Properties

Label 1003.2.a.h
Level 1003
Weight 2
Character orbit 1003.a
Self dual yes
Analytic conductor 8.009
Analytic rank 1
Dimension 16
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1003 = 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1003.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.00899532273\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} -\beta_{10} q^{3} + ( 1 + \beta_{2} ) q^{4} + ( -1 - \beta_{4} ) q^{5} + ( -1 + \beta_{1} - \beta_{2} - \beta_{5} ) q^{6} + ( -1 - \beta_{1} + \beta_{3} + \beta_{5} + \beta_{10} - \beta_{11} ) q^{7} + ( -2 \beta_{1} + \beta_{6} - \beta_{7} ) q^{8} + ( 1 + \beta_{4} + \beta_{11} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} -\beta_{10} q^{3} + ( 1 + \beta_{2} ) q^{4} + ( -1 - \beta_{4} ) q^{5} + ( -1 + \beta_{1} - \beta_{2} - \beta_{5} ) q^{6} + ( -1 - \beta_{1} + \beta_{3} + \beta_{5} + \beta_{10} - \beta_{11} ) q^{7} + ( -2 \beta_{1} + \beta_{6} - \beta_{7} ) q^{8} + ( 1 + \beta_{4} + \beta_{11} ) q^{9} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{14} ) q^{10} + ( -1 + 2 \beta_{1} - \beta_{3} - \beta_{5} - \beta_{9} + \beta_{11} + \beta_{13} ) q^{11} + ( -1 + 2 \beta_{1} + \beta_{5} + \beta_{7} + \beta_{11} ) q^{12} + ( -2 + \beta_{1} + \beta_{5} + \beta_{7} + \beta_{10} - \beta_{11} - \beta_{13} ) q^{13} + ( 1 + 2 \beta_{1} - \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{11} - \beta_{13} - \beta_{14} ) q^{14} + ( \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} + \beta_{10} + \beta_{11} - \beta_{14} ) q^{15} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{9} - \beta_{12} - \beta_{14} ) q^{16} + q^{17} + ( 1 + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{9} - \beta_{14} ) q^{18} + ( -1 + \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} + \beta_{9} + \beta_{10} - \beta_{11} + 2 \beta_{12} ) q^{19} + ( -3 + \beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{8} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{20} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} + \beta_{9} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{21} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{12} + 2 \beta_{14} ) q^{22} + ( \beta_{1} + \beta_{5} - \beta_{8} - \beta_{10} + \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{23} + ( -4 + 3 \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{6} + \beta_{7} + 2 \beta_{10} - \beta_{11} + \beta_{12} ) q^{24} + ( 1 + \beta_{2} + 2 \beta_{4} - \beta_{8} + \beta_{11} ) q^{25} + ( -3 + 2 \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} + 2 \beta_{11} + \beta_{13} - \beta_{14} ) q^{26} + ( -2 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} + \beta_{10} - 2 \beta_{11} - \beta_{13} + \beta_{14} ) q^{27} + ( -1 - 3 \beta_{1} + 2 \beta_{3} + \beta_{5} + 3 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} - \beta_{12} + 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{28} + ( -2 - \beta_{2} - \beta_{5} - \beta_{6} + \beta_{9} - \beta_{13} + \beta_{15} ) q^{29} + ( 2 - 4 \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} + 4 \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{11} - 2 \beta_{12} - \beta_{13} + \beta_{15} ) q^{30} + ( \beta_{2} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} - \beta_{12} - \beta_{13} ) q^{31} + ( 3 - 5 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{10} - \beta_{13} ) q^{32} + ( -1 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{6} - 2 \beta_{7} + 3 \beta_{10} - 3 \beta_{11} + 2 \beta_{12} + \beta_{14} - \beta_{15} ) q^{33} -\beta_{1} q^{34} + ( -1 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{13} - \beta_{15} ) q^{35} + ( 1 - 5 \beta_{1} + \beta_{3} + 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{11} - \beta_{13} + \beta_{14} ) q^{36} + ( -2 + \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{13} - \beta_{14} ) q^{37} + ( -3 + \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{9} + \beta_{11} + 2 \beta_{13} + \beta_{14} ) q^{38} + ( -1 - \beta_{1} + \beta_{2} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{13} ) q^{39} + ( -1 + 8 \beta_{1} - \beta_{3} - 3 \beta_{6} + 4 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{13} + \beta_{15} ) q^{40} + ( -1 - \beta_{3} + \beta_{6} + 2 \beta_{8} - 2 \beta_{10} - \beta_{12} ) q^{41} + ( -1 + \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{8} - \beta_{10} + \beta_{12} ) q^{42} + ( 2 - 3 \beta_{1} - \beta_{3} + \beta_{4} + \beta_{6} + \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - \beta_{13} + \beta_{15} ) q^{43} + ( -2 + 5 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + \beta_{11} - \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{44} + ( -4 - 3 \beta_{4} - \beta_{6} - 3 \beta_{11} - \beta_{13} ) q^{45} + ( -3 + \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{11} - \beta_{14} - \beta_{15} ) q^{46} + ( -5 - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{47} + ( -4 - 4 \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{6} + 2 \beta_{9} + 2 \beta_{10} - 3 \beta_{11} + \beta_{12} + \beta_{13} + 3 \beta_{14} ) q^{48} + ( 1 + 6 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + 4 \beta_{11} + \beta_{12} + \beta_{13} - 2 \beta_{14} ) q^{49} + ( 2 - 5 \beta_{1} + 2 \beta_{2} + \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{50} -\beta_{10} q^{51} + ( -2 + 5 \beta_{1} - 3 \beta_{2} + 3 \beta_{4} - \beta_{5} - 3 \beta_{6} - \beta_{8} + \beta_{9} - \beta_{11} + \beta_{14} ) q^{52} + ( -2 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{9} + \beta_{15} ) q^{53} + ( 1 - \beta_{1} - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} + \beta_{12} + 2 \beta_{13} - \beta_{15} ) q^{54} + ( 4 - 5 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{11} - 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{55} + ( 1 + 2 \beta_{1} - 3 \beta_{3} + \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{10} - \beta_{11} - 3 \beta_{13} - \beta_{14} + \beta_{15} ) q^{56} + ( 2 - \beta_{1} + 3 \beta_{2} - \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 3 \beta_{11} + \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{57} + ( 5 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} + 3 \beta_{7} + 2 \beta_{8} - 2 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{58} + q^{59} + ( 4 - 5 \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + \beta_{6} - 4 \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{60} + ( -2 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} - 3 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{13} - \beta_{15} ) q^{61} + ( 2 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{8} - \beta_{9} - 3 \beta_{10} + 2 \beta_{11} - \beta_{12} - \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{62} + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} + 4 \beta_{5} - 4 \beta_{6} + 3 \beta_{7} - \beta_{8} - 2 \beta_{9} + 4 \beta_{10} - 3 \beta_{11} - 4 \beta_{13} + 2 \beta_{15} ) q^{63} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{11} - \beta_{12} - \beta_{13} - 2 \beta_{14} ) q^{64} + ( -2 \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} - 3 \beta_{10} + 3 \beta_{11} - \beta_{12} + 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{65} + ( 3 + 3 \beta_{2} + \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} - 3 \beta_{8} - \beta_{9} + 2 \beta_{11} - 2 \beta_{12} + \beta_{13} - \beta_{14} ) q^{66} + ( 1 + \beta_{1} - 3 \beta_{3} + \beta_{5} + \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} - 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{67} + ( 1 + \beta_{2} ) q^{68} + ( 2 - 2 \beta_{1} + \beta_{2} + 3 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} - \beta_{12} - 2 \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{69} + ( 2 - 5 \beta_{1} + \beta_{2} + 4 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} - \beta_{7} - 3 \beta_{10} + \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{70} + ( -1 + 2 \beta_{4} - \beta_{5} - 2 \beta_{9} + \beta_{10} + 2 \beta_{11} + 2 \beta_{13} + \beta_{14} ) q^{71} + ( 1 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} + 2 \beta_{11} - \beta_{12} + 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{72} + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{10} + 3 \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{73} + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{7} - \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + \beta_{12} - 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{74} + ( -1 - 2 \beta_{1} + \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} + 4 \beta_{7} - 2 \beta_{8} - 3 \beta_{11} + \beta_{12} - \beta_{13} + 3 \beta_{14} ) q^{75} + ( -2 + 9 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 2 \beta_{5} - 4 \beta_{6} + 3 \beta_{7} + \beta_{11} - 2 \beta_{13} + \beta_{14} ) q^{76} + ( -3 - 3 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{8} - 2 \beta_{10} - \beta_{11} - \beta_{13} ) q^{77} + ( 3 - 4 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + 3 \beta_{10} - 3 \beta_{11} + 2 \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{78} + ( -2 - \beta_{1} + 2 \beta_{3} - 2 \beta_{5} - \beta_{6} - 3 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{14} ) q^{79} + ( -9 - 2 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} + 2 \beta_{12} + \beta_{13} + 3 \beta_{14} - \beta_{15} ) q^{80} + ( -6 + 6 \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{8} + \beta_{10} + 3 \beta_{11} + 3 \beta_{13} - 2 \beta_{14} ) q^{81} + ( 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} - 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{15} ) q^{82} + ( -5 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + 2 \beta_{7} + \beta_{9} + \beta_{10} + \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{83} + ( \beta_{1} + \beta_{3} + 2 \beta_{4} + 3 \beta_{5} - 4 \beta_{6} + 4 \beta_{7} - \beta_{8} - \beta_{9} + 4 \beta_{10} - 3 \beta_{11} - 2 \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{84} + ( -1 - \beta_{4} ) q^{85} + ( 5 + \beta_{1} + 5 \beta_{2} - 3 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 3 