Properties

Label 201.2.e.c
Level 201
Weight 2
Character orbit 201.e
Analytic conductor 1.605
Analytic rank 0
Dimension 10
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 201 = 3 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 201.e (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(1.60499308063\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: 10.0.3665654523963.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{9} q^{2} + q^{3} + ( \beta_{1} - 2 \beta_{4} - 2 \beta_{5} ) q^{4} + ( 1 + \beta_{2} + \beta_{3} ) q^{5} + \beta_{9} q^{6} + ( \beta_{1} - \beta_{5} - \beta_{8} ) q^{7} + ( 1 + 2 \beta_{2} + \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{8} + q^{9} +O(q^{10})\) \( q + \beta_{9} q^{2} + q^{3} + ( \beta_{1} - 2 \beta_{4} - 2 \beta_{5} ) q^{4} + ( 1 + \beta_{2} + \beta_{3} ) q^{5} + \beta_{9} q^{6} + ( \beta_{1} - \beta_{5} - \beta_{8} ) q^{7} + ( 1 + 2 \beta_{2} + \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{8} + q^{9} + ( -3 - \beta_{1} - \beta_{3} + 3 \beta_{5} - \beta_{7} + \beta_{9} ) q^{10} + ( -\beta_{2} + 2 \beta_{4} + 2 \beta_{5} + \beta_{8} - \beta_{9} ) q^{11} + ( \beta_{1} - 2 \beta_{4} - 2 \beta_{5} ) q^{12} + ( \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{13} + ( -1 - \beta_{6} ) q^{14} + ( 1 + \beta_{2} + \beta_{3} ) q^{15} + ( -5 + 5 \beta_{5} + \beta_{6} - 3 \beta_{7} - \beta_{8} ) q^{16} + ( 1 - \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{17} + \beta_{9} q^{18} + ( -2 - 3 \beta_{1} - 3 \beta_{3} + 2 \beta_{5} + \beta_{7} - \beta_{9} ) q^{19} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - \beta_{8} - 2 \beta_{9} ) q^{20} + ( \beta_{1} - \beta_{5} - \beta_{8} ) q^{21} + ( 2 - 3 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{22} + ( -3 - 2 \beta_{1} - 2 \beta_{3} + 3 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} + 2 \beta_{9} ) q^{23} + ( 1 + 2 \beta_{2} + \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{24} + ( -1 + 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{25} + ( \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - 5 \beta_{5} - 2 \beta_{8} - 2 \beta_{9} ) q^{26} + q^{27} + ( 2 + 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{28} + ( 3 \beta_{1} - \beta_{2} + 2 \beta_{4} - 4 \beta_{5} - 2 \beta_{8} - \beta_{9} ) q^{29} + ( -3 - \beta_{1} - \beta_{3} + 3 \beta_{5} - \beta_{7} + \beta_{9} ) q^{30} + ( -\beta_{2} - 2 \beta_{4} - 4 \beta_{5} - 3 \beta_{8} - \beta_{9} ) q^{31} + ( -5 \beta_{2} - \beta_{5} - 5 \beta_{9} ) q^{32} + ( -\beta_{2} + 2 \beta_{4} + 2 \beta_{5} + \beta_{8} - \beta_{9} ) q^{33} + ( 4 \beta_{2} - \beta_{4} + \beta_{5} + 3 \beta_{8} + 4 \beta_{9} ) q^{34} + ( -\beta_{1} + \beta_{4} + \beta_{8} ) q^{35} + ( \beta_{1} - 2 \beta_{4} - 2 \beta_{5} ) q^{36} + ( -6 - 2 \beta_{1} - 2 \beta_{3} + 6 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} + 3 \beta_{8} - \beta_{9} ) q^{37} + ( -\beta_{1} - \beta_{2} + 6 \beta_{4} + 8 \beta_{5} + \beta_{8} - \beta_{9} ) q^{38} + ( \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{39} + ( 5 + \beta_{2} + 6 \beta_{4} + \beta_{6} + 6 \beta_{7} ) q^{40} + ( -4 \beta_{1} + \beta_{2} + \beta_{4} + 4 \beta_{5} - \beta_{8} + \beta_{9} ) q^{41} + ( -1 - \beta_{6} ) q^{42} + ( 1 + \beta_{2} + \beta_{4} + 3 \beta_{6} + \beta_{7} ) q^{43} + ( 10 + 3 \beta_{1} + 3 \beta_{3} - 10 \beta_{5} - 2 \beta_{6} + 5 \beta_{7} + 2 \beta_{8} ) q^{44} + ( 1 + \beta_{2} + \beta_{3} ) q^{45} + ( 2 \beta_{1} - \beta_{2} - 4 \beta_{4} - 6 \beta_{5} + 2 \beta_{8} - \beta_{9} ) q^{46} + ( -\beta_{2} - \beta_{4} - 4 \beta_{5} + 3 \beta_{8} - \beta_{9} ) q^{47} + ( -5 + 5 \beta_{5} + \beta_{6} - 3 \beta_{7} - \beta_{8} ) q^{48} + ( 3 + \beta_{1} + \beta_{3} - 3 \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{49} + ( -6 - 2 \beta_{1} - 2 \beta_{3} + 6 \beta_{5} - \beta_{7} - \beta_{9} ) q^{50} + ( 1 - \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{51} + ( 9 + 3 \beta_{2} + 2 \beta_{3} + 5 \beta_{4} + 2 \beta_{6} + 5 \beta_{7} ) q^{52} + ( 1 + 3 \beta_{2} + 2 \beta_{3} + \beta_{4} + 3 \beta_{6} + \beta_{7} ) q^{53} + \beta_{9} q^{54} + ( 2 \beta_{1} + 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{8} ) q^{55} + ( 2 \beta_{2} - 3 \beta_{5} - 2 \beta_{8} + 2 \beta_{9} ) q^{56} + ( -2 - 3 \beta_{1} - 3 \beta_{3} + 2 \beta_{5} + \beta_{7} - \beta_{9} ) q^{57} + ( -1 + \beta_{3} - \beta_{4} - 4 \beta_{6} - \beta_{7} ) q^{58} + ( -1 - \beta_{2} + 3 \beta_{3} - \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{59} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - \beta_{8} - 2 \beta_{9} ) q^{60} + ( 3 - \beta_{1} - \beta_{3} - 3 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} + 2 \beta_{9} ) q^{61} + ( 6 + 3 \beta_{2} + \beta_{3} + 7 \beta_{4} - \beta_{6} + 7 \beta_{7} ) q^{62} + ( \beta_{1} - \beta_{5} - \beta_{8} ) q^{63} + ( 10 + \beta_{2} + 5 \beta_{3} + 4 \beta_{4} + 2 \beta_{6} + 4 \beta_{7} ) q^{64} + ( -4 - \beta_{1} - \beta_{3} + 4 \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{65} + ( 2 - 3 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{66} + ( 1 - 2 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} - 3 \beta_{7} + \beta_{8} - 3 \beta_{9} ) q^{67} + ( -13 + 3 \beta_{2} - 4 \beta_{3} - 8 \beta_{4} - 8 \beta_{7} ) q^{68} + ( -3 - 2 \beta_{1} - 2 \beta_{3} + 3 