Properties

Label 560.2.g.c
Level 560
Weight 2
Character orbit 560.g
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.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.47162251319\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + ( 1 + 2 i ) q^{5} -i q^{7} + 3 q^{9} +O(q^{10})\) \( q + ( 1 + 2 i ) q^{5} -i q^{7} + 3 q^{9} -4 i q^{13} + 4 i q^{17} + 4 q^{19} + 8 i q^{23} + ( -3 + 4 i ) q^{25} -2 q^{29} + 8 q^{31} + ( 2 - i ) q^{35} -8 i q^{37} + 6 q^{41} + 8 i q^{43} + ( 3 + 6 i ) q^{45} -8 i q^{47} - q^{49} -4 q^{59} -6 q^{61} -3 i q^{63} + ( 8 - 4 i ) q^{65} -8 i q^{67} -12 q^{71} + 4 i q^{73} -4 q^{79} + 9 q^{81} + ( -8 + 4 i ) q^{85} + 10 q^{89} -4 q^{91} + ( 4 + 8 i ) q^{95} -12 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} + 6q^{9} + O(q^{10}) \) \( 2q + 2q^{5} + 6q^{9} + 8q^{19} - 6q^{25} - 4q^{29} + 16q^{31} + 4q^{35} + 12q^{41} + 6q^{45} - 2q^{49} - 8q^{59} - 12q^{61} + 16q^{65} - 24q^{71} - 8q^{79} + 18q^{81} - 16q^{85} + 20q^{89} - 8q^{91} + 8q^{95} + O(q^{100}) \)

Character values

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

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

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 0 0 1.00000 2.00000i 0 1.00000i 0 3.00000 0
449.2 0 0 0 1.00000 + 2.00000i 0 1.00000i 0 3.00000 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.c 2
3.b odd 2 1 5040.2.t.g 2
4.b odd 2 1 140.2.e.b 2
5.b even 2 1 inner 560.2.g.c 2
5.c odd 4 1 2800.2.a.o 1
5.c odd 4 1 2800.2.a.s 1
8.b even 2 1 2240.2.g.c 2
8.d odd 2 1 2240.2.g.d 2
12.b even 2 1 1260.2.k.b 2
15.d odd 2 1 5040.2.t.g 2
20.d odd 2 1 140.2.e.b 2
20.e even 4 1 700.2.a.f 1
20.e even 4 1 700.2.a.h 1
28.d even 2 1 980.2.e.a 2
28.f even 6 2 980.2.q.e 4
28.g odd 6 2 980.2.q.d 4
40.e odd 2 1 2240.2.g.d 2
40.f even 2 1 2240.2.g.c 2
60.h even 2 1 1260.2.k.b 2
60.l odd 4 1 6300.2.a.g 1
60.l odd 4 1 6300.2.a.y 1
140.c even 2 1 980.2.e.a 2
140.j odd 4 1 4900.2.a.l 1
140.j odd 4 1 4900.2.a.m 1
140.p odd 6 2 980.2.q.d 4
140.s even 6 2 980.2.q.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.e.b 2 4.b odd 2 1
140.2.e.b 2 20.d odd 2 1
560.2.g.c 2 1.a even 1 1 trivial
560.2.g.c 2 5.b even 2 1 inner
700.2.a.f 1 20.e even 4 1
700.2.a.h 1 20.e even 4 1
980.2.e.a 2 28.d even 2 1
980.2.e.a 2 140.c even 2 1
980.2.q.d 4 28.g odd 6 2
980.2.q.d 4 140.p odd 6 2
980.2.q.e 4 28.f even 6 2
980.2.q.e 4 140.s even 6 2
1260.2.k.b 2 12.b even 2 1
1260.2.k.b 2 60.h even 2 1
2240.2.g.c 2 8.b even 2 1
2240.2.g.c 2 40.f even 2 1
2240.2.g.d 2 8.d odd 2 1
2240.2.g.d 2 40.e odd 2 1
2800.2.a.o 1 5.c odd 4 1
2800.2.a.s 1 5.c odd 4 1
4900.2.a.l 1 140.j odd 4 1
4900.2.a.m 1 140.j odd 4 1
5040.2.t.g 2 3.b odd 2 1
5040.2.t.g 2 15.d odd 2 1
6300.2.a.g 1 60.l odd 4 1
6300.2.a.y 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 \(S_{2}^{\mathrm{new}}(560, [\chi])\):

\( T_{3} \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 3 T^{2} )^{2} \)
$5$ \( 1 - 2 T + 5 T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( 1 + 11 T^{2} )^{2} \)
$13$ \( ( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \)
$17$ \( 1 - 18 T^{2} + 289 T^{4} \)
$19$ \( ( 1 - 4 T + 19 T^{2} )^{2} \)
$23$ \( 1 + 18 T^{2} + 529 T^{4} \)
$29$ \( ( 1 + 2 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 8 T + 31 T^{2} )^{2} \)
$37$ \( 1 - 10 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 - 6 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 22 T^{2} + 1849 T^{4} \)
$47$ \( 1 - 30 T^{2} + 2209 T^{4} \)
$53$ \( ( 1 - 53 T^{2} )^{2} \)
$59$ \( ( 1 + 4 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 6 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 70 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + 12 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 130 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 + 4 T + 79 T^{2} )^{2} \)
$83$ \( ( 1 - 83 T^{2} )^{2} \)
$89$ \( ( 1 - 10 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 50 T^{2} + 9409 T^{4} \)
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