Properties

Label 1875.4.a.g
Level $1875$
Weight $4$
Character orbit 1875.a
Self dual yes
Analytic conductor $110.629$
Analytic rank $1$
Dimension $14$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1875.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(110.628581261\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - 81 x^{12} - 7 x^{11} + 2512 x^{10} + 517 x^{9} - 36970 x^{8} - 12987 x^{7} + 257291 x^{6} + 125779 x^{5} - 718713 x^{4} - 371750 x^{3} + 579848 x^{2} + 394896 x + 42064\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 5^{3} \)
Twist minimal: no (minimal twist has level 75)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + 3 q^{3} + ( 4 + \beta_{2} ) q^{4} + 3 \beta_{1} q^{6} + ( -1 - 2 \beta_{1} + \beta_{3} + \beta_{5} + \beta_{8} ) q^{7} + ( 3 + 3 \beta_{1} + 3 \beta_{3} + \beta_{6} ) q^{8} + 9 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + 3 q^{3} + ( 4 + \beta_{2} ) q^{4} + 3 \beta_{1} q^{6} + ( -1 - 2 \beta_{1} + \beta_{3} + \beta_{5} + \beta_{8} ) q^{7} + ( 3 + 3 \beta_{1} + 3 \beta_{3} + \beta_{6} ) q^{8} + 9 q^{9} + ( -4 - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{13} ) q^{11} + ( 12 + 3 \beta_{2} ) q^{12} + ( -15 + \beta_{1} - 5 \beta_{3} - \beta_{4} + \beta_{6} + 2 \beta_{7} - 2 \beta_{12} + \beta_{13} ) q^{13} + ( -26 + 2 \beta_{1} - 2 \beta_{2} - 10 \beta_{3} + \beta_{4} - 3 \beta_{5} - 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + 3 \beta_{11} - \beta_{13} ) q^{14} + ( 5 - 3 \beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{11} - \beta_{13} ) q^{16} + ( -15 - \beta_{2} - 7 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} - 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} + \beta_{13} ) q^{17} + 9 \beta_{1} q^{18} + ( -14 - 7 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - 9 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + 4 \beta_{11} + \beta_{12} - 2 \beta_{13} ) q^{19} + ( -3 - 6 \beta_{1} + 3 \beta_{3} + 3 \beta_{5} + 3 \beta_{8} ) q^{21} + ( -15 - \beta_{1} - 5 \beta_{2} + 15 \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 3 \beta_{9} + \beta_{10} - 4 \beta_{11} + 2 \beta_{12} + 4 \beta_{13} ) q^{22} + ( -12 - 9 \beta_{1} - \beta_{3} - 3 \beta_{4} + 6 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} + 4 \beta_{9} + \beta_{10} - \beta_{11} + 3 \beta_{12} + \beta_{13} ) q^{23} + ( 9 + 9 \beta_{1} + 9 \beta_{3} + 3 \beta_{6} ) q^{24} + ( 5 - 39 \beta_{1} + 3 \beta_{2} - 6 \beta_{3} - 3 \beta_{4} + 16 \beta_{5} - \beta_{6} - 5 \beta_{7} - 7 \beta_{8} + 7 \beta_{9} - 3 \beta_{10} - 4 \beta_{11} + 6 \beta_{12} + 4 \beta_{13} ) q^{26} + 27 q^{27} + ( 27 - 40 \beta_{1} - 4 \beta_{2} + 25 \beta_{3} - \beta_{4} + 15 \beta_{5} - 3 \beta_{6} - 6 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} - \beta_{10} - 3 \beta_{11} - \beta_{13} ) q^{28} + ( 35 - 4 \beta_{1} + 35 \beta_{3} - 2 \beta_{4} - 7 \beta_{5} - \beta_{6} - 4 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{12} - \beta_{13} ) q^{29} + ( -71 - 11 \beta_{1} - 9 \beta_{2} - 8 \beta_{3} - \beta_{4} - 5 \beta_{5} - 6 \beta_{7} - \beta_{8} - 5 \beta_{10} - 4 \beta_{11} + 6 \beta_{12} + 4 \beta_{13} ) q^{31} + ( -36 - 16 \beta_{1} - 2 \beta_{2} - 18 \beta_{3} - 4 \beta_{4} - 15 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} - 5 \beta_{10} + 3 \beta_{11} + \beta_{12} + 3 \beta_{13} ) q^{32} + ( -12 - 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} - 3 \beta_{8} - 6 \beta_{9} - 3 \beta_{13} ) q^{33} + ( -30 - 10 \beta_{1} - 6 \beta_{2} - 12 \beta_{3} + 11 \beta_{4} + 10 \beta_{5} - 3 \beta_{6} + \beta_{7} - 4 \beta_{8} - 7 \beta_{9} + 3 \beta_{10} - 6 \beta_{11} - \beta_{12} + 3 \beta_{13} ) q^{34} + ( 36 + 9 \beta_{2} ) q^{36} + ( -44 - 6 \beta_{1} + 4 \beta_{2} - 7 \beta_{3} - 13 \beta_{5} + 11 \beta_{6} - 8 \beta_{7} + \beta_{8} + 12 \beta_{9} - 5 \beta_{10} + 4 \beta_{11} + 6 \beta_{12} + 2 \beta_{13} ) q^{37} + ( -89 - 50 \beta_{1} - 19 \beta_{2} + 91 \beta_{3} - 4 \beta_{4} + 11 \beta_{5} + \beta_{6} - 2 \beta_{7} + 10 \beta_{8} - 2 \beta_{10} - 7 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} ) q^{38} + ( -45 + 3 \beta_{1} - 15 \beta_{3} - 3 \beta_{4} + 3 \beta_{6} + 6 \beta_{7} - 6 \beta_{12} + 3 \beta_{13} ) q^{39} + ( -23 + 16 \beta_{1} - 17 \beta_{2} + \beta_{3} - \beta_{4} - 17 \beta_{5} - 13 \beta_{6} + 4 \beta_{7} - \beta_{8} - \beta_{9} + 7 \beta_{10} - 6 \beta_{11} - \beta_{12} - 7 \beta_{13} ) q^{41} + ( -78 + 6 \beta_{1} - 6 \beta_{2} - 30 \beta_{3} + 3 \beta_{4} - 9 \beta_{5} - 6 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} - 3 \beta_{10} + 9 \beta_{11} - 3 \beta_{13} ) q^{42} + ( -52 + 14 \beta_{1} - 9 \beta_{2} + 8 \beta_{3} - 6 \beta_{4} + 7 \beta_{5} + \beta_{6} - 6 \beta_{7} - 6 \beta_{8} + 8 \beta_{9} + 6 \beta_{10} - 4 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} ) q^{43} + ( -24 - 14 \beta_{1} + \beta_{2} - 38 \beta_{3} + 5 \beta_{4} - 14 \beta_{5} - 9 \beta_{6} + 19 \beta_{7} - 13 \beta_{8} - 5 \beta_{9} + 12 \beta_{10} + \beta_{11} - 8 \beta_{12} + 2 \beta_{13} ) q^{44} + ( -68 - 9 \beta_{1} + 17 \beta_{2} - 91 \beta_{3} - 8 \beta_{4} - 10 \beta_{5} + \beta_{6} + 10 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 16 \beta_{11} - 5 \beta_{12} - 8 \beta_{13} ) q^{46} + ( 23 - 17 \beta_{1} + 25 \beta_{3} - \beta_{4} + 19 \beta_{5} - 9 \beta_{6} - 4 \beta_{7} - 4 \beta_{8} + 13 \beta_{9} - 3 \beta_{10} + 13 \beta_{11} + 3 \beta_{13} ) q^{47} + ( 15 - 9 \beta_{1} - 3 \beta_{2} - 6 \beta_{4} - 3 \beta_{5} - 6 \beta_{6} - 3 \beta_{8} + 6 \beta_{9} + 3 \beta_{10} - 3 \beta_{11} - 3 \beta_{13} ) q^{48} + ( 38 - 19 \beta_{1} + 10 \beta_{2} + 76 \beta_{3} + 12 \beta_{4} + 57 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} - 12 \beta_{8} - 8 \beta_{9} - 9 \beta_{11} + 3 \beta_{12} - 3 \beta_{13} ) q^{49} + ( -45 - 3 \beta_{2} - 21 \beta_{3} + 6 \beta_{4} + 6 \beta_{5} + 6 \beta_{7} - 6 \beta_{9} + 6 \beta_{10} - 6 \beta_{11} - 6 \beta_{12} + 3 \beta_{13} ) q^{51} + ( -238 + 53 \beta_{1} - 11 \beta_{2} - 175 \beta_{3} + \beta_{4} - 29 \beta_{5} - 3 \beta_{6} + 16 \beta_{7} - 2 \beta_{8} - 19 \beta_{9} + 8 \beta_{10} + 26 \beta_{11} - 2 \beta_{12} - 18 \beta_{13} ) q^{52} + ( -102 - 50 \beta_{1} - 16 \beta_{2} - 61 \beta_{3} + 3 \beta_{4} + 34 \beta_{5} + 13 \beta_{6} + 3 \beta_{7} + 5 \beta_{8} - 3 \beta_{9} - 5 \beta_{10} - 15 \beta_{11} - 10 \beta_{12} + 7 \beta_{13} ) q^{53} + 27 \beta_{1} q^{54} + ( -218 + 49 \beta_{1} - 24 \beta_{2} - 91 \beta_{3} - 3 \beta_{4} - 52 \beta_{5} + 11 \beta_{6} + 6 \beta_{7} + 16 \beta_{8} - 7 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 12 \beta_{12} + 4 \beta_{13} ) q^{56} + ( -42 - 21 \beta_{1} - 9 \beta_{2} + 6 \beta_{3} - 27 \beta_{5} - 6 \beta_{6} - 6 \beta_{7} - 3 \beta_{8} + 3 \beta_{9} - 3 \beta_{10} + 12 \beta_{11} + 3 \beta_{12} - 6 \beta_{13} ) q^{57} + ( -42 + 85 \beta_{1} - 13 \beta_{2} + 79 \beta_{3} + 3 \beta_{4} - 60 \beta_{5} + 2 \beta_{6} + 7 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} - 7 \beta_{12} - 8 \beta_{13} ) q^{58} + ( 37 - 39 \beta_{1} + 20 \beta_{2} + 55 \beta_{3} - 19 \beta_{4} + 14 \beta_{6} - 20 \beta_{7} - 14 \beta_{8} + 30 \beta_{9} - 11 \beta_{10} - 4 \beta_{11} - 2 \beta_{12} + 17 \beta_{13} ) q^{59} + ( -107 + 30 \beta_{1} + 28 \beta_{2} - 181 \beta_{3} - 7 \beta_{4} - 29 \beta_{5} + 2 \beta_{6} - 23 \beta_{7} + 5 \beta_{8} + 9 \beta_{9} - 5 \beta_{10} + 35 \beta_{11} + 6 \beta_{12} + 2 \beta_{13} ) q^{61} + ( -112 - 61 \beta_{1} - 20 \beta_{2} + 15 \beta_{3} + 5 \beta_{4} - 37 \beta_{5} - 18 \beta_{6} + 33 \beta_{7} - 5 \beta_{8} - 19 \beta_{9} + 11 \beta_{10} + 6 \beta_{11} - 20 \beta_{12} - 16 \beta_{13} ) q^{62} + ( -9 - 18 \beta_{1} + 9 \beta_{3} + 9 \beta_{5} + 9 \beta_{8} ) q^{63} + ( -214 - 44 \beta_{1} - 21 \beta_{2} + 141 \beta_{3} + 8 \beta_{4} + 31 \beta_{5} + 6 \beta_{6} + \beta_{7} + 6 \beta_{8} - 4 \beta_{9} - 12 \beta_{10} + 2 \beta_{11} - \beta_{13} ) q^{64} + ( -45 - 3 \beta_{1} - 15 \beta_{2} + 45 \beta_{3} + 3 \beta_{4} + 6 \beta_{5} + 3 \beta_{6} + 6 \beta_{7} - 6 \beta_{8} - 9 \beta_{9} + 3 \beta_{10} - 12 \beta_{11} + 6 \beta_{12} + 12 \beta_{13} ) q^{66} + ( -201 + 28 \beta_{1} - 2 \beta_{2} - 104 \beta_{3} + 2 \beta_{4} + 47 \beta_{5} - 18 \beta_{6} - \beta_{7} - 13 \beta_{8} - 2 \beta_{9} + 19 \beta_{10} + 15 \beta_{11} - \beta_{12} - 6 \beta_{13} ) q^{67} + ( -89 + 12 \beta_{1} - 33 \beta_{2} - 29 \beta_{3} + 29 \beta_{4} - 6 \beta_{5} - 17 \beta_{6} + 14 \beta_{7} - 35 \beta_{9} + 7 \beta_{10} + \beta_{11} - 4 \beta_{12} - 4 \beta_{13} ) q^{68} + ( -36 - 27 \beta_{1} - 3 \beta_{3} - 9 \beta_{4} + 18 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} - 6 \beta_{8} + 12 \beta_{9} + 3 \beta_{10} - 3 \beta_{11} + 9 \beta_{12} + 3 \beta_{13} ) q^{69} + ( 179 - 66 \beta_{1} + 40 \beta_{2} - 33 \beta_{3} + 19 \beta_{4} + \beta_{5} + 2 \beta_{6} + 4 \beta_{7} + 4 \beta_{8} - 7 \beta_{9} - 11 \beta_{10} + 2 \beta_{11} + 15 \beta_{12} - 3 \beta_{13} ) q^{71} + ( 27 + 27 \beta_{1} + 27 \beta_{3} + 9 \beta_{6} ) q^{72} + ( 36 - 22 \beta_{2} - 25 \beta_{3} + 25 \beta_{4} + 4 \beta_{5} - 6 \beta_{6} + 10 \beta_{7} - 14 \beta_{8} - \beta_{9} - 6 \beta_{10} - 48 \beta_{11} - 7 \beta_{12} - 6 \beta_{13} ) q^{73} + ( 45 - 99 \beta_{1} + 31 \beta_{2} + 155 \beta_{3} - 29 \beta_{4} + 28 \beta_{5} - 20 \beta_{6} + \beta_{7} + 20 \beta_{8} + 19 \beta_{9} + 9 \beta_{10} - 15 \beta_{12} - 29 \beta_{13} ) q^{74} + ( -506 - 77 \beta_{1} - 17 \beta_{2} - 194 \beta_{3} + 4 \beta_{4} - 98 \beta_{5} + 8 \beta_{7} + 7 \beta_{8} + 4 \beta_{9} - \beta_{10} + 6 \beta_{11} - 17 \beta_{12} - \beta_{13} ) q^{76} + ( -392 - 22 \beta_{1} + 6 \beta_{2} - 54 \beta_{3} - 29 \beta_{4} - 23 \beta_{5} + 3 \beta_{6} - 6 \beta_{7} + 17 \beta_{8} + 18 \beta_{9} - 3 \beta_{10} + 19 \beta_{11} - 3 \beta_{12} + 9 \beta_{13} ) q^{77} + ( 15 - 117 \beta_{1} + 9 \beta_{2} - 18 \beta_{3} - 9 \beta_{4} + 48 \beta_{5} - 3 \beta_{6} - 15 \beta_{7} - 21 \beta_{8} + 21 \beta_{9} - 9 \beta_{10} - 12 \beta_{11} + 18 \beta_{12} + 12 \beta_{13} ) q^{78} + ( 7 - 28 \beta_{1} + 6 \beta_{2} + 122 \beta_{3} + 20 \beta_{4} - 10 \beta_{5} + 3 \beta_{6} + 15 \beta_{7} + 3 \beta_{8} - 15 \beta_{9} - 5 \beta_{10} - 40 \beta_{11} - 21 \beta_{12} - 2 \beta_{13} ) q^{79} + 81 q^{81} + ( 154 - 35 \beta_{1} - 28 \beta_{2} + 189 \beta_{3} + 13 \beta_{4} - 73 \beta_{5} + 30 \beta_{6} - 17 \beta_{7} + 15 \beta_{8} - 13 \beta_{9} - 17 \beta_{10} - 5 \beta_{11} + 4 \beta_{12} + 20 \beta_{13} ) q^{82} + ( -422 + 156 \beta_{1} - 37 \beta_{2} - 60 \beta_{3} - 7 \beta_{4} - 58 \beta_{5} - 3 \beta_{6} - 11 \beta_{7} + 23 \beta_{8} + 7 \beta_{9} - 7 \beta_{10} - 11 \beta_{11} + 12 \beta_{12} - 18 \beta_{13} ) q^{83} + ( 81 - 120 \beta_{1} - 12 \beta_{2} + 75 \beta_{3} - 3 \beta_{4} + 45 \beta_{5} - 9 \beta_{6} - 18 \beta_{7} + 9 \beta_{8} + 9 \beta_{9} - 3 \beta_{10} - 9 \beta_{11} - 3 \beta_{13} ) q^{84} + ( 240 - 71 \beta_{1} + 51 \beta_{2} - 131 \beta_{3} - 4 \beta_{4} - 38 \beta_{5} + \beta_{6} + 20 \beta_{7} + 3 \beta_{8} - 12 \beta_{9} + 9 \beta_{10} + 8 \beta_{11} - 20 \beta_{12} - 13 \beta_{13} ) q^{86} + ( 105 - 12 \beta_{1} + 105 \beta_{3} - 6 \beta_{4} - 21 \beta_{5} - 3 \beta_{6} - 12 \beta_{7} + 6 \beta_{8} - 6 \beta_{9} + 6 \beta_{12} - 3 \beta_{13} ) q^{87} + ( -210 - 75 \beta_{1} - 15 \beta_{2} + 45 \beta_{3} + 23 \beta_{4} + 119 \beta_{5} + 4 \beta_{6} - 5 \beta_{7} - 22 \beta_{8} - 17 \beta_{9} - 7 \beta_{10} - 14 \beta_{11} + 12 \beta_{12} + 11 \beta_{13} ) q^{88} + ( 16 - 91 \beta_{1} + 26 \beta_{2} + 196 \beta_{3} - \beta_{4} + 76 \beta_{5} + 18 \beta_{6} - 24 \beta_{7} - 13 \beta_{8} + 16 \beta_{9} - 12 \beta_{10} + 52 \beta_{11} + 30 \beta_{12} + 16 \beta_{13} ) q^{89} + ( -16 + 70 \beta_{1} - 55 \beta_{2} + 217 \beta_{3} + 15 \beta_{4} - 116 \beta_{5} + 23 \beta_{6} + 18 \beta_{7} + \beta_{8} - 