Properties

Label 201.1.g.a
Level 201
Weight 1
Character orbit 201.g
Analytic conductor 0.100
Analytic rank 0
Dimension 4
Projective image \(A_{4}\)
CM/RM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 201 = 3 \cdot 67 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 201.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.100312067539\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(A_{4}\)
Projective field Galois closure of 4.0.40401.1
Artin image $\SL(2,3):C_2$
Artin field Galois closure of \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{12}^{5} q^{2} + \zeta_{12}^{3} q^{3} -\zeta_{12}^{2} q^{6} + \zeta_{12}^{4} q^{7} -\zeta_{12}^{3} q^{8} - q^{9} +O(q^{10})\) \( q + \zeta_{12}^{5} q^{2} + \zeta_{12}^{3} q^{3} -\zeta_{12}^{2} q^{6} + \zeta_{12}^{4} q^{7} -\zeta_{12}^{3} q^{8} - q^{9} + \zeta_{12} q^{11} -\zeta_{12}^{2} q^{13} -\zeta_{12}^{3} q^{14} + \zeta_{12}^{2} q^{16} + \zeta_{12}^{5} q^{17} -\zeta_{12}^{5} q^{18} + \zeta_{12}^{2} q^{19} -\zeta_{12} q^{21} - q^{22} -\zeta_{12}^{5} q^{23} + q^{24} + q^{25} + \zeta_{12} q^{26} -\zeta_{12}^{3} q^{27} + \zeta_{12} q^{29} -\zeta_{12}^{4} q^{31} + \zeta_{12}^{4} q^{33} -\zeta_{12}^{4} q^{34} -\zeta_{12}^{2} q^{37} -\zeta_{12} q^{38} -\zeta_{12}^{5} q^{39} -\zeta_{12} q^{41} + q^{42} + \zeta_{12}^{4} q^{46} + \zeta_{12} q^{47} + \zeta_{12}^{5} q^{48} + \zeta_{12}^{5} q^{50} -\zeta_{12}^{2} q^{51} + \zeta_{12}^{2} q^{54} + \zeta_{12} q^{56} + \zeta_{12}^{5} q^{57} - q^{58} -\zeta_{12}^{2} q^{61} + \zeta_{12}^{3} q^{62} -\zeta_{12}^{4} q^{63} - q^{64} -\zeta_{12}^{3} q^{66} - q^{67} + \zeta_{12}^{2} q^{69} -\zeta_{12} q^{71} + \zeta_{12}^{3} q^{72} + \zeta_{12}^{2} q^{73} + \zeta_{12} q^{74} + \zeta_{12}^{3} q^{75} + \zeta_{12}^{5} q^{77} + \zeta_{12}^{4} q^{78} + \zeta_{12}^{4} q^{79} + q^{81} + q^{82} + \zeta_{12}^{5} q^{83} + \zeta_{12}^{4} q^{87} -\zeta_{12}^{4} q^{88} -2 \zeta_{12}^{3} q^{89} + q^{91} + \zeta_{12} q^{93} - q^{94} -\zeta_{12}^{2} q^{97} -\zeta_{12} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{6} - 2q^{7} - 4q^{9} + O(q^{10}) \) \( 4q - 2q^{6} - 2q^{7} - 4q^{9} - 2q^{13} + 2q^{16} + 2q^{19} - 4q^{22} + 4q^{24} + 4q^{25} + 2q^{31} - 2q^{33} + 2q^{34} - 2q^{37} + 4q^{42} - 2q^{46} - 2q^{51} + 2q^{54} - 4q^{58} - 2q^{61} + 2q^{63} - 4q^{64} - 4q^{67} + 2q^{69} + 2q^{73} - 2q^{78} - 2q^{79} + 4q^{81} + 4q^{82} - 2q^{87} + 2q^{88} + 4q^{91} - 4q^{94} - 2q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(68\) \(136\)
\(\chi(n)\) \(-1\) \(\zeta_{12}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 0.500000i 1.00000i 0 0 −0.500000 + 0.866025i −0.500000 0.866025i 1.00000i −1.00000 0
29.2 0.866025 + 0.500000i 1.00000i 0 0 −0.500000 + 0.866025i −0.500000 0.866025i 1.00000i −1.00000 0
104.1 −0.866025 + 0.500000i 1.00000i 0 0 −0.500000 0.866025i −0.500000 + 0.866025i 1.00000i −1.00000 0
104.2 0.866025 0.500000i 1.00000i 0 0 −0.500000 0.866025i −0.500000 + 0.866025i 1.00000i −1.00000 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 inner
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 201.1.g.a 4
3.b odd 2 1 inner 201.1.g.a 4
4.b odd 2 1 3216.1.bb.b 4
12.b even 2 1 3216.1.bb.b 4
67.c even 3 1 inner 201.1.g.a 4
201.g odd 6 1 inner 201.1.g.a 4
268.g odd 6 1 3216.1.bb.b 4
804.l even 6 1 3216.1.bb.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.1.g.a 4 1.a even 1 1 trivial
201.1.g.a 4 3.b odd 2 1 inner
201.1.g.a 4 67.c even 3 1 inner
201.1.g.a 4 201.g odd 6 1 inner
3216.1.bb.b 4 4.b odd 2 1
3216.1.bb.b 4 12.b even 2 1
3216.1.bb.b 4 268.g odd 6 1
3216.1.bb.b 4 804.l even 6 1

Hecke kernels

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

Hecke Characteristic Polynomials

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