Properties

Label 585.2.a.b
Level $585$
Weight $2$
Character orbit 585.a
Self dual yes
Analytic conductor $4.671$
Analytic rank $0$
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2 q^{2} + 2 q^{4} - q^{5} + 3 q^{7} + O(q^{10}) \) \( q - 2 q^{2} + 2 q^{4} - q^{5} + 3 q^{7} + 2 q^{10} + q^{11} - q^{13} - 6 q^{14} - 4 q^{16} + q^{17} - 2 q^{19} - 2 q^{20} - 2 q^{22} + 3 q^{23} + q^{25} + 2 q^{26} + 6 q^{28} + 2 q^{29} - 6 q^{31} + 8 q^{32} - 2 q^{34} - 3 q^{35} + 11 q^{37} + 4 q^{38} + 5 q^{41} + 4 q^{43} + 2 q^{44} - 6 q^{46} + 10 q^{47} + 2 q^{49} - 2 q^{50} - 2 q^{52} - 11 q^{53} - q^{55} - 4 q^{58} - 8 q^{59} + 13 q^{61} + 12 q^{62} - 8 q^{64} + q^{65} + 12 q^{67} + 2 q^{68} + 6 q^{70} + 5 q^{71} + 10 q^{73} - 22 q^{74} - 4 q^{76} + 3 q^{77} - 3 q^{79} + 4 q^{80} - 10 q^{82} + 12 q^{83} - q^{85} - 8 q^{86} + 15 q^{89} - 3 q^{91} + 6 q^{92} - 20 q^{94} + 2 q^{95} + 17 q^{97} - 4 q^{98} + O(q^{100}) \)

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
−2.00000 0 2.00000 −1.00000 0 3.00000 0 0 2.00000
\(n\): 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.b 1
3.b odd 2 1 195.2.a.b 1
4.b odd 2 1 9360.2.a.d 1
5.b even 2 1 2925.2.a.q 1
5.c odd 4 2 2925.2.c.c 2
12.b even 2 1 3120.2.a.u 1
13.b even 2 1 7605.2.a.u 1
15.d odd 2 1 975.2.a.c 1
15.e even 4 2 975.2.c.a 2
21.c even 2 1 9555.2.a.v 1
39.d odd 2 1 2535.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.a.b 1 3.b odd 2 1
585.2.a.b 1 1.a even 1 1 trivial
975.2.a.c 1 15.d odd 2 1
975.2.c.a 2 15.e even 4 2
2535.2.a.a 1 39.d odd 2 1
2925.2.a.q 1 5.b even 2 1
2925.2.c.c 2 5.c odd 4 2
3120.2.a.u 1 12.b even 2 1
7605.2.a.u 1 13.b even 2 1
9360.2.a.d 1 4.b odd 2 1
9555.2.a.v 1 21.c even 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( T \)
$5$ \( 1 + T \)
$7$ \( -3 + T \)
$11$ \( -1 + T \)
$13$ \( 1 + T \)
$17$ \( -1 + T \)
$19$ \( 2 + T \)
$23$ \( -3 + T \)
$29$ \( -2 + T \)
$31$ \( 6 + T \)
$37$ \( -11 + T \)
$41$ \( -5 + T \)
$43$ \( -4 + T \)
$47$ \( -10 + T \)
$53$ \( 11 + T \)
$59$ \( 8 + T \)
$61$ \( -13 + T \)
$67$ \( -12 + T \)
$71$ \( -5 + T \)
$73$ \( -10 + T \)
$79$ \( 3 + T \)
$83$ \( -12 + T \)
$89$ \( -15 + T \)
$97$ \( -17 + T \)
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