47 \beta_{9} + 6 \beta_{10} + 10 \beta_{11} - 9 \beta_{12} - 22 \beta_{13} ) q^{91} + ( -41 - 87 \beta_{1} + 21 \beta_{2} + 107 \beta_{3} - 26 \beta_{4} + 157 \beta_{5} + 18 \beta_{6} - 27 \beta_{7} - 8 \beta_{8} + 20 \beta_{9} - 30 \beta_{10} - 10 \beta_{11} + 31 \beta_{12} + 21 \beta_{13} ) q^{92} + ( -213 - 33 \beta_{1} - 27 \beta_{2} - 24 \beta_{3} - 3 \beta_{4} - 15 \beta_{5} - 18 \beta_{7} - 3 \beta_{8} - 15 \beta_{10} - 12 \beta_{11} + 18 \beta_{12} + 12 \beta_{13} ) q^{93} + ( -176 - 60 \beta_{1} - 28 \beta_{2} - 275 \beta_{3} + 6 \beta_{4} + 52 \beta_{5} + 13 \beta_{6} - 8 \beta_{7} + 31 \beta_{8} - 14 \beta_{9} - 15 \beta_{10} + 30 \beta_{11} + 6 \beta_{12} + 9 \beta_{13} ) q^{94} + ( -108 - 48 \beta_{1} - 6 \beta_{2} - 54 \beta_{3} - 12 \beta_{4} - 45 \beta_{5} + 6 \beta_{6} - 12 \beta_{7} + 9 \beta_{8} + 6 \beta_{9} - 15 \beta_{10} + 9 \beta_{11} + 3 \beta_{12} + 9 \beta_{13} ) q^{96} + ( 151 + 233 \beta_{1} - 19 \beta_{2} + 356 \beta_{3} - 10 \beta_{4} + 2 \beta_{5} + \beta_{6} - 3 \beta_{7} - 8 \beta_{8} - 4 \beta_{9} + 11 \beta_{10} + 22 \beta_{11} - 16 \beta_{12} + 13 \beta_{13} ) q^{97} + ( -218 + 161 \beta_{1} + 26 \beta_{2} - 600 \beta_{3} + 11 \beta_{4} - 102 \beta_{5} + 6 \beta_{6} + 36 \beta_{7} - 7 \beta_{8} - 41 \beta_{9} + 26 \beta_{10} + 28 \beta_{11} - 13 \beta_{12} + 15 \beta_{13} ) q^{98} + ( -36 - 9 \beta_{3} + 9 \beta_{4} - 9 \beta_{5} - 9 \beta_{6} + 9 \beta_{7} - 9 \beta_{8} - 18 \beta_{9} - 9 \beta_{13} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 42 q^{3} + 50 q^{4} - 27 q^{7} + 21 q^{8} + 126 q^{9} + O(q^{10}) \) \( 14 q + 42 q^{3} + 50 q^{4} - 27 q^{7} + 21 q^{8} + 126 q^{9} - 33 q^{11} + 150 q^{12} - 188 q^{13} - 287 q^{14} + 82 q^{16} - 146 q^{17} - 184 q^{19} - 81 q^{21} - 277 q^{22} - 164 q^{23} + 63 q^{24} + 103 q^{26} + 378 q^{27} + 224 q^{28} + 252 q^{29} - 889 q^{31} - 422 q^{32} - 99 q^{33} - 230 q^{34} + 450 q^{36} - 642 q^{37} - 1833 q^{38} - 564 q^{39} - 164 q^{41} - 861 q^{42} - 696 q^{43} - 6 q^{44} - 419 q^{46} + 92 q^{47} + 246 q^{48} + 81 q^{49} - 438 q^{51} - 1956 q^{52} - 949 q^{53} - 2380 q^{56} - 552 q^{57} - 1041 q^{58} - 81 q^{59} - 496 q^{61} - 1454 q^{62} - 243 q^{63} - 3903 q^{64} - 831 q^{66} - 1926 q^{67} - 685 q^{68} - 492 q^{69} + 2498 q^{71} + 189 q^{72} + 1026 q^{73} - 707 q^{74} - 5704 q^{76} - 5434 q^{77} + 309 q^{78} - 695 q^{79} + 1134 q^{81} + 886 q^{82} - 5315 q^{83} + 672 q^{84} + 3997 q^{86} + 756 q^{87} - 2969 q^{88} - 1424 q^{89} - 1194 q^{91} - 1607 q^{92} - 2667 q^{93} - 629 q^{94} - 1266 q^{96} - 291 q^{97} + 1018 q^{98} - 297 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14} - 81 x^{12} - 7 x^{11} + 2512 x^{10} + 517 x^{9} - 36970 x^{8} - 12987 x^{7} + 257291 x^{6} + 125779 x^{5} - 718713 x^{4} - 371750 x^{3} + 579848 x^{2} + 394896 x + 42064\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 12 \)
\(\beta_{3}\)\(=\)\((\)\(-154777191 \nu^{13} + 214678293 \nu^{12} + 12768694032 \nu^{11} - 14915595999 \nu^{10} - 404069408515 \nu^{9} + 370272399598 \nu^{8} + 6092827974316 \nu^{7} - 3872467992151 \nu^{6} - 43818200690808 \nu^{5} + 15486244836395 \nu^{4} + 128798752266598 \nu^{3} - 19532525709704 \nu^{2} - 107932745742176 \nu - 29392108806288\)\()/ 9573422384000 \)
\(\beta_{4}\)\(=\)\((\)\(-39842017881 \nu^{13} + 18793858963 \nu^{12} + 3381917381312 \nu^{11} + 617153824991 \nu^{10} - 107039813741365 \nu^{9} - 86380604207582 \nu^{8} + 1577495446704356 \nu^{7} + 2255658862839559 \nu^{6} - 11311893366723728 \nu^{5} - 22364379485677555 \nu^{4} + 41656166382191418 \nu^{3} + 80521422793290536 \nu^{2} - 85954039584517216 \nu - 61602820198807408\)\()/ 2039138967792000 \)
\(\beta_{5}\)\(=\)\((\)\(-214678293 \nu^{13} - 231741561 \nu^{12} + 15999036336 \nu^{11} + 15269104723 \nu^{10} - 450292207345 \nu^{9} - 370715223046 \nu^{8} + 5882559371668 \nu^{7} + 3995422441227 \nu^{6} - 34953965143184 \nu^{5} - 17558372991415 \nu^{4} + 77070946463954 \nu^{3} + 18185501095208 \nu^{2} - 31728784810848 \nu - 6510547762224\)\()/ 9573422384000 \)
