Properties

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

Related objects

Downloads

Learn more about

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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} - \beta_{2} ) q^{3} + \beta_{2} q^{5} + ( 1 + \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{7} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} - \beta_{2} ) q^{3} + \beta_{2} q^{5} + ( 1 + \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{7} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{9} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{11} + 2 q^{13} + ( 1 + \beta_{3} ) q^{15} + ( 2 + 4 \beta_{1} + 2 \beta_{2} ) q^{17} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{19} + ( -4 + 2 \beta_{1} - 3 \beta_{2} ) q^{21} + ( -\beta_{1} + 7 \beta_{2} - \beta_{3} ) q^{23} + ( -1 - \beta_{2} ) q^{25} + ( 1 - \beta_{3} ) q^{27} + ( -5 + 2 \beta_{3} ) q^{29} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{31} + ( -4 \beta_{1} + 6 \beta_{2} - 4 \beta_{3} ) q^{33} + ( -1 - 2 \beta_{1} - \beta_{3} ) q^{35} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{37} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{39} + ( 3 - 2 \beta_{3} ) q^{41} + ( -3 - 7 \beta_{3} ) q^{43} + 2 \beta_{1} q^{45} + ( -4 \beta_{1} - 6 \beta_{2} - 4 \beta_{3} ) q^{47} + ( -2 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} ) q^{49} + ( -2 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} ) q^{51} + 4 \beta_{1} q^{53} + ( 2 + 2 \beta_{3} ) q^{55} + ( -8 - 4 \beta_{3} ) q^{57} + ( -4 - 4 \beta_{2} ) q^{59} + ( 4 \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{61} + ( 4 + 8 \beta_{2} - 2 \beta_{3} ) q^{63} + 2 \beta_{2} q^{65} + ( -3 - 7 \beta_{1} - 3 \beta_{2} ) q^{67} + ( 9 + 8 \beta_{3} ) q^{69} + 12 q^{71} + ( 2 - 4 \beta_{1} + 2 \beta_{2} ) q^{73} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{75} + ( -8 + 4 \beta_{1} - 6 \beta_{2} ) q^{77} + 4 \beta_{2} q^{79} + ( 1 - 6 \beta_{1} + \beta_{2} ) q^{81} + ( 9 + 3 \beta_{3} ) q^{83} + ( -2 + 4 \beta_{3} ) q^{85} + ( 1 - 3 \beta_{1} + \beta_{2} ) q^{87} + ( 4 \beta_{1} - 11 \beta_{2} + 4 \beta_{3} ) q^{89} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{91} + ( -4 \beta_{1} + 6 \beta_{2} - 4 \beta_{3} ) q^{93} -4 \beta_{1} q^{95} + 6 q^{97} + ( 8 + 4 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{3} - 2q^{5} + 2q^{7} + O(q^{10}) \) \( 4q - 2q^{3} - 2q^{5} + 2q^{7} - 4q^{11} + 8q^{13} + 4q^{15} + 4q^{17} - 10q^{21} - 14q^{23} - 2q^{25} + 4q^{27} - 20q^{29} - 4q^{31} - 12q^{33} - 4q^{35} - 4q^{39} + 12q^{41} - 12q^{43} + 12q^{47} + 10q^{49} - 12q^{51} + 8q^{55} - 32q^{57} - 8q^{59} - 2q^{61} - 4q^{65} - 6q^{67} + 36q^{69} + 48q^{71} + 4q^{73} - 2q^{75} - 20q^{77} - 8q^{79} + 2q^{81} + 36q^{83} - 8q^{85} + 2q^{87} + 22q^{89} + 4q^{91} - 12q^{93} + 24q^{97} + 32q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 2 x^{2} + 4\):

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

Character values

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

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(\beta_{2}\) \(1\) \(1\) \(1\)

