Properties

Label 17.4.d.a
Level 17
Weight 4
Character orbit 17.d
Analytic conductor 1.003
Analytic rank 0
Dimension 12
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 17.d (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.00303247010\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -\beta_{3} - \beta_{10} ) q^{2} + \beta_{8} q^{3} + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{9} ) q^{4} + ( -2 - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} - \beta_{11} ) q^{5} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{6} + ( -2 + \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{10} - \beta_{11} ) q^{7} + ( 2 + 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{7} - 3 \beta_{8} + 2 \beta_{9} + 3 \beta_{11} ) q^{8} + ( -4 + 2 \beta_{1} + 5 \beta_{2} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} - 4 \beta_{9} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{3} - \beta_{10} ) q^{2} + \beta_{8} q^{3} + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{9} ) q^{4} + ( -2 - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} - \beta_{11} ) q^{5} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{6} + ( -2 + \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{10} - \beta_{11} ) q^{7} + ( 2 + 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{7} - 3 \beta_{8} + 2 \beta_{9} + 3 \beta_{11} ) q^{8} + ( -4 + 2 \beta_{1} + 5 \beta_{2} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} - 4 \beta_{9} ) q^{9} + ( -10 - 4 \beta_{1} + 3 \beta_{2} + 10 \beta_{3} - 4 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + 3 \beta_{8} - 3 \beta_{9} + 4 \beta_{10} + 4 \beta_{11} ) q^{10} + ( 2 - 7 \beta_{1} - 8 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{6} - 5 \beta_{7} - \beta_{8} - 8 \beta_{9} + \beta_{10} - 7 \beta_{11} ) q^{11} + ( 5 + 4 \beta_{1} + 5 \beta_{2} - 19 \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{6} - 3 \beta_{7} + 19 \beta_{9} - 4 \beta_{10} - \beta_{11} ) q^{12} + ( 6 \beta_{1} - 11 \beta_{2} - 11 \beta_{3} + 3 \beta_{5} - 3 \beta_{6} + 4 \beta_{7} + 4 \beta_{8} + 2 \beta_{9} - 6 \beta_{10} - 6 \beta_{11} ) q^{13} + ( -7 - 4 \beta_{1} + 7 \beta_{2} + 11 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{7} + 6 \beta_{8} + 11 \beta_{9} - 4 \beta_{10} + 2 \beta_{11} ) q^{14} + ( 20 - 5 \beta_{1} + 12 \beta_{3} + 5 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} - 20 \beta_{9} + 2 \beta_{10} ) q^{15} + ( 21 + 9 \beta_{2} - 9 \beta_{3} - 5 \beta_{4} + 5 \beta_{5} + 5 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 8 \beta_{10} + 8 \beta_{11} ) q^{16} + ( 12 - 4 \beta_{1} - 6 \beta_{2} - 10 \beta_{3} + \beta_{4} - 8 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} + \beta_{8} - 31 \beta_{9} - 15 \beta_{10} + 8 \beta_{11} ) q^{17} + ( 7 + 16 \beta_{2} - 16 \beta_{3} + 13 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - \beta_{10} + \beta_{11} ) q^{18} + ( -14 + 19 \beta_{1} - 16 \beta_{3} - 19 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} - 3 \beta_{8} + 14 \beta_{9} + 14 \beta_{10} ) q^{19} + ( 38 + 8 \beta_{1} - 38 \beta_{2} + 29 \beta_{3} + 12 \beta_{4} - 6 \beta_{5} + 6 \beta_{7} - 9 \beta_{8} + 29 \beta_{9} + 8 \beta_{10} - 12 \beta_{11} ) q^{20} + ( -16 \beta_{1} + 40 \beta_{2} + 40 \beta_{3} + \beta_{5} - \beta_{6} - 5 \beta_{7} - 5 \beta_{8} + 10 \beta_{9} - 5 \beta_{10} - 5 \beta_{11} ) q^{21} + ( -65 - 7 \beta_{1} - 65 \beta_{2} - 13 \beta_{3} - 16 \beta_{4} - 4 \beta_{5} - \beta_{6} - 4 \beta_{7} + 13 \beta_{9} + 7 \beta_{10} - 16 \beta_{11} ) q^{22} + ( -20 + 9 \beta_{1} + 24 \beta_{2} - 20 \beta_{3} + 24 \beta_{4} + 24 \beta_{9} + 24 \beta_{10} + 9 \beta_{11} ) q^{23} + ( -21 - 4 \beta_{1} + 47 \beta_{2} + 21 \beta_{3} - 13 \beta_{4} + 5 \beta_{5} + \beta_{6} - \beta_{8} - 47 \beta_{9} + 13 \beta_{10} + 4 \beta_{11} ) q^{24} + ( -27 - 7 \beta_{1} - 47 \beta_{2} - 7 \beta_{4} + 10 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} + 10 \beta_{8} - 27 \beta_{9} + 22 \beta_{11} ) q^{25} + ( -57 + 4 \beta_{1} + 64 \beta_{2} + 4 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} - 57 \beta_{9} - 2 \beta_{11} ) q^{26} + ( -60 - 4 \beta_{1} - 26 \beta_{2} + 60 \beta_{3} - 18 \beta_{5} + 11 \beta_{6} - 11 \beta_{8} + 26 \beta_{9} + 4 \beta_{11} ) q^{27} + ( 35 - 2 \beta_{1} - 31 \beta_{2} + 35 \beta_{3} + 12 \beta_{4} + 3 \beta_{6} + 12 \beta_{7} + 3 \beta_{8} - 31 \beta_{9} + 12 \beta_{10} - 2 \beta_{11} ) q^{28} + ( 51 + 10 \beta_{1} + 51 \beta_{2} - 88 \beta_{3} - 3 \beta_{4} + 14 \beta_{5} - 9 \beta_{6} + 14 \beta_{7} + 88 \beta_{9} - 10 \beta_{10} - 3 \beta_{11} ) q^{29} + ( 20 \beta_{1} - 62 \beta_{2} - 62 \beta_{3} - 11 \beta_{5} + 11 \beta_{6} - 7 \beta_{7} - 7 \beta_{8} + 42 \beta_{9} - 22 \beta_{10} - 22 \beta_{11} ) q^{30} + ( 6 + 3 \beta_{1} - 6 \beta_{2} + 74 \beta_{3} - 4 \beta_{4} - 