Properties

Label 201.4.a.e
Level 201
Weight 4
Character orbit 201.a
Self dual yes
Analytic conductor 11.859
Analytic rank 0
Dimension 11
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 201 = 3 \cdot 67 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 201.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.8593839112\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + 3 q^{3} + ( 6 + \beta_{2} ) q^{4} + ( 1 - \beta_{3} ) q^{5} + 3 \beta_{1} q^{6} + ( 7 + \beta_{1} - \beta_{7} ) q^{7} + ( 7 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{8} + \beta_{9} ) q^{8} + 9 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + 3 q^{3} + ( 6 + \beta_{2} ) q^{4} + ( 1 - \beta_{3} ) q^{5} + 3 \beta_{1} q^{6} + ( 7 + \beta_{1} - \beta_{7} ) q^{7} + ( 7 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{8} + \beta_{9} ) q^{8} + 9 q^{9} + ( 4 - 4 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{10} + ( 9 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{8} - \beta_{10} ) q^{11} + ( 18 + 3 \beta_{2} ) q^{12} + ( 16 - 2 \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{13} + ( 10 + 10 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{9} ) q^{14} + ( 3 - 3 \beta_{3} ) q^{15} + ( 41 + 4 \beta_{2} + 4 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{10} ) q^{16} + ( -4 + \beta_{1} - 3 \beta_{2} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{17} + 9 \beta_{1} q^{18} + ( 18 - \beta_{1} - 5 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - \beta_{6} - 4 \beta_{7} + 2 \beta_{9} ) q^{19} + ( -46 + 3 \beta_{1} - 7 \beta_{2} - 2 \beta_{3} - 5 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + \beta_{10} ) q^{20} + ( 21 + 3 \beta_{1} - 3 \beta_{7} ) q^{21} + ( -15 + 8 \beta_{1} - \beta_{2} + 3 \beta_{4} + \beta_{5} + 6 \beta_{6} + 5 \beta_{7} + 2 \beta_{8} - 3 \beta_{10} ) q^{22} + ( 13 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} + 4 \beta_{7} - \beta_{8} - 3 \beta_{9} + 3 \beta_{10} ) q^{23} + ( 21 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{8} + 3 \beta_{9} ) q^{24} + ( 35 + 4 \beta_{1} - 3 \beta_{2} + \beta_{4} + 3 \beta_{5} - 3 \beta_{6} - 5 \beta_{7} + \beta_{9} + 2 \beta_{10} ) q^{25} + ( -24 + 11 \beta_{1} - 8 \beta_{2} - 4 \beta_{4} + 2 \beta_{5} - 5 \beta_{6} - 7 \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} ) q^{26} + 27 q^{27} + ( 83 + 9 \beta_{1} + 10 \beta_{2} + 5 \beta_{3} + 6 \beta_{4} - 5 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} + \beta_{8} - 3 \beta_{10} ) q^{28} + ( 4 - 9 \beta_{1} - 3 \beta_{2} - 5 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} - 8 \beta_{7} - 5 \beta_{8} + \beta_{9} + \beta_{10} ) q^{29} + ( 12 - 12 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} + 3 \beta_{10} ) q^{30} + ( 50 - 6 \beta_{1} - 9 \beta_{2} + 5 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} + 5 \beta_{8} + 2 \beta_{9} - \beta_{10} ) q^{31} + ( -21 + 39 \beta_{1} + 3 \beta_{3} + 9 \beta_{4} - \beta_{5} + 8 \beta_{6} + 9 \beta_{7} + 9 \beta_{8} - 3 \beta_{9} - 3 \beta_{10} ) q^{32} + ( 27 - 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{8} - 3 \beta_{10} ) q^{33} + ( 18 - 42 \beta_{1} + 7 \beta_{2} - 9 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} - 5 \beta_{8} - \beta_{9} ) q^{34} + ( -16 - 35 \beta_{1} + 9 \beta_{2} - 18 \beta_{3} + \beta_{4} - 2 \beta_{5} + 8 \beta_{6} + 10 \beta_{7} + \beta_{8} - 8 \beta_{9} + \beta_{10} ) q^{35} + ( 54 + 9 \beta_{2} ) q^{36} + ( 87 - 40 \beta_{1} + 7 \beta_{2} - \beta_{3} + 2 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} + 3 \beta_{7} - \beta_{8} + \beta_{9} + 3 \beta_{10} ) q^{37} + ( -32 + 10 \beta_{1} + 3 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} - 4 \beta_{10} ) q^{38} + ( 48 - 6 \beta_{1} + 3 \beta_{3} - 3 \beta_{5} + 3 \beta_{6} + 6 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} - 3 \beta_{10} ) q^{39} + ( 25 - 86 \beta_{1} + 2 \beta_{2} - 15 \beta_{3} - 14 \beta_{4} + 3 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - 5 \beta_{8} + 5 \beta_{9} ) q^{40} + ( 68 - 25 \beta_{1} - 5 \beta_{2} + 4 \beta_{3} - 7 \beta_{4} + 10 \beta_{5} - 