Properties

Label 20.4.e.b
Level 20
Weight 4
Character orbit 20.e
Analytic conductor 1.180
Analytic rank 0
Dimension 12
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 20.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.18003820011\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{5} ) q^{2} -\beta_{9} q^{3} + ( 2 \beta_{1} + \beta_{2} + \beta_{5} - \beta_{11} ) q^{4} + ( -1 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{5} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} ) q^{6} + ( -1 + \beta_{1} + \beta_{4} + \beta_{7} - 2 \beta_{8} + \beta_{10} + 2 \beta_{11} ) q^{7} + ( 1 + \beta_{1} + 4 \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{6} + 3 \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{11} ) q^{8} + ( -10 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{9} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{5} ) q^{2} -\beta_{9} q^{3} + ( 2 \beta_{1} + \beta_{2} + \beta_{5} - \beta_{11} ) q^{4} + ( -1 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{5} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} ) q^{6} + ( -1 + \beta_{1} + \beta_{4} + \beta_{7} - 2 \beta_{8} + \beta_{10} + 2 \beta_{11} ) q^{7} + ( 1 + \beta_{1} + 4 \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{6} + 3 \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{11} ) q^{8} + ( -10 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{9} + ( -10 - 4 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + \beta_{9} + 3 \beta_{10} - \beta_{11} ) q^{10} + ( -2 - 4 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} - \beta_{6} + \beta_{7} + 4 \beta_{8} + \beta_{9} - \beta_{10} ) q^{11} + ( -10 + 8 \beta_{1} + 5 \beta_{3} + 5 \beta_{4} + 2 \beta_{5} + 5 \beta_{6} + 5 \beta_{7} - 2 \beta_{8} - 2 \beta_{10} + 2 \beta_{11} ) q^{12} + ( 10 + 10 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + \beta_{4} + 4 \beta_{6} - \beta_{7} ) q^{13} + ( -12 \beta_{1} - 5 \beta_{2} - 6 \beta_{3} + 6 \beta_{4} - 5 \beta_{5} - 7 \beta_{6} - 7 \beta_{7} - 3 \beta_{9} - 3 \beta_{10} - \beta_{11} ) q^{14} + ( 7 + 5 \beta_{1} + 12 \beta_{2} + 4 \beta_{3} + 5 \beta_{4} - 4 \beta_{5} - 7 \beta_{7} - 2 \beta_{8} + \beta_{9} - 2 \beta_{10} - 6 \beta_{11} ) q^{15} + ( 26 + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{8} - 6 \beta_{9} + 6 \beta_{10} ) q^{16} + ( -27 + 29 \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{17} + ( 15 + 15 \beta_{1} - 7 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{6} + 3 \beta_{7} + 3 \beta_{8} - 6 \beta_{9} + 3 \beta_{11} ) q^{18} + ( -2 \beta_{1} - 12 \beta_{2} + 7 \beta_{3} - 7 \beta_{4} - 12 \beta_{5} + 5 \beta_{6} + 5 \beta_{7} + 2 \beta_{9} + 2 \beta_{10} - 4 \beta_{11} ) q^{19} + ( 9 - 33 \beta_{1} + 5 \beta_{2} + \beta_{3} - 7 \beta_{4} + 5 \beta_{5} - 9 \beta_{6} + 7 \beta_{7} - 3 \beta_{8} - 6 \beta_{9} - 8 \beta_{10} - 4 \beta_{11} ) q^{20} + ( 8 + 19 \beta_{1} + 19 \beta_{2} - 11 \beta_{3} - 11 \beta_{4} - 19 \beta_{5} - 8 \beta_{6} + 8 \beta_{7} ) q^{21} + ( 27 - 29 \beta_{1} + 13 \beta_{3} + \beta_{4} + 2 \beta_{5} + 13 \beta_{6} + \beta_{7} - 5 \beta_{8} + 6 \beta_{10} + 5 \beta_{11} ) q^{22} + ( 7 + 7 \beta_{1} + 8 \beta_{2} - 3 \beta_{3} + 4 \beta_{4} + 3 \beta_{6} - 4 \beta_{7} + 6 \beta_{8} + 7 \beta_{9} + 6 \beta_{11} ) q^{23} + ( 60 \beta_{1} - 12 \beta_{3} + 12 \beta_{4} - 12 \beta_{6} - 12 \beta_{7} + 8 \beta_{9} + 8 \beta_{10} + 4 \beta_{11} ) q^{24} + ( 5 - 35 \beta_{1} - 25 \beta_{2} + 15 \beta_{3} + 