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} + 2 \beta_{9} ) q^{69} + ( -\beta_{4} - \beta_{7} ) q^{70} + ( 2 \beta_{1} + 5 \beta_{5} ) q^{71} + ( 1 + 2 \beta_{2} + \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{72} + ( 3 + 2 \beta_{1} + 2 \beta_{3} - 3 \beta_{5} - \beta_{7} ) q^{73} + ( -\beta_{1} - 6 \beta_{2} - 2 \beta_{4} + 3 \beta_{5} - 6 \beta_{9} ) q^{74} + ( -1 + 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{75} + ( -5 - 11 \beta_{2} - 5 \beta_{3} - 2 \beta_{4} - 5 \beta_{6} - 2 \beta_{7} ) q^{76} + ( 2 - 2 \beta_{5} + 3 \beta_{6} - 3 \beta_{8} + \beta_{9} ) q^{77} + ( \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - 5 \beta_{5} - 2 \beta_{8} - 2 \beta_{9} ) q^{78} + ( 4 \beta_{1} - 3 \beta_{2} - 5 \beta_{4} - 4 \beta_{5} - 3 \beta_{8} - 3 \beta_{9} ) q^{79} + ( -6 - 3 \beta_{1} - 3 \beta_{3} + 6 \beta_{5} - 3 \beta_{6} - 5 \beta_{7} + 3 \beta_{8} + 6 \beta_{9} ) q^{80} + q^{81} + ( -1 - 6 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{82} + ( 1 + 2 \beta_{1} + 2 \beta_{3} - \beta_{5} + 2 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} ) q^{83} + ( 2 + 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{84} + ( 3 + 2 \beta_{1} + 2 \beta_{3} - 3 \beta_{5} + \beta_{6} - \beta_{8} - 3 \beta_{9} ) q^{85} + ( -5 - \beta_{1} - \beta_{3} + 5 \beta_{5} + 2 \beta_{6} - 6 \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{86} + ( 3 \beta_{1} - \beta_{2} + 2 \beta_{4} - 4 \beta_{5} - 2 \beta_{8} - \beta_{9} ) q^{87} + ( -2 \beta_{1} + 11 \beta_{2} - 2 \beta_{4} + 6 \beta_{5} + 5 \beta_{8} + 11 \beta_{9} ) q^{88} + ( -4 - 2 \beta_{2} - 7 \beta_{3} - 6 \beta_{4} + \beta_{6} - 6 \beta_{7} ) q^{89} + ( -3 - \beta_{1} - \beta_{3} + 3 \beta_{5} - \beta_{7} + \beta_{9} ) q^{90} + ( -6 + \beta_{2} - \beta_{3} - 4 \beta_{6} ) q^{91} + ( 8 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{92} + ( -\beta_{2} - 2 \beta_{4} - 4 \beta_{5} - 3 \beta_{8} - \beta_{9} ) q^{93} + ( 5 + 8 \beta_{2} + \beta_{3} + 4 \beta_{6} ) q^{94} + ( 1 + \beta_{1} + \beta_{3} - \beta_{5} + 4 \beta_{6} + 9 \beta_{7} - 4 \beta_{8} + \beta_{9} ) q^{95} + ( -5 \beta_{2} - \beta_{5} - 5 \beta_{9} ) q^{96} + ( 3 + \beta_{1} + \beta_{3} - 3 \beta_{5} + \beta_{6} - \beta_{8} - 2 \beta_{9} ) q^{97} + ( -\beta_{1} + 6 \beta_{2} + 4 \beta_{5} + 3 \beta_{8} + 6 \beta_{9} ) q^{98} + ( -\beta_{2} + 2 \beta_{4} + 2 \beta_{5} + \beta_{8} - \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 10q^{3} - 6q^{4} + 6q^{5} - q^{7} + 10q^{9} + O(q^{10}) \) \( 10q + 10q^{3} - 6q^{4} + 6q^{5} - q^{7} + 10q^{9} - 12q^{10} + 6q^{11} - 6q^{12} - q^{13} - 6q^{14} + 6q^{15} - 24q^{16} + 8q^{17} - 5q^{19} - 