Properties

Label 201.4.a.d
Level 201
Weight 4
Character orbit 201.a
Self dual yes
Analytic conductor 11.859
Analytic rank 0
Dimension 9
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: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} -3 q^{3} + ( 5 + \beta_{1} + \beta_{2} ) q^{4} + ( 1 + \beta_{1} + \beta_{2} + \beta_{7} ) q^{5} -3 \beta_{1} q^{6} + ( \beta_{1} + \beta_{3} - \beta_{5} + 2 \beta_{8} ) q^{7} + ( 5 + 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{8} + 9 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} -3 q^{3} + ( 5 + \beta_{1} + \beta_{2} ) q^{4} + ( 1 + \beta_{1} + \beta_{2} + \beta_{7} ) q^{5} -3 \beta_{1} q^{6} + ( \beta_{1} + \beta_{3} - \beta_{5} + 2 \beta_{8} ) q^{7} + ( 5 + 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{8} + 9 q^{9} + ( 4 + 8 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{6} - 3 \beta_{8} ) q^{10} + ( 7 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} ) q^{11} + ( -15 - 3 \beta_{1} - 3 \beta_{2} ) q^{12} + ( -23 + 4 \beta_{1} - 3 \beta_{2} + \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} ) q^{13} + ( 11 - 2 \beta_{1} - \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - \beta_{5} - 5 \beta_{7} - \beta_{8} ) q^{14} + ( -3 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{7} ) q^{15} + ( 18 + 13 \beta_{1} + \beta_{2} - 6 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - 4 \beta_{8} ) q^{16} + ( 13 - 3 \beta_{2} + 6 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} + 6 \beta_{8} ) q^{17} + 9 \beta_{1} q^{18} + ( 15 + 10 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} + 5 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} - 2 \beta_{8} ) q^{19} + ( 84 + 16 \beta_{1} + 3 \beta_{2} - 7 \beta_{3} - 8 \beta_{4} + 8 \beta_{5} + \beta_{6} + 4 \beta_{7} - 3 \beta_{8} ) q^{20} + ( -3 \beta_{1} - 3 \beta_{3} + 3 \beta_{5} - 6 \beta_{8} ) q^{21} + ( 23 + 10 \beta_{1} + 9 \beta_{4} - \beta_{5} + 2 \beta_{6} - 4 \beta_{7} + 6 \beta_{8} ) q^{22} + ( 51 + \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} - 3 \beta_{6} + 2 \beta_{7} + 4 \beta_{8} ) q^{23} + ( -15 - 15 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{24} + ( 51 + 15 \beta_{1} - 7 \beta_{4} + 9 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} ) q^{25} + ( 61 - 26 \beta_{1} + 4 \beta_{2} + 7 \beta_{3} + 18 \beta_{4} - 11 \beta_{5} - \beta_{6} - 9 \beta_{7} + 6 \beta_{8} ) q^{26} -27 q^{27} + ( -28 - 2 \beta_{1} - 10 \beta_{2} - 11 \beta_{5} - 12 \beta_{6} + 11 \beta_{7} + 7 \beta_{8} ) q^{28} + ( 64 - 22 \beta_{1} + 2 \beta_{2} - 7 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 9 \beta_{6} - 9 \beta_{7} ) q^{29} + ( -12 - 24 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 9 \beta_{5} - 3 \beta_{6} + 9 \beta_{8} ) q^{30} + ( 25 - 9 \beta_{1} - 7 \beta_{2} + 11 \beta_{3} - 11 \beta_{4} + 5 \beta_{5} - 6 \beta_{6} + 8 \beta_{7} - 8 \beta_{8} ) q^{31} + ( 138 + 9 \beta_{1} + 10 \beta_{2} - 8 \beta_{3} - 10 \beta_{4} + 4 \beta_{5} + 8 \beta_{6} - 4 \beta_{8} ) q^{32} + ( -21 - 6 \beta_{1} + 3 \beta_{2} - 6 \beta_{3} + 9 \beta_{4} + 6 \beta_{5} ) q^{33} + ( 11 - 22 \beta_{1} - 14 \beta_{2} + 28 \beta_{3} - \beta_{4} - 13 \beta_{5} - 6 \beta_{6} - 6 \beta_{7} + 4 \beta_{8} ) q^{34} + ( 24 - 41 \beta_{1} + 2 \beta_{2} + 13 \beta_{4} - 2 \beta_{5} - 10 \beta_{6} + 13 \beta_{7} - 6 \beta_{8} ) q^{35} + ( 45 + 9 \beta_{1} + 9 \beta_{2} ) q^{36} + ( -24 - 19 \beta_{1} + 8 \beta_{3} + 11 \beta_{4} - 11 \beta_{5} + 9 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} ) q^{37} + ( 129 + 22 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} - 29 \beta_{4} + 13 \beta_{5} - 4 \beta_{6} + 20 \beta_{7} - 24 \beta_{8} ) q^{38} + ( 69 - 12 \beta_{1} + 9 \beta_{2} - 3 \beta_{3} + 6 \beta_{4} + 12 \beta_{5} + 9 \beta_{6} - 9 \beta_{7} ) q^{39} + ( 167 + 64 \beta_{1} + 24 \beta_{2} - 27 \beta_{3} - 8 \beta_{4} + 26 \beta_{5} + 25 \beta_{6} - 12 \beta_{7} - 7 \beta_{8} ) q^{40} + ( 44 - 33 \beta_{1} - 10 \beta_{2} - 4 \beta_{3} - 9 \beta_{4} - 8 \beta_{5} - 12 \beta_{6} + 21 \beta_{7} - 2 \beta_{8} ) q^{41} + ( -33 + 6 \beta_{1} + 3 \beta_{2} - 12 \beta_{3} - 6 \beta_{4} + 3 \beta_{5} + 15 \beta_{7} + 3 \beta_{8} ) q^{42} + ( 23 + \beta_{1} + 13 \beta_{2} - 8 \beta_{3} + 4 \beta_{4} - 9 \beta_{5} + \beta_{6} - 5 \beta_{7} ) q^{43} + ( 97 - 14 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} - \beta_{4} - 3 \beta_{5} - 18 \beta_{6} + 4 \beta_{7} - 2 \beta_{8} ) q^{44} + ( 9 + 9 \beta_{1} + 9 \beta_{2} + 9 \beta_{7} ) q^{45} + ( 28 + 46 \beta_{1} + 10 \beta_{2} + 10 \beta_{4} + 3 \beta_{5} + 8 \beta_{6} - 23 \beta_{7} - 11 \beta_{8} ) q^{46} + ( 61 - 34 \beta_{1} + \beta_{2} + 9 \beta_{3} + 28 \beta_{4} + 6 \beta_{5} - 3 \beta_{6} - 11 \beta_{7} + 6 \beta_{8} ) q^{47} + ( -54 - 39 \beta_{1} - 3 \beta_{2} + 18 \beta_{3} + 9 \beta_{4} - 9 \beta_{5} + 12 \beta_{8} ) q^{48} + ( 74 - 39 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} + 8 \beta_{4} + 9 \beta_{5} + 13 \beta_{6} - 29 \beta_{7} + 2 \beta_{8} ) q^{49} + ( 199 + 61 \beta_{1} + 25 \beta_{2} - 15 \beta_{3} - 7 \beta_{4} + 43 \beta_{5} + 31 \beta_{6} - 14 \beta_{7} - 19 \beta_{8} ) q^{50} + ( -39 + 9 \beta_{2} - 18 \beta_{3} - 9 \beta_{4} + 6 \beta_{5} + 6 \beta_{7} - 18 \beta_{8} ) q^{51} + ( -183 + 6 \beta_{1} - 32 \beta_{2} + 31 \beta_{3} + 6 \beta_{4} - 47 \beta_{5} - 37 \beta_{6} - \beta_{7} + 46 \beta_{8} ) q^{52} + ( -34 - 21 \beta_{1} - 8 \beta_{2} - 10 \beta_{3} + 11 \beta_{4} + 4 \beta_{5} + 8 \beta_{6} - 33 \beta_{7} + 20 \beta_{8} ) q^{53} -27 \beta_{1} q^{54} + ( -51 - 18 \beta_{1} - 7 \beta_{2} - 2 \beta_{3} - 9 \beta_{4} + 8 \beta_{5} - 34 \beta_{6} + 34 \beta_{7} - 28 \beta_{8} ) q^{55} + ( -62 - 62 \beta_{1} + 9 \beta_{2} + 6 \beta_{3} + 33 \beta_{4} - 30 \beta_{5} - 4 \beta_{6} - 5 \beta_{7} + 7 \beta_{8} ) q^{56} + ( -45 - 30 \beta_{1} - 9 \beta_{2} + 12 \beta_{3} - 15 \beta_{4} - 6 \beta_{5} - 12 \beta_{6} - 12 \beta_{7} + 6 \beta_{8} ) q^{57} + ( -250 + 46 \beta_{1} - 30 \beta_{2} - 33 \beta_{3} - 33 \beta_{4} + 2 \beta_{5} - 7 \beta_{6} + 5 \beta_{7} + 6 \beta_{8} ) q^{58} + ( 98 - 43 \beta_{1} - 24 \beta_{2} - 18 \beta_{3} - 39 \beta_{4} - 8 \beta_{5} + 14 \beta_{6} + 17 \beta_{7} + 4 \beta_{8} ) q^{59} + ( -252 - 48 \beta_{1} - 9 \beta_{2} + 21 \beta_{3} + 24 \beta_{4} - 24 \beta_{5} - 3 \beta_{6} - 12 \beta_{7} + 9 \beta_{8} ) q^{60} + ( -89 + 14 \beta_{1} + 13 \beta_{2} - 37 \beta_{3} - 34 \beta_{4} + 2 \beta_{5} + 7 \beta_{6} + 19 \beta_{7} - 8 \beta_{8} ) q^{61} + ( -144 + 10 \beta_{1} + \beta_{2} + 14 \beta_{3} + 45 \beta_{4} + 26 \beta_{5} + 32 \beta_{6} + \beta_{7} + 5 \beta_{8} ) q^{62} + ( 9 \beta_{1} + 9 \beta_{3} - 9 \beta_{5} + 18 \beta_{8} ) q^{63} + ( -73 + 113 \beta_{1} + 