Properties

Label 315.2.i.f
Level 315
Weight 2
Character orbit 315.i
Analytic conductor 2.515
Analytic rank 0
Dimension 16
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 315.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( \beta_{2} + \beta_{13} ) q^{2} + \beta_{5} q^{3} + ( -2 - 2 \beta_{3} + \beta_{7} - \beta_{10} + \beta_{12} ) q^{4} + ( -1 - \beta_{3} ) q^{5} + ( -\beta_{3} + \beta_{4} - \beta_{7} ) q^{6} -\beta_{3} q^{7} + ( \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{8} + ( 1 + \beta_{3} + \beta_{6} - \beta_{12} - \beta_{13} ) q^{9} +O(q^{10})\) \( q + ( \beta_{2} + \beta_{13} ) q^{2} + \beta_{5} q^{3} + ( -2 - 2 \beta_{3} + \beta_{7} - \beta_{10} + \beta_{12} ) q^{4} + ( -1 - \beta_{3} ) q^{5} + ( -\beta_{3} + \beta_{4} - \beta_{7} ) q^{6} -\beta_{3} q^{7} + ( \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{8} + ( 1 + \beta_{3} + \beta_{6} - \beta_{12} - \beta_{13} ) q^{9} -\beta_{2} q^{10} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{11} + ( -1 + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{8} - \beta_{10} + \beta_{12} + 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{12} + ( -1 + \beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{8} - \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{13} + \beta_{13} q^{14} -\beta_{1} q^{15} + ( 4 + 5 \beta_{3} - 3 \beta_{4} - \beta_{5} + 4 \beta_{6} - \beta_{8} + 2 \beta_{10} - \beta_{11} - 3 \beta_{12} - \beta_{13} + 2 \beta_{15} ) q^{16} + ( \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{17} + ( 2 - 2 \beta_{1} + \beta_{3} + \beta_{5} - \beta_{9} - \beta_{11} - \beta_{13} - \beta_{14} ) q^{18} + ( 1 - 2 \beta_{1} + \beta_{3} + 2 \beta_{5} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{19} + ( 1 + 2 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{10} - \beta_{12} ) q^{20} + ( -\beta_{1} + \beta_{5} ) q^{21} + ( -4 - \beta_{2} - 3 \beta_{3} + \beta_{5} + 2 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{12} - \beta_{13} + \beta_{15} ) q^{22} + ( 2 + 2 \beta_{3} - \beta_{7} + \beta_{10} - \beta_{11} - \beta_{12} ) q^{23} + ( -6 - \beta_{1} - 2 \beta_{2} - 5 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + 4 \beta_{7} + \beta_{8} - 2 \beta_{9} - 3 \beta_{10} - \beta_{11} + 4 \beta_{12} - \beta_{13} + 2 \beta_{14} ) q^{24} + \beta_{3} q^{25} + ( -2 \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} - \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{26} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{7} - \beta_{9} + \beta_{10} + \beta_{14} ) q^{27} + ( -1 - \beta_{4} + \beta_{6} + \beta_{7} ) q^{28} + ( 2 + \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{8} + 2 \beta_{10} - \beta_{12} + 2 \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{29} + ( -2 - \beta_{3} - \beta_{6} + \beta_{7} - \beta_{10} + \beta_{12} - \beta_{15} ) q^{30} + ( \beta_{2} - \beta_{3} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{11} - \beta_{12} - \beta_{15} ) q^{31} + ( 4 - 3 \beta_{1} + 4 \beta_{3} + \beta_{5} + 3 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + 3 \beta_{10} - 4 \beta_{12} - 4 \beta_{13} - 4 \beta_{14} + 3 \beta_{15} ) q^{32} + ( -3 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{9} + \beta_{10} + 2 \beta_{11} - \beta_{13} ) q^{33} + ( -4 + \beta_{1} + \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + \beta_{5} - 4 \beta_{6} - 2 \beta_{7} + 4 \beta_{8} - 2 \beta_{10} + \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + \beta_{14} - 4 \beta_{15} ) q^{34} - q^{35} + ( 2 + 2 \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} - 3 \beta_{6} - \beta_{7} + \beta_{9} + \beta_{11} + 2 \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{36} + ( 3 - \beta_{2} - \beta_{4} + \beta_{7} + \beta_{9} ) q^{37} + ( -2 - 2 \beta_{1} - 4 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{10} + 3 \beta_{12} + \beta_{13} - \beta_{14} ) q^{38} + ( 3 - \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{14} + 2 \beta_{15} ) q^{39} + ( 1 - \beta_{1} + \beta_{3} + 2 \beta_{5} - \beta_{7} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{40} + ( -3 - \beta_{1} + \beta_{2} - 4 \beta_{3} + \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{41} + ( -2 - 2 \beta_{3} + \beta_{4} - \beta_{6} - \beta_{10} + \beta_{12} - \beta_{15} ) q^{42} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{8} + 2 \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{43} + ( 1 + 4 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{4} - 4 \beta_{5} - 2 \beta_{8} + \beta_{10} + \beta_{11} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{44} + ( \beta_{2} - \beta_{3} - \beta_{6} - \beta_{7} + \beta_{13} - \beta_{15} ) q^{45} + ( 1 - \beta_{1} + 3 