Properties

Label 105.4.b.b
Level $105$
Weight $4$
Character orbit 105.b
Analytic conductor $6.195$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 105.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.19520055060\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 96 x^{14} + 3618 x^{12} + 68560 x^{10} + 697017 x^{8} + 3791184 x^{6} + 10461796 x^{4} + 13209792 x^{2} + 5760000\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} -\beta_{3} q^{3} + ( -4 + \beta_{2} ) q^{4} + 5 q^{5} + ( 2 - \beta_{5} ) q^{6} -\beta_{10} q^{7} + ( -4 \beta_{1} - \beta_{3} + \beta_{4} ) q^{8} + ( -1 + \beta_{1} + \beta_{2} - \beta_{12} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} -\beta_{3} q^{3} + ( -4 + \beta_{2} ) q^{4} + 5 q^{5} + ( 2 - \beta_{5} ) q^{6} -\beta_{10} q^{7} + ( -4 \beta_{1} - \beta_{3} + \beta_{4} ) q^{8} + ( -1 + \beta_{1} + \beta_{2} - \beta_{12} ) q^{9} + 5 \beta_{1} q^{10} + ( \beta_{1} + \beta_{3} - \beta_{13} ) q^{11} + ( 4 + 3 \beta_{1} - \beta_{2} + 4 \beta_{3} - \beta_{4} + \beta_{8} ) q^{12} + ( -1 + 4 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{9} + \beta_{12} ) q^{13} + ( 7 - \beta_{1} + 5 \beta_{3} - \beta_{4} - 2 \beta_{6} + \beta_{8} - \beta_{9} + \beta_{11} ) q^{14} -5 \beta_{3} q^{15} + ( 25 - 4 \beta_{2} + 3 \beta_{3} - 3 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{11} - \beta_{12} + \beta_{14} ) q^{16} + ( 4 - 2 \beta_{2} - 3 \beta_{3} - \beta_{5} + \beta_{6} + \beta_{14} - \beta_{15} ) q^{17} + ( -12 - 6 \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{6} - \beta_{7} - 2 \beta_{11} + \beta_{14} ) q^{18} + ( -1 - 5 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} ) q^{19} + ( -20 + 5 \beta_{2} ) q^{20} + ( -4 + 6 \beta_{1} - \beta_{2} + 2 \beta_{3} + 3 \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{9} - \beta_{10} - 2 \beta_{11} + \beta_{12} - \beta_{14} + \beta_{15} ) q^{21} + ( -18 + 3 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + \beta_{14} ) q^{22} + ( -2 - 7 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} - \beta_{15} ) q^{23} + ( -34 + 11 \beta_{1} + 8 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{7} - 2 \beta_{9} + 3 \beta_{10} - 3 \beta_{11} + 2 \beta_{13} - \beta_{15} ) q^{24} + 25 q^{25} + ( -44 + 4 \beta_{2} - 3 \beta_{3} - \beta_{5} + 2 \beta_{6} - 3 \beta_{10} + 3 \beta_{11} - 2 \beta_{14} - \beta_{15} ) q^{26} + ( 10 - 16 \beta_{1} + \beta_{2} + 4 \beta_{3} - \beta_{5} - 4 \beta_{6} + 2 \beta_{8} + 4 \beta_{10} + 2 \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} + 3 \beta_{15} ) q^{27} + ( 1 + 9 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + 6 \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{9} + \beta_{10} - 2 \beta_{11} - 2 \beta_{12} + 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{28} + ( 2 - 3 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} + \beta_{4} - 3 \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + \beta_{15} ) q^{29} + ( 10 - 5 \beta_{5} ) q^{30} + ( -1 - 8 \beta_{1} - \beta_{2} - 5 \beta_{3} + 3 \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - 5 \beta_{12} - 3 \beta_{13} - 3 \beta_{15} ) q^{31} + ( 27 \beta_{1} + 21 \beta_{3} - 4 \beta_{4} + 3 \beta_{5} + 4 \beta_{6} - 3 \beta_{7} + 3 \beta_{8} + 4 \beta_{10} + 4 \beta_{11} + 4 \beta_{13} ) q^{32} + ( 32 + 4 \beta_{1} - \beta_{2} - 4 \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{8} - 2 \beta_{9} + 4 \beta_{10} - 2 \beta_{11} + \beta_{12} - 3 \beta_{13} + 2 \beta_{15} ) q^{33} + ( 21 \beta_{1} + 3 \beta_{3} - 3 \beta_{4} - 4 \beta_{5} + \beta_{7} - \beta_{8} + \beta_{10} + \beta_{11} + 6 \beta_{12} + 3 \beta_{15} ) q^{34} -5 \beta_{10} q^{35} + ( 62 - 11 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} - \beta_{8} - 5 \beta_{10} - 7 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} + \beta_{14} + 3 \beta_{15} ) q^{36} + ( -52 + 6 \beta_{2} + 6 \beta_{3} - 9 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + 3 \beta_{10} - 3 \beta_{11} - 3 \beta_{14} - 7 \beta_{15} ) q^{37} + ( 78 - 20 \beta_{2} + 