Properties

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

Related objects

Downloads

Learn more about

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: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + 3 \zeta_{12} q^{3} + ( 2 \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{5} + ( -\zeta_{12} - 2 \zeta_{12}^{3} ) q^{7} + 6 \zeta_{12}^{2} q^{9} +O(q^{10})\) \( q + 3 \zeta_{12} q^{3} + ( 2 \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{5} + ( -\zeta_{12} - 2 \zeta_{12}^{3} ) q^{7} + 6 \zeta_{12}^{2} q^{9} + 2 \zeta_{12}^{3} q^{13} + ( 6 - 3 \zeta_{12}^{3} ) q^{15} + 2 \zeta_{12} q^{17} + 2 \zeta_{12}^{2} q^{19} + ( 6 - 9 \zeta_{12}^{2} ) q^{21} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{23} + ( 3 - 4 \zeta_{12} - 3 \zeta_{12}^{2} ) q^{25} + 9 \zeta_{12}^{3} q^{27} + q^{29} + ( 10 - 10 \zeta_{12}^{2} ) q^{31} + ( -2 - 2 \zeta_{12} - 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{35} + ( -8 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{37} + ( -6 + 6 \zeta_{12}^{2} ) q^{39} -3 q^{41} + 5 \zeta_{12}^{3} q^{43} + ( 6 + 12 \zeta_{12} - 6 \zeta_{12}^{2} ) q^{45} + ( -8 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{47} + ( -8 + 5 \zeta_{12}^{2} ) q^{49} + 6 \zeta_{12}^{2} q^{51} -6 \zeta_{12} q^{53} + 6 \zeta_{12}^{3} q^{57} + ( -2 + 2 \zeta_{12}^{2} ) q^{59} + 9 \zeta_{12}^{2} q^{61} + ( 12 \zeta_{12} - 18 \zeta_{12}^{3} ) q^{63} + ( 2 \zeta_{12} + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{65} -7 \zeta_{12} q^{67} -3 q^{69} -6 q^{71} -10 \zeta_{12} q^{73} + ( 9 \zeta_{12} - 12 \zeta_{12}^{2} - 9 \zeta_{12}^{3} ) q^{75} + 10 \zeta_{12}^{2} q^{79} + ( -9 + 9 \zeta_{12}^{2} ) q^{81} -9 \zeta_{12}^{3} q^{83} + ( 4 - 2 \zeta_{12}^{3} ) q^{85} + 3 \zeta_{12} q^{87} -7 \zeta_{12}^{2} q^{89} + ( 6 - 2 \zeta_{12}^{2} ) q^{91} + ( 30 \zeta_{12} - 30 \zeta_{12}^{3} ) q^{93} + ( 2 + 4 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{5} + 12q^{9} + O(q^{10}) \) \( 4q - 2q^{5} + 12q^{9} + 24q^{15} + 4q^{19} + 6q^{21} + 6q^{25} + 4q^{29} + 20q^{31} - 16q^{35} - 12q^{39} - 12q^{41} + 12q^{45} - 22q^{49} + 12q^{51} - 4q^{59} + 18q^{61} + 8q^{65} - 12q^{69} - 24q^{71} - 24q^{75} + 20q^{79} - 18q^{81} + 16q^{85} - 14q^{89} + 20q^{91} + 4q^{95} + O(q^{100}) \)

