Properties

Label 560.2.q.e
Level 560
Weight 2
Character orbit 560.q
Analytic conductor 4.472
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.47162251319\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + ( 1 - 3 \zeta_{6} ) q^{7} + 2 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( -1 + \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + ( 1 - 3 \zeta_{6} ) q^{7} + 2 \zeta_{6} q^{9} + ( -2 + 2 \zeta_{6} ) q^{11} + 4 q^{13} - q^{15} + 6 \zeta_{6} q^{19} + ( 2 + \zeta_{6} ) q^{21} + 3 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} -5 q^{27} -3 q^{29} -2 \zeta_{6} q^{33} + ( 3 - 2 \zeta_{6} ) q^{35} + 12 \zeta_{6} q^{37} + ( -4 + 4 \zeta_{6} ) q^{39} -7 q^{41} + 9 q^{43} + ( -2 + 2 \zeta_{6} ) q^{45} + ( -8 + 3 \zeta_{6} ) q^{49} + ( 6 - 6 \zeta_{6} ) q^{53} -2 q^{55} -6 q^{57} + ( -10 + 10 \zeta_{6} ) q^{59} -5 \zeta_{6} q^{61} + ( 6 - 4 \zeta_{6} ) q^{63} + 4 \zeta_{6} q^{65} + ( 11 - 11 \zeta_{6} ) q^{67} -3 q^{69} + 10 q^{71} + ( 8 - 8 \zeta_{6} ) q^{73} -\zeta_{6} q^{75} + ( 4 + 2 \zeta_{6} ) q^{77} + 6 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} + 3 q^{83} + ( 3 - 3 \zeta_{6} ) q^{87} -17 \zeta_{6} q^{89} + ( 4 - 12 \zeta_{6} ) q^{91} + ( -6 + 6 \zeta_{6} ) q^{95} -2 q^{97} -4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} + q^{5} - q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - q^{3} + q^{5} - q^{7} + 2q^{9} - 2q^{11} + 8q^{13} - 2q^{15} + 6q^{19} + 5q^{21} + 3q^{23} - q^{25} - 10q^{27} - 6q^{29} - 2q^{33} + 4q^{35} + 12q^{37} - 4q^{39} - 14q^{41} + 18q^{43} - 2q^{45} - 13q^{49} + 6q^{53} - 4q^{55} - 12q^{57} - 10q^{59} - 5q^{61} + 8q^{63} + 4q^{65} + 11q^{67} - 6q^{69} + 20q^{71} + 8q^{73} - q^{75} + 10q^{77} + 6q^{79} - q^{81} + 6q^{83} + 3q^{87} - 17q^{89} - 4q^{91} - 6q^{95} - 4q^{97} - 8q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 −0.500000 2.59808i 0 1.00000 + 1.73205i 0
401.1 0 −0.500000 0.866025i 0 0.500000 0.866025i 0 −0.500000 + 2.59808i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.q.e 2
4.b odd 2 1 280.2.q.b 2
7.c even 3 1 inner 560.2.q.e 2
7.c even 3 1 3920.2.a.v 1
7.d odd 6 1 3920.2.a.q 1
12.b even 2 1 2520.2.bi.d 2
20.d odd 2 1 1400.2.q.c 2
20.e even 4 2 1400.2.bh.c 4
28.d even 2 1 1960.2.q.d 2
28.f even 6 1 1960.2.a.l 1
28.f even 6 1 1960.2.q.d 2
28.g odd 6 1 280.2.q.b 2
28.g odd 6 1 1960.2.a.c 1
84.n even 6 1 2520.2.bi.d 2
140.p odd 6 1 1400.2.q.c 2
140.p odd 6 1 9800.2.a.z 1
140.s even 6 1 9800.2.a.o 1
140.w even 12 2 1400.2.bh.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.b 2 4.b odd 2 1
280.2.q.b 2 28.g odd 6 1
560.2.q.e 2 1.a even 1 1 trivial
560.2.q.e 2 7.c even 3 1 inner
1400.2.q.c 2 20.d odd 2 1
1400.2.q.c 2 140.p odd 6 1
1400.2.bh.c 4 20.e even 4 2
1400.2.bh.c 4 140.w even 12 2
1960.2.a.c 1 28.g odd 6 1
1960.2.a.l 1 28.f even 6 1
1960.2.q.d 2 28.d even 2 1
1960.2.q.d 2 28.f even 6 1
2520.2.bi.d 2 12.b even 2 1
2520.2.bi.d 2 84.n even 6 1
3920.2.a.q 1 7.d odd 6 1
3920.2.a.v 1 7.c even 3 1
9800.2.a.o 1 140.s even 6 1
9800.2.a.z 1 140.p odd 6 1

Hecke kernels

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

\( T_{3}^{2} + T_{3} + 1 \)
\( T_{11}^{2} + 2 T_{11} + 4 \)
\( T_{13} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4} \)
$5$ \( 1 - T + T^{2} \)
$7$ \( 1 + T + 7 T^{2} \)
$11$ \( 1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 4 T + 13 T^{2} )^{2} \)
$17$ \( 1 - 17 T^{2} + 289 T^{4} \)
$19$ \( 1 - 6 T + 17 T^{2} - 114 T^{3} + 361 T^{4} \)
$23$ \( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \)
$29$ \( ( 1 + 3 T + 29 T^{2} )^{2} \)
$31$ \( 1 - 31 T^{2} + 961 T^{4} \)
$37$ \( 1 - 12 T + 107 T^{2} - 444 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + 7 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 9 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 47 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 10 T + 41 T^{2} + 590 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 5 T - 36 T^{2} + 305 T^{3} + 3721 T^{4} \)
$67$ \( ( 1 - 16 T + 67 T^{2} )( 1 + 5 T + 67 T^{2} ) \)
$71$ \( ( 1 - 10 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 8 T - 9 T^{2} - 584 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 6 T - 43 T^{2} - 474 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 - 3 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 17 T + 200 T^{2} + 1513 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 + 2 T + 97 T^{2} )^{2} \)
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