6 \beta_{3} + \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{8} - 6 \beta_{10} + 6 \beta_{11} - \beta_{14} ) q^{38} + ( -38 - 8 \beta_{1} - 3 \beta_{2} - \beta_{4} - 2 \beta_{5} + 8 \beta_{6} - 2 \beta_{8} - 6 \beta_{10} - \beta_{12} - 3 \beta_{14} ) q^{39} + ( -20 \beta_{1} - 5 \beta_{3} + 5 \beta_{4} ) q^{40} + ( 65 - 3 \beta_{2} + 3 \beta_{3} - \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + \beta_{9} - 4 \beta_{10} + 6 \beta_{11} - \beta_{12} - 2 \beta_{14} + 6 \beta_{15} ) q^{41} + ( -63 + 7 \beta_{1} - 2 \beta_{2} - 7 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} - 6 \beta_{6} + 3 \beta_{7} - \beta_{9} - 2 \beta_{10} - 9 \beta_{11} - 4 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{42} + ( -36 - 12 \beta_{2} + 9 \beta_{5} - 2 \beta_{9} - 6 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - 5 \beta_{14} + \beta_{15} ) q^{43} + ( -3 - 13 \beta_{1} - 3 \beta_{2} - 21 \beta_{3} - \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + \beta_{7} - \beta_{8} + 3 \beta_{9} + 10 \beta_{10} + 10 \beta_{11} - 5 \beta_{12} - 4 \beta_{15} ) q^{44} + ( -5 + 5 \beta_{1} + 5 \beta_{2} - 5 \beta_{12} ) q^{45} + ( 73 - 7 \beta_{2} - 33 \beta_{3} - 7 \beta_{5} + 13 \beta_{6} - 3 \beta_{7} - 3 \beta_{8} + \beta_{9} - 2 \beta_{10} + 4 \beta_{11} - \beta_{12} - \beta_{14} ) q^{46} + ( -51 + 15 \beta_{2} + 3 \beta_{5} - 4 \beta_{6} + \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} + 7 \beta_{14} + 7 \beta_{15} ) q^{47} + ( -125 - 65 \beta_{1} + 14 \beta_{2} - 19 \beta_{3} + 9 \beta_{4} + \beta_{5} + 7 \beta_{6} - \beta_{7} - 4 \beta_{8} + 3 \beta_{9} + 4 \beta_{10} - 12 \beta_{11} + 5 \beta_{12} - 2 \beta_{13} - \beta_{14} ) q^{48} + ( 24 - 25 \beta_{1} + \beta_{2} + 17 \beta_{3} + 13 \beta_{5} - 8 \beta_{6} + 2 \beta_{7} - 4 \beta_{8} - 3 \beta_{9} - 4 \beta_{10} - 4 \beta_{11} + \beta_{12} - 3 \beta_{13} - 3 \beta_{14} + 3 \beta_{15} ) q^{49} + 25 \beta_{1} q^{50} + ( 43 - 4 \beta_{1} - 8 \beta_{2} - 9 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} - 4 \beta_{8} + 5 \beta_{9} + \beta_{10} - 7 \beta_{11} - 4 \beta_{12} - \beta_{13} - 2 \beta_{14} - 8 \beta_{15} ) q^{51} + ( 2 - 64 \beta_{1} + 2 \beta_{2} + 8 \beta_{3} + 4 \beta_{5} - 4 \beta_{6} - 5 \beta_{7} + 5 \beta_{8} - 2 \beta_{9} + 3 \beta_{10} + 3 \beta_{11} + \beta_{15} ) q^{52} + ( -2 + 17 \beta_{1} - 2 \beta_{2} + 14 \beta_{3} + \beta_{4} - 5 \beta_{5} + 12 \beta_{6} + 6 \beta_{7} - 6 \beta_{8} + 2 \beta_{9} + 4 \beta_{10} + 4 \beta_{11} - 2 \beta_{13} - \beta_{15} ) q^{53} + ( 179 + 20 \beta_{1} - 27 \beta_{2} + 5 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} - \beta_{8} + 5 \beta_{9} + 6 \beta_{10} - 6 \beta_{11} - \beta_{12} + 4 \beta_{13} - 2 \beta_{15} ) q^{54} + ( 5 \beta_{1} + 5 \beta_{3} - 5 \beta_{13} ) q^{55} + ( -81 - 22 \beta_{1} + 26 \beta_{2} - 54 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} - 7 \beta_{8} - 5 \beta_{9} + 4 \beta_{10} - 3 \beta_{11} + 8 \beta_{12} - 2 \beta_{13} ) q^{56} + ( 22 + 20 \beta_{1} + 8 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 7 \beta_{5} + 8 \beta_{6} - 4 \beta_{7} - 2 \beta_{9} - 6 \beta_{10} - 6 \beta_{11} + 6 \beta_{12} + 8 \beta_{13} - 3 \beta_{14} - \beta_{15} ) q^{57} + ( 48 - 4 \beta_{2} + 27 \beta_{3} - 20 \beta_{5} - 6 \beta_{6} + \beta_{7} + \beta_{8} + 6 \beta_{9} + 3 \beta_{10} + 9 \beta_{11} - 6 \beta_{12} - \beta_{14} + 3 \beta_{15} ) q^{58} + ( 33 - 25 \beta_{2} + 3 \beta_{3} + 16 \beta_{5} - \beta_{6} - 4 \beta_{7} - 4 \beta_{8} - 5 \beta_{9} - 6 \beta_{10} - 4 \beta_{11} + 5 \beta_{12} - 6 \beta_{14} ) q^{59} + ( 20 + 15 \beta_{1} - 5 \beta_{2} + 20 \beta_{3} - 5 \beta_{4} + 5 \beta_{8} ) q^{60} + ( 2 - 13 \beta_{1} + 2 \beta_{2} - 40 \beta_{3} + 3 \beta_{4} - 5 \beta_{5} - 10 \beta_{6} + 4 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} + 13 \beta_{10} + 13 \beta_{11} + \beta_{15} ) q^{61} + ( 119 - 17 \beta_{2} - 15 \beta_{3} - 16 \beta_{5} + 5 \beta_{6} + 5 \beta_{7} + 5 \beta_{8} + 5 \beta_{9} - 5 \beta_{10} + 15 \beta_{11} - 5 \beta_{12} + 