6 \beta_{5} + 6 \beta_{7} - 14 \beta_{8} + 74 \beta_{9} + 3 \beta_{10} + 4 \beta_{11} ) q^{31} + ( 72 - 6 \beta_{1} - 3 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} + 10 \beta_{6} - 10 \beta_{7} + 2 \beta_{8} - 72 \beta_{9} - 15 \beta_{10} ) q^{32} + ( 130 + 56 \beta_{2} - 56 \beta_{3} + 6 \beta_{4} - 11 \beta_{5} - 11 \beta_{6} - 16 \beta_{7} + 16 \beta_{8} - 12 \beta_{10} + 12 \beta_{11} ) q^{33} + ( 56 - 13 \beta_{1} - 45 \beta_{2} + 10 \beta_{3} - 18 \beta_{4} + 25 \beta_{5} - 21 \beta_{6} + 2 \beta_{7} - \beta_{8} - 122 \beta_{9} - 2 \beta_{10} - 8 \beta_{11} ) q^{34} + ( -44 + 48 \beta_{2} - 48 \beta_{3} - 12 \beta_{4} + 6 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 11 \beta_{10} + 11 \beta_{11} ) q^{35} + ( 45 - 5 \beta_{1} - 67 \beta_{3} + 5 \beta_{4} - 13 \beta_{5} - \beta_{6} + \beta_{7} + 13 \beta_{8} - 45 \beta_{9} - \beta_{10} ) q^{36} + ( 98 - 26 \beta_{1} - 98 \beta_{2} + 73 \beta_{3} + 15 \beta_{4} + 12 \beta_{5} - 12 \beta_{7} + 9 \beta_{8} + 73 \beta_{9} - 26 \beta_{10} - 15 \beta_{11} ) q^{37} + ( -10 \beta_{1} + 146 \beta_{2} + 146 \beta_{3} - 4 \beta_{5} + 4 \beta_{6} + 16 \beta_{7} + 16 \beta_{8} + 58 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{38} + ( -86 - 4 \beta_{1} - 86 \beta_{2} - 32 \beta_{3} + 14 \beta_{4} + 26 \beta_{5} - 4 \beta_{6} + 26 \beta_{7} + 32 \beta_{9} + 4 \beta_{10} + 14 \beta_{11} ) q^{39} + ( -139 + 6 \beta_{1} + 20 \beta_{2} - 139 \beta_{3} - 30 \beta_{4} + 11 \beta_{6} - 21 \beta_{7} + 11 \beta_{8} + 20 \beta_{9} - 30 \beta_{10} + 6 \beta_{11} ) q^{40} + ( -1 + 21 \beta_{1} + 44 \beta_{2} + \beta_{3} - 7 \beta_{4} - 12 \beta_{5} - 23 \beta_{6} + 23 \beta_{8} - 44 \beta_{9} + 7 \beta_{10} - 21 \beta_{11} ) q^{41} + ( 6 + 26 \beta_{1} - 120 \beta_{2} + 26 \beta_{4} - 8 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} - 8 \beta_{8} + 6 \beta_{9} - 4 \beta_{11} ) q^{42} + ( -136 + 6 \beta_{1} + 116 \beta_{2} + 6 \beta_{4} - 28 \beta_{5} - \beta_{6} - \beta_{7} - 28 \beta_{8} - 136 \beta_{9} - 18 \beta_{11} ) q^{43} + ( -115 - 7 \beta_{1} - 51 \beta_{2} + 115 \beta_{3} - 22 \beta_{4} + 33 \beta_{5} - 4 \beta_{6} + 4 \beta_{8} + 51 \beta_{9} + 22 \beta_{10} + 7 \beta_{11} ) q^{44} + ( -11 - 10 \beta_{1} - 54 \beta_{2} - 11 \beta_{3} - 13 \beta_{4} - 3 \beta_{6} + 9 \beta_{7} - 3 \beta_{8} - 54 \beta_{9} - 13 \beta_{10} - 10 \beta_{11} ) q^{45} + ( 63 - 20 \beta_{1} + 63 \beta_{2} - 187 \beta_{3} + 24 \beta_{4} - 15 \beta_{5} + 48 \beta_{6} - 15 \beta_{7} + 187 \beta_{9} + 20 \beta_{10} + 24 \beta_{11} ) q^{46} + ( -30 \beta_{1} + 6 \beta_{5} - 6 \beta_{6} - 22 \beta_{7} - 22 \beta_{8} - 112 \beta_{9} + 52 \beta_{10} + 52 \beta_{11} ) q^{47} + ( -51 + 5 \beta_{1} + 51 \beta_{2} + 149 \beta_{3} + 24 \beta_{4} + 3 \beta_{5} - 3 \beta_{7} - 19 \beta_{8} + 149 \beta_{9} + 5 \beta_{10} - 24 \beta_{11} ) q^{48} + ( 58 + 43 \beta_{1} + 125 \beta_{3} - 43 \beta_{4} - \beta_{5} - 5 \beta_{6} + 5 \beta_{7} + \beta_{8} - 58 \beta_{9} + 34 \beta_{10} ) q^{49} + ( 157 - 111 \beta_{2} + 111 \beta_{3} - 13 \beta_{4} - 17 \beta_{5} - 17 \beta_{6} + 52 \beta_{7} - 52 \beta_{8} + 46 \beta_{10} - 46 \beta_{11} ) q^{50} + ( 34 + 34 \beta_{1} - 34 \beta_{2} + 68 \beta_{3} + 34 \beta_{4} + 34 \beta_{6} + 17 \beta_{7} - 136 \beta_{9} + 51 \beta_{10} + 17 \beta_{11} ) q^{51} + ( 22 + 59 \beta_{2} - 59 \beta_{3} + 18 \beta_{4} - 30 \beta_{5} - 30 \beta_{6} + 25 \beta_{7} - 25 \beta_{8} + 20 \beta_{10} - 20 \beta_{11} ) q^{52} + ( 29 - 39 \beta_{1} - 236 \beta_{3} + 39 \beta_{4} + 13 \beta_{5} + 40 \beta_{6} - 40 \beta_{7} - 13 \beta_{8} - 29 \beta_{9} - 58 \beta_{10} ) q^{53} + ( -8 + 45 \beta_{1} + 8 \beta_{2} + 60 \beta_{3} - 73 \beta_{4} + 25 \beta_{5} - 25 \beta_{7} + 32 \beta_{8} + 60 \beta_{9} + 45 \beta_{10} + 73 \beta_{11} ) q^{54} + ( 66 \beta_{1} + 122 \beta_{2} + 122 \beta_{3} - 22 \beta_{5} + 22 \beta_{6} - 6 \beta_{7} - 6 \beta_{8} + 132 \beta_{9} + 15 \beta_{10} + 15 \beta_{11} ) q^{55} + ( -127 + 40 \beta_{1} - 127 \beta_{2} - 101 \beta_{3} + 22 \beta_{4} - 15 \beta_{5} - 18 \beta_{6} - 15 \beta_{7} + 101 \beta_{9} - 40 \beta_{10} + 22 \beta_{11} ) q^{56} + ( -56 - 71 \beta_{1} + 14 \beta_{2} - 56 \beta_{3} - 21 \beta_{4} - 6 \beta_{6} + 48 \beta_{7} - 6 \beta_{8} + 14 \beta_{9} - 21 \beta_{10} - 71 \beta_{11} ) q^{57} + ( 39 - 56 \beta_{1} + 172 \beta_{2} - 39 \beta_{3} + 84 \beta_{4} - 17 \beta_{5} + 10 \beta_{6} - 10 \beta_{8} - 172 \beta_{9} - 84 \beta_{10} + 56 \beta_{11} ) q^{58} + ( -94 - 54 \beta_{1} - 356 \beta_{2} - 54 \beta_{4} + 14 \beta_{5} + 15 \beta_{6} + 15 \beta_{7} + 14 \beta_{8} - 94 \beta_{9} - 110 \beta_{11} ) q^{59} + ( -112 - 54 \beta_{1} + 158 \beta_{2} - 54 \beta_{4} + 55 \beta_{5} - 33 \beta_{6} - 33 \beta_{7} + 55 \beta_{8} - 112 \beta_{9} + 56 \beta_{11} ) q^{60} + ( -54 + 68 \beta_{1} - 51 \beta_{2} + 54 \beta_{3} + 85 \beta_{4} - \beta_{5} - 29 \beta_{6} + 29 \beta_{8} + 51 \beta_{9} - 85 \beta_{10} - 68 \beta_{11} ) q^{61} + ( 13 + 78 \beta_{1} + 89 \beta_{2} + 13 \beta_{3} - 40 \beta_{4} + 19 \beta_{6} - 18 \beta_{7} + 19 \beta_{8} + 89 \beta_{9} - 40 \beta_{10} + 78 \beta_{11} ) q^{62} + ( 106 - 8 \beta_{1} + 106 \beta_{2} - 54 \beta_{3} - 35 \beta_{4} - 18 \beta_{5} - 42 \beta_{6} - 18 \beta_{7} + 54 \beta_{9} + 8 \beta_{10} - 35 \beta_{11} ) q^{63} + ( -79 \beta_{1} - 29 \beta_{2} - 29 \beta_{3} + 15 \beta_{5} - 15 \beta_{6} + 44 \beta_{7} + 44 \beta_{8} + 41 \beta_{9} + 8 \beta_{10} + 8 \beta_{11} ) q^{64} + ( 69 - 12 \beta_{1} - 69 \beta_{2} + 23 \beta_{3} - 36 \beta_{4} + 23 \beta_{5} - 23 \beta_{7} + 48 \beta_{8} + 23 \beta_{9} - 12 \beta_{10} + 36 \beta_{11} ) q^{65} + ( 124 - 39 \beta_{1} - 100 \beta_{3} + 39 \beta_{4} + 28 \beta_{5} - 27 \beta_{6} + 27 \beta_{7} - 28 \beta_{8} - 124 \beta_{9} - 44 \beta_{10} ) q^{66} + ( 144 + 128 \beta_{2} - 128 \beta_{3} + 50 \beta_{4} + 25 \beta_{5} + 25 \beta_{6} - 66 \beta_{7} + 66 \beta_{8} + 35 \beta_{10} - 35 \beta_{11} ) q^{67} + ( 175 + 38 \beta_{1} - 181 \beta_{2} + 27 \beta_{3} + 16 \beta_{4} - 43 \beta_{5} - 4 \beta_{6} - 32 \beta_{7} - 18 \beta_{8} + 82 \beta_{9} - 2 \beta_{10} - 93 \beta_{11} ) q^{68} + ( -66 - 18 \beta_{2} + 18 \beta_{3} - 114 \beta_{4} + 9 \beta_{5} + 9 \beta_{6} + 5 \beta_{7} - 5 \beta_{8} + 3 \beta_{10} - 3 \beta_{11} ) q^{69} + ( 160 - 68 \beta_{1} + 154 \beta_{3} + 68 \beta_{4} + 13 \beta_{5} - 33 \beta_{6} + 33 \beta_{7} - 13 \beta_{8} - 160 \beta_{9} + 40 \beta_{10} ) q^{70} + ( -16 - 43 \beta_{1} + 16 \beta_{2} + 50 \beta_{3} - 44 \beta_{4} - 42 \beta_{5} + 42 \beta_{7} - 14 \beta_{8} + 50 \beta_{9} - 43 \beta_{10} + 44 \beta_{11} ) q^{71} + ( 67 \beta_{1} + 70 \beta_{2} + 70 \beta_{3} + 23 \beta_{5} - 23 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 19 \beta_{9} + \beta_{10} + \beta_{11} ) q^{72} + ( 139 - 5 \beta_{1} + 139 \beta_{2} - 42 \beta_{3} + 73 \beta_{4} - 41 \beta_{5} + 70 \beta_{6} - 41 \beta_{7} + 42 \beta_{9} + 5 \beta_{10} + 73 \beta_{11} ) q^{73} + ( -198 + 46 \beta_{1} - 109 \beta_{2} - 198 \beta_{3} - 68 \beta_{4} - 32 \beta_{6} + 3 \beta_{7} - 32 \beta_{8} - 109 \beta_{9} - 68 \beta_{10} + 46 \beta_{11} ) q^{74} + ( 30 + 77 \beta_{1} - 190 \beta_{2} - 30 \beta_{3} + 13 \beta_{4} + 41 \beta_{5} + 50 \beta_{6} - 50 \beta_{8} + 190 \beta_{9} - 13 \beta_{10} - 77 \beta_{11} ) q^{75} + ( 92 + 38 \beta_{1} + 142 \beta_{2} + 38 \beta_{4} - 44 \beta_{5} + 22 \beta_{6} + 22 \beta_{7} - 44 \beta_{8} + 92 \beta_{9} - 6 \beta_{11} ) q^{76} + ( -88 + 35 \beta_{1} + 14 \beta_{2} + 35 \beta_{4} - 9 \beta_{5} + 67 \beta_{6} + 67 \beta_{7} - 9 \beta_{8} - 88 \beta_{9} + 32 \beta_{11} ) q^{77} + ( 64 + 2 \beta_{1} - 22 \beta_{2} - 64 \beta_{3} - 12 \beta_{4} - 64 \beta_{5} + 48 \beta_{6} - 48 \beta_{8} + 22 \beta_{9} + 12 \beta_{10} - 2 \beta_{11} ) q^{78} + ( -42 + 27 \beta_{1} + 6 \beta_{2} - 42 \beta_{3} + 6 \beta_{4} - 64 \beta_{6} - 24 \beta_{7} - 64 \beta_{8} + 6 \beta_{9} + 6 \beta_{10} + 27 \beta_{11} ) q^{79} + ( -139 - 96 \beta_{1} - 139 \beta_{2} + 132 \beta_{3} - 32 \beta_{4} + 20 \beta_{5} - 31 \beta_{6} + 20 \beta_{7} - 132 \beta_{9} + 96 \beta_{10} - 32 \beta_{11} ) q^{80} + ( -28 \beta_{1} + 62 \beta_{2} + 62 \beta_{3} - 12 \beta_{5} + 12 \beta_{6} - 7 \beta_{7} - 7 \beta_{8} - 165 \beta_{9} + 86 \beta_{10} + 86 \beta_{11} ) q^{81} + ( -150 + 82 \beta_{1} + 150 \beta_{2} + 87 \beta_{3} + 43 \beta_{4} + 3 \beta_{5} - 3 \beta_{7} + 8 \beta_{8} + 87 \beta_{9} + 82 \beta_{10} - 43 \beta_{11} ) q^{82} + ( -118 - 16 \beta_{1} + 272 \beta_{3} + 16 \beta_{4} - 19 \beta_{5} - 12 \beta_{6} + 12 \beta_{7} + 19 \beta_{8} + 118 \beta_{9} + 94 \beta_{10} ) q^{83} + ( -308 - 110 \beta_{2} + 110 \beta_{3} - 32 \beta_{4} + 74 \beta_{5} + 74 \beta_{6} - 34 \beta_{7} + 34 \beta_{8} - 42 \beta_{10} + 42 \beta_{11} ) q^{84} + ( -279 - 60 \beta_{1} + 165 \beta_{2} + 122 \beta_{3} - 53 \beta_{4} - \beta_{5} - 42 \beta_{6} + 21 \beta_{7} + 15 \beta_{8} - 108 \beta_{9} + 98 \beta_{10} + 69 \beta_{11} ) q^{85} + ( -98 - 72 \beta_{2} + 72 \beta_{3} + 2 \beta_{4} + 34 \beta_{5} + 34 \beta_{6} - 79 \beta_{7} + 79 \beta_{8} + 111 \beta_{10} - 111 \beta_{11} ) q^{86} + ( -266 - \beta_{1} - 72 \beta_{3} + \beta_{4} - 20 \beta_{5} - 96 \beta_{6} + 96 \beta_{7} + 20 \beta_{8} + 266 \beta_{9} - 22 \beta_{10} ) q^{87} + ( -37 - 34 \beta_{1} + 37 \beta_{2} - 269 \beta_{3} + 75 \beta_{4} - 34 \beta_{5} + 34 \beta_{7} - 63 \beta_{8} - 269 \beta_{9} - 34 \beta_{10} - 75 \beta_{11} ) q^{88} + ( 96 \beta_{1} - 249 \beta_{2} - 249 \beta_{3} + 21 \beta_{5} - 21 \beta_{6} - 53 \beta_{7} - 53 \beta_{8} + 194 \beta_{9} + 7 \beta_{10} + 7 \beta_{11} ) q^{89} + ( -108 + 10 \beta_{1} - 108 \beta_{2} + 109 \beta_{3} - 54 \beta_{4} - 9 \beta_{5} - 11 \beta_{6} - 9 \beta_{7} - 109 \beta_{9} - 10 \beta_{10} - 54 \beta_{11} ) q^{90} + ( 242 - 34 \beta_{1} + 140 \beta_{2} + 242 \beta_{3} - 26 \beta_{4} + 22 \beta_{6} - 102 \beta_{7} + 22 \beta_{8} + 140 \beta_{9} - 26 \beta_{10} - 34 \beta_{11} ) q^{91} + ( 77 - 106 \beta_{1} + 191 \beta_{2} - 77 \beta_{3} - 6 \beta_{4} + 38 \beta_{5} - 19 \beta_{6} + 19 \beta_{8} - 191 \beta_{9} + 6 \beta_{10} + 106 \beta_{11} ) q^{92} + ( -82 - 33 \beta_{1} + 246 \beta_{2} - 33 \beta_{4} - 3 \beta_{5} - 77 \beta_{6} - 77 \beta_{7} - 3 \beta_{8} - 82 \beta_{9} + 2 \beta_{11} ) q^{93} + ( 402 - 60 \beta_{1} - 512 \beta_{2} - 60 \beta_{4} - 72 \beta_{5} + 58 \beta_{6} + 58 \beta_{7} - 72 \beta_{8} + 402 \beta_{9} - 180 \beta_{11} ) q^{94} + ( 244 - 55 \beta_{1} + 64 \beta_{2} - 244 \beta_{3} + 27 \beta_{4} + 80 \beta_{5} - 28 \beta_{6} + 28 \beta_{8} - 64 \beta_{9} - 27 \beta_{10} + 55 \beta_{11} ) q^{95} + ( 187 + 17 \beta_{1} + 15 \beta_{2} + 187 \beta_{3} + 48 \beta_{4} + 53 \beta_{6} - 21 \beta_{7} + 53 \beta_{8} + 15 \beta_{9} + 48 \beta_{10} + 17 \beta_{11} ) q^{96} + ( 22 + 104 \beta_{1} + 22 \beta_{2} + 331 \beta_{3} + 108 \beta_{4} + 75 \beta_{5} - 28 \beta_{6} + 75 \beta_{7} - 331 \beta_{9} - 104 \beta_{10} + 108 \beta_{11} ) q^{97} + ( 147 \beta_{1} + 254 \beta_{2} + 254 \beta_{3} + 38 \beta_{7} + 38 \beta_{8} + 323 \beta_{9} - 60 \beta_{10} - 60 \beta_{11} ) q^{98} + ( 272 - 93 \beta_{1} - 272 \beta_{2} - 536 \beta_{3} - 17 \beta_{4} - 72 \beta_{5} + 72 \beta_{7} + 47 \beta_{8} - 536 \beta_{9} - 93 \beta_{10} + 17 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 4q^{2} - 4q^{3} - 20q^{5} + 20q^{6} - 4q^{7} + 28q^{8} - 64q^{9} + O(q^{10}) \) \( 12q - 4q^{2} - 4q^{3} - 20q^{5} + 20q^{6} - 4q^{7} + 28q^{8} - 64q^{9} - 116q^{10} + 40q^{11} + 56q^{12} - 132q^{14} + 244q^{15} + 184q^{16} + 52q^{17} - 12q^{19} + 572q^{20} - 620q^{22} - 276q^{23} - 184q^{24} - 464q^{25} - 708q^{26} - 664q^{27} + 452q^{28} + 632q^{29} + 188q^{31} + 700q^{32} + 1400q^{33} + 764q^{34} - 632q^{35} + 524q^{36} + 940q^{37} - 1112q^{39} - 1864q^{40} + 176q^{41} + 48q^{42} - 1360q^{43} - 1364q^{44} - 32q^{45} + 452q^{46} - 540q^{48} + 1044q^{49} + 2856q^{50} + 340q^{51} + 792q^{52} - 360q^{53} - 244q^{54} - 1788q^{56} - 148q^{57} - 360q^{58} - 584q^{59} - 1792q^{60} - 1052q^{61} - 380q^{62} + 1752q^{63} + 404q^{65} + 1372q^{66} + 1080q^{67} + 2532q^{68} - 344q^{69} + 2072q^{70} + 28q^{71} + 824q^{73} - 2292q^{74} + 400q^{75} + 1328q^{76} - 1252q^{77} + 1128q^{78} - 196q^{79} - 904q^{80} - 1528q^{82} - 1008q^{83} - 4768q^{84} - 2824q^{85} - 1200q^{86} - 2516q^{87} - 56q^{88} - 860q^{90} + 2456q^{91} + 396q^{92} - 836q^{93} + 6360q^{94} + 2172q^{95} + 1668q^{96} - 904q^{97} + 3280q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} + 54 x^{10} + 1085 x^{8} + 9836 x^{6} + 38276 x^{4} + 49664 x^{2} + 16384\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{11} - 408 \nu^{10} + 10 \nu^{9} - 17680 \nu^{8} - 1725 \nu^{7} - 268600 \nu^{6} - 73516 \nu^{5} - 1728288 \nu^{4} - 729732 \nu^{3} - 4623456 \nu^{2} - 1741056 \nu - 2889728 \)\()/1392640\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{11} + 408 \nu^{10} + 10 \nu^{9} + 17680 \nu^{8} - 1725 \nu^{7} + 268600 \nu^{6} - 73516 \nu^{5} + 1728288 \nu^{4} - 729732 \nu^{3} + 4623456 \nu^{2} - 1741056 \nu + 2889728 \)\()/1392640\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{10} + 41 \nu^{8} + 569 \nu^{6} + 3051 \nu^{4} + 5498 \nu^{2} + 2432 \)\()/544\)
\(\beta_{5}\)\(=\)\((\)\( 241 \nu^{11} - 280 \nu^{10} + 19350 \nu^{9} + 2800 \nu^{8} + 502765 \nu^{7} + 387400 \nu^{6} + 5292396 \nu^{5} + 5701600 \nu^{4} + 20629572 \nu^{3} + 23023520 \nu^{2} + 17121536 \nu + 12462080 \)\()/1392640\)
\(\beta_{6}\)\(=\)\((\)\( -241 \nu^{11} - 280 \nu^{10} - 19350 \nu^{9} + 2800 \nu^{8} - 502765 \nu^{7} + 387400 \nu^{6} - 5292396 \nu^{5} + 5701600 \nu^{4} - 20629572 \nu^{3} + 23023520 \nu^{2} - 17121536 \nu + 12462080 \)\()/1392640\)
\(\beta_{7}\)\(=\)\((\)\( -325 \nu^{11} + 872 \nu^{10} - 18510 \nu^{9} + 34800 \nu^{8} - 386545 \nu^{7} + 459720 \nu^{6} - 3460060 \nu^{5} + 2176992 \nu^{4} - 10989460 \nu^{3} + 1717664 \nu^{2} - 1301760 \nu - 5347328 \)\()/1392640\)
\(\beta_{8}\)\(=\)\((\)\(-325 \nu^{11} - 872 \nu^{10} - 18510 \nu^{9} - 34800 \nu^{8} - 386545 \nu^{7} - 459720 \nu^{6} - 3460060 \nu^{5} - 2176992 \nu^{4} - 10989460 \nu^{3} - 1717664 \nu^{2} - 1301760 \nu + 5347328\)\()/1392640\)
\(\beta_{9}\)\(=\)\((\)\( 19 \nu^{11} + 898 \nu^{9} + 15367 \nu^{7} + 114052 \nu^{5} + 336716 \nu^{3} + 239872 \nu \)\()/69632\)
\(\beta_{10}\)\(=\)\((\)\( -51 \nu^{11} + 8 \nu^{10} - 2210 \nu^{9} - 80 \nu^{8} - 33575 \nu^{7} - 7960 \nu^{6} - 216036 \nu^{5} - 86432 \nu^{4} - 577932 \nu^{3} - 211424 \nu^{2} - 361216 \nu + 2048 \)\()/174080\)