5 \beta_{4} + 25 \beta_{5} + 10 \beta_{6} - 20 \beta_{7} ) q^{25} + ( -10 + 7 \beta_{1} + 7 \beta_{2} + 8 \beta_{3} + 8 \beta_{4} - 7 \beta_{5} - 2 \beta_{8} + 8 \beta_{9} - 8 \beta_{10} ) q^{26} + ( -12 - 12 \beta_{1} - 12 \beta_{3} + 24 \beta_{5} - 12 \beta_{6} - 8 \beta_{10} ) q^{27} + ( -82 - 82 \beta_{1} - 22 \beta_{2} - 9 \beta_{3} - \beta_{4} + 9 \beta_{6} + \beta_{7} - 2 \beta_{8} + 6 \beta_{9} - 2 \beta_{11} ) q^{28} + ( -46 \beta_{1} + 20 \beta_{2} + \beta_{3} - \beta_{4} + 20 \beta_{5} + 19 \beta_{6} + 19 \beta_{7} ) q^{29} + ( -101 + 55 \beta_{1} + 9 \beta_{2} - 7 \beta_{3} - 15 \beta_{4} + 7 \beta_{5} - 10 \beta_{6} + 26 \beta_{7} + 11 \beta_{8} + 7 \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{30} + ( -22 - 16 \beta_{1} - 16 \beta_{2} - 5 \beta_{3} - 5 \beta_{4} + 16 \beta_{5} - 11 \beta_{6} + 11 \beta_{7} - 12 \beta_{8} - 9 \beta_{9} + 9 \beta_{10} ) q^{31} + ( -18 + 26 \beta_{1} + 10 \beta_{3} - 22 \beta_{4} - 8 \beta_{5} + 10 \beta_{6} - 22 \beta_{7} + 6 \beta_{8} - 4 \beta_{10} - 6 \beta_{11} ) q^{32} + ( -12 - 12 \beta_{1} - 22 \beta_{2} + 23 \beta_{3} + 11 \beta_{4} - 23 \beta_{6} - 11 \beta_{7} ) q^{33} + ( -37 \beta_{1} + 27 \beta_{2} + 27 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{34} + ( 20 + 30 \beta_{1} + 20 \beta_{2} + 5 \beta_{3} + 15 \beta_{4} - 20 \beta_{5} + 15 \beta_{6} - 5 \beta_{7} - 10 \beta_{9} + 15 \beta_{10} + 20 \beta_{11} ) q^{35} + ( 51 + 7 \beta_{1} + 7 \beta_{2} + 6 \beta_{3} + 6 \beta_{4} - 7 \beta_{5} + 24 \beta_{6} - 24 \beta_{7} - \beta_{8} + 6 \beta_{9} - 6 \beta_{10} ) q^{36} + ( 74 - 6 \beta_{1} - 34 \beta_{3} + 7 \beta_{4} - 68 \beta_{5} - 34 \beta_{6} + 7 \beta_{7} ) q^{37} + ( 146 + 146 \beta_{1} + 40 \beta_{2} + 8 \beta_{3} - 16 \beta_{4} - 8 \beta_{6} + 16 \beta_{7} - 14 \beta_{8} - 14 \beta_{11} ) q^{38} + ( 6 \beta_{1} - 32 \beta_{2} + 13 \beta_{3} - 13 \beta_{4} - 32 \beta_{5} + 19 \beta_{6} + 19 \beta_{7} - 17 \beta_{9} - 17 \beta_{10} + 12 \beta_{11} ) q^{39} + ( 131 - 31 \beta_{1} - 32 \beta_{2} - 37 \beta_{3} - 9 \beta_{4} - 36 \beta_{5} + 7 \beta_{6} + 11 \beta_{7} + 11 \beta_{8} + 12 \beta_{9} + 6 \beta_{10} + 13 \beta_{11} ) q^{40} + ( -86 - 29 \beta_{1} - 29 \beta_{2} + 18 \beta_{3} + 18 \beta_{4} + 29 \beta_{5} + 11 \beta_{6} - 11 \beta_{7} ) q^{41} + ( 95 - 109 \beta_{1} + 3 \beta_{3} + 3 \beta_{4} + 14 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} + 19 \beta_{8} - 6 \beta_{10} - 19 \beta_{11} ) q^{42} + ( 6 + 6 \beta_{1} + 32 \beta_{2} + 10 \beta_{3} + 16 \beta_{4} - 10 \beta_{6} - 16 \beta_{7} - 20 \beta_{8} - 17 \beta_{9} - 20 \beta_{11} ) q^{43} + ( 182 \beta_{1} - 26 \beta_{2} - 8 \beta_{3} + 8 \beta_{4} - 26 \beta_{5} - 18 \beta_{6} - 18 \beta_{7} - 14 \beta_{9} - 14 \beta_{10} - 4 \beta_{11} ) q^{44} + ( 111 + 122 \beta_{1} + 2 \beta_{2} - 30 \beta_{3} + 13 \beta_{4} - 19 \beta_{5} + 11 \beta_{6} + 15 \beta_{7} ) q^{45} + ( -143 - 19 \beta_{1} - 19 \beta_{2} + 25 \beta_{3} + 25 \beta_{4} + 19 \beta_{5} + 10 \beta_{6} - 10 \beta_{7} + \beta_{8} - 13 \beta_{9} + 13 \beta_{10} ) q^{46} + ( -7 - 33 \beta_{1} - 20 \beta_{3} - 13 \beta_{4} + 40 \beta_{5} - 20 \beta_{6} - 13 \beta_{7} + 26 \beta_{8} + 25 \beta_{10} - 26 \beta_{11} ) q^{47} + ( -216 - 216 \beta_{1} + 40 \beta_{2} + 20 \beta_{3} + 4 \beta_{4} - 20 \beta_{6} - 4 \beta_{7} - 8 \beta_{8} - 24 \beta_{9} - 8 \beta_{11} ) q^{48} + ( -6 \beta_{1} + 37 \beta_{2} + 39 \beta_{3} - 39 \beta_{4} + 37 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{49} + ( -125 + 150 \beta_{1} - 5 \beta_{3} - 5 \beta_{4} - 25 \beta_{5} + 15 \beta_{6} + 15 \beta_{7} - 25 \beta_{8} - 20 \beta_{9} - 10 \beta_{10} + 25 \beta_{11} ) q^{50} + ( 8 + 8 \beta_{1} + 8 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - 8 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} + 31 \beta_{9} - 31 \beta_{10} ) q^{51} + ( -85 + 83 \beta_{1} - 22 \beta_{3} + 18 \beta_{4} + 2 \beta_{5} - 22 \beta_{6} + 18 \beta_{7} - \beta_{8} + 28 \beta_{10} + \beta_{11} ) q^{52} + ( -32 - 32 \beta_{1} + 16 \beta_{2} - 41 \beta_{3} - 8 \beta_{4} + 41 \beta_{6} + 8 \beta_{7} ) q^{53} + ( -224 \beta_{1} - 32 \beta_{2} - 16 \beta_{3} + 16 \beta_{4} - 32 \beta_{5} + 8 \beta_{6} + 8 \beta_{7} + 8 \beta_{9} + 8 \beta_{10} + 32 \beta_{11} ) q^{54} + ( 2 - 10 \beta_{1} - 28 \beta_{2} - 6 \beta_{3} - 20 \beta_{4} - 4 \beta_{5} + 10 \beta_{6} + 8 \beta_{7} + 28 \beta_{8} + 41 \beta_{9} - 37 \beta_{10} + 4 \beta_{11} ) q^{55} + ( 92 - 56 \beta_{1} - 56 \beta_{2} + 20 \beta_{3} + 20 \beta_{4} + 56 \beta_{5} - 20 \beta_{6} + 20 \beta_{7} - 28 \beta_{8} + 16 \beta_{9} - 16 \beta_{10} ) q^{56} + ( -98 + 30 \beta_{1} + 34 \beta_{3} - 48 \beta_{4} + 68 \beta_{5} + 34 \beta_{6} - 48 \beta_{7} ) q^{57} + ( 100 + 100 \beta_{1} - 64 \beta_{2} + 18 \beta_{3} + 18 \beta_{4} - 18 \beta_{6} - 18 \beta_{7} + 20 \beta_{8} + 36 \beta_{9} + 20 \beta_{11} ) q^{58} + ( 2 \beta_{1} + 20 \beta_{2} - 11 \beta_{3} + 11 \beta_{4} + 20 \beta_{5} - 9 \beta_{6} - 9 \beta_{7} + 54 \beta_{9} + 54 \beta_{10} + 4 \beta_{11} ) q^{59} + ( 258 - 200 \beta_{1} + 68 \beta_{2} + 41 \beta_{3} - 15 \beta_{4} + 94 \beta_{5} + 5 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} + 4 \beta_{9} + 42 \beta_{10} + 6 \beta_{11} ) q^{60} + ( -26 + 51 \beta_{1} + 51 \beta_{2} - 22 \beta_{3} - 22 \beta_{4} - 51 \beta_{5} - 29 \beta_{6} + 29 \beta_{7} ) q^{61} + ( 233 - 183 \beta_{1} - 33 \beta_{3} - 21 \beta_{4} - 50 \beta_{5} - 33 \beta_{6} - 21 \beta_{7} - 7 \beta_{8} - 30 \beta_{10} + 7 \beta_{11} ) q^{62} + ( -17 - 17 \beta_{1} - 24 \beta_{2} + 5 \beta_{3} - 12 \beta_{4} - 5 \beta_{6} + 12 \beta_{7} - 10 \beta_{8} + 19 \beta_{9} - 10 \beta_{11} ) q^{63} + ( 164 \beta_{1} + 72 \beta_{2} + 16 \beta_{3} - 16 \beta_{4} + 72 \beta_{5} + 60 \beta_{6} + 60 \beta_{7} - 28 \beta_{9} - 28 \beta_{10} - 4 \beta_{11} ) q^{64} + ( -221 - 102 \beta_{1} + 43 \beta_{2} + 30 \beta_{3} - 8 \beta_{4} - \beta_{5} - 31 \beta_{6} + 35 \beta_{7} ) q^{65} + ( -110 + 22 \beta_{1} + 22 \beta_{2} - 46 \beta_{3} - 46 \beta_{4} - 22 \beta_{5} - 22 \beta_{8} - 46 \beta_{9} + 46 \beta_{10} ) q^{66} + ( 30 + 50 \beta_{1} + 40 \beta_{3} + 10 \beta_{4} - 80 \beta_{5} + 40 \beta_{6} + 10 \beta_{7} - 20 \beta_{8} - 43 \beta_{10} + 20 \beta_{11} ) q^{67} + ( -91 - 91 \beta_{1} - 54 \beta_{2} + 6 \beta_{3} - 10 \beta_{4} - 6 \beta_{6} + 10 \beta_{7} + 25 \beta_{8} - 4 \beta_{9} + 25 \beta_{11} ) q^{68} + ( 205 \beta_{1} - 85 \beta_{2} - 64 \beta_{3} + 64 \beta_{4} - 85 \beta_{5} - 21 \beta_{6} - 21 \beta_{7} ) q^{69} + ( -290 + 60 \beta_{1} - 55 \beta_{2} + 10 \beta_{3} + 30 \beta_{4} + 65 \beta_{5} + 15 \beta_{6} - 105 \beta_{7} + 30 \beta_{8} - 25 \beta_{9} - 25 \beta_{10} - 35 \beta_{11} ) q^{70} + ( 50 + 48 \beta_{1} + 48 \beta_{2} + 23 \beta_{3} + 23 \beta_{4} - 48 \beta_{5} + 25 \beta_{6} - 25 \beta_{7} + 4 \beta_{8} - 45 \beta_{9} + 45 \beta_{10} ) q^{71} + ( -263 + 267 \beta_{1} + 9 \beta_{3} + 37 \beta_{4} - 4 \beta_{5} + 9 \beta_{6} + 