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} + \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{86} + ( 1 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} - 3 \beta_{12} - 2 \beta_{13} - 3 \beta_{14} ) q^{87} + ( -2 \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{88} + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + 2 \beta_{12} + \beta_{13} + 3 \beta_{14} - 2 \beta_{15} ) q^{89} + ( -3 + \beta_{1} - 3 \beta_{2} + \beta_{3} + 3 \beta_{4} + 4 \beta_{5} + \beta_{6} + 3 \beta_{9} - \beta_{12} + \beta_{13} + 3 \beta_{14} ) q^{90} + ( 4 - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{8} - 2 \beta_{10} + 3 \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{15} ) q^{91} + ( -3 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 3 \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{12} - \beta_{13} + \beta_{15} ) q^{92} + ( -3 - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{13} - \beta_{14} ) q^{93} + ( -1 + 8 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{14} + \beta_{15} ) q^{94} + ( 2 - 5 \beta_{1} + 3 \beta_{2} + \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} - 2 \beta_{10} + \beta_{11} - 2 \beta_{12} + \beta_{13} ) q^{95} + ( 9 \beta_{1} - \beta_{2} - 4 \beta_{3} - \beta_{5} - \beta_{6} + 3 \beta_{7} + \beta_{8} + \beta_{9} - 4 \beta_{10} + 4 \beta_{11} - \beta_{12} + \beta_{13} ) q^{96} + ( -1 - 3 \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} - 4 \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} + 3 \beta_{13} - 2 \beta_{15} ) q^{97} + ( -6 - 7 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} + \beta_{4} + 4 \beta_{5} + 2 \beta_{7} - \beta_{8} + \beta_{9} + 4 \beta_{10} - 5 \beta_{11} + \beta_{12} - \beta_{13} + 4 \beta_{14} + \beta_{15} ) q^{98} + ( -4 + 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + 3 \beta_{11} + 3 \beta_{13} - \beta_{14} - \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 6q^{2} - 7q^{3} + 14q^{4} - 21q^{5} - 5q^{6} - 11q^{7} - 12q^{8} + 17q^{9} + O(q^{10}) \) \( 16q - 6q^{2} - 7q^{3} + 14q^{4} - 21q^{5} - 5q^{6} - 11q^{7} - 12q^{8} + 17q^{9} + 12q^{10} - 7q^{11} - 4q^{12} - 16q^{13} + 11q^{14} + 7q^{15} + 22q^{16} + 16q^{17} + 11q^{18} + q^{19} - 43q^{20} - 8q^{21} - 18q^{22} - 8q^{23} - 39q^{24} + 23q^{25} - 49q^{26} - 7q^{27} - 15q^{28} - 39q^{29} + q^{30} - 3q^{31} - 15q^{33} - 6q^{34} + 9q^{35} - 17q^{36} - 28q^{37} - 27q^{38} - 4q^{39} + 26q^{40} - 31q^{41} - 45q^{42} + 5q^{43} - 19q^{44} - 79q^{45} - 39q^{46} - 47q^{47} - 31q^{48} + 35q^{49} - 13q^{50} - 7q^{51} + 9q^{52} - 36q^{53} + 9q^{54} + 32q^{55} + 3q^{56} + 6q^{57} + 22q^{58} + 16q^{59} + 6q^{60} - 22q^{61} + q^{62} - 19q^{63} + 32q^{64} - 19q^{65} + 48q^{66} + 4q^{67} + 14q^{68} + 14q^{69} - 11q^{70} - 5q^{71} + 40q^{72} - 35q^{73} + 31q^{74} - 12q^{75} + q^{76} - 79q^{77} + 31q^{78} - 48q^{79} - 127q^{80} - 28q^{81} - 7q^{82} - 42q^{83} + 28q^{84} - 21q^{85} + 58q^{86} - 16q^{87} - 2q^{88} + 20q^{89} - 14q^{90} + 49q^{91} - 21q^{92} - 55q^{93} - 22q^{95} + 24q^{96} - 2q^{97} - 78q^{98} - 35q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 6 x^{15} - 5 x^{14} + 90 x^{13} - 82 x^{12} - 456 x^{11} + 723 x^{10} + 951 x^{9} - 2105 x^{8} - 695 x^{7} + 2641 x^{6} - 151 x^{5} - 1323 x^{4} + 301 x^{3} + 179 x^{2} - 50 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\(-21607 \nu^{15} + 206853 \nu^{14} - 210704 \nu^{13} - 2846302 \nu^{12} + 6450296 \nu^{11} + 12562022 \nu^{10} - 38927099 \nu^{9} - 22572604 \nu^{8} + 96459971 \nu^{7} + 21890310 \nu^{6} - 111632131 \nu^{5} - 20768470 \nu^{4} + 58636299 \nu^{3} + 12947770 \nu^{2} - 12584729 \nu - 1629531\)\()/796522\)
\(\beta_{4}\)\(=\)\((\)\(-46479 \nu^{15} + 448281 \nu^{14} - 494240 \nu^{13} - 6086430 \nu^{12} + 14620862 \nu^{11} + 25713044 \nu^{10} - 88715715 \nu^{9} - 38866840 \nu^{8} + 222583253 \nu^{7} + 17845342 \nu^{6} - 259608919 \nu^{5} - 6096132 \nu^{4} + 129950349 \nu^{3} + 6021412 \nu^{2} - 18751633 \nu - 252191\)\()/796522\)
\(\beta_{5}\)\(=\)\((\)\(-73855 \nu^{15} + 331382 \nu^{14} + 929938 \nu^{13} - 5685137 \nu^{12} - 2256821 \nu^{11} + 35569812 \nu^{10} - 11490419 \nu^{9} - 102911265 \nu^{8} + 63824178 \nu^{7} + 140772231 \nu^{6} - 102838342 \nu^{5} - 81535535 \nu^{4} + 58146626 \nu^{3} + 13601600 \nu^{2} - 8034099 \nu + 621523\)\()/398261\)
\(\beta_{6}\)\(=\)\((\)\(169225 \nu^{15} - 821071 \nu^{14} - 1814188 \nu^{13} + 13610466 \nu^{12} + 173262 \nu^{11} - 80818546 \nu^{10} + 54087255 \nu^{9} + 216056280 \nu^{8} - 215179431 \nu^{7} - 258192220 \nu^{6} + 314110313 \nu^{5} + 109309352 \nu^{4} - 170098381 \nu^{3} - 1162256 \nu^{2} + 26785741 \nu - 1684399\)\()/796522\)
\(\beta_{7}\)\(=\)\((\)\(169225 \nu^{15} - 821071 \nu^{14} - 1814188 \nu^{13} + 13610466 \nu^{12} + 173262 \nu^{11} - 80818546 \nu^{10} + 54087255 \nu^{9} + 216056280 \nu^{8} - 215179431 \nu^{7} - 258192220 \nu^{6} + 314110313 \nu^{5} + 109309352 \nu^{4} - 169301859 \nu^{3} - 1162256 \nu^{2} + 22006609 \nu - 1684399\)\()/796522\)
