Properties

Label 585.2.a.g
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: no (minimal twist has level 195)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - q^{4} - q^{5} - 3q^{8} + O(q^{10}) \) \( q + q^{2} - q^{4} - q^{5} - 3q^{8} - q^{10} - 4q^{11} + q^{13} - q^{16} - 2q^{17} - 4q^{19} + q^{20} - 4q^{22} - 8q^{23} + q^{25} + q^{26} + 2q^{29} - 8q^{31} + 5q^{32} - 2q^{34} + 6q^{37} - 4q^{38} + 3q^{40} + 6q^{41} - 4q^{43} + 4q^{44} - 8q^{46} + 8q^{47} - 7q^{49} + q^{50} - q^{52} - 6q^{53} + 4q^{55} + 2q^{58} + 12q^{59} - 2q^{61} - 8q^{62} + 7q^{64} - q^{65} - 4q^{67} + 2q^{68} - 6q^{73} + 6q^{74} + 4q^{76} + 16q^{79} + q^{80} + 6q^{82} + 4q^{83} + 2q^{85} - 4q^{86} + 12q^{88} - 10q^{89} + 8q^{92} + 8q^{94} + 4q^{95} + 18q^{97} - 7q^{98} + O(q^{100}) \)

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
1.00000 0 −1.00000 −1.00000 0 0 −3.00000 0 −1.00000
\(n\): 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.g 1
3.b odd 2 1 195.2.a.a 1
4.b odd 2 1 9360.2.a.o 1
5.b even 2 1 2925.2.a.d 1
5.c odd 4 2 2925.2.c.f 2
12.b even 2 1 3120.2.a.k 1
13.b even 2 1 7605.2.a.h 1
15.d odd 2 1 975.2.a.i 1
15.e even 4 2 975.2.c.e 2
21.c even 2 1 9555.2.a.b 1
39.d odd 2 1 2535.2.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.a.a 1 3.b odd 2 1
585.2.a.g 1 1.a even 1 1 trivial
975.2.a.i 1 15.d odd 2 1
975.2.c.e 2 15.e even 4 2
2535.2.a.k 1 39.d odd 2 1
2925.2.a.d 1 5.b even 2 1
2925.2.c.f 2 5.c odd 4 2
3120.2.a.k 1 12.b even 2 1
7605.2.a.h 1 13.b even 2 1
9360.2.a.o 1 4.b odd 2 1
9555.2.a.b 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(585))\):

\( T_{2} - 1 \)
\( T_{7} \)
\( T_{11} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( T \)
$5$ \( 1 + T \)
$7$ \( T \)
$11$ \( 4 + T \)
$13$ \( -1 + T \)
$17$ \( 2 + T \)
$19$ \( 4 + T \)
$23$ \( 8 + T \)
$29$ \( -2 + T \)
$31$ \( 8 + T \)
$37$ \( -6 + T \)
$41$ \( -6 + T \)
$43$ \( 4 + T \)
$47$ \( -8 + T \)
$53$ \( 6 + T \)
$59$ \( -12 + T \)
$61$ \( 2 + T \)
$67$ \( 4 + T \)
$71$ \( T \)
$73$ \( 6 + T \)
$79$ \( -16 + T \)
$83$ \( -4 + T \)
$89$ \( 10 + T \)
$97$ \( -18 + T \)
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