37 \beta_{7} + \beta_{8} - 26 \beta_{10} - \beta_{11} ) q^{72} + ( 95 + 95 \beta_{1} - 52 \beta_{2} - 4 \beta_{3} + 26 \beta_{4} + 4 \beta_{6} - 26 \beta_{7} ) q^{73} + ( -293 \beta_{1} - 47 \beta_{2} - 47 \beta_{5} - 14 \beta_{6} - 14 \beta_{7} + 14 \beta_{9} + 14 \beta_{10} - 68 \beta_{11} ) q^{74} + ( -70 - 90 \beta_{1} - 40 \beta_{2} - 20 \beta_{3} - 30 \beta_{4} + 80 \beta_{5} - 60 \beta_{6} + 10 \beta_{7} - 20 \beta_{8} - 85 \beta_{9} + 20 \beta_{10} - 60 \beta_{11} ) q^{75} + ( 248 + 172 \beta_{1} + 172 \beta_{2} - 44 \beta_{3} - 44 \beta_{4} - 172 \beta_{5} - 56 \beta_{6} + 56 \beta_{7} + 40 \beta_{8} + 12 \beta_{9} - 12 \beta_{10} ) q^{76} + ( 210 - 280 \beta_{1} + 35 \beta_{3} + 45 \beta_{4} + 70 \beta_{5} + 35 \beta_{6} + 45 \beta_{7} ) q^{77} + ( 321 + 321 \beta_{1} - 6 \beta_{2} - 57 \beta_{3} + 59 \beta_{4} + 57 \beta_{6} - 59 \beta_{7} - 15 \beta_{8} + 22 \beta_{9} - 15 \beta_{11} ) q^{78} + ( -8 \beta_{1} + 112 \beta_{2} - 52 \beta_{3} + 52 \beta_{4} + 112 \beta_{5} - 60 \beta_{6} - 60 \beta_{7} - 72 \beta_{9} - 72 \beta_{10} - 16 \beta_{11} ) q^{79} + ( 148 - 42 \beta_{1} - 48 \beta_{2} + 88 \beta_{3} + 72 \beta_{4} - 104 \beta_{5} + 14 \beta_{6} - 34 \beta_{7} - 44 \beta_{8} + 2 \beta_{9} - 14 \beta_{10} - 42 \beta_{11} ) q^{80} + ( 437 - 57 \beta_{1} - 57 \beta_{2} + 36 \beta_{3} + 36 \beta_{4} + 57 \beta_{5} + 21 \beta_{6} - 21 \beta_{7} ) q^{81} + ( -145 + 95 \beta_{1} - 7 \beta_{3} - 7 \beta_{4} + 50 \beta_{5} - 7 \beta_{6} - 7 \beta_{7} - 29 \beta_{8} + 14 \beta_{10} + 29 \beta_{11} ) q^{82} + ( -28 - 28 \beta_{1} - 136 \beta_{2} - 40 \beta_{3} - 68 \beta_{4} + 40 \beta_{6} + 68 \beta_{7} + 80 \beta_{8} - 33 \beta_{9} + 80 \beta_{11} ) q^{83} + ( 222 \beta_{1} - 58 \beta_{2} + 64 \beta_{3} - 64 \beta_{4} - 58 \beta_{5} + 38 \beta_{6} + 38 \beta_{7} + 50 \beta_{9} + 50 \beta_{10} + 20 \beta_{11} ) q^{84} + ( 96 - 3 \beta_{1} + 67 \beta_{2} + 5 \beta_{3} - 82 \beta_{4} - 19 \beta_{5} - 24 \beta_{6} - 15 \beta_{7} ) q^{85} + ( -175 + 11 \beta_{1} + 11 \beta_{2} - 77 \beta_{3} - 77 \beta_{4} - 11 \beta_{5} - 46 \beta_{6} + 46 \beta_{7} + 49 \beta_{8} + 37 \beta_{9} - 37 \beta_{10} ) q^{86} + ( 34 + 110 \beta_{1} + 72 \beta_{3} + 38 \beta_{4} - 144 \beta_{5} + 72 \beta_{6} + 38 \beta_{7} - 76 \beta_{8} + 68 \beta_{10} + 76 \beta_{11} ) q^{87} + ( -100 - 100 \beta_{1} + 176 \beta_{2} - 48 \beta_{3} - 8 \beta_{4} + 48 \beta_{6} + 8 \beta_{7} - 12 \beta_{8} + 16 \beta_{9} - 12 \beta_{11} ) q^{88} + ( -224 \beta_{1} - 148 \beta_{2} - 66 \beta_{3} + 66 \beta_{4} - 148 \beta_{5} - 82 \beta_{6} - 82 \beta_{7} ) q^{89} + ( 10 - \beta_{1} + 77 \beta_{2} + 41 \beta_{3} + 41 \beta_{4} - 94 \beta_{5} - 28 \beta_{6} - 28 \beta_{7} + 2 \beta_{8} + 69 \beta_{9} - 13 \beta_{10} - 19 \beta_{11} ) q^{90} + ( -4 - 16 \beta_{1} - 16 \beta_{2} - 14 \beta_{3} - 14 \beta_{4} + 16 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 24 \beta_{8} + 7 \beta_{9} - 7 \beta_{10} ) q^{91} + ( -214 + 60 \beta_{1} + \beta_{3} - 55 \beta_{4} + 154 \beta_{5} + \beta_{6} - 55 \beta_{7} - 6 \beta_{8} + 6 \beta_{10} + 6 \beta_{11} ) q^{92} + ( -196 - 196 \beta_{1} + 214 \beta_{2} + 89 \beta_{3} - 107 \beta_{4} - 89 \beta_{6} + 107 \beta_{7} ) q^{93} + ( -320 \beta_{1} + 63 \beta_{2} + 154 \beta_{3} - 154 \beta_{4} + 63 \beta_{5} + 53 \beta_{6} + 53 \beta_{7} + \beta_{9} + \beta_{10} + 15 \beta_{11} ) q^{94} + ( -96 - 30 \beta_{1} - 36 \beta_{2} - 7 \beta_{3} + 15 \beta_{4} + 52 \beta_{5} - 45 \beta_{6} + 