Properties

Label 560.2.a.e
Level $560$
Weight $2$
Character orbit 560.a
Self dual yes
Analytic conductor $4.472$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.47162251319\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} - q^{5} + q^{7} - 2q^{9} + O(q^{10}) \) \( q + q^{3} - q^{5} + q^{7} - 2q^{9} + 5q^{11} + q^{13} - q^{15} + 3q^{17} + 6q^{19} + q^{21} + 6q^{23} + q^{25} - 5q^{27} - 9q^{29} + 5q^{33} - q^{35} + 6q^{37} + q^{39} + 8q^{41} - 6q^{43} + 2q^{45} - 3q^{47} + q^{49} + 3q^{51} - 12q^{53} - 5q^{55} + 6q^{57} - 8q^{59} - 4q^{61} - 2q^{63} - q^{65} + 4q^{67} + 6q^{69} - 8q^{71} + 10q^{73} + q^{75} + 5q^{77} + 3q^{79} + q^{81} + 12q^{83} - 3q^{85} - 9q^{87} - 16q^{89} + q^{91} - 6q^{95} + 7q^{97} - 10q^{99} + O(q^{100}) \)

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 1.00000 0 −1.00000 0 1.00000 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(7\) \(-1\)

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 560.2.a.e 1
3.b odd 2 1 5040.2.a.be 1
4.b odd 2 1 280.2.a.b 1
5.b even 2 1 2800.2.a.i 1
5.c odd 4 2 2800.2.g.m 2
7.b odd 2 1 3920.2.a.r 1
8.b even 2 1 2240.2.a.j 1
8.d odd 2 1 2240.2.a.v 1
12.b even 2 1 2520.2.a.p 1
20.d odd 2 1 1400.2.a.k 1
20.e even 4 2 1400.2.g.e 2
28.d even 2 1 1960.2.a.k 1
28.f even 6 2 1960.2.q.e 2
28.g odd 6 2 1960.2.q.m 2
140.c even 2 1 9800.2.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.b 1 4.b odd 2 1
560.2.a.e 1 1.a even 1 1 trivial
1400.2.a.k 1 20.d odd 2 1
1400.2.g.e 2 20.e even 4 2
1960.2.a.k 1 28.d even 2 1
1960.2.q.e 2 28.f even 6 2
1960.2.q.m 2 28.g odd 6 2
2240.2.a.j 1 8.b even 2 1
2240.2.a.v 1 8.d odd 2 1
2520.2.a.p 1 12.b even 2 1
2800.2.a.i 1 5.b even 2 1
2800.2.g.m 2 5.c odd 4 2
3920.2.a.r 1 7.b odd 2 1
5040.2.a.be 1 3.b odd 2 1
9800.2.a.n 1 140.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(560))\):

\( T_{3} - 1 \)
\( T_{11} - 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -1 + T \)
$5$ \( 1 + T \)
$7$ \( -1 + T \)
$11$ \( -5 + T \)
$13$ \( -1 + T \)
$17$ \( -3 + T \)
$19$ \( -6 + T \)
$23$ \( -6 + T \)
$29$ \( 9 + T \)
$31$ \( T \)
$37$ \( -6 + T \)
$41$ \( -8 + T \)
$43$ \( 6 + T \)
$47$ \( 3 + T \)
$53$ \( 12 + T \)
$59$ \( 8 + T \)
$61$ \( 4 + T \)
$67$ \( -4 + T \)
$71$ \( 8 + T \)
$73$ \( -10 + T \)
$79$ \( -3 + T \)
$83$ \( -12 + T \)
$89$ \( 16 + T \)
$97$ \( -7 + T \)
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