\(\beta_{6}\)\(=\)\((\)\(464331573 \nu^{13} - 644034879 \nu^{12} - 38306082096 \nu^{11} + 44746787997 \nu^{10} + 1212208225545 \nu^{9} - 1110817198794 \nu^{8} - 18278483922948 \nu^{7} + 11617403976453 \nu^{6} + 131454602072424 \nu^{5} - 46458734509185 \nu^{4} - 376822834415794 \nu^{3} + 58597577129112 \nu^{2} + 141903211930528 \nu + 59456059266864\)\()/ 9573422384000 \)
\(\beta_{7}\)\(=\)\((\)\(52774069869 \nu^{13} - 146787187287 \nu^{12} - 4038749648088 \nu^{11} + 10590731904141 \nu^{10} + 114153083031385 \nu^{9} - 284994761799082 \nu^{8} - 1435508388868644 \nu^{7} + 3540851664674509 \nu^{6} + 7351605029930072 \nu^{5} - 20698866949726305 \nu^{4} - 7399996906970082 \nu^{3} + 52601762509172536 \nu^{2} - 16843498300706016 \nu - 24134699084790608\)\()/ 679712989264000 \)
\(\beta_{8}\)\(=\)\((\)\(198268707541 \nu^{13} - 330612632143 \nu^{12} - 15446080755832 \nu^{11} + 24130842178949 \nu^{10} + 453486628821265 \nu^{9} - 626111554905898 \nu^{8} - 6126575299424516 \nu^{7} + 6864964367337301 \nu^{6} + 36372690966037208 \nu^{5} - 30340650774791145 \nu^{4} - 67233637278192898 \nu^{3} + 60972601507328504 \nu^{2} + 9290772958676576 \nu - 30074258777775312\)\()/ 2039138967792000 \)
\(\beta_{9}\)\(=\)\((\)\(-24723383429 \nu^{13} - 19698303673 \nu^{12} + 1994905817228 \nu^{11} + 1654834981079 \nu^{10} - 61648333032485 \nu^{9} - 54212075098438 \nu^{8} + 907965942281524 \nu^{7} + 849773703313171 \nu^{6} - 6436658577450832 \nu^{5} - 6141712440159795 \nu^{4} + 19640680670547962 \nu^{3} + 15485072554819544 \nu^{2} - 21750605688750304 \nu - 9141952716611952\)\()/ 203913896779200 \)
\(\beta_{10}\)\(=\)\((\)\(-8146109293 \nu^{13} + 6363639039 \nu^{12} + 663666222536 \nu^{11} - 440450615677 \nu^{10} - 20765197179345 \nu^{9} + 10887849198954 \nu^{8} + 310790823668868 \nu^{7} - 117002587471373 \nu^{6} - 2247892398399984 \nu^{5} + 586318929748585 \nu^{4} + 6973604604781954 \nu^{3} - 1764109520445592 \nu^{2} - 7322318299380448 \nu - 212680826375024\)\()/ 53661551784000 \)
\(\beta_{11}\)\(=\)\((\)\(-2089067853 \nu^{13} + 1186234119 \nu^{12} + 166990685056 \nu^{11} - 90007487317 \nu^{10} - 5108559447745 \nu^{9} + 2389171674634 \nu^{8} + 74321331141828 \nu^{7} - 26188788364733 \nu^{6} - 516374760265864 \nu^{5} + 108613804503785 \nu^{4} + 1483963872871634 \nu^{3} - 151211736871832 \nu^{2} - 1216927897475808 \nu - 228794009350704\)\()/ 9573422384000 \)
\(\beta_{12}\)\(=\)\((\)\(268898217 \nu^{13} - 60162011 \nu^{12} - 21662421664 \nu^{11} + 3229526593 \nu^{10} + 662126520605 \nu^{9} - 31650082226 \nu^{8} - 9449742817332 \nu^{7} - 684436038823 \nu^{6} + 61605242598856 \nu^{5} + 10543225997835 \nu^{4} - 145774591408826 \nu^{3} - 13288458871592 \nu^{2} + 70339095459552 \nu + 16216865165616\)\()/ 957342238400 \)
\(\beta_{13}\)\(=\)\((\)\(-629711814493 \nu^{13} + 211929067839 \nu^{12} + 51141259110536 \nu^{11} - 12148429797277 \nu^{10} - 1593816456351345 \nu^{9} + 171646433410554 \nu^{8} + 23644160022864468 \nu^{7} + 951266497072627 \nu^{6} - 166468628891531184 \nu^{5} - 27127244705617415 \nu^{4} + 469758003849131554 \nu^{3} + 70922058848184008 \nu^{2} - 351247214799828448 \nu - 108688968627954224\)\()/ 2039138967792000 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 12\)
\(\nu^{3}\)\(=\)\(\beta_{6} + 3 \beta_{3} + 19 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(-\beta_{13} - \beta_{11} + \beta_{10} + 2 \beta_{9} - \beta_{8} - 2 \beta_{6} - \beta_{5} - 2 \beta_{4} + 23 \beta_{2} - 3 \beta_{1} + 229\)
\(\nu^{5}\)\(=\)\(3 \beta_{13} + \beta_{12} + 3 \beta_{11} - 5 \beta_{10} + 2 \beta_{9} + 3 \beta_{8} - 4 \beta_{7} + 34 \beta_{6} - 15 \beta_{5} - 4 \beta_{4} + 78 \beta_{3} - 2 \beta_{2} + 400 \beta_{1} + 60\)
\(\nu^{6}\)\(=\)\(-41 \beta_{13} - 38 \beta_{11} + 28 \beta_{10} + 76 \beta_{9} - 34 \beta_{8} + \beta_{7} - 74 \beta_{6} - 9 \beta_{5} - 72 \beta_{4} + 141 \beta_{3} + 515 \beta_{2} - 164 \beta_{1} + 4850\)