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 −2.59808 + 1.50000i 0 −2.23205 0.133975i 0 0.866025 2.50000i 0 3.00000 5.19615i 0
289.2 0 2.59808 1.50000i 0 1.23205 + 1.86603i 0 −0.866025 + 2.50000i 0 3.00000 5.19615i 0
529.1 0 −2.59808 1.50000i 0 −2.23205 + 0.133975i 0 0.866025 + 2.50000i 0 3.00000 + 5.19615i 0
529.2 0 2.59808 + 1.50000i 0 1.23205 1.86603i 0 −0.866025 2.50000i 0 3.00000 + 5.19615i 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.c 4
4.b odd 2 1 70.2.i.a 4
5.b even 2 1 inner 560.2.bw.c 4
7.c even 3 1 inner 560.2.bw.c 4
12.b even 2 1 630.2.u.b 4
20.d odd 2 1 70.2.i.a 4
20.e even 4 1 350.2.e.f 2
20.e even 4 1 350.2.e.g 2
28.d even 2 1 490.2.i.b 4
28.f even 6 1 490.2.c.b 2
28.f even 6 1 490.2.i.b 4
28.g odd 6 1 70.2.i.a 4
28.g odd 6 1 490.2.c.c 2
35.j even 6 1 inner 560.2.bw.c 4
60.h even 2 1 630.2.u.b 4
84.n even 6 1 630.2.u.b 4
140.c even 2 1 490.2.i.b 4
140.p odd 6 1 70.2.i.a 4
140.p odd 6 1 490.2.c.c 2
140.s even 6 1 490.2.c.b 2
140.s even 6 1 490.2.i.b 4
140.w even 12 1 350.2.e.f 2
140.w even 12 1 350.2.e.g 2
140.w even 12 1 2450.2.a.r 1
140.w even 12 1 2450.2.a.s 1
140.x odd 12 1 2450.2.a.c 1
140.x odd 12 1 2450.2.a.bh 1
420.ba even 6 1 630.2.u.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.i.a 4 4.b odd 2 1
70.2.i.a 4 20.d odd 2 1
70.2.i.a 4 28.g odd 6 1
70.2.i.a 4 140.p odd 6 1
350.2.e.f 2 20.e even 4 1
350.2.e.f 2 140.w even 12 1
350.2.e.g 2 20.e even 4 1
350.2.e.g 2 140.w even 12 1
490.2.c.b 2 28.f even 6 1
490.2.c.b 2 140.s even 6 1
490.2.c.c 2 28.g odd 6 1
490.2.c.c 2 140.p odd 6 1
490.2.i.b 4 28.d even 2 1
490.2.i.b 4 28.f even 6 1
490.2.i.b 4 140.c even 2 1
490.2.i.b 4 140.s even 6 1
560.2.bw.c 4 1.a even 1 1 trivial
560.2.bw.c 4 5.b even 2 1 inner
560.2.bw.c 4 7.c even 3 1 inner
560.2.bw.c 4 35.j even 6 1 inner
630.2.u.b 4 12.b even 2 1
630.2.u.b 4 60.h even 2 1
630.2.u.b 4 84.n even 6 1
630.2.u.b 4 420.ba even 6 1
2450.2.a.c 1 140.x odd 12 1
2450.2.a.r 1 140.w even 12 1
2450.2.a.s 1 140.w even 12 1
2450.2.a.bh 1 140.x odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 9 T_{3}^{2} + 81 \) acting on \(S_{2}^{\mathrm{new}}(560, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 3 T^{2} )^{2}( 1 + 3 T^{2} + 9 T^{4} ) \)
$5$ \( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \)
$7$ \( 1 + 11 T^{2} + 49 T^{4} \)
$11$ \( ( 1 - 11 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 22 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 - 8 T + 47 T^{2} - 136 T^{3} + 289 T^{4} )( 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} ) \)
$19$ \( ( 1 - 2 T - 15 T^{2} - 38 T^{3} + 361 T^{4} )^{2} \)
$23$ \( 1 + 45 T^{2} + 1496 T^{4} + 23805 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 - T + 29 T^{2} )^{4} \)
$31$ \( ( 1 - 10 T + 69 T^{2} - 310 T^{3} + 961 T^{4} )^{2} \)
$37$ \( 1 + 10 T^{2} - 1269 T^{4} + 13690 T^{6} + 1874161 T^{8} \)
$41$ \( ( 1 + 3 T + 41 T^{2} )^{4} \)
$43$ \( ( 1 - 61 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( 1 + 30 T^{2} - 1309 T^{4} + 66270 T^{6} + 4879681 T^{8} \)
$53$ \( 1 + 70 T^{2} + 2091 T^{4} + 196630 T^{6} + 7890481 T^{8} \)
$59$ \( ( 1 + 2 T - 55 T^{2} + 118 T^{3} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 9 T + 20 T^{2} - 549 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( 1 + 85 T^{2} + 2736 T^{4} + 381565 T^{6} + 20151121 T^{8} \)
$71$ \( ( 1 + 6 T + 71 T^{2} )^{4} \)
$73$ \( ( 1 - 97 T^{2} + 5329 T^{4} )( 1 + 143 T^{2} + 5329 T^{4} ) \)
$79$ \( ( 1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 - 85 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 + 7 T - 40 T^{2} + 623 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 97 T^{2} )^{4} \)
show more
show less