\(\beta_{8}\)\(=\)\((\)\(171663 \nu^{15} - 912757 \nu^{14} - 1423874 \nu^{13} + 14140202 \nu^{12} - 4762104 \nu^{11} - 76771974 \nu^{10} + 68975265 \nu^{9} + 188185360 \nu^{8} - 212042053 \nu^{7} - 220121182 \nu^{6} + 263860137 \nu^{5} + 112170862 \nu^{4} - 126473895 \nu^{3} - 17602844 \nu^{2} + 12942671 \nu + 623575\)\()/796522\)
\(\beta_{9}\)\(=\)\((\)\(-252191 \nu^{15} + 1466667 \nu^{14} + 1709236 \nu^{13} - 23191430 \nu^{12} + 14593232 \nu^{11} + 129619958 \nu^{10} - 156621049 \nu^{9} - 328549356 \nu^{8} + 491995215 \nu^{7} + 397855998 \nu^{6} - 648191089 \nu^{5} - 221528078 \nu^{4} + 327552561 \nu^{3} + 54040858 \nu^{2} - 39120777 \nu - 5345561\)\()/796522\)
\(\beta_{10}\)\(=\)\((\)\(174999 \nu^{15} - 976139 \nu^{14} - 1206377 \nu^{13} + 14819972 \nu^{12} - 8664781 \nu^{11} - 77542723 \nu^{10} + 90954465 \nu^{9} + 177914468 \nu^{8} - 265461630 \nu^{7} - 185448483 \nu^{6} + 321400128 \nu^{5} + 76413493 \nu^{4} - 149988142 \nu^{3} - 5471927 \nu^{2} + 17324960 \nu - 317590\)\()/398261\)
\(\beta_{11}\)\(=\)\((\)\(454985 \nu^{15} - 2624923 \nu^{14} - 2947570 \nu^{13} + 40413834 \nu^{12} - 26534880 \nu^{11} - 216678478 \nu^{10} + 267361123 \nu^{9} + 518142928 \nu^{8} - 795429433 \nu^{7} - 580720748 \nu^{6} + 991791603 \nu^{5} + 280288612 \nu^{4} - 474483529 \nu^{3} - 38353282 \nu^{2} + 54032887 \nu - 180195\)\()/796522\)
\(\beta_{12}\)\(=\)\((\)\(-493455 \nu^{15} + 2751753 \nu^{14} + 3520236 \nu^{13} - 42280306 \nu^{12} + 22964484 \nu^{11} + 226708120 \nu^{10} - 252272051 \nu^{9} - 548042556 \nu^{8} + 755318627 \nu^{7} + 642026498 \nu^{6} - 949139277 \nu^{5} - 351211644 \nu^{4} + 472849827 \nu^{3} + 68918706 \nu^{2} - 60979649 \nu - 1226399\)\()/796522\)
\(\beta_{13}\)\(=\)\((\)\(-710147 \nu^{15} + 3849819 \nu^{14} + 5665410 \nu^{13} - 60313288 \nu^{12} + 25543250 \nu^{11} + 332167660 \nu^{10} - 336831221 \nu^{9} - 822852954 \nu^{8} + 1066670989 \nu^{7} + 954414268 \nu^{6} - 1385054993 \nu^{5} - 466744348 \nu^{4} + 695675295 \nu^{3} + 66001372 \nu^{2} - 87539399 \nu + 303563\)\()/796522\)
\(\beta_{14}\)\(=\)\((\)\(360387 \nu^{15} - 1988496 \nu^{14} - 2756504 \nu^{13} + 31115804 \nu^{12} - 14693575 \nu^{11} - 171588528 \nu^{10} + 179552242 \nu^{9} + 430148838 \nu^{8} - 560595280 \nu^{7} - 521963732 \nu^{6} + 724676789 \nu^{5} + 293307769 \nu^{4} - 364544169 \nu^{3} - 62155134 \nu^{2} + 47365022 \nu + 1585084\)\()/398261\)
\(\beta_{15}\)\(=\)\((\)\(-488289 \nu^{15} + 2675091 \nu^{14} + 3747450 \nu^{13} - 41578626 \nu^{12} + 19291176 \nu^{11} + 226475122 \nu^{10} - 236146200 \nu^{9} - 554950973 \nu^{8} + 728495661 \nu^{7} + 644160968 \nu^{6} - 927386318 \nu^{5} - 329011326 \nu^{4} + 461460405 \nu^{3} + 57412699 \nu^{2} - 59282695 \nu - 1545655\)\()/398261\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{7} - \beta_{6} + 6 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{14} - \beta_{12} - \beta_{9} + \beta_{4} - \beta_{3} + 7 \beta_{2} + \beta_{1} + 15\)
\(\nu^{5}\)\(=\)\(\beta_{13} + \beta_{10} + 10 \beta_{7} - 9 \beta_{6} - \beta_{5} - 2 \beta_{3} - \beta_{2} + 41 \beta_{1} - 3\)
\(\nu^{6}\)\(=\)\(-12 \beta_{14} - \beta_{13} - 11 \beta_{12} + \beta_{11} - 10 \beta_{9} - 2 \beta_{8} - \beta_{7} - \beta_{6} - 3 \beta_{5} + 10 \beta_{4} - 11 \beta_{3} + 47 \beta_{2} + 11 \beta_{1} + 88\)
\(\nu^{7}\)\(=\)\(2 \beta_{15} - \beta_{14} + 9 \beta_{13} + 14 \beta_{10} - \beta_{9} - \beta_{8} + 81 \beta_{7} - 73 \beta_{6} - 15 \beta_{5} + \beta_{4} - 26 \beta_{3} - 13 \beta_{2} + 291 \beta_{1} - 38\)
\(\nu^{8}\)\(=\)\(\beta_{15} - 107 \beta_{14} - 17 \beta_{13} - 91 \beta_{12} + 11 \beta_{11} + 4 \beta_{10} - 81 \beta_{9} - 30 \beta_{8} - 14 \beta_{7} - 19 \beta_{6} - 45 \beta_{5} + 80 \beta_{4} - 97 \beta_{3} + 317 \beta_{2} + 97 \beta_{1} + 560\)
\(\nu^{9}\)\(=\)\(30 \beta_{15} - 19 \beta_{14} + 54 \beta_{13} + \beta_{12} - \beta_{11} + 137 \beta_{10} - 13 \beta_{9} - 22 \beta_{8} + 612 \beta_{7} - 574 \beta_{6} - 160 \beta_{5} + 14 \beta_{4} - 251 \beta_{3} - 127 \beta_{2} + 2094 \beta_{1} - 348\)
\(\nu^{10}\)\(=\)\(22 \beta_{15} - 865 \beta_{14} - 201 \beta_{13} - 688 \beta_{12} + 92 \beta_{11} + 67 \beta_{10} - 613 \beta_{9} - 314 \beta_{8} - 132 \beta_{7} - 237 \beta_{6} - 473 \beta_{5} + 605 \beta_{4} - 806 \beta_{3} + 2160 \beta_{2} + 810 \beta_{1} + 3732\)
\(\nu^{11}\)\(=\)\(314 \beta_{15} - 237 \beta_{14} + 232 \beta_{13} + 19 \beta_{12} - 12 \beta_{11} + 1173 \beta_{10} - 119 \beta_{9} - 299 \beta_{8} + 4496 \beta_{7} - 4445 \beta_{6} - 1484 \beta_{5} + 152 \beta_{4} - 2178 \beta_{3} - 1105 \beta_{2} + 15170 \beta_{1} - 2828\)
\(\nu^{12}\)\(=\)\(299 \beta_{15} - 6703 \beta_{14} - 2027 \beta_{13} - 5027 \beta_{12} + 711 \beta_{11} + 754 \beta_{10} - 4506 \beta_{9} - 2868 \beta_{8} - 1052 \beta_{7} - 2474 \beta_{6} - 4314 \beta_{5} + 4514 \beta_{4} - 6531 \beta_{3} + 14860 \beta_{2} + 6644 \beta_{1} + 25562\)
\(\nu^{13}\)\(=\)\(2868 \beta_{15} - 2464 \beta_{14} + 304 \beta_{13} + 211 \beta_{12} - 87 \beta_{11} + 9428 \beta_{10} - 960 \beta_{9} - 3294 \beta_{8} + 32663 \beta_{7} - 34104 \beta_{6} - 12770 \beta_{5} + 1525 \beta_{4} - 17978 \beta_{3} - 9022 \beta_{2} + 110347 \beta_{1} - 21700\)