8q^{20} - q^{21} + 22q^{22} - 7q^{23} - 8q^{25} - 17q^{26} + 10q^{27} + 3q^{28} - 12q^{29} - 12q^{30} - 12q^{31} - 5q^{32} + 6q^{33} - 5q^{35} - 6q^{36} - 17q^{37} + 30q^{38} - q^{39} + 34q^{40} + 13q^{41} - 6q^{42} - 4q^{43} + 43q^{44} + 6q^{45} - 26q^{46} - 25q^{47} - 24q^{48} + 16q^{49} - 25q^{50} + 8q^{51} + 64q^{52} - 12q^{53} - 14q^{55} - 11q^{56} - 5q^{57} + 4q^{58} - 12q^{59} - 8q^{60} + 9q^{61} + 46q^{62} - q^{63} + 64q^{64} - 14q^{65} + 22q^{66} + 2q^{67} - 98q^{68} - 7q^{69} + 2q^{70} + 29q^{71} + 12q^{73} + 15q^{74} - 8q^{75} - 6q^{76} + 4q^{77} - 17q^{78} - q^{79} - 13q^{80} + 10q^{81} - 2q^{82} - 6q^{83} + 3q^{84} + 9q^{85} - 21q^{86} - 12q^{87} + 18q^{88} - 4q^{89} - 12q^{90} - 40q^{91} - 12q^{93} + 30q^{94} - 14q^{95} - 5q^{96} + 11q^{97} + 12q^{98} + 6q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 2 x^{9} + 8 x^{8} + 21 x^{6} - 5 x^{5} + 26 x^{4} + 4 x^{3} + 13 x^{2} - 3 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -1127 \nu^{9} + 7711 \nu^{8} - 25937 \nu^{7} + 48871 \nu^{6} - 53276 \nu^{5} + 58183 \nu^{4} - 144911 \nu^{3} + 39957 \nu^{2} - 9814 \nu - 47489 \)\()/44248\)
\(\beta_{3}\)\(=\)\((\)\( 1899 \nu^{9} - 6083 \nu^{8} + 20461 \nu^{7} - 17723 \nu^{6} + 42028 \nu^{5} - 45899 \nu^{4} + 89131 \nu^{3} - 31521 \nu^{2} + 7742 \nu - 12403 \)\()/44248\)
\(\beta_{4}\)\(=\)\((\)\( -3473 \nu^{9} + 2993 \nu^{8} - 22135 \nu^{7} - 25035 \nu^{6} - 95096 \nu^{5} - 47811 \nu^{4} - 116057 \nu^{3} - 66953 \nu^{2} - 90294 \nu - 6827 \)\()/22124\)
\(\beta_{5}\)\(=\)\((\)\( 12403 \nu^{9} - 22907 \nu^{8} + 93141 \nu^{7} + 20461 \nu^{6} + 242740 \nu^{5} - 19987 \nu^{4} + 276579 \nu^{3} + 138743 \nu^{2} + 129718 \nu + 14781 \)\()/44248\)
\(\beta_{6}\)\(=\)\((\)\( -14781 \nu^{9} + 41965 \nu^{8} - 141155 \nu^{7} + 93141 \nu^{6} - 289940 \nu^{5} + 316645 \nu^{4} - 404293 \nu^{3} + 217455 \nu^{2} - 53410 \nu + 129813 \)\()/44248\)
\(\beta_{7}\)\(=\)\((\)\( 3911 \nu^{9} - 7629 \nu^{8} + 31695 \nu^{7} - 1369 \nu^{6} + 89918 \nu^{5} - 22367 \nu^{4} + 104757 \nu^{3} + 1699 \nu^{2} + 52952 \nu - 12251 \)\()/11062\)
\(\beta_{8}\)\(=\)\((\)\( 2133 \nu^{9} - 2953 \nu^{8} + 14961 \nu^{7} + 9085 \nu^{6} + 49566 \nu^{5} + 14424 \nu^{4} + 58697 \nu^{3} + 31963 \nu^{2} + 49012 \nu + 3317 \)\()/5531\)
\(\beta_{9}\)\(=\)\((\)\( -9205 \nu^{9} + 14061 \nu^{8} - 61375 \nu^{7} - 41255 \nu^{6} - 165532 \nu^{5} - 38715 \nu^{4} - 146785 \nu^{3} - 130277 \nu^{2} - 56110 \nu + 12005 \)\()/22124\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{3} + 2 \beta_{1} - 1\)
\(\nu^{3}\)\(=\)\(2 \beta_{7} + 2 \beta_{6} + 2 \beta_{4} + 7 \beta_{3} + \beta_{2}\)
\(\nu^{4}\)\(=\)\(2 \beta_{9} + 7 \beta_{8} - 2 \beta_{5} + 8 \beta_{4} + 2 \beta_{2} - 18 \beta_{1}\)