\(\beta_{11}\)\(=\)\((\)\( -51 \nu^{11} - 8 \nu^{10} - 2210 \nu^{9} + 80 \nu^{8} - 33575 \nu^{7} + 7960 \nu^{6} - 216036 \nu^{5} + 86432 \nu^{4} - 577932 \nu^{3} + 211424 \nu^{2} - 361216 \nu - 2048 \)\()/174080\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{8} - \beta_{7} + \beta_{4} - \beta_{3} + \beta_{2} - 8\)
\(\nu^{3}\)\(=\)\(2 \beta_{11} + 2 \beta_{10} + 6 \beta_{9} + \beta_{8} + \beta_{7} - \beta_{3} - \beta_{2} - 13 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-16 \beta_{8} + 16 \beta_{7} - \beta_{6} - \beta_{5} - 21 \beta_{4} + 31 \beta_{3} - 31 \beta_{2} + 106\)
\(\nu^{5}\)\(=\)\(-50 \beta_{11} - 50 \beta_{10} - 138 \beta_{9} - 15 \beta_{8} - 15 \beta_{7} - 4 \beta_{6} + 4 \beta_{5} + 19 \beta_{3} + 19 \beta_{2} + 189 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-28 \beta_{11} + 28 \beta_{10} + 250 \beta_{8} - 250 \beta_{7} + 31 \beta_{6} + 31 \beta_{5} + 373 \beta_{4} - 619 \beta_{3} + 619 \beta_{2} - 1574\)
\(\nu^{7}\)\(=\)\(990 \beta_{11} + 990 \beta_{10} + 2602 \beta_{9} + 191 \beta_{8} + 191 \beta_{7} + 120 \beta_{6} - 120 \beta_{5} - 535 \beta_{3} - 535 \beta_{2} - 2885 \beta_{1}\)
\(\nu^{8}\)\(=\)\(1072 \beta_{11} - 1072 \beta_{10} - 3946 \beta_{8} + 3946 \beta_{7} - 679 \beta_{6} - 679 \beta_{5} - 6349 \beta_{4} + 11187 \beta_{3} - 11187 \beta_{2} + 24438\)
\(\nu^{9}\)\(=\)\(-18242 \beta_{11} - 18242 \beta_{10} - 46402 \beta_{9} - 2195 \beta_{8} - 2195 \beta_{7} - 2796 \beta_{6} + 2796 \beta_{5} + 13567 \beta_{3} + 13567 \beta_{2} + 45213 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-28020 \beta_{11} + 28020 \beta_{10} + 62854 \beta_{8} - 62854 \beta_{7} + 13251 \beta_{6} + 13251 \beta_{5} + 107189 \beta_{4} - 195539 \beta_{3} + 195539 \beta_{2} - 388206\)
\(\nu^{11}\)\(=\)\(326166 \beta_{11} + 326166 \beta_{10} + 814346 \beta_{9} + 21583 \beta_{8} + 21583 \beta_{7} + 59104 \beta_{6} - 59104 \beta_{5} - 304847 \beta_{3} - 304847 \beta_{2} - 720309 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/17\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
3.86166i
0.705468i
4.15292i
3.68604i
1.22788i
2.49971i
3.86166i
0.705468i
4.15292i
3.68604i
1.22788i
2.49971i
−3.43772 3.43772i −4.67995 + 1.93850i 15.6358i −7.10390 17.1503i 22.7523 + 9.42432i 5.36561 12.9537i 26.2496 26.2496i −0.947753 + 0.947753i −34.5367 + 83.3791i
2.2 −1.20595 1.20595i 4.10553 1.70057i 5.09138i 2.60601 + 6.29147i −7.00185 2.90026i −5.31013 + 12.8198i −15.7875 + 15.7875i −5.12843 + 5.12843i 4.44447 10.7299i
2.3 2.22945 + 2.22945i −1.83980 + 0.762069i 1.94089i −1.91633 4.62643i −5.80073 2.40274i 1.06584 2.57316i 13.5085 13.5085i −16.2878 + 16.2878i 6.04203 14.5867i
8.1 −1.89932 + 1.89932i −1.65755 + 4.00167i 0.785167i 1.92782 + 0.798529i −4.45224 10.7487i 23.0956 9.56650i −16.6858 16.6858i 5.82599 + 5.82599i −5.17821 + 2.14488i
8.2 −0.161134 + 0.161134i 3.15299 7.61199i 7.94807i 2.54200 + 1.05293i 0.718496 + 1.73460i −19.8837 + 8.23610i −2.56978 2.56978i −28.9092 28.9092i −0.579266 + 0.239940i
8.3 2.47467 2.47467i −1.08123 + 2.61032i 4.24796i −8.05561 3.33674i 3.78400 + 9.13537i −6.33320 + 2.62330i 9.28506 + 9.28506i 13.4472 + 13.4472i −28.1923 + 11.6776i
9.1 −3.43772 + 3.43772i −4.67995 1.93850i 15.6358i −7.10390 + 17.1503i 22.7523 9.42432i 5.36561 + 12.9537i 26.2496 + 26.2496i −0.947753 0.947753i −34.5367 83.3791i
9.2 −1.20595 + 1.20595i 4.10553 + 1.70057i 5.09138i 2.60601 6.29147i −7.00185 + 2.90026i −5.31013 12.8198i −15.7875 15.7875i −5.12843 5.12843i 4.44447 + 10.7299i
9.3 2.22945 2.22945i −1.83980 0.762069i 1.94089i −1.91633 + 4.62643i −5.80073 + 2.40274i 1.06584 + 2.57316i 13.5085 + 13.5085i −16.2878 16.2878i 6.04203 + 14.5867i
15.1 −1.89932 1.89932i −1.65755 4.00167i 0.785167i 1.92782 0.798529i −4.45224 + 10.7487i 23.0956 + 9.56650i −16.6858 + 16.6858i 5.82599 5.82599i −5.17821 2.14488i
15.2 −0.161134 0.161134i 3.15299 + 7.61199i 7.94807i 2.54200 1.05293i 0.718496 1.73460i −19.8837 8.23610i −2.56978 + 2.56978i −28.9092 + 28.9092i −0.579266 0.239940i
15.3 2.47467 + 2.47467i −1.08123 2.61032i 4.24796i −8.05561 + 3.33674i 3.78400 9.13537i −6.33320 2.62330i 9.28506 9.28506i 13.4472 13.4472i −28.1923 11.6776i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 17.4.d.a 12
3.b odd 2 1 153.4.l.a 12
17.d even 8 1 inner 17.4.d.a 12
17.e odd 16 2 289.4.a.g 12
17.e odd 16 2 289.4.b.e 12
51.g odd 8 1 153.4.l.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.4.d.a 12 1.a even 1 1 trivial
17.4.d.a 12 17.d even 8 1 inner
153.4.l.a 12 3.b odd 2 1
153.4.l.a 12 51.g odd 8 1
289.4.a.g 12 17.e odd 16 2
289.4.b.e 12 17.e odd 16 2

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(17, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T + 8 T^{2} + 12 T^{3} - 62 T^{4} - 388 T^{5} - 984 T^{6} - 3700 T^{7} - 6783 T^{8} + 6520 T^{9} + 50208 T^{10} + 211072 T^{11} + 853440 T^{12} + 1688576 T^{13} + 3213312 T^{14} + 3338240 T^{15} - 27783168 T^{16} - 121241600 T^{17} - 257949696 T^{18} - 813694976 T^{19} - 1040187392 T^{20} + 1610612736 T^{21} + 8589934592 T^{22} + 34359738368 T^{23} + 68719476736 T^{24} \)
