Properties

Label 560.2.g.e
Level $560$
Weight $2$
Character orbit 560.g
Analytic conductor $4.472$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} + \beta_{3} ) q^{3} + ( 1 + \beta_{1} - \beta_{2} ) q^{5} -\beta_{2} q^{7} -3 q^{9} +O(q^{10})\) \( q + ( \beta_{1} + \beta_{3} ) q^{3} + ( 1 + \beta_{1} - \beta_{2} ) q^{5} -\beta_{2} q^{7} -3 q^{9} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{11} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{13} + ( -3 + 2 \beta_{1} + 3 \beta_{2} ) q^{15} -2 \beta_{2} q^{17} + ( 4 - \beta_{1} + \beta_{3} ) q^{19} + ( \beta_{1} - \beta_{3} ) q^{21} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{23} + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{25} + ( -2 - 2 \beta_{1} + 2 \beta_{3} ) q^{29} + ( -4 - 2 \beta_{1} + 2 \beta_{3} ) q^{31} + 12 \beta_{2} q^{33} + ( -1 - \beta_{2} - \beta_{3} ) q^{35} -2 \beta_{2} q^{37} + ( -6 - 2 \beta_{1} + 2 \beta_{3} ) q^{39} + ( -6 + 2 \beta_{1} - 2 \beta_{3} ) q^{41} + ( 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{43} + ( -3 - 3 \beta_{1} + 3 \beta_{2} ) q^{45} + ( 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{47} - q^{49} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{51} + ( -2 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} ) q^{53} + ( 6 + 6 \beta_{2} - 4 \beta_{3} ) q^{55} + ( 4 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{57} + ( -4 + \beta_{1} - \beta_{3} ) q^{59} + ( 6 - \beta_{1} + \beta_{3} ) q^{61} + 3 \beta_{2} q^{63} + ( -1 + 2 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} ) q^{65} -8 \beta_{2} q^{67} + ( 12 - 2 \beta_{1} + 2 \beta_{3} ) q^{69} + ( 6 - 2 \beta_{1} + 2 \beta_{3} ) q^{71} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{73} + ( -\beta_{1} + 12 \beta_{2} + \beta_{3} ) q^{75} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{77} + ( 2 + 2 \beta_{1} - 2 \beta_{3} ) q^{79} -9 q^{81} + ( \beta_{1} + \beta_{3} ) q^{83} + ( -2 - 2 \beta_{2} - 2 \beta_{3} ) q^{85} + ( -2 \beta_{1} - 12 \beta_{2} - 2 \beta_{3} ) q^{87} + 10 q^{89} + ( 2 + \beta_{1} - \beta_{3} ) q^{91} + ( -4 \beta_{1} - 12 \beta_{2} - 4 \beta_{3} ) q^{93} + ( 1 + 4 \beta_{1} - 7 \beta_{2} + 2 \beta_{3} ) q^{95} + ( -4 \beta_{1} - 6 \beta_{2} - 4 \beta_{3} ) q^{97} + ( -6 \beta_{1} + 6 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5} - 12 q^{9} + O(q^{10}) \) \( 4 q + 4 q^{5} - 12 q^{9} - 12 q^{15} + 16 q^{19} - 8 q^{29} - 16 q^{31} - 4 q^{35} - 24 q^{39} - 24 q^{41} - 12 q^{45} - 4 q^{49} + 24 q^{55} - 16 q^{59} + 24 q^{61} - 4 q^{65} + 48 q^{69} + 24 q^{71} + 8 q^{79} - 36 q^{81} - 8 q^{85} + 40 q^{89} + 8 q^{91} + 4 q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/3\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(3 \beta_{2}\)
\(\nu^{3}\)\(=\)\(3 \beta_{3}\)