17 \beta_{2} - 12 \beta_{3} + 10 \beta_{4} + 30 \beta_{5} + 24 \beta_{6} + 8 \beta_{8} ) q^{64} + ( -51 - 84 \beta_{1} - 49 \beta_{2} + 46 \beta_{3} - 19 \beta_{4} - 16 \beta_{5} - 14 \beta_{6} - 30 \beta_{7} + 6 \beta_{8} ) q^{65} + ( -69 - 30 \beta_{1} - 27 \beta_{4} + 3 \beta_{5} - 6 \beta_{6} + 12 \beta_{7} - 18 \beta_{8} ) q^{66} -67 q^{67} + ( -393 - 78 \beta_{1} - 40 \beta_{2} + 32 \beta_{3} + 29 \beta_{4} - 55 \beta_{5} - 26 \beta_{6} + 18 \beta_{7} + 28 \beta_{8} ) q^{68} + ( -153 - 3 \beta_{1} + 3 \beta_{2} + 9 \beta_{3} + 3 \beta_{4} + 9 \beta_{6} - 6 \beta_{7} - 12 \beta_{8} ) q^{69} + ( -575 + 2 \beta_{1} - 34 \beta_{2} + 29 \beta_{3} + 18 \beta_{4} - 30 \beta_{5} - 23 \beta_{6} + 24 \beta_{7} - 3 \beta_{8} ) q^{70} + ( 353 - 26 \beta_{1} + 17 \beta_{2} - 9 \beta_{3} + 10 \beta_{4} + 6 \beta_{5} + 23 \beta_{6} + 3 \beta_{7} + 6 \beta_{8} ) q^{71} + ( 45 + 45 \beta_{1} + 18 \beta_{2} - 18 \beta_{3} + 9 \beta_{4} + 9 \beta_{5} ) q^{72} + ( -123 - 21 \beta_{1} + 7 \beta_{2} + 20 \beta_{3} - 6 \beta_{4} + 19 \beta_{5} + 15 \beta_{6} - 51 \beta_{7} - 4 \beta_{8} ) q^{73} + ( -251 - 88 \beta_{1} - 67 \beta_{2} + 43 \beta_{3} - 39 \beta_{4} - 29 \beta_{5} - 39 \beta_{6} + 42 \beta_{7} - \beta_{8} ) q^{74} + ( -153 - 45 \beta_{1} + 21 \beta_{4} - 27 \beta_{5} - 9 \beta_{6} + 9 \beta_{7} - 6 \beta_{8} ) q^{75} + ( 45 + 198 \beta_{1} + 56 \beta_{2} - 46 \beta_{3} + 25 \beta_{4} + 95 \beta_{5} + 48 \beta_{6} - 28 \beta_{7} - 16 \beta_{8} ) q^{76} + ( 155 + 52 \beta_{1} - 51 \beta_{2} + 22 \beta_{3} - 47 \beta_{4} + 12 \beta_{5} - 26 \beta_{6} - 22 \beta_{7} + 6 \beta_{8} ) q^{77} + ( -183 + 78 \beta_{1} - 12 \beta_{2} - 21 \beta_{3} - 54 \beta_{4} + 33 \beta_{5} + 3 \beta_{6} + 27 \beta_{7} - 18 \beta_{8} ) q^{78} + ( 294 - 24 \beta_{1} + 14 \beta_{2} - 39 \beta_{3} - 21 \beta_{4} - 10 \beta_{5} + 5 \beta_{6} + 5 \beta_{7} + 16 \beta_{8} ) q^{79} + ( 123 + 212 \beta_{1} + 75 \beta_{2} - 81 \beta_{3} - 49 \beta_{4} + 73 \beta_{5} + 41 \beta_{6} - 4 \beta_{7} - 55 \beta_{8} ) q^{80} + 81 q^{81} + ( -389 - 14 \beta_{1} - 14 \beta_{2} + 13 \beta_{3} + 60 \beta_{4} + 21 \beta_{6} - 40 \beta_{7} - 23 \beta_{8} ) q^{82} + ( 132 - 49 \beta_{1} + 4 \beta_{2} - 12 \beta_{3} + 17 \beta_{4} + 26 \beta_{6} + 17 \beta_{7} - 40 \beta_{8} ) q^{83} + ( 84 + 6 \beta_{1} + 30 \beta_{2} + 33 \beta_{5} + 36 \beta_{6} - 33 \beta_{7} - 21 \beta_{8} ) q^{84} + ( -329 - 108 \beta_{1} + 11 \beta_{2} + 58 \beta_{3} + 87 \beta_{4} - 52 \beta_{5} - 30 \beta_{6} + 60 \beta_{7} ) q^{85} + ( -48 + 118 \beta_{1} + 7 \beta_{2} - 39 \beta_{3} + 2 \beta_{4} - 30 \beta_{5} - 31 \beta_{6} + 2 \beta_{7} + 23 \beta_{8} ) q^{86} + ( -192 + 66 \beta_{1} - 6 \beta_{2} + 21 \beta_{3} + 9 \beta_{4} + 6 \beta_{5} - 27 \beta_{6} + 27 \beta_{7} ) q^{87} + ( -457 + 90 \beta_{1} + 20 \beta_{2} + 13 \beta_{4} - 25 \beta_{5} - 18 \beta_{6} + 4 \beta_{7} - 10 \beta_{8} ) q^{88} + ( 85 - 2 \beta_{1} - 31 \beta_{2} - 36 \beta_{3} - 3 \beta_{4} - 18 \beta_{5} - 38 \beta_{6} + 34 \beta_{7} + 26 \beta_{8} ) q^{89} + ( 36 + 72 \beta_{1} + 18 \beta_{2} - 9 \beta_{3} + 9 \beta_{4} + 27 \beta_{5} + 9 \beta_{6} - 27 \beta_{8} ) q^{90} + ( 235 + 20 \beta_{1} + 53 \beta_{2} + 8 \beta_{3} + \beta_{4} + 30 \beta_{5} - 18 \beta_{6} - 26 \beta_{7} - 82 \beta_{8} ) q^{91} + ( 148 + 134 \beta_{1} + 41 \beta_{2} - 10 \beta_{3} - 37 \beta_{4} - 42 \beta_{5} - 24 \beta_{6} + 53 \beta_{7} + 37 \beta_{8} ) q^{92} + ( -75 + 27 \beta_{1} + 21 \beta_{2} - 33 \beta_{3} + 33 \beta_{4} - 15 \beta_{5} + 18 \beta_{6} - 24 \beta_{7} + 24 \beta_{8} ) q^{93} + ( -447 - 20 \beta_{1} - 58 \beta_{2} + 67 \beta_{3} - 40 \beta_{4} - 59 \beta_{5} - 49 \beta_{6} + 41 \beta_{7} + 24 \beta_{8} ) q^{94} + ( 655 + 218 \beta_{1} + 57 \beta_{2} - 12 \beta_{3} + 67 \beta_{4} + 82 \beta_{5} + 66 \beta_{6} - 62 \beta_{7} + 18 \beta_{8} ) q^{95} + ( -414 - 27 \beta_{1} - 30 \beta_{2} + 24 \beta_{3} + 30 \beta_{4} - 12 \beta_{5} - 24 \beta_{6} + 12 \beta_{8} ) q^{96} + ( -181 + 4 \beta_{1} + 31 \beta_{2} + 32 \beta_{3} + 49 \beta_{4} - 30 \beta_{5} + 16 \beta_{6} - 18 \beta_{7} + 104 \beta_{8} ) q^{97} + ( -482 + 17 \beta_{1} - 65 \beta_{2} - 9 \beta_{3} - 70 \beta_{4} - 16 \beta_{5} - 25 \beta_{6} + 40 \beta_{7} + 43 \beta_{8} ) q^{98} + ( 63 + 18 \beta_{1} - 9 \beta_{2} + 18 \beta_{3} - 27 \beta_{4} - 18 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q + 3q^{2} - 27q^{3} + 45q^{4} + 12q^{5} - 9q^{6} + 8q^{7} + 51q^{8} + 81q^{9} + O(q^{10}) \) \( 9q + 3q^{2} - 27q^{3} + 45q^{4} + 12q^{5} - 9q^{6} + 8q^{7} + 51q^{8} + 81q^{9} + 45q^{10} + 72q^{11} - 135q^{12} - 166q^{13} + 93q^{14} - 36q^{15} + 173q^{16} + 146q^{17} + 27q^{18} + 154q^{19} + 763q^{20} - 24q^{21} + 244q^{22} + 476q^{23} - 153q^{24} + 465q^{25} + 502q^{26} - 243q^{27} - 141q^{28} + 432q^{29} - 135q^{30} + 248q^{31} + 1171q^{32} - 216q^{33} + 146q^{34} + 178q^{35} + 405q^{36} - 240q^{37} + 1182q^{38} + 498q^{39} + 1409q^{40} + 406q^{41} - 279q^{42} + 154q^{43} + 892q^{44} + 108q^{45} + 273q^{46} + 494q^{47} - 519q^{48} + 431q^{49} + 1658q^{50} - 438q^{51} - 1258q^{52} - 450q^{53} - 81q^{54} - 346q^{55} - 659q^{56} - 462q^{57} - 2114q^{58} + 732q^{59} - 2289q^{60} - 914q^{61} - 1265q^{62} + 72q^{63} - 467q^{64} - 536q^{65} - 732q^{66} - 603q^{67} - 3314q^{68} - 1428q^{69} - 4805q^{70} + 2990q^{71} + 459q^{72} - 1384q^{73} - 2043q^{74} - 1395q^{75} + 450q^{76} + 1660q^{77} - 1506q^{78} + 2438q^{79} + 995q^{80} + 729q^{81} - 3561q^{82} + 972q^{83} + 423q^{84} - 2706q^{85} - 21q^{86} - 1296q^{87} - 3796q^{88} + 1034q^{89} + 405q^{90} + 1898q^{91} + 1827q^{92} - 744q^{93} - 3502q^{94} + 6040q^{95} - 3513q^{96} - 1516q^{97} - 3996q^{98} + 648q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 3 x^{8} - 54 x^{7} + 138 x^{6} + 949 x^{5} - 2039 x^{4} - 5472 x^{3} + 10352 x^{2} + 3808 x - 6656\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 13 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{8} - 6 \nu^{7} - 16 \nu^{6} + 146 \nu^{5} - 289 \nu^{4} - 492 \nu^{3} + 4784 \nu^{2} - 800 \nu - 7552 \)\()/640\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{8} - 18 \nu^{7} - 88 \nu^{6} + 558 \nu^{5} + 533 \nu^{4} - 3996 \nu^{3} + 832 \nu^{2} + 1920 \nu + 1024 \)\()/320\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{8} + 6 \nu^{7} + 36 \nu^{6} - 206 \nu^{5} - 411 \nu^{4} + 1912 \nu^{3} + 1656 \nu^{2} - 4400 \nu - 928 \)\()/160\)
\(\beta_{6}\)\(=\)\((\)\( 3 \nu^{8} - 18 \nu^{7} - 88 \nu^{6} + 598 \nu^{5} + 493 \nu^{4} - 5316 \nu^{3} + 1512 \nu^{2} + 10880 \nu - 2496 \)\()/320\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{8} + 16 \nu^{7} - 24 \nu^{6} - 486 \nu^{5} + 1409 \nu^{4} + 3382 \nu^{3} - 11144 \nu^{2} - 640 \nu + 8192 \)\()/320\)