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{46} + ( 1 + \beta_{1} - \beta_{3} + \beta_{6} - \beta_{7} - \beta_{10} - \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{47} + ( 3 + \beta_{1} - 4 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} - 3 \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} + \beta_{15} ) q^{48} + ( -1 - \beta_{3} ) q^{49} -\beta_{13} q^{50} + ( -1 - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{51} + ( 3 + \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - 2 \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{11} - 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{52} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{7} + \beta_{8} - \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{53} + ( 4 + 2 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + \beta_{5} + 3 \beta_{6} - \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} - 2 \beta_{12} - \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{54} + ( 1 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{9} + \beta_{12} + \beta_{14} ) q^{55} + ( 1 - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{56} + ( 4 + 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} + 3 \beta_{10} + \beta_{11} - \beta_{12} + 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{57} + ( -3 + \beta_{2} - 4 \beta_{3} + 2 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} - 3 \beta_{12} - 3 \beta_{13} - 3 \beta_{14} + 2 \beta_{15} ) q^{58} + ( -1 + 2 \beta_{1} - \beta_{3} - 3 \beta_{5} + \beta_{6} + 2 \beta_{10} + 2 \beta_{11} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{59} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} - \beta_{13} + \beta_{14} ) q^{60} + ( -3 - 2 \beta_{2} + \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} + \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{11} + \beta_{12} - 2 \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{61} + ( -1 - \beta_{1} + \beta_{3} + 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{10} - \beta_{11} - 2 \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{62} + ( 1 + \beta_{2} - \beta_{7} - \beta_{12} - \beta_{15} ) q^{63} + ( 5 - 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 7 \beta_{7} + \beta_{8} + 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{64} + ( -\beta_{2} + \beta_{3} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} - \beta_{13} + \beta_{15} ) q^{65} + ( 1 + 2 \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} + 3 \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} + 2 \beta_{12} + \beta_{13} + 3 \beta_{14} ) q^{66} + ( -3 + \beta_{1} - 3 \beta_{3} - 2 \beta_{5} + 2 \beta_{7} - \beta_{10} + 2 \beta_{12} + \beta_{14} ) q^{67} + ( -5 - \beta_{1} - \beta_{2} - 4 \beta_{3} - \beta_{5} - 4 \beta_{6} + 6 \beta_{7} + \beta_{8} - \beta_{9} - 6 \beta_{10} - \beta_{11} + 6 \beta_{12} + 3 \beta_{14} - 3 \beta_{15} ) q^{68} + ( 4 - \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + \beta_{7} - 4 \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} - 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} + 3 \beta_{15} ) q^{69} + ( -\beta_{2} - \beta_{13} ) q^{70} + ( -2 \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{6} - \beta_{7} + \beta_{9} ) q^{71} + ( -1 - 2 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{5} + 3 \beta_{7} + 2 \beta_{8} - 2 \beta_{10} + \beta_{11} + 4 \beta_{12} + 4 \beta_{13} + 3 \beta_{14} ) q^{72} + ( 1 + 3 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{7} - \beta_{8} + \beta_{10} + \beta_{11} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{73} + ( 1 + 2 \beta_{1} + \beta_{2} + 4 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{8} + \beta_{10} - 2 \beta_{12} + \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{74} + ( \beta_{1} - \beta_{5} ) q^{75} + ( -4 + 5 \beta_{1} + \beta_{2} - 5 \beta_{3} - 4 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + 3 \beta_{11} + \beta_{12} + 5 \beta_{13} + 4 \beta_{14} - 2 \beta_{15} ) q^{76} + ( \beta_{1} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{11} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{77} + ( -5 + \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - \beta_{4} - 5 \beta_{5} + 2 \beta_{6} + \beta_{7} - 3 \beta_{8} + \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + 3 \beta_{13} - 3 \beta_{14} + \beta_{15} ) q^{78} + ( -2 + \beta_{1} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{10} - 2 \beta_{15} ) q^{79} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} - 2 \beta_{6} - 3 \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{80} + ( -1 - \beta_{1} - 2 \beta_{2} - 5 \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} + \beta_{11} + \beta_{12} - 