$3$ \( 1 + 4 T + 40 T^{2} + 324 T^{3} + 1520 T^{4} + 17004 T^{5} + 93168 T^{6} + 623820 T^{7} + 3542815 T^{8} + 19106488 T^{9} + 116779528 T^{10} + 611518776 T^{11} + 3634702400 T^{12} + 16511006952 T^{13} + 85132275912 T^{14} + 376073003304 T^{15} + 1882797146415 T^{16} + 8951135164740 T^{17} + 36095192119152 T^{18} + 177867845863812 T^{19} + 429292895451120 T^{20} + 2470693585135788 T^{21} + 8235645283785960 T^{22} + 22236242266222092 T^{23} + 150094635296999121 T^{24} \)
$5$ \( 1 + 20 T + 432 T^{2} + 5812 T^{3} + 76752 T^{4} + 671740 T^{5} + 5496800 T^{6} + 5471500 T^{7} - 426046641 T^{8} - 9209620720 T^{9} - 139560868176 T^{10} - 1662905986592 T^{11} - 19002322495072 T^{12} - 207863248324000 T^{13} - 2180638565250000 T^{14} - 17987540468750000 T^{15} - 104015293212890625 T^{16} + 166976928710937500 T^{17} + 20968627929687500000 T^{18} + \)\(32\!\cdots\!00\)\( T^{19} + \)\(45\!\cdots\!00\)\( T^{20} + \)\(43\!\cdots\!00\)\( T^{21} + \)\(40\!\cdots\!00\)\( T^{22} + \)\(23\!\cdots\!00\)\( T^{23} + \)\(14\!\cdots\!25\)\( T^{24} \)
$7$ \( 1 + 4 T - 514 T^{2} - 6116 T^{3} + 115922 T^{4} + 1116940 T^{5} - 8154266 T^{6} + 477986852 T^{7} + 5461415235 T^{8} - 238144861440 T^{9} - 5762877749244 T^{10} + 45937354375984 T^{11} + 1745889417070628 T^{12} + 15756512550962512 T^{13} - 677996804320807356 T^{14} - 9610004147619214080 T^{15} + 75593016791551907235 T^{16} + \)\(22\!\cdots\!36\)\( T^{17} - \)\(13\!\cdots\!34\)\( T^{18} + \)\(62\!\cdots\!80\)\( T^{19} + \)\(22\!\cdots\!22\)\( T^{20} - \)\(40\!\cdots\!88\)\( T^{21} - \)\(11\!\cdots\!86\)\( T^{22} + \)\(30\!\cdots\!28\)\( T^{23} + \)\(26\!\cdots\!01\)\( T^{24} \)
$11$ \( 1 - 40 T + 4148 T^{2} - 162224 T^{3} + 9095112 T^{4} - 332343560 T^{5} + 18025759684 T^{6} - 645155164752 T^{7} + 32705329538127 T^{8} - 1145801951121704 T^{9} + 50413798046517864 T^{10} - 1580384424194772536 T^{11} + 69762836759754864688 T^{12} - \)\(21\!\cdots\!16\)\( T^{13} + \)\(89\!\cdots\!04\)\( T^{14} - \)\(27\!\cdots\!64\)\( T^{15} + \)\(10\!\cdots\!67\)\( T^{16} - \)\(26\!\cdots\!52\)\( T^{17} + \)\(10\!\cdots\!04\)\( T^{18} - \)\(24\!\cdots\!60\)\( T^{19} + \)\(89\!\cdots\!92\)\( T^{20} - \)\(21\!\cdots\!04\)\( T^{21} + \)\(72\!\cdots\!48\)\( T^{22} - \)\(92\!\cdots\!40\)\( T^{23} + \)\(30\!\cdots\!61\)\( T^{24} \)
$13$ \( 1 - 14448 T^{2} + 107666638 T^{4} - 546851229616 T^{6} + 2094008234358095 T^{8} - 6327905324283989088 T^{10} + \)\(15\!\cdots\!00\)\( T^{12} - \)\(30\!\cdots\!92\)\( T^{14} + \)\(48\!\cdots\!95\)\( T^{16} - \)\(61\!\cdots\!64\)\( T^{18} + \)\(58\!\cdots\!18\)\( T^{20} - \)\(37\!\cdots\!52\)\( T^{22} + \)\(12\!\cdots\!41\)\( T^{24} \)
$17$ \( 1 - 52 T - 1224 T^{2} + 491300 T^{3} - 28461009 T^{4} - 1278873552 T^{5} + 118745480624 T^{6} - 6283105760976 T^{7} - 686979568547121 T^{8} + 58262223722976100 T^{9} - 713129618369227464 T^{10} - \)\(14\!\cdots\!36\)\( T^{11} + \)\(14\!\cdots\!09\)\( T^{12} \)
$19$ \( 1 + 12 T + 72 T^{2} + 1356220 T^{3} + 80627506 T^{4} + 1256173316 T^{5} + 928935243560 T^{6} + 122232243190932 T^{7} + 4866957296223951 T^{8} + 356984531389493592 T^{9} + 95598460087921452240 T^{10} + \)\(61\!\cdots\!04\)\( T^{11} + \)\(12\!\cdots\!48\)\( T^{12} + \)\(42\!\cdots\!36\)\( T^{13} + \)\(44\!\cdots\!40\)\( T^{14} + \)\(11\!\cdots\!68\)\( T^{15} + \)\(10\!\cdots\!11\)\( T^{16} + \)\(18\!\cdots\!68\)\( T^{17} + \)\(96\!\cdots\!60\)\( T^{18} + \)\(89\!\cdots\!04\)\( T^{19} + \)\(39\!\cdots\!26\)\( T^{20} + \)\(45\!\cdots\!80\)\( T^{21} + \)\(16\!\cdots\!72\)\( T^{22} + \)\(18\!\cdots\!08\)\( T^{23} + \)\(10\!\cdots\!81\)\( T^{24} \)
$23$ \( 1 + 276 T + 17214 T^{2} - 1142692 T^{3} - 27820014 T^{4} + 28497242460 T^{5} + 1982704026150 T^{6} - 65861887634364 T^{7} + 2296978221239043 T^{8} - 875617922313069328 T^{9} - \)\(39\!\cdots\!16\)\( T^{10} + \)\(14\!\cdots\!48\)\( T^{11} + \)\(80\!\cdots\!56\)\( T^{12} + \)\(17\!\cdots\!16\)\( T^{13} - \)\(58\!\cdots\!24\)\( T^{14} - \)\(15\!\cdots\!64\)\( T^{15} + \)\(50\!\cdots\!03\)\( T^{16} - \)\(17\!\cdots\!48\)\( T^{17} + \)\(64\!\cdots\!50\)\( T^{18} + \)\(11\!\cdots\!80\)\( T^{19} - \)\(13\!\cdots\!74\)\( T^{20} - \)\(66\!\cdots\!24\)\( T^{21} + \)\(12\!\cdots\!86\)\( T^{22} + \)\(23\!\cdots\!08\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \)
$29$ \( 1 - 632 T + 193012 T^{2} - 31241744 T^{3} + 1162856136 T^{4} + 678887482016 T^{5} - 143704921685972 T^{6} + 1869446104826552 T^{7} + 4704974888482939551 T^{8} - \)\(10\!\cdots\!44\)\( T^{9} + \)\(72\!\cdots\!24\)\( T^{10} + \)\(12\!\cdots\!52\)\( T^{11} - \)\(36\!\cdots\!96\)\( T^{12} + \)\(29\!\cdots\!28\)\( T^{13} + \)\(42\!\cdots\!04\)\( T^{14} - \)\(14\!\cdots\!36\)\( T^{15} + \)\(16\!\cdots\!91\)\( T^{16} + \)\(16\!\cdots\!48\)\( T^{17} - \)\(30\!\cdots\!92\)\( T^{18} + \)\(34\!\cdots\!64\)\( T^{19} + \)\(14\!\cdots\!16\)\( T^{20} - \)\(95\!\cdots\!96\)\( T^{21} + \)\(14\!\cdots\!12\)\( T^{22} - \)\(11\!\cdots\!48\)\( T^{23} + \)\(44\!\cdots\!21\)\( T^{24} \)