51 \beta_{7} - 104 \beta_{8} + 82 \beta_{9} + 26 \beta_{10} + 28 \beta_{11} ) q^{95} + ( -96 - 240 \beta_{1} - 240 \beta_{2} - 80 \beta_{3} - 80 \beta_{4} + 240 \beta_{5} + 16 \beta_{6} - 16 \beta_{7} + 64 \beta_{8} ) q^{96} + ( -487 + 379 \beta_{1} + 54 \beta_{3} + 106 \beta_{4} + 108 \beta_{5} + 54 \beta_{6} + 106 \beta_{7} ) q^{97} + ( 185 + 185 \beta_{1} + 35 \beta_{2} - 41 \beta_{3} - 41 \beta_{4} + 41 \beta_{6} + 41 \beta_{7} + 37 \beta_{8} - 82 \beta_{9} + 37 \beta_{11} ) q^{98} + ( -14 \beta_{1} + 76 \beta_{2} - 31 \beta_{3} + 31 \beta_{4} + 76 \beta_{5} - 45 \beta_{6} - 45 \beta_{7} + 29 \beta_{9} + 29 \beta_{10} - 28 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 6q^{2} + 8q^{6} - 12q^{8} + O(q^{10}) \) \( 12q - 6q^{2} + 8q^{6} - 12q^{8} - 110q^{10} - 80q^{12} + 116q^{13} + 312q^{16} - 332q^{17} + 198q^{18} + 140q^{20} - 144q^{21} + 360q^{22} + 340q^{25} - 164q^{26} - 880q^{28} - 1240q^{30} - 376q^{32} + 80q^{33} + 460q^{36} + 508q^{37} + 1600q^{38} + 1420q^{40} - 656q^{41} + 1160q^{42} + 1180q^{45} - 1432q^{46} - 2720q^{48} - 1570q^{50} - 932q^{52} - 644q^{53} + 2048q^{56} - 960q^{57} + 1576q^{58} + 3280q^{60} - 896q^{61} + 2440q^{62} - 2740q^{65} - 1680q^{66} - 844q^{68} - 3040q^{70} - 3036q^{72} + 1436q^{73} + 800q^{76} + 3120q^{77} + 3720q^{78} + 1840q^{80} + 5988q^{81} - 1352q^{82} + 500q^{85} - 2552q^{86} - 2400q^{88} - 750q^{90} - 1840q^{92} - 3280q^{93} + 1088q^{96} - 4772q^{97} + 1698q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 7 x^{10} + 44 x^{8} - 156 x^{6} + 704 x^{4} - 1792 x^{2} + 4096\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{11} - 8 \nu^{9} + 27 \nu^{7} - 128 \nu^{5} + 412 \nu^{3} - 1504 \nu \)\()/640\)
\(\beta_{2}\)\(=\)\((\)\( 5 \nu^{11} + 32 \nu^{10} - 35 \nu^{9} - 256 \nu^{8} + 220 \nu^{7} + 864 \nu^{6} - 780 \nu^{5} - 4096 \nu^{4} + 3520 \nu^{3} + 13184 \nu^{2} - 8960 \nu - 48128 \)\()/5120\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{11} + 26 \nu^{10} + 23 \nu^{9} - 198 \nu^{8} - 92 \nu^{7} + 872 \nu^{6} + 668 \nu^{5} - 3608 \nu^{4} - 1152 \nu^{3} + 11072 \nu^{2} + 8704 \nu - 34304 \)\()/2560\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{11} + 26 \nu^{10} - 23 \nu^{9} - 198 \nu^{8} + 92 \nu^{7} + 872 \nu^{6} - 668 \nu^{5} - 3608 \nu^{4} + 1152 \nu^{3} + 11072 \nu^{2} - 8704 \nu - 34304 \)\()/2560\)
\(\beta_{5}\)\(=\)\((\)\( 13 \nu^{11} - 32 \nu^{10} - 99 \nu^{9} + 256 \nu^{8} + 436 \nu^{7} - 864 \nu^{6} - 1804 \nu^{5} + 4096 \nu^{4} + 6816 \nu^{3} - 13184 \nu^{2} - 20992 \nu + 48128 \)\()/5120\)
\(\beta_{6}\)\(=\)\((\)\( 5 \nu^{11} - 34 \nu^{10} - 55 \nu^{9} + 62 \nu^{8} + 200 \nu^{7} - 648 \nu^{6} - 1180 \nu^{5} + 1272 \nu^{4} + 2800 \nu^{3} - 8768 \nu^{2} - 9600 \nu - 1024 \)\()/2560\)
\(\beta_{7}\)\(=\)\((\)\( 5 \nu^{11} + 34 \nu^{10} - 55 \nu^{9} - 62 \nu^{8} + 200 \nu^{7} + 648 \nu^{6} - 1180 \nu^{5} - 1272 \nu^{4} + 2800 \nu^{3} + 8768 \nu^{2} - 9600 \nu + 1024 \)\()/2560\)
\(\beta_{8}\)\(=\)\((\)\( 13 \nu^{10} - 99 \nu^{8} + 436 \nu^{6} - 1804 \nu^{4} + 6816 \nu^{2} - 19072 \)\()/640\)
\(\beta_{9}\)\(=\)\((\)\( 45 \nu^{11} - 12 \nu^{10} - 115 \nu^{9} - 44 \nu^{8} + 900 \nu^{7} - 144 \nu^{6} - 1740 \nu^{5} - 2224 \nu^{4} + 10720 \nu^{3} + 256 \nu^{2} - 2560 \nu - 28672 \)\()/5120\)
\(\beta_{10}\)\(=\)\((\)\( 45 \nu^{11} + 12 \nu^{10} - 115 \nu^{9} + 44 \nu^{8} + 900 \nu^{7} + 144 \nu^{6} - 1740 \nu^{5} + 2224 \nu^{4} + 10720 \nu^{3} - 256 \nu^{2} - 2560 \nu + 28672 \)\()/5120\)
