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 disc. -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: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

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

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 CM by \(\Q(\sqrt{-3}) \) yes

Atkin-Lehner signs

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

Hecke kernels

There are no other newforms in \(S_{4}^{\mathrm{new}}(\Gamma_0(9))\).