\(\beta_{8}\)\(=\)\((\)\( -9 \nu^{8} + 54 \nu^{7} + 264 \nu^{6} - 1674 \nu^{5} - 1759 \nu^{4} + 12468 \nu^{3} + 544 \nu^{2} - 12960 \nu - 2432 \)\()/640\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 13\)
\(\nu^{3}\)\(=\)\(\beta_{5} + \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 21 \beta_{1} + 5\)
\(\nu^{4}\)\(=\)\(-4 \beta_{8} + 3 \beta_{5} - 3 \beta_{4} - 6 \beta_{3} + 25 \beta_{2} + 37 \beta_{1} + 266\)
\(\nu^{5}\)\(=\)\(-4 \beta_{8} + 8 \beta_{6} + 36 \beta_{5} + 22 \beta_{4} - 72 \beta_{3} + 74 \beta_{2} + 489 \beta_{1} + 298\)
\(\nu^{6}\)\(=\)\(-152 \beta_{8} + 24 \beta_{6} + 150 \beta_{5} - 110 \beta_{4} - 252 \beta_{3} + 633 \beta_{2} + 1209 \beta_{1} + 6087\)
\(\nu^{7}\)\(=\)\(-296 \beta_{8} + 32 \beta_{7} + 368 \beta_{6} + 1199 \beta_{5} + 355 \beta_{4} - 2142 \beta_{3} + 2306 \beta_{2} + 12029 \beta_{1} + 11447\)
\(\nu^{8}\)\(=\)\(-4780 \beta_{8} + 192 \beta_{7} + 1424 \beta_{6} + 5697 \beta_{5} - 3217 \beta_{4} - 8450 \beta_{3} + 16585 \beta_{2} + 37165 \beta_{1} + 147260\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−4.46691
−4.44127
−3.19321
−0.874823
0.837618
1.74668
3.11334
4.92710
5.35147
−4.46691 −3.00000 11.9533 −5.77925 13.4007 −31.2412 −17.6590 9.00000 25.8154
1.2 −4.44127 −3.00000 11.7249 14.0824 13.3238 12.5267 −16.5432 9.00000 −62.5436
1.3 −3.19321 −3.00000 2.19659 10.4341 9.57963 21.8629 18.5315 9.00000 −33.3182
1.4 −0.874823 −3.00000 −7.23468 −14.3103 2.62447 −20.3511 13.3277 9.00000 12.5190
1.5 0.837618 −3.00000 −7.29840 −4.06696 −2.51285 −13.0667 −12.8142 9.00000 −3.40656
1.6 1.74668 −3.00000 −4.94910 −20.3979 −5.24005 28.2703 −22.6180 9.00000 −35.6287
1.7 3.11334 −3.00000 1.69289 13.7648 −9.34002 11.2339 −19.6362 9.00000 42.8546
1.8 4.92710 −3.00000 16.2764 −2.16819 −14.7813 13.4538 40.7785 9.00000 −10.6829
1.9 5.35147 −3.00000 20.6382 20.4413 −16.0544 −14.6886 67.6328 9.00000 109.391
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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.d 9
3.b odd 2 1 603.4.a.f 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.4.a.d 9 1.a even 1 1 trivial
603.4.a.f 9 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 \(T_{2}^{9} - \cdots\) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(201))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T + 18 T^{2} - 54 T^{3} + 229 T^{4} - 791 T^{5} + 2920 T^{6} - 8432 T^{7} + 28256 T^{8} - 71040 T^{9} + 226048 T^{10} - 539648 T^{11} + 1495040 T^{12} - 3239936 T^{13} + 7503872 T^{14} - 14155776 T^{15} + 37748736 T^{16} - 50331648 T^{17} + 134217728 T^{18} \)
$3$ \( ( 1 + 3 T )^{9} \)
$5$ \( 1 - 12 T + 402 T^{2} - 3878 T^{3} + 83536 T^{4} - 592678 T^{5} + 10993915 T^{6} - 72898286 T^{7} + 1332330526 T^{8} - 8377438196 T^{9} + 166541315750 T^{10} - 1139035718750 T^{11} + 21472490234375 T^{12} - 144696777343750 T^{13} + 2549316406250000 T^{14} - 14793395996093750 T^{15} + 191688537597656250 T^{16} - 715255737304687500 T^{17} + 7450580596923828125 T^{18} \)
$7$ \( 1 - 8 T + 1360 T^{2} - 6146 T^{3} + 1030114 T^{4} - 2845284 T^{5} + 539455139 T^{6} - 605952564 T^{7} + 222804332978 T^{8} - 163555664060 T^{9} + 76421886211454 T^{10} - 71289713202036 T^{11} + 21768960673336373 T^{12} - 39382393012410084 T^{13} + 4890529577253423502 T^{14} - 10008229972757619554 T^{15} + \)\(75\!\cdots\!20\)\( T^{16} - \)\(15\!\cdots\!08\)\( T^{17} + \)\(65\!\cdots\!43\)\( T^{18} \)
