Properties

Label 201.2.f.a
Level 201
Weight 2
Character orbit 201.f
Analytic conductor 1.605
Analytic rank 0
Dimension 2
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.f (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(1.60499308063\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - 2 \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{4} + ( 4 - 2 \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})\) \( q + ( 1 - 2 \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{4} + ( 4 - 2 \zeta_{6} ) q^{7} -3 q^{9} + ( 4 - 2 \zeta_{6} ) q^{12} + ( 1 + \zeta_{6} ) q^{13} + ( -4 + 4 \zeta_{6} ) q^{16} + ( 8 - 8 \zeta_{6} ) q^{19} -6 \zeta_{6} q^{21} -5 q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} + ( 4 + 4 \zeta_{6} ) q^{28} + ( -10 + 5 \zeta_{6} ) q^{31} -6 \zeta_{6} q^{36} + ( -10 + 10 \zeta_{6} ) q^{37} + ( 3 - 3 \zeta_{6} ) q^{39} + ( -7 + 14 \zeta_{6} ) q^{43} + ( 4 + 4 \zeta_{6} ) q^{48} + ( 5 - 5 \zeta_{6} ) q^{49} + ( -2 + 4 \zeta_{6} ) q^{52} + ( -8 - 8 \zeta_{6} ) q^{57} + ( -5 - 5 \zeta_{6} ) q^{61} + ( -12 + 6 \zeta_{6} ) q^{63} -8 q^{64} + ( 2 - 9 \zeta_{6} ) q^{67} + ( 17 - 17 \zeta_{6} ) q^{73} + ( -5 + 10 \zeta_{6} ) q^{75} + 16 q^{76} + ( 6 - 3 \zeta_{6} ) q^{79} + 9 q^{81} + ( 12 - 12 \zeta_{6} ) q^{84} + 6 q^{91} + 15 \zeta_{6} q^{93} + ( -3 - 3 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{4} + 6q^{7} - 6q^{9} + O(q^{10}) \) \( 2q + 2q^{4} + 6q^{7} - 6q^{9} + 6q^{12} + 3q^{13} - 4q^{16} + 8q^{19} - 6q^{21} - 10q^{25} + 12q^{28} - 15q^{31} - 6q^{36} - 10q^{37} + 3q^{39} + 12q^{48} + 5q^{49} - 24q^{57} - 15q^{61} - 18q^{63} - 16q^{64} - 5q^{67} + 17q^{73} + 32q^{76} + 9q^{79} + 18q^{81} + 12q^{84} + 12q^{91} + 15q^{93} - 9q^{97} + O(q^{100}) \)

Character Values

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

\(n\) \(68\) \(136\)
\(\chi(n)\) \(-1\) \(\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
38.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.73205i 1.00000 + 1.73205i 0 0 3.00000 1.73205i 0 −3.00000 0
164.1 0 1.73205i 1.00000 1.73205i 0 0 3.00000 + 1.73205i 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
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.d Odd 1 yes
201.f Even 1 yes

Hecke kernels

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