6 \beta_{14} - \beta_{15} ) q^{62} + ( 32 + 45 \beta_{1} - 21 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - \beta_{5} - 5 \beta_{6} - 2 \beta_{8} - 8 \beta_{9} - \beta_{11} - \beta_{12} + 5 \beta_{13} + 2 \beta_{14} + 5 \beta_{15} ) q^{63} + ( -258 + 31 \beta_{2} - 42 \beta_{3} + 24 \beta_{5} + 8 \beta_{6} - 8 \beta_{7} - 8 \beta_{8} - 6 \beta_{9} + 6 \beta_{10} - 18 \beta_{11} + 6 \beta_{12} + 6 \beta_{14} + 8 \beta_{15} ) q^{64} + ( -5 + 20 \beta_{1} - 5 \beta_{2} - 10 \beta_{3} + 5 \beta_{9} + 5 \beta_{12} ) q^{65} + ( -91 + 65 \beta_{1} + 27 \beta_{2} + 18 \beta_{3} + 4 \beta_{4} + \beta_{5} - 5 \beta_{6} + 7 \beta_{7} + 5 \beta_{9} - 5 \beta_{10} - 13 \beta_{11} + 5 \beta_{12} - 4 \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{66} + ( 42 - 2 \beta_{2} - 42 \beta_{3} - \beta_{5} + 14 \beta_{6} + 6 \beta_{7} + 6 \beta_{8} + 6 \beta_{10} - 6 \beta_{11} + \beta_{14} - 7 \beta_{15} ) q^{67} + ( -237 + 23 \beta_{2} + 21 \beta_{3} - \beta_{5} - 5 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - \beta_{9} - 7 \beta_{10} + 5 \beta_{11} + \beta_{12} - 3 \beta_{14} - 7 \beta_{15} ) q^{68} + ( -126 - 73 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} + 3 \beta_{6} + 4 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - \beta_{10} + 5 \beta_{11} - 6 \beta_{12} + \beta_{13} - 6 \beta_{14} - 4 \beta_{15} ) q^{69} + ( 35 - 5 \beta_{1} + 25 \beta_{3} - 5 \beta_{4} - 10 \beta_{6} + 5 \beta_{8} - 5 \beta_{9} + 5 \beta_{11} ) q^{70} + ( 4 - 32 \beta_{1} + 4 \beta_{2} + 48 \beta_{3} - 9 \beta_{4} - \beta_{5} + 3 \beta_{6} - 6 \beta_{7} + 6 \beta_{8} - 4 \beta_{9} + 4 \beta_{10} + 4 \beta_{11} + 10 \beta_{12} + 3 \beta_{13} + 7 \beta_{15} ) q^{71} + ( 130 + 47 \beta_{1} - 32 \beta_{2} + 19 \beta_{3} - 16 \beta_{5} - 12 \beta_{6} + 9 \beta_{7} + 11 \beta_{8} + 12 \beta_{9} - 11 \beta_{10} + 11 \beta_{11} - 8 \beta_{12} + 4 \beta_{13} + 7 \beta_{14} - 9 \beta_{15} ) q^{72} + ( 9 - 28 \beta_{1} + 9 \beta_{2} + 29 \beta_{3} - 10 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - 9 \beta_{9} + \beta_{10} + \beta_{11} - 5 \beta_{12} + 10 \beta_{13} + 2 \beta_{15} ) q^{73} + ( -1 - 129 \beta_{1} - \beta_{2} + 63 \beta_{3} + 15 \beta_{4} + 3 \beta_{5} + 25 \beta_{6} + \beta_{9} + \beta_{10} + \beta_{11} - 5 \beta_{12} - 8 \beta_{13} - 3 \beta_{15} ) q^{74} -25 \beta_{3} q^{75} + ( 3 + 177 \beta_{1} + 3 \beta_{2} + 45 \beta_{3} - 15 \beta_{4} + 7 \beta_{5} + 5 \beta_{6} - 7 \beta_{7} + 7 \beta_{8} - 3 \beta_{9} + 4 \beta_{10} + 4 \beta_{11} - 3 \beta_{12} + 12 \beta_{13} ) q^{76} + ( -71 + 32 \beta_{1} + 37 \beta_{2} - 26 \beta_{3} - 2 \beta_{4} + 5 \beta_{5} - 20 \beta_{6} + 4 \beta_{7} + 8 \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} + 9 \beta_{12} + 8 \beta_{13} + \beta_{14} + 13 \beta_{15} ) q^{77} + ( 106 - 28 \beta_{1} + 16 \beta_{2} + 63 \beta_{3} - 7 \beta_{4} + 16 \beta_{5} - 12 \beta_{6} - 5 \beta_{7} + 8 \beta_{8} - 16 \beta_{9} - 14 \beta_{10} - 4 \beta_{11} - 4 \beta_{12} - 4 \beta_{13} - 2 \beta_{14} + 4 \beta_{15} ) q^{78} + ( 103 - 15 \beta_{2} + 54 \beta_{3} - 18 \beta_{5} - 18 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + 5 \beta_{9} + 9 \beta_{10} + \beta_{11} - 5 \beta_{12} + 8 \beta_{14} + 4 \beta_{15} ) q^{79} + ( 125 - 20 \beta_{2} + 15 \beta_{3} - 15 \beta_{5} - 5 \beta_{6} + 5 \beta_{7} + 5 \beta_{8} + 5 \beta_{9} + 10 \beta_{11} - 5 \beta_{12} + 5 \beta_{14} ) q^{80} + ( -206 + 8 \beta_{2} - 20 \beta_{3} + 12 \beta_{4} + 5 \beta_{5} + 25 \beta_{6} + 10 \beta_{7} + 5 \beta_{9} + 11 \beta_{10} - 11 \beta_{11} - 2 \beta_{12} + 3 \beta_{13} + 5 \beta_{14} - 5 \beta_{15} ) q^{81} + ( 11 + 104 \beta_{1} + 11 \beta_{2} + 59 \beta_{3} - 16 \beta_{4} + 6 \beta_{5} - 7 \beta_{6} - 10 \beta_{7} + 10 \beta_{8} - 11 \beta_{9} - 16 \beta_{10} - 16 \beta_{11} - 3 \beta_{12} - 8 \beta_{13} + 4 \beta_{15} ) q^{82} + ( 103 + \beta_{2} + 39 \beta_{3} + 16 \beta_{5} - 13 \beta_{6} - 7 \beta_{9} + 7 \beta_{10} - 21 \beta_{11} + 7 \beta_{12} - 2 \beta_{14} - 12 \beta_{15} ) q^{83} + ( -24 + 2 \beta_{1} - 52 \beta_{2} - 14 \beta_{3} + 9 \beta_{4} - 9 \beta_{5} - 23 \beta_{6} + 8 \beta_{7} + 6 \beta_{8} + 14 \beta_{9} - \beta_{10} + 18 \beta_{11} - \beta_{12} + 6 \beta_{13} - 2 \beta_{14} - 5 \beta_{15} ) q^{84} + ( 20 - 10 \beta_{2} - 15 \beta_{3} - 5 \beta_{5} + 5 \beta_{6} + 5 \beta_{14} - 5 \beta_{15} ) q^{85} + ( 13 + 25 \beta_{1} + 13 \beta_{2} - 29 \beta_{3} - 5 \beta_{4} + 9 \beta_{5} - 21 \beta_{6} - 3 \beta_{7} + 3 \beta_{8} - 13 \beta_{9} + 6 \beta_{10} + 6 \beta_{11} - 25 \beta_{12} - 6 \beta_{15} ) q^{86} + ( -122 + 50 \beta_{1} + 45 \beta_{2} - 10 \beta_{3} + 4 \beta_{4} - 18 \beta_{5} + 9 \beta_{6} - 2 \beta_{7} + 4 \beta_{8} + 6 \beta_{9} + 13 \beta_{10} + 19 \beta_{11} - 5 \beta_{12} - 5 \beta_{13} + 6 \beta_{14} - 12 \beta_{15} ) q^{87} + ( -78 + 32 \beta_{2} - 114 \beta_{3} - 15 \beta_{5} + 38 \beta_{6} - 3 \beta_{7} - 3 \beta_{8} + 4 \beta_{9} + 10 \beta_{10} - 2 \beta_{11} - 4 \beta_{12} + 7 \beta_{14} + 4 \beta_{15} ) q^{88} + ( -385 + 31 \beta_{2} - 21 \beta_{3} - 12 \beta_{5} + 15 \beta_{6} - 10 \beta_{7} - 10 \beta_{8} - \beta_{9} + 2 \beta_{10} - 4 \beta_{11} + \beta_{12} - 10 \beta_{14} - 6 \beta_{15} ) q^{89} + ( -60 - 30 \beta_{1} + 10 \beta_{2} - 5 \beta_{3} + 10 \beta_{6} - 5 \beta_{7} - 10 \beta_{11} + 5 \beta_{14} ) q^{90} + ( -28 - 101 \beta_{1} + 4 \beta_{2} + 85 \beta_{3} - 4 \beta_{4} + 13 \beta_{5} - 8 \beta_{6} - 6 \beta_{7} + 10 \beta_{8} + 6 \beta_{9} + \beta_{10} + 7 \beta_{11} + 14 \beta_{12} + \beta_{13} + 3 \beta_{14} + 11 \beta_{15} ) q^{91} + ( 4 + 77 \beta_{1} + 4 \beta_{2} + 50 \beta_{3} - 9 \beta_{4} - 5 \beta_{5} - 4 \beta_{6} - 7 \beta_{7} + 7 \beta_{8} - 4 \beta_{9} - 8 \beta_{10} - 8 \beta_{11} + 20 \beta_{12} - 12 \beta_{13} + 12 \beta_{15} ) q^{92} + ( -52 - 29 \beta_{1} + 40 \beta_{2} + 20 \beta_{3} - 19 \beta_{4} + 2 \beta_{5} - 34 \beta_{6} - 8 \beta_{7} - 4 \beta_{9} + 21 \beta_{10} + 15 \beta_{11} + 12 \beta_{12} - 2 \beta_{13} - 3 \beta_{14} + 16 \beta_{15} ) q^{93} + ( -11 - 105 \beta_{1} - 11 \beta_{2} - 54 \beta_{3} + 3 \beta_{4} - 14 \beta_{5} - 9 \beta_{6} + 4 \beta_{7} - 4 \beta_{8} + 11 \beta_{9} - 12 \beta_{10} - 12 \beta_{11} + 31 \beta_{12} + 10 \beta_{15} ) q^{94} + ( -5 - 25 \beta_{1} - 5 \beta_{2} + 5 \beta_{3} + 10 \beta_{4} + 10 \beta_{6} + 5 \beta_{9} - 5 \beta_{10} - 5 \beta_{11} + 5 \beta_{12} + 5 \beta_{13} ) q^{95} + ( 592 - 160 \beta_{1} - 82 \beta_{2} - 9 \beta_{3} + 25 \beta_{4} - 15 \beta_{5} - 4 \beta_{6} + 18 \beta_{7} + 5 \beta_{8} + 4 \beta_{9} - 17 \beta_{10} - 5 \beta_{11} + 6 \beta_{13} - 13 \beta_{15} ) q^{96} + ( 2 + 98 \beta_{1} + 2 \beta_{2} - 55 \beta_{3} + 24 \beta_{4} + 6 \beta_{5} - 7 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 6 \beta_{10} - 6 \beta_{11} - 18 \beta_{12} - 8 \beta_{15} ) q^{97} + ( 303 + 12 \beta_{1} - 19 \beta_{2} - 97 \beta_{3} + 19 \beta_{4} - 8 \beta_{5} - 21 \beta_{6} + 11 \beta_{7} - \beta_{8} + 11 \beta_{9} - 3 \beta_{10} + 19 \beta_{11} - 25 \beta_{12} - 4 \beta_{13} - \beta_{14} - 13 \beta_{15} ) q^{98} + ( 158 + 25 \beta_{1} - 19 \beta_{2} - 55 \beta_{3} - 23 \beta_{5} + 15 \beta_{6} + 6 \beta_{7} - 8 \beta_{8} + 12 \beta_{9} + 14 \beta_{10} + 28 \beta_{11} - 7 \beta_{12} - 16 \beta_{13} + 11 \beta_{14} + 3 \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 2q^{3} - 64q^{4} + 80q^{5} + 28q^{6} - 4q^{7} - 22q^{9} + O(q^{10}) \) \( 16q + 2q^{3} - 64q^{4} + 80q^{5} + 28q^{6} - 4q^{7} - 22q^{9} + 66q^{12} + 90q^{14} + 10q^{15} + 376q^{16} + 72q^{17} - 182q^{18} - 320q^{20} - 70q^{21} - 276q^{22} - 526q^{24} + 400q^{25} - 696q^{26} + 128q^{27} + 10q^{28} + 140q^{30} + 502q^{33} - 20q^{35} + 996q^{36} - 812q^{37} + 1200q^{38} - 594q^{39} + 936q^{41} - 974q^{42} - 548q^{43} - 110q^{45} + 1224q^{46} - 912q^{47} - 1850q^{48} + 328q^{49} + 750q^{51} + 2950q^{54} - 1254q^{56} + 432q^{57} + 576q^{58} + 552q^{59} + 330q^{60} + 1860q^{62} + 362q^{63} - 4000q^{64} - 1378q^{66} + 1004q^{67} - 3828q^{68} - 1988q^{69} + 450q^{70} + 1988q^{72} + 50q^{75} - 1152q^{77} + 1446q^{78} + 1292q^{79} + 1880q^{80} - 2950q^{81} + 1752q^{83} - 420q^{84} + 360q^{85} - 1910q^{87} - 912q^{88} - 6096q^{89} - 910q^{90} - 552q^{91} - 1080q^{93} + 9546q^{96} + 4824q^{98} + 2530q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 96 x^{14} + 3618 x^{12} + 68560 x^{10} + 697017 x^{8} + 3791184 x^{6} + 10461796 x^{4} + 13209792 x^{2} + 5760000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 12 \)