$11$ \( 1 - 72 T + 8415 T^{2} - 371036 T^{3} + 26502200 T^{4} - 881101256 T^{5} + 55609614160 T^{6} - 1700130869828 T^{7} + 98424777319954 T^{8} - 2696859907348576 T^{9} + 131003378612858774 T^{10} - 3011885543883361508 T^{11} + \)\(13\!\cdots\!60\)\( T^{12} - \)\(27\!\cdots\!76\)\( T^{13} + \)\(11\!\cdots\!00\)\( T^{14} - \)\(20\!\cdots\!16\)\( T^{15} + \)\(62\!\cdots\!65\)\( T^{16} - \)\(70\!\cdots\!52\)\( T^{17} + \)\(13\!\cdots\!71\)\( T^{18} \)
$13$ \( 1 + 166 T + 19227 T^{2} + 1700084 T^{3} + 130220014 T^{4} + 8636685096 T^{5} + 524787877386 T^{6} + 29001087157964 T^{7} + 1506839569534976 T^{8} + 72656475062787556 T^{9} + 3310526534268342272 T^{10} + \)\(13\!\cdots\!76\)\( T^{11} + \)\(55\!\cdots\!78\)\( T^{12} + \)\(20\!\cdots\!76\)\( T^{13} + \)\(66\!\cdots\!98\)\( T^{14} + \)\(19\!\cdots\!36\)\( T^{15} + \)\(47\!\cdots\!51\)\( T^{16} + \)\(90\!\cdots\!26\)\( T^{17} + \)\(11\!\cdots\!17\)\( T^{18} \)
$17$ \( 1 - 146 T + 25393 T^{2} - 2988060 T^{3} + 312043948 T^{4} - 28772370960 T^{5} + 2416866553132 T^{6} - 184170173953316 T^{7} + 13896304963125990 T^{8} - 961479618638920764 T^{9} + 68272546283837988870 T^{10} - \)\(44\!\cdots\!04\)\( T^{11} + \)\(28\!\cdots\!04\)\( T^{12} - \)\(16\!\cdots\!60\)\( T^{13} + \)\(89\!\cdots\!64\)\( T^{14} - \)\(42\!\cdots\!40\)\( T^{15} + \)\(17\!\cdots\!81\)\( T^{16} - \)\(49\!\cdots\!66\)\( T^{17} + \)\(16\!\cdots\!73\)\( T^{18} \)
$19$ \( 1 - 154 T + 25071 T^{2} - 3294396 T^{3} + 427051432 T^{4} - 40197104720 T^{5} + 4147025596304 T^{6} - 377879862349732 T^{7} + 34156542879579354 T^{8} - 2626769419926025836 T^{9} + \)\(23\!\cdots\!86\)\( T^{10} - \)\(17\!\cdots\!92\)\( T^{11} + \)\(13\!\cdots\!16\)\( T^{12} - \)\(88\!\cdots\!20\)\( T^{13} + \)\(64\!\cdots\!68\)\( T^{14} - \)\(34\!\cdots\!36\)\( T^{15} + \)\(17\!\cdots\!49\)\( T^{16} - \)\(75\!\cdots\!34\)\( T^{17} + \)\(33\!\cdots\!39\)\( T^{18} \)
$23$ \( 1 - 476 T + 184170 T^{2} - 47979806 T^{3} + 10912553666 T^{4} - 2004773142082 T^{5} + 332699111283739 T^{6} - 47337566744182662 T^{7} + 6192689463326208976 T^{8} - \)\(71\!\cdots\!04\)\( T^{9} + \)\(75\!\cdots\!92\)\( T^{10} - \)\(70\!\cdots\!18\)\( T^{11} + \)\(59\!\cdots\!57\)\( T^{12} - \)\(43\!\cdots\!22\)\( T^{13} + \)\(29\!\cdots\!62\)\( T^{14} - \)\(15\!\cdots\!14\)\( T^{15} + \)\(72\!\cdots\!10\)\( T^{16} - \)\(22\!\cdots\!16\)\( T^{17} + \)\(58\!\cdots\!47\)\( T^{18} \)
$29$ \( 1 - 432 T + 163239 T^{2} - 37685132 T^{3} + 7516592690 T^{4} - 996267521000 T^{5} + 96567376225078 T^{6} + 928165322969836 T^{7} - 1889580242874161708 T^{8} + \)\(44\!\cdots\!36\)\( T^{9} - \)\(46\!\cdots\!12\)\( T^{10} + \)\(55\!\cdots\!56\)\( T^{11} + \)\(14\!\cdots\!82\)\( T^{12} - \)\(35\!\cdots\!00\)\( T^{13} + \)\(64\!\cdots\!10\)\( T^{14} - \)\(79\!\cdots\!52\)\( T^{15} + \)\(83\!\cdots\!31\)\( T^{16} - \)\(54\!\cdots\!92\)\( T^{17} + \)\(30\!\cdots\!09\)\( T^{18} \)
$31$ \( 1 - 248 T + 116084 T^{2} - 26140994 T^{3} + 6510341658 T^{4} - 1387341258952 T^{5} + 289781191846555 T^{6} - 57893685453153312 T^{7} + 11470480361208499942 T^{8} - \)\(19\!\cdots\!48\)\( T^{9} + \)\(34\!\cdots\!22\)\( T^{10} - \)\(51\!\cdots\!72\)\( T^{11} + \)\(76\!\cdots\!05\)\( T^{12} - \)\(10\!\cdots\!72\)\( T^{13} + \)\(15\!\cdots\!58\)\( T^{14} - \)\(18\!\cdots\!54\)\( T^{15} + \)\(24\!\cdots\!04\)\( T^{16} - \)\(15\!\cdots\!08\)\( T^{17} + \)\(18\!\cdots\!11\)\( T^{18} \)
