Properties

Label 560.2.a.f
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

\(f(q)\) \(=\) \( q + 3q^{3} + q^{5} - q^{7} + 6q^{9} + O(q^{10}) \) \( q + 3q^{3} + q^{5} - q^{7} + 6q^{9} + 5q^{11} - 5q^{13} + 3q^{15} - 7q^{17} + 2q^{19} - 3q^{21} + 2q^{23} + q^{25} + 9q^{27} + 7q^{29} - 4q^{31} + 15q^{33} - q^{35} - 6q^{37} - 15q^{39} - 12q^{41} + 2q^{43} + 6q^{45} - q^{47} + q^{49} - 21q^{51} + 5q^{55} + 6q^{57} + 4q^{59} + 4q^{61} - 6q^{63} - 5q^{65} - 8q^{67} + 6q^{69} + 6q^{73} + 3q^{75} - 5q^{77} + 3q^{79} + 9q^{81} + 4q^{83} - 7q^{85} + 21q^{87} + 5q^{91} - 12q^{93} + 2q^{95} + 13q^{97} + 30q^{99} + O(q^{100}) \)

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 3.00000 0 1.00000 0 −1.00000 0 6.00000 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.f 1
3.b odd 2 1 5040.2.a.a 1
4.b odd 2 1 280.2.a.a 1
5.b even 2 1 2800.2.a.c 1
5.c odd 4 2 2800.2.g.b 2
7.b odd 2 1 3920.2.a.c 1
8.b even 2 1 2240.2.a.a 1
8.d odd 2 1 2240.2.a.z 1
12.b even 2 1 2520.2.a.i 1
20.d odd 2 1 1400.2.a.n 1
20.e even 4 2 1400.2.g.a 2
28.d even 2 1 1960.2.a.o 1
28.f even 6 2 1960.2.q.a 2
28.g odd 6 2 1960.2.q.o 2
140.c even 2 1 9800.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.a 1 4.b odd 2 1
560.2.a.f 1 1.a even 1 1 trivial
1400.2.a.n 1 20.d odd 2 1
1400.2.g.a 2 20.e even 4 2
1960.2.a.o 1 28.d even 2 1
1960.2.q.a 2 28.f even 6 2
1960.2.q.o 2 28.g odd 6 2
2240.2.a.a 1 8.b even 2 1
2240.2.a.z 1 8.d odd 2 1
2520.2.a.i 1 12.b even 2 1
2800.2.a.c 1 5.b even 2 1
2800.2.g.b 2 5.c odd 4 2
3920.2.a.c 1 7.b odd 2 1
5040.2.a.a 1 3.b odd 2 1
9800.2.a.a 1 140.c even 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(560))\):

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

Hecke characteristic polynomials

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