Properties

Label 2151.2.a.i
Level $2151$
Weight $2$
Character orbit 2151.a
Self dual yes
Analytic conductor $17.176$
Analytic rank $0$
Dimension $17$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2151 = 3^{2} \cdot 239 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2151.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.1758214748\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Defining polynomial: \(x^{17} - 28 x^{15} - x^{14} + 319 x^{13} + 17 x^{12} - 1903 x^{11} - 91 x^{10} + 6377 x^{9} + 125 x^{8} - 11967 x^{7} + 233 x^{6} + 11733 x^{5} - 503 x^{4} - 5015 x^{3} + 94 x^{2} + 609 x - 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 239)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{17} - 28 x^{15} - x^{14} + 319 x^{13} + 17 x^{12} - 1903 x^{11} - 91 x^{10} + 6377 x^{9} + 125 x^{8} - 11967 x^{7} + 233 x^{6} + 11733 x^{5} - 503 x^{4} - 5015 x^{3} + 94 x^{2} + 609 x - 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\(2861757 \nu^{16} + 2069683 \nu^{15} - 74529028 \nu^{14} - 59373989 \nu^{13} + 768331898 \nu^{12} + 655783718 \nu^{11} - 3980000819 \nu^{10} - 3512167813 \nu^{9} + 10882996999 \nu^{8} + 9463241113 \nu^{7} - 15192765978 \nu^{6} - 12170546998 \nu^{5} + 9552821155 \nu^{4} + 6369419080 \nu^{3} - 1813232220 \nu^{2} - 749884686 \nu + 128997701\)\()/11107271\)
\(\beta_{4}\)\(=\)\((\)\(3427187 \nu^{16} + 6181042 \nu^{15} - 92146964 \nu^{14} - 163673784 \nu^{13} + 979124437 \nu^{12} + 1701355701 \nu^{11} - 5213496021 \nu^{10} - 8760380237 \nu^{9} + 14585493979 \nu^{8} + 23242888500 \nu^{7} - 20569733321 \nu^{6} - 30210320397 \nu^{5} + 12366485951 \nu^{4} + 16402549220 \nu^{3} - 1494402472 \nu^{2} - 2139931357 \nu + 187483100\)\()/11107271\)
\(\beta_{5}\)\(=\)\((\)\(550174 \nu^{16} + 924456 \nu^{15} - 14665692 \nu^{14} - 24610776 \nu^{13} + 154256901 \nu^{12} + 256737372 \nu^{11} - 810750554 \nu^{10} - 1322960680 \nu^{9} + 2226408486 \nu^{8} + 3496299358 \nu^{7} - 3045198493 \nu^{6} - 4486645884 \nu^{5} + 1712101699 \nu^{4} + 2352732831 \nu^{3} - 121863657 \nu^{2} - 260979730 \nu + 12767993\)\()/1586753\)
\(\beta_{6}\)\(=\)\((\)\(-641596 \nu^{16} - 912943 \nu^{15} + 17130733 \nu^{14} + 24430526 \nu^{13} - 181095707 \nu^{12} - 255776258 \nu^{11} + 962203455 \nu^{10} + 1319913690 \nu^{9} - 2699858556 \nu^{8} - 3483484570 \nu^{7} + 3858416872 \nu^{6} + 4450468047 \nu^{5} - 2420001141 \nu^{4} - 2324850172 \nu^{3} + 368577018 \nu^{2} + 272751448 \nu - 27498884\)\()/1586753\)
\(\beta_{7}\)\(=\)\((\)\(-6696568 \nu^{16} - 8516319 \nu^{15} + 176926190 \nu^{14} + 230617909 \nu^{13} - 1848010432 \nu^{12} - 2437664584 \nu^{11} + 9677318143 \nu^{10} + 12666479224 \nu^{9} - 26646052279 \nu^{8} - 33568330057 \nu^{7} + 37110029161 \nu^{6} + 42992988989 \nu^{5} - 22457204081 \nu^{4} - 22592657238 \nu^{3} + 3180700269 \nu^{2} + 2752840722 \nu - 261422000\)\()/11107271\)