\(\nu^{14}\)\(=\)\(3294 \beta_{15} - 50861 \beta_{14} - 18715 \beta_{13} - 36261 \beta_{12} + 5341 \beta_{11} + 7218 \beta_{10} - 32672 \beta_{9} - 24544 \beta_{8} - 7599 \beta_{7} - 23460 \beta_{6} - 36587 \beta_{5} + 33654 \beta_{4} - 52104 \beta_{3} + 103054 \beta_{2} + 54037 \beta_{1} + 178066\)
\(\nu^{15}\)\(=\)\(24544 \beta_{15} - 23195 \beta_{14} - 8241 \beta_{13} + 1770 \beta_{12} - 452 \beta_{11} + 73300 \beta_{10} - 7352 \beta_{9} - 32375 \beta_{8} + 236445 \beta_{7} - 260023 \beta_{6} - 105031 \beta_{5} + 14633 \beta_{4} - 144381 \beta_{3} - 70781 \beta_{2} + 804918 \beta_{1} - 161242\)

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
2.74400
2.52760
2.33579
1.98811
1.79724
1.62876
0.886486
0.431919
0.377947
−0.0187805
−0.446420
−0.896165
−1.35393
−1.38820
−1.94258
−2.67178
−2.74400 −0.794276 5.52956 −3.63969 2.17950 −4.59580 −9.68512 −2.36913 9.98732
1.2 −2.52760 −0.131380 4.38874 −1.05573 0.332076 3.96203 −6.03776 −2.98274 2.66845
1.3 −2.33579 2.58736 3.45592 −4.30332 −6.04353 1.79793 −3.40073 3.69442 10.0517
1.4 −1.98811 1.88888 1.95260 1.43913 −3.75530 −4.79587 0.0942457 0.567851 −2.86115
1.5 −1.79724 −2.09283 1.23006 −2.26932 3.76130 −0.408229 1.38377 1.37993 4.07850
1.6 −1.62876 −1.50871 0.652867 1.81397 2.45732 −1.88099 2.19416 −0.723806 −2.95453
1.7 −0.886486 −3.35617 −1.21414 −3.79855 2.97520 −4.56031 2.84929 8.26386 3.36736
1.8 −0.431919 0.385290 −1.81345 −0.726091 −0.166414 1.71462 1.64710 −2.85155 0.313613
1.9 −0.377947 −1.60628 −1.85716 0.658559 0.607088 −1.52100 1.45780 −0.419863 −0.248900
1.10 0.0187805 1.61675 −1.99965 −1.12710 0.0303633 −0.878091 −0.0751153 −0.386118 −0.0211674
1.11 0.446420 −3.01975 −1.80071 −2.85809 −1.34808 4.83764 −1.69671 6.11887 −1.27591
1.12 0.896165 1.99385 −1.19689 −0.516029 1.78681 −1.20609 −2.86494 0.975426 −0.462447
1.13 1.35393 −1.79865 −0.166881 1.77529 −2.43525 2.41510 −2.93380 0.235159 2.40362
1.14 1.38820 2.64276 −0.0729129 −3.88583 3.66866 −3.47551 −2.87761 3.98417 −5.39429
1.15 1.94258 −1.53769 1.77364 1.30751 −2.98709 −3.17249 −0.439731 −0.635509 2.53995
1.16 2.67178 −2.26915 5.13841 −3.81472 −6.06267 0.767070 8.38515 2.14904 −10.1921
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

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 1003.2.a.h 16
3.b odd 2 1 9027.2.a.n 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1003.2.a.h 16 1.a even 1 1 trivial
9027.2.a.n 16 3.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(17\) \(-1\)
\(59\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1003))\):

\(T_{2}^{16} + \cdots\)
\(T_{3}^{16} + \cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 6 T + 27 T^{2} + 90 T^{3} + 258 T^{4} + 636 T^{5} + 1415 T^{6} + 2841 T^{7} + 5267 T^{8} + 9017 T^{9} + 14477 T^{10} + 21833 T^{11} + 31441 T^{12} + 43557 T^{13} + 59447 T^{14} + 81092 T^{15} + 113283 T^{16} + 162184 T^{17} + 237788 T^{18} + 348456 T^{19} + 503056 T^{20} + 698656 T^{21} + 926528 T^{22} + 1154176 T^{23} + 1348352 T^{24} + 1454592 T^{25} + 1448960 T^{26} + 1302528 T^{27} + 1056768 T^{28} + 737280 T^{29} + 442368 T^{30} + 196608 T^{31} + 65536 T^{32} \)
$3$ \( 1 + 7 T + 40 T^{2} + 161 T^{3} + 587 T^{4} + 1837 T^{5} + 5398 T^{6} + 14467 T^{7} + 36819 T^{8} + 87678 T^{9} + 199691 T^{10} + 431612 T^{11} + 896496 T^{12} + 1779482 T^{13} + 3402090 T^{14} + 6235902 T^{15} + 11024208 T^{16} + 18707706 T^{17} + 30618810 T^{18} + 48046014 T^{19} + 72616176 T^{20} + 104881716 T^{21} + 145574739 T^{22} + 191751786 T^{23} + 241569459 T^{24} + 284753961 T^{25} + 318746502 T^{26} + 325419039 T^{27} + 311955867 T^{28} + 256686003 T^{29} + 191318760 T^{30} + 100442349 T^{31} + 43046721 T^{32} \)
$5$ \( 1 + 21 T + 249 T^{2} + 2143 T^{3} + 14807 T^{4} + 86535 T^{5} + 441605 T^{6} + 2009432 T^{7} + 8272917 T^{8} + 31143873 T^{9} + 108048711 T^{10} + 347499761 T^{11} + 1040653133 T^{12} + 2911461051 T^{13} + 7628016320 T^{14} + 18746200712 T^{15} + 43253965373 T^{16} + 93731003560 T^{17} + 190700408000 T^{18} + 363932631375 T^{19} + 650408208125 T^{20} + 1085936753125 T^{21} + 1688261109375 T^{22} + 2433115078125 T^{23} + 3231608203125 T^{24} + 3924671875000 T^{25} + 4312548828125 T^{26} + 4225341796875 T^{27} + 3614990234375 T^{28} + 2615966796875 T^{29} + 1519775390625 T^{30} + 640869140625 T^{31} + 152587890625 T^{32} \)
$7$ \( 1 + 11 T + 99 T^{2} + 631 T^{3} + 3529 T^{4} + 16644 T^{5} + 71610 T^{6} + 275534 T^{7} + 992543 T^{8} + 3302021 T^{9} + 10512336 T^{10} + 31647691 T^{11} + 92926579 T^{12} + 262444709 T^{13} + 731975367 T^{14} + 1980226443 T^{15} + 5314535269 T^{16} + 13861585101 T^{17} + 35866792983 T^{18} + 90018535187 T^{19} + 223116716179 T^{20} + 531902742637 T^{21} + 1236765818064 T^{22} + 2719356280403 T^{23} + 5721812878943 T^{24} + 11118790751138 T^{25} + 20228052580890 T^{26} + 32910626310492 T^{27} + 48845902532329 T^{28} + 61136965566817 T^{29} + 67144084212051 T^{30} + 52223176609373 T^{31} + 33232930569601 T^{32} \)