4 \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{81} + ( 4 + \beta_{1} - 4 \beta_{2} + 4 \beta_{4} - 2 \beta_{5} - 3 \beta_{7} - \beta_{9} - 2 \beta_{12} - 2 \beta_{14} ) q^{82} + ( -2 \beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{5} + \beta_{8} - 2 \beta_{10} - \beta_{11} - 2 \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{83} + ( \beta_{1} - \beta_{2} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{10} + \beta_{12} + \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{84} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{12} + 2 \beta_{14} - \beta_{15} ) q^{85} + ( 1 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} - 3 \beta_{10} + 2 \beta_{12} - \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{86} + ( -1 + 2 \beta_{1} - 3 \beta_{2} + \beta_{3} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{12} + \beta_{15} ) q^{87} + ( 5 + 3 \beta_{1} + 3 \beta_{2} + 11 \beta_{3} - 5 \beta_{4} - 4 \beta_{5} + 5 \beta_{6} - 2 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} + 7 \beta_{10} + 2 \beta_{11} - 4 \beta_{12} + 3 \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{88} + ( 3 - \beta_{1} + 4 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} + 2 \beta_{9} + \beta_{10} + \beta_{11} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{89} + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{13} + \beta_{14} ) q^{90} + ( -1 + \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} + \beta_{12} + \beta_{14} ) q^{91} + ( -6 + 3 \beta_{2} - 15 \beta_{3} + 5 \beta_{4} - 6 \beta_{6} + \beta_{8} - 4 \beta_{10} + \beta_{11} + 6 \beta_{12} + 4 \beta_{13} - \beta_{15} ) q^{92} + ( -3 - \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} + \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{93} + ( 5 - 3 \beta_{1} - \beta_{2} + 6 \beta_{3} + \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + 3 \beta_{10} - \beta_{11} - 3 \beta_{12} + \beta_{13} - 3 \beta_{14} + 3 \beta_{15} ) q^{94} + ( \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} + \beta_{13} ) q^{95} + ( 5 + \beta_{1} + 11 \beta_{3} - 7 \beta_{4} + 7 \beta_{6} - 3 \beta_{7} + 7 \beta_{10} - 7 \beta_{12} - 3 \beta_{13} + \beta_{15} ) q^{96} + ( -4 \beta_{1} + 4 \beta_{3} + \beta_{5} + 3 \beta_{7} - 3 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} + 3 \beta_{12} + 2 \beta_{15} ) q^{97} -\beta_{2} q^{98} + ( -5 + \beta_{1} - \beta_{2} - 6 \beta_{3} + \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{10} + 3 \beta_{11} - 4 \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + q^{2} + q^{3} - 11q^{4} - 8q^{5} + 8q^{6} + 8q^{7} - 6q^{8} + 3q^{9} + O(q^{10}) \) \( 16q + q^{2} + q^{3} - 11q^{4} - 8q^{5} + 8q^{6} + 8q^{7} - 6q^{8} + 3q^{9} - 2q^{10} - 4q^{11} - 3q^{12} - 5q^{13} - q^{14} - 2q^{15} - 21q^{16} + 8q^{17} + 26q^{18} + 6q^{19} - 11q^{20} - q^{21} - 23q^{22} + 8q^{23} - 38q^{24} - 8q^{25} - 6q^{26} + 10q^{27} - 22q^{28} - 19q^{29} - 13q^{30} + 12q^{32} - 21q^{33} - 9q^{34} - 16q^{35} + 70q^{36} + 42q^{37} + 28q^{38} + 32q^{39} + 3q^{40} - 20q^{41} - 5q^{42} - 13q^{43} + 30q^{44} + 9q^{45} + 34q^{46} + 11q^{47} - 18q^{48} - 8q^{49} + q^{50} - 14q^{51} - 13q^{52} - 16q^{53} + 8q^{55} - 3q^{56} + 8q^{57} - 37q^{58} - 7q^{59} + 9q^{60} - 24q^{61} - 30q^{62} + 12q^{63} + 110q^{64} - 5q^{65} - 11q^{66} - 16q^{67} - 5q^{68} + 21q^{69} - q^{70} - 10q^{71} + 17q^{72} + 20q^{73} - 21q^{74} + q^{75} - 25q^{76} + 4q^{77} - 61q^{78} - 27q^{79} + 42q^{80} + 11q^{81} + 72q^{82} - 5q^{83} + 6q^{84} - 4q^{85} + 27q^{86} - 46q^{87} - 67q^{88} + 54q^{89} - 7q^{90} - 10q^{91} + 93q^{92} - 9q^{93} + 17q^{94} - 3q^{95} - 98q^{96} - 27q^{97} - 2q^{98} - 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 2 x^{15} + 8 x^{14} - 10 x^{13} + 40 x^{12} - 45 x^{11} + 159 x^{10} - 180 x^{9} + 576 x^{8} - 540 x^{7} + 1431 x^{6} - 1215 x^{5} + 3240 x^{4} - 2430 x^{3} + 5832 x^{2} - 4374 x + 6561\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{15} - 54 \nu^{14} - 13 \nu^{13} - 285 \nu^{12} - 92 \nu^{11} - 1376 \nu^{10} - 564 \nu^{9} - 5133 \nu^{8} - 729 \nu^{7} - 15138 \nu^{6} - 5130 \nu^{5} - 35586 \nu^{4} - 6399 \nu^{3} - 66015 \nu^{2} - 6804 \nu - 105705 \)\()/5832\)
\(\beta_{3}\)\(=\)\((\)\(-331 \nu^{15} - 384 \nu^{14} - 1549 \nu^{13} - 3459 \nu^{12} - 7034 \nu^{11} - 20030 \nu^{10} - 31770 \nu^{9} - 69753 \nu^{8} - 77643 \nu^{7} - 230976 \nu^{6} - 219564 \nu^{5} - 553176 \nu^{4} - 364743 \nu^{3} - 944217 \nu^{2} - 642006 \nu - 2009853\)\()/763992\)
\(\beta_{4}\)\(=\)\((\)\( 23 \nu^{15} + 98 \nu^{14} + 139 \nu^{13} + 571 \nu^{12} + 668 \nu^{11} + 2808 \nu^{10} + 3036 \nu^{9} + 9171 \nu^{8} + 7335 \nu^{7} + 29214 \nu^{6} + 21006 \nu^{5} + 66582 \nu^{4} + 34425 \nu^{3} + 109593 \nu^{2} + 55404 \nu + 185895 \)\()/17496\)