\(\beta_{8}\)\(=\)\((\)\(-1212120 \nu^{16} - 1662081 \nu^{15} + 32007121 \nu^{14} + 44956652 \nu^{13} - 333745650 \nu^{12} - 475783550 \nu^{11} + 1741052488 \nu^{10} + 2484349237 \nu^{9} - 4756496095 \nu^{8} - 6655433366 \nu^{7} + 6517851116 \nu^{6} + 8698751597 \nu^{5} - 3801020746 \nu^{4} - 4722308086 \nu^{3} + 463201158 \nu^{2} + 587074092 \nu - 51585352\)\()/1586753\)
\(\beta_{9}\)\(=\)\((\)\(11677912 \nu^{16} + 15595482 \nu^{15} - 308905793 \nu^{14} - 422390585 \nu^{13} + 3229351808 \nu^{12} + 4475719293 \nu^{11} - 16913750167 \nu^{10} - 23396485147 \nu^{9} + 46505505592 \nu^{8} + 62740182049 \nu^{7} - 64415924668 \nu^{6} - 82079193243 \nu^{5} + 38301793083 \nu^{4} + 44622861417 \nu^{3} - 4964271980 \nu^{2} - 5586773826 \nu + 462319543\)\()/11107271\)
\(\beta_{10}\)\(=\)\((\)\(-12203491 \nu^{16} - 13792352 \nu^{15} + 320507355 \nu^{14} + 380091963 \nu^{13} - 3326297544 \nu^{12} - 4086391628 \nu^{11} + 17288741324 \nu^{10} + 21611882159 \nu^{9} - 47127874535 \nu^{8} - 58489609578 \nu^{7} + 64589333427 \nu^{6} + 77121550865 \nu^{5} - 37934311750 \nu^{4} - 42271965217 \nu^{3} + 4929621218 \nu^{2} + 5327637605 \nu - 476302848\)\()/11107271\)
\(\beta_{11}\)\(=\)\((\)\(-16501139 \nu^{16} - 18838666 \nu^{15} + 433888308 \nu^{14} + 516984745 \nu^{13} - 4509017669 \nu^{12} - 5533746020 \nu^{11} + 23474771890 \nu^{10} + 29118660662 \nu^{9} - 64137341975 \nu^{8} - 78287038350 \nu^{7} + 88210240855 \nu^{6} + 102245655105 \nu^{5} - 52077281946 \nu^{4} - 55254001989 \nu^{3} + 6741029188 \nu^{2} + 6789612993 \nu - 531232758\)\()/11107271\)
\(\beta_{12}\)\(=\)\((\)\(-16771351 \nu^{16} - 20065815 \nu^{15} + 442373694 \nu^{14} + 548454202 \nu^{13} - 4613893796 \nu^{12} - 5855599700 \nu^{11} + 24126751696 \nu^{10} + 30789210039 \nu^{9} - 66289587616 \nu^{8} - 82906055202 \nu^{7} + 91822850183 \nu^{6} + 108744026520 \nu^{5} - 54627655140 \nu^{4} - 59185678764 \nu^{3} + 7102994828 \nu^{2} + 7384450585 \nu - 608340411\)\()/11107271\)
\(\beta_{13}\)\(=\)\((\)\(-22511799 \nu^{16} - 28856065 \nu^{15} + 595209258 \nu^{14} + 783888478 \nu^{13} - 6223382488 \nu^{12} - 8326382581 \nu^{11} + 32632212856 \nu^{10} + 43603144382 \nu^{9} - 89969920460 \nu^{8} - 117052738164 \nu^{7} + 125297104388 \nu^{6} + 153180936380 \nu^{5} - 75340597103 \nu^{4} - 83236570496 \nu^{3} + 10250253852 \nu^{2} + 10451957825 \nu - 928678597\)\()/11107271\)
\(\beta_{14}\)\(=\)\((\)\(28123016 \nu^{16} + 35294287 \nu^{15} - 741907502 \nu^{14} - 960763074 \nu^{13} + 7735764280 \nu^{12} + 10220413435 \nu^{11} - 40413877625 \nu^{10} - 53564805294 \nu^{9} + 110840105488 \nu^{8} + 143790237565 \nu^{7} - 153109931974 \nu^{6} - 187990306900 \nu^{5} + 90771044929 \nu^{4} + 101944835473 \nu^{3} - 11762161370 \nu^{2} - 12671613695 \nu + 1092357616\)\()/11107271\)