$37$ \( 1 + 240 T + 222132 T^{2} + 51591910 T^{3} + 29345935528 T^{4} + 6379672578708 T^{5} + 2654244823067435 T^{6} + 515928894717883916 T^{7} + \)\(17\!\cdots\!20\)\( T^{8} + \)\(30\!\cdots\!52\)\( T^{9} + \)\(90\!\cdots\!60\)\( T^{10} + \)\(13\!\cdots\!44\)\( T^{11} + \)\(34\!\cdots\!95\)\( T^{12} + \)\(41\!\cdots\!48\)\( T^{13} + \)\(97\!\cdots\!04\)\( T^{14} + \)\(87\!\cdots\!90\)\( T^{15} + \)\(19\!\cdots\!84\)\( T^{16} + \)\(10\!\cdots\!40\)\( T^{17} + \)\(21\!\cdots\!33\)\( T^{18} \)
$41$ \( 1 - 406 T + 443078 T^{2} - 169007214 T^{3} + 95833068084 T^{4} - 33603463540144 T^{5} + 13170271004672971 T^{6} - 4135158820090704122 T^{7} + \)\(12\!\cdots\!38\)\( T^{8} - \)\(34\!\cdots\!52\)\( T^{9} + \)\(86\!\cdots\!98\)\( T^{10} - \)\(19\!\cdots\!02\)\( T^{11} + \)\(43\!\cdots\!31\)\( T^{12} - \)\(75\!\cdots\!64\)\( T^{13} + \)\(14\!\cdots\!84\)\( T^{14} - \)\(18\!\cdots\!94\)\( T^{15} + \)\(32\!\cdots\!98\)\( T^{16} - \)\(20\!\cdots\!66\)\( T^{17} + \)\(35\!\cdots\!81\)\( T^{18} \)
$43$ \( 1 - 154 T + 500942 T^{2} - 78351160 T^{3} + 119493446954 T^{4} - 18638281797594 T^{5} + 18123972363860097 T^{6} - 2694706711311237100 T^{7} + \)\(19\!\cdots\!88\)\( T^{8} - \)\(25\!\cdots\!24\)\( T^{9} + \)\(15\!\cdots\!16\)\( T^{10} - \)\(17\!\cdots\!00\)\( T^{11} + \)\(91\!\cdots\!71\)\( T^{12} - \)\(74\!\cdots\!94\)\( T^{13} + \)\(37\!\cdots\!78\)\( T^{14} - \)\(19\!\cdots\!40\)\( T^{15} + \)\(10\!\cdots\!06\)\( T^{16} - \)\(24\!\cdots\!54\)\( T^{17} + \)\(12\!\cdots\!07\)\( T^{18} \)
$47$ \( 1 - 494 T + 671533 T^{2} - 282294988 T^{3} + 219372065914 T^{4} - 79086646685320 T^{5} + 45200567442112378 T^{6} - 14064789473590519764 T^{7} + \)\(64\!\cdots\!20\)\( T^{8} - \)\(17\!\cdots\!60\)\( T^{9} + \)\(67\!\cdots\!60\)\( T^{10} - \)\(15\!\cdots\!56\)\( T^{11} + \)\(50\!\cdots\!26\)\( T^{12} - \)\(91\!\cdots\!20\)\( T^{13} + \)\(26\!\cdots\!02\)\( T^{14} - \)\(35\!\cdots\!32\)\( T^{15} + \)\(87\!\cdots\!51\)\( T^{16} - \)\(66\!\cdots\!14\)\( T^{17} + \)\(14\!\cdots\!63\)\( T^{18} \)
$53$ \( 1 + 450 T + 769898 T^{2} + 289396366 T^{3} + 273430432984 T^{4} + 88115920345792 T^{5} + 60836319355642539 T^{6} + 17453599155816128270 T^{7} + \)\(10\!\cdots\!26\)\( T^{8} + \)\(27\!\cdots\!40\)\( T^{9} + \)\(15\!\cdots\!02\)\( T^{10} + \)\(38\!\cdots\!30\)\( T^{11} + \)\(20\!\cdots\!87\)\( T^{12} + \)\(43\!\cdots\!72\)\( T^{13} + \)\(19\!\cdots\!88\)\( T^{14} + \)\(31\!\cdots\!74\)\( T^{15} + \)\(12\!\cdots\!94\)\( T^{16} + \)\(10\!\cdots\!50\)\( T^{17} + \)\(35\!\cdots\!37\)\( T^{18} \)
$59$ \( 1 - 732 T + 945340 T^{2} - 612879532 T^{3} + 513207430894 T^{4} - 282220552064066 T^{5} + 184384670746115453 T^{6} - 88565394268901556874 T^{7} + \)\(49\!\cdots\!34\)\( T^{8} - \)\(20\!\cdots\!52\)\( T^{9} + \)\(10\!\cdots\!86\)\( T^{10} - \)\(37\!\cdots\!34\)\( T^{11} + \)\(15\!\cdots\!67\)\( T^{12} - \)\(50\!\cdots\!46\)\( T^{13} + \)\(18\!\cdots\!06\)\( T^{14} - \)\(45\!\cdots\!72\)\( T^{15} + \)\(14\!\cdots\!60\)\( T^{16} - \)\(23\!\cdots\!52\)\( T^{17} + \)\(65\!\cdots\!19\)\( T^{18} \)
$61$ \( 1 + 914 T + 1384791 T^{2} + 944091564 T^{3} + 820017012594 T^{4} + 466805168615128 T^{5} + 303065212323191990 T^{6} + \)\(15\!\cdots\!96\)\( T^{7} + \)\(83\!\cdots\!04\)\( T^{8} + \)\(38\!\cdots\!76\)\( T^{9} + \)\(18\!\cdots\!24\)\( T^{10} + \)\(79\!\cdots\!56\)\( T^{11} + \)\(35\!\cdots\!90\)\( T^{12} + \)\(12\!\cdots\!88\)\( T^{13} + \)\(49\!\cdots\!94\)\( T^{14} + \)\(12\!\cdots\!84\)\( T^{15} + \)\(42\!\cdots\!51\)\( T^{16} + \)\(64\!\cdots\!74\)\( T^{17} + \)\(15\!\cdots\!21\)\( T^{18} \)