\(\beta_{11}\)\(=\)\((\)\( -3 \nu^{11} + 7 \nu^{9} - 50 \nu^{7} + 156 \nu^{5} - 824 \nu^{3} + 448 \nu \)\()/256\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} - 2 \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{8} - \beta_{4} - \beta_{3} + 3\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{11} - \beta_{10} - \beta_{9} + 3 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} - 2 \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{10} - 3 \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_{1} - 11\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(3 \beta_{11} + \beta_{10} + \beta_{9} - 5 \beta_{7} - 5 \beta_{6} + 5 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} + 5 \beta_{2} + 14 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(7 \beta_{10} - 7 \beta_{9} - \beta_{8} - 3 \beta_{7} + 3 \beta_{6} + 19 \beta_{5} + 15 \beta_{4} + 15 \beta_{3} - 19 \beta_{2} - 19 \beta_{1} - 61\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(25 \beta_{11} + 19 \beta_{10} + 19 \beta_{9} + 3 \beta_{7} + 3 \beta_{6} + 15 \beta_{5} + 13 \beta_{4} - 13 \beta_{3} + 15 \beta_{2} - 74 \beta_{1}\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(-35 \beta_{10} + 35 \beta_{9} + 9 \beta_{8} + 23 \beta_{7} - 23 \beta_{6} + 65 \beta_{5} + 9 \beta_{4} + 9 \beta_{3} - 65 \beta_{2} - 65 \beta_{1} - 339\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(-49 \beta_{11} - 3 \beta_{10} - 3 \beta_{9} - 47 \beta_{7} - 47 \beta_{6} + 33 \beta_{5} + 135 \beta_{4} - 135 \beta_{3} + 33 \beta_{2} - 358 \beta_{1}\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(-85 \beta_{10} + 85 \beta_{9} - 185 \beta_{8} + 137 \beta_{7} - 137 \beta_{6} - 281 \beta_{5} - 49 \beta_{4} - 49 \beta_{3} + 281 \beta_{2} + 281 \beta_{1} - 701\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(-271 \beta_{11} + 3 \beta_{10} + 3 \beta_{9} - 345 \beta_{7} - 345 \beta_{6} - 737 \beta_{5} + 417 \beta_{4} - 417 \beta_{3} - 737 \beta_{2} + 1526 \beta_{1}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(17\)
\(\chi(n)\) \(-1\) \(-\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
−1.13579 1.64620i
−1.83244 0.801352i
−1.76129 + 0.947553i
1.13579 1.64620i
1.83244 0.801352i
1.76129 + 0.947553i
−1.13579 + 1.64620i
−1.83244 + 0.801352i
−1.76129 0.947553i
1.13579 + 1.64620i
1.83244 + 0.801352i
1.76129 0.947553i
−2.78199 + 0.510409i 4.02923 4.02923i 7.47897 2.83991i 10.9349 + 2.32970i −9.15273 + 13.2658i −14.4440 14.4440i −19.3569 + 11.7179i 5.46937i −31.6100 0.899920i
3.2 −2.63379 1.03109i −5.55970 + 5.55970i 5.87372 + 5.43134i −10.4994 + 3.84216i 20.3756 8.91056i 1.14202 + 1.14202i −9.86997 20.3613i 34.8205i 31.6149 + 0.706375i
3.3 −0.813737 2.70884i 2.61822 2.61822i −6.67566 + 4.40857i −0.435501 11.1719i −9.22289 4.96181i 17.7783 + 17.7783i 17.3744 + 14.4959i 13.2899i −29.9084 + 10.2707i
3.4 −0.510409 + 2.78199i −4.02923 + 4.02923i −7.47897 2.83991i 10.9349 + 2.32970i −9.15273 13.2658i 14.4440 + 14.4440i 11.7179 19.3569i 5.46937i −12.0625 + 29.2318i
3.5 1.03109 + 2.63379i 5.55970 5.55970i −5.87372 + 5.43134i −10.4994 + 3.84216i 20.3756 + 8.91056i −1.14202 1.14202i −20.3613 9.86997i 34.8205i −20.9453 23.6917i
3.6 2.70884 + 0.813737i −2.61822 + 2.61822i 6.67566 + 4.40857i −0.435501 11.1719i −9.22289 + 4.96181i −17.7783 17.7783i 14.4959 + 17.3744i 13.2899i 7.91125 30.6172i
7.1 −2.78199 0.510409i 4.02923 + 4.02923i 7.47897 + 2.83991i 10.9349 2.32970i −9.15273 13.2658i −14.4440 + 14.4440i −19.3569 11.7179i 5.46937i −31.6100 + 0.899920i