$11$ \( 1 + 7 T + 99 T^{2} + 573 T^{3} + 4855 T^{4} + 24373 T^{5} + 158595 T^{6} + 708606 T^{7} + 3877077 T^{8} + 15687965 T^{9} + 75519819 T^{10} + 280177541 T^{11} + 1218630412 T^{12} + 4181500767 T^{13} + 16705740691 T^{14} + 53294814856 T^{15} + 197433330166 T^{16} + 586242963416 T^{17} + 2021394623611 T^{18} + 5565577520877 T^{19} + 17841967862092 T^{20} + 45122873155591 T^{21} + 133787966067459 T^{22} + 305714056597015 T^{23} + 831085887270837 T^{24} + 1670855881528746 T^{25} + 4113545854595595 T^{26} + 6953901347801903 T^{27} + 15237069768980455 T^{28} + 19781514058472463 T^{29} + 37595233524740859 T^{30} + 29240737185909557 T^{31} + 45949729863572161 T^{32} \)
$13$ \( 1 + 16 T + 203 T^{2} + 1792 T^{3} + 14275 T^{4} + 95415 T^{5} + 600564 T^{6} + 3371571 T^{7} + 18169721 T^{8} + 89901243 T^{9} + 431538912 T^{10} + 1930538895 T^{11} + 8424397174 T^{12} + 34540053626 T^{13} + 138565115677 T^{14} + 524517883170 T^{15} + 1945586553346 T^{16} + 6818732481210 T^{17} + 23417504549413 T^{18} + 75884497816322 T^{19} + 240609207686614 T^{20} + 716795577941235 T^{21} + 2082955904291808 T^{22} + 5641169674706631 T^{23} + 14821599611698841 T^{24} + 35753822555524983 T^{25} + 82792847298802836 T^{26} + 170998983997040355 T^{27} + 332580165123416275 T^{28} + 542752191013317376 T^{29} + 799287406296955667 T^{30} + 818974288225452112 T^{31} + 665416609183179841 T^{32} \)
$17$ \( ( 1 - T )^{16} \)
$19$ \( 1 - T + 107 T^{2} - 107 T^{3} + 6348 T^{4} - 4850 T^{5} + 273578 T^{6} - 134840 T^{7} + 9529325 T^{8} - 2760589 T^{9} + 282243356 T^{10} - 46032983 T^{11} + 7281865981 T^{12} - 636489901 T^{13} + 165561919843 T^{14} - 8211582005 T^{15} + 3336711850579 T^{16} - 156020058095 T^{17} + 59767853063323 T^{18} - 4365684230959 T^{19} + 948980056509901 T^{20} - 113982223173317 T^{21} + 13278387339416636 T^{22} - 2467612490094271 T^{23} + 161841891875677325 T^{24} - 43511209168520360 T^{25} + 1677324844676681978 T^{26} - 564977755656362150 T^{27} + 14050123106231990028 T^{28} - 4499669230461505313 T^{29} + 85493715378768600947 T^{30} - 15181127029874798299 T^{31} + \)\(28\!\cdots\!81\)\( T^{32} \)
$23$ \( 1 + 8 T + 160 T^{2} + 1156 T^{3} + 13275 T^{4} + 89694 T^{5} + 775926 T^{6} + 4914858 T^{7} + 35792094 T^{8} + 211598797 T^{9} + 1371736394 T^{10} + 7558519108 T^{11} + 44803889428 T^{12} + 230539116504 T^{13} + 1264759463636 T^{14} + 6088439654145 T^{15} + 31112640363596 T^{16} + 140034112045335 T^{17} + 669057756263444 T^{18} + 2804969430504168 T^{19} + 12537965221420948 T^{20} + 48649221551142044 T^{21} + 203066216559444266 T^{22} + 720456968580187259 T^{23} + 2802914146410168414 T^{24} + 8852409567412717254 T^{25} + 32143907139961813974 T^{26} + 85461318426331768338 T^{27} + \)\(29\!\cdots\!75\)\( T^{28} + \)\(58\!\cdots\!48\)\( T^{29} + \)\(18\!\cdots\!40\)\( T^{30} + \)\(21\!\cdots\!56\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \)
$29$ \( 1 + 39 T + 928 T^{2} + 16208 T^{3} + 230056 T^{4} + 2780632 T^{5} + 29604156 T^{6} + 283587370 T^{7} + 2485169966 T^{8} + 20154979173 T^{9} + 152664366097 T^{10} + 1087056903986 T^{11} + 7313935686038 T^{12} + 46666449986350 T^{13} + 283154391474588 T^{14} + 1636759340906011 T^{15} + 9023892718674587 T^{16} + 47466020886274319 T^{17} + 238132843230128508 T^{18} + 1138148048717090150 T^{19} + 5173007745956642678 T^{20} + 22296786129135539914 T^{21} + 90808325240177348137 T^{22} + \)\(34\!\cdots\!57\)\( T^{23} + \)\(12\!\cdots\!26\)\( T^{24} + \)\(41\!\cdots\!30\)\( T^{25} + \)\(12\!\cdots\!56\)\( T^{26} + \)\(33\!\cdots\!28\)\( T^{27} + \)\(81\!\cdots\!96\)\( T^{28} + \)\(16\!\cdots\!12\)\( T^{29} + \)\(27\!\cdots\!68\)\( T^{30} + \)\(33\!\cdots\!11\)\( T^{31} + \)\(25\!\cdots\!21\)\( T^{32} \)
$31$ \( 1 + 3 T + 255 T^{2} + 791 T^{3} + 32526 T^{4} + 101121 T^{5} + 2753424 T^{6} + 8468886 T^{7} + 174562090 T^{8} + 528154096 T^{9} + 8901907102 T^{10} + 26393054295 T^{11} + 382768477555 T^{12} + 1104116824605 T^{13} + 14294540542319 T^{14} + 39577703446301 T^{15} + 470796472164880 T^{16} + 1226908806835331 T^{17} + 13737053461168559 T^{18} + 32892744321807555 T^{19} + 353494727160071155 T^{20} + 755610736762753545 T^{21} + 7900475320945042462 T^{22} + 14530899834392048656 T^{23} + \)\(14\!\cdots\!90\)\( T^{24} + \)\(22\!\cdots\!06\)\( T^{25} + \)\(22\!\cdots\!24\)\( T^{26} + \)\(25\!\cdots\!51\)\( T^{27} + \)\(25\!\cdots\!86\)\( T^{28} + \)\(19\!\cdots\!81\)\( T^{29} + \)\(19\!\cdots\!55\)\( T^{30} + \)\(70\!\cdots\!53\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \)
$37$ \( 1 + 28 T + 659 T^{2} + 11230 T^{3} + 168029 T^{4} + 2160736 T^{5} + 25241316 T^{6} + 267548264 T^{7} + 2629412336 T^{8} + 24030058442 T^{9} + 206141237141 T^{10} + 1665515462176 T^{11} + 12724942102618 T^{12} + 92203962435862 T^{13} + 634461000765648 T^{14} + 4155167012371534 T^{15} + 25893210569519688 T^{16} + 153741179457746758 T^{17} + 868577110048172112 T^{18} + 4670407309263717886 T^{19} + 23848590215984653498 T^{20} + \)\(11\!\cdots\!32\)\( T^{21} + \)\(52\!\cdots\!69\)\( T^{22} + \)\(22\!\cdots\!86\)\( T^{23} + \)\(92\!\cdots\!56\)\( T^{24} + \)\(34\!\cdots\!28\)\( T^{25} + \)\(12\!\cdots\!84\)\( T^{26} + \)\(38\!\cdots\!68\)\( T^{27} + \)\(11\!\cdots\!49\)\( T^{28} + \)\(27\!\cdots\!10\)\( T^{29} + \)\(59\!\cdots\!51\)\( T^{30} + \)\(93\!\cdots\!04\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \)