\(\beta_{15}\)\(=\)\((\)\(-31952759 \nu^{16} - 40216701 \nu^{15} + 843349731 \nu^{14} + 1093768675 \nu^{13} - 8798123778 \nu^{12} - 11623749134 \nu^{11} + 45991437255 \nu^{10} + 60846133922 \nu^{9} - 126231112560 \nu^{8} - 163062418821 \nu^{7} + 174557837156 \nu^{6} + 212628276688 \nu^{5} - 103692004443 \nu^{4} - 114823385333 \nu^{3} + 13514995932 \nu^{2} + 14180166325 \nu - 1197538062\)\()/11107271\)
\(\beta_{16}\)\(=\)\((\)\(48244866 \nu^{16} + 61658282 \nu^{15} - 1274476952 \nu^{14} - 1675542503 \nu^{13} + 13308856210 \nu^{12} + 17798594570 \nu^{11} - 69651539155 \nu^{10} - 93176463520 \nu^{9} + 191452483894 \nu^{8} + 249907127633 \nu^{7} - 265277916826 \nu^{6} - 326475454491 \nu^{5} + 158000743346 \nu^{4} + 176893301114 \nu^{3} - 20677366018 \nu^{2} - 22048442793 \nu + 1848670313\)\()/11107271\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{15} - \beta_{12} - \beta_{8} - \beta_{7} + \beta_{2} + 6 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{16} + \beta_{14} - \beta_{10} - \beta_{6} + \beta_{4} + 8 \beta_{2} + 15\)
\(\nu^{5}\)\(=\)\(\beta_{16} + 11 \beta_{15} - 9 \beta_{12} - \beta_{11} + \beta_{10} - 10 \beta_{8} - 8 \beta_{7} + \beta_{5} + 2 \beta_{3} + 9 \beta_{2} + 39 \beta_{1} + 4\)
\(\nu^{6}\)\(=\)\(-13 \beta_{16} - 2 \beta_{15} + 9 \beta_{14} - 3 \beta_{13} + \beta_{12} - 2 \beta_{11} - 11 \beta_{10} + 2 \beta_{9} + \beta_{8} - \beta_{7} - 10 \beta_{6} + 10 \beta_{4} - 3 \beta_{3} + 55 \beta_{2} - \beta_{1} + 83\)
\(\nu^{7}\)\(=\)\(13 \beta_{16} + 94 \beta_{15} - \beta_{14} - 70 \beta_{12} - 12 \beta_{11} + 14 \beta_{10} + 2 \beta_{9} - 80 \beta_{8} - 58 \beta_{7} + 4 \beta_{6} + 13 \beta_{5} + \beta_{4} + 27 \beta_{3} + 67 \beta_{2} + 263 \beta_{1} + 48\)
\(\nu^{8}\)\(=\)\(-124 \beta_{16} - 31 \beta_{15} + 62 \beta_{14} - 45 \beta_{13} + 15 \beta_{12} - 29 \beta_{11} - 95 \beta_{10} + 30 \beta_{9} + 13 \beta_{8} - 13 \beta_{7} - 79 \beta_{6} + \beta_{5} + 81 \beta_{4} - 40 \beta_{3} + 366 \beta_{2} - 20 \beta_{1} + 483\)
\(\nu^{9}\)\(=\)\(124 \beta_{16} + 740 \beta_{15} - 19 \beta_{14} - \beta_{13} - 527 \beta_{12} - 105 \beta_{11} + 140 \beta_{10} + 35 \beta_{9} - 602 \beta_{8} - 418 \beta_{7} + 65 \beta_{6} + 127 \beta_{5} + 11 \beta_{4} + 266 \beta_{3} + 469 \beta_{2} + 1809 \beta_{1} + 419\)
\(\nu^{10}\)\(=\)\(-1055 \beta_{16} - 336 \beta_{15} + 386 \beta_{14} - 475 \beta_{13} + 156 \beta_{12} - 298 \beta_{11} - 754 \beta_{10} + 313 \beta_{9} + 124 \beta_{8} - 121 \beta_{7} - 575 \beta_{6} + 21 \beta_{5} + 620 \beta_{4} - 381 \beta_{3} + 2424 \beta_{2} - 258 \beta_{1} + 2901\)
