Properties

Label 560.2.e.b
Level 560
Weight 2
Character orbit 560.e
Analytic conductor 4.472
Analytic rank 0
Dimension 4
CM discriminant -20
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: \(\Q(\sqrt{-2}, \sqrt{-5})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} + \beta_{3} q^{5} + ( \beta_{1} - 2 \beta_{2} ) q^{7} + q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} + \beta_{3} q^{5} + ( \beta_{1} - 2 \beta_{2} ) q^{7} + q^{9} + ( 2 \beta_{1} - \beta_{2} ) q^{15} + ( -3 - \beta_{3} ) q^{21} + ( 6 \beta_{1} - 3 \beta_{2} ) q^{23} -5 q^{25} -4 \beta_{2} q^{27} + 6 q^{29} + ( 3 \beta_{1} + \beta_{2} ) q^{35} -2 \beta_{3} q^{41} + ( -2 \beta_{1} + \beta_{2} ) q^{43} + \beta_{3} q^{45} -7 \beta_{2} q^{47} + ( -2 - 3 \beta_{3} ) q^{49} + 6 \beta_{3} q^{61} + ( \beta_{1} - 2 \beta_{2} ) q^{63} + ( -10 \beta_{1} + 5 \beta_{2} ) q^{67} -6 \beta_{3} q^{69} + 5 \beta_{2} q^{75} -5 q^{81} + 11 \beta_{2} q^{83} -6 \beta_{2} q^{87} + 8 \beta_{3} q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{9} - 12q^{21} - 20q^{25} + 24q^{29} - 8q^{49} - 20q^{81} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 4 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - \nu \)\()/3\)
\(\beta_{3}\)\(=\)\( \nu^{2} - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 2\)
\(\nu^{3}\)\(=\)\(3 \beta_{2} + \beta_{1}\)

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
−1.58114 + 0.707107i
1.58114 + 0.707107i
1.58114 0.707107i
−1.58114 0.707107i
0 1.41421i 0 2.23607i 0 −1.58114 2.12132i 0 1.00000 0
559.2 0 1.41421i 0 2.23607i 0 1.58114 2.12132i 0 1.00000 0
559.3 0 1.41421i 0 2.23607i 0 1.58114 + 2.12132i 0 1.00000 0
559.4 0 1.41421i 0 2.23607i 0 −1.58114 + 2.12132i 0 1.00000 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
28.d even 2 1 inner
35.c odd 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.b 4
4.b odd 2 1 inner 560.2.e.b 4
5.b even 2 1 inner 560.2.e.b 4
5.c odd 4 2 2800.2.k.g 4
7.b odd 2 1 inner 560.2.e.b 4
8.b even 2 1 2240.2.e.c 4
8.d odd 2 1 2240.2.e.c 4
20.d odd 2 1 CM 560.2.e.b 4
20.e even 4 2 2800.2.k.g 4
28.d even 2 1 inner 560.2.e.b 4
35.c odd 2 1 inner 560.2.e.b 4
35.f even 4 2 2800.2.k.g 4
40.e odd 2 1 2240.2.e.c 4
40.f even 2 1 2240.2.e.c 4
56.e even 2 1 2240.2.e.c 4
56.h odd 2 1 2240.2.e.c 4
140.c even 2 1 inner 560.2.e.b 4
140.j odd 4 2 2800.2.k.g 4
280.c odd 2 1 2240.2.e.c 4
280.n even 2 1 2240.2.e.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.e.b 4 1.a even 1 1 trivial
560.2.e.b 4 4.b odd 2 1 inner
560.2.e.b 4 5.b even 2 1 inner
560.2.e.b 4 7.b odd 2 1 inner
560.2.e.b 4 20.d odd 2 1 CM
560.2.e.b 4 28.d even 2 1 inner
560.2.e.b 4 35.c odd 2 1 inner
560.2.e.b 4 140.c even 2 1 inner
2240.2.e.c 4 8.b even 2 1
2240.2.e.c 4 8.d odd 2 1
2240.2.e.c 4 40.e odd 2 1
2240.2.e.c 4 40.f even 2 1
2240.2.e.c 4 56.e even 2 1
2240.2.e.c 4 56.h odd 2 1
2240.2.e.c 4 280.c odd 2 1
2240.2.e.c 4 280.n even 2 1
2800.2.k.g 4 5.c odd 4 2
2800.2.k.g 4 20.e even 4 2
2800.2.k.g 4 35.f even 4 2
2800.2.k.g 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 \(S_{2}^{\mathrm{new}}(560, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 4 T^{2} + 9 T^{4} )^{2} \)
$5$ \( ( 1 + 5 T^{2} )^{2} \)
$7$ \( 1 + 4 T^{2} + 49 T^{4} \)
$11$ \( ( 1 - 11 T^{2} )^{4} \)
$13$ \( ( 1 + 13 T^{2} )^{4} \)
$17$ \( ( 1 + 17 T^{2} )^{4} \)
$19$ \( ( 1 + 19 T^{2} )^{4} \)
$23$ \( ( 1 - 44 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 6 T + 29 T^{2} )^{4} \)
$31$ \( ( 1 + 31 T^{2} )^{4} \)
$37$ \( ( 1 - 37 T^{2} )^{4} \)
$41$ \( ( 1 - 12 T + 41 T^{2} )^{2}( 1 + 12 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 + 76 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 + 4 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 - 53 T^{2} )^{4} \)
$59$ \( ( 1 + 59 T^{2} )^{4} \)
$61$ \( ( 1 - 8 T + 61 T^{2} )^{2}( 1 + 8 T + 61 T^{2} )^{2} \)
$67$ \( ( 1 - 116 T^{2} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 - 71 T^{2} )^{4} \)
$73$ \( ( 1 + 73 T^{2} )^{4} \)
$79$ \( ( 1 - 79 T^{2} )^{4} \)
$83$ \( ( 1 + 76 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 6 T + 89 T^{2} )^{2}( 1 + 6 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 + 97 T^{2} )^{4} \)
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