Properties

Label 560.2.bw.d
Level $560$
Weight $2$
Character orbit 560.bw
Analytic conductor $4.472$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.bw (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.47162251319\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{12} + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{5} + ( -2 \zeta_{12} - \zeta_{12}^{3} ) q^{7} -3 \zeta_{12}^{2} q^{9} +O(q^{10})\) \( q + ( \zeta_{12} + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{5} + ( -2 \zeta_{12} - \zeta_{12}^{3} ) q^{7} -3 \zeta_{12}^{2} q^{9} + ( 3 - 3 \zeta_{12}^{2} ) q^{11} -5 \zeta_{12}^{3} q^{13} -2 \zeta_{12} q^{17} + 5 \zeta_{12}^{2} q^{19} + ( 7 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{23} + ( -3 + 4 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{25} + 4 q^{29} + ( -2 + 2 \zeta_{12}^{2} ) q^{31} + ( -2 + 2 \zeta_{12} - \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{35} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{37} + 3 q^{41} -2 \zeta_{12}^{3} q^{43} + ( 6 - 3 \zeta_{12} - 6 \zeta_{12}^{2} ) q^{45} + ( -7 \zeta_{12} + 7 \zeta_{12}^{3} ) q^{47} + ( -5 + 8 \zeta_{12}^{2} ) q^{49} -9 \zeta_{12} q^{53} + ( 6 - 3 \zeta_{12}^{3} ) q^{55} + ( 4 - 4 \zeta_{12}^{2} ) q^{59} -6 \zeta_{12}^{2} q^{61} + ( -3 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{63} + ( 10 \zeta_{12} - 5 \zeta_{12}^{2} - 10 \zeta_{12}^{3} ) q^{65} -2 \zeta_{12} q^{67} + 6 q^{71} + 16 \zeta_{12} q^{73} + ( -9 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{77} -14 \zeta_{12}^{2} q^{79} + ( -9 + 9 \zeta_{12}^{2} ) q^{81} + 6 \zeta_{12}^{3} q^{83} + ( -2 - 4 \zeta_{12}^{3} ) q^{85} + 2 \zeta_{12}^{2} q^{89} + ( -15 + 10 \zeta_{12}^{2} ) q^{91} + ( -10 + 5 \zeta_{12} + 10 \zeta_{12}^{2} ) q^{95} + 12 \zeta_{12}^{3} q^{97} -9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5} - 6 q^{9} + O(q^{10}) \) \( 4 q + 4 q^{5} - 6 q^{9} + 6 q^{11} + 10 q^{19} - 6 q^{25} + 16 q^{29} - 4 q^{31} - 10 q^{35} + 12 q^{41} + 12 q^{45} - 4 q^{49} + 24 q^{55} + 8 q^{59} - 12 q^{61} - 10 q^{65} + 24 q^{71} - 28 q^{79} - 18 q^{81} - 8 q^{85} + 4 q^{89} - 40 q^{91} - 20 q^{95} - 36 q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
0 0 0 0.133975 2.23205i 0 1.73205 2.00000i 0 −1.50000 + 2.59808i 0
289.2 0 0 0 1.86603 1.23205i 0 −1.73205 + 2.00000i 0 −1.50000 + 2.59808i 0
529.1 0 0 0 0.133975 + 2.23205i 0 1.73205 + 2.00000i 0 −1.50000 2.59808i 0
529.2 0 0 0 1.86603 + 1.23205i 0 −1.73205 2.00000i 0 −1.50000 2.59808i 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