\(\nu^{7}\)\(=\)\(122 \beta_{13} + 49 \beta_{12} + 146 \beta_{11} - 202 \beta_{10} + 106 \beta_{9} + 128 \beta_{8} - 180 \beta_{7} + 931 \beta_{6} - 724 \beta_{5} - 174 \beta_{4} + 1565 \beta_{3} - 104 \beta_{2} + 8783 \beta_{1} + 636\)
\(\nu^{8}\)\(=\)\(-1257 \beta_{13} - 9 \beta_{12} - 1152 \beta_{11} + 672 \beta_{10} + 2204 \beta_{9} - 872 \beta_{8} + 23 \beta_{7} - 2125 \beta_{6} + 421 \beta_{5} - 2046 \beta_{4} + 6944 \beta_{3} + 11581 \beta_{2} - 6028 \beta_{1} + 107016\)
\(\nu^{9}\)\(=\)\(3623 \beta_{13} + 1534 \beta_{12} + 5048 \beta_{11} - 6120 \beta_{10} + 3642 \beta_{9} + 4046 \beta_{8} - 5813 \beta_{7} + 23756 \beta_{6} - 24641 \beta_{5} - 5368 \beta_{4} + 27815 \beta_{3} - 3748 \beta_{2} + 197465 \beta_{1} - 5982\)
\(\nu^{10}\)\(=\)\(-34186 \beta_{13} - 247 \beta_{12} - 32506 \beta_{11} + 15550 \beta_{10} + 58426 \beta_{9} - 20316 \beta_{8} - 212 \beta_{7} - 56078 \beta_{6} + 26260 \beta_{5} - 53402 \beta_{4} + 239954 \beta_{3} + 262069 \beta_{2} - 188740 \beta_{1} + 2413393\)
\(\nu^{11}\)\(=\)\(95312 \beta_{13} + 39355 \beta_{12} + 152581 \beta_{11} - 166261 \beta_{10} + 104498 \beta_{9} + 114853 \beta_{8} - 163757 \beta_{7} + 587178 \beta_{6} - 732762 \beta_{5} - 144636 \beta_{4} + 438203 \beta_{3} - 118030 \beta_{2} + 4511346 \beta_{1} - 620070\)
\(\nu^{12}\)\(=\)\(-873549 \beta_{13} + 1297 \beta_{12} - 886792 \beta_{11} + 355354 \beta_{10} + 1487098 \beta_{9} - 455066 \beta_{8} - 37205 \beta_{7} - 1425812 \beta_{6} + 998693 \beta_{5} - 1336102 \beta_{4} + 7225185 \beta_{3} + 5964281 \beta_{2} - 5445306 \beta_{1} + 55193599\)
\(\nu^{13}\)\(=\)\(2359274 \beta_{13} + 891798 \beta_{12} + 4304259 \beta_{11} - 4273639 \beta_{10} + 2722752 \beta_{9} + 3092949 \beta_{8} - 4288261 \beta_{7} + 14271698 \beta_{6} - 20388722 \beta_{5} - 3635478 \beta_{4} + 5475971 \beta_{3} - 3467652 \beta_{2} + 104295128 \beta_{1} - 25133913\)

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
−4.95189
−4.79301
−4.01755
−2.93781
−2.33957
−0.955230
−0.574607
−0.134967
1.33976
1.73740
3.90684
4.24928
4.65416
4.81720
−4.95189 3.00000 16.5212 0 −14.8557 28.2766 −42.1962 9.00000 0
1.2 −4.79301 3.00000 14.9730 0 −14.3790 0.140520 −33.4216 9.00000 0
1.3 −4.01755 3.00000 8.14069 0 −12.0526 1.75849 −0.565245 9.00000 0
1.4 −2.93781 3.00000 0.630743 0 −8.81344 18.9115 21.6495 9.00000 0
1.5 −2.33957 3.00000 −2.52642 0 −7.01870 −32.9322 24.6273 9.00000 0
1.6 −0.955230 3.00000 −7.08754 0 −2.86569 12.4836 14.4121 9.00000 0
1.7 −0.574607 3.00000 −7.66983 0 −1.72382 −2.67744 9.00399 9.00000 0
1.8 −0.134967 3.00000 −7.98178 0 −0.404902 −17.3099 2.15702 9.00000 0
1.9 1.33976 3.00000 −6.20504 0 4.01928 −12.2101 −19.0313 9.00000 0
1.10 1.73740 3.00000 −4.98145 0 5.21219 7.66213 −22.5539 9.00000 0
1.11 3.90684 3.00000 7.26339 0 11.7205 22.0918 −2.87782 9.00000 0
1.12 4.24928 3.00000 10.0564 0 12.7478 −28.2853 8.73812 9.00000 0
1.13 4.65416 3.00000 13.6612 0 13.9625 −26.0445 26.3483 9.00000 0
1.14 4.81720 3.00000 15.2054 0 14.4516 1.13492 34.7100 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1875.4.a.g 14
5.b even 2 1 1875.4.a.f 14
25.d even 5 2 75.4.g.b 28
75.j odd 10 2 225.4.h.a 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.4.g.b 28 25.d even 5 2
225.4.h.a 28 75.j odd 10 2
1875.4.a.f 14 5.b even 2 1
1875.4.a.g 14 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{14} - \cdots\) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1875))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 42064 + 394896 T + 579848 T^{2} - 371750 T^{3} - 718713 T^{4} + 125779 T^{5} + 257291 T^{6} - 12987 T^{7} - 36970 T^{8} + 517 T^{9} + 2512 T^{10} - 7 T^{11} - 81 T^{12} + T^{14} \)
$3$ \( ( -3 + T )^{14} \)
$5$ \( T^{14} \)
$7$ \( 4350562367600 - 36079490460000 T + 36651152619480 T^{2} - 663734760500 T^{3} - 6596379692525 T^{4} + 586108666845 T^{5} + 102232594276 T^{6} - 7398605073 T^{7} - 597524497 T^{8} + 30372004 T^{9} + 1621610 T^{10} - 49146 T^{11} - 2077 T^{12} + 27 T^{13} + T^{14} \)