12 \beta_{6} - 10 \beta_{7} - 5 \beta_{8} + 4 \beta_{9} + 5 \beta_{10} ) q^{41} + ( 30 + 30 \beta_{1} - 3 \beta_{2} + 6 \beta_{3} + 3 \beta_{4} + 6 \beta_{5} - 3 \beta_{6} - 3 \beta_{9} ) q^{42} + ( 79 - 41 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} - 9 \beta_{5} + \beta_{6} + 9 \beta_{7} + 5 \beta_{8} + \beta_{9} + 3 \beta_{10} ) q^{43} + ( 35 - 40 \beta_{1} + 19 \beta_{2} - 2 \beta_{3} + \beta_{4} + 3 \beta_{5} - 2 \beta_{6} - 17 \beta_{7} - 16 \beta_{8} + 6 \beta_{9} - \beta_{10} ) q^{44} + ( 9 - 9 \beta_{3} ) q^{45} + ( 52 - 15 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + 3 \beta_{4} - 3 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 3 \beta_{9} + 6 \beta_{10} ) q^{46} + ( 11 - 22 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} - 9 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 6 \beta_{8} + 9 \beta_{9} ) q^{47} + ( 123 + 12 \beta_{2} + 12 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - 6 \beta_{6} - 3 \beta_{7} - 6 \beta_{8} + 3 \beta_{10} ) q^{48} + ( 130 - 9 \beta_{1} + 12 \beta_{2} + 7 \beta_{3} + 14 \beta_{4} - 9 \beta_{5} + 9 \beta_{6} + 11 \beta_{7} + 7 \beta_{8} - 7 \beta_{9} - 5 \beta_{10} ) q^{49} + ( 41 + 40 \beta_{1} + 14 \beta_{2} + 14 \beta_{3} + 12 \beta_{4} - 5 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - 8 \beta_{9} + 4 \beta_{10} ) q^{50} + ( -12 + 3 \beta_{1} - 9 \beta_{2} - 3 \beta_{5} + 3 \beta_{6} + 6 \beta_{7} ) q^{51} + ( 22 - 45 \beta_{1} - 6 \beta_{2} - 8 \beta_{3} - 10 \beta_{4} + 8 \beta_{5} - \beta_{6} - 5 \beta_{7} + 6 \beta_{8} - 3 \beta_{9} + 5 \beta_{10} ) q^{52} + ( -18 - 11 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} + \beta_{8} + 8 \beta_{9} - 7 \beta_{10} ) q^{53} + 27 \beta_{1} q^{54} + ( 29 - 9 \beta_{1} - 21 \beta_{2} + 5 \beta_{3} + 5 \beta_{4} + 8 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} + 6 \beta_{9} - \beta_{10} ) q^{55} + ( -17 + 112 \beta_{1} - 8 \beta_{2} + 44 \beta_{3} + 12 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + 9 \beta_{7} + 24 \beta_{8} + 5 \beta_{9} - \beta_{10} ) q^{56} + ( 54 - 3 \beta_{1} - 15 \beta_{2} + 6 \beta_{3} - 6 \beta_{4} + 9 \beta_{5} - 3 \beta_{6} - 12 \beta_{7} + 6 \beta_{9} ) q^{57} + ( -118 - 15 \beta_{1} - 11 \beta_{2} - 11 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 19 \beta_{6} + 21 \beta_{7} + 11 \beta_{8} - 8 \beta_{9} - 3 \beta_{10} ) q^{58} + ( 3 - 25 \beta_{1} + 17 \beta_{2} - \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - \beta_{6} + 2 \beta_{7} - 4 \beta_{9} - 6 \beta_{10} ) q^{59} + ( -138 + 9 \beta_{1} - 21 \beta_{2} - 6 \beta_{3} - 15 \beta_{4} - 6 \beta_{5} + 6 \beta_{6} + 3 \beta_{7} + 6 \beta_{8} + 6 \beta_{9} + 3 \beta_{10} ) q^{60} + ( 40 - 28 \beta_{1} - 20 \beta_{2} + 7 \beta_{3} - 6 \beta_{4} + 13 \beta_{5} - 5 \beta_{6} - 18 \beta_{7} - 7 \beta_{8} + 5 \beta_{9} - 7 \beta_{10} ) q^{61} + ( -91 + 20 \beta_{1} + 10 \beta_{3} - 12 \beta_{4} - 11 \beta_{5} - 17 \beta_{6} - 21 \beta_{7} - 6 \beta_{8} - 3 \beta_{9} + 3 \beta_{10} ) q^{62} + ( 63 + 9 \beta_{1} - 9 \beta_{7} ) q^{63} + ( 200 - 40 \beta_{1} + 31 \beta_{2} - 4 \beta_{3} + 14 \beta_{4} + 14 \beta_{5} - 26 \beta_{6} - 48 \beta_{7} - 28 \beta_{8} - 2 \beta_{9} - 6 \beta_{10} ) q^{64} + ( -81 + 29 \beta_{1} + 9 \beta_{2} + 17 \beta_{3} - \beta_{4} - 20 \beta_{5} + 4 \beta_{6} + 6 \beta_{7} + 11 \beta_{8} + 2 \beta_{9} - \beta_{10} ) q^{65} + ( -45 + 24 \beta_{1} - 3 \beta_{2} + 9 \beta_{4} + 3 \beta_{5} + 18 \beta_{6} + 15 \beta_{7} + 6 \beta_{8} - 9 \beta_{10} ) q^{66} + 67 q^{67} + ( -503 + 4 \beta_{1} - 21 \beta_{2} - 38 \beta_{3} - 13 \beta_{4} + 9 \beta_{5} + 16 \beta_{6} + 7 \beta_{7} + 4 \beta_{8} + 8 \beta_{9} - 3 \beta_{10} ) q^{68} + ( 39 + 9 \beta_{1} - 3 \beta_{2} + 6 \beta_{3} + 6 \beta_{4} - 12 \beta_{5} + 12 \beta_{7} - 3 \beta_{8} - 9 \beta_{9} + 9 \beta_{10} ) q^{69} + ( -361 - 82 \beta_{1} - 36 \beta_{2} - 45 \beta_{3} - 26 \beta_{4} + 7 \beta_{5} - 20 \beta_{6} - 20 \beta_{7} - 35 \beta_{8} - 3 \beta_{9} + 16 \beta_{10} ) q^{70} + ( 4 - 2 \beta_{1} + 52 \beta_{2} - 5 \beta_{3} + 30 \beta_{4} - 5 \beta_{5} + 5 \beta_{6} + 40 \beta_{7} - 5 \beta_{8} - 19 \beta_{9} + \beta_{10} ) q^{71} + ( 63 \beta_{1} - 9 \beta_{2} + 9 \beta_{3} + 9 \beta_{8} + 9 \beta_{9} ) q^{72} + ( 18 + 31 \beta_{1} - 20 \beta_{2} + 26 \beta_{3} - \beta_{4} - 20 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} + \beta_{9} ) q^{73} + ( -570 + 115 \beta_{1} - 24 \beta_{2} - 9 \beta_{3} + 9 \beta_{4} + 6 \beta_{5} + 8 \beta_{6} + 17 \beta_{7} - 5 \beta_{8} + 9 \beta_{9} + 5 \beta_{10} ) q^{74} + ( 105 + 12 \beta_{1} - 9 \beta_{2} + 3 \beta_{4} + 9 \beta_{5} - 9 \beta_{6} - 15 \beta_{7} + 3 \beta_{9} + 6 \beta_{10} ) q^{75} + ( -23 - 16 \beta_{1} + 31 \beta_{2} - 8 \beta_{3} + 21 \beta_{4} - 7 \beta_{5} + 12 \beta_{6} + 37 \beta_{7} + 2 \beta_{8} - 18 \beta_{9} - 9 \beta_{10} ) q^{76} + ( -17 - 63 \beta_{1} + 29 \beta_{2} - 13 \beta_{3} - 5 \beta_{4} + 24 \beta_{5} - 6 \beta_{6} - 24 \beta_{7} - 15 \beta_{8} - 10 \beta_{9} - 3 \beta_{10} ) q^{77} + ( -72 + 33 \beta_{1} - 24 \beta_{2} - 12 \beta_{4} + 6 \beta_{5} - 15 \beta_{6} - 21 \beta_{7} - 6 \beta_{8} + 3 \beta_{9} - 3 \beta_{10} ) q^{78} + ( 39 - 5 \beta_{1} + 7 \beta_{2} - 6 \beta_{3} + 3 \beta_{4} + 5 \beta_{5} - 17 \beta_{6} - 2 \beta_{7} - 24 \beta_{8} - 13 \beta_{9} + 14 \beta_{10} ) q^{79} + ( -739 - 29 \beta_{1} - 36 \beta_{2} - 46 \beta_{3} - 20 \beta_{4} + \beta_{5} + 22 \beta_{6} + 42 \beta_{7} + 4 \beta_{8} - 8 \beta_{9} - 2 \beta_{10} ) q^{80} + 81 q^{81} + ( -349 + 128 \beta_{1} - 22 \beta_{2} - \beta_{3} - 2 \beta_{4} - 33 \beta_{5} + 22 \beta_{6} + 14 \beta_{7} + 37 \beta_{8} + 3 \beta_{9} - 2 \beta_{10} ) q^{82} + ( -226 + 5 \beta_{1} + \beta_{2} + 14 \beta_{3} - 11 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} + 6 \beta_{7} - 5 \beta_{8} + 6 \beta_{9} + 15 \beta_{10} ) q^{83} + ( 249 + 27 \beta_{1} + 30 \beta_{2} + 15 \beta_{3} + 18 \beta_{4} - 15 \beta_{5} - 9 \beta_{6} - 9 \beta_{7} + 3 \beta_{8} - 9 \beta_{10} ) q^{84} + ( 229 + 41 \beta_{1} + 49 \beta_{2} + 5 \beta_{3} + 27 \beta_{4} - 2 \beta_{6} + 6 \beta_{7} - 5 \beta_{8} - 10 \beta_{9} - 7 \beta_{10} ) q^{85} + ( -556 + 25 \beta_{1} - 37 \beta_{2} - 8 \beta_{3} - 31 \beta_{4} - 2 \beta_{5} - 16 \beta_{6} + 7 \beta_{7} - 4 \beta_{8} + 21 \beta_{10} ) q^{86} + ( 12 - 27 \beta_{1} - 9 \beta_{2} - 15 \beta_{3} - 6 \beta_{4} + 12 \beta_{5} - 12 \beta_{6} - 24 \beta_{7} - 15 \beta_{8} + 3 \beta_{9} + 3 \beta_{10} ) q^{87} + ( -525 + 152 \beta_{1} - 49 \beta_{2} - 6 \beta_{3} + 17 \beta_{4} + 11 \beta_{5} + 34 \beta_{6} + 43 \beta_{7} + 36 \beta_{8} + 22 \beta_{9} - 9 \beta_{10} ) q^{88} + ( -222 + 53 \beta_{1} + 21 \beta_{2} - 4 \beta_{3} + 8 \beta_{4} + 7 \beta_{5} + 19 \beta_{6} + 14 \beta_{8} - 16 \beta_{9} - 10 \beta_{10} ) q^{89} + ( 36 - 36 \beta_{1} + 9 \beta_{2} - 9 \beta_{3} - 9 \beta_{4} + 9 \beta_{7} - 9 \beta_{8} - 9 \beta_{9} + 9 \beta_{10} ) q^{90} + ( -79 + 29 \beta_{1} - 17 \beta_{2} - 25 \beta_{3} - 27 \beta_{4} + 16 \beta_{5} + 6 \beta_{6} - 56 \beta_{7} + 13 \beta_{8} + 30 \beta_{9} - 21 \beta_{10} ) q^{91} + ( -275 + 32 \beta_{1} + 8 \beta_{2} - 2 \beta_{3} - 12 \beta_{4} + 19 \beta_{5} - 23 \beta_{6} - 45 \beta_{7} + 2 \beta_{8} + 13 \beta_{9} - \beta_{10} ) q^{92} + ( 150 - 18 \beta_{1} - 27 \beta_{2} + 15 \beta_{3} - 9 \beta_{4} + 6 \beta_{5} - 6 \beta_{6} + 3 \beta_{7} + 15 \beta_{8} + 6 \beta_{9} - 3 \beta_{10} ) q^{93} + ( -325 + 27 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 29 \beta_{4} - 21 \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} + 14 \beta_{9} + 6 \beta_{10} ) q^{94} + ( -231 - 17 \beta_{1} + 17 \beta_{2} - 43 \beta_{3} + 7 \beta_{4} + 16 \beta_{5} - 16 \beta_{6} - 20 \beta_{7} - 5 \beta_{8} + 14 \beta_{9} - 13 \beta_{10} ) q^{95} + ( -63 + 117 \beta_{1} + 9 \beta_{3} + 27 \beta_{4} - 3 \beta_{5} + 24 \beta_{6} + 27 \beta_{7} + 27 \beta_{8} - 9 \beta_{9} - 9 \beta_{10} ) q^{96} + ( 157 + 39 \beta_{1} - 7 \beta_{2} + 55 \beta_{3} - 23 \beta_{4} - 8 \beta_{5} - 2 \beta_{6} + 22 \beta_{7} + 5 \beta_{8} - 6 \beta_{9} + 25 \beta_{10} ) q^{97} + ( -172 + 190 \beta_{1} - 23 \beta_{2} + 54 \beta_{3} + 33 \beta_{4} + 26 \beta_{5} - 40 \beta_{6} - 45 \beta_{7} - 22 \beta_{8} + 2 \beta_{9} - 3 \beta_{10} ) q^{98} + ( 