$31$ \( 1 - 188 T + 22934 T^{2} + 18386644 T^{3} - 3691984606 T^{4} + 776536429588 T^{5} + 99645628308078 T^{6} - 25120109404874988 T^{7} + 8682201249054683203 T^{8} - \)\(15\!\cdots\!80\)\( T^{9} + \)\(20\!\cdots\!44\)\( T^{10} + \)\(40\!\cdots\!64\)\( T^{11} - \)\(34\!\cdots\!20\)\( T^{12} + \)\(11\!\cdots\!24\)\( T^{13} + \)\(17\!\cdots\!64\)\( T^{14} - \)\(41\!\cdots\!80\)\( T^{15} + \)\(68\!\cdots\!83\)\( T^{16} - \)\(58\!\cdots\!88\)\( T^{17} + \)\(69\!\cdots\!98\)\( T^{18} + \)\(16\!\cdots\!28\)\( T^{19} - \)\(22\!\cdots\!26\)\( T^{20} + \)\(33\!\cdots\!84\)\( T^{21} + \)\(12\!\cdots\!34\)\( T^{22} - \)\(30\!\cdots\!08\)\( T^{23} + \)\(48\!\cdots\!81\)\( T^{24} \)
$37$ \( 1 - 940 T + 400776 T^{2} - 69989748 T^{3} - 12837369392 T^{4} + 10952154351924 T^{5} - 2855667555891144 T^{6} + 200153198828812748 T^{7} + \)\(10\!\cdots\!87\)\( T^{8} - \)\(40\!\cdots\!08\)\( T^{9} + \)\(48\!\cdots\!68\)\( T^{10} + \)\(89\!\cdots\!52\)\( T^{11} - \)\(42\!\cdots\!88\)\( T^{12} + \)\(45\!\cdots\!56\)\( T^{13} + \)\(12\!\cdots\!12\)\( T^{14} - \)\(53\!\cdots\!16\)\( T^{15} + \)\(71\!\cdots\!47\)\( T^{16} + \)\(66\!\cdots\!64\)\( T^{17} - \)\(48\!\cdots\!76\)\( T^{18} + \)\(93\!\cdots\!88\)\( T^{19} - \)\(55\!\cdots\!12\)\( T^{20} - \)\(15\!\cdots\!84\)\( T^{21} + \)\(44\!\cdots\!24\)\( T^{22} - \)\(52\!\cdots\!80\)\( T^{23} + \)\(28\!\cdots\!41\)\( T^{24} \)
$41$ \( 1 - 176 T - 88034 T^{2} + 34173648 T^{3} + 827249858 T^{4} - 2621758340656 T^{5} + 280349260544702 T^{6} + 147031456973046352 T^{7} - 22298918693016177249 T^{8} - \)\(82\!\cdots\!96\)\( T^{9} + \)\(33\!\cdots\!96\)\( T^{10} + \)\(28\!\cdots\!24\)\( T^{11} - \)\(30\!\cdots\!88\)\( T^{12} + \)\(19\!\cdots\!04\)\( T^{13} + \)\(15\!\cdots\!36\)\( T^{14} - \)\(26\!\cdots\!56\)\( T^{15} - \)\(50\!\cdots\!69\)\( T^{16} + \)\(22\!\cdots\!52\)\( T^{17} + \)\(30\!\cdots\!42\)\( T^{18} - \)\(19\!\cdots\!96\)\( T^{19} + \)\(42\!\cdots\!38\)\( T^{20} + \)\(11\!\cdots\!88\)\( T^{21} - \)\(21\!\cdots\!34\)\( T^{22} - \)\(29\!\cdots\!96\)\( T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \)
$43$ \( 1 + 1360 T + 924800 T^{2} + 445088144 T^{3} + 187950043870 T^{4} + 76345837363280 T^{5} + 29065866207767168 T^{6} + 10133858719746215760 T^{7} + \)\(33\!\cdots\!95\)\( T^{8} + \)\(10\!\cdots\!56\)\( T^{9} + \)\(34\!\cdots\!40\)\( T^{10} + \)\(10\!\cdots\!60\)\( T^{11} + \)\(29\!\cdots\!16\)\( T^{12} + \)\(81\!\cdots\!20\)\( T^{13} + \)\(21\!\cdots\!60\)\( T^{14} + \)\(54\!\cdots\!08\)\( T^{15} + \)\(13\!\cdots\!95\)\( T^{16} + \)\(32\!\cdots\!20\)\( T^{17} + \)\(73\!\cdots\!32\)\( T^{18} + \)\(15\!\cdots\!40\)\( T^{19} + \)\(30\!\cdots\!70\)\( T^{20} + \)\(56\!\cdots\!08\)\( T^{21} + \)\(93\!\cdots\!00\)\( T^{22} + \)\(10\!\cdots\!80\)\( T^{23} + \)\(63\!\cdots\!01\)\( T^{24} \)
$47$ \( 1 - 580412 T^{2} + 182778426498 T^{4} - 40216713491295692 T^{6} + \)\(68\!\cdots\!79\)\( T^{8} - \)\(94\!\cdots\!00\)\( T^{10} + \)\(10\!\cdots\!68\)\( T^{12} - \)\(10\!\cdots\!00\)\( T^{14} + \)\(79\!\cdots\!39\)\( T^{16} - \)\(50\!\cdots\!88\)\( T^{18} + \)\(24\!\cdots\!38\)\( T^{20} - \)\(84\!\cdots\!88\)\( T^{22} + \)\(15\!\cdots\!21\)\( T^{24} \)
$53$ \( 1 + 360 T + 64800 T^{2} + 135249864 T^{3} + 60322424078 T^{4} + 11373697219528 T^{5} + 9331900774784928 T^{6} + 6170118859401097768 T^{7} + \)\(17\!\cdots\!63\)\( T^{8} + \)\(52\!\cdots\!24\)\( T^{9} + \)\(41\!\cdots\!00\)\( T^{10} + \)\(14\!\cdots\!20\)\( T^{11} + \)\(24\!\cdots\!08\)\( T^{12} + \)\(21\!\cdots\!40\)\( T^{13} + \)\(91\!\cdots\!00\)\( T^{14} + \)\(17\!\cdots\!92\)\( T^{15} + \)\(84\!\cdots\!83\)\( T^{16} + \)\(45\!\cdots\!76\)\( T^{17} + \)\(10\!\cdots\!92\)\( T^{18} + \)\(18\!\cdots\!84\)\( T^{19} + \)\(14\!\cdots\!18\)\( T^{20} + \)\(48\!\cdots\!68\)\( T^{21} + \)\(34\!\cdots\!00\)\( T^{22} + \)\(28\!\cdots\!80\)\( T^{23} + \)\(11\!\cdots\!21\)\( T^{24} \)
$59$ \( 1 + 584 T + 170528 T^{2} - 175550056 T^{3} - 94146703170 T^{4} + 18580823027848 T^{5} + 42314760727238560 T^{6} + 18131962503470660184 T^{7} - \)\(19\!\cdots\!73\)\( T^{8} - \)\(38\!\cdots\!88\)\( T^{9} - \)\(96\!\cdots\!00\)\( T^{10} + \)\(60\!\cdots\!68\)\( T^{11} + \)\(52\!\cdots\!08\)\( T^{12} + \)\(12\!\cdots\!72\)\( T^{13} - \)\(40\!\cdots\!00\)\( T^{14} - \)\(33\!\cdots\!32\)\( T^{15} - \)\(34\!\cdots\!13\)\( T^{16} + \)\(66\!\cdots\!16\)\( T^{17} + \)\(31\!\cdots\!60\)\( T^{18} + \)\(28\!\cdots\!32\)\( T^{19} - \)\(29\!\cdots\!70\)\( T^{20} - \)\(11\!\cdots\!64\)\( T^{21} + \)\(22\!\cdots\!28\)\( T^{22} + \)\(16\!\cdots\!36\)\( T^{23} + \)\(56\!\cdots\!41\)\( T^{24} \)
