Properties

Label 585.2.a.f
Level $585$
Weight $2$
Character orbit 585.a
Self dual yes
Analytic conductor $4.671$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.67124851824\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2 q^{4} + q^{5} - q^{7} + O(q^{10}) \) \( q - 2 q^{4} + q^{5} - q^{7} - 3 q^{11} + q^{13} + 4 q^{16} - 3 q^{17} - 4 q^{19} - 2 q^{20} - 9 q^{23} + q^{25} + 2 q^{28} - 6 q^{29} + 2 q^{31} - q^{35} - q^{37} - 3 q^{41} + 2 q^{43} + 6 q^{44} - 6 q^{47} - 6 q^{49} - 2 q^{52} + 9 q^{53} - 3 q^{55} - 12 q^{59} + 5 q^{61} - 8 q^{64} + q^{65} - 4 q^{67} + 6 q^{68} + 9 q^{71} + 14 q^{73} + 8 q^{76} + 3 q^{77} - 7 q^{79} + 4 q^{80} - 3 q^{85} + 15 q^{89} - q^{91} + 18 q^{92} - 4 q^{95} + 5 q^{97} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(13\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.a.f yes 1
3.b odd 2 1 585.2.a.e 1
4.b odd 2 1 9360.2.a.bu 1
5.b even 2 1 2925.2.a.i 1
5.c odd 4 2 2925.2.c.l 2
12.b even 2 1 9360.2.a.r 1
13.b even 2 1 7605.2.a.j 1
15.d odd 2 1 2925.2.a.k 1
15.e even 4 2 2925.2.c.m 2
39.d odd 2 1 7605.2.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.a.e 1 3.b odd 2 1
585.2.a.f yes 1 1.a even 1 1 trivial
2925.2.a.i 1 5.b even 2 1
2925.2.a.k 1 15.d odd 2 1
2925.2.c.l 2 5.c odd 4 2
2925.2.c.m 2 15.e even 4 2
7605.2.a.j 1 13.b even 2 1
7605.2.a.m 1 39.d odd 2 1
9360.2.a.r 1 12.b even 2 1
9360.2.a.bu 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(585))\):

\( T_{2} \)
\( T_{7} + 1 \)
\( T_{11} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( -1 + T \)
$7$ \( 1 + T \)
$11$ \( 3 + T \)
$13$ \( -1 + T \)
$17$ \( 3 + T \)
$19$ \( 4 + T \)
$23$ \( 9 + T \)
$29$ \( 6 + T \)
$31$ \( -2 + T \)
$37$ \( 1 + T \)
$41$ \( 3 + T \)
$43$ \( -2 + T \)
$47$ \( 6 + T \)
$53$ \( -9 + T \)
$59$ \( 12 + T \)
$61$ \( -5 + T \)
$67$ \( 4 + T \)
$71$ \( -9 + T \)
$73$ \( -14 + T \)
$79$ \( 7 + T \)
$83$ \( T \)
$89$ \( -15 + T \)
$97$ \( -5 + T \)
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