\(\beta_{3}\)\(=\)\((\)\(547 \nu^{15} + 1777 \nu^{14} + 50246 \nu^{13} + 180054 \nu^{12} + 1775248 \nu^{11} + 7185508 \nu^{10} + 30263338 \nu^{9} + 143671962 \nu^{8} + 250516321 \nu^{7} + 1511305167 \nu^{6} + 824026304 \nu^{5} + 8004483344 \nu^{4} + 59746156 \nu^{3} + 18319341788 \nu^{2} - 1655412336 \nu + 12630768000\)\()/ 380788800 \)
\(\beta_{4}\)\(=\)\((\)\(547 \nu^{15} + 1777 \nu^{14} + 50246 \nu^{13} + 180054 \nu^{12} + 1775248 \nu^{11} + 7185508 \nu^{10} + 30263338 \nu^{9} + 143671962 \nu^{8} + 250516321 \nu^{7} + 1511305167 \nu^{6} + 824026304 \nu^{5} + 8004483344 \nu^{4} + 440534956 \nu^{3} + 18319341788 \nu^{2} + 5960363664 \nu + 12630768000\)\()/ 380788800 \)
\(\beta_{5}\)\(=\)\((\)\(1777 \nu^{15} - 2266 \nu^{14} + 180054 \nu^{13} - 203798 \nu^{12} + 7185508 \nu^{11} - 7238982 \nu^{10} + 143671962 \nu^{9} - 130751978 \nu^{8} + 1511305167 \nu^{7} - 1249751344 \nu^{6} + 8004483344 \nu^{5} - 5662856256 \nu^{4} + 18319341788 \nu^{3} - 8881168560 \nu^{2} + 12630768000 \nu - 2389142400\)\()/ 380788800 \)
\(\beta_{6}\)\(=\)\((\)\(547 \nu^{15} - 1777 \nu^{14} + 50246 \nu^{13} - 180054 \nu^{12} + 1775248 \nu^{11} - 7185508 \nu^{10} + 30263338 \nu^{9} - 143671962 \nu^{8} + 250516321 \nu^{7} - 1511305167 \nu^{6} + 824026304 \nu^{5} - 8004483344 \nu^{4} + 59746156 \nu^{3} - 18319341788 \nu^{2} - 1655412336 \nu - 12630768000\)\()/ 126929600 \)
\(\beta_{7}\)\(=\)\((\)\(2521 \nu^{15} - 25944 \nu^{14} + 219100 \nu^{13} - 2400552 \nu^{12} + 6878486 \nu^{11} - 86563888 \nu^{10} + 88736116 \nu^{9} - 1552833632 \nu^{8} + 245139341 \nu^{7} - 14607742056 \nu^{6} - 3779023952 \nu^{5} - 69427453304 \nu^{4} - 26864345276 \nu^{3} - 144212081680 \nu^{2} - 42119845632 \nu - 90153456000\)\()/ 380788800 \)
\(\beta_{8}\)\(=\)\((\)\(-3751 \nu^{15} - 29009 \nu^{14} - 348908 \nu^{13} - 2736916 \nu^{12} - 12288746 \nu^{11} - 100881430 \nu^{10} - 202144740 \nu^{9} - 1853097540 \nu^{8} - 1505928187 \nu^{7} - 17891906213 \nu^{6} - 3401433088 \nu^{5} - 87778047080 \nu^{4} + 8604749644 \nu^{3} - 190288938484 \nu^{2} + 27833665296 \nu - 125656617600\)\()/ 380788800 \)
\(\beta_{9}\)\(=\)\((\)\(-17137 \nu^{15} - 46928 \nu^{14} - 1503096 \nu^{13} - 4329248 \nu^{12} - 49505042 \nu^{11} - 153385008 \nu^{10} - 757860320 \nu^{9} - 2632690032 \nu^{8} - 5303568969 \nu^{7} - 22705763584 \nu^{6} - 12874508776 \nu^{5} - 93023998704 \nu^{4} + 10431012108 \nu^{3} - 150358937344 \nu^{2} + 38974522752 \nu - 60084528000\)\()/ 761577600 \)
\(\beta_{10}\)\(=\)\((\)\(27779 \nu^{15} - 35226 \nu^{14} + 2607108 \nu^{13} - 3211404 \nu^{12} + 94836522 \nu^{11} - 112663080 \nu^{10} + 1696532852 \nu^{9} - 1929050340 \nu^{8} + 15633450711 \nu^{7} - 16923098022 \nu^{6} + 71244147816 \nu^{5} - 74019685200 \nu^{4} + 139040974460 \nu^{3} - 144818899416 \nu^{2} + 85968016416 \nu - 93289344000\)\()/ 761577600 \)
\(\beta_{11}\)\(=\)\((\)\(27779 \nu^{15} + 35226 \nu^{14} + 2607108 \nu^{13} + 3211404 \nu^{12} + 94836522 \nu^{11} + 112663080 \nu^{10} + 1696532852 \nu^{9} + 1929050340 \nu^{8} + 15633450711 \nu^{7} + 16923098022 \nu^{6} + 71244147816 \nu^{5} + 74019685200 \nu^{4} + 139040974460 \nu^{3} + 144818899416 \nu^{2} + 85968016416 \nu + 93289344000\)\()/ 761577600 \)
\(\beta_{12}\)\(=\)\((\)\(26393 \nu^{15} + 54036 \nu^{14} + 2471672 \nu^{13} + 5049464 \nu^{12} + 89784930 \nu^{11} + 182127040 \nu^{10} + 1606883040 \nu^{9} + 3207377880 \nu^{8} + 14874956401 \nu^{7} + 28750984252 \nu^{6} + 68874025320 \nu^{5} + 125041932080 \nu^{4} + 142197211348 \nu^{3} + 224397882096 \nu^{2} + 103242762240 \nu + 120508108800\)\()/ 761577600 \)