\(\nu^{11}\)\(=\)\(1050 \beta_{16} + 5616 \beta_{15} - 233 \beta_{14} - 21 \beta_{13} - 3921 \beta_{12} - 823 \beta_{11} + 1230 \beta_{10} + 411 \beta_{9} - 4429 \beta_{8} - 3025 \beta_{7} + 720 \beta_{6} + 1105 \beta_{5} + 78 \beta_{4} + 2311 \beta_{3} + 3192 \beta_{2} + 12588 \beta_{1} + 3239\)
\(\nu^{12}\)\(=\)\(-8482 \beta_{16} - 3145 \beta_{15} + 2272 \beta_{14} - 4351 \beta_{13} + 1416 \beta_{12} - 2668 \beta_{11} - 5750 \beta_{10} + 2822 \beta_{9} + 1068 \beta_{8} - 986 \beta_{7} - 4033 \beta_{6} + 275 \beta_{5} + 4641 \beta_{4} - 3195 \beta_{3} + 16103 \beta_{2} - 2745 \beta_{1} + 17820\)
\(\nu^{13}\)\(=\)\(8382 \beta_{16} + 41805 \beta_{15} - 2363 \beta_{14} - 277 \beta_{13} - 28983 \beta_{12} - 6155 \beta_{11} + 10139 \beta_{10} + 4053 \beta_{9} - 32272 \beta_{8} - 21955 \beta_{7} + 6799 \beta_{6} + 9033 \beta_{5} + 415 \beta_{4} + 18816 \beta_{3} + 21392 \beta_{2} + 88239 \beta_{1} + 23594\)
\(\nu^{14}\)\(=\)\(-66014 \beta_{16} - 27261 \beta_{15} + 12831 \beta_{14} - 36980 \beta_{13} + 12039 \beta_{12} - 22214 \beta_{11} - 42905 \beta_{10} + 23589 \beta_{9} + 8822 \beta_{8} - 7502 \beta_{7} - 27780 \beta_{6} + 2900 \beta_{5} + 34318 \beta_{4} - 25229 \beta_{3} + 107542 \beta_{2} - 26258 \beta_{1} + 111347\)
\(\nu^{15}\)\(=\)\(64775 \beta_{16} + 307691 \beta_{15} - 21616 \beta_{14} - 2954 \beta_{13} - 213234 \beta_{12} - 45015 \beta_{11} + 80639 \beta_{10} + 36244 \beta_{9} - 234090 \beta_{8} - 159489 \beta_{7} + 59010 \beta_{6} + 71087 \beta_{5} + 1348 \beta_{4} + 147501 \beta_{3} + 142006 \beta_{2} + 621634 \beta_{1} + 166311\)
\(\nu^{16}\)\(=\)\(-503410 \beta_{16} - 225817 \beta_{15} + 69648 \beta_{14} - 300336 \beta_{13} + 98643 \beta_{12} - 176974 \beta_{11} - 316062 \beta_{10} + 188545 \beta_{9} + 71421 \beta_{8} - 54780 \beta_{7} - 189753 \beta_{6} + 27008 \beta_{5} + 251657 \beta_{4} - 193114 \beta_{3} + 722348 \beta_{2} - 235139 \beta_{1} + 705176\)

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
−2.70108
−2.33063
−2.12688
−2.08085
−1.88923
−1.39112
−0.685793
−0.535519
0.0842488
0.364980
1.17471
1.25955
1.26402
1.82989
2.49907
2.61209
2.65254
−2.70108 0 5.29584 −1.77568 0 −4.04078 −8.90233 0 4.79626
1.2 −2.33063 0 3.43183 3.31496 0 2.77554 −3.33707 0 −7.72595
1.3 −2.12688 0 2.52362 −0.256394 0 0.961358 −1.11367 0 0.545320
1.4 −2.08085 0 2.32992 −2.29912 0 3.55134 −0.686506 0 4.78411
1.5 −1.88923 0 1.56918 1.09185 0 −4.47828 0.813913 0 −2.06276
1.6 −1.39112 0 −0.0647804 −4.19763 0 −0.214893 2.87236 0 5.83941
1.7 −0.685793 0 −1.52969 0.233223 0 1.65270 2.42063 0 −0.159942
1.8 −0.535519 0 −1.71322 −0.933854 0 4.14877 1.98850 0 0.500097
1.9 0.0842488 0 −1.99290 −2.54226 0 −1.62505 −0.336397 0 −0.214182
1.10 0.364980 0 −1.86679 2.96035 0 5.11564 −1.41130 0 1.08047
