Properties

Label 201.3.g.a
Level 201
Weight 3
Character orbit 201.g
Analytic conductor 5.477
Analytic rank 0
Dimension 2
CM discriminant -3
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 201 = 3 \cdot 67 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 201.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.47685331364\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q -3 q^{3} + ( -4 + 4 \zeta_{6} ) q^{4} + ( -2 + 2 \zeta_{6} ) q^{7} + 9 q^{9} +O(q^{10})\) \( q -3 q^{3} + ( -4 + 4 \zeta_{6} ) q^{4} + ( -2 + 2 \zeta_{6} ) q^{7} + 9 q^{9} + ( 12 - 12 \zeta_{6} ) q^{12} -23 \zeta_{6} q^{13} -16 \zeta_{6} q^{16} -26 \zeta_{6} q^{19} + ( 6 - 6 \zeta_{6} ) q^{21} + 25 q^{25} -27 q^{27} -8 \zeta_{6} q^{28} + ( 13 - 13 \zeta_{6} ) q^{31} + ( -36 + 36 \zeta_{6} ) q^{36} -26 \zeta_{6} q^{37} + 69 \zeta_{6} q^{39} -61 q^{43} + 48 \zeta_{6} q^{48} + 45 \zeta_{6} q^{49} + 92 q^{52} + 78 \zeta_{6} q^{57} -47 \zeta_{6} q^{61} + ( -18 + 18 \zeta_{6} ) q^{63} + 64 q^{64} + ( -77 + 45 \zeta_{6} ) q^{67} -143 \zeta_{6} q^{73} -75 q^{75} + 104 q^{76} + ( -131 + 131 \zeta_{6} ) q^{79} + 81 q^{81} + 24 \zeta_{6} q^{84} + 46 q^{91} + ( -39 + 39 \zeta_{6} ) q^{93} -167 \zeta_{6} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{3} - 4q^{4} - 2q^{7} + 18q^{9} + O(q^{10}) \) \( 2q - 6q^{3} - 4q^{4} - 2q^{7} + 18q^{9} + 12q^{12} - 23q^{13} - 16q^{16} - 26q^{19} + 6q^{21} + 50q^{25} - 54q^{27} - 8q^{28} + 13q^{31} - 36q^{36} - 26q^{37} + 69q^{39} - 122q^{43} + 48q^{48} + 45q^{49} + 184q^{52} + 78q^{57} - 47q^{61} - 18q^{63} + 128q^{64} - 109q^{67} - 143q^{73} - 150q^{75} + 208q^{76} - 131q^{79} + 162q^{81} + 24q^{84} + 92q^{91} - 39q^{93} - 167q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −3.00000 −2.00000 3.46410i 0 0 −1.00000 1.73205i 0 9.00000 0
104.1 0 −3.00000 −2.00000 + 3.46410i 0 0 −1.00000 + 1.73205i 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
67.c even 3 1 inner
201.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.3.g.a 2
3.b odd 2 1 CM 201.3.g.a 2
67.c even 3 1 inner 201.3.g.a 2
201.g odd 6 1 inner 201.3.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.3.g.a 2 1.a even 1 1 trivial
201.3.g.a 2 3.b odd 2 1 CM
201.3.g.a 2 67.c even 3 1 inner
201.3.g.a 2 201.g odd 6 1 inner

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T + 4 T^{2} )( 1 + 2 T + 4 T^{2} ) \)
$3$ \( ( 1 + 3 T )^{2} \)
$5$ \( ( 1 - 5 T )^{2}( 1 + 5 T )^{2} \)
$7$ \( ( 1 - 11 T + 49 T^{2} )( 1 + 13 T + 49 T^{2} ) \)
$11$ \( ( 1 - 11 T + 121 T^{2} )( 1 + 11 T + 121 T^{2} ) \)
$13$ \( ( 1 + T + 169 T^{2} )( 1 + 22 T + 169 T^{2} ) \)
$17$ \( ( 1 - 17 T + 289 T^{2} )( 1 + 17 T + 289 T^{2} ) \)
$19$ \( ( 1 - 11 T + 361 T^{2} )( 1 + 37 T + 361 T^{2} ) \)
$23$ \( ( 1 - 23 T + 529 T^{2} )( 1 + 23 T + 529 T^{2} ) \)
$29$ \( ( 1 - 29 T + 841 T^{2} )( 1 + 29 T + 841 T^{2} ) \)
$31$ \( ( 1 - 59 T + 961 T^{2} )( 1 + 46 T + 961 T^{2} ) \)
$37$ \( ( 1 - 47 T + 1369 T^{2} )( 1 + 73 T + 1369 T^{2} ) \)
$41$ \( ( 1 - 41 T + 1681 T^{2} )( 1 + 41 T + 1681 T^{2} ) \)
$43$ \( ( 1 + 61 T + 1849 T^{2} )^{2} \)
$47$ \( ( 1 - 47 T + 2209 T^{2} )( 1 + 47 T + 2209 T^{2} ) \)
$53$ \( ( 1 - 53 T )^{2}( 1 + 53 T )^{2} \)
$59$ \( ( 1 - 59 T )^{2}( 1 + 59 T )^{2} \)
$61$ \( ( 1 - 74 T + 3721 T^{2} )( 1 + 121 T + 3721 T^{2} ) \)
$67$ \( 1 + 109 T + 4489 T^{2} \)
$71$ \( ( 1 - 71 T + 5041 T^{2} )( 1 + 71 T + 5041 T^{2} ) \)
$73$ \( ( 1 + 46 T + 5329 T^{2} )( 1 + 97 T + 5329 T^{2} ) \)
$79$ \( ( 1 - 11 T + 6241 T^{2} )( 1 + 142 T + 6241 T^{2} ) \)
$83$ \( ( 1 - 83 T + 6889 T^{2} )( 1 + 83 T + 6889 T^{2} ) \)
$89$ \( ( 1 - 89 T )^{2}( 1 + 89 T )^{2} \)
$97$ \( ( 1 - 2 T + 9409 T^{2} )( 1 + 169 T + 9409 T^{2} ) \)
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