$41$ \( 1 + 31 T + 845 T^{2} + 16144 T^{3} + 274419 T^{4} + 3937303 T^{5} + 51468151 T^{6} + 602401175 T^{7} + 6533515132 T^{8} + 65168277381 T^{9} + 609165944944 T^{10} + 5312237628797 T^{11} + 43729918253400 T^{12} + 338547209812076 T^{13} + 2484799560505024 T^{14} + 17223431233458923 T^{15} + 113409862522256899 T^{16} + 706160680571815843 T^{17} + 4176948061208945344 T^{18} + 23333012247458089996 T^{19} + \)\(12\!\cdots\!00\)\( T^{20} + \)\(61\!\cdots\!97\)\( T^{21} + \)\(28\!\cdots\!04\)\( T^{22} + \)\(12\!\cdots\!61\)\( T^{23} + \)\(52\!\cdots\!72\)\( T^{24} + \)\(19\!\cdots\!75\)\( T^{25} + \)\(69\!\cdots\!51\)\( T^{26} + \)\(21\!\cdots\!23\)\( T^{27} + \)\(61\!\cdots\!39\)\( T^{28} + \)\(14\!\cdots\!24\)\( T^{29} + \)\(32\!\cdots\!45\)\( T^{30} + \)\(48\!\cdots\!31\)\( T^{31} + \)\(63\!\cdots\!41\)\( T^{32} \)
$43$ \( 1 - 5 T + 197 T^{2} - 481 T^{3} + 23722 T^{4} - 45125 T^{5} + 2315260 T^{6} - 3018587 T^{7} + 179938455 T^{8} - 163326585 T^{9} + 12170024380 T^{10} - 8524437672 T^{11} + 711460104377 T^{12} - 359409231654 T^{13} + 36587132230511 T^{14} - 16088275772689 T^{15} + 1675798794529458 T^{16} - 691795858225627 T^{17} + 67649607494214839 T^{18} - 28575549781114578 T^{19} + 2432340516304191977 T^{20} - 1253164309611264696 T^{21} + 76931142421161134620 T^{22} - 44395205491549379595 T^{23} + \)\(21\!\cdots\!55\)\( T^{24} - \)\(15\!\cdots\!41\)\( T^{25} + \)\(50\!\cdots\!40\)\( T^{26} - \)\(41\!\cdots\!75\)\( T^{27} + \)\(94\!\cdots\!22\)\( T^{28} - \)\(82\!\cdots\!83\)\( T^{29} + \)\(14\!\cdots\!53\)\( T^{30} - \)\(15\!\cdots\!35\)\( T^{31} + \)\(13\!\cdots\!01\)\( T^{32} \)
$47$ \( 1 + 47 T + 1521 T^{2} + 35695 T^{3} + 695344 T^{4} + 11443167 T^{5} + 166215515 T^{6} + 2150890354 T^{7} + 25336473439 T^{8} + 273240068043 T^{9} + 2733546016518 T^{10} + 25462180643677 T^{11} + 222818061560779 T^{12} + 1835643848459763 T^{13} + 14325453981515050 T^{14} + 105939294919881198 T^{15} + 745268954307368866 T^{16} + 4979146861234416306 T^{17} + 31644927845166745450 T^{18} + \)\(19\!\cdots\!49\)\( T^{19} + \)\(10\!\cdots\!99\)\( T^{20} + \)\(58\!\cdots\!39\)\( T^{21} + \)\(29\!\cdots\!22\)\( T^{22} + \)\(13\!\cdots\!09\)\( T^{23} + \)\(60\!\cdots\!79\)\( T^{24} + \)\(24\!\cdots\!18\)\( T^{25} + \)\(87\!\cdots\!35\)\( T^{26} + \)\(28\!\cdots\!01\)\( T^{27} + \)\(80\!\cdots\!04\)\( T^{28} + \)\(19\!\cdots\!65\)\( T^{29} + \)\(39\!\cdots\!49\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \)
$53$ \( 1 + 36 T + 1175 T^{2} + 26167 T^{3} + 526553 T^{4} + 8814464 T^{5} + 135423948 T^{6} + 1846719868 T^{7} + 23401307600 T^{8} + 271068847869 T^{9} + 2943032349289 T^{10} + 29666205870053 T^{11} + 281889792560510 T^{12} + 2508334485133314 T^{13} + 21112835718481295 T^{14} + 167153704971172239 T^{15} + 1253870241859978341 T^{16} + 8859146363472128667 T^{17} + 59305955533213957655 T^{18} + \)\(37\!\cdots\!78\)\( T^{19} + \)\(22\!\cdots\!10\)\( T^{20} + \)\(12\!\cdots\!29\)\( T^{21} + \)\(65\!\cdots\!81\)\( T^{22} + \)\(31\!\cdots\!53\)\( T^{23} + \)\(14\!\cdots\!00\)\( T^{24} + \)\(60\!\cdots\!44\)\( T^{25} + \)\(23\!\cdots\!52\)\( T^{26} + \)\(81\!\cdots\!08\)\( T^{27} + \)\(25\!\cdots\!73\)\( T^{28} + \)\(68\!\cdots\!91\)\( T^{29} + \)\(16\!\cdots\!75\)\( T^{30} + \)\(26\!\cdots\!52\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \)
$59$ \( ( 1 - T )^{16} \)
$61$ \( 1 + 22 T + 726 T^{2} + 11788 T^{3} + 229971 T^{4} + 2989207 T^{5} + 44054827 T^{6} + 482781207 T^{7} + 5931009252 T^{8} + 57058208894 T^{9} + 619281836577 T^{10} + 5407165081688 T^{11} + 53761513199754 T^{12} + 435350040127477 T^{13} + 4029639992006474 T^{14} + 30479162475907733 T^{15} + 263314323349656692 T^{16} + 1859228911030371713 T^{17} + 14994290410256089754 T^{18} + 98816187458174856937 T^{19} + \)\(74\!\cdots\!14\)\( T^{20} + \)\(45\!\cdots\!88\)\( T^{21} + \)\(31\!\cdots\!97\)\( T^{22} + \)\(17\!\cdots\!74\)\( T^{23} + \)\(11\!\cdots\!12\)\( T^{24} + \)\(56\!\cdots\!87\)\( T^{25} + \)\(31\!\cdots\!27\)\( T^{26} + \)\(13\!\cdots\!27\)\( T^{27} + \)\(61\!\cdots\!91\)\( T^{28} + \)\(19\!\cdots\!28\)\( T^{29} + \)\(71\!\cdots\!66\)\( T^{30} + \)\(13\!\cdots\!22\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \)
$67$ \( 1 - 4 T + 508 T^{2} - 486 T^{3} + 121581 T^{4} + 261148 T^{5} + 19273613 T^{6} + 99314231 T^{7} + 2421011347 T^{8} + 17808146326 T^{9} + 267088396674 T^{10} + 2154454341593 T^{11} + 26510155079768 T^{12} + 200163741901303 T^{13} + 2300638246781225 T^{14} + 15443017906683695 T^{15} + 168758840060292406 T^{16} + 1034682199747807565 T^{17} + 10327565089800919025 T^{18} + 60201847505461594189 T^{19} + \)\(53\!\cdots\!28\)\( T^{20} + \)\(29\!\cdots\!51\)\( T^{21} + \)\(24\!\cdots\!06\)\( T^{22} + \)\(10\!\cdots\!98\)\( T^{23} + \)\(98\!\cdots\!27\)\( T^{24} + \)\(27\!\cdots\!57\)\( T^{25} + \)\(35\!\cdots\!37\)\( T^{26} + \)\(31\!\cdots\!84\)\( T^{27} + \)\(99\!\cdots\!41\)\( T^{28} - \)\(26\!\cdots\!82\)\( T^{29} + \)\(18\!\cdots\!32\)\( T^{30} - \)\(98\!\cdots\!72\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \)
