Properties

Label 540.4.a.d
Level $540$
Weight $4$
Character orbit 540.a
Self dual yes
Analytic conductor $31.861$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 540 = 2^{2} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 540.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.8610314031\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q + 5 q^{5} + 17 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 q^{5} + 17 q^{7} - 30 q^{11} - 61 q^{13} - 120 q^{17} - 43 q^{19} + 90 q^{23} + 25 q^{25} - 90 q^{29} + 8 q^{31} + 85 q^{35} + 317 q^{37} - 30 q^{41} - 220 q^{43} - 180 q^{47} - 54 q^{49} - 630 q^{53} - 150 q^{55} - 840 q^{59} + 599 q^{61} - 305 q^{65} + 107 q^{67} - 210 q^{71} - 421 q^{73} - 510 q^{77} + 353 q^{79} - 1350 q^{83} - 600 q^{85} + 1020 q^{89} - 1037 q^{91} - 215 q^{95} - 997 q^{97}+O(q^{100}) \) 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 5.00000 0 17.0000 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 540.4.a.d yes 1
3.b odd 2 1 540.4.a.b 1
4.b odd 2 1 2160.4.a.k 1
9.c even 3 2 1620.4.i.b 2
9.d odd 6 2 1620.4.i.h 2
12.b even 2 1 2160.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.4.a.b 1 3.b odd 2 1
540.4.a.d yes 1 1.a even 1 1 trivial
1620.4.i.b 2 9.c even 3 2
1620.4.i.h 2 9.d odd 6 2
2160.4.a.a 1 12.b even 2 1
2160.4.a.k 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(540))\):

\( T_{7} - 17 \) Copy content Toggle raw display
\( T_{11} + 30 \) 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
$5$ \( T - 5 \) Copy content Toggle raw display
$7$ \( T - 17 \) Copy content Toggle raw display
$11$ \( T + 30 \) Copy content Toggle raw display
$13$ \( T + 61 \) Copy content Toggle raw display
$17$ \( T + 120 \) Copy content Toggle raw display
$19$ \( T + 43 \) Copy content Toggle raw display
$23$ \( T - 90 \) Copy content Toggle raw display
$29$ \( T + 90 \) Copy content Toggle raw display
$31$ \( T - 8 \) Copy content Toggle raw display
$37$ \( T - 317 \) Copy content Toggle raw display
$41$ \( T + 30 \) Copy content Toggle raw display
$43$ \( T + 220 \) Copy content Toggle raw display
$47$ \( T + 180 \) Copy content Toggle raw display
$53$ \( T + 630 \) Copy content Toggle raw display
$59$ \( T + 840 \) Copy content Toggle raw display
$61$ \( T - 599 \) Copy content Toggle raw display
$67$ \( T - 107 \) Copy content Toggle raw display
$71$ \( T + 210 \) Copy content Toggle raw display
$73$ \( T + 421 \) Copy content Toggle raw display
$79$ \( T - 353 \) Copy content Toggle raw display
$83$ \( T + 1350 \) Copy content Toggle raw display
$89$ \( T - 1020 \) Copy content Toggle raw display
$97$ \( T + 997 \) Copy content Toggle raw display
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