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 $1$

Related objects

Downloads

Learn more

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

\(f(q)\) \(=\) \( q + q^{2} - q^{4} - q^{5} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{4} - q^{5} - 3 q^{8} - q^{10} - 4 q^{11} + q^{13} - q^{16} - 2 q^{17} - 4 q^{19} + q^{20} - 4 q^{22} - 8 q^{23} + q^{25} + q^{26} + 2 q^{29} - 8 q^{31} + 5 q^{32} - 2 q^{34} + 6 q^{37} - 4 q^{38} + 3 q^{40} + 6 q^{41} - 4 q^{43} + 4 q^{44} - 8 q^{46} + 8 q^{47} - 7 q^{49} + q^{50} - q^{52} - 6 q^{53} + 4 q^{55} + 2 q^{58} + 12 q^{59} - 2 q^{61} - 8 q^{62} + 7 q^{64} - q^{65} - 4 q^{67} + 2 q^{68} - 6 q^{73} + 6 q^{74} + 4 q^{76} + 16 q^{79} + q^{80} + 6 q^{82} + 4 q^{83} + 2 q^{85} - 4 q^{86} + 12 q^{88} - 10 q^{89} + 8 q^{92} + 8 q^{94} + 4 q^{95} + 18 q^{97} - 7 q^{98}+O(q^{100}) \) 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
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(585))\):

\( T_{2} - 1 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{11} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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