$71$ \( 1 + 5 T + 707 T^{2} + 3199 T^{3} + 248382 T^{4} + 1037845 T^{5} + 57673238 T^{6} + 225467389 T^{7} + 9923775540 T^{8} + 36556228657 T^{9} + 1343843822580 T^{10} + 4670089592113 T^{11} + 148323449480723 T^{12} + 484195559141675 T^{13} + 13622265401829407 T^{14} + 41396926211404761 T^{15} + 1052601653972946460 T^{16} + 2939181761009738031 T^{17} + 68669839890622040687 T^{18} + \)\(17\!\cdots\!25\)\( T^{19} + \)\(37\!\cdots\!63\)\( T^{20} + \)\(84\!\cdots\!63\)\( T^{21} + \)\(17\!\cdots\!80\)\( T^{22} + \)\(33\!\cdots\!87\)\( T^{23} + \)\(64\!\cdots\!40\)\( T^{24} + \)\(10\!\cdots\!59\)\( T^{25} + \)\(18\!\cdots\!38\)\( T^{26} + \)\(23\!\cdots\!95\)\( T^{27} + \)\(40\!\cdots\!62\)\( T^{28} + \)\(37\!\cdots\!89\)\( T^{29} + \)\(58\!\cdots\!67\)\( T^{30} + \)\(29\!\cdots\!55\)\( T^{31} + \)\(41\!\cdots\!21\)\( T^{32} \)
$73$ \( 1 + 35 T + 1035 T^{2} + 20128 T^{3} + 344201 T^{4} + 4528714 T^{5} + 52898365 T^{6} + 472574263 T^{7} + 3575301535 T^{8} + 14472242265 T^{9} - 10259759258 T^{10} - 1413670638428 T^{11} - 13344230940186 T^{12} - 98999228688970 T^{13} + 14056056520030 T^{14} + 4087388912498153 T^{15} + 69628091750516746 T^{16} + 298379390612365169 T^{17} + 74904725195239870 T^{18} - 38512382946897042490 T^{19} - \)\(37\!\cdots\!26\)\( T^{20} - \)\(29\!\cdots\!04\)\( T^{21} - \)\(15\!\cdots\!62\)\( T^{22} + \)\(15\!\cdots\!05\)\( T^{23} + \)\(28\!\cdots\!35\)\( T^{24} + \)\(27\!\cdots\!19\)\( T^{25} + \)\(22\!\cdots\!85\)\( T^{26} + \)\(14\!\cdots\!78\)\( T^{27} + \)\(78\!\cdots\!21\)\( T^{28} + \)\(33\!\cdots\!24\)\( T^{29} + \)\(12\!\cdots\!15\)\( T^{30} + \)\(31\!\cdots\!95\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \)
$79$ \( 1 + 48 T + 1716 T^{2} + 43733 T^{3} + 949683 T^{4} + 17347139 T^{5} + 284281513 T^{6} + 4151599168 T^{7} + 55961588988 T^{8} + 692772349249 T^{9} + 8070707342734 T^{10} + 88091360119644 T^{11} + 919032606186396 T^{12} + 9123573520975185 T^{13} + 87713721926917761 T^{14} + 811771645641552513 T^{15} + 7336995592941878765 T^{16} + 64129960005682648527 T^{17} + \)\(54\!\cdots\!01\)\( T^{18} + \)\(44\!\cdots\!15\)\( T^{19} + \)\(35\!\cdots\!76\)\( T^{20} + \)\(27\!\cdots\!56\)\( T^{21} + \)\(19\!\cdots\!14\)\( T^{22} + \)\(13\!\cdots\!91\)\( T^{23} + \)\(84\!\cdots\!68\)\( T^{24} + \)\(49\!\cdots\!92\)\( T^{25} + \)\(26\!\cdots\!13\)\( T^{26} + \)\(12\!\cdots\!81\)\( T^{27} + \)\(56\!\cdots\!03\)\( T^{28} + \)\(20\!\cdots\!87\)\( T^{29} + \)\(63\!\cdots\!96\)\( T^{30} + \)\(13\!\cdots\!52\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \)
$83$ \( 1 + 42 T + 1656 T^{2} + 45425 T^{3} + 1121429 T^{4} + 23462170 T^{5} + 447315865 T^{6} + 7661927885 T^{7} + 121285518000 T^{8} + 1767394289294 T^{9} + 24044579507563 T^{10} + 304858972200110 T^{11} + 3632307914546706 T^{12} + 40607544102297339 T^{13} + 428241004835045976 T^{14} + 4252453091155187749 T^{15} + 39902891433324960408 T^{16} + \)\(35\!\cdots\!67\)\( T^{17} + \)\(29\!\cdots\!64\)\( T^{18} + \)\(23\!\cdots\!93\)\( T^{19} + \)\(17\!\cdots\!26\)\( T^{20} + \)\(12\!\cdots\!30\)\( T^{21} + \)\(78\!\cdots\!47\)\( T^{22} + \)\(47\!\cdots\!38\)\( T^{23} + \)\(27\!\cdots\!00\)\( T^{24} + \)\(14\!\cdots\!55\)\( T^{25} + \)\(69\!\cdots\!85\)\( T^{26} + \)\(30\!\cdots\!90\)\( T^{27} + \)\(11\!\cdots\!69\)\( T^{28} + \)\(40\!\cdots\!75\)\( T^{29} + \)\(12\!\cdots\!24\)\( T^{30} + \)\(25\!\cdots\!94\)\( T^{31} + \)\(50\!\cdots\!81\)\( T^{32} \)
$89$ \( 1 - 20 T + 597 T^{2} - 8420 T^{3} + 158284 T^{4} - 1939051 T^{5} + 29478345 T^{6} - 339122373 T^{7} + 4443220742 T^{8} - 48716866782 T^{9} + 567385758156 T^{10} - 5969911399957 T^{11} + 63612180884481 T^{12} - 645883424308806 T^{13} + 6449935124831842 T^{14} - 63212766860278727 T^{15} + 599655028685899152 T^{16} - 5625936250564806703 T^{17} + 51089936123793020482 T^{18} - \)\(45\!\cdots\!14\)\( T^{19} + \)\(39\!\cdots\!21\)\( T^{20} - \)\(33\!\cdots\!93\)\( T^{21} + \)\(28\!\cdots\!16\)\( T^{22} - \)\(21\!\cdots\!78\)\( T^{23} + \)\(17\!\cdots\!02\)\( T^{24} - \)\(11\!\cdots\!57\)\( T^{25} + \)\(91\!\cdots\!45\)\( T^{26} - \)\(53\!\cdots\!39\)\( T^{27} + \)\(39\!\cdots\!64\)\( T^{28} - \)\(18\!\cdots\!80\)\( T^{29} + \)\(11\!\cdots\!77\)\( T^{30} - \)\(34\!\cdots\!80\)\( T^{31} + \)\(15\!\cdots\!61\)\( T^{32} \)
$97$ \( 1 + 2 T + 621 T^{2} - 213 T^{3} + 205664 T^{4} - 437598 T^{5} + 49093119 T^{6} - 163737942 T^{7} + 9352075548 T^{8} - 38413772588 T^{9} + 1489316155695 T^{10} - 6708588826879 T^{11} + 203631678495233 T^{12} - 934785974558696 T^{13} + 24183348030921161 T^{14} - 107966335249417574 T^{15} + 2507492762870435228 T^{16} - 10472734519193504678 T^{17} + \)\(22\!\cdots\!49\)\( T^{18} - \)\(85\!\cdots\!08\)\( T^{19} + \)\(18\!\cdots\!73\)\( T^{20} - \)\(57\!\cdots\!03\)\( T^{21} + \)\(12\!\cdots\!55\)\( T^{22} - \)\(31\!\cdots\!44\)\( T^{23} + \)\(73\!\cdots\!28\)\( T^{24} - \)\(12\!\cdots\!14\)\( T^{25} + \)\(36\!\cdots\!31\)\( T^{26} - \)\(31\!\cdots\!94\)\( T^{27} + \)\(14\!\cdots\!24\)\( T^{28} - \)\(14\!\cdots\!01\)\( T^{29} + \)\(40\!\cdots\!49\)\( T^{30} + \)\(12\!\cdots\!86\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \)
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