Properties

Label 201.1.k.a
Level 201
Weight 1
Character orbit 201.k
Analytic conductor 0.100
Analytic rank 0
Dimension 10
Projective image \(D_{11}\)
CM discriminant -3
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 201 = 3 \cdot 67 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 201.k (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.100312067539\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\Q(\zeta_{22})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{11}\)
Projective field Galois closure of \(\mathbb{Q}[x]/(x^{11} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{22}^{7} q^{3} + \zeta_{22}^{6} q^{4} + ( \zeta_{22}^{8} - \zeta_{22}^{9} ) q^{7} -\zeta_{22}^{3} q^{9} +O(q^{10})\) \( q -\zeta_{22}^{7} q^{3} + \zeta_{22}^{6} q^{4} + ( \zeta_{22}^{8} - \zeta_{22}^{9} ) q^{7} -\zeta_{22}^{3} q^{9} + \zeta_{22}^{2} q^{12} + ( -\zeta_{22}^{5} + \zeta_{22}^{10} ) q^{13} -\zeta_{22} q^{16} + ( -\zeta_{22} + \zeta_{22}^{4} ) q^{19} + ( \zeta_{22}^{4} - \zeta_{22}^{5} ) q^{21} + \zeta_{22}^{2} q^{25} + \zeta_{22}^{10} q^{27} + ( -\zeta_{22}^{3} + \zeta_{22}^{4} ) q^{28} + ( \zeta_{22}^{8} + \zeta_{22}^{10} ) q^{31} -\zeta_{22}^{9} q^{36} + ( \zeta_{22}^{2} - \zeta_{22}^{9} ) q^{37} + ( -\zeta_{22} + \zeta_{22}^{6} ) q^{39} + ( -\zeta_{22}^{3} - \zeta_{22}^{7} ) q^{43} + \zeta_{22}^{8} q^{48} + ( -\zeta_{22}^{5} + \zeta_{22}^{6} - \zeta_{22}^{7} ) q^{49} + ( 1 - \zeta_{22}^{5} ) q^{52} + ( 1 + \zeta_{22}^{8} ) q^{57} + ( -\zeta_{22}^{3} + \zeta_{22}^{6} ) q^{61} + ( 1 - \zeta_{22} ) q^{63} -\zeta_{22}^{7} q^{64} + \zeta_{22}^{2} q^{67} + ( 1 - \zeta_{22}^{9} ) q^{73} -\zeta_{22}^{9} q^{75} + ( -\zeta_{22}^{7} + \zeta_{22}^{10} ) q^{76} + ( 1 + \zeta_{22}^{4} ) q^{79} + \zeta_{22}^{6} q^{81} + ( 1 + \zeta_{22}^{10} ) q^{84} + ( \zeta_{22}^{2} - \zeta_{22}^{3} - \zeta_{22}^{7} + \zeta_{22}^{8} ) q^{91} + ( \zeta_{22}^{4} + \zeta_{22}^{6} ) q^{93} + ( \zeta_{22}^{4} - \zeta_{22}^{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - q^{3} - q^{4} - 2q^{7} - q^{9} + O(q^{10}) \) \( 10q - q^{3} - q^{4} - 2q^{7} - q^{9} - q^{12} - 2q^{13} - q^{16} - 2q^{19} - 2q^{21} - q^{25} - q^{27} - 2q^{28} - 2q^{31} - q^{36} - 2q^{37} - 2q^{39} - 2q^{43} - q^{48} - 3q^{49} + 9q^{52} + 9q^{57} - 2q^{61} + 9q^{63} - q^{64} - q^{67} + 9q^{73} - q^{75} - 2q^{76} + 9q^{79} - q^{81} + 9q^{84} - 4q^{91} - 2q^{93} - 2q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(68\) \(136\)
\(\chi(n)\) \(-1\) \(\zeta_{22}^{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1
0.142315 + 0.989821i
−0.841254 0.540641i
0.959493 0.281733i
−0.415415 + 0.909632i
−0.841254 + 0.540641i
0.959493 + 0.281733i
−0.415415 0.909632i
0.654861 0.755750i
0.654861 + 0.755750i
0.142315 0.989821i
0 0.841254 + 0.540641i −0.654861 + 0.755750i 0 0 −0.544078 1.19136i 0 0.415415 + 0.909632i 0
59.1 0 −0.654861 0.755750i −0.959493 0.281733i 0 0 0.273100 1.89945i 0 −0.142315 + 0.989821i 0
62.1 0 0.415415 + 0.909632i −0.142315 0.989821i 0 0 0.186393 0.215109i 0 −0.654861 + 0.755750i 0
89.1 0 −0.142315 0.989821i 0.841254 0.540641i 0 0 −1.61435 + 0.474017i 0 −0.959493 + 0.281733i 0
92.1 0 −0.654861 + 0.755750i −0.959493 + 0.281733i 0 0 0.273100 + 1.89945i 0 −0.142315 0.989821i 0
107.1 0 0.415415 0.909632i −0.142315 + 0.989821i 0 0 0.186393 + 0.215109i 0 −0.654861 0.755750i 0
131.1 0 −0.142315 + 0.989821i 0.841254 + 0.540641i 0 0 −1.61435 0.474017i 0 −0.959493 0.281733i 0
143.1 0 −0.959493 0.281733i 0.415415 + 0.909632i 0 0 0.698939 + 0.449181i 0 0.841254 + 0.540641i 0
149.1 0 −0.959493 + 0.281733i 0.415415 0.909632i 0 0 0.698939 0.449181i 0 0.841254 0.540641i 0
158.1 0 0.841254 0.540641i −0.654861 0.755750i 0 0 −0.544078 + 1.19136i 0 0.415415 0.909632i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 158.1
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.e even 11 1 inner
201.k odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.1.k.a 10
3.b odd 2 1 CM 201.1.k.a 10
4.b odd 2 1 3216.1.cg.a 10
12.b even 2 1 3216.1.cg.a 10
67.e even 11 1 inner 201.1.k.a 10
201.k odd 22 1 inner 201.1.k.a 10
268.k odd 22 1 3216.1.cg.a 10
804.w even 22 1 3216.1.cg.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.1.k.a 10 1.a even 1 1 trivial
201.1.k.a 10 3.b odd 2 1 CM
201.1.k.a 10 67.e even 11 1 inner
201.1.k.a 10 201.k odd 22 1 inner
3216.1.cg.a 10 4.b odd 2 1
3216.1.cg.a 10 12.b even 2 1
3216.1.cg.a 10 268.k odd 22 1
3216.1.cg.a 10 804.w even 22 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(201, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
$3$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
$5$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
$7$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
$11$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
$13$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
$17$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
$19$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
$23$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
$29$ \( ( 1 - T )^{10}( 1 + T )^{10} \)
$31$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
$37$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
$41$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
$43$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
$47$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
$53$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
$59$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
$61$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
$67$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
$71$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
$73$ \( ( 1 - T )^{10}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
$79$ \( ( 1 - T )^{10}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
$83$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
$89$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
$97$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
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