Properties

Label 108.2.a.a
Level 108
Weight 2
Character orbit 108.a
Self dual yes
Analytic conductor 0.862
Analytic rank 0
Dimension 1
CM discriminant -3
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(0.862384341830\)
Analytic rank: \(0\)
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

\(f(q)\) \(=\) \( q + 5q^{7} + O(q^{10}) \) \( q + 5q^{7} - 7q^{13} - q^{19} - 5q^{25} - 4q^{31} - q^{37} + 8q^{43} + 18q^{49} - 13q^{61} + 11q^{67} + 17q^{73} - 13q^{79} - 35q^{91} + 5q^{97} + O(q^{100}) \)

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 5.00000 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 108.2.a.a 1
3.b odd 2 1 CM 108.2.a.a 1
4.b odd 2 1 432.2.a.d 1
5.b even 2 1 2700.2.a.b 1
5.c odd 4 2 2700.2.d.g 2
7.b odd 2 1 5292.2.a.j 1
8.b even 2 1 1728.2.a.p 1
8.d odd 2 1 1728.2.a.m 1
9.c even 3 2 324.2.e.b 2
9.d odd 6 2 324.2.e.b 2
12.b even 2 1 432.2.a.d 1
15.d odd 2 1 2700.2.a.b 1
15.e even 4 2 2700.2.d.g 2
21.c even 2 1 5292.2.a.j 1
24.f even 2 1 1728.2.a.m 1
24.h odd 2 1 1728.2.a.p 1
36.f odd 6 2 1296.2.i.j 2
36.h even 6 2 1296.2.i.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.2.a.a 1 1.a even 1 1 trivial
108.2.a.a 1 3.b odd 2 1 CM
324.2.e.b 2 9.c even 3 2
324.2.e.b 2 9.d odd 6 2
432.2.a.d 1 4.b odd 2 1
432.2.a.d 1 12.b even 2 1
1296.2.i.j 2 36.f odd 6 2
1296.2.i.j 2 36.h even 6 2
1728.2.a.m 1 8.d odd 2 1
1728.2.a.m 1 24.f even 2 1
1728.2.a.p 1 8.b even 2 1
1728.2.a.p 1 24.h odd 2 1
2700.2.a.b 1 5.b even 2 1
2700.2.a.b 1 15.d odd 2 1
2700.2.d.g 2 5.c odd 4 2
2700.2.d.g 2 15.e even 4 2
5292.2.a.j 1 7.b odd 2 1
5292.2.a.j 1 21.c even 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(\Gamma_0(108))\).

Hecke Characteristic Polynomials

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