7.2 −2.63379 + 1.03109i −5.55970 5.55970i 5.87372 5.43134i −10.4994 3.84216i 20.3756 + 8.91056i 1.14202 1.14202i −9.86997 + 20.3613i 34.8205i 31.6149 0.706375i
7.3 −0.813737 + 2.70884i 2.61822 + 2.61822i −6.67566 4.40857i −0.435501 + 11.1719i −9.22289 + 4.96181i 17.7783 17.7783i 17.3744 14.4959i 13.2899i −29.9084 10.2707i
7.4 −0.510409 2.78199i −4.02923 4.02923i −7.47897 + 2.83991i 10.9349 2.32970i −9.15273 + 13.2658i 14.4440 14.4440i 11.7179 + 19.3569i 5.46937i −12.0625 29.2318i
7.5 1.03109 2.63379i 5.55970 + 5.55970i −5.87372 5.43134i −10.4994 3.84216i 20.3756 8.91056i −1.14202 + 1.14202i −20.3613 + 9.86997i 34.8205i −20.9453 + 23.6917i
7.6 2.70884 0.813737i −2.61822 2.61822i 6.67566 4.40857i −0.435501 + 11.1719i −9.22289 4.96181i −17.7783 + 17.7783i 14.4959 17.3744i 13.2899i 7.91125 + 30.6172i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 20.4.e.b 12
3.b odd 2 1 180.4.k.e 12
4.b odd 2 1 inner 20.4.e.b 12
5.b even 2 1 100.4.e.e 12
5.c odd 4 1 inner 20.4.e.b 12
5.c odd 4 1 100.4.e.e 12
8.b even 2 1 320.4.n.k 12
8.d odd 2 1 320.4.n.k 12
12.b even 2 1 180.4.k.e 12
15.e even 4 1 180.4.k.e 12
20.d odd 2 1 100.4.e.e 12
20.e even 4 1 inner 20.4.e.b 12
20.e even 4 1 100.4.e.e 12
40.i odd 4 1 320.4.n.k 12
40.k even 4 1 320.4.n.k 12
60.l odd 4 1 180.4.k.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.4.e.b 12 1.a even 1 1 trivial
20.4.e.b 12 4.b odd 2 1 inner
20.4.e.b 12 5.c odd 4 1 inner
20.4.e.b 12 20.e even 4 1 inner
100.4.e.e 12 5.b even 2 1
100.4.e.e 12 5.c odd 4 1
100.4.e.e 12 20.d odd 2 1
100.4.e.e 12 20.e even 4 1
180.4.k.e 12 3.b odd 2 1
180.4.k.e 12 12.b even 2 1
180.4.k.e 12 15.e even 4 1
180.4.k.e 12 60.l odd 4 1
320.4.n.k 12 8.b even 2 1
320.4.n.k 12 8.d odd 2 1
320.4.n.k 12 40.i odd 4 1
320.4.n.k 12 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 5064 T_{3}^{8} + 4945680 T_{3}^{4} + 757350400 \) acting on \(S_{4}^{\mathrm{new}}(20, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 6 T + 18 T^{2} + 40 T^{3} - 256 T^{5} - 736 T^{6} - 2048 T^{7} + 20480 T^{9} + 73728 T^{10} + 196608 T^{11} + 262144 T^{12} \)
$3$ \( 1 - 2226 T^{4} + 2588175 T^{8} - 2136402140 T^{12} + 1375462310175 T^{16} - 628688148206706 T^{20} + 150094635296999121 T^{24} \)
$5$ \( ( 1 - 85 T^{2} - 400 T^{3} - 10625 T^{4} + 1953125 T^{6} )^{2} \)
$7$ \( 1 - 163746 T^{4} - 12219131425 T^{8} + 4249089576371140 T^{12} - \)\(16\!\cdots\!25\)\( T^{16} - \)\(31\!\cdots\!46\)\( T^{20} + \)\(26\!\cdots\!01\)\( T^{24} \)
$11$ \( ( 1 - 4986 T^{2} + 12380375 T^{4} - 19918383340 T^{6} + 21932589515375 T^{8} - 15648203886330906 T^{10} + 5559917313492231481 T^{12} )^{2} \)
$13$ \( ( 1 - 58 T + 1682 T^{2} - 72450 T^{3} + 7507335 T^{4} - 469328492 T^{5} + 17218216316 T^{6} - 1031114696924 T^{7} + 36236472144015 T^{8} - 768295979573850 T^{9} + 39187379176013042 T^{10} - 2968781794817263906 T^{11} + \)\(11\!\cdots\!29\)\( T^{12} )^{2} \)
$17$ \( ( 1 + 166 T + 13778 T^{2} + 1478470 T^{3} + 180009375 T^{4} + 13146146484 T^{5} + 795027918044 T^{6} + 64587017675892 T^{7} + 4344988709709375 T^{8} + 175328617764519590 T^{9} + 8027369184551647058 T^{10} + \)\(47\!\cdots\!38\)\( T^{11} + \)\(14\!\cdots\!09\)\( T^{12} )^{2} \)
$19$ \( ( 1 + 16994 T^{2} + 156289815 T^{4} + 1042328090300 T^{6} + 7352792038002015 T^{8} + 37613073734610340034 T^{10} + \)\(10\!\cdots\!41\)\( T^{12} )^{2} \)
$23$ \( 1 + 86914334 T^{4} + 54372416869500575 T^{8} + \)\(33\!\cdots\!60\)\( T^{12} + \)\(11\!\cdots\!75\)\( T^{16} + \)\(41\!\cdots\!94\)\( T^{20} + \)\(10\!\cdots\!61\)\( T^{24} \)
