} + 18 T^{16} + T^{17} \)
$89$ \( -770084080128 + 1628927697408 T - 794173892480 T^{2} - 255516173760 T^{3} + 301243166656 T^{4} - 46874416944 T^{5} - 24064949680 T^{6} + 8843326416 T^{7} - 101842301 T^{8} - 351302994 T^{9} + 41546430 T^{10} + 4046243 T^{11} - 955231 T^{12} + 7989 T^{13} + 7384 T^{14} - 303 T^{15} - 18 T^{16} + T^{17} \)
$97$ \( -4404611986402304 - 12959554831435776 T + 3452622908628224 T^{2} + 1533252250475008 T^{3} - 405835181764864 T^{4} - 42640025670016 T^{5} + 14359066017472 T^{6} + 384374586096 T^{7} - 236700781836 T^{8} + 1052379264 T^{9} + 2085618573 T^{10} - 41733303 T^{11} - 10120670 T^{12} + 292102 T^{13} + 25485 T^{14} - 879 T^{15} - 26 T^{16} + T^{17} \)
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