Properties

Label 560.2.k.b
Level 560
Weight 2
Character orbit 560.k
Analytic conductor 4.472
Analytic rank 0
Dimension 8
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.k (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.47162251319\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.116319195136.7
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{3} + \beta_{3} q^{5} + ( \beta_{4} - \beta_{6} ) q^{7} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{7} q^{3} + \beta_{3} q^{5} + ( \beta_{4} - \beta_{6} ) q^{7} + ( 1 + \beta_{2} ) q^{9} + ( \beta_{1} + \beta_{4} + \beta_{6} ) q^{11} -\beta_{5} q^{13} + \beta_{6} q^{15} + \beta_{5} q^{17} + ( -2 \beta_{1} + 2 \beta_{4} + 2 \beta_{7} ) q^{19} + ( -1 - 3 \beta_{3} + \beta_{5} ) q^{21} + ( \beta_{1} + \beta_{4} - 2 \beta_{6} ) q^{23} - q^{25} + ( -\beta_{1} + \beta_{4} + 3 \beta_{7} ) q^{27} -\beta_{2} q^{29} + 2 \beta_{7} q^{31} + ( 6 \beta_{3} - \beta_{5} ) q^{33} + ( -\beta_{1} + \beta_{7} ) q^{35} + ( -2 - 2 \beta_{2} ) q^{37} + ( -\beta_{1} - \beta_{4} + 5 \beta_{6} ) q^{39} -6 \beta_{3} q^{41} + ( 3 \beta_{1} + 3 \beta_{4} ) q^{43} + ( \beta_{3} - \beta_{5} ) q^{45} + ( 2 \beta_{1} - 2 \beta_{4} - \beta_{7} ) q^{47} + ( -2 - \beta_{2} - 3 \beta_{3} - \beta_{5} ) q^{49} + ( \beta_{1} + \beta_{4} - 5 \beta_{6} ) q^{51} -6 q^{53} + ( -\beta_{1} + \beta_{4} - \beta_{7} ) q^{55} + ( 4 + 2 \beta_{2} ) q^{57} + ( -2 \beta_{1} + 2 \beta_{4} - 2 \beta_{7} ) q^{59} + ( 6 \beta_{3} + 2 \beta_{5} ) q^{61} + ( \beta_{1} - 2 \beta_{4} - 5 \beta_{6} - \beta_{7} ) q^{63} -\beta_{2} q^{65} + ( \beta_{1} + \beta_{4} + 4 \beta_{6} ) q^{67} + ( -6 \beta_{3} + 2 \beta_{5} ) q^{69} + ( -2 \beta_{1} - 2 \beta_{4} - 2 \beta_{6} ) q^{71} + 6 \beta_{3} q^{73} -\beta_{7} q^{75} + ( 2 \beta_{2} - 6 \beta_{3} - \beta_{5} ) q^{77} + ( -3 \beta_{1} - 3 \beta_{4} + 3 \beta_{6} ) q^{79} + 7 q^{81} + ( 3 \beta_{1} - 3 \beta_{4} ) q^{83} + \beta_{2} q^{85} + ( \beta_{1} - \beta_{4} - 5 \beta_{7} ) q^{87} + 12 \beta_{3} q^{89} + ( 3 \beta_{1} + \beta_{4} - \beta_{6} + 4 \beta_{7} ) q^{91} + ( 8 + 2 \beta_{2} ) q^{93} + ( -2 \beta_{1} - 2 \beta_{4} + 2 \beta_{6} ) q^{95} + ( -12 \beta_{3} + \beta_{5} ) q^{97} + ( -4 \beta_{1} - 4 \beta_{4} + 8 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 12q^{9} + O(q^{10}) \) \( 8q + 12q^{9} - 8q^{21} - 8q^{25} - 4q^{29} - 24q^{37} - 20q^{49} - 48q^{53} + 40q^{57} - 4q^{65} + 8q^{77} + 56q^{81} + 4q^{85} + 72q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 77 x^{4} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{4} + 43 \)\()/9\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} + 79 \nu^{2} \)\()/18\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} + 79 \nu^{3} \)\()/18\)
\(\beta_{5}\)\(=\)\((\)\( -5 \nu^{6} - 377 \nu^{2} \)\()/18\)
\(\beta_{6}\)\(=\)\((\)\( 9 \nu^{7} + 2 \nu^{5} + 693 \nu^{3} + 158 \nu \)\()/36\)
\(\beta_{7}\)\(=\)\((\)\( -9 \nu^{7} + 2 \nu^{5} - 693 \nu^{3} + 158 \nu \)\()/36\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} + 5 \beta_{3}\)
\(\nu^{3}\)\(=\)\(\beta_{7} - \beta_{6} + 9 \beta_{4}\)
\(\nu^{4}\)\(=\)\(9 \beta_{2} - 43\)
\(\nu^{5}\)\(=\)\(9 \beta_{7} + 9 \beta_{6} - 79 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-79 \beta_{5} - 377 \beta_{3}\)
\(\nu^{7}\)\(=\)\(-79 \beta_{7} + 79 \beta_{6} - 693 \beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
111.1
−0.337637 + 0.337637i
−0.337637 0.337637i
−2.09428 + 2.09428i
−2.09428 2.09428i
2.09428 2.09428i
2.09428 + 2.09428i
0.337637 0.337637i
0.337637 + 0.337637i
0 −2.96176 0 1.00000i 0 0.337637 2.62412i 0 5.77200 0
111.2 0 −2.96176 0 1.00000i 0 0.337637 + 2.62412i 0 5.77200 0
111.3 0 −0.477491 0 1.00000i 0 2.09428 + 1.61679i 0 −2.77200 0