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.bw.d 4
4.b odd 2 1 70.2.i.b 4
5.b even 2 1 inner 560.2.bw.d 4
7.c even 3 1 inner 560.2.bw.d 4
12.b even 2 1 630.2.u.a 4
20.d odd 2 1 70.2.i.b 4
20.e even 4 1 350.2.e.c 2
20.e even 4 1 350.2.e.j 2
28.d even 2 1 490.2.i.a 4
28.f even 6 1 490.2.c.d 2
28.f even 6 1 490.2.i.a 4
28.g odd 6 1 70.2.i.b 4
28.g odd 6 1 490.2.c.a 2
35.j even 6 1 inner 560.2.bw.d 4
60.h even 2 1 630.2.u.a 4
84.n even 6 1 630.2.u.a 4
140.c even 2 1 490.2.i.a 4
140.p odd 6 1 70.2.i.b 4
140.p odd 6 1 490.2.c.a 2
140.s even 6 1 490.2.c.d 2
140.s even 6 1 490.2.i.a 4
140.w even 12 1 350.2.e.c 2
140.w even 12 1 350.2.e.j 2
140.w even 12 1 2450.2.a.k 1
140.w even 12 1 2450.2.a.ba 1
140.x odd 12 1 2450.2.a.j 1
140.x odd 12 1 2450.2.a.bb 1
420.ba even 6 1 630.2.u.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.i.b 4 4.b odd 2 1
70.2.i.b 4 20.d odd 2 1
70.2.i.b 4 28.g odd 6 1
70.2.i.b 4 140.p odd 6 1
350.2.e.c 2 20.e even 4 1
350.2.e.c 2 140.w even 12 1
350.2.e.j 2 20.e even 4 1
350.2.e.j 2 140.w even 12 1
490.2.c.a 2 28.g odd 6 1
490.2.c.a 2 140.p odd 6 1
490.2.c.d 2 28.f even 6 1
490.2.c.d 2 140.s even 6 1
490.2.i.a 4 28.d even 2 1
490.2.i.a 4 28.f even 6 1
490.2.i.a 4 140.c even 2 1
490.2.i.a 4 140.s even 6 1
560.2.bw.d 4 1.a even 1 1 trivial
560.2.bw.d 4 5.b even 2 1 inner
560.2.bw.d 4 7.c even 3 1 inner
560.2.bw.d 4 35.j even 6 1 inner
630.2.u.a 4 12.b even 2 1
630.2.u.a 4 60.h even 2 1
630.2.u.a 4 84.n even 6 1
630.2.u.a 4 420.ba even 6 1
2450.2.a.j 1 140.x odd 12 1
2450.2.a.k 1 140.w even 12 1
2450.2.a.ba 1 140.w even 12 1
2450.2.a.bb 1 140.x odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{2}^{\mathrm{new}}(560, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 25 - 20 T + 11 T^{2} - 4 T^{3} + T^{4} \)
$7$ \( 49 + 2 T^{2} + T^{4} \)
$11$ \( ( 9 - 3 T + T^{2} )^{2} \)
$13$ \( ( 25 + T^{2} )^{2} \)
$17$ \( 16 - 4 T^{2} + T^{4} \)
$19$ \( ( 25 - 5 T + T^{2} )^{2} \)
$23$ \( 2401 - 49 T^{2} + T^{4} \)
$29$ \( ( -4 + T )^{4} \)
$31$ \( ( 4 + 2 T + T^{2} )^{2} \)
$37$ \( 1 - T^{2} + T^{4} \)
$41$ \( ( -3 + T )^{4} \)
$43$ \( ( 4 + T^{2} )^{2} \)
$47$ \( 2401 - 49 T^{2} + T^{4} \)
$53$ \( 6561 - 81 T^{2} + T^{4} \)
$59$ \( ( 16 - 4 T + T^{2} )^{2} \)
$61$ \( ( 36 + 6 T + T^{2} )^{2} \)
$67$ \( 16 - 4 T^{2} + T^{4} \)
$71$ \( ( -6 + T )^{4} \)
$73$ \( 65536 - 256 T^{2} + T^{4} \)
$79$ \( ( 196 + 14 T + T^{2} )^{2} \)
$83$ \( ( 36 + T^{2} )^{2} \)
$89$ \( ( 4 - 2 T + T^{2} )^{2} \)
$97$ \( ( 144 + T^{2} )^{2} \)
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