$11$ \( 5764077430622613296 + 1507677062132541840 T - 855681654260066124 T^{2} - 242733983656176984 T^{3} - 9835909487812913 T^{4} + 1066884930962511 T^{5} + 48959766721035 T^{6} - 1499447927724 T^{7} - 65299078964 T^{8} + 956751342 T^{9} + 37613893 T^{10} - 287805 T^{11} - 9962 T^{12} + 33 T^{13} + T^{14} \)
$13$ \( -\)\(21\!\cdots\!44\)\( - \)\(42\!\cdots\!60\)\( T - 47032161251868398799 T^{2} + 25646783046552680586 T^{3} + 1085862714403958147 T^{4} - 21077930977537104 T^{5} - 1696532315540885 T^{6} - 8673418046574 T^{7} + 824144814991 T^{8} + 12120157862 T^{9} - 102059102 T^{10} - 2825270 T^{11} - 4812 T^{12} + 188 T^{13} + T^{14} \)
$17$ \( \)\(29\!\cdots\!76\)\( + \)\(49\!\cdots\!26\)\( T - \)\(23\!\cdots\!19\)\( T^{2} - 32816113701709161254 T^{3} + 610038988052245576 T^{4} + 70982228024492948 T^{5} - 610333855436152 T^{6} - 62821236487212 T^{7} + 101517499983 T^{8} + 24024114548 T^{9} + 85408251 T^{10} - 3304874 T^{11} - 19019 T^{12} + 146 T^{13} + T^{14} \)
$19$ \( \)\(37\!\cdots\!00\)\( + \)\(31\!\cdots\!00\)\( T - \)\(14\!\cdots\!00\)\( T^{2} - \)\(16\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!00\)\( T^{4} + 2841198965954291950 T^{5} - 34726605217517945 T^{6} - 1002862023481670 T^{7} + 1337747491471 T^{8} + 141420598654 T^{9} + 362266825 T^{10} - 8638140 T^{11} - 38355 T^{12} + 184 T^{13} + T^{14} \)
$23$ \( \)\(33\!\cdots\!00\)\( + \)\(11\!\cdots\!00\)\( T - \)\(10\!\cdots\!20\)\( T^{2} - \)\(30\!\cdots\!60\)\( T^{3} + \)\(54\!\cdots\!40\)\( T^{4} + 18982846228407905300 T^{5} - 24221173103274669 T^{6} - 3695309631784384 T^{7} - 11277611988098 T^{8} + 309631197358 T^{9} + 1489359480 T^{10} - 11709918 T^{11} - 66263 T^{12} + 164 T^{13} + T^{14} \)
$29$ \( \)\(69\!\cdots\!00\)\( + \)\(14\!\cdots\!00\)\( T + \)\(93\!\cdots\!25\)\( T^{2} + \)\(10\!\cdots\!00\)\( T^{3} - \)\(68\!\cdots\!10\)\( T^{4} - 14227824879253520150 T^{5} + 175882989856245770 T^{6} + 4255425669002550 T^{7} - 21908974669911 T^{8} - 458873919812 T^{9} + 1757372715 T^{10} + 19647770 T^{11} - 75255 T^{12} - 252 T^{13} + T^{14} \)
$31$ \( -\)\(98\!\cdots\!00\)\( - \)\(36\!\cdots\!00\)\( T - \)\(12\!\cdots\!00\)\( T^{2} + \)\(71\!\cdots\!00\)\( T^{3} + \)\(55\!\cdots\!95\)\( T^{4} - \)\(43\!\cdots\!55\)\( T^{5} - 50841138013235448069 T^{6} + 63060464302951726 T^{7} + 1947041427227922 T^{8} + 2826632875268 T^{9} - 31605230405 T^{10} - 102564443 T^{11} + 121202 T^{12} + 889 T^{13} + T^{14} \)
$37$ \( -\)\(50\!\cdots\!00\)\( - \)\(17\!\cdots\!00\)\( T - \)\(10\!\cdots\!75\)\( T^{2} + \)\(22\!\cdots\!00\)\( T^{3} + \)\(30\!\cdots\!60\)\( T^{4} + \)\(59\!\cdots\!50\)\( T^{5} - \)\(79\!\cdots\!74\)\( T^{6} - 3617847087770206858 T^{7} + 5524289989739563 T^{8} + 52188079800694 T^{9} + 19913955355 T^{10} - 305458496 T^{11} - 351137 T^{12} + 642 T^{13} + T^{14} \)
$41$ \( \)\(73\!\cdots\!00\)\( + \)\(52\!\cdots\!50\)\( T - \)\(37\!\cdots\!95\)\( T^{2} + \)\(32\!\cdots\!80\)\( T^{3} + \)\(10\!\cdots\!15\)\( T^{4} - \)\(20\!\cdots\!20\)\( T^{5} + 93787510249608512791 T^{6} + 413959096240895356 T^{7} - 4089875850891158 T^{8} - 426567022962 T^{9} + 60232611955 T^{10} - 35417028 T^{11} - 402873 T^{12} + 164 T^{13} + T^{14} \)
$43$ \( -\)\(55\!\cdots\!16\)\( - \)\(37\!\cdots\!24\)\( T - \)\(13\!\cdots\!60\)\( T^{2} - \)\(20\!\cdots\!24\)\( T^{3} + \)\(40\!\cdots\!56\)\( T^{4} + \)\(30\!\cdots\!30\)\( T^{5} - 40743873922886994185 T^{6} - 396796010890961440 T^{7} + 1393430274212340 T^{8} + 17425316928200 T^{9} - 540658811 T^{10} - 212665434 T^{11} - 219410 T^{12} + 696 T^{13} + T^{14} \)
$47$ \( -\)\(35\!\cdots\!84\)\( - \)\(24\!\cdots\!32\)\( T + \)\(14\!\cdots\!40\)\( T^{2} + \)\(50\!\cdots\!12\)\( T^{3} - \)\(18\!\cdots\!76\)\( T^{4} - \)\(15\!\cdots\!70\)\( T^{5} + \)\(78\!\cdots\!35\)\( T^{6} + 308484145814548700 T^{7} - 15106377976408115 T^{8} - 4805344868220 T^{9} + 139953474234 T^{10} + 36737852 T^{11} - 610010 T^{12} - 92 T^{13} + T^{14} \)