\(\beta_{5}\)\(=\)\((\)\(1046 \nu^{15} - 1099 \nu^{14} + 6769 \nu^{13} - 6206 \nu^{12} + 34925 \nu^{11} - 20859 \nu^{10} + 129333 \nu^{9} - 113013 \nu^{8} + 409716 \nu^{7} - 254097 \nu^{6} + 955341 \nu^{5} - 707697 \nu^{4} + 1748547 \nu^{3} - 1288386 \nu^{2} + 3457647 \nu - 2171691\)\()/763992\)
\(\beta_{6}\)\(=\)\((\)\(3181 \nu^{15} + 19087 \nu^{14} + 19856 \nu^{13} + 110357 \nu^{12} + 111133 \nu^{11} + 545253 \nu^{10} + 450105 \nu^{9} + 1921554 \nu^{8} + 1030491 \nu^{7} + 5968809 \nu^{6} + 3475683 \nu^{5} + 13723749 \nu^{4} + 5623344 \nu^{3} + 24917463 \nu^{2} + 8374023 \nu + 38027556\)\()/2291976\)
\(\beta_{7}\)\(=\)\((\)\(4417 \nu^{15} - 21707 \nu^{14} + 18134 \nu^{13} - 111667 \nu^{12} + 85891 \nu^{11} - 556257 \nu^{10} + 222711 \nu^{9} - 2202156 \nu^{8} + 1032759 \nu^{7} - 6435693 \nu^{6} + 881901 \nu^{5} - 15973281 \nu^{4} + 2504682 \nu^{3} - 29565081 \nu^{2} + 3849849 \nu - 48914442\)\()/2291976\)
\(\beta_{8}\)\(=\)\((\)\(5084 \nu^{15} - 28999 \nu^{14} + 24811 \nu^{13} - 145040 \nu^{12} + 102185 \nu^{11} - 728211 \nu^{10} + 259437 \nu^{9} - 2833947 \nu^{8} + 1367514 \nu^{7} - 8426889 \nu^{6} + 1382049 \nu^{5} - 20449341 \nu^{4} + 4608657 \nu^{3} - 38228760 \nu^{2} + 7169715 \nu - 64820493\)\()/2291976\)
\(\beta_{9}\)\(=\)\((\)\(-7037 \nu^{15} - 4493 \nu^{14} - 37522 \nu^{13} - 33481 \nu^{12} - 196979 \nu^{11} - 182583 \nu^{10} - 785487 \nu^{9} - 500112 \nu^{8} - 2386251 \nu^{7} - 2109699 \nu^{6} - 6031773 \nu^{5} - 4314951 \nu^{4} - 10866150 \nu^{3} - 7488531 \nu^{2} - 19893681 \nu - 11822922\)\()/2291976\)
\(\beta_{10}\)\(=\)\((\)\(7210 \nu^{15} - 8453 \nu^{14} + 41831 \nu^{13} - 44614 \nu^{12} + 178159 \nu^{11} - 172845 \nu^{10} + 627531 \nu^{9} - 856971 \nu^{8} + 1982160 \nu^{7} - 2270403 \nu^{6} + 3962331 \nu^{5} - 5366655 \nu^{4} + 6340437 \nu^{3} - 11517714 \nu^{2} + 11935917 \nu - 19925757\)\()/2291976\)
\(\beta_{11}\)\(=\)\((\)\(-9662 \nu^{15} + 17977 \nu^{14} - 46603 \nu^{13} + 104798 \nu^{12} - 193703 \nu^{11} + 470901 \nu^{10} - 641559 \nu^{9} + 2068767 \nu^{8} - 2112984 \nu^{7} + 5830515 \nu^{6} - 3921075 \nu^{5} + 14304843 \nu^{4} - 7126137 \nu^{3} + 28572426 \nu^{2} - 14735277 \nu + 46493433\)\()/2291976\)
\(\beta_{12}\)\(=\)\((\)\(3765 \nu^{15} - 343 \nu^{14} + 21146 \nu^{13} - 1619 \nu^{12} + 99331 \nu^{11} + 9839 \nu^{10} + 355491 \nu^{9} - 51720 \nu^{8} + 1071351 \nu^{7} + 70947 \nu^{6} + 2488941 \nu^{5} - 75789 \nu^{4} + 4524174 \nu^{3} - 54513 \nu^{2} + 7656201 \nu - 1106622\)\()/763992\)
\(\beta_{13}\)\(=\)\((\)\(-12086 \nu^{15} + 23761 \nu^{14} - 60991 \nu^{13} + 125582 \nu^{12} - 286595 \nu^{11} + 566589 \nu^{10} - 937479 \nu^{9} + 2445795 \nu^{8} - 3348720 \nu^{7} + 6687495 \nu^{6} - 6339411 \nu^{5} + 17213067 \nu^{4} - 12803589 \nu^{3} + 32468202 \nu^{2} - 23731137 \nu + 58198257\)\()/2291976\)
\(\beta_{14}\)\(=\)\((\)\(-5017 \nu^{15} + 6145 \nu^{14} - 26580 \nu^{13} + 31991 \nu^{12} - 129525 \nu^{11} + 144563 \nu^{10} - 454785 \nu^{9} + 650802 \nu^{8} - 1497951 \nu^{7} + 1722375 \nu^{6} - 3160539 \nu^{5} + 4473819 \nu^{4} - 6304716 \nu^{3} + 8279901 \nu^{2} - 10487151 \nu + 14521680\)\()/763992\)
\(\beta_{15}\)\(=\)\((\)\(-16105 \nu^{15} + 1514 \nu^{14} - 86681 \nu^{13} + 4519 \nu^{12} - 420040 \nu^{11} - 14028 \nu^{10} - 1510836 \nu^{9} + 340047 \nu^{8} - 4533309 \nu^{7} + 273618 \nu^{6} - 10364814 \nu^{5} + 2215350 \nu^{4} - 18592011 \nu^{3} + 6076701 \nu^{2} - 29492424 \nu + 12984219\)\()/2291976\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{15} + \beta_{12} + \beta_{7} - \beta_{2} - 1\)
\(\nu^{3}\)\(=\)\(-\beta_{14} - \beta_{10} + \beta_{9} - \beta_{7} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(-3 \beta_{14} + 2 \beta_{13} - 3 \beta_{12} - \beta_{11} + 2 \beta_{10} - \beta_{9} - \beta_{8} - 2 \beta_{7} + 3 \beta_{6} - 2 \beta_{5} - \beta_{4} + 3 \beta_{3} + 4 \beta_{2} - \beta_{1} - 1\)
\(\nu^{5}\)\(=\)\(4 \beta_{14} + \beta_{13} - 3 \beta_{11} - 4 \beta_{9} + 3 \beta_{8} + 2 \beta_{6} + 3 \beta_{5} - 5 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} - 5 \beta_{1} + 3\)
\(\nu^{6}\)\(=\)\(-7 \beta_{15} + 10 \beta_{14} - \beta_{13} + 7 \beta_{12} + 2 \beta_{11} - 7 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} - 9 \beta_{6} + 4 \beta_{5} + 3 \beta_{4} + 2 \beta_{2} - 5\)
\(\nu^{7}\)\(=\)\(4 \beta_{15} - 5 \beta_{14} - 11 \beta_{13} - 4 \beta_{12} + 9 \beta_{11} + 10 \beta_{10} - \beta_{9} - 6 \beta_{8} - \beta_{7} - 4 \beta_{6} - 15 \beta_{5} + 5 \beta_{4} + 12 \beta_{3} - 2 \beta_{2} + \beta_{1} + 20\)
\(\nu^{8}\)\(=\)\(-11 \beta_{15} - 5 \beta_{14} - 12 \beta_{13} - 13 \beta_{12} + 12 \beta_{11} + 5 \beta_{10} + 14 \beta_{9} - 9 \beta_{7} - 12 \beta_{6} - 6 \beta_{5} - 6 \beta_{4} - 37 \beta_{3} - 15 \beta_{2} + 25 \beta_{1}\)
