Properties

Label 201.2.p.a
Level 201
Weight 2
Character orbit 201.p
Analytic conductor 1.605
Analytic rank 0
Dimension 20
CM disc. -3
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 201 = 3 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 201.p (of order \(66\) and degree \(20\))

Newform invariants

Self dual: No
Analytic conductor: \(1.60499308063\)
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.

\(f(q)\) \(=\) \( q + ( 2 \zeta_{33} + \zeta_{33}^{12} ) q^{3} -2 \zeta_{33}^{4} q^{4} + ( 3 - 3 \zeta_{33} + 3 \zeta_{33}^{3} - 3 \zeta_{33}^{4} + 6 \zeta_{33}^{6} - 3 \zeta_{33}^{7} + 2 \zeta_{33}^{9} - 3 \zeta_{33}^{10} + 3 \zeta_{33}^{11} - 3 \zeta_{33}^{13} + 3 \zeta_{33}^{14} - 3 \zeta_{33}^{16} + 4 \zeta_{33}^{17} - 3 \zeta_{33}^{19} ) q^{7} + ( 3 \zeta_{33}^{2} + 3 \zeta_{33}^{13} ) q^{9} +O(q^{10})\) \( q + ( 2 \zeta_{33} + \zeta_{33}^{12} ) q^{3} -2 \zeta_{33}^{4} q^{4} + ( 3 - 3 \zeta_{33} + 3 \zeta_{33}^{3} - 3 \zeta_{33}^{4} + 6 \zeta_{33}^{6} - 3 \zeta_{33}^{7} + 2 \zeta_{33}^{9} - 3 \zeta_{33}^{10} + 3 \zeta_{33}^{11} - 3 \zeta_{33}^{13} + 3 \zeta_{33}^{14} - 3 \zeta_{33}^{16} + 4 \zeta_{33}^{17} - 3 \zeta_{33}^{19} ) q^{7} + ( 3 \zeta_{33}^{2} + 3 \zeta_{33}^{13} ) q^{9} + ( -4 \zeta_{33}^{5} - 2 \zeta_{33}^{16} ) q^{12} + ( \zeta_{33}^{3} - 4 \zeta_{33}^{7} - 3 \zeta_{33}^{14} - 3 \zeta_{33}^{18} ) q^{13} + 4 \zeta_{33}^{8} q^{16} + ( -5 + 5 \zeta_{33}^{2} - 5 \zeta_{33}^{3} + 5 \zeta_{33}^{5} - 5 \zeta_{33}^{6} + 2 \zeta_{33}^{8} - 5 \zeta_{33}^{9} + 3 \zeta_{33}^{10} - 5 \zeta_{33}^{12} + 5 \zeta_{33}^{13} - 5 \zeta_{33}^{15} + 5 \zeta_{33}^{16} - 5 \zeta_{33}^{18} + 7 \zeta_{33}^{19} ) q^{19} + ( 4 - 4 \zeta_{33}^{2} + 4 \zeta_{33}^{3} - 4 \zeta_{33}^{5} + 4 \zeta_{33}^{6} + 5 \zeta_{33}^{7} - 4 \zeta_{33}^{8} + 4 \zeta_{33}^{9} + \zeta_{33}^{10} + 4 \zeta_{33}^{12} - 4 \zeta_{33}^{13} + 4 \zeta_{33}^{15} - 4 \zeta_{33}^{16} + 8 \zeta_{33}^{18} - 4 \zeta_{33}^{19} ) q^{21} + ( 5 \zeta_{33}^{5} + 5 \zeta_{33}^{16} ) q^{25} + ( 3 \zeta_{33}^{3} + 6 \zeta_{33}^{14} ) q^{27} + ( 2 - 8 \zeta_{33}^{2} + 2 \zeta_{33}^{3} - 2 \zeta_{33}^{5} + 2 \zeta_{33}^{6} - 2 \zeta_{33}^{8} + 2 \zeta_{33}^{9} - 6 \zeta_{33}^{10} + 2 \zeta_{33}^{12} - 6 \zeta_{33}^{13} + 2 \zeta_{33}^{15} - 2 \zeta_{33}^{16} + 2 \zeta_{33}^{18} - 2 \zeta_{33}^{19} ) q^{28} + ( -6 + 6 \zeta_{33} - 12 \zeta_{33}^{3} + 6 \zeta_{33}^{4} - 6 \zeta_{33}^{6} + 6 \zeta_{33}^{7} - 5 \zeta_{33}^{9} + 6 \zeta_{33}^{10} - 6 \zeta_{33}^{11} + 6 \zeta_{33}^{13} - 7 \zeta_{33}^{14} + 6 \zeta_{33}^{16} - 6 \zeta_{33}^{17} + 6 \zeta_{33}^{19} ) q^{31} + ( -6 \zeta_{33}^{6} - 6 \zeta_{33}^{17} ) q^{36} + ( 3 \zeta_{33}^{5} - 4 \zeta_{33}^{6} - 4 \zeta_{33}^{16} + 3 \zeta_{33}^{17} ) q^{37} + ( 5 \zeta_{33}^{4} - 5 \zeta_{33}^{8} - 2 \zeta_{33}^{15} - 7 \zeta_{33}^{19} ) q^{39} + ( -7 \zeta_{33} + \zeta_{33}^{2} - 6 \zeta_{33}^{12} - 6 \zeta_{33}^{13} ) q^{43} + ( -4 + 4 \zeta_{33} - 4 \zeta_{33}^{3} + 4 \zeta_{33}^{4} - 4 \zeta_{33}^{6} + 4 \zeta_{33}^{7} + 4 \zeta_{33}^{9} + 4 \zeta_{33}^{10} - 4 \zeta_{33}^{11} + 4 \zeta_{33}^{13} - 4 \zeta_{33}^{14} + 4 \zeta_{33}^{16} - 4 \zeta_{33}^{17} + 4 \zeta_{33}^{19} ) q^{48} + ( -5 \zeta_{33} + 7 \zeta_{33}^{4} + 3 \zeta_{33}^{7} + 3 \zeta_{33}^{12} + 7 \zeta_{33}^{15} - 5 \zeta_{33}^{18} ) q^{49} + ( -6 - 2 \zeta_{33}^{7} + 2 \zeta_{33}^{11} + 6 \zeta_{33}^{18} ) q^{52} + ( -7 - \zeta_{33} + \zeta_{33}^{3} - \zeta_{33}^{4} + \zeta_{33}^{6} - \zeta_{33}^{7} - 7 \zeta_{33}^{9} - \zeta_{33}^{10} - 6 \zeta_{33}^{11} - \zeta_{33}^{13} + \zeta_{33}^{14} - \zeta_{33}^{16} + \zeta_{33}^{17} - \zeta_{33}^{19} ) q^{57} + ( -4 \zeta_{33}^{2} - 4 \zeta_{33}^{4} + 5 \zeta_{33}^{13} - 9 \zeta_{33}^{15} ) q^{61} + ( 3 + 6 \zeta_{33}^{8} + 9 \zeta_{33}^{11} + 9 \zeta_{33}^{19} ) q^{63} -8 \zeta_{33}^{12} q^{64} + ( -2 \zeta_{33}^{5} + 7 \zeta_{33}^{16} ) q^{67} + ( -1 + \zeta_{33}^{6} - 9 \zeta_{33}^{11} - 8 \zeta_{33}^{17} ) q^{73} + ( 5 \zeta_{33}^{6} + 10 \zeta_{33}^{17} ) q^{75} + ( 4 \zeta_{33} + 10 \zeta_{33}^{3} + 10 \zeta_{33}^{12} + 4 \zeta_{33}^{14} ) q^{76} + ( 7 + 3 \zeta_{33}^{2} - 3 \zeta_{33}^{3} + 3 \zeta_{33}^{5} - 3 \zeta_{33}^{6} + 3 \zeta_{33}^{8} - 3 \zeta_{33}^{9} + 10 \zeta_{33}^{10} + 7 \zeta_{33}^{11} - 3 \zeta_{33}^{12} + 3 \zeta_{33}^{13} - 3 \zeta_{33}^{15} + 3 \zeta_{33}^{16} - 3 \zeta_{33}^{18} + 3 \zeta_{33}^{19} ) q^{79} + 9 \zeta_{33}^{15} q^{81} + ( 8 - 8 \zeta_{33}^{3} - 2 \zeta_{33}^{11} - 10 \zeta_{33}^{14} ) q^{84} + ( 5 + 4 \zeta_{33} + 10 \zeta_{33}^{2} + 5 \zeta_{33}^{3} - 5 \zeta_{33}^{4} - 6 \zeta_{33}^{5} + 5 \zeta_{33}^{6} - 5 \zeta_{33}^{7} + 11 \zeta_{33}^{9} - 5 \zeta_{33}^{10} + 5 \zeta_{33}^{11} - \zeta_{33}^{12} - 4 \zeta_{33}^{13} + 5 \zeta_{33}^{14} - 16 \zeta_{33}^{16} + 5 \zeta_{33}^{17} - 5 \zeta_{33}^{19} ) q^{91} + ( -7 + 7 \zeta_{33}^{2} - 7 \zeta_{33}^{3} - 11 \zeta_{33}^{4} + 7 \zeta_{33}^{5} - 7 \zeta_{33}^{6} + 7 \zeta_{33}^{8} - 7 \zeta_{33}^{9} - 4 \zeta_{33}^{10} - 7 \zeta_{33}^{12} + 7 \zeta_{33}^{13} - 14 \zeta_{33}^{15} + 7 \zeta_{33}^{16} - 7 \zeta_{33}^{18} + 7 \zeta_{33}^{19} ) q^{93} + ( 11 + 11 \zeta_{33} - 11 \zeta_{33}^{2} + 11 \zeta_{33}^{3} - 11 \zeta_{33}^{5} + 11 \zeta_{33}^{6} - 11 \zeta_{33}^{8} + 11 \zeta_{33}^{9} - 8 \zeta_{33}^{10} + 19 \zeta_{33}^{12} - 11 \zeta_{33}^{13} + 11 \zeta_{33}^{15} - 11 \zeta_{33}^{16} + 11 \zeta_{33}^{18} - 11 \zeta_{33}^{19} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 2q^{4} - 6q^{7} + 6q^{9} + O(q^{10}) \) \( 20q - 2q^{4} - 6q^{7} + 6q^{9} - 6q^{12} - 3q^{13} + 4q^{16} - 8q^{19} + 6q^{21} + 10q^{25} - 12q^{28} + 15q^{31} + 6q^{36} + 10q^{37} - 3q^{39} - 12q^{48} - 5q^{49} - 154q^{52} - 75q^{57} + 15q^{61} - 15q^{63} + 16q^{64} + 5q^{67} + 60q^{73} - 32q^{76} + 134q^{79} - 18q^{81} + 186q^{84} - 12q^{91} - 15q^{93} + 9q^{97} + O(q^{100}) \)