81 - 9 \beta_{1} + 9 \beta_{2} - 9 \beta_{3} + 9 \beta_{4} - 9 \beta_{8} - 9 \beta_{10} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q + 3q^{2} + 33q^{3} + 69q^{4} + 8q^{5} + 9q^{6} + 78q^{7} + 21q^{8} + 99q^{9} + O(q^{10}) \) \( 11q + 3q^{2} + 33q^{3} + 69q^{4} + 8q^{5} + 9q^{6} + 78q^{7} + 21q^{8} + 99q^{9} + 29q^{10} + 104q^{11} + 207q^{12} + 172q^{13} + 143q^{14} + 24q^{15} + 485q^{16} - 48q^{17} + 27q^{18} + 180q^{19} - 539q^{20} + 234q^{21} - 144q^{22} + 156q^{23} + 63q^{24} + 383q^{25} - 252q^{26} + 297q^{27} + 1011q^{28} - 4q^{29} + 87q^{30} + 514q^{31} - 119q^{32} + 312q^{33} + 72q^{34} - 338q^{35} + 621q^{36} + 854q^{37} - 308q^{38} + 516q^{39} - 15q^{40} + 674q^{41} + 429q^{42} + 738q^{43} + 356q^{44} + 72q^{45} + 507q^{46} + 54q^{47} + 1455q^{48} + 1465q^{49} + 656q^{50} - 144q^{51} - 12q^{52} - 190q^{53} + 81q^{54} + 262q^{55} + 239q^{56} + 540q^{57} - 1466q^{58} + 18q^{59} - 1617q^{60} + 328q^{61} - 915q^{62} + 702q^{63} + 2253q^{64} - 732q^{65} - 432q^{66} + 737q^{67} - 5746q^{68} + 468q^{69} - 4451q^{70} + 264q^{71} + 189q^{72} + 330q^{73} - 5975q^{74} + 1149q^{75} - 178q^{76} - 368q^{77} - 756q^{78} + 456q^{79} - 8515q^{80} + 891q^{81} - 3629q^{82} - 2432q^{83} + 3033q^{84} + 2882q^{85} - 6225q^{86} - 12q^{87} - 5492q^{88} - 2340q^{89} + 261q^{90} - 994q^{91} - 2939q^{92} + 1542q^{93} - 3506q^{94} - 2568q^{95} - 357q^{96} + 1892q^{97} - 1078q^{98} + 936q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11} - 3 x^{10} - 74 x^{9} + 208 x^{8} + 1913 x^{7} - 4831 x^{6} - 20432 x^{5} + 42994 x^{4} + 79648 x^{3} - 112288 x^{2} - 85440 x + 3072\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 14 \)
\(\beta_{3}\)\(=\)\((\)\( 37 \nu^{10} - 10337 \nu^{9} - 910 \nu^{8} + 803028 \nu^{7} - 138167 \nu^{6} - 21118529 \nu^{5} + 6977192 \nu^{4} + 209612366 \nu^{3} - 81613224 \nu^{2} - 566622048 \nu - 17164864 \)\()/17885888\)
\(\beta_{4}\)\(=\)\((\)\( -1563 \nu^{10} - 1415 \nu^{9} + 68654 \nu^{8} - 54128 \nu^{7} + 5581 \nu^{6} + 7020509 \nu^{5} - 24456864 \nu^{4} - 137118310 \nu^{3} + 147600512 \nu^{2} + 475227072 \nu + 161452352 \)\()/17885888\)
\(\beta_{5}\)\(=\)\((\)\( -999 \nu^{10} - 368 \nu^{9} + 24570 \nu^{8} + 116670 \nu^{7} + 935839 \nu^{6} - 5222270 \nu^{5} - 29087994 \nu^{4} + 67862816 \nu^{3} + 191394648 \nu^{2} - 226155488 \nu - 260927136 \)\()/8942944\)
\(\beta_{6}\)\(=\)\((\)\(2205 \nu^{10} - 41989 \nu^{9} - 190188 \nu^{8} + 2763750 \nu^{7} + 5709131 \nu^{6} - 59395043 \nu^{5} - 69381996 \nu^{4} + 462023874 \nu^{3} + 270932640 \nu^{2} - 963079888 \nu - 284052576\)\()/8942944\)
\(\beta_{7}\)\(=\)\((\)\(-27763 \nu^{10} + 74816 \nu^{9} + 1906432 \nu^{8} - 5007380 \nu^{7} - 44730735 \nu^{6} + 109869896 \nu^{5} + 424133218 \nu^{4} - 896074196 \nu^{3} - 1432652256 \nu^{2} + 1973665984 \nu + 1141218944\)\()/35771776\)
\(\beta_{8}\)\(=\)\((\)\(28189 \nu^{10} + 47870 \nu^{9} - 1856484 \nu^{8} - 2968156 \nu^{7} + 40420809 \nu^{6} + 59988542 \nu^{5} - 335341682 \nu^{4} - 421088000 \nu^{3} + 998152784 \nu^{2} + 706988096 \nu - 713324160\)\()/35771776\)
\(\beta_{9}\)\(=\)\((\)\(-28263 \nu^{10} - 27196 \nu^{9} + 1858304 \nu^{8} + 1362100 \nu^{7} - 40144475 \nu^{6} - 17751484 \nu^{5} + 321387298 \nu^{4} + 37635044 \nu^{3} - 799154560 \nu^{2} - 396494848 \nu + 246849024\)\()/35771776\)
\(\beta_{10}\)\(=\)\((\)\(45089 \nu^{10} - 81302 \nu^{9} - 3359788 \nu^{8} + 5317020 \nu^{7} + 86621461 \nu^{6} - 111295230 \nu^{5} - 889090122 \nu^{4} + 826729752 \nu^{3} + 2852788064 \nu^{2} - 1639096640 \nu - 961174016\)\()/35771776\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 14\)
\(\nu^{3}\)\(=\)\(\beta_{9} + \beta_{8} + \beta_{3} - \beta_{2} + 23 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{10} - 2 \beta_{8} - \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{4} + 4 \beta_{3} + 28 \beta_{2} + 313\)
\(\nu^{5}\)\(=\)\(-3 \beta_{10} + 29 \beta_{9} + 41 \beta_{8} + 9 \beta_{7} + 8 \beta_{6} - \beta_{5} + 9 \beta_{4} + 35 \beta_{3} - 32 \beta_{2} + 583 \beta_{1} - 21\)
\(\nu^{6}\)\(=\)\(34 \beta_{10} - 2 \beta_{9} - 108 \beta_{8} - 88 \beta_{7} - 106 \beta_{6} - 26 \beta_{5} + 54 \beta_{4} + 156 \beta_{3} + 767 \beta_{2} - 40 \beta_{1} + 7856\)
