Properties

Label 560.2.a.c
Level 560
Weight 2
Character orbit 560.a
Self dual yes
Analytic conductor 4.472
Analytic rank 1
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} + q^{5} - q^{7} - 2q^{9} + O(q^{10}) \) \( q - q^{3} + q^{5} - q^{7} - 2q^{9} - 3q^{11} - q^{13} - q^{15} - 3q^{17} - 2q^{19} + q^{21} + 6q^{23} + q^{25} + 5q^{27} - 9q^{29} - 8q^{31} + 3q^{33} - q^{35} - 10q^{37} + q^{39} - 2q^{43} - 2q^{45} + 3q^{47} + q^{49} + 3q^{51} - 3q^{55} + 2q^{57} - 12q^{59} + 8q^{61} + 2q^{63} - q^{65} - 8q^{67} - 6q^{69} + 14q^{73} - q^{75} + 3q^{77} - 5q^{79} + q^{81} + 12q^{83} - 3q^{85} + 9q^{87} + 12q^{89} + q^{91} + 8q^{93} - 2q^{95} + 17q^{97} + 6q^{99} + O(q^{100}) \)

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:

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.c 1
3.b odd 2 1 5040.2.a.h 1
4.b odd 2 1 140.2.a.a 1
5.b even 2 1 2800.2.a.y 1
5.c odd 4 2 2800.2.g.j 2
7.b odd 2 1 3920.2.a.u 1
8.b even 2 1 2240.2.a.r 1
8.d odd 2 1 2240.2.a.g 1
12.b even 2 1 1260.2.a.c 1
20.d odd 2 1 700.2.a.d 1
20.e even 4 2 700.2.e.c 2
28.d even 2 1 980.2.a.c 1
28.f even 6 2 980.2.i.h 2
28.g odd 6 2 980.2.i.d 2
60.h even 2 1 6300.2.a.d 1
60.l odd 4 2 6300.2.k.c 2
84.h odd 2 1 8820.2.a.r 1
140.c even 2 1 4900.2.a.p 1
140.j odd 4 2 4900.2.e.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.a.a 1 4.b odd 2 1
560.2.a.c 1 1.a even 1 1 trivial
700.2.a.d 1 20.d odd 2 1
700.2.e.c 2 20.e even 4 2
980.2.a.c 1 28.d even 2 1
980.2.i.d 2 28.g odd 6 2
980.2.i.h 2 28.f even 6 2
1260.2.a.c 1 12.b even 2 1
2240.2.a.g 1 8.d odd 2 1
2240.2.a.r 1 8.b even 2 1
2800.2.a.y 1 5.b even 2 1
2800.2.g.j 2 5.c odd 4 2
3920.2.a.u 1 7.b odd 2 1
4900.2.a.p 1 140.c even 2 1
4900.2.e.l 2 140.j odd 4 2
5040.2.a.h 1 3.b odd 2 1
6300.2.a.d 1 60.h even 2 1
6300.2.k.c 2 60.l odd 4 2
8820.2.a.r 1 84.h odd 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(560))\):

\( T_{3} + 1 \)
\( T_{11} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + T + 3 T^{2} \)
$5$ \( 1 - T \)
$7$ \( 1 + T \)
$11$ \( 1 + 3 T + 11 T^{2} \)
$13$ \( 1 + T + 13 T^{2} \)
$17$ \( 1 + 3 T + 17 T^{2} \)
$19$ \( 1 + 2 T + 19 T^{2} \)
$23$ \( 1 - 6 T + 23 T^{2} \)
$29$ \( 1 + 9 T + 29 T^{2} \)
$31$ \( 1 + 8 T + 31 T^{2} \)
$37$ \( 1 + 10 T + 37 T^{2} \)
$41$ \( 1 + 41 T^{2} \)
$43$ \( 1 + 2 T + 43 T^{2} \)
$47$ \( 1 - 3 T + 47 T^{2} \)
$53$ \( 1 + 53 T^{2} \)
$59$ \( 1 + 12 T + 59 T^{2} \)
$61$ \( 1 - 8 T + 61 T^{2} \)
$67$ \( 1 + 8 T + 67 T^{2} \)
$71$ \( 1 + 71 T^{2} \)
$73$ \( 1 - 14 T + 73 T^{2} \)
$79$ \( 1 + 5 T + 79 T^{2} \)
$83$ \( 1 - 12 T + 83 T^{2} \)
$89$ \( 1 - 12 T + 89 T^{2} \)
$97$ \( 1 - 17 T + 97 T^{2} \)
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