Properties

Label 560.2.e.a
Level 560
Weight 2
Character orbit 560.e
Analytic conductor 4.472
Analytic rank 0
Dimension 4
CM discriminant -35
Inner twists 8

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Newspace parameters

Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.47162251319\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{5}, \sqrt{-7})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} + \beta_{1} q^{5} + \beta_{2} q^{7} -4 q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} + \beta_{1} q^{5} + \beta_{2} q^{7} -4 q^{9} + \beta_{3} q^{11} + 3 \beta_{1} q^{13} + \beta_{3} q^{15} -\beta_{1} q^{17} + 7 q^{21} + 5 q^{25} + \beta_{2} q^{27} -9 q^{29} -7 \beta_{1} q^{33} -\beta_{3} q^{35} + 3 \beta_{3} q^{39} -4 \beta_{1} q^{45} + 3 \beta_{2} q^{47} -7 q^{49} -\beta_{3} q^{51} -5 \beta_{2} q^{55} -4 \beta_{2} q^{63} + 15 q^{65} + 2 \beta_{3} q^{71} + 6 \beta_{1} q^{73} -5 \beta_{2} q^{75} + 7 \beta_{1} q^{77} -3 \beta_{3} q^{79} -5 q^{81} + 6 \beta_{2} q^{83} -5 q^{85} + 9 \beta_{2} q^{87} -3 \beta_{3} q^{91} + 3 \beta_{1} q^{97} -4 \beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 16q^{9} + O(q^{10}) \) \( 4q - 16q^{9} + 28q^{21} + 20q^{25} - 36q^{29} - 28q^{49} + 60q^{65} - 20q^{81} - 20q^{85} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 5 x^{2} + 2 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + \nu^{2} + \nu + 7 \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( -2 \nu^{3} + 4 \nu^{2} - 8 \nu + 1 \)\()/3\)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} + 7 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - \beta_{1} + 1\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 3 \beta_{2} + 3 \beta_{1} - 9\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{3} - 2 \beta_{2} + 5 \beta_{1} - 10\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
559.1
0.809017 + 2.14046i
−0.309017 0.817582i
0.809017 2.14046i
−0.309017 + 0.817582i
0 2.64575i 0 −2.23607 0 2.64575i 0 −4.00000 0
559.2 0 2.64575i 0 2.23607 0 2.64575i 0 −4.00000 0
559.3 0 2.64575i 0 −2.23607 0 2.64575i 0 −4.00000 0
559.4 0 2.64575i 0 2.23607 0 2.64575i 0 −4.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
20.d odd 2 1 inner
28.d even 2 1 inner
140.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.e.a 4
4.b odd 2 1 inner 560.2.e.a 4
5.b even 2 1 inner 560.2.e.a 4
5.c odd 4 2 2800.2.k.j 4
7.b odd 2 1 inner 560.2.e.a 4
8.b even 2 1 2240.2.e.b 4
8.d odd 2 1 2240.2.e.b 4
20.d odd 2 1 inner 560.2.e.a 4
20.e even 4 2 2800.2.k.j 4
28.d even 2 1 inner 560.2.e.a 4
35.c odd 2 1 CM 560.2.e.a 4
35.f even 4 2 2800.2.k.j 4
40.e odd 2 1 2240.2.e.b 4
40.f even 2 1 2240.2.e.b 4
56.e even 2 1 2240.2.e.b 4
56.h odd 2 1 2240.2.e.b 4
140.c even 2 1 inner 560.2.e.a 4
140.j odd 4 2 2800.2.k.j 4
280.c odd 2 1 2240.2.e.b 4
280.n even 2 1 2240.2.e.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.e.a 4 1.a even 1 1 trivial
560.2.e.a 4 4.b odd 2 1 inner
560.2.e.a 4 5.b even 2 1 inner
560.2.e.a 4 7.b odd 2 1 inner
560.2.e.a 4 20.d odd 2 1 inner
560.2.e.a 4 28.d even 2 1 inner
560.2.e.a 4 35.c odd 2 1 CM
560.2.e.a 4 140.c even 2 1 inner
2240.2.e.b 4 8.b even 2 1
2240.2.e.b 4 8.d odd 2 1
2240.2.e.b 4 40.e odd 2 1
2240.2.e.b 4 40.f even 2 1
2240.2.e.b 4 56.e even 2 1
2240.2.e.b 4 56.h odd 2 1
2240.2.e.b 4 280.c odd 2 1
2240.2.e.b 4 280.n even 2 1
2800.2.k.j 4 5.c odd 4 2
2800.2.k.j 4 20.e even 4 2
2800.2.k.j 4 35.f even 4 2
2800.2.k.j 4 140.j odd 4 2

Hecke kernels

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

\( T_{3}^{2} + 7 \)
\( T_{11}^{2} + 35 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + T^{2} + 9 T^{4} )^{2} \)
$5$ \( ( 1 - 5 T^{2} )^{2} \)
$7$ \( ( 1 + 7 T^{2} )^{2} \)
$11$ \( ( 1 - 3 T + 11 T^{2} )^{2}( 1 + 3 T + 11 T^{2} )^{2} \)
$13$ \( ( 1 - 19 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 + 29 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 + 19 T^{2} )^{4} \)
$23$ \( ( 1 + 23 T^{2} )^{4} \)
$29$ \( ( 1 + 9 T + 29 T^{2} )^{4} \)
$31$ \( ( 1 + 31 T^{2} )^{4} \)
$37$ \( ( 1 - 37 T^{2} )^{4} \)
$41$ \( ( 1 - 41 T^{2} )^{4} \)
$43$ \( ( 1 + 43 T^{2} )^{4} \)
$47$ \( ( 1 - 31 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 - 53 T^{2} )^{4} \)
$59$ \( ( 1 + 59 T^{2} )^{4} \)
$61$ \( ( 1 - 61 T^{2} )^{4} \)
$67$ \( ( 1 + 67 T^{2} )^{4} \)
$71$ \( ( 1 - 12 T + 71 T^{2} )^{2}( 1 + 12 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 34 T^{2} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - T + 79 T^{2} )^{2}( 1 + T + 79 T^{2} )^{2} \)
$83$ \( ( 1 + 86 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 89 T^{2} )^{4} \)
$97$ \( ( 1 + 149 T^{2} + 9409 T^{4} )^{2} \)
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