\(\nu^{7}\)\(=\)\(-172 \beta_{10} + 755 \beta_{9} + 1395 \beta_{8} + 452 \beta_{7} + 412 \beta_{6} - 16 \beta_{5} + 468 \beta_{4} + 1139 \beta_{3} - 1043 \beta_{2} + 15591 \beta_{1} - 1548\)
\(\nu^{8}\)\(=\)\(911 \beta_{10} - 164 \beta_{9} - 4210 \beta_{8} - 4063 \beta_{7} - 4086 \beta_{6} - 631 \beta_{5} + 2143 \beta_{4} + 5128 \beta_{3} + 21442 \beta_{2} - 3168 \beta_{1} + 208851\)
\(\nu^{9}\)\(=\)\(-7009 \beta_{10} + 19693 \beta_{9} + 44825 \beta_{8} + 17035 \beta_{7} + 15584 \beta_{6} + 469 \beta_{5} + 18003 \beta_{4} + 36263 \beta_{3} - 34870 \beta_{2} + 431911 \beta_{1} - 81047\)
\(\nu^{10}\)\(=\)\(23316 \beta_{10} - 8654 \beta_{9} - 146472 \beta_{8} - 153782 \beta_{7} - 141002 \beta_{6} - 16524 \beta_{5} + 75152 \beta_{4} + 160420 \beta_{3} + 612557 \beta_{2} - 165272 \beta_{1} + 5762126\)

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
−5.56558
−4.47941
−3.89104
−2.12976
−0.638765
0.0344354
1.96751
2.85564
4.59496
4.82380
5.42821
−5.56558 3.00000 22.9757 −9.37616 −16.6967 18.3374 −83.3483 9.00000 52.1838
1.2 −4.47941 3.00000 12.0651 −12.5434 −13.4382 −24.1947 −18.2093 9.00000 56.1868
1.3 −3.89104 3.00000 7.14021 11.5975 −11.6731 32.2920 3.34547 9.00000 −45.1263
1.4 −2.12976 3.00000 −3.46413 17.7773 −6.38927 −9.52178 24.4158 9.00000 −37.8613
1.5 −0.638765 3.00000 −7.59198 −13.5481 −1.91629 17.9442 9.95961 9.00000 8.65405
1.6 0.0344354 3.00000 −7.99881 3.05553 0.103306 −26.7198 −0.550926 9.00000 0.105218
1.7 1.96751 3.00000 −4.12891 17.2496 5.90252 22.3999 −23.8637 9.00000 33.9387
1.8 2.85564 3.00000 0.154694 −2.88345 8.56693 19.3750 −22.4034 9.00000 −8.23409
1.9 4.59496 3.00000 13.1136 11.7140 13.7849 4.20913 23.4970 9.00000 53.8252
1.10 4.82380 3.00000 15.2690 4.99073 14.4714 −10.0496 35.0643 9.00000 24.0743
1.11 5.42821 3.00000 21.4655 −20.0336 16.2846 33.9282 73.0934 9.00000 −108.746
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

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 201.4.a.e 11
3.b odd 2 1 603.4.a.g 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.4.a.e 11 1.a even 1 1 trivial
603.4.a.g 11 3.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(67\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{11} - \cdots\) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(201))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T + 14 T^{2} - 32 T^{3} + 105 T^{4} - 159 T^{5} + 680 T^{6} - 478 T^{7} + 2528 T^{8} + 9056 T^{9} - 21696 T^{10} + 111872 T^{11} - 173568 T^{12} + 579584 T^{13} + 1294336 T^{14} - 1957888 T^{15} + 22282240 T^{16} - 41680896 T^{17} + 220200960 T^{18} - 536870912 T^{19} + 1879048192 T^{20} - 3221225472 T^{21} + 8589934592 T^{22} \)
$3$ \( ( 1 - 3 T )^{11} \)
$5$ \( 1 - 8 T + 528 T^{2} - 3294 T^{3} + 153449 T^{4} - 758250 T^{5} + 31698985 T^{6} - 114311740 T^{7} + 5141388104 T^{8} - 13113139694 T^{9} + 715855958205 T^{10} - 1489367624492 T^{11} + 89481994775625 T^{12} - 204892807718750 T^{13} + 10041773640625000 T^{14} - 27908139648437500 T^{15} + 967376251220703125 T^{16} - 2892494201660156250 T^{17} + 73170185089111328125 T^{18} - \)\(19\!\cdots\!50\)\( T^{19} + \)\(39\!\cdots\!00\)\( T^{20} - \)\(74\!\cdots\!00\)\( T^{21} + \)\(11\!\cdots\!25\)\( T^{22} \)
$7$ \( 1 - 78 T + 4196 T^{2} - 165064 T^{3} + 5579811 T^{4} - 163930200 T^{5} + 4407732639 T^{6} - 108402451456 T^{7} + 2501162383708 T^{8} - 53838264847882 T^{9} + 1095562407971021 T^{10} - 20868481983780240 T^{11} + 375777905934060203 T^{12} - 6334018021088469418 T^{13} + \)\(10\!\cdots\!56\)\( T^{14} - \)\(15\!\cdots\!56\)\( T^{15} + \)\(20\!\cdots\!77\)\( T^{16} - \)\(26\!\cdots\!00\)\( T^{17} + \)\(31\!\cdots\!77\)\( T^{18} - \)\(31\!\cdots\!64\)\( T^{19} + \)\(27\!\cdots\!28\)\( T^{20} - \)\(17\!\cdots\!22\)\( T^{21} + \)\(77\!\cdots\!07\)\( T^{22} \)
$11$ \( 1 - 104 T + 10789 T^{2} - 701204 T^{3} + 44873803 T^{4} - 2179988344 T^{5} + 105602689991 T^{6} - 4129236939728 T^{7} + 167073931700570 T^{8} - 5656574969046752 T^{9} + 214830362371995074 T^{10} - 7173906879300840120 T^{11} + \)\(28\!\cdots\!94\)\( T^{12} - \)\(10\!\cdots\!72\)\( T^{13} + \)\(39\!\cdots\!70\)\( T^{14} - \)\(12\!\cdots\!88\)\( T^{15} + \)\(44\!\cdots\!41\)\( T^{16} - \)\(12\!\cdots\!64\)\( T^{17} + \)\(33\!\cdots\!33\)\( T^{18} - \)\(69\!\cdots\!64\)\( T^{19} + \)\(14\!\cdots\!19\)\( T^{20} - \)\(18\!\cdots\!04\)\( T^{21} + \)\(23\!\cdots\!31\)\( T^{22} \)