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
−1.22474 1.22474i
1.22474 1.22474i
−1.22474 + 1.22474i
1.22474 + 1.22474i
0 2.44949i 0 −0.224745 2.22474i 0 1.00000i 0 −3.00000 0
449.2 0 2.44949i 0 2.22474 0.224745i 0 1.00000i 0 −3.00000 0
449.3 0 2.44949i 0 −0.224745 + 2.22474i 0 1.00000i 0 −3.00000 0
449.4 0 2.44949i 0 2.22474 + 0.224745i 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.e 4
3.b odd 2 1 5040.2.t.t 4
4.b odd 2 1 70.2.c.a 4
5.b even 2 1 inner 560.2.g.e 4
5.c odd 4 1 2800.2.a.bl 2
5.c odd 4 1 2800.2.a.bm 2
8.b even 2 1 2240.2.g.i 4
8.d odd 2 1 2240.2.g.j 4
12.b even 2 1 630.2.g.g 4
15.d odd 2 1 5040.2.t.t 4
20.d odd 2 1 70.2.c.a 4
20.e even 4 1 350.2.a.g 2
20.e even 4 1 350.2.a.h 2
28.d even 2 1 490.2.c.e 4
28.f even 6 2 490.2.i.f 8
28.g odd 6 2 490.2.i.c 8
40.e odd 2 1 2240.2.g.j 4
40.f even 2 1 2240.2.g.i 4
60.h even 2 1 630.2.g.g 4
60.l odd 4 1 3150.2.a.bs 2
60.l odd 4 1 3150.2.a.bt 2
140.c even 2 1 490.2.c.e 4
140.j odd 4 1 2450.2.a.bl 2
140.j odd 4 1 2450.2.a.bq 2
140.p odd 6 2 490.2.i.c 8
140.s even 6 2 490.2.i.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.c.a 4 4.b odd 2 1
70.2.c.a 4 20.d odd 2 1
350.2.a.g 2 20.e even 4 1
350.2.a.h 2 20.e even 4 1
490.2.c.e 4 28.d even 2 1
490.2.c.e 4 140.c even 2 1
490.2.i.c 8 28.g odd 6 2
490.2.i.c 8 140.p odd 6 2
490.2.i.f 8 28.f even 6 2
490.2.i.f 8 140.s even 6 2
560.2.g.e 4 1.a even 1 1 trivial
560.2.g.e 4 5.b even 2 1 inner
630.2.g.g 4 12.b even 2 1
630.2.g.g 4 60.h even 2 1
2240.2.g.i 4 8.b even 2 1
2240.2.g.i 4 40.f even 2 1
2240.2.g.j 4 8.d odd 2 1
2240.2.g.j 4 40.e odd 2 1
2450.2.a.bl 2 140.j odd 4 1
2450.2.a.bq 2 140.j odd 4 1
2800.2.a.bl 2 5.c odd 4 1
2800.2.a.bm 2 5.c odd 4 1
3150.2.a.bs 2 60.l odd 4 1
3150.2.a.bt 2 60.l odd 4 1
5040.2.t.t 4 3.b odd 2 1
5040.2.t.t 4 15.d odd 2 1

Hecke kernels

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

\( T_{3}^{2} + 6 \)
\( T_{11}^{2} - 24 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 6 + T^{2} )^{2} \)
$5$ \( 25 - 20 T + 8 T^{2} - 4 T^{3} + T^{4} \)
$7$ \( ( 1 + T^{2} )^{2} \)
$11$ \( ( -24 + T^{2} )^{2} \)
$13$ \( 4 + 20 T^{2} + T^{4} \)
$17$ \( ( 4 + T^{2} )^{2} \)
$19$ \( ( 10 - 8 T + T^{2} )^{2} \)
$23$ \( 400 + 56 T^{2} + T^{4} \)
$29$ \( ( -20 + 4 T + T^{2} )^{2} \)
$31$ \( ( -8 + 8 T + T^{2} )^{2} \)
$37$ \( ( 4 + T^{2} )^{2} \)
$41$ \( ( 12 + 12 T + T^{2} )^{2} \)
$43$ \( 64 + 80 T^{2} + T^{4} \)
$47$ \( 64 + 80 T^{2} + T^{4} \)
$53$ \( 144 + 120 T^{2} + T^{4} \)
$59$ \( ( 10 + 8 T + T^{2} )^{2} \)
$61$ \( ( 30 - 12 T + T^{2} )^{2} \)
$67$ \( ( 64 + T^{2} )^{2} \)
$71$ \( ( 12 - 12 T + T^{2} )^{2} \)
$73$ \( 400 + 56 T^{2} + T^{4} \)
$79$ \( ( -20 - 4 T + T^{2} )^{2} \)
$83$ \( ( 6 + T^{2} )^{2} \)
$89$ \( ( -10 + T )^{4} \)
$97$ \( 3600 + 264 T^{2} + T^{4} \)
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