Properties

Label 108.6.a.a
Level 108
Weight 6
Character orbit 108.a
Self dual yes
Analytic conductor 17.321
Analytic rank 1
Dimension 1
CM discriminant -3
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) = \( 6 \)
Character orbit: \([\chi]\) = 108.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.3214525398\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q - 25q^{7} + O(q^{10}) \) \( q - 25q^{7} - 427q^{13} - 1711q^{19} - 3125q^{25} - 10324q^{31} - 6661q^{37} - 3352q^{43} - 16182q^{49} + 56927q^{61} - 37939q^{67} + 79577q^{73} + 90857q^{79} + 10675q^{91} + 177725q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 0 0 0 0 −25.0000 0 0 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 \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.6.a.a 1
3.b odd 2 1 CM 108.6.a.a 1
4.b odd 2 1 432.6.a.e 1
9.c even 3 2 324.6.e.c 2
9.d odd 6 2 324.6.e.c 2
12.b even 2 1 432.6.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.6.a.a 1 1.a even 1 1 trivial
108.6.a.a 1 3.b odd 2 1 CM
324.6.e.c 2 9.c even 3 2
324.6.e.c 2 9.d odd 6 2
432.6.a.e 1 4.b odd 2 1
432.6.a.e 1 12.b even 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(108))\):

\( T_{5} \)
\( T_{11} \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + 3125 T^{2} \)
$7$ \( 1 + 25 T + 16807 T^{2} \)
$11$ \( 1 + 161051 T^{2} \)
$13$ \( 1 + 427 T + 371293 T^{2} \)
$17$ \( 1 + 1419857 T^{2} \)
$19$ \( 1 + 1711 T + 2476099 T^{2} \)
$23$ \( 1 + 6436343 T^{2} \)
$29$ \( 1 + 20511149 T^{2} \)
$31$ \( 1 + 10324 T + 28629151 T^{2} \)
$37$ \( 1 + 6661 T + 69343957 T^{2} \)
$41$ \( 1 + 115856201 T^{2} \)
$43$ \( 1 + 3352 T + 147008443 T^{2} \)
$47$ \( 1 + 229345007 T^{2} \)
$53$ \( 1 + 418195493 T^{2} \)
$59$ \( 1 + 714924299 T^{2} \)
$61$ \( 1 - 56927 T + 844596301 T^{2} \)
$67$ \( 1 + 37939 T + 1350125107 T^{2} \)
$71$ \( 1 + 1804229351 T^{2} \)
$73$ \( 1 - 79577 T + 2073071593 T^{2} \)
$79$ \( 1 - 90857 T + 3077056399 T^{2} \)
$83$ \( 1 + 3939040643 T^{2} \)
$89$ \( 1 + 5584059449 T^{2} \)
$97$ \( 1 - 177725 T + 8587340257 T^{2} \)
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