Properties

Label 201.3.o.a
Level 201
Weight 3
Character orbit 201.o
Analytic conductor 5.477
Analytic rank 0
Dimension 20
CM disc. -3
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 201 = 3 \cdot 67 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 201.o (of order \(66\) and degree \(20\))

Newform invariants

Self dual: No
Analytic conductor: \(5.47685331364\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\Q(\zeta_{33})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{U}(1)[D_{66}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{33}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 3 \zeta_{33}^{2} + 3 \zeta_{33}^{13} ) q^{3} + 4 \zeta_{33}^{8} q^{4} + ( -5 \zeta_{33} + 3 \zeta_{33}^{7} + 3 \zeta_{33}^{12} - 5 \zeta_{33}^{18} ) q^{7} + 9 \zeta_{33}^{15} q^{9} +O(q^{10})\) \( q + ( 3 \zeta_{33}^{2} + 3 \zeta_{33}^{13} ) q^{3} + 4 \zeta_{33}^{8} q^{4} + ( -5 \zeta_{33} + 3 \zeta_{33}^{7} + 3 \zeta_{33}^{12} - 5 \zeta_{33}^{18} ) q^{7} + 9 \zeta_{33}^{15} q^{9} + ( -12 + 12 \zeta_{33}^{2} - 12 \zeta_{33}^{3} + 12 \zeta_{33}^{5} - 12 \zeta_{33}^{6} + 12 \zeta_{33}^{8} - 12 \zeta_{33}^{9} + 12 \zeta_{33}^{10} - 12 \zeta_{33}^{12} + 12 \zeta_{33}^{13} - 12 \zeta_{33}^{15} + 12 \zeta_{33}^{16} - 12 \zeta_{33}^{18} + 12 \zeta_{33}^{19} ) q^{12} + ( -15 \zeta_{33}^{3} - 8 \zeta_{33}^{6} - 8 \zeta_{33}^{14} - 15 \zeta_{33}^{17} ) q^{13} + 16 \zeta_{33}^{16} q^{16} + ( 21 - 21 \zeta_{33} + 21 \zeta_{33}^{3} - 21 \zeta_{33}^{4} + 16 \zeta_{33}^{5} + 21 \zeta_{33}^{6} - 21 \zeta_{33}^{7} + 16 \zeta_{33}^{9} - 21 \zeta_{33}^{10} + 21 \zeta_{33}^{11} - 21 \zeta_{33}^{13} + 21 \zeta_{33}^{14} + 21 \zeta_{33}^{17} - 21 \zeta_{33}^{19} ) q^{19} + ( -9 + 9 \zeta_{33} - 33 \zeta_{33}^{3} + 9 \zeta_{33}^{4} - 9 \zeta_{33}^{6} + 9 \zeta_{33}^{7} + 15 \zeta_{33}^{9} + 9 \zeta_{33}^{10} - 9 \zeta_{33}^{11} + 9 \zeta_{33}^{13} - 24 \zeta_{33}^{14} + 9 \zeta_{33}^{16} - 9 \zeta_{33}^{17} + 9 \zeta_{33}^{19} ) q^{21} + ( -25 + 25 \zeta_{33}^{2} - 25 \zeta_{33}^{3} + 25 \zeta_{33}^{5} - 25 \zeta_{33}^{6} + 25 \zeta_{33}^{8} - 25 \zeta_{33}^{9} - 25 \zeta_{33}^{12} + 25 \zeta_{33}^{13} - 25 \zeta_{33}^{15} + 25 \zeta_{33}^{16} - 25 \zeta_{33}^{18} + 25 \zeta_{33}^{19} ) q^{25} -27 \zeta_{33}^{6} q^{27} + ( -12 + 12 \zeta_{33} - 12 \zeta_{33}^{3} + 32 \zeta_{33}^{4} - 12 \zeta_{33}^{6} + 12 \zeta_{33}^{7} - 32 \zeta_{33}^{9} + 12 \zeta_{33}^{10} - 12 \zeta_{33}^{11} + 12 \zeta_{33}^{13} - 12 \zeta_{33}^{14} + 32 \zeta_{33}^{15} + 12 \zeta_{33}^{16} - 12 \zeta_{33}^{17} + 12 \zeta_{33}^{19} ) q^{28} + ( 35 \zeta_{33}^{6} + 24 \zeta_{33}^{7} + 11 \zeta_{33}^{17} - 11 \zeta_{33}^{18} ) q^{31} + ( -36 \zeta_{33} - 36 \zeta_{33}^{12} ) q^{36} + ( 40 + 33 \zeta_{33} - 40 \zeta_{33}^{2} + 40 \zeta_{33}^{3} - 40 \zeta_{33}^{5} + 40 \zeta_{33}^{6} - 40 \zeta_{33}^{8} + 40 \zeta_{33}^{9} - 7 \zeta_{33}^{10} + 80 \zeta_{33}^{12} - 40 \zeta_{33}^{13} + 40 \zeta_{33}^{15} - 40 \zeta_{33}^{16} + 40 \zeta_{33}^{18} - 40 \zeta_{33}^{19} ) q^{37} + ( -21 \zeta_{33}^{5} + 21 \zeta_{33}^{8} - 45 \zeta_{33}^{16} - 24 \zeta_{33}^{19} ) q^{39} + ( 13 \zeta_{33}^{2} - 35 \zeta_{33}^{4} + 48 \zeta_{33}^{13} - 48 \zeta_{33}^{15} ) q^{43} -48 \zeta_{33}^{7} q^{48} + ( 16 \zeta_{33}^{2} + 55 \zeta_{33}^{3} - 39 \zeta_{33}^{13} + 39 \zeta_{33}^{14} + 49 \zeta_{33}^{19} ) q^{49} + ( 32 + 60 \zeta_{33}^{3} - 28 \zeta_{33}^{11} + 28 \zeta_{33}^{14} ) q^{52} + ( 15 - 15 \zeta_{33}^{7} + 63 \zeta_{33}^{11} + 48 \zeta_{33}^{18} ) q^{57} + ( -9 \zeta_{33}^{4} - 65 \zeta_{33}^{8} - 65 \zeta_{33}^{15} - 9 \zeta_{33}^{19} ) q^{61} + ( -72 - 27 \zeta_{33}^{5} - 27 \zeta_{33}^{11} - 72 \zeta_{33}^{16} ) q^{63} + ( -64 \zeta_{33}^{2} - 64 \zeta_{33}^{13} ) q^{64} + ( 77 - 77 \zeta_{33}^{2} + 77 \zeta_{33}^{3} - 77 \zeta_{33}^{5} + 77 \zeta_{33}^{6} - 77 \zeta_{33}^{8} + 77 \zeta_{33}^{9} - 45 \zeta_{33}^{10} + 77 \zeta_{33}^{12} - 77 \zeta_{33}^{13} + 77 \zeta_{33}^{15} - 77 \zeta_{33}^{16} + 77 \zeta_{33}^{18} - 77 \zeta_{33}^{19} ) q^{67} + ( -80 + 80 \zeta_{33} - 63 \zeta_{33}^{11} + 17 \zeta_{33}^{12} ) q^{73} -75 \zeta_{33}^{12} q^{75} + ( -84 \zeta_{33}^{2} + 84 \zeta_{33}^{6} - 20 \zeta_{33}^{13} + 64 \zeta_{33}^{17} ) q^{76} + ( 11 + 40 \zeta_{33} - 40 \zeta_{33}^{3} + 40 \zeta_{33}^{4} - 40 \zeta_{33}^{6} + 40 \zeta_{33}^{7} - 91 \zeta_{33}^{9} + 40 \zeta_{33}^{10} + 51 \zeta_{33}^{11} + 40 \zeta_{33}^{13} - 40 \zeta_{33}^{14} + 40 \zeta_{33}^{16} - 40 \zeta_{33}^{17} + 40 \zeta_{33}^{19} ) q^{79} + ( -81 \zeta_{33}^{8} - 81 \zeta_{33}^{19} ) q^{81} + ( 60 - 36 \zeta_{33}^{6} - 36 \zeta_{33}^{11} + 60 \zeta_{33}^{17} ) q^{84} + ( -11 + 91 \zeta_{33}^{2} - 11 \zeta_{33}^{3} + 99 \zeta_{33}^{4} + 11 \zeta_{33}^{5} - 11 \zeta_{33}^{6} + 85 \zeta_{33}^{7} + 11 \zeta_{33}^{8} - 11 \zeta_{33}^{9} - 85 \zeta_{33}^{10} - 11 \zeta_{33}^{12} - 8 \zeta_{33}^{13} + 8 \zeta_{33}^{15} + 11 \zeta_{33}^{16} + 85 \zeta_{33}^{18} + 11 \zeta_{33}^{19} ) q^{91} + ( -72 + 72 \zeta_{33} - 72 \zeta_{33}^{3} + 72 \zeta_{33}^{4} - 72 \zeta_{33}^{6} + 72 \zeta_{33}^{7} + 72 \zeta_{33}^{8} + 33 \zeta_{33}^{9} + 72 \zeta_{33}^{10} - 72 \zeta_{33}^{11} + 72 \zeta_{33}^{13} - 72 \zeta_{33}^{14} + 72 \zeta_{33}^{16} - 72 \zeta_{33}^{17} + 177 \zeta_{33}^{19} ) q^{93} + ( 112 - 112 \zeta_{33} + 57 \zeta_{33}^{2} + 112 \zeta_{33}^{3} - 112 \zeta_{33}^{4} + 112 \zeta_{33}^{6} - 112 \zeta_{33}^{7} + 57 \zeta_{33}^{9} - 112 \zeta_{33}^{10} + 112 \zeta_{33}^{11} + 112 \zeta_{33}^{14} - 112 \zeta_{33}^{16} + 112 \zeta_{33}^{17} - 112 \zeta_{33}^{19} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 6q^{3} + 4q^{4} + 2q^{7} - 18q^{9} + O(q^{10}) \) \( 20q + 6q^{3} + 4q^{4} + 2q^{7} - 18q^{9} - 12q^{12} + 23q^{13} + 16q^{16} + 26q^{19} - 6q^{21} - 50q^{25} + 54q^{27} + 8q^{28} - 13q^{31} + 36q^{36} + 26q^{37} - 69q^{39} + 122q^{43} - 48q^{48} - 45q^{49} + 828q^{52} - 441q^{57} + 47q^{61} - 1269q^{63} - 128q^{64} + 109q^{67} - 924q^{73} + 150q^{75} - 208q^{76} + 252q^{79} - 162q^{81} + 1692q^{84} - 92q^{91} + 39q^{93} + 167q^{97} + O(q^{100}) \)