Character Values

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

\(n\) \(68\) \(136\)
\(\chi(n)\) \(-1\) \(-\zeta_{33}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
0.723734 0.690079i
0.580057 0.814576i
0.928368 0.371662i
−0.786053 + 0.618159i
−0.888835 0.458227i
−0.786053 0.618159i
0.0475819 + 0.998867i
0.235759 + 0.971812i
−0.327068 0.945001i
0.981929 0.189251i
−0.327068 + 0.945001i
0.723734 + 0.690079i
−0.995472 + 0.0950560i
0.580057 + 0.814576i
0.981929 + 0.189251i
−0.888835 + 0.458227i
0.235759 0.971812i
−0.995472 0.0950560i
0.928368 + 0.371662i
0.0475819 0.998867i
0 0.487975 1.66189i 1.99094 + 0.190112i 0 0 2.32668 + 4.51312i 0 −2.52376 1.62192i 0
11.1 0 1.57553 0.719520i 1.57211 1.23632i 0 0 −0.656000 + 3.40365i 0 1.96458 2.26725i 0
20.1 0 1.71442 + 0.246497i −0.0951638 + 1.99773i 0 0 −0.730437 + 0.177202i 0 2.87848 + 0.845198i 0
32.1 0 −1.71442 + 0.246497i 1.77767 + 0.916453i 0 0 −3.50336 + 3.67422i 0 2.87848 0.845198i 0
41.1 0 −0.936417 1.45709i 0.654136 1.89000i 0 0 −0.0717192 0.751078i 0 −1.24625 + 2.72890i 0
44.1 0 −1.71442 0.246497i 1.77767 0.916453i 0 0 −3.50336 3.67422i 0 2.87848 + 0.845198i 0
50.1 0 0.936417 + 1.45709i −1.96386 + 0.378502i 0 0 −4.31033 + 3.06937i 0 −1.24625 + 2.72890i 0
74.1 0 −0.487975 + 1.66189i −1.16011 + 1.62915i 0 0 −2.19700 0.104656i 0 −2.52376 1.62192i 0
80.1 0 −1.30900 1.13425i −0.471518 + 1.94362i 0 0 −0.816487 + 2.03949i 0 0.426945 + 2.96946i 0
95.1 0 1.30900 1.13425i −1.44747 + 1.38016i 0 0 2.75125 3.49850i 0 0.426945 2.96946i 0
98.1 0 −1.30900 + 1.13425i −0.471518 1.94362i 0 0 −0.816487 2.03949i 0 0.426945 2.96946i 0
101.1 0 0.487975 + 1.66189i 1.99094 0.190112i 0 0 2.32668 4.51312i 0 −2.52376 + 1.62192i 0
113.1 0 −1.57553 0.719520i −1.85674 + 0.743325i 0 0 4.20741 + 1.45620i 0 1.96458 + 2.26725i 0
128.1 0 1.57553 + 0.719520i 1.57211 + 1.23632i 0 0 −0.656000 3.40365i 0 1.96458 + 2.26725i 0
146.1 0 1.30900 + 1.13425i −1.44747 1.38016i 0 0 2.75125 + 3.49850i 0 0.426945 + 2.96946i 0
152.1 0 −0.936417 + 1.45709i 0.654136 + 1.89000i 0 0 −0.0717192 + 0.751078i 0 −1.24625 2.72890i 0
182.1 0 −0.487975 1.66189i −1.16011 1.62915i 0 0 −2.19700 + 0.104656i 0 −2.52376 + 1.62192i 0
185.1 0 −1.57553 + 0.719520i −1.85674 0.743325i 0 0 4.20741 1.45620i 0 1.96458 2.26725i 0
191.1 0 1.71442 0.246497i −0.0951638 1.99773i 0 0 −0.730437 0.177202i 0 2.87848 0.845198i 0
197.1 0 0.936417 1.45709i −1.96386 0.378502i 0 0 −4.31033 3.06937i 0 −1.24625 2.72890i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.1
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.h Odd 1 yes
201.p Even 1 yes

Hecke kernels

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