Properties

Label 1620.2.a.a
Level $1620$
Weight $2$
Character orbit 1620.a
Self dual yes
Analytic conductor $12.936$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1620.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.9357651274\)
Analytic rank: \(0\)
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

\(f(q)\) \(=\) \( q - q^{5} - 4q^{7} + O(q^{10}) \) \( q - q^{5} - 4q^{7} - 3q^{11} - 4q^{13} + 5q^{19} + 6q^{23} + q^{25} + 9q^{29} + 5q^{31} + 4q^{35} + 2q^{37} + 9q^{41} - 10q^{43} + 6q^{47} + 9q^{49} + 12q^{53} + 3q^{55} - 9q^{59} - 10q^{61} + 4q^{65} + 2q^{67} - 3q^{71} - 4q^{73} + 12q^{77} - 4q^{79} - 6q^{83} + 9q^{89} + 16q^{91} - 5q^{95} + 2q^{97} + O(q^{100}) \)

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 0 −1.00000 0 −4.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)

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 1620.2.a.a 1
3.b odd 2 1 1620.2.a.d yes 1
4.b odd 2 1 6480.2.a.m 1
5.b even 2 1 8100.2.a.m 1
5.c odd 4 2 8100.2.d.b 2
9.c even 3 2 1620.2.i.l 2
9.d odd 6 2 1620.2.i.e 2
12.b even 2 1 6480.2.a.y 1
15.d odd 2 1 8100.2.a.n 1
15.e even 4 2 8100.2.d.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.2.a.a 1 1.a even 1 1 trivial
1620.2.a.d yes 1 3.b odd 2 1
1620.2.i.e 2 9.d odd 6 2
1620.2.i.l 2 9.c even 3 2
6480.2.a.m 1 4.b odd 2 1
6480.2.a.y 1 12.b even 2 1
8100.2.a.m 1 5.b even 2 1
8100.2.a.n 1 15.d odd 2 1
8100.2.d.b 2 5.c odd 4 2
8100.2.d.g 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1620))\):

\( T_{7} + 4 \)
\( T_{11} + 3 \)
\( T_{17} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( 1 + T \)
$7$ \( 4 + T \)
$11$ \( 3 + T \)
$13$ \( 4 + T \)
$17$ \( T \)
$19$ \( -5 + T \)
$23$ \( -6 + T \)
$29$ \( -9 + T \)
$31$ \( -5 + T \)
$37$ \( -2 + T \)
$41$ \( -9 + T \)
$43$ \( 10 + T \)
$47$ \( -6 + T \)
$53$ \( -12 + T \)
$59$ \( 9 + T \)
$61$ \( 10 + T \)
$67$ \( -2 + T \)
$71$ \( 3 + T \)
$73$ \( 4 + T \)
$79$ \( 4 + T \)
$83$ \( 6 + T \)
$89$ \( -9 + T \)
$97$ \( -2 + T \)
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