$13$ \( 1 - 172 T + 25059 T^{2} - 2734188 T^{3} + 261093971 T^{4} - 21607677036 T^{5} + 1611315015081 T^{6} - 108815924458864 T^{7} + 6729575428051490 T^{8} - 383051696698780936 T^{9} + 20143515895825358598 T^{10} - \)\(98\!\cdots\!28\)\( T^{11} + \)\(44\!\cdots\!06\)\( T^{12} - \)\(18\!\cdots\!24\)\( T^{13} + \)\(71\!\cdots\!70\)\( T^{14} - \)\(25\!\cdots\!84\)\( T^{15} + \)\(82\!\cdots\!17\)\( T^{16} - \)\(24\!\cdots\!44\)\( T^{17} + \)\(64\!\cdots\!23\)\( T^{18} - \)\(14\!\cdots\!68\)\( T^{19} + \)\(29\!\cdots\!03\)\( T^{20} - \)\(45\!\cdots\!28\)\( T^{21} + \)\(57\!\cdots\!53\)\( T^{22} \)
$17$ \( 1 + 48 T + 36584 T^{2} + 1761206 T^{3} + 637150548 T^{4} + 30407284932 T^{5} + 7089769267281 T^{6} + 328664427271504 T^{7} + 56992758015663398 T^{8} + 2496371415942949452 T^{9} + \)\(35\!\cdots\!64\)\( T^{10} + \)\(14\!\cdots\!08\)\( T^{11} + \)\(17\!\cdots\!32\)\( T^{12} + \)\(60\!\cdots\!88\)\( T^{13} + \)\(67\!\cdots\!06\)\( T^{14} + \)\(19\!\cdots\!44\)\( T^{15} + \)\(20\!\cdots\!33\)\( T^{16} + \)\(42\!\cdots\!88\)\( T^{17} + \)\(44\!\cdots\!16\)\( T^{18} + \)\(59\!\cdots\!26\)\( T^{19} + \)\(61\!\cdots\!32\)\( T^{20} + \)\(39\!\cdots\!52\)\( T^{21} + \)\(40\!\cdots\!37\)\( T^{22} \)
$19$ \( 1 - 180 T + 65990 T^{2} - 9444778 T^{3} + 1955735718 T^{4} - 230637470136 T^{5} + 34989106381707 T^{6} - 3497485904948176 T^{7} + 429087495405719902 T^{8} - 37105843792049732116 T^{9} + \)\(38\!\cdots\!48\)\( T^{10} - \)\(29\!\cdots\!12\)\( T^{11} + \)\(26\!\cdots\!32\)\( T^{12} - \)\(17\!\cdots\!96\)\( T^{13} + \)\(13\!\cdots\!58\)\( T^{14} - \)\(77\!\cdots\!36\)\( T^{15} + \)\(53\!\cdots\!93\)\( T^{16} - \)\(24\!\cdots\!76\)\( T^{17} + \)\(13\!\cdots\!42\)\( T^{18} - \)\(46\!\cdots\!38\)\( T^{19} + \)\(22\!\cdots\!10\)\( T^{20} - \)\(41\!\cdots\!80\)\( T^{21} + \)\(15\!\cdots\!59\)\( T^{22} \)
$23$ \( 1 - 156 T + 73431 T^{2} - 11373248 T^{3} + 2687398847 T^{4} - 394055328278 T^{5} + 65406470497063 T^{6} - 8767598228575096 T^{7} + 1194914341933969043 T^{8} - \)\(14\!\cdots\!68\)\( T^{9} + \)\(17\!\cdots\!93\)\( T^{10} - \)\(19\!\cdots\!20\)\( T^{11} + \)\(21\!\cdots\!31\)\( T^{12} - \)\(21\!\cdots\!52\)\( T^{13} + \)\(21\!\cdots\!09\)\( T^{14} - \)\(19\!\cdots\!16\)\( T^{15} + \)\(17\!\cdots\!41\)\( T^{16} - \)\(12\!\cdots\!82\)\( T^{17} + \)\(10\!\cdots\!81\)\( T^{18} - \)\(54\!\cdots\!68\)\( T^{19} + \)\(42\!\cdots\!57\)\( T^{20} - \)\(11\!\cdots\!44\)\( T^{21} + \)\(86\!\cdots\!83\)\( T^{22} \)
$29$ \( 1 + 4 T + 168672 T^{2} - 2671458 T^{3} + 14166871544 T^{4} - 422348015844 T^{5} + 783634871333089 T^{6} - 29815119693721768 T^{7} + 31717856367449893154 T^{8} - \)\(12\!\cdots\!44\)\( T^{9} + \)\(98\!\cdots\!20\)\( T^{10} - \)\(37\!\cdots\!84\)\( T^{11} + \)\(24\!\cdots\!80\)\( T^{12} - \)\(76\!\cdots\!24\)\( T^{13} + \)\(46\!\cdots\!26\)\( T^{14} - \)\(10\!\cdots\!88\)\( T^{15} + \)\(67\!\cdots\!61\)\( T^{16} - \)\(88\!\cdots\!84\)\( T^{17} + \)\(72\!\cdots\!76\)\( T^{18} - \)\(33\!\cdots\!98\)\( T^{19} + \)\(51\!\cdots\!48\)\( T^{20} + \)\(29\!\cdots\!04\)\( T^{21} + \)\(18\!\cdots\!89\)\( T^{22} \)
$31$ \( 1 - 514 T + 273008 T^{2} - 91018212 T^{3} + 30204604243 T^{4} - 8006812338532 T^{5} + 2113688769044763 T^{6} - 479616467203094800 T^{7} + \)\(10\!\cdots\!64\)\( T^{8} - \)\(21\!\cdots\!90\)\( T^{9} + \)\(41\!\cdots\!65\)\( T^{10} - \)\(71\!\cdots\!84\)\( T^{11} + \)\(12\!\cdots\!15\)\( T^{12} - \)\(18\!\cdots\!90\)\( T^{13} + \)\(28\!\cdots\!44\)\( T^{14} - \)\(37\!\cdots\!00\)\( T^{15} + \)\(49\!\cdots\!13\)\( T^{16} - \)\(55\!\cdots\!12\)\( T^{17} + \)\(62\!\cdots\!33\)\( T^{18} - \)\(56\!\cdots\!52\)\( T^{19} + \)\(50\!\cdots\!88\)\( T^{20} - \)\(28\!\cdots\!14\)\( T^{21} + \)\(16\!\cdots\!91\)\( T^{22} \)