\(\nu^{9}\)\(=\)\(-7 \beta_{15} - 6 \beta_{14} + 23 \beta_{13} + 13 \beta_{12} + 2 \beta_{11} - 17 \beta_{10} + 17 \beta_{9} - 7 \beta_{8} + 23 \beta_{7} - 36 \beta_{6} + 28 \beta_{5} + 28 \beta_{4} - 64 \beta_{3} - 6 \beta_{2} + 23 \beta_{1} - 117\)
\(\nu^{10}\)\(=\)\(79 \beta_{15} - 26 \beta_{14} - 48 \beta_{13} - 19 \beta_{12} - 22 \beta_{11} + 15 \beta_{10} + 34 \beta_{9} - 22 \beta_{8} + 21 \beta_{7} + 65 \beta_{6} + 40 \beta_{5} + 22 \beta_{4} - 45 \beta_{3} - 25 \beta_{2} - 93 \beta_{1} + 46\)
\(\nu^{11}\)\(=\)\(-20 \beta_{15} - 58 \beta_{14} + 34 \beta_{13} - 94 \beta_{12} - 21 \beta_{11} + 32 \beta_{10} + 10 \beta_{9} - 45 \beta_{8} - 3 \beta_{7} + 194 \beta_{6} + 9 \beta_{5} - 104 \beta_{4} + 127 \beta_{3} + 126 \beta_{2} + 5 \beta_{1} + 165\)
\(\nu^{12}\)\(=\)\(-8 \beta_{15} + 109 \beta_{14} + 232 \beta_{13} + 104 \beta_{12} - 68 \beta_{11} - 15 \beta_{10} - 189 \beta_{9} + 193 \beta_{8} + 119 \beta_{7} + 87 \beta_{6} - 22 \beta_{5} - 62 \beta_{4} + 275 \beta_{3} - 19 \beta_{2} + 137 \beta_{1} + 37\)
\(\nu^{13}\)\(=\)\(-124 \beta_{15} + 458 \beta_{14} + \beta_{13} + 508 \beta_{12} + 65 \beta_{11} - 137 \beta_{10} + 45 \beta_{9} + 419 \beta_{8} - 181 \beta_{7} - 425 \beta_{6} + 163 \beta_{5} + 192 \beta_{4} + 51 \beta_{3} - 216 \beta_{2} + 124 \beta_{1} + 111\)
\(\nu^{14}\)\(=\)\(-112 \beta_{15} - 393 \beta_{14} - 134 \beta_{13} - 71 \beta_{12} + 48 \beta_{11} + 4 \beta_{10} + 96 \beta_{9} - 501 \beta_{8} - 276 \beta_{7} - 424 \beta_{6} - 531 \beta_{5} + 292 \beta_{4} + 1464 \beta_{3} + 180 \beta_{2} + 108 \beta_{1} + 438\)
\(\nu^{15}\)\(=\)\(522 \beta_{15} - 734 \beta_{14} - 498 \beta_{13} - 528 \beta_{12} - 438 \beta_{11} + 82 \beta_{10} - 346 \beta_{9} - 744 \beta_{8} + 232 \beta_{7} + 378 \beta_{6} - 1872 \beta_{5} - 514 \beta_{4} - 1208 \beta_{3} - 298 \beta_{2} - 176 \beta_{1} - 113\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
106.1
1.06614 1.36504i
−1.41596 0.997525i
−1.26474 + 1.18340i
1.40613 + 1.01134i
−0.216795 1.71843i
0.803782 + 1.53425i
1.14603 1.29869i
−0.524589 + 1.65070i
1.06614 + 1.36504i
−1.41596 + 0.997525i
−1.26474 1.18340i
1.40613 1.01134i
−0.216795 + 1.71843i
0.803782 1.53425i
1.14603 + 1.29869i
−0.524589 1.65070i
−1.32864 + 2.30128i −0.649093 1.60583i −2.53058 4.38310i −0.500000 0.866025i 4.55786 + 0.639826i 0.500000 0.866025i 8.13440 −2.15736 + 2.08466i 2.65729
106.2 −0.978244 + 1.69437i −1.57186 + 0.727495i −0.913922 1.58296i −0.500000 0.866025i 0.305020 3.37498i 0.500000 0.866025i −0.336819 1.94150 2.28704i 1.95649
106.3 −0.522039 + 0.904198i 0.392487 + 1.68700i 0.454951 + 0.787999i −0.500000 0.866025i −1.73027 0.525791i 0.500000 0.866025i −3.03816 −2.69191 + 1.32425i 1.04408
106.4 −0.219523 + 0.380224i 1.57891 0.712071i 0.903620 + 1.56512i −0.500000 0.866025i −0.0758596 + 0.756655i 0.500000 0.866025i −1.67155 1.98591 2.24859i 0.439045
106.5 0.441371 0.764477i −1.59660 0.671465i 0.610383 + 1.05721i −0.500000 0.866025i −1.21801 + 0.924200i 0.500000 0.866025i 2.84311 2.09827 + 2.14412i −0.882742
106.6 0.627726 1.08725i 1.73059 + 0.0710311i 0.211920 + 0.367057i −0.500000 0.866025i 1.16357 1.83701i 0.500000 0.866025i 3.04302 2.98991 + 0.245852i −1.25545
106.7 1.09035 1.88855i −0.551686 1.64184i −1.37775 2.38633i −0.500000 0.866025i −3.70223 0.748302i 0.500000 0.866025i −1.64751 −2.39128 + 1.81156i −2.18071
106.8 1.38900 2.40581i 1.16725 + 1.27966i −2.85862 4.95128i −0.500000 0.866025i 4.69992 1.03075i 0.500000 0.866025i −10.3265 −0.275041 + 2.98737i −2.77799
211.1 −1.32864 2.30128i −0.649093 + 1.60583i −2.53058 + 4.38310i −0.500000 + 0.866025i 4.55786 0.639826i 0.500000 + 0.866025i 8.13440 −2.15736 2.08466i 2.65729
211.2 −0.978244 1.69437i −1.57186 0.727495i −0.913922 + 1.58296i −0.500000 + 0.866025i 0.305020 + 3.37498i 0.500000 + 0.866025i −0.336819 1.94150 + 2.28704i 1.95649
211.3 −0.522039 0.904198i 0.392487 1.68700i 0.454951 0.787999i −0.500000 + 0.866025i −1.73027 + 0.525791i 0.500000 + 0.866025i −3.03816 −2.69191 1.32425i 1.04408
211.4 −0.219523 0.380224i 1.57891 + 0.712071i 0.903620 1.56512i −0.500000 + 0.866025i −0.0758596 0.756655i 0.500000 + 0.866025i −1.67155 1.98591 + 2.24859i 0.439045
211.5 0.441371 + 0.764477i −1.59660 + 0.671465i 0.610383 1.05721i −0.500000 + 0.866025i −1.21801 0.924200i 0.500000 + 0.866025i 2.84311 2.09827 2.14412i −0.882742
211.6 0.627726 + 1.08725i 1.73059 0.0710311i 0.211920 0.367057i −0.500000 + 0.866025i 1.16357 + 1.83701i 0.500000 + 0.866025i 3.04302 2.98991 0.245852i −1.25545
211.7 1.09035 + 1.88855i −0.551686 + 1.64184i −1.37775 + 2.38633i −0.500000 + 0.866025i −3.70223 + 0.748302i 0.500000 + 0.866025i −1.64751 −2.39128 1.81156i −2.18071
211.8 1.38900 + 2.40581i 1.16725 1.27966i −2.85862 + 4.95128i −0.500000 + 0.866025i 4.69992 + 1.03075i 0.500000 + 0.866025i −10.3265 −0.275041 2.98737i −2.77799
