Properties

Label 560.2.g.a
Level 560
Weight 2
Character orbit 560.g
Analytic conductor 4.472
Analytic rank 1
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.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.47162251319\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 i q^{3} + ( -2 - i ) q^{5} -i q^{7} -6 q^{9} +O(q^{10})\) \( q + 3 i q^{3} + ( -2 - i ) q^{5} -i q^{7} -6 q^{9} -3 q^{11} -i q^{13} + ( 3 - 6 i ) q^{15} -5 i q^{17} -8 q^{19} + 3 q^{21} + 2 i q^{23} + ( 3 + 4 i ) q^{25} -9 i q^{27} + q^{29} + 2 q^{31} -9 i q^{33} + ( -1 + 2 i ) q^{35} + 10 i q^{37} + 3 q^{39} -6 q^{41} -4 i q^{43} + ( 12 + 6 i ) q^{45} -11 i q^{47} - q^{49} + 15 q^{51} -6 i q^{53} + ( 6 + 3 i ) q^{55} -24 i q^{57} -10 q^{59} + 6 i q^{63} + ( -1 + 2 i ) q^{65} + 10 i q^{67} -6 q^{69} + 10 i q^{73} + ( -12 + 9 i ) q^{75} + 3 i q^{77} -7 q^{79} + 9 q^{81} + 12 i q^{83} + ( -5 + 10 i ) q^{85} + 3 i q^{87} -8 q^{89} - q^{91} + 6 i q^{93} + ( 16 + 8 i ) q^{95} + 3 i q^{97} + 18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{5} - 12q^{9} + O(q^{10}) \) \( 2q - 4q^{5} - 12q^{9} - 6q^{11} + 6q^{15} - 16q^{19} + 6q^{21} + 6q^{25} + 2q^{29} + 4q^{31} - 2q^{35} + 6q^{39} - 12q^{41} + 24q^{45} - 2q^{49} + 30q^{51} + 12q^{55} - 20q^{59} - 2q^{65} - 12q^{69} - 24q^{75} - 14q^{79} + 18q^{81} - 10q^{85} - 16q^{89} - 2q^{91} + 32q^{95} + 36q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
1.00000i
1.00000i
0 3.00000i 0 −2.00000 + 1.00000i 0 1.00000i 0 −6.00000 0
449.2 0 3.00000i 0 −2.00000 1.00000i 0 1.00000i 0 −6.00000 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.g.a 2
3.b odd 2 1 5040.2.t.s 2
4.b odd 2 1 140.2.e.a 2
5.b even 2 1 inner 560.2.g.a 2
5.c odd 4 1 2800.2.a.a 1
5.c odd 4 1 2800.2.a.bf 1
8.b even 2 1 2240.2.g.f 2
8.d odd 2 1 2240.2.g.e 2
12.b even 2 1 1260.2.k.c 2
15.d odd 2 1 5040.2.t.s 2
20.d odd 2 1 140.2.e.a 2
20.e even 4 1 700.2.a.a 1
20.e even 4 1 700.2.a.j 1
28.d even 2 1 980.2.e.b 2
28.f even 6 2 980.2.q.c 4
28.g odd 6 2 980.2.q.f 4
40.e odd 2 1 2240.2.g.e 2
40.f even 2 1 2240.2.g.f 2
60.h even 2 1 1260.2.k.c 2
60.l odd 4 1 6300.2.a.c 1
60.l odd 4 1 6300.2.a.t 1
140.c even 2 1 980.2.e.b 2
140.j odd 4 1 4900.2.a.b 1
140.j odd 4 1 4900.2.a.w 1
140.p odd 6 2 980.2.q.f 4
140.s even 6 2 980.2.q.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.e.a 2 4.b odd 2 1
140.2.e.a 2 20.d odd 2 1
560.2.g.a 2 1.a even 1 1 trivial
560.2.g.a 2 5.b even 2 1 inner
700.2.a.a 1 20.e even 4 1
700.2.a.j 1 20.e even 4 1
980.2.e.b 2 28.d even 2 1
980.2.e.b 2 140.c even 2 1
980.2.q.c 4 28.f even 6 2
980.2.q.c 4 140.s even 6 2
980.2.q.f 4 28.g odd 6 2
980.2.q.f 4 140.p odd 6 2
1260.2.k.c 2 12.b even 2 1
1260.2.k.c 2 60.h even 2 1
2240.2.g.e 2 8.d odd 2 1
2240.2.g.e 2 40.e odd 2 1
2240.2.g.f 2 8.b even 2 1
2240.2.g.f 2 40.f even 2 1
2800.2.a.a 1 5.c odd 4 1
2800.2.a.bf 1 5.c odd 4 1
4900.2.a.b 1 140.j odd 4 1
4900.2.a.w 1 140.j odd 4 1
5040.2.t.s 2 3.b odd 2 1
5040.2.t.s 2 15.d odd 2 1
6300.2.a.c 1 60.l odd 4 1
6300.2.a.t 1 60.l odd 4 1

Hecke kernels

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

\( T_{3}^{2} + 9 \)
\( T_{11} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 3 T^{2} + 9 T^{4} \)
$5$ \( 1 + 4 T + 5 T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( 1 + 3 T + 11 T^{2} )^{2} \)
$13$ \( 1 - 25 T^{2} + 169 T^{4} \)
$17$ \( 1 - 9 T^{2} + 289 T^{4} \)
$19$ \( ( 1 + 8 T + 19 T^{2} )^{2} \)
$23$ \( 1 - 42 T^{2} + 529 T^{4} \)
$29$ \( ( 1 - T + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 2 T + 31 T^{2} )^{2} \)
$37$ \( 1 + 26 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 + 6 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 70 T^{2} + 1849 T^{4} \)
$47$ \( 1 + 27 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 70 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 + 10 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 61 T^{2} )^{2} \)
$67$ \( 1 - 34 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( 1 - 46 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 + 7 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 22 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 + 8 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 185 T^{2} + 9409 T^{4} \)
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