$37$ \( 1 - 854 T + 571647 T^{2} - 266727976 T^{3} + 108319230249 T^{4} - 36878041821690 T^{5} + 11563161668890667 T^{6} - 3266843702881482060 T^{7} + \)\(88\!\cdots\!89\)\( T^{8} - \)\(22\!\cdots\!66\)\( T^{9} + \)\(53\!\cdots\!51\)\( T^{10} - \)\(12\!\cdots\!28\)\( T^{11} + \)\(27\!\cdots\!03\)\( T^{12} - \)\(56\!\cdots\!94\)\( T^{13} + \)\(11\!\cdots\!53\)\( T^{14} - \)\(21\!\cdots\!60\)\( T^{15} + \)\(38\!\cdots\!31\)\( T^{16} - \)\(62\!\cdots\!10\)\( T^{17} + \)\(92\!\cdots\!13\)\( T^{18} - \)\(11\!\cdots\!36\)\( T^{19} + \)\(12\!\cdots\!51\)\( T^{20} - \)\(94\!\cdots\!46\)\( T^{21} + \)\(56\!\cdots\!97\)\( T^{22} \)
$41$ \( 1 - 674 T + 505392 T^{2} - 248120982 T^{3} + 113138864141 T^{4} - 41842480021342 T^{5} + 14451356447123773 T^{6} - 4252961104909941832 T^{7} + \)\(12\!\cdots\!44\)\( T^{8} - \)\(30\!\cdots\!08\)\( T^{9} + \)\(80\!\cdots\!45\)\( T^{10} - \)\(20\!\cdots\!72\)\( T^{11} + \)\(55\!\cdots\!45\)\( T^{12} - \)\(14\!\cdots\!28\)\( T^{13} + \)\(39\!\cdots\!84\)\( T^{14} - \)\(95\!\cdots\!92\)\( T^{15} + \)\(22\!\cdots\!73\)\( T^{16} - \)\(44\!\cdots\!82\)\( T^{17} + \)\(83\!\cdots\!81\)\( T^{18} - \)\(12\!\cdots\!02\)\( T^{19} + \)\(17\!\cdots\!52\)\( T^{20} - \)\(16\!\cdots\!74\)\( T^{21} + \)\(16\!\cdots\!21\)\( T^{22} \)
$43$ \( 1 - 738 T + 678320 T^{2} - 336098816 T^{3} + 188122192295 T^{4} - 73683204831676 T^{5} + 32094290901486523 T^{6} - 10693441771300133020 T^{7} + \)\(39\!\cdots\!16\)\( T^{8} - \)\(11\!\cdots\!82\)\( T^{9} + \)\(38\!\cdots\!09\)\( T^{10} - \)\(10\!\cdots\!72\)\( T^{11} + \)\(30\!\cdots\!63\)\( T^{12} - \)\(73\!\cdots\!18\)\( T^{13} + \)\(19\!\cdots\!88\)\( T^{14} - \)\(42\!\cdots\!20\)\( T^{15} + \)\(10\!\cdots\!61\)\( T^{16} - \)\(18\!\cdots\!24\)\( T^{17} + \)\(37\!\cdots\!85\)\( T^{18} - \)\(53\!\cdots\!16\)\( T^{19} + \)\(86\!\cdots\!40\)\( T^{20} - \)\(74\!\cdots\!62\)\( T^{21} + \)\(80\!\cdots\!43\)\( T^{22} \)
$47$ \( 1 - 54 T + 697182 T^{2} + 26080052 T^{3} + 221385365174 T^{4} + 31282600679358 T^{5} + 43485131295521671 T^{6} + 11069890776990630472 T^{7} + \)\(61\!\cdots\!54\)\( T^{8} + \)\(21\!\cdots\!80\)\( T^{9} + \)\(71\!\cdots\!96\)\( T^{10} + \)\(27\!\cdots\!44\)\( T^{11} + \)\(74\!\cdots\!08\)\( T^{12} + \)\(23\!\cdots\!20\)\( T^{13} + \)\(69\!\cdots\!18\)\( T^{14} + \)\(12\!\cdots\!52\)\( T^{15} + \)\(52\!\cdots\!53\)\( T^{16} + \)\(39\!\cdots\!62\)\( T^{17} + \)\(28\!\cdots\!78\)\( T^{18} + \)\(35\!\cdots\!12\)\( T^{19} + \)\(97\!\cdots\!66\)\( T^{20} - \)\(78\!\cdots\!46\)\( T^{21} + \)\(15\!\cdots\!27\)\( T^{22} \)
$53$ \( 1 + 190 T + 1185100 T^{2} + 241452606 T^{3} + 662303481849 T^{4} + 138577956455522 T^{5} + 233792199385929361 T^{6} + 48211804718728645108 T^{7} + \)\(58\!\cdots\!84\)\( T^{8} + \)\(11\!\cdots\!64\)\( T^{9} + \)\(11\!\cdots\!37\)\( T^{10} + \)\(19\!\cdots\!40\)\( T^{11} + \)\(16\!\cdots\!49\)\( T^{12} + \)\(25\!\cdots\!56\)\( T^{13} + \)\(19\!\cdots\!72\)\( T^{14} + \)\(23\!\cdots\!28\)\( T^{15} + \)\(17\!\cdots\!77\)\( T^{16} + \)\(15\!\cdots\!58\)\( T^{17} + \)\(10\!\cdots\!97\)\( T^{18} + \)\(58\!\cdots\!86\)\( T^{19} + \)\(42\!\cdots\!00\)\( T^{20} + \)\(10\!\cdots\!10\)\( T^{21} + \)\(79\!\cdots\!73\)\( T^{22} \)
$59$ \( 1 - 18 T + 1688415 T^{2} - 55601626 T^{3} + 1368203079403 T^{4} - 59175589303606 T^{5} + 704311947568826655 T^{6} - 34184170031622487006 T^{7} + \)\(25\!\cdots\!31\)\( T^{8} - \)\(12\!\cdots\!98\)\( T^{9} + \)\(69\!\cdots\!53\)\( T^{10} - \)\(31\!\cdots\!36\)\( T^{11} + \)\(14\!\cdots\!87\)\( T^{12} - \)\(52\!\cdots\!18\)\( T^{13} + \)\(22\!\cdots\!09\)\( T^{14} - \)\(60\!\cdots\!86\)\( T^{15} + \)\(25\!\cdots\!45\)\( T^{16} - \)\(44\!\cdots\!26\)\( T^{17} + \)\(21\!\cdots\!77\)\( T^{18} - \)\(17\!\cdots\!86\)\( T^{19} + \)\(10\!\cdots\!85\)\( T^{20} - \)\(24\!\cdots\!18\)\( T^{21} + \)\(27\!\cdots\!79\)\( T^{22} \)
$61$ \( 1 - 328 T + 1630931 T^{2} - 660209892 T^{3} + 1301904162907 T^{4} - 602122605777864 T^{5} + 673101019700605745 T^{6} - \)\(33\!\cdots\!96\)\( T^{7} + \)\(25\!\cdots\!22\)\( T^{8} - \)\(12\!\cdots\!92\)\( T^{9} + \)\(72\!\cdots\!54\)\( T^{10} - \)\(33\!\cdots\!24\)\( T^{11} + \)\(16\!\cdots\!74\)\( T^{12} - \)\(64\!\cdots\!12\)\( T^{13} + \)\(29\!\cdots\!02\)\( T^{14} - \)\(88\!\cdots\!16\)\( T^{15} + \)\(40\!\cdots\!45\)\( T^{16} - \)\(82\!\cdots\!84\)\( T^{17} + \)\(40\!\cdots\!27\)\( T^{18} - \)\(46\!\cdots\!72\)\( T^{19} + \)\(26\!\cdots\!51\)\( T^{20} - \)\(11\!\cdots\!28\)\( T^{21} + \)\(82\!\cdots\!81\)\( T^{22} \)