111.4 0 −0.477491 0 1.00000i 0 2.09428 1.61679i 0 −2.77200 0
111.5 0 0.477491 0 1.00000i 0 −2.09428 1.61679i 0 −2.77200 0
111.6 0 0.477491 0 1.00000i 0 −2.09428 + 1.61679i 0 −2.77200 0
111.7 0 2.96176 0 1.00000i 0 −0.337637 + 2.62412i 0 5.77200 0
111.8 0 2.96176 0 1.00000i 0 −0.337637 2.62412i 0 5.77200 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 111.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d 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.k.b 8
3.b odd 2 1 5040.2.d.d 8
4.b odd 2 1 inner 560.2.k.b 8
5.b even 2 1 2800.2.k.m 8
5.c odd 4 1 2800.2.e.g 8
5.c odd 4 1 2800.2.e.h 8
7.b odd 2 1 inner 560.2.k.b 8
8.b even 2 1 2240.2.k.d 8
8.d odd 2 1 2240.2.k.d 8
12.b even 2 1 5040.2.d.d 8
20.d odd 2 1 2800.2.k.m 8
20.e even 4 1 2800.2.e.g 8
20.e even 4 1 2800.2.e.h 8
21.c even 2 1 5040.2.d.d 8
28.d even 2 1 inner 560.2.k.b 8
35.c odd 2 1 2800.2.k.m 8
35.f even 4 1 2800.2.e.g 8
35.f even 4 1 2800.2.e.h 8
56.e even 2 1 2240.2.k.d 8
56.h odd 2 1 2240.2.k.d 8
84.h odd 2 1 5040.2.d.d 8
140.c even 2 1 2800.2.k.m 8
140.j odd 4 1 2800.2.e.g 8
140.j odd 4 1 2800.2.e.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.k.b 8 1.a even 1 1 trivial
560.2.k.b 8 4.b odd 2 1 inner
560.2.k.b 8 7.b odd 2 1 inner
560.2.k.b 8 28.d even 2 1 inner
2240.2.k.d 8 8.b even 2 1
2240.2.k.d 8 8.d odd 2 1
2240.2.k.d 8 56.e even 2 1
2240.2.k.d 8 56.h odd 2 1
2800.2.e.g 8 5.c odd 4 1
2800.2.e.g 8 20.e even 4 1
2800.2.e.g 8 35.f even 4 1
2800.2.e.g 8 140.j odd 4 1
2800.2.e.h 8 5.c odd 4 1
2800.2.e.h 8 20.e even 4 1
2800.2.e.h 8 35.f even 4 1
2800.2.e.h 8 140.j odd 4 1
2800.2.k.m 8 5.b even 2 1
2800.2.k.m 8 20.d odd 2 1
2800.2.k.m 8 35.c odd 2 1
2800.2.k.m 8 140.c even 2 1
5040.2.d.d 8 3.b odd 2 1
5040.2.d.d 8 12.b even 2 1
5040.2.d.d 8 21.c even 2 1
5040.2.d.d 8 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 9 T_{3}^{2} + 2 \) acting on \(S_{2}^{\mathrm{new}}(560, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + 3 T^{2} + 2 T^{4} + 27 T^{6} + 81 T^{8} )^{2} \)
$5$ \( ( 1 + T^{2} )^{4} \)
$7$ \( 1 + 10 T^{2} + 50 T^{4} + 490 T^{6} + 2401 T^{8} \)
$11$ \( ( 1 - 9 T^{2} + 244 T^{4} - 1089 T^{6} + 14641 T^{8} )^{2} \)
$13$ \( ( 1 - 15 T^{2} + 376 T^{4} - 2535 T^{6} + 28561 T^{8} )^{2} \)
$17$ \( ( 1 - 31 T^{2} + 800 T^{4} - 8959 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 + 430 T^{4} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 - 54 T^{2} + 1714 T^{4} - 28566 T^{6} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 + T + 40 T^{2} + 29 T^{3} + 841 T^{4} )^{4} \)
$31$ \( ( 1 + 88 T^{2} + 3566 T^{4} + 84568 T^{6} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 + 6 T + 10 T^{2} + 222 T^{3} + 1369 T^{4} )^{4} \)
$41$ \( ( 1 - 46 T^{2} + 1681 T^{4} )^{4} \)
$43$ \( ( 1 - 10 T^{2} - 2190 T^{4} - 18490 T^{6} + 3418801 T^{8} )^{2} \)
$47$ \( ( 1 + 123 T^{2} + 7306 T^{4} + 271707 T^{6} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 + 6 T + 53 T^{2} )^{8} \)
$59$ \( ( 1 + 96 T^{2} + 8974 T^{4} + 334176 T^{6} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 48 T^{2} + 718 T^{4} - 178608 T^{6} + 13845841 T^{8} )^{2} \)
$67$ \( ( 1 - 74 T^{2} + 6770 T^{4} - 332186 T^{6} + 20151121 T^{8} )^{2} \)
$71$ \( ( 1 - 144 T^{2} + 14974 T^{4} - 725904 T^{6} + 25411681 T^{8} )^{2} \)
$73$ \( ( 1 - 16 T + 73 T^{2} )^{4}( 1 + 16 T + 73 T^{2} )^{4} \)
$79$ \( ( 1 - 145 T^{2} + 16260 T^{4} - 904945 T^{6} + 38950081 T^{8} )^{2} \)
$83$ \( ( 1 + 170 T^{2} + 15090 T^{4} + 1171130 T^{6} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 - 34 T^{2} + 7921 T^{4} )^{4} \)
$97$ \( ( 1 - 39 T^{2} + 7792 T^{4} - 366951 T^{6} + 88529281 T^{8} )^{2} \)
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