Properties

Label 9.4.a.a
Level $9$
Weight $4$
Character orbit 9.a
Self dual yes
Analytic conductor $0.531$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.531017190052\)
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

\(f(q)\) \(=\) \( q - 8 q^{4} + 20 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 8 q^{4} + 20 q^{7} - 70 q^{13} + 64 q^{16} + 56 q^{19} - 125 q^{25} - 160 q^{28} + 308 q^{31} + 110 q^{37} - 520 q^{43} + 57 q^{49} + 560 q^{52} + 182 q^{61} - 512 q^{64} - 880 q^{67} + 1190 q^{73} - 448 q^{76} + 884 q^{79} - 1400 q^{91} - 1330 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Expression as an eta quotient

\(f(z) = \eta(3z)^{8}=q\prod_{n=1}^\infty(1 - q^{3n})^{8}\)

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 −8.00000 0 0 20.0000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)

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 9.4.a.a 1
3.b odd 2 1 CM 9.4.a.a 1
4.b odd 2 1 144.4.a.d 1
5.b even 2 1 225.4.a.d 1
5.c odd 4 2 225.4.b.g 2
7.b odd 2 1 441.4.a.f 1
7.c even 3 2 441.4.e.i 2
7.d odd 6 2 441.4.e.j 2
8.b even 2 1 576.4.a.m 1
8.d odd 2 1 576.4.a.l 1
9.c even 3 2 81.4.c.b 2
9.d odd 6 2 81.4.c.b 2
11.b odd 2 1 1089.4.a.g 1
12.b even 2 1 144.4.a.d 1
13.b even 2 1 1521.4.a.g 1
15.d odd 2 1 225.4.a.d 1
15.e even 4 2 225.4.b.g 2
21.c even 2 1 441.4.a.f 1
21.g even 6 2 441.4.e.j 2
21.h odd 6 2 441.4.e.i 2
24.f even 2 1 576.4.a.l 1
24.h odd 2 1 576.4.a.m 1
33.d even 2 1 1089.4.a.g 1
39.d odd 2 1 1521.4.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.a.a 1 1.a even 1 1 trivial
9.4.a.a 1 3.b odd 2 1 CM
81.4.c.b 2 9.c even 3 2
81.4.c.b 2 9.d odd 6 2
144.4.a.d 1 4.b odd 2 1
144.4.a.d 1 12.b even 2 1
225.4.a.d 1 5.b even 2 1
225.4.a.d 1 15.d odd 2 1
225.4.b.g 2 5.c odd 4 2
225.4.b.g 2 15.e even 4 2
441.4.a.f 1 7.b odd 2 1
441.4.a.f 1 21.c even 2 1
441.4.e.i 2 7.c even 3 2
441.4.e.i 2 21.h odd 6 2
441.4.e.j 2 7.d odd 6 2
441.4.e.j 2 21.g even 6 2
576.4.a.l 1 8.d odd 2 1
576.4.a.l 1 24.f even 2 1
576.4.a.m 1 8.b even 2 1
576.4.a.m 1 24.h odd 2 1
1089.4.a.g 1 11.b odd 2 1
1089.4.a.g 1 33.d even 2 1
1521.4.a.g 1 13.b even 2 1
1521.4.a.g 1 39.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(\Gamma_0(9))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 20 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 70 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T - 56 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T - 308 \) Copy content Toggle raw display
$37$ \( T - 110 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T + 520 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 182 \) Copy content Toggle raw display
$67$ \( T + 880 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 1190 \) Copy content Toggle raw display
$79$ \( T - 884 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T + 1330 \) Copy content Toggle raw display
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