$67$ \( ( 1 - 67 T )^{11} \)
$71$ \( 1 - 264 T + 786269 T^{2} - 265960708 T^{3} + 375768138543 T^{4} - 139813100971560 T^{5} + 172540296176047659 T^{6} - 51391817085993895472 T^{7} + \)\(77\!\cdots\!38\)\( T^{8} - \)\(21\!\cdots\!12\)\( T^{9} + \)\(30\!\cdots\!74\)\( T^{10} - \)\(93\!\cdots\!16\)\( T^{11} + \)\(11\!\cdots\!14\)\( T^{12} - \)\(27\!\cdots\!52\)\( T^{13} + \)\(35\!\cdots\!78\)\( T^{14} - \)\(84\!\cdots\!52\)\( T^{15} + \)\(10\!\cdots\!09\)\( T^{16} - \)\(29\!\cdots\!60\)\( T^{17} + \)\(28\!\cdots\!53\)\( T^{18} - \)\(71\!\cdots\!48\)\( T^{19} + \)\(75\!\cdots\!79\)\( T^{20} - \)\(91\!\cdots\!64\)\( T^{21} + \)\(12\!\cdots\!11\)\( T^{22} \)
$73$ \( 1 - 330 T + 2575219 T^{2} - 419660560 T^{3} + 2965582712909 T^{4} + 76152339817278 T^{5} + 2040405011688266531 T^{6} + \)\(52\!\cdots\!92\)\( T^{7} + \)\(97\!\cdots\!61\)\( T^{8} + \)\(50\!\cdots\!30\)\( T^{9} + \)\(37\!\cdots\!79\)\( T^{10} + \)\(25\!\cdots\!20\)\( T^{11} + \)\(14\!\cdots\!43\)\( T^{12} + \)\(76\!\cdots\!70\)\( T^{13} + \)\(57\!\cdots\!93\)\( T^{14} + \)\(11\!\cdots\!32\)\( T^{15} + \)\(18\!\cdots\!67\)\( T^{16} + \)\(26\!\cdots\!82\)\( T^{17} + \)\(39\!\cdots\!57\)\( T^{18} - \)\(22\!\cdots\!60\)\( T^{19} + \)\(52\!\cdots\!43\)\( T^{20} - \)\(26\!\cdots\!70\)\( T^{21} + \)\(30\!\cdots\!33\)\( T^{22} \)
$79$ \( 1 - 456 T + 3551589 T^{2} - 1005960608 T^{3} + 5377288299895 T^{4} - 317445479239976 T^{5} + 4484252851245209195 T^{6} + \)\(12\!\cdots\!48\)\( T^{7} + \)\(22\!\cdots\!58\)\( T^{8} + \)\(18\!\cdots\!00\)\( T^{9} + \)\(85\!\cdots\!22\)\( T^{10} + \)\(12\!\cdots\!92\)\( T^{11} + \)\(42\!\cdots\!58\)\( T^{12} + \)\(44\!\cdots\!00\)\( T^{13} + \)\(27\!\cdots\!02\)\( T^{14} + \)\(72\!\cdots\!68\)\( T^{15} + \)\(13\!\cdots\!05\)\( T^{16} - \)\(45\!\cdots\!36\)\( T^{17} + \)\(38\!\cdots\!05\)\( T^{18} - \)\(35\!\cdots\!48\)\( T^{19} + \)\(61\!\cdots\!51\)\( T^{20} - \)\(38\!\cdots\!56\)\( T^{21} + \)\(41\!\cdots\!39\)\( T^{22} \)
$83$ \( 1 + 2432 T + 7739218 T^{2} + 13306850120 T^{3} + 24669236287699 T^{4} + 32962222670044834 T^{5} + 44909190536655960389 T^{6} + \)\(48\!\cdots\!26\)\( T^{7} + \)\(52\!\cdots\!22\)\( T^{8} + \)\(47\!\cdots\!86\)\( T^{9} + \)\(43\!\cdots\!79\)\( T^{10} + \)\(32\!\cdots\!28\)\( T^{11} + \)\(24\!\cdots\!73\)\( T^{12} + \)\(15\!\cdots\!34\)\( T^{13} + \)\(99\!\cdots\!66\)\( T^{14} + \)\(52\!\cdots\!86\)\( T^{15} + \)\(27\!\cdots\!23\)\( T^{16} + \)\(11\!\cdots\!06\)\( T^{17} + \)\(49\!\cdots\!17\)\( T^{18} + \)\(15\!\cdots\!20\)\( T^{19} + \)\(50\!\cdots\!86\)\( T^{20} + \)\(90\!\cdots\!68\)\( T^{21} + \)\(21\!\cdots\!63\)\( T^{22} \)
$89$ \( 1 + 2340 T + 6296568 T^{2} + 10689382786 T^{3} + 18332612588476 T^{4} + 24911509233643624 T^{5} + 33287596323009666953 T^{6} + \)\(37\!\cdots\!04\)\( T^{7} + \)\(42\!\cdots\!14\)\( T^{8} + \)\(41\!\cdots\!76\)\( T^{9} + \)\(39\!\cdots\!88\)\( T^{10} + \)\(33\!\cdots\!12\)\( T^{11} + \)\(27\!\cdots\!72\)\( T^{12} + \)\(20\!\cdots\!36\)\( T^{13} + \)\(14\!\cdots\!26\)\( T^{14} + \)\(93\!\cdots\!84\)\( T^{15} + \)\(57\!\cdots\!97\)\( T^{16} + \)\(30\!\cdots\!44\)\( T^{17} + \)\(15\!\cdots\!64\)\( T^{18} + \)\(65\!\cdots\!26\)\( T^{19} + \)\(27\!\cdots\!72\)\( T^{20} + \)\(70\!\cdots\!40\)\( T^{21} + \)\(21\!\cdots\!69\)\( T^{22} \)
$97$ \( 1 - 1892 T + 7005955 T^{2} - 11480999900 T^{3} + 24150834766071 T^{4} - 33848527356974084 T^{5} + 53665754963792945309 T^{6} - \)\(64\!\cdots\!28\)\( T^{7} + \)\(85\!\cdots\!54\)\( T^{8} - \)\(89\!\cdots\!28\)\( T^{9} + \)\(10\!\cdots\!74\)\( T^{10} - \)\(93\!\cdots\!12\)\( T^{11} + \)\(92\!\cdots\!02\)\( T^{12} - \)\(74\!\cdots\!12\)\( T^{13} + \)\(64\!\cdots\!18\)\( T^{14} - \)\(44\!\cdots\!48\)\( T^{15} + \)\(33\!\cdots\!37\)\( T^{16} - \)\(19\!\cdots\!76\)\( T^{17} + \)\(12\!\cdots\!87\)\( T^{18} - \)\(55\!\cdots\!00\)\( T^{19} + \)\(30\!\cdots\!15\)\( T^{20} - \)\(75\!\cdots\!08\)\( T^{21} + \)\(36\!\cdots\!77\)\( T^{22} \)
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