Character Values

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

\(n\) \(68\) \(136\)
\(\chi(n)\) \(-1\) \(\zeta_{33}^{8}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
0.723734 + 0.690079i
−0.327068 + 0.945001i
0.235759 0.971812i
−0.327068 0.945001i
0.981929 0.189251i
−0.888835 + 0.458227i
0.928368 0.371662i
0.723734 0.690079i
0.981929 + 0.189251i
0.0475819 0.998867i
−0.786053 + 0.618159i
0.235759 + 0.971812i
−0.995472 0.0950560i
−0.888835 0.458227i
0.0475819 + 0.998867i
0.928368 + 0.371662i
0.580057 + 0.814576i
−0.995472 + 0.0950560i
0.580057 0.814576i
−0.786053 0.618159i
0 −2.52376 + 1.62192i 3.92771 0.757005i 0 0 −6.83405 9.59709i 0 3.73874 8.18669i 0
23.1 0 0.426945 2.96946i −3.55534 + 1.83291i 0 0 6.63942 6.33068i 0 −8.63544 2.53559i 0
26.1 0 −2.52376 + 1.62192i −1.30827 + 3.78000i 0 0 −9.12076 + 0.870927i 0 3.73874 8.18669i 0
35.1 0 0.426945 + 2.96946i −3.55534 1.83291i 0 0 6.63942 + 6.33068i 0 −8.63544 + 2.53559i 0
47.1 0 0.426945 2.96946i 0.190328 3.99547i 0 0 −1.36948 5.64509i 0 −8.63544 2.53559i 0
56.1 0 −1.24625 2.72890i −3.14421 + 2.47264i 0 0 13.1880 + 2.54178i 0 −5.89375 + 6.80175i 0
65.1 0 2.87848 + 0.845198i −3.98189 0.380224i 0 0 −11.9416 + 6.15630i 0 7.57128 + 4.86577i 0
71.1 0 −2.52376 1.62192i 3.92771 + 0.757005i 0 0 −6.83405 + 9.59709i 0 3.73874 + 8.18669i 0
77.1 0 0.426945 + 2.96946i 0.190328 + 3.99547i 0 0 −1.36948 + 5.64509i 0 −8.63544 + 2.53559i 0
83.1 0 −1.24625 2.72890i 3.71347 + 1.48665i 0 0 4.57895 + 13.2300i 0 −5.89375 + 6.80175i 0
86.1 0 2.87848 0.845198i 2.32023 + 3.25830i 0 0 −0.560201 11.7601i 0 7.57128 4.86577i 0
116.1 0 −2.52376 1.62192i −1.30827 3.78000i 0 0 −9.12076 0.870927i 0 3.73874 + 8.18669i 0
122.1 0 1.96458 2.26725i 2.89494 + 2.76032i 0 0 4.57702 3.59941i 0 −1.28083 8.90839i 0
140.1 0 −1.24625 + 2.72890i −3.14421 2.47264i 0 0 13.1880 2.54178i 0 −5.89375 6.80175i 0
155.1 0 −1.24625 + 2.72890i 3.71347 1.48665i 0 0 4.57895 13.2300i 0 −5.89375 6.80175i 0
167.1 0 2.87848 0.845198i −3.98189 + 0.380224i 0 0 −11.9416 6.15630i 0 7.57128 4.86577i 0
170.1 0 1.96458 + 2.26725i 0.943036 + 3.88725i 0 0 1.84264 0.737681i 0 −1.28083 + 8.90839i 0
173.1 0 1.96458 + 2.26725i 2.89494 2.76032i 0 0 4.57702 + 3.59941i 0 −1.28083 + 8.90839i 0
188.1 0 1.96458 2.26725i 0.943036 3.88725i 0 0 1.84264 + 0.737681i 0 −1.28083 8.90839i 0
194.1 0 2.87848 + 0.845198i 2.32023 3.25830i 0 0 −0.560201 + 11.7601i 0 7.57128 + 4.86577i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 194.1
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 CM by \(\Q(\sqrt{-3}) \) yes
67.g Even 1 yes
201.o Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{3}^{\mathrm{new}}(201, [\chi])\).