$67$ \( ( 1 + 67 T )^{9} \)
$71$ \( 1 - 2990 T + 6605845 T^{2} - 10318863444 T^{3} + 13414035645594 T^{4} - 14430255958701560 T^{5} + 13495843927433373994 T^{6} - \)\(10\!\cdots\!32\)\( T^{7} + \)\(78\!\cdots\!16\)\( T^{8} - \)\(49\!\cdots\!48\)\( T^{9} + \)\(28\!\cdots\!76\)\( T^{10} - \)\(14\!\cdots\!72\)\( T^{11} + \)\(61\!\cdots\!14\)\( T^{12} - \)\(23\!\cdots\!60\)\( T^{13} + \)\(78\!\cdots\!94\)\( T^{14} - \)\(21\!\cdots\!84\)\( T^{15} + \)\(49\!\cdots\!95\)\( T^{16} - \)\(80\!\cdots\!90\)\( T^{17} + \)\(96\!\cdots\!91\)\( T^{18} \)
$73$ \( 1 + 1384 T + 3034984 T^{2} + 2812890514 T^{3} + 3642043401336 T^{4} + 2629469468687432 T^{5} + 2653195541634116599 T^{6} + \)\(16\!\cdots\!40\)\( T^{7} + \)\(13\!\cdots\!40\)\( T^{8} + \)\(74\!\cdots\!00\)\( T^{9} + \)\(54\!\cdots\!80\)\( T^{10} + \)\(24\!\cdots\!60\)\( T^{11} + \)\(15\!\cdots\!87\)\( T^{12} + \)\(60\!\cdots\!72\)\( T^{13} + \)\(32\!\cdots\!52\)\( T^{14} + \)\(97\!\cdots\!66\)\( T^{15} + \)\(40\!\cdots\!32\)\( T^{16} + \)\(72\!\cdots\!44\)\( T^{17} + \)\(20\!\cdots\!97\)\( T^{18} \)
$79$ \( 1 - 2438 T + 5772541 T^{2} - 8299448616 T^{3} + 11415392087258 T^{4} - 11837332411608056 T^{5} + 11976906545341109578 T^{6} - \)\(98\!\cdots\!48\)\( T^{7} + \)\(81\!\cdots\!76\)\( T^{8} - \)\(56\!\cdots\!04\)\( T^{9} + \)\(40\!\cdots\!64\)\( T^{10} - \)\(24\!\cdots\!08\)\( T^{11} + \)\(14\!\cdots\!82\)\( T^{12} - \)\(69\!\cdots\!96\)\( T^{13} + \)\(33\!\cdots\!42\)\( T^{14} - \)\(11\!\cdots\!76\)\( T^{15} + \)\(40\!\cdots\!39\)\( T^{16} - \)\(85\!\cdots\!78\)\( T^{17} + \)\(17\!\cdots\!59\)\( T^{18} \)
$83$ \( 1 - 972 T + 3617968 T^{2} - 3355067444 T^{3} + 6562635805982 T^{4} - 5506694628270722 T^{5} + 7609104727353806845 T^{6} - \)\(55\!\cdots\!18\)\( T^{7} + \)\(61\!\cdots\!58\)\( T^{8} - \)\(38\!\cdots\!72\)\( T^{9} + \)\(34\!\cdots\!46\)\( T^{10} - \)\(18\!\cdots\!42\)\( T^{11} + \)\(14\!\cdots\!35\)\( T^{12} - \)\(58\!\cdots\!42\)\( T^{13} + \)\(40\!\cdots\!74\)\( T^{14} - \)\(11\!\cdots\!96\)\( T^{15} + \)\(72\!\cdots\!44\)\( T^{16} - \)\(11\!\cdots\!12\)\( T^{17} + \)\(65\!\cdots\!27\)\( T^{18} \)
$89$ \( 1 - 1034 T + 4128177 T^{2} - 2752260700 T^{3} + 7010341794268 T^{4} - 2580866299893328 T^{5} + 6687942370385771884 T^{6} - \)\(59\!\cdots\!80\)\( T^{7} + \)\(46\!\cdots\!22\)\( T^{8} + \)\(29\!\cdots\!24\)\( T^{9} + \)\(32\!\cdots\!18\)\( T^{10} - \)\(29\!\cdots\!80\)\( T^{11} + \)\(23\!\cdots\!56\)\( T^{12} - \)\(63\!\cdots\!88\)\( T^{13} + \)\(12\!\cdots\!32\)\( T^{14} - \)\(33\!\cdots\!00\)\( T^{15} + \)\(35\!\cdots\!53\)\( T^{16} - \)\(63\!\cdots\!94\)\( T^{17} + \)\(43\!\cdots\!29\)\( T^{18} \)
$97$ \( 1 + 1516 T + 4926113 T^{2} + 4305288356 T^{3} + 8347290572836 T^{4} + 2319749075206136 T^{5} + 4784892495212791012 T^{6} - \)\(57\!\cdots\!44\)\( T^{7} - \)\(17\!\cdots\!82\)\( T^{8} - \)\(99\!\cdots\!64\)\( T^{9} - \)\(16\!\cdots\!86\)\( T^{10} - \)\(48\!\cdots\!76\)\( T^{11} + \)\(36\!\cdots\!04\)\( T^{12} + \)\(16\!\cdots\!76\)\( T^{13} + \)\(52\!\cdots\!48\)\( T^{14} + \)\(24\!\cdots\!84\)\( T^{15} + \)\(25\!\cdots\!61\)\( T^{16} + \)\(72\!\cdots\!96\)\( T^{17} + \)\(43\!\cdots\!13\)\( T^{18} \)
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