Properties

Label 560.2.q.d
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: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -2 + 2 \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + ( 1 + 2 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( -2 + 2 \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + ( 1 + 2 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( 3 - 3 \zeta_{6} ) q^{11} + 5 q^{13} -2 q^{15} + ( -6 + 6 \zeta_{6} ) q^{17} -\zeta_{6} q^{19} + ( -6 + 2 \zeta_{6} ) q^{21} + 3 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} -4 q^{27} -6 q^{29} + ( -4 + 4 \zeta_{6} ) q^{31} + 6 \zeta_{6} q^{33} + ( -2 + 3 \zeta_{6} ) q^{35} -11 \zeta_{6} q^{37} + ( -10 + 10 \zeta_{6} ) q^{39} + 3 q^{41} + 10 q^{43} + ( 1 - \zeta_{6} ) q^{45} + 3 \zeta_{6} q^{47} + ( -3 + 8 \zeta_{6} ) q^{49} -12 \zeta_{6} q^{51} + ( -3 + 3 \zeta_{6} ) q^{53} + 3 q^{55} + 2 q^{57} + 4 \zeta_{6} q^{61} + ( 2 - 3 \zeta_{6} ) q^{63} + 5 \zeta_{6} q^{65} + ( -4 + 4 \zeta_{6} ) q^{67} -6 q^{69} -12 q^{71} + ( 4 - 4 \zeta_{6} ) q^{73} -2 \zeta_{6} q^{75} + ( 9 - 3 \zeta_{6} ) q^{77} -10 \zeta_{6} q^{79} + ( 11 - 11 \zeta_{6} ) q^{81} + 12 q^{83} -6 q^{85} + ( 12 - 12 \zeta_{6} ) q^{87} -6 \zeta_{6} q^{89} + ( 5 + 10 \zeta_{6} ) q^{91} -8 \zeta_{6} q^{93} + ( 1 - \zeta_{6} ) q^{95} + 14 q^{97} -3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + q^{5} + 4q^{7} - q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + q^{5} + 4q^{7} - q^{9} + 3q^{11} + 10q^{13} - 4q^{15} - 6q^{17} - q^{19} - 10q^{21} + 3q^{23} - q^{25} - 8q^{27} - 12q^{29} - 4q^{31} + 6q^{33} - q^{35} - 11q^{37} - 10q^{39} + 6q^{41} + 20q^{43} + q^{45} + 3q^{47} + 2q^{49} - 12q^{51} - 3q^{53} + 6q^{55} + 4q^{57} + 4q^{61} + q^{63} + 5q^{65} - 4q^{67} - 12q^{69} - 24q^{71} + 4q^{73} - 2q^{75} + 15q^{77} - 10q^{79} + 11q^{81} + 24q^{83} - 12q^{85} + 12q^{87} - 6q^{89} + 20q^{91} - 8q^{93} + q^{95} + 28q^{97} - 6q^{99} + O(q^{100}) \)

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 −1.00000 + 1.73205i 0 0.500000 + 0.866025i 0 2.00000 + 1.73205i 0 −0.500000 0.866025i 0
401.1 0 −1.00000 1.73205i 0 0.500000 0.866025i 0 2.00000 1.73205i 0 −0.500000 + 0.866025i 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.d 2
4.b odd 2 1 70.2.e.b 2
7.c even 3 1 inner 560.2.q.d 2
7.c even 3 1 3920.2.a.be 1
7.d odd 6 1 3920.2.a.g 1
12.b even 2 1 630.2.k.e 2
20.d odd 2 1 350.2.e.h 2
20.e even 4 2 350.2.j.a 4
28.d even 2 1 490.2.e.a 2
28.f even 6 1 490.2.a.j 1
28.f even 6 1 490.2.e.a 2
28.g odd 6 1 70.2.e.b 2
28.g odd 6 1 490.2.a.g 1
84.j odd 6 1 4410.2.a.c 1
84.n even 6 1 630.2.k.e 2
84.n even 6 1 4410.2.a.m 1
140.p odd 6 1 350.2.e.h 2
140.p odd 6 1 2450.2.a.p 1
140.s even 6 1 2450.2.a.f 1
140.w even 12 2 350.2.j.a 4
140.w even 12 2 2450.2.c.f 2
140.x odd 12 2 2450.2.c.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.b 2 4.b odd 2 1
70.2.e.b 2 28.g odd 6 1
350.2.e.h 2 20.d odd 2 1
350.2.e.h 2 140.p odd 6 1
350.2.j.a 4 20.e even 4 2
350.2.j.a 4 140.w even 12 2
490.2.a.g 1 28.g odd 6 1
490.2.a.j 1 28.f even 6 1
490.2.e.a 2 28.d even 2 1
490.2.e.a 2 28.f even 6 1
560.2.q.d 2 1.a even 1 1 trivial
560.2.q.d 2 7.c even 3 1 inner
630.2.k.e 2 12.b even 2 1
630.2.k.e 2 84.n even 6 1
2450.2.a.f 1 140.s even 6 1
2450.2.a.p 1 140.p odd 6 1
2450.2.c.f 2 140.w even 12 2
2450.2.c.p 2 140.x odd 12 2
3920.2.a.g 1 7.d odd 6 1
3920.2.a.be 1 7.c even 3 1
4410.2.a.c 1 84.j odd 6 1
4410.2.a.m 1 84.n even 6 1

Hecke kernels

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

\( T_{3}^{2} + 2 T_{3} + 4 \)
\( T_{11}^{2} - 3 T_{11} + 9 \)
\( T_{13} - 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 4 + 2 T + T^{2} \)
$5$ \( 1 - T + T^{2} \)
$7$ \( 7 - 4 T + T^{2} \)
$11$ \( 9 - 3 T + T^{2} \)
$13$ \( ( -5 + T )^{2} \)
$17$ \( 36 + 6 T + T^{2} \)
$19$ \( 1 + T + T^{2} \)
$23$ \( 9 - 3 T + T^{2} \)
$29$ \( ( 6 + T )^{2} \)
$31$ \( 16 + 4 T + T^{2} \)
$37$ \( 121 + 11 T + T^{2} \)
$41$ \( ( -3 + T )^{2} \)
$43$ \( ( -10 + T )^{2} \)
$47$ \( 9 - 3 T + T^{2} \)
$53$ \( 9 + 3 T + T^{2} \)
$59$ \( T^{2} \)
$61$ \( 16 - 4 T + T^{2} \)
$67$ \( 16 + 4 T + T^{2} \)
$71$ \( ( 12 + T )^{2} \)
$73$ \( 16 - 4 T + T^{2} \)
$79$ \( 100 + 10 T + T^{2} \)
$83$ \( ( -12 + T )^{2} \)
$89$ \( 36 + 6 T + T^{2} \)
$97$ \( ( -14 + T )^{2} \)
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