Properties

Label 1620.2.a.b
Level $1620$
Weight $2$
Character orbit 1620.a
Self dual yes
Analytic conductor $12.936$
Analytic rank $1$
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 180)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{5} - q^{7} + O(q^{10}) \) \( q - q^{5} - q^{7} - 4q^{13} + 6q^{17} + 2q^{19} + 3q^{23} + q^{25} - 3q^{29} - 10q^{31} + q^{35} - 10q^{37} - 9q^{41} - 4q^{43} - 9q^{47} - 6q^{49} + 6q^{53} + 6q^{59} - q^{61} + 4q^{65} + 11q^{67} - 12q^{71} - 4q^{73} - 10q^{79} + 9q^{83} - 6q^{85} - 9q^{89} + 4q^{91} - 2q^{95} - 10q^{97} + O(q^{100}) \)

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 −1.00000 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.b 1
3.b odd 2 1 1620.2.a.e 1
4.b odd 2 1 6480.2.a.h 1
5.b even 2 1 8100.2.a.h 1
5.c odd 4 2 8100.2.d.f 2
9.c even 3 2 540.2.i.a 2
9.d odd 6 2 180.2.i.a 2
12.b even 2 1 6480.2.a.t 1
15.d odd 2 1 8100.2.a.i 1
15.e even 4 2 8100.2.d.e 2
36.f odd 6 2 2160.2.q.e 2
36.h even 6 2 720.2.q.a 2
45.h odd 6 2 900.2.i.a 2
45.j even 6 2 2700.2.i.a 2
45.k odd 12 4 2700.2.s.a 4
45.l even 12 4 900.2.s.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.2.i.a 2 9.d odd 6 2
540.2.i.a 2 9.c even 3 2
720.2.q.a 2 36.h even 6 2
900.2.i.a 2 45.h odd 6 2
900.2.s.a 4 45.l even 12 4
1620.2.a.b 1 1.a even 1 1 trivial
1620.2.a.e 1 3.b odd 2 1
2160.2.q.e 2 36.f odd 6 2
2700.2.i.a 2 45.j even 6 2
2700.2.s.a 4 45.k odd 12 4
6480.2.a.h 1 4.b odd 2 1
6480.2.a.t 1 12.b even 2 1
8100.2.a.h 1 5.b even 2 1
8100.2.a.i 1 15.d odd 2 1
8100.2.d.e 2 15.e even 4 2
8100.2.d.f 2 5.c odd 4 2

Hecke kernels

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

\( T_{7} + 1 \)
\( T_{11} \)
\( T_{17} - 6 \)

Hecke characteristic polynomials

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