\(\beta_{13}\)\(=\)\((\)\(-56747 \nu^{15} + 3554 \nu^{14} - 5360852 \nu^{13} + 360108 \nu^{12} - 197441554 \nu^{11} + 14371016 \nu^{10} - 3611070764 \nu^{9} + 287343924 \nu^{8} - 34539744535 \nu^{7} + 3022610334 \nu^{6} - 166819393952 \nu^{5} + 16008966688 \nu^{4} - 352776724844 \nu^{3} + 36638683576 \nu^{2} - 225162344352 \nu + 25261536000\)\()/ 761577600 \)
\(\beta_{14}\)\(=\)\((\)\(547 \nu^{15} + 52749 \nu^{14} + 50246 \nu^{13} + 4923702 \nu^{12} + 1775248 \nu^{11} + 177708268 \nu^{10} + 30263338 \nu^{9} + 3142627082 \nu^{8} + 250516321 \nu^{7} + 28487839131 \nu^{6} + 824026304 \nu^{5} + 127584100544 \nu^{4} + 59746156 \nu^{3} + 251163060460 \nu^{2} - 1655412336 \nu + 162993638400\)\()/ 380788800 \)
\(\beta_{15}\)\(=\)\((\)\(-1230 \nu^{15} - 58079 \nu^{14} - 129808 \nu^{13} - 5433316 \nu^{12} - 5410260 \nu^{11} - 196551530 \nu^{10} - 113408624 \nu^{9} - 3481801820 \nu^{8} - 1260788846 \nu^{7} - 31512040763 \nu^{6} - 7180457040 \nu^{5} - 138709271680 \nu^{4} - 18259595632 \nu^{3} - 251598392444 \nu^{2} - 14286180336 \nu - 135528019200\)\()/ 380788800 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 12\)
\(\nu^{3}\)\(=\)\(\beta_{4} - \beta_{3} - 20 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{14} - \beta_{12} + 2 \beta_{11} + \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} - 3 \beta_{5} + 3 \beta_{3} - 28 \beta_{2} + 249\)
\(\nu^{5}\)\(=\)\(4 \beta_{13} + 4 \beta_{11} + 4 \beta_{10} + 3 \beta_{8} - 3 \beta_{7} + 4 \beta_{6} + 3 \beta_{5} - 36 \beta_{4} + 53 \beta_{3} + 475 \beta_{1}\)
\(\nu^{6}\)\(=\)\(8 \beta_{15} - 34 \beta_{14} + 46 \beta_{12} - 98 \beta_{11} + 6 \beta_{10} - 46 \beta_{9} - 48 \beta_{8} - 48 \beta_{7} + 48 \beta_{6} + 144 \beta_{5} - 162 \beta_{3} + 767 \beta_{2} - 6122\)
\(\nu^{7}\)\(=\)\(24 \beta_{15} - 192 \beta_{13} + 70 \beta_{12} - 216 \beta_{11} - 216 \beta_{10} + 22 \beta_{9} - 156 \beta_{8} + 156 \beta_{7} - 198 \beta_{6} - 180 \beta_{5} + 1135 \beta_{4} - 2025 \beta_{3} - 22 \beta_{2} - 12238 \beta_{1} - 22\)
\(\nu^{8}\)\(=\)\(-436 \beta_{15} + 923 \beta_{14} - 1645 \beta_{12} + 3734 \beta_{11} - 444 \beta_{10} + 1645 \beta_{9} + 1783 \beta_{8} + 1783 \beta_{7} - 1853 \beta_{6} - 5233 \beta_{5} + 6579 \beta_{3} - 21612 \beta_{2} + 162433\)
\(\nu^{9}\)\(=\)\(-1484 \beta_{15} + 7132 \beta_{13} - 4370 \beta_{12} + 8656 \beta_{11} + 8656 \beta_{10} - 1402 \beta_{9} + 6101 \beta_{8} - 6101 \beta_{7} + 7218 \beta_{6} + 7585 \beta_{5} - 34938 \beta_{4} + 69083 \beta_{3} + 1402 \beta_{2} + 331825 \beta_{1} + 1402\)
\(\nu^{10}\)\(=\)\(17276 \beta_{15} - 24096 \beta_{14} + 54358 \beta_{12} - 129530 \beta_{11} + 20814 \beta_{10} - 54358 \beta_{9} - 60826 \beta_{8} - 60826 \beta_{7} + 65396 \beta_{6} + 173882 \beta_{5} - 237258 \beta_{3} + 624523 \beta_{2} - 4509374\)
\(\nu^{11}\)\(=\)\(63684 \beta_{15} - 243304 \beta_{13} + 188220 \beta_{12} - 309708 \beta_{11} - 309708 \beta_{10} + 60852 \beta_{9} - 214150 \beta_{8} + 214150 \beta_{7} - 238720 \beta_{6} - 277834 \beta_{5} + 1070753 \beta_{4} - 2244079 \beta_{3} - 60852 \beta_{2} - 9326360 \beta_{1} - 60852\)
\(\nu^{12}\)\(=\)\(-609280 \beta_{15} + 636685 \beta_{14} - 1737773 \beta_{12} + 4284758 \beta_{11} - 809212 \beta_{10} + 1737773 \beta_{9} + 1994437 \beta_{8} + 1994437 \beta_{7} - 2198457 \beta_{6} - 5565935 \beta_{5} + 8049075 \beta_{3} - 18391960 \beta_{2} + 129075333\)
\(\nu^{13}\)\(=\)\(-2366976 \beta_{15} + 7977748 \beta_{13} - 6992780 \beta_{12} + 10466204 \beta_{11} + 10466204 \beta_{10} - 2258828 \beta_{9} + 7123619 \beta_{8} - 7123619 \beta_{7} + 7599672 \beta_{6} + 9490595 \beta_{5} - 32811480 \beta_{4} + 71154285 \beta_{3} + 2258828 \beta_{2} + 269098919 \beta_{1} + 2258828\)
\(\nu^{14}\)\(=\)\(20324704 \beta_{15} - 17290382 \beta_{14} + 54658390 \beta_{12} - 137940986 \beta_{11} + 28624206 \beta_{10} - 54658390 \beta_{9} - 63967132 \beta_{8} - 63967132 \beta_{7} + 71783912 \beta_{6} + 174991132 \beta_{5} - 263735706 \beta_{3} + 549074015 \beta_{2} - 3775554106\)
