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

Related objects

Downloads

Learn more about

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: \( 2^{5} \)
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.

Atkin-Lehner signs

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

Hecke kernels

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