1.11 1.17471 0 −0.620061 −4.24400 0 −3.49274 −3.07781 0 −4.98546
1.12 1.25955 0 −0.413530 3.10986 0 −4.47710 −3.03997 0 3.91703
1.13 1.26402 0 −0.402263 −1.81204 0 1.66736 −3.03650 0 −2.29044
1.14 1.82989 0 1.34850 3.32555 0 −0.206452 −1.19217 0 6.08539
1.15 2.49907 0 4.24535 −1.73010 0 5.13009 5.61129 0 −4.32364
1.16 2.61209 0 4.82304 3.37708 0 1.51525 7.37403 0 8.82124
1.17 2.65254 0 5.03596 −3.62180 0 −2.98274 8.05299 0 −9.60695
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(239\) \(1\)

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 2151.2.a.i 17
3.b odd 2 1 239.2.a.b 17
12.b even 2 1 3824.2.a.p 17
15.d odd 2 1 5975.2.a.g 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
239.2.a.b 17 3.b odd 2 1
2151.2.a.i 17 1.a even 1 1 trivial
3824.2.a.p 17 12.b even 2 1
5975.2.a.g 17 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2151))\):

\(T_{2}^{17} - \cdots\)
\(T_{5}^{17} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -49 + 609 T + 94 T^{2} - 5015 T^{3} - 503 T^{4} + 11733 T^{5} + 233 T^{6} - 11967 T^{7} + 125 T^{8} + 6377 T^{9} - 91 T^{10} - 1903 T^{11} + 17 T^{12} + 319 T^{13} - T^{14} - 28 T^{15} + T^{17} \)
$3$ \( T^{17} \)
$5$ \( -43871 - 64969 T + 775724 T^{2} + 1466245 T^{3} - 168922 T^{4} - 1708498 T^{5} - 707930 T^{6} + 500506 T^{7} + 345487 T^{8} - 47987 T^{9} - 66664 T^{10} - 1715 T^{11} + 6439 T^{12} + 647 T^{13} - 311 T^{14} - 44 T^{15} + 6 T^{16} + T^{17} \)
$7$ \( -262144 - 1900544 T - 540672 T^{2} + 10784768 T^{3} - 8086528 T^{4} - 5164544 T^{5} + 6211328 T^{6} + 251392 T^{7} - 1590464 T^{8} + 166432 T^{9} + 193664 T^{10} - 31088 T^{11} - 12292 T^{12} + 2276 T^{13} + 393 T^{14} - 77 T^{15} - 5 T^{16} + T^{17} \)
$11$ \( -151629817 - 115295798 T + 305552905 T^{2} + 50388863 T^{3} - 195019206 T^{4} + 15048164 T^{5} + 53477248 T^{6} - 10803586 T^{7} - 6787983 T^{8} + 1893660 T^{9} + 427543 T^{10} - 149697 T^{11} - 13619 T^{12} + 6056 T^{13} + 202 T^{14} - 123 T^{15} - T^{16} + T^{17} \)
$13$ \( 11583488 + 9224192 T - 224333824 T^{2} + 144123904 T^{3} + 180070400 T^{4} - 147311616 T^{5} - 25163008 T^{6} + 38764288 T^{7} - 2052928 T^{8} - 4205376 T^{9} + 647712 T^{10} + 199728 T^{11} - 47272 T^{12} - 3250 T^{13} + 1407 T^{14} - 30 T^{15} - 15 T^{16} + T^{17} \)
$17$ \( 207461296 + 133688896 T - 405970976 T^{2} - 226358836 T^{3} + 285355545 T^{4} + 129367089 T^{5} - 90233331 T^{6} - 32480912 T^{7} + 13043990 T^{8} + 4052487 T^{9} - 926590 T^{10} - 263220 T^{11} + 33533 T^{12} + 8983 T^{13} - 591 T^{14} - 152 T^{15} + 4 T^{16} + T^{17} \)
$19$ \( 7399800832 + 5990776832 T - 12219383808 T^{2} - 3217784832 T^{3} + 5870931968 T^{4} - 134083584 T^{5} - 1049447680 T^{6} + 167344512 T^{7} + 78598592 T^{8} - 21557376 T^{9} - 1855152 T^{10} + 1079696 T^{11} - 45600 T^{12} - 20812 T^{13} + 2387 T^{14} + 79 T^{15} - 24 T^{16} + T^{17} \)