\(\nu^{5}\)\(=\)\(8 \beta_{9} + 18 \beta_{8} - 20 \beta_{7} - 18 \beta_{6} - \beta_{5} - 53 \beta_{3} - 53 \beta_{1} + 1\)
\(\nu^{6}\)\(=\)\(-61 \beta_{7} - 53 \beta_{6} - 61 \beta_{4} - 145 \beta_{3} - 20 \beta_{2} + 7\)
\(\nu^{7}\)\(=\)\(-61 \beta_{9} - 145 \beta_{8} + 11 \beta_{5} - 165 \beta_{4} - 61 \beta_{2} + 411 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-165 \beta_{9} - 411 \beta_{8} + 472 \beta_{7} + 411 \beta_{6} + 40 \beta_{5} + 1143 \beta_{3} + 1143 \beta_{1} - 40\)
\(\nu^{9}\)\(=\)\(1308 \beta_{7} + 1143 \beta_{6} + 1308 \beta_{4} + 3209 \beta_{3} + 472 \beta_{2} - 95\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/201\mathbb{Z}\right)^\times\).

\(n\) \(68\) \(136\)
\(\chi(n)\) \(1\) \(-\beta_{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
0.586579 1.01598i
−0.725148 + 1.25599i
1.39901 2.42316i
0.133354 0.230976i
−0.393795 + 0.682072i
0.586579 + 1.01598i
−0.725148 1.25599i
1.39901 + 2.42316i
0.133354 + 0.230976i
−0.393795 0.682072i
−1.32873 2.30142i 1.00000 −2.53104 + 4.38389i 2.48430 −1.32873 2.30142i 0.160379 0.277784i 8.13733 1.00000 −3.30096 5.71743i
37.2 −0.453162 0.784899i 1.00000 0.589289 1.02068i 3.35662 −0.453162 0.784899i −0.380391 + 0.658856i −2.88082 1.00000 −1.52109 2.63461i
37.3 −0.0732181 0.126817i 1.00000 0.989278 1.71348i −1.65158 −0.0732181 0.126817i 1.22031 2.11364i −0.582605 1.00000 0.120926 + 0.209450i
37.4 0.538290 + 0.932346i 1.00000 0.420487 0.728305i −0.343289 0.538290 + 0.932346i −1.74135 + 3.01611i 3.05854 1.00000 −0.184789 0.320064i
37.5 1.31682 + 2.28079i 1.00000 −2.46802 + 4.27473i −0.846046 1.31682 + 2.28079i 0.241054 0.417518i −7.73244 1.00000 −1.11409 1.92966i
163.1 −1.32873 + 2.30142i 1.00000 −2.53104 4.38389i 2.48430 −1.32873 + 2.30142i 0.160379 + 0.277784i 8.13733 1.00000 −3.30096 + 5.71743i
163.2 −0.453162 + 0.784899i 1.00000 0.589289 + 1.02068i 3.35662 −0.453162 + 0.784899i −0.380391 0.658856i −2.88082 1.00000 −1.52109 + 2.63461i
163.3 −0.0732181 + 0.126817i 1.00000 0.989278 + 1.71348i −1.65158 −0.0732181 + 0.126817i 1.22031 + 2.11364i −0.582605 1.00000 0.120926 0.209450i
163.4 0.538290 0.932346i 1.00000 0.420487 + 0.728305i −0.343289 0.538290 0.932346i −1.74135 3.01611i 3.05854 1.00000 −0.184789 + 0.320064i
163.5 1.31682 2.28079i 1.00000 −2.46802 4.27473i −0.846046 1.31682 2.28079i 0.241054 + 0.417518i −7.73244 1.00000 −1.11409 + 1.92966i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.5
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
67.c Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{10} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(201, [\chi])\).