$29$ \( ( 1 - 80686 T^{2} + 2793486135 T^{4} - 69052346800420 T^{6} + 1661630699988154335 T^{8} - \)\(28\!\cdots\!26\)\( T^{10} + \)\(21\!\cdots\!61\)\( T^{12} )^{2} \)
$31$ \( ( 1 - 103106 T^{2} + 5864634815 T^{4} - 214293224466300 T^{6} + 5204884986033254015 T^{8} - \)\(81\!\cdots\!66\)\( T^{10} + \)\(69\!\cdots\!41\)\( T^{12} )^{2} \)
$37$ \( ( 1 - 254 T + 32258 T^{2} + 4711770 T^{3} - 385957065 T^{4} - 961296674116 T^{5} + 267719946494684 T^{6} - 48692560433997748 T^{7} - 990260234410629585 T^{8} + \)\(61\!\cdots\!90\)\( T^{9} + \)\(21\!\cdots\!98\)\( T^{10} - \)\(84\!\cdots\!22\)\( T^{11} + \)\(16\!\cdots\!29\)\( T^{12} )^{2} \)
$41$ \( ( 1 + 164 T + 188335 T^{2} + 20815080 T^{3} + 12980236535 T^{4} + 779017095524 T^{5} + 327381934393961 T^{6} )^{4} \)
$43$ \( 1 + 9995711534 T^{4} - 4601157531678236945 T^{8} - \)\(44\!\cdots\!80\)\( T^{12} - \)\(18\!\cdots\!45\)\( T^{16} + \)\(15\!\cdots\!34\)\( T^{20} + \)\(63\!\cdots\!01\)\( T^{24} \)
$47$ \( 1 - 14480422466 T^{4} + \)\(18\!\cdots\!35\)\( T^{8} - \)\(27\!\cdots\!80\)\( T^{12} + \)\(21\!\cdots\!35\)\( T^{16} - \)\(19\!\cdots\!46\)\( T^{20} + \)\(15\!\cdots\!21\)\( T^{24} \)
$53$ \( ( 1 + 322 T + 51842 T^{2} + 24976890 T^{3} + 1487618135 T^{4} + 291824210108 T^{5} + 328768813336156 T^{6} + 43445912928248716 T^{7} + 32972105566189474415 T^{8} + \)\(82\!\cdots\!70\)\( T^{9} + \)\(25\!\cdots\!22\)\( T^{10} + \)\(23\!\cdots\!54\)\( T^{11} + \)\(10\!\cdots\!89\)\( T^{12} )^{2} \)
$59$ \( ( 1 + 543794 T^{2} + 146689289095 T^{4} + 30962597079669980 T^{6} + \)\(61\!\cdots\!95\)\( T^{8} + \)\(96\!\cdots\!14\)\( T^{10} + \)\(75\!\cdots\!21\)\( T^{12} )^{2} \)
$61$ \( ( 1 + 224 T + 625475 T^{2} + 89988720 T^{3} + 141970940975 T^{4} + 11540563856864 T^{5} + 11694146092834141 T^{6} )^{4} \)
$67$ \( 1 + 152118288974 T^{4} - \)\(67\!\cdots\!65\)\( T^{8} - \)\(22\!\cdots\!80\)\( T^{12} - \)\(55\!\cdots\!65\)\( T^{16} + \)\(10\!\cdots\!54\)\( T^{20} + \)\(54\!\cdots\!81\)\( T^{24} \)
$71$ \( ( 1 - 1431506 T^{2} + 1032027741935 T^{4} - 460759845466007580 T^{6} + \)\(13\!\cdots\!35\)\( T^{8} - \)\(23\!\cdots\!46\)\( T^{10} + \)\(21\!\cdots\!61\)\( T^{12} )^{2} \)
$73$ \( ( 1 - 718 T + 257762 T^{2} - 319815230 T^{3} + 531457295055 T^{4} - 223706294377092 T^{5} + 74772514744761596 T^{6} - 87025551519693198564 T^{7} + \)\(80\!\cdots\!95\)\( T^{8} - \)\(18\!\cdots\!90\)\( T^{9} + \)\(59\!\cdots\!02\)\( T^{10} - \)\(63\!\cdots\!26\)\( T^{11} + \)\(34\!\cdots\!69\)\( T^{12} )^{2} \)
$79$ \( ( 1 + 574874 T^{2} + 335775077615 T^{4} + 287028695918457900 T^{6} + \)\(81\!\cdots\!15\)\( T^{8} + \)\(33\!\cdots\!34\)\( T^{10} + \)\(14\!\cdots\!61\)\( T^{12} )^{2} \)
$83$ \( 1 - 1040598079986 T^{4} + \)\(60\!\cdots\!75\)\( T^{8} - \)\(23\!\cdots\!40\)\( T^{12} + \)\(64\!\cdots\!75\)\( T^{16} - \)\(11\!\cdots\!06\)\( T^{20} + \)\(12\!\cdots\!81\)\( T^{24} \)
$89$ \( ( 1 - 2798646 T^{2} + 3790036251455 T^{4} - 3254227731213032180 T^{6} + \)\(18\!\cdots\!55\)\( T^{8} - \)\(69\!\cdots\!66\)\( T^{10} + \)\(12\!\cdots\!81\)\( T^{12} )^{2} \)
$97$ \( ( 1 + 2386 T + 2846498 T^{2} + 2507112050 T^{3} + 1418732317695 T^{4} + 602523284624124 T^{5} + 542007267886579004 T^{6} + \)\(54\!\cdots\!52\)\( T^{7} + \)\(11\!\cdots\!55\)\( T^{8} + \)\(19\!\cdots\!50\)\( T^{9} + \)\(19\!\cdots\!18\)\( T^{10} + \)\(15\!\cdots\!98\)\( T^{11} + \)\(57\!\cdots\!89\)\( T^{12} )^{2} \)
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