$23$ \( -44744704 + 299687936 T - 617107456 T^{2} + 385914880 T^{3} + 230717440 T^{4} - 365522432 T^{5} + 65933568 T^{6} + 76211840 T^{7} - 30197504 T^{8} - 3644768 T^{9} + 2910240 T^{10} - 76336 T^{11} - 108648 T^{12} + 8334 T^{13} + 1675 T^{14} - 166 T^{15} - 9 T^{16} + T^{17} \)
$29$ \( 117964056107 + 159446889949 T - 155139854168 T^{2} - 105602645079 T^{3} + 33927640430 T^{4} + 21789989174 T^{5} - 2766604430 T^{6} - 2046593758 T^{7} + 91562821 T^{8} + 100075099 T^{9} - 764236 T^{10} - 2701703 T^{11} - 19235 T^{12} + 40629 T^{13} + 403 T^{14} - 318 T^{15} - 2 T^{16} + T^{17} \)
$31$ \( 2326460584 - 8804785252 T + 10170134456 T^{2} - 1523610965 T^{3} - 4262952656 T^{4} + 2499741368 T^{5} + 738880 T^{6} - 387364142 T^{7} + 101848444 T^{8} + 8634002 T^{9} - 7817764 T^{10} + 940291 T^{11} + 118728 T^{12} - 38474 T^{13} + 2520 T^{14} + 163 T^{15} - 28 T^{16} + T^{17} \)
$37$ \( 491454464 + 22415851520 T - 5820583936 T^{2} - 17874927616 T^{3} + 1578812416 T^{4} + 5254270976 T^{5} + 166509824 T^{6} - 657221888 T^{7} - 56346816 T^{8} + 39889664 T^{9} + 4200736 T^{10} - 1296560 T^{11} - 135688 T^{12} + 23392 T^{13} + 2009 T^{14} - 228 T^{15} - 11 T^{16} + T^{17} \)
$41$ \( 65418838016 - 52417683456 T - 85114331136 T^{2} + 62529579008 T^{3} + 32843495424 T^{4} - 19639664128 T^{5} - 5627808256 T^{6} + 2168443776 T^{7} + 534217984 T^{8} - 88828832 T^{9} - 23772464 T^{10} + 1360800 T^{11} + 513604 T^{12} - 494 T^{13} - 5229 T^{14} - 161 T^{15} + 20 T^{16} + T^{17} \)
$43$ \( -36073422848 - 69611462656 T + 200707547136 T^{2} + 463332448256 T^{3} + 158549216256 T^{4} - 38491359744 T^{5} - 21983748864 T^{6} + 470870144 T^{7} + 1150844992 T^{8} + 49358464 T^{9} - 30453536 T^{10} - 2227584 T^{11} + 432332 T^{12} + 39220 T^{13} - 3125 T^{14} - 320 T^{15} + 9 T^{16} + T^{17} \)
$47$ \( 170160357376 - 71265517568 T - 169164386304 T^{2} + 100526370816 T^{3} + 30316536832 T^{4} - 28413413888 T^{5} + 523226624 T^{6} + 2822249344 T^{7} - 379113216 T^{8} - 106621920 T^{9} + 22658656 T^{10} + 1396648 T^{11} - 531508 T^{12} + 4762 T^{13} + 5309 T^{14} - 215 T^{15} - 18 T^{16} + T^{17} \)
$53$ \( -527283519488 + 316866560 T + 834801287168 T^{2} + 24783900672 T^{3} - 307558466560 T^{4} + 6610594816 T^{5} + 40222431232 T^{6} - 1726466944 T^{7} - 2254024384 T^{8} + 133665504 T^{9} + 60487120 T^{10} - 4269112 T^{11} - 802128 T^{12} + 62084 T^{13} + 5055 T^{14} - 411 T^{15} - 12 T^{16} + T^{17} \)
$59$ \( -19084961644544 + 14248361918464 T + 2750319464448 T^{2} - 3558175051776 T^{3} - 21410101248 T^{4} + 352772897280 T^{5} - 12804441600 T^{6} - 18449862400 T^{7} + 758621632 T^{8} + 556963584 T^{9} - 17863552 T^{10} - 9843704 T^{11} + 190808 T^{12} + 98100 T^{13} - 843 T^{14} - 500 T^{15} + T^{16} + T^{17} \)