\(\nu^{15}\)\(=\)\(81854816 \beta_{15} - 255868528 \beta_{13} + 241200530 \beta_{12} - 342267664 \beta_{11} - 342267664 \beta_{10} + 77490898 \beta_{9} - 229967384 \beta_{8} + 229967384 \beta_{7} - 237910602 \beta_{6} - 311822200 \beta_{5} + 1006221539 \beta_{4} - 2227113949 \beta_{3} - 77490898 \beta_{2} - 7914840770 \beta_{1} - 77490898\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\) \(71\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1
5.54840i
4.61386i
4.47752i
3.20688i
2.73921i
1.80256i
1.39379i
0.948735i
0.948735i
1.39379i
1.80256i
2.73921i
3.20688i
4.47752i
4.61386i
5.54840i
5.54840i −3.35452 + 3.96828i −22.7847 5.00000 22.0176 + 18.6122i 9.21642 + 16.0642i 82.0315i −4.49445 26.6233i 27.7420i
41.2 4.61386i 3.98691 + 3.33234i −13.2877 5.00000 15.3750 18.3951i 0.582685 18.5111i 24.3969i 4.79097 + 26.5715i 23.0693i
41.3 4.47752i −0.787490 5.13613i −12.0482 5.00000 −22.9971 + 3.52600i −18.2591 + 3.09943i 18.1258i −25.7597 + 8.08931i 22.3876i
41.4 3.20688i 2.93464 4.28811i −2.28410 5.00000 −13.7515 9.41106i 18.2867 + 2.93186i 18.3302i −9.77574 25.1681i 16.0344i
41.5 2.73921i −4.32728 + 2.87658i 0.496728 5.00000 7.87955 + 11.8533i −18.4646 1.43529i 23.2743i 10.4506 24.8955i 13.6961i
41.6 1.80256i 1.60858 + 4.94090i 4.75076 5.00000 8.90629 2.89956i 2.01165 + 18.4107i 22.9841i −21.8250 + 15.8956i 9.01282i
41.7 1.39379i 5.17944 0.416449i 6.05735 5.00000 −0.580442 7.21904i −10.8253 15.0271i 19.5930i 26.6531 4.31394i 6.96894i
41.8 0.948735i −4.24029 3.00332i 7.09990 5.00000 −2.84936 + 4.02291i 15.4514 + 10.2104i 14.3258i 8.96014 + 25.4699i 4.74368i
41.9 0.948735i −4.24029 + 3.00332i 7.09990 5.00000 −2.84936 4.02291i 15.4514 10.2104i 14.3258i 8.96014 25.4699i 4.74368i
41.10 1.39379i 5.17944 + 0.416449i 6.05735 5.00000 −0.580442 + 7.21904i −10.8253 + 15.0271i 19.5930i 26.6531 + 4.31394i 6.96894i
41.11 1.80256i 1.60858 4.94090i 4.75076 5.00000 8.90629 + 2.89956i 2.01165 18.4107i 22.9841i −21.8250 15.8956i 9.01282i
41.12 2.73921i −4.32728 2.87658i 0.496728 5.00000 7.87955 11.8533i −18.4646 + 1.43529i 23.2743i 10.4506 + 24.8955i 13.6961i
41.13 3.20688i 2.93464 + 4.28811i −2.28410 5.00000 −13.7515 + 9.41106i 18.2867 2.93186i 18.3302i −9.77574 + 25.1681i 16.0344i
41.14 4.47752i −0.787490 + 5.13613i −12.0482 5.00000 −22.9971 3.52600i −18.2591 3.09943i 18.1258i −25.7597 8.08931i 22.3876i
41.15 4.61386i 3.98691 3.33234i −13.2877 5.00000 15.3750 + 18.3951i 0.582685 + 18.5111i 24.3969i 4.79097 26.5715i 23.0693i
41.16 5.54840i −3.35452 3.96828i −22.7847 5.00000 22.0176 18.6122i 9.21642 16.0642i 82.0315i −4.49445 + 26.6233i 27.7420i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.4.b.b yes 16
3.b odd 2 1 105.4.b.a 16
7.b odd 2 1 105.4.b.a 16
21.c even 2 1 inner 105.4.b.b yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.b.a 16 3.b odd 2 1
105.4.b.a 16 7.b odd 2 1
105.4.b.b yes 16 1.a even 1 1 trivial
105.4.b.b yes 16 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(26\!\cdots\!16\)\( T_{17} - \)\(14\!\cdots\!20\)\( \)">\(T_{17}^{8} - \cdots\) acting on \(S_{4}^{\mathrm{new}}(105, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 5760000 + 13209792 T^{2} + 10461796 T^{4} + 3791184 T^{6} + 697017 T^{8} + 68560 T^{10} + 3618 T^{12} + 96 T^{14} + T^{16} \)
$3$ \( 282429536481 - 20920706406 T + 5036466357 T^{2} - 947027862 T^{3} + 481485546 T^{4} - 125065782 T^{5} + 16290963 T^{6} - 4576662 T^{7} + 544266 T^{8} - 169506 T^{9} + 22347 T^{10} - 6354 T^{11} + 906 T^{12} - 66 T^{13} + 13 T^{14} - 2 T^{15} + T^{16} \)
$5$ \( ( -5 + T )^{16} \)
$7$ \( \)\(19\!\cdots\!01\)\( + 2234183456333136028 T - 254032521274030044 T^{2} + 29112047178970476 T^{3} - 1455826587801180 T^{4} - 39195135650244 T^{5} + 3285636800348 T^{6} - 96922540372 T^{7} + 781972086 T^{8} - 282573004 T^{9} + 27927452 T^{10} - 971292 T^{11} - 105180 T^{12} + 6132 T^{13} - 156 T^{14} + 4 T^{15} + T^{16} \)
$11$ \( \)\(15\!\cdots\!00\)\( + \)\(30\!\cdots\!00\)\( T^{2} + 21312676928409491200 T^{4} + 73239720004993152 T^{6} + 131700393212400 T^{8} + 124231198492 T^{10} + 58703601 T^{12} + 12690 T^{14} + T^{16} \)
$13$ \( \)\(35\!\cdots\!00\)\( + 76099399903047426048 T^{2} + 3665783202760233216 T^{4} + 55293969691628928 T^{6} + 157504313285424 T^{8} + 166589510904 T^{10} + 73621641 T^{12} + 14238 T^{14} + T^{16} \)