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 211.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.i.f 16
3.b odd 2 1 945.2.i.f 16
9.c even 3 1 inner 315.2.i.f 16
9.c even 3 1 2835.2.a.x 8
9.d odd 6 1 945.2.i.f 16
9.d odd 6 1 2835.2.a.y 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.i.f 16 1.a even 1 1 trivial
315.2.i.f 16 9.c even 3 1 inner
945.2.i.f 16 3.b odd 2 1
945.2.i.f 16 9.d odd 6 1
2835.2.a.x 8 9.c even 3 1
2835.2.a.y 8 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{16} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T - 2 T^{2} + 3 T^{3} + T^{4} - 5 T^{5} - 2 T^{6} + 6 T^{7} - 2 T^{9} + 13 T^{10} - 34 T^{11} + 77 T^{12} + 46 T^{13} - 264 T^{14} - 24 T^{15} + 500 T^{16} - 48 T^{17} - 1056 T^{18} + 368 T^{19} + 1232 T^{20} - 1088 T^{21} + 832 T^{22} - 256 T^{23} + 3072 T^{25} - 2048 T^{26} - 10240 T^{27} + 4096 T^{28} + 24576 T^{29} - 32768 T^{30} - 32768 T^{31} + 65536 T^{32} \)
$3$ \( 1 - T - T^{2} - 2 T^{3} + T^{4} + 6 T^{5} + 6 T^{6} - 45 T^{7} + 90 T^{8} - 135 T^{9} + 54 T^{10} + 162 T^{11} + 81 T^{12} - 486 T^{13} - 729 T^{14} - 2187 T^{15} + 6561 T^{16} \)
$5$ \( ( 1 + T + T^{2} )^{8} \)
$7$ \( ( 1 - T + T^{2} )^{8} \)
$11$ \( 1 + 4 T - 8 T^{2} - 252 T^{3} - 884 T^{4} + 761 T^{5} + 29542 T^{6} + 109554 T^{7} + 21618 T^{8} - 2194876 T^{9} - 9004202 T^{10} - 8688626 T^{11} + 112135271 T^{12} + 523246502 T^{13} + 801407592 T^{14} - 4004853117 T^{15} - 22200406036 T^{16} - 44053384287 T^{17} + 96970318632 T^{18} + 696441094162 T^{19} + 1641772502711 T^{20} - 1399311905926 T^{21} - 15951493099322 T^{22} - 42771923935796 T^{23} + 4634010289458 T^{24} + 258322601339814 T^{25} + 766243397562742 T^{26} + 217122181334971 T^{27} - 2774370685021364 T^{28} - 8699723460270612 T^{29} - 3037998668665928 T^{30} + 16708992677662604 T^{31} + 45949729863572161 T^{32} \)
$13$ \( 1 + 5 T - 24 T^{2} - 129 T^{3} + 240 T^{4} + 1174 T^{5} - 1462 T^{6} - 4299 T^{7} - 36 T^{8} + 124186 T^{9} + 631224 T^{10} - 2988076 T^{11} - 15052803 T^{12} + 29465898 T^{13} + 184401460 T^{14} - 98215947 T^{15} - 1918362832 T^{16} - 1276807311 T^{17} + 31163846740 T^{18} + 64736577906 T^{19} - 429923106483 T^{20} - 1109451702268 T^{21} + 3046797684216 T^{22} + 7792487332162 T^{23} - 29366305956 T^{24} - 45588742804527 T^{25} - 201549115083238 T^{26} + 2103996302599438 T^{27} + 5591540429395440 T^{28} - 39070888750400637 T^{29} - 94497033256782936 T^{30} + 255929465070453785 T^{31} + 665416609183179841 T^{32} \)
$17$ \( ( 1 - 4 T + 48 T^{2} - 60 T^{3} + 722 T^{4} + 693 T^{5} + 14099 T^{6} - 6718 T^{7} + 352974 T^{8} - 114206 T^{9} + 4074611 T^{10} + 3404709 T^{11} + 60302162 T^{12} - 85191420 T^{13} + 1158603312 T^{14} - 1641354692 T^{15} + 6975757441 T^{16} )^{2} \)
$19$ \( ( 1 - 3 T + 51 T^{2} - 156 T^{3} + 1837 T^{4} - 5706 T^{5} + 46629 T^{6} - 140085 T^{7} + 980820 T^{8} - 2661615 T^{9} + 16833069 T^{10} - 39137454 T^{11} + 239399677 T^{12} - 386271444 T^{13} + 2399339931 T^{14} - 2681615217 T^{15} + 16983563041 T^{16} )^{2} \)
$23$ \( 1 - 8 T - 56 T^{2} + 636 T^{3} + 781 T^{4} - 20716 T^{5} + 6724 T^{6} + 401256 T^{7} - 165759 T^{8} - 5155168 T^{9} - 11298644 T^{10} + 13905856 T^{11} + 907093574 T^{12} + 431304176 T^{13} - 30428900436 T^{14} - 606821352 T^{15} + 718124779886 T^{16} - 13956891096 T^{17} - 16096888330644 T^{18} + 5247677909392 T^{19} + 253841972841734 T^{20} + 89502858924608 T^{21} - 1672604809034516 T^{22} - 17552447189960096 T^{23} - 12980750609193279 T^{24} + 722723312327997528 T^{25} + 278551861400575876 T^{26} - 19738406944944911732 T^{27} + 17115321681407870701 T^{28} + \)\(32\!\cdots\!88\)\( T^{29} - \)\(64\!\cdots\!04\)\( T^{30} - \)\(21\!\cdots\!56\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \)
$29$ \( 1 + 19 T + 128 T^{2} + 803 T^{3} + 9164 T^{4} + 53572 T^{5} + 135245 T^{6} + 1457003 T^{7} + 11306929 T^{8} + 8567345 T^{9} + 77536775 T^{10} + 2101449919 T^{11} + 3528685908 T^{12} - 3024827910 T^{13} + 382687393437 T^{14} + 1623090782861 T^{15} - 344149034518 T^{16} + 47069632702969 T^{17} + 321840097880517 T^{18} - 73772527896990 T^{19} + 2495772497696148 T^{20} + 43103152404646931 T^{21} + 46120682005129775 T^{22} + 147785641546529605 T^{23} + 5656250673854706769 T^{24} + 21136955208279060607 T^{25} + 56898549767685684245 T^{26} + \)\(65\!\cdots\!88\)\( T^{27} + \)\(32\!\cdots\!24\)\( T^{28} + \)\(82\!\cdots\!67\)\( T^{29} + \)\(38\!\cdots\!68\)\( T^{30} + \)\(16\!\cdots\!31\)\( T^{31} + \)\(25\!\cdots\!21\)\( T^{32} \)