$61$ \( 58961351552 - 24800619392 T - 129393161888 T^{2} + 106557228880 T^{3} + 7995198056 T^{4} - 25199734720 T^{5} + 2481600130 T^{6} + 2310209445 T^{7} - 347174032 T^{8} - 101816867 T^{9} + 17974572 T^{10} + 2117498 T^{11} - 453008 T^{12} - 15038 T^{13} + 5442 T^{14} - 79 T^{15} - 24 T^{16} + T^{17} \)
$67$ \( -11993903104 - 38658688000 T + 41951832128 T^{2} + 117268359136 T^{3} - 101830715089 T^{4} - 34017276029 T^{5} + 29214612841 T^{6} + 3962964864 T^{7} - 2653566154 T^{8} - 104716797 T^{9} + 100289634 T^{10} - 2167240 T^{11} - 1444485 T^{12} + 67257 T^{13} + 8229 T^{14} - 472 T^{15} - 16 T^{16} + T^{17} \)
$71$ \( -30800934780608 + 49772205598296 T + 9121864973643 T^{2} - 12037087655969 T^{3} - 405386579543 T^{4} + 1068087411894 T^{5} - 26773856050 T^{6} - 45361051067 T^{7} + 2292958505 T^{8} + 1044641799 T^{9} - 61570898 T^{10} - 14012615 T^{11} + 784533 T^{12} + 111699 T^{13} - 4881 T^{14} - 502 T^{15} + 12 T^{16} + T^{17} \)
$73$ \( -552452096 - 5545435136 T + 14912864256 T^{2} + 15965069312 T^{3} - 14055530496 T^{4} - 9084362752 T^{5} + 3804327168 T^{6} + 1815461376 T^{7} - 413644736 T^{8} - 139411072 T^{9} + 23853024 T^{10} + 4294088 T^{11} - 710604 T^{12} - 41648 T^{13} + 8869 T^{14} - 73 T^{15} - 30 T^{16} + T^{17} \)
$79$ \( -1721480118272 - 3167862128640 T + 783985672192 T^{2} + 3341205667840 T^{3} + 584358608896 T^{4} - 667393114112 T^{5} - 173581430784 T^{6} + 27377273856 T^{7} + 10432968704 T^{8} - 55051008 T^{9} - 220668416 T^{10} - 9545728 T^{11} + 1968736 T^{12} + 130416 T^{13} - 7568 T^{14} - 624 T^{15} + 10 T^{16} + T^{17} \)
$83$ \( -16034570219864 + 31956807913252 T - 22466737725124 T^{2} + 4827724743967 T^{3} + 1573937038232 T^{4} - 899123761152 T^{5} + 62151285660 T^{6} + 36464811486 T^{7} - 6229269412 T^{8} - 492130250 T^{9} + 160847696 T^{10} - 667869 T^{11} - 1819116 T^{12} + 66874 T^{13} + 9132 T^{14} - 485 T^{15} - 16 T^{16} + T^{17} \)
$89$ \( -6261514240000 + 12748941312000 T - 2096476672000 T^{2} - 5178784634880 T^{3} + 589404695552 T^{4} + 784484703232 T^{5} - 5998902272 T^{6} - 54115337088 T^{7} - 4407249984 T^{8} + 1535500000 T^{9} + 254693792 T^{10} - 6758256 T^{11} - 4186196 T^{12} - 292854 T^{13} + 3987 T^{14} + 1405 T^{15} + 65 T^{16} + T^{17} \)
$97$ \( 111392870170624 - 489372455059456 T + 445102724706304 T^{2} - 70809334319104 T^{3} - 41718091262976 T^{4} + 11759087050240 T^{5} + 1004676712960 T^{6} - 536421150592 T^{7} + 13850972480 T^{8} + 9942474080 T^{9} - 891967104 T^{10} - 54847560 T^{11} + 11187916 T^{12} - 354762 T^{13} - 29283 T^{14} + 2766 T^{15} - 87 T^{16} + T^{17} \)
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