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

Related objects

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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

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

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 −8.00000 0 0 20.0000 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 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

Atkin-Lehner signs

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

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 8 T^{2} \)
$3$ 1
$5$ \( 1 + 125 T^{2} \)
$7$ \( 1 - 20 T + 343 T^{2} \)
$11$ \( 1 + 1331 T^{2} \)
$13$ \( 1 + 70 T + 2197 T^{2} \)
$17$ \( 1 + 4913 T^{2} \)
$19$ \( 1 - 56 T + 6859 T^{2} \)
$23$ \( 1 + 12167 T^{2} \)
$29$ \( 1 + 24389 T^{2} \)
$31$ \( 1 - 308 T + 29791 T^{2} \)
$37$ \( 1 - 110 T + 50653 T^{2} \)
$41$ \( 1 + 68921 T^{2} \)
$43$ \( 1 + 520 T + 79507 T^{2} \)
$47$ \( 1 + 103823 T^{2} \)
$53$ \( 1 + 148877 T^{2} \)
$59$ \( 1 + 205379 T^{2} \)
$61$ \( 1 - 182 T + 226981 T^{2} \)
$67$ \( 1 + 880 T + 300763 T^{2} \)
$71$ \( 1 + 357911 T^{2} \)
$73$ \( 1 - 1190 T + 389017 T^{2} \)
$79$ \( 1 - 884 T + 493039 T^{2} \)
$83$ \( 1 + 571787 T^{2} \)
$89$ \( 1 + 704969 T^{2} \)
$97$ \( 1 + 1330 T + 912673 T^{2} \)
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Additional information

This cusp form has an eta product $\eta(3z)^8=q\prod_{n=1}^\infty (1-q^{3n})^8$ where $q=\exp(2\pi i z)$.