Properties

Label 540.2.a.e
Level $540$
Weight $2$
Character orbit 540.a
Self dual yes
Analytic conductor $4.312$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 540 = 2^{2} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 540.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.31192170915\)
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

\(f(q)\) \(=\) \( q + q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} - q^{7} + 6 q^{11} - q^{13} - q^{19} + 6 q^{23} + q^{25} + 6 q^{29} + 8 q^{31} - q^{35} - 7 q^{37} - 6 q^{41} - 4 q^{43} + 12 q^{47} - 6 q^{49} - 6 q^{53} + 6 q^{55} + 11 q^{61} - q^{65} - 7 q^{67} - 6 q^{71} + 11 q^{73} - 6 q^{77} - q^{79} + 6 q^{83} - 12 q^{89} + q^{91} - q^{95} - 13 q^{97}+O(q^{100}) \) 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
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 540.2.a.e yes 1
3.b odd 2 1 540.2.a.b 1
4.b odd 2 1 2160.2.a.s 1
5.b even 2 1 2700.2.a.m 1
5.c odd 4 2 2700.2.d.k 2
8.b even 2 1 8640.2.a.l 1
8.d odd 2 1 8640.2.a.s 1
9.c even 3 2 1620.2.i.d 2
9.d odd 6 2 1620.2.i.j 2
12.b even 2 1 2160.2.a.h 1
15.d odd 2 1 2700.2.a.k 1
15.e even 4 2 2700.2.d.a 2
24.f even 2 1 8640.2.a.bu 1
24.h odd 2 1 8640.2.a.br 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.2.a.b 1 3.b odd 2 1
540.2.a.e yes 1 1.a even 1 1 trivial
1620.2.i.d 2 9.c even 3 2
1620.2.i.j 2 9.d odd 6 2
2160.2.a.h 1 12.b even 2 1
2160.2.a.s 1 4.b odd 2 1
2700.2.a.k 1 15.d odd 2 1
2700.2.a.m 1 5.b even 2 1
2700.2.d.a 2 15.e even 4 2
2700.2.d.k 2 5.c odd 4 2
8640.2.a.l 1 8.b even 2 1
8640.2.a.s 1 8.d odd 2 1
8640.2.a.br 1 24.h odd 2 1
8640.2.a.bu 1 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(540))\):

\( T_{7} + 1 \) Copy content Toggle raw display
\( T_{11} - 6 \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display

Hecke characteristic polynomials

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