$17$ \( ( -14231625894720 + 2613586192416 T - 99699071472 T^{2} - 2696813856 T^{3} + 80245500 T^{4} + 673446 T^{5} - 17439 T^{6} - 36 T^{7} + T^{8} )^{2} \)
$19$ \( \)\(15\!\cdots\!00\)\( + \)\(16\!\cdots\!00\)\( T^{2} + \)\(40\!\cdots\!00\)\( T^{4} + 20031736244328557568 T^{6} + 17669582808795648 T^{8} + 5827449233088 T^{10} + 803627040 T^{12} + 47508 T^{14} + T^{16} \)
$23$ \( \)\(82\!\cdots\!00\)\( + \)\(28\!\cdots\!72\)\( T^{2} + \)\(66\!\cdots\!36\)\( T^{4} + \)\(47\!\cdots\!76\)\( T^{6} + 152491196098857216 T^{8} + 24857218625344 T^{10} + 2049203904 T^{12} + 77076 T^{14} + T^{16} \)
$29$ \( \)\(66\!\cdots\!00\)\( + \)\(92\!\cdots\!12\)\( T^{2} + \)\(26\!\cdots\!68\)\( T^{4} + \)\(32\!\cdots\!08\)\( T^{6} + 21355521440644356240 T^{8} + 809323696323004 T^{10} + 17675473473 T^{12} + 206658 T^{14} + T^{16} \)
$31$ \( \)\(10\!\cdots\!00\)\( + \)\(19\!\cdots\!00\)\( T^{2} + \)\(10\!\cdots\!00\)\( T^{4} + \)\(20\!\cdots\!08\)\( T^{6} + 19457429187958391808 T^{8} + 885388778034048 T^{10} + 20574555600 T^{12} + 232728 T^{14} + T^{16} \)
$37$ \( ( 701394437334784000 + 93939842508428800 T + 1623562622557120 T^{2} + 9026793332768 T^{3} + 1789385552 T^{4} - 125409280 T^{5} - 233492 T^{6} + 406 T^{7} + T^{8} )^{2} \)
$41$ \( ( 6578222266765900800 - 216053191075984128 T + 2389336748432064 T^{2} - 9342547363008 T^{3} - 9995711760 T^{4} + 155048112 T^{5} - 224988 T^{6} - 468 T^{7} + T^{8} )^{2} \)
$43$ \( ( -666624855021731840 - 6284591104729088 T + 220826612908288 T^{2} + 2771188130432 T^{3} - 1794810736 T^{4} - 99121720 T^{5} - 244736 T^{6} + 274 T^{7} + T^{8} )^{2} \)
$47$ \( ( 95108965237210982400 - 730245757410809856 T - 2794130106527808 T^{2} + 22337546899536 T^{3} + 46205699604 T^{4} - 217787226 T^{5} - 447207 T^{6} + 456 T^{7} + T^{8} )^{2} \)
$53$ \( \)\(10\!\cdots\!00\)\( + \)\(86\!\cdots\!00\)\( T^{2} + \)\(72\!\cdots\!00\)\( T^{4} + \)\(21\!\cdots\!00\)\( T^{6} + \)\(27\!\cdots\!36\)\( T^{8} + 186724587839773312 T^{10} + 685218029424 T^{12} + 1299852 T^{14} + T^{16} \)
$59$ \( ( 32333016885926400000 + 510377933594880000 T - 3694367062563840 T^{2} - 14851114314624 T^{3} + 81659372208 T^{4} + 127733616 T^{5} - 563664 T^{6} - 276 T^{7} + T^{8} )^{2} \)
$61$ \( \)\(29\!\cdots\!00\)\( + \)\(29\!\cdots\!00\)\( T^{2} + \)\(10\!\cdots\!00\)\( T^{4} + \)\(15\!\cdots\!72\)\( T^{6} + \)\(11\!\cdots\!96\)\( T^{8} + 528004601408332800 T^{10} + 1354175905296 T^{12} + 1827996 T^{14} + T^{16} \)
$67$ \( ( -\)\(18\!\cdots\!60\)\( + 4907247297878571008 T - 25192160408625152 T^{2} - 95193919625728 T^{3} + 317830459120 T^{4} + 448865672 T^{5} - 1063784 T^{6} - 502 T^{7} + T^{8} )^{2} \)
$71$ \( \)\(43\!\cdots\!00\)\( + \)\(88\!\cdots\!00\)\( T^{2} + \)\(37\!\cdots\!00\)\( T^{4} + \)\(70\!\cdots\!28\)\( T^{6} + \)\(71\!\cdots\!88\)\( T^{8} + 410211272958651520 T^{10} + 1280280572304 T^{12} + 1941672 T^{14} + T^{16} \)
$73$ \( \)\(16\!\cdots\!24\)\( + \)\(20\!\cdots\!44\)\( T^{2} + \)\(67\!\cdots\!32\)\( T^{4} + \)\(85\!\cdots\!56\)\( T^{6} + \)\(53\!\cdots\!00\)\( T^{8} + 1802852843163152256 T^{10} + 3208219701168 T^{12} + 2857944 T^{14} + T^{16} \)
$79$ \( ( -\)\(13\!\cdots\!00\)\( + \)\(10\!\cdots\!80\)\( T - 62500410900413696 T^{2} - 764576674485952 T^{3} + 870620077696 T^{4} + 1489511132 T^{5} - 1961927 T^{6} - 646 T^{7} + T^{8} )^{2} \)
$83$ \( ( -\)\(26\!\cdots\!00\)\( - 55584393130700421120 T - 316856318724260352 T^{2} - 213919992874368 T^{3} + 1321755786672 T^{4} + 1055769696 T^{5} - 1944540 T^{6} - 876 T^{7} + T^{8} )^{2} \)
$89$ \( ( \)\(38\!\cdots\!00\)\( - 19284986938452480000 T + 192397192816170240 T^{2} + 391676655410496 T^{3} - 1250006468880 T^{4} - 2058534576 T^{5} + 1582032 T^{6} + 3048 T^{7} + T^{8} )^{2} \)
$97$ \( \)\(43\!\cdots\!36\)\( + \)\(76\!\cdots\!08\)\( T^{2} + \)\(26\!\cdots\!48\)\( T^{4} + \)\(23\!\cdots\!88\)\( T^{6} + \)\(85\!\cdots\!28\)\( T^{8} + 15752097847324563048 T^{10} + 14633076197721 T^{12} + 6322590 T^{14} + T^{16} \)
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