$53$ \( -\)\(14\!\cdots\!75\)\( - \)\(45\!\cdots\!25\)\( T + \)\(52\!\cdots\!55\)\( T^{2} + \)\(13\!\cdots\!30\)\( T^{3} + \)\(18\!\cdots\!60\)\( T^{4} - \)\(70\!\cdots\!50\)\( T^{5} - \)\(34\!\cdots\!99\)\( T^{6} + 6516415660530495771 T^{7} + 83652929362473172 T^{8} + 172206630603353 T^{9} - 139006754380 T^{10} - 838353583 T^{11} - 515118 T^{12} + 949 T^{13} + T^{14} \)
$59$ \( -\)\(44\!\cdots\!00\)\( + \)\(52\!\cdots\!00\)\( T + \)\(25\!\cdots\!00\)\( T^{2} - \)\(18\!\cdots\!00\)\( T^{3} - \)\(54\!\cdots\!65\)\( T^{4} + \)\(26\!\cdots\!85\)\( T^{5} + \)\(56\!\cdots\!30\)\( T^{6} - 18955981845711729605 T^{7} - 319440000157563414 T^{8} + 70290908330611 T^{9} + 988116611030 T^{10} - 123683635 T^{11} - 1571520 T^{12} + 81 T^{13} + T^{14} \)
$61$ \( -\)\(10\!\cdots\!96\)\( + \)\(50\!\cdots\!66\)\( T + \)\(28\!\cdots\!45\)\( T^{2} - \)\(19\!\cdots\!84\)\( T^{3} - \)\(81\!\cdots\!69\)\( T^{4} + \)\(64\!\cdots\!00\)\( T^{5} + \)\(19\!\cdots\!55\)\( T^{6} - 75123341009989901440 T^{7} - 189058860776957830 T^{8} + 370797690312350 T^{9} + 832117192799 T^{10} - 746199964 T^{11} - 1566105 T^{12} + 496 T^{13} + T^{14} \)
$67$ \( -\)\(22\!\cdots\!44\)\( - \)\(36\!\cdots\!04\)\( T - \)\(38\!\cdots\!04\)\( T^{2} + \)\(19\!\cdots\!76\)\( T^{3} + \)\(86\!\cdots\!76\)\( T^{4} - \)\(13\!\cdots\!12\)\( T^{5} - \)\(12\!\cdots\!17\)\( T^{6} - 78732851925826520092 T^{7} + 623848358428442748 T^{8} + 914735750632898 T^{9} - 960346639119 T^{10} - 2416444354 T^{11} - 206854 T^{12} + 1926 T^{13} + T^{14} \)
$71$ \( \)\(16\!\cdots\!04\)\( - \)\(14\!\cdots\!08\)\( T - \)\(30\!\cdots\!84\)\( T^{2} + \)\(58\!\cdots\!48\)\( T^{3} - \)\(39\!\cdots\!76\)\( T^{4} + \)\(11\!\cdots\!56\)\( T^{5} - \)\(59\!\cdots\!37\)\( T^{6} - \)\(56\!\cdots\!76\)\( T^{7} + 1317932177744904797 T^{8} - 165607672396014 T^{9} - 2769213009754 T^{10} + 2865738358 T^{11} + 788384 T^{12} - 2498 T^{13} + T^{14} \)
$73$ \( \)\(33\!\cdots\!00\)\( + \)\(12\!\cdots\!00\)\( T - \)\(44\!\cdots\!95\)\( T^{2} + \)\(35\!\cdots\!10\)\( T^{3} + \)\(35\!\cdots\!05\)\( T^{4} - \)\(88\!\cdots\!50\)\( T^{5} + \)\(57\!\cdots\!86\)\( T^{6} + \)\(67\!\cdots\!46\)\( T^{7} - 624462397473435103 T^{8} - 1965405949272162 T^{9} + 1914818572830 T^{10} + 2391655842 T^{11} - 2338603 T^{12} - 1026 T^{13} + T^{14} \)
$79$ \( \)\(54\!\cdots\!00\)\( - \)\(38\!\cdots\!00\)\( T - \)\(24\!\cdots\!00\)\( T^{2} + \)\(25\!\cdots\!00\)\( T^{3} - \)\(37\!\cdots\!75\)\( T^{4} - \)\(17\!\cdots\!25\)\( T^{5} + \)\(48\!\cdots\!25\)\( T^{6} + \)\(34\!\cdots\!50\)\( T^{7} - 1904085718063679350 T^{8} + 125390838374850 T^{9} + 3386089411915 T^{10} - 988881545 T^{11} - 2858260 T^{12} + 695 T^{13} + T^{14} \)
$83$ \( \)\(23\!\cdots\!96\)\( + \)\(71\!\cdots\!12\)\( T + \)\(28\!\cdots\!48\)\( T^{2} - \)\(31\!\cdots\!00\)\( T^{3} - \)\(31\!\cdots\!67\)\( T^{4} - \)\(30\!\cdots\!87\)\( T^{5} + \)\(84\!\cdots\!31\)\( T^{6} + \)\(18\!\cdots\!14\)\( T^{7} + 3200839981290608140 T^{8} - 24485804351626516 T^{9} - 25435026056408 T^{10} - 3615562806 T^{11} + 8404291 T^{12} + 5315 T^{13} + T^{14} \)
$89$ \( -\)\(11\!\cdots\!00\)\( + \)\(30\!\cdots\!00\)\( T - \)\(14\!\cdots\!75\)\( T^{2} - \)\(87\!\cdots\!00\)\( T^{3} + \)\(80\!\cdots\!85\)\( T^{4} + \)\(60\!\cdots\!10\)\( T^{5} + \)\(97\!\cdots\!15\)\( T^{6} - \)\(14\!\cdots\!80\)\( T^{7} - 6827947173373727299 T^{8} + 15289936318450534 T^{9} + 9274914893730 T^{10} - 7588677530 T^{11} - 5079790 T^{12} + 1424 T^{13} + T^{14} \)
$97$ \( \)\(17\!\cdots\!71\)\( - \)\(80\!\cdots\!49\)\( T - \)\(71\!\cdots\!84\)\( T^{2} + \)\(39\!\cdots\!91\)\( T^{3} + \)\(11\!\cdots\!26\)\( T^{4} - \)\(83\!\cdots\!67\)\( T^{5} + \)\(99\!\cdots\!73\)\( T^{6} + \)\(10\!\cdots\!18\)\( T^{7} - 24106759266949406857 T^{8} - 3242530120073727 T^{9} + 20318494053146 T^{10} - 654354449 T^{11} - 7439184 T^{12} + 291 T^{13} + T^{14} \)
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