Properties

Label 560.1.bt.a
Level 560
Weight 1
Character orbit 560.bt
Analytic conductor 0.279
Analytic rank 0
Dimension 4
Projective image \(D_{6}\)
CM discriminant -20
Inner twists 8

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Newspace parameters

Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 560.bt (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.279476407074\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{6}\)
Projective field Galois closure of 6.0.3841600.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
0 −0.866025 1.50000i 0 0.500000 0.866025i 0 0.866025 0.500000i 0 −1.00000 + 1.73205i 0
79.2 0 0.866025 + 1.50000i 0 0.500000 0.866025i 0 −0.866025 + 0.500000i 0 −1.00000 + 1.73205i 0
319.1 0 −0.866025 + 1.50000i 0 0.500000 + 0.866025i 0 0.866025 + 0.500000i 0 −1.00000 1.73205i 0
319.2 0 0.866025 1.50000i 0 0.500000 + 0.866025i 0 −0.866025 0.500000i 0 −1.00000 1.73205i 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.c even 3 1 inner
28.g odd 6 1 inner
35.j even 6 1 inner
140.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.1.bt.a 4
4.b odd 2 1 inner 560.1.bt.a 4
5.b even 2 1 inner 560.1.bt.a 4
5.c odd 4 1 2800.1.ce.a 2
5.c odd 4 1 2800.1.ce.b 2
7.b odd 2 1 3920.1.bt.c 4
7.c even 3 1 inner 560.1.bt.a 4
7.c even 3 1 3920.1.j.b 2
7.d odd 6 1 3920.1.j.d 2
7.d odd 6 1 3920.1.bt.c 4
8.b even 2 1 2240.1.bt.c 4
8.d odd 2 1 2240.1.bt.c 4
20.d odd 2 1 CM 560.1.bt.a 4
20.e even 4 1 2800.1.ce.a 2
20.e even 4 1 2800.1.ce.b 2
28.d even 2 1 3920.1.bt.c 4
28.f even 6 1 3920.1.j.d 2
28.f even 6 1 3920.1.bt.c 4
28.g odd 6 1 inner 560.1.bt.a 4
28.g odd 6 1 3920.1.j.b 2
35.c odd 2 1 3920.1.bt.c 4
35.i odd 6 1 3920.1.j.d 2
35.i odd 6 1 3920.1.bt.c 4
35.j even 6 1 inner 560.1.bt.a 4
35.j even 6 1 3920.1.j.b 2
35.l odd 12 1 2800.1.ce.a 2
35.l odd 12 1 2800.1.ce.b 2
40.e odd 2 1 2240.1.bt.c 4
40.f even 2 1 2240.1.bt.c 4
56.k odd 6 1 2240.1.bt.c 4
56.p even 6 1 2240.1.bt.c 4
140.c even 2 1 3920.1.bt.c 4
140.p odd 6 1 inner 560.1.bt.a 4
140.p odd 6 1 3920.1.j.b 2
140.s even 6 1 3920.1.j.d 2
140.s even 6 1 3920.1.bt.c 4
140.w even 12 1 2800.1.ce.a 2
140.w even 12 1 2800.1.ce.b 2
280.bf even 6 1 2240.1.bt.c 4
280.bi odd 6 1 2240.1.bt.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.1.bt.a 4 1.a even 1 1 trivial
560.1.bt.a 4 4.b odd 2 1 inner
560.1.bt.a 4 5.b even 2 1 inner
560.1.bt.a 4 7.c even 3 1 inner
560.1.bt.a 4 20.d odd 2 1 CM
560.1.bt.a 4 28.g odd 6 1 inner
560.1.bt.a 4 35.j even 6 1 inner
560.1.bt.a 4 140.p odd 6 1 inner
2240.1.bt.c 4 8.b even 2 1
2240.1.bt.c 4 8.d odd 2 1
2240.1.bt.c 4 40.e odd 2 1
2240.1.bt.c 4 40.f even 2 1
2240.1.bt.c 4 56.k odd 6 1
2240.1.bt.c 4 56.p even 6 1
2240.1.bt.c 4 280.bf even 6 1
2240.1.bt.c 4 280.bi odd 6 1
2800.1.ce.a 2 5.c odd 4 1
2800.1.ce.a 2 20.e even 4 1
2800.1.ce.a 2 35.l odd 12 1
2800.1.ce.a 2 140.w even 12 1
2800.1.ce.b 2 5.c odd 4 1
2800.1.ce.b 2 20.e even 4 1
2800.1.ce.b 2 35.l odd 12 1
2800.1.ce.b 2 140.w even 12 1
3920.1.j.b 2 7.c even 3 1
3920.1.j.b 2 28.g odd 6 1
3920.1.j.b 2 35.j even 6 1
3920.1.j.b 2 140.p odd 6 1
3920.1.j.d 2 7.d odd 6 1
3920.1.j.d 2 28.f even 6 1
3920.1.j.d 2 35.i odd 6 1
3920.1.j.d 2 140.s even 6 1
3920.1.bt.c 4 7.b odd 2 1
3920.1.bt.c 4 7.d odd 6 1
3920.1.bt.c 4 28.d even 2 1
3920.1.bt.c 4 28.f even 6 1
3920.1.bt.c 4 35.c odd 2 1
3920.1.bt.c 4 35.i odd 6 1
3920.1.bt.c 4 140.c even 2 1
3920.1.bt.c 4 140.s even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(560, [\chi])\).

Hecke characteristic polynomials

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