$31$ \( 1 - 140 T^{2} + 126 T^{3} + 9063 T^{4} - 13239 T^{5} - 402475 T^{6} + 353781 T^{7} + 16648256 T^{8} + 3630141 T^{9} - 737222427 T^{10} - 31707297 T^{11} + 29744064802 T^{12} - 15980009589 T^{13} - 962711725271 T^{14} + 358072466157 T^{15} + 28652584678461 T^{16} + 11100246450867 T^{17} - 925165967985431 T^{18} - 476060465665899 T^{19} + 27469268470007842 T^{20} - 907752993614847 T^{21} - 654287617678253787 T^{22} + 99874668501519651 T^{23} + 14199148331423352896 T^{24} + 9353835967624347051 T^{25} - \)\(32\!\cdots\!75\)\( T^{26} - \)\(33\!\cdots\!09\)\( T^{27} + \)\(71\!\cdots\!43\)\( T^{28} + \)\(30\!\cdots\!66\)\( T^{29} - \)\(10\!\cdots\!40\)\( T^{30} + \)\(72\!\cdots\!81\)\( T^{32} \)
$37$ \( ( 1 - 21 T + 401 T^{2} - 5016 T^{3} + 57403 T^{4} - 523584 T^{5} + 4405763 T^{6} - 31274655 T^{7} + 205649488 T^{8} - 1157162235 T^{9} + 6031489547 T^{10} - 26521100352 T^{11} + 107582463883 T^{12} - 347829288312 T^{13} + 1028856290009 T^{14} - 1993569419793 T^{15} + 3512479453921 T^{16} )^{2} \)
$41$ \( 1 + 20 T + 118 T^{2} - 150 T^{3} - 4985 T^{4} - 42197 T^{5} - 220145 T^{6} + 1270731 T^{7} + 23848134 T^{8} + 110309977 T^{9} + 134825125 T^{10} - 2724509833 T^{11} - 47046106654 T^{12} - 288572614127 T^{13} - 46076715579 T^{14} + 8512166036907 T^{15} + 65404558839995 T^{16} + 348998807513187 T^{17} - 77454958888299 T^{18} - 19888713138246967 T^{19} - 132941053384713694 T^{20} - 315651358838524433 T^{21} + 640433398055855125 T^{22} + 21483339472464810737 T^{23} + \)\(19\!\cdots\!14\)\( T^{24} + \)\(41\!\cdots\!91\)\( T^{25} - \)\(29\!\cdots\!45\)\( T^{26} - \)\(23\!\cdots\!77\)\( T^{27} - \)\(11\!\cdots\!85\)\( T^{28} - \)\(13\!\cdots\!50\)\( T^{29} + \)\(44\!\cdots\!98\)\( T^{30} + \)\(31\!\cdots\!20\)\( T^{31} + \)\(63\!\cdots\!41\)\( T^{32} \)
$43$ \( 1 + 13 T - 93 T^{2} - 1248 T^{3} + 9810 T^{4} + 69431 T^{5} - 827740 T^{6} - 2844144 T^{7} + 45303354 T^{8} + 67606073 T^{9} - 1695183309 T^{10} + 999801391 T^{11} + 52470440688 T^{12} - 81486657567 T^{13} - 1021133493305 T^{14} + 1151633415525 T^{15} + 13207491736631 T^{16} + 49520236867575 T^{17} - 1888075829120945 T^{18} - 6478759683179469 T^{19} + 179385995094575088 T^{20} + 146979245800144213 T^{21} - 10715869130794149141 T^{22} + 18376588865258452811 T^{23} + \)\(52\!\cdots\!54\)\( T^{24} - \)\(14\!\cdots\!92\)\( T^{25} - \)\(17\!\cdots\!60\)\( T^{26} + \)\(64\!\cdots\!17\)\( T^{27} + \)\(39\!\cdots\!10\)\( T^{28} - \)\(21\!\cdots\!64\)\( T^{29} - \)\(68\!\cdots\!57\)\( T^{30} + \)\(41\!\cdots\!91\)\( T^{31} + \)\(13\!\cdots\!01\)\( T^{32} \)
$47$ \( 1 - 11 T - 184 T^{2} + 2519 T^{3} + 18578 T^{4} - 328940 T^{5} - 1113262 T^{6} + 30708371 T^{7} + 18431464 T^{8} - 2157799474 T^{9} + 3826585394 T^{10} + 115855042036 T^{11} - 510755934975 T^{12} - 4424444319954 T^{13} + 38393050326972 T^{14} + 80181278073827 T^{15} - 2076366795569416 T^{16} + 3768520069469869 T^{17} + 84810248172281148 T^{18} - 459359082630584142 T^{19} - 2492326031534742975 T^{20} + 26570775426731714252 T^{21} + 41247587936732304626 T^{22} - \)\(10\!\cdots\!62\)\( T^{23} + \)\(43\!\cdots\!04\)\( T^{24} + \)\(34\!\cdots\!57\)\( T^{25} - \)\(58\!\cdots\!38\)\( T^{26} - \)\(81\!\cdots\!20\)\( T^{27} + \)\(21\!\cdots\!98\)\( T^{28} + \)\(13\!\cdots\!13\)\( T^{29} - \)\(47\!\cdots\!96\)\( T^{30} - \)\(13\!\cdots\!73\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \)
$53$ \( ( 1 + 8 T + 306 T^{2} + 2229 T^{3} + 44063 T^{4} + 295215 T^{5} + 3976442 T^{6} + 23818424 T^{7} + 249123624 T^{8} + 1262376472 T^{9} + 11169825578 T^{10} + 43950723555 T^{11} + 347678264303 T^{12} + 932157753897 T^{13} + 6782294505474 T^{14} + 9397689118696 T^{15} + 62259690411361 T^{16} )^{2} \)
$59$ \( 1 + 7 T - 253 T^{2} - 1432 T^{3} + 34994 T^{4} + 154249 T^{5} - 3340312 T^{6} - 13153948 T^{7} + 235615798 T^{8} + 1028459735 T^{9} - 13624035805 T^{10} - 71911153079 T^{11} + 749305929396 T^{12} + 3788827420407 T^{13} - 43280647002525 T^{14} - 92100685540933 T^{15} + 2571044091919703 T^{16} - 5433940446915047 T^{17} - 150659932215789525 T^{18} + 778145586775769253 T^{19} + 9079610445931843956 T^{20} - 51411030705285766621 T^{21} - \)\(57\!\cdots\!05\)\( T^{22} + \)\(25\!\cdots\!65\)\( T^{23} + \)\(34\!\cdots\!58\)\( T^{24} - \)\(11\!\cdots\!72\)\( T^{25} - \)\(17\!\cdots\!12\)\( T^{26} + \)\(46\!\cdots\!91\)\( T^{27} + \)\(62\!\cdots\!14\)\( T^{28} - \)\(15\!\cdots\!28\)\( T^{29} - \)\(15\!\cdots\!33\)\( T^{30} + \)\(25\!\cdots\!93\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \)
$61$ \( 1 + 24 T + 19 T^{2} - 3276 T^{3} - 8256 T^{4} + 259110 T^{5} - 422515 T^{6} - 22871148 T^{7} + 156657818 T^{8} + 2168029494 T^{9} - 15387409002 T^{10} - 138383636634 T^{11} + 1199692712956 T^{12} + 4463711557584 T^{13} - 105028925164004 T^{14} - 58126091327262 T^{15} + 7596603299571471 T^{16} - 3545691570962982 T^{17} - 390812630535258884 T^{18} + 1013177713051973904 T^{19} + 16610754552447415996 T^{20} - \)\(11\!\cdots\!34\)\( T^{21} - \)\(79\!\cdots\!22\)\( T^{22} + \)\(68\!\cdots\!74\)\( T^{23} + \)\(30\!\cdots\!58\)\( T^{24} - \)\(26\!\cdots\!68\)\( T^{25} - \)\(30\!\cdots\!15\)\( T^{26} + \)\(11\!\cdots\!10\)\( T^{27} - \)\(21\!\cdots\!76\)\( T^{28} - \)\(53\!\cdots\!56\)\( T^{29} + \)\(18\!\cdots\!79\)\( T^{30} + \)\(14\!\cdots\!24\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \)