$61$ \( 1 + 1052 T + 1006376 T^{2} + 412979636 T^{3} + 133288994960 T^{4} - 72852168865956 T^{5} - 77682489128187272 T^{6} - 64250295850085738220 T^{7} - \)\(24\!\cdots\!17\)\( T^{8} - \)\(69\!\cdots\!76\)\( T^{9} + \)\(27\!\cdots\!44\)\( T^{10} + \)\(28\!\cdots\!40\)\( T^{11} + \)\(20\!\cdots\!96\)\( T^{12} + \)\(65\!\cdots\!40\)\( T^{13} + \)\(13\!\cdots\!84\)\( T^{14} - \)\(81\!\cdots\!16\)\( T^{15} - \)\(65\!\cdots\!57\)\( T^{16} - \)\(38\!\cdots\!20\)\( T^{17} - \)\(10\!\cdots\!32\)\( T^{18} - \)\(22\!\cdots\!16\)\( T^{19} + \)\(93\!\cdots\!60\)\( T^{20} + \)\(66\!\cdots\!56\)\( T^{21} + \)\(36\!\cdots\!76\)\( T^{22} + \)\(86\!\cdots\!12\)\( T^{23} + \)\(18\!\cdots\!61\)\( T^{24} \)
$67$ \( ( 1 - 540 T + 1237196 T^{2} - 437516948 T^{3} + 687431155711 T^{4} - 191267075797288 T^{5} + 250196846973981368 T^{6} - 57526059518019730744 T^{7} + \)\(62\!\cdots\!59\)\( T^{8} - \)\(11\!\cdots\!56\)\( T^{9} + \)\(10\!\cdots\!56\)\( T^{10} - \)\(13\!\cdots\!20\)\( T^{11} + \)\(74\!\cdots\!09\)\( T^{12} )^{2} \)
$71$ \( 1 - 28 T - 195882 T^{2} + 194007508 T^{3} + 13906393890 T^{4} - 24274230169564 T^{5} + 39419381006302414 T^{6} + 24734369575425164292 T^{7} - \)\(95\!\cdots\!37\)\( T^{8} - \)\(41\!\cdots\!16\)\( T^{9} + \)\(42\!\cdots\!20\)\( T^{10} + \)\(13\!\cdots\!48\)\( T^{11} - \)\(63\!\cdots\!80\)\( T^{12} + \)\(49\!\cdots\!28\)\( T^{13} + \)\(54\!\cdots\!20\)\( T^{14} - \)\(19\!\cdots\!96\)\( T^{15} - \)\(15\!\cdots\!17\)\( T^{16} + \)\(14\!\cdots\!92\)\( T^{17} + \)\(82\!\cdots\!54\)\( T^{18} - \)\(18\!\cdots\!44\)\( T^{19} + \)\(37\!\cdots\!90\)\( T^{20} + \)\(18\!\cdots\!28\)\( T^{21} - \)\(67\!\cdots\!82\)\( T^{22} - \)\(34\!\cdots\!08\)\( T^{23} + \)\(44\!\cdots\!21\)\( T^{24} \)
$73$ \( 1 - 824 T + 588222 T^{2} + 9262472 T^{3} - 118766234814 T^{4} + 230361433132360 T^{5} - 186489223297230882 T^{6} + \)\(15\!\cdots\!92\)\( T^{7} - \)\(70\!\cdots\!77\)\( T^{8} + \)\(10\!\cdots\!48\)\( T^{9} + \)\(24\!\cdots\!96\)\( T^{10} - \)\(26\!\cdots\!64\)\( T^{11} + \)\(16\!\cdots\!68\)\( T^{12} - \)\(10\!\cdots\!88\)\( T^{13} + \)\(37\!\cdots\!44\)\( T^{14} + \)\(61\!\cdots\!24\)\( T^{15} - \)\(16\!\cdots\!17\)\( T^{16} + \)\(13\!\cdots\!44\)\( T^{17} - \)\(64\!\cdots\!58\)\( T^{18} + \)\(31\!\cdots\!80\)\( T^{19} - \)\(62\!\cdots\!74\)\( T^{20} + \)\(18\!\cdots\!84\)\( T^{21} + \)\(46\!\cdots\!78\)\( T^{22} - \)\(25\!\cdots\!92\)\( T^{23} + \)\(12\!\cdots\!61\)\( T^{24} \)
$79$ \( 1 + 196 T + 788046 T^{2} + 351355900 T^{3} + 349469377586 T^{4} + 376015608150652 T^{5} + 21556020940992630 T^{6} + \)\(18\!\cdots\!52\)\( T^{7} + \)\(20\!\cdots\!43\)\( T^{8} + \)\(54\!\cdots\!52\)\( T^{9} + \)\(53\!\cdots\!64\)\( T^{10} - \)\(49\!\cdots\!12\)\( T^{11} + \)\(41\!\cdots\!72\)\( T^{12} - \)\(24\!\cdots\!68\)\( T^{13} + \)\(13\!\cdots\!44\)\( T^{14} + \)\(64\!\cdots\!88\)\( T^{15} + \)\(12\!\cdots\!63\)\( T^{16} + \)\(54\!\cdots\!48\)\( T^{17} + \)\(30\!\cdots\!30\)\( T^{18} + \)\(26\!\cdots\!08\)\( T^{19} + \)\(12\!\cdots\!66\)\( T^{20} + \)\(60\!\cdots\!00\)\( T^{21} + \)\(66\!\cdots\!46\)\( T^{22} + \)\(82\!\cdots\!44\)\( T^{23} + \)\(20\!\cdots\!21\)\( T^{24} \)
$83$ \( 1 + 1008 T + 508032 T^{2} + 608566144 T^{3} + 1311140396574 T^{4} + 925961362981888 T^{5} + 452444151744975104 T^{6} + \)\(51\!\cdots\!40\)\( T^{7} + \)\(71\!\cdots\!79\)\( T^{8} + \)\(40\!\cdots\!08\)\( T^{9} + \)\(20\!\cdots\!72\)\( T^{10} + \)\(22\!\cdots\!60\)\( T^{11} + \)\(25\!\cdots\!56\)\( T^{12} + \)\(12\!\cdots\!20\)\( T^{13} + \)\(65\!\cdots\!68\)\( T^{14} + \)\(76\!\cdots\!24\)\( T^{15} + \)\(76\!\cdots\!19\)\( T^{16} + \)\(31\!\cdots\!80\)\( T^{17} + \)\(15\!\cdots\!36\)\( T^{18} + \)\(18\!\cdots\!04\)\( T^{19} + \)\(14\!\cdots\!54\)\( T^{20} + \)\(39\!\cdots\!88\)\( T^{21} + \)\(18\!\cdots\!68\)\( T^{22} + \)\(21\!\cdots\!04\)\( T^{23} + \)\(12\!\cdots\!81\)\( T^{24} \)
$89$ \( 1 - 5719348 T^{2} + 16180825465546 T^{4} - 29845545554168255940 T^{6} + \)\(39\!\cdots\!99\)\( T^{8} - \)\(40\!\cdots\!04\)\( T^{10} + \)\(32\!\cdots\!60\)\( T^{12} - \)\(20\!\cdots\!44\)\( T^{14} + \)\(98\!\cdots\!79\)\( T^{16} - \)\(36\!\cdots\!40\)\( T^{18} + \)\(98\!\cdots\!86\)\( T^{20} - \)\(17\!\cdots\!48\)\( T^{22} + \)\(15\!\cdots\!61\)\( T^{24} \)
$97$ \( 1 + 904 T + 307166 T^{2} - 2157443544 T^{3} - 1987213960190 T^{4} - 448060583775704 T^{5} + 3336323390652175454 T^{6} + \)\(24\!\cdots\!08\)\( T^{7} + \)\(10\!\cdots\!15\)\( T^{8} - \)\(37\!\cdots\!80\)\( T^{9} - \)\(19\!\cdots\!56\)\( T^{10} + \)\(70\!\cdots\!36\)\( T^{11} + \)\(35\!\cdots\!84\)\( T^{12} + \)\(64\!\cdots\!28\)\( T^{13} - \)\(15\!\cdots\!24\)\( T^{14} - \)\(28\!\cdots\!60\)\( T^{15} + \)\(74\!\cdots\!15\)\( T^{16} + \)\(15\!\cdots\!44\)\( T^{17} + \)\(19\!\cdots\!06\)\( T^{18} - \)\(23\!\cdots\!88\)\( T^{19} - \)\(95\!\cdots\!90\)\( T^{20} - \)\(94\!\cdots\!72\)\( T^{21} + \)\(12\!\cdots\!34\)\( T^{22} + \)\(33\!\cdots\!08\)\( T^{23} + \)\(33\!\cdots\!21\)\( T^{24} \)
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