Properties

Label 1521.2.a.b
Level $1521$
Weight $2$
Character orbit 1521.a
Self dual yes
Analytic conductor $12.145$
Analytic rank $1$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.1452461474\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 117)
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q - 2 q^{4} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{4} - q^{7} + 4 q^{16} + 8 q^{19} - 5 q^{25} + 2 q^{28} - 7 q^{31} - 10 q^{37} - 13 q^{43} - 6 q^{49} - 13 q^{61} - 8 q^{64} + 11 q^{67} + 17 q^{73} - 16 q^{76} - 13 q^{79} + 5 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 −2.00000 0 0 −1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(13\) \(1\)

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 1521.2.a.b 1
3.b odd 2 1 CM 1521.2.a.b 1
13.b even 2 1 1521.2.a.c 1
13.c even 3 2 117.2.g.a 2
13.d odd 4 2 1521.2.b.e 2
39.d odd 2 1 1521.2.a.c 1
39.f even 4 2 1521.2.b.e 2
39.i odd 6 2 117.2.g.a 2
52.j odd 6 2 1872.2.t.g 2
156.p even 6 2 1872.2.t.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.2.g.a 2 13.c even 3 2
117.2.g.a 2 39.i odd 6 2
1521.2.a.b 1 1.a even 1 1 trivial
1521.2.a.b 1 3.b odd 2 1 CM
1521.2.a.c 1 13.b even 2 1
1521.2.a.c 1 39.d odd 2 1
1521.2.b.e 2 13.d odd 4 2
1521.2.b.e 2 39.f even 4 2
1872.2.t.g 2 52.j odd 6 2
1872.2.t.g 2 156.p even 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1521))\):

\( T_{2} \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{7} + 1 \) Copy content Toggle raw display

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
$7$ \( T + 1 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T - 8 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T + 7 \) Copy content Toggle raw display
$37$ \( T + 10 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T + 13 \) 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
$61$ \( T + 13 \) Copy content Toggle raw display
$67$ \( T - 11 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 17 \) Copy content Toggle raw display
$79$ \( T + 13 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T - 5 \) Copy content Toggle raw display
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