$67$ \( 1 + 16 T - 303 T^{2} - 5196 T^{3} + 65028 T^{4} + 1020572 T^{5} - 10522045 T^{6} - 137230818 T^{7} + 1450070274 T^{8} + 14061798986 T^{9} - 167697342162 T^{10} - 1081437670724 T^{11} + 16651518835248 T^{12} + 59637358262580 T^{13} - 1404138817994528 T^{14} - 1533447489642366 T^{15} + 101851891881704651 T^{16} - 102740981806038522 T^{17} - 6303179153977436192 T^{18} + 17936710783128348540 T^{19} + \)\(33\!\cdots\!08\)\( T^{20} - \)\(14\!\cdots\!68\)\( T^{21} - \)\(15\!\cdots\!78\)\( T^{22} + \)\(85\!\cdots\!78\)\( T^{23} + \)\(58\!\cdots\!34\)\( T^{24} - \)\(37\!\cdots\!46\)\( T^{25} - \)\(19\!\cdots\!05\)\( T^{26} + \)\(12\!\cdots\!76\)\( T^{27} + \)\(53\!\cdots\!08\)\( T^{28} - \)\(28\!\cdots\!52\)\( T^{29} - \)\(11\!\cdots\!87\)\( T^{30} + \)\(39\!\cdots\!88\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \)
$71$ \( ( 1 + 5 T + 278 T^{2} + 1065 T^{3} + 37904 T^{4} + 109648 T^{5} + 3610954 T^{6} + 7649988 T^{7} + 277532887 T^{8} + 543149148 T^{9} + 18202819114 T^{10} + 39244225328 T^{11} + 963204356624 T^{12} + 1921504258815 T^{13} + 35611878930038 T^{14} + 45475600791955 T^{15} + 645753531245761 T^{16} )^{2} \)
$73$ \( ( 1 - 10 T + 496 T^{2} - 3839 T^{3} + 108764 T^{4} - 681709 T^{5} + 14326022 T^{6} - 74456369 T^{7} + 1261625443 T^{8} - 5435314937 T^{9} + 76343371238 T^{10} - 265196390053 T^{11} + 3088706284124 T^{12} - 7958521845527 T^{13} + 75061776239344 T^{14} - 110473985190970 T^{15} + 806460091894081 T^{16} )^{2} \)
$79$ \( 1 + 27 T - 135 T^{2} - 7902 T^{3} + 39995 T^{4} + 1914831 T^{5} - 6362838 T^{6} - 289531215 T^{7} + 1468553911 T^{8} + 37012370742 T^{9} - 224794814526 T^{10} - 3282480268776 T^{11} + 32122316535770 T^{12} + 227052617209830 T^{13} - 3341971702380213 T^{14} - 6208633901935233 T^{15} + 304269296557139782 T^{16} - 490482078252883407 T^{17} - 20857245394554909333 T^{18} + \)\(11\!\cdots\!70\)\( T^{19} + \)\(12\!\cdots\!70\)\( T^{20} - \)\(10\!\cdots\!24\)\( T^{21} - \)\(54\!\cdots\!46\)\( T^{22} + \)\(71\!\cdots\!78\)\( T^{23} + \)\(22\!\cdots\!71\)\( T^{24} - \)\(34\!\cdots\!85\)\( T^{25} - \)\(60\!\cdots\!38\)\( T^{26} + \)\(14\!\cdots\!49\)\( T^{27} + \)\(23\!\cdots\!95\)\( T^{28} - \)\(36\!\cdots\!78\)\( T^{29} - \)\(49\!\cdots\!35\)\( T^{30} + \)\(78\!\cdots\!73\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \)
$83$ \( 1 + 5 T - 412 T^{2} - 1919 T^{3} + 85319 T^{4} + 351005 T^{5} - 12811975 T^{6} - 43639307 T^{7} + 1623479161 T^{8} + 4403947840 T^{9} - 179310486826 T^{10} - 369180098536 T^{11} + 17626003433352 T^{12} + 22874863773276 T^{13} - 1609250765486967 T^{14} - 692777265967880 T^{15} + 138259235603031206 T^{16} - 57500513075334040 T^{17} - 11086128523439715663 T^{18} + 13079549732330164212 T^{19} + \)\(83\!\cdots\!92\)\( T^{20} - \)\(14\!\cdots\!48\)\( T^{21} - \)\(58\!\cdots\!94\)\( T^{22} + \)\(11\!\cdots\!80\)\( T^{23} + \)\(36\!\cdots\!01\)\( T^{24} - \)\(81\!\cdots\!21\)\( T^{25} - \)\(19\!\cdots\!75\)\( T^{26} + \)\(45\!\cdots\!35\)\( T^{27} + \)\(91\!\cdots\!59\)\( T^{28} - \)\(17\!\cdots\!97\)\( T^{29} - \)\(30\!\cdots\!48\)\( T^{30} + \)\(30\!\cdots\!35\)\( T^{31} + \)\(50\!\cdots\!81\)\( T^{32} \)
$89$ \( ( 1 - 27 T + 659 T^{2} - 11088 T^{3} + 177613 T^{4} - 2313648 T^{5} + 28600757 T^{6} - 303824349 T^{7} + 3074549044 T^{8} - 27040367061 T^{9} + 226546596197 T^{10} - 1631050116912 T^{11} + 11143837650733 T^{12} - 61916051170512 T^{13} + 327510670743299 T^{14} - 1194246042179283 T^{15} + 3936588805702081 T^{16} )^{2} \)
$97$ \( 1 + 27 T + 15 T^{2} - 9096 T^{3} - 108991 T^{4} + 887943 T^{5} + 28936428 T^{6} + 99887319 T^{7} - 3490694579 T^{8} - 39893502474 T^{9} + 151428852180 T^{10} + 5743284567084 T^{11} + 21023505312788 T^{12} - 484348205982834 T^{13} - 5169233095099629 T^{14} + 18335501604402927 T^{15} + 619275518773189270 T^{16} + 1778543655627083919 T^{17} - 48637314191792409261 T^{18} - \)\(44\!\cdots\!82\)\( T^{19} + \)\(18\!\cdots\!28\)\( T^{20} + \)\(49\!\cdots\!88\)\( T^{21} + \)\(12\!\cdots\!20\)\( T^{22} - \)\(32\!\cdots\!62\)\( T^{23} - \)\(27\!\cdots\!19\)\( T^{24} + \)\(75\!\cdots\!23\)\( T^{25} + \)\(21\!\cdots\!72\)\( T^{26} + \)\(63\!\cdots\!79\)\( T^{27} - \)\(75\!\cdots\!31\)\( T^{28} - \)\(61\!\cdots\!92\)\( T^{29} + \)\(97\!\cdots\!35\)\( T^{30} + \)\(17\!\cdots\!11\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \)
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