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
−0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
0.707107 + 1.22474i
0 −1.20711 + 2.09077i 0 −0.500000 0.866025i 0 2.62132 + 0.358719i 0 −1.41421 2.44949i 0
81.2 0 0.207107 0.358719i 0 −0.500000 0.866025i 0 −1.62132 2.09077i 0 1.41421 + 2.44949i 0
401.1 0 −1.20711 2.09077i 0 −0.500000 + 0.866025i 0 2.62132 0.358719i 0 −1.41421 + 2.44949i 0
401.2 0 0.207107 + 0.358719i 0 −0.500000 + 0.866025i 0 −1.62132 + 2.09077i 0 1.41421 2.44949i 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.j 4
4.b odd 2 1 280.2.q.d 4
7.c even 3 1 inner 560.2.q.j 4
7.c even 3 1 3920.2.a.bz 2
7.d odd 6 1 3920.2.a.bp 2
12.b even 2 1 2520.2.bi.k 4
20.d odd 2 1 1400.2.q.h 4
20.e even 4 2 1400.2.bh.g 8
28.d even 2 1 1960.2.q.q 4
28.f even 6 1 1960.2.a.t 2
28.f even 6 1 1960.2.q.q 4
28.g odd 6 1 280.2.q.d 4
28.g odd 6 1 1960.2.a.p 2
84.n even 6 1 2520.2.bi.k 4
140.p odd 6 1 1400.2.q.h 4
140.p odd 6 1 9800.2.a.bz 2
140.s even 6 1 9800.2.a.br 2
140.w even 12 2 1400.2.bh.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.d 4 4.b odd 2 1
280.2.q.d 4 28.g odd 6 1
560.2.q.j 4 1.a even 1 1 trivial
560.2.q.j 4 7.c even 3 1 inner
1400.2.q.h 4 20.d odd 2 1
1400.2.q.h 4 140.p odd 6 1
1400.2.bh.g 8 20.e even 4 2
1400.2.bh.g 8 140.w even 12 2
1960.2.a.p 2 28.g odd 6 1
1960.2.a.t 2 28.f even 6 1
1960.2.q.q 4 28.d even 2 1
1960.2.q.q 4 28.f even 6 1
2520.2.bi.k 4 12.b even 2 1
2520.2.bi.k 4 84.n even 6 1
3920.2.a.bp 2 7.d odd 6 1
3920.2.a.bz 2 7.c even 3 1
9800.2.a.br 2 140.s even 6 1
9800.2.a.bz 2 140.p odd 6 1

Hecke kernels

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

\( T_{3}^{4} + 2 T_{3}^{3} + 5 T_{3}^{2} - 2 T_{3} + 1 \)
\( T_{11}^{4} + 4 T_{11}^{3} + 20 T_{11}^{2} - 16 T_{11} + 16 \)
\( T_{13} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 2 T - T^{2} - 2 T^{3} + 4 T^{4} - 6 T^{5} - 9 T^{6} + 54 T^{7} + 81 T^{8} \)
$5$ \( ( 1 + T + T^{2} )^{2} \)
$7$ \( 1 - 2 T - 3 T^{2} - 14 T^{3} + 49 T^{4} \)
$11$ \( 1 + 4 T - 2 T^{2} - 16 T^{3} + 27 T^{4} - 176 T^{5} - 242 T^{6} + 5324 T^{7} + 14641 T^{8} \)
$13$ \( ( 1 - 2 T + 13 T^{2} )^{4} \)
$17$ \( 1 - 4 T + 10 T^{2} + 112 T^{3} - 525 T^{4} + 1904 T^{5} + 2890 T^{6} - 19652 T^{7} + 83521 T^{8} \)
$19$ \( 1 - 6 T^{2} - 325 T^{4} - 2166 T^{6} + 130321 T^{8} \)
$23$ \( 1 + 14 T + 103 T^{2} + 658 T^{3} + 3612 T^{4} + 15134 T^{5} + 54487 T^{6} + 170338 T^{7} + 279841 T^{8} \)
$29$ \( ( 1 + 10 T + 75 T^{2} + 290 T^{3} + 841 T^{4} )^{2} \)
$31$ \( 1 + 4 T - 42 T^{2} - 16 T^{3} + 1907 T^{4} - 496 T^{5} - 40362 T^{6} + 119164 T^{7} + 923521 T^{8} \)
$37$ \( 1 - 42 T^{2} + 395 T^{4} - 57498 T^{6} + 1874161 T^{8} \)
$41$ \( ( 1 - 6 T + 83 T^{2} - 246 T^{3} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 + 6 T - 3 T^{2} + 258 T^{3} + 1849 T^{4} )^{2} \)
$47$ \( 1 - 12 T + 46 T^{2} - 48 T^{3} + 627 T^{4} - 2256 T^{5} + 101614 T^{6} - 1245876 T^{7} + 4879681 T^{8} \)
$53$ \( 1 - 74 T^{2} + 2667 T^{4} - 207866 T^{6} + 7890481 T^{8} \)
$59$ \( ( 1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4} )^{2} \)
$61$ \( 1 + 2 T - 87 T^{2} - 62 T^{3} + 4316 T^{4} - 3782 T^{5} - 323727 T^{6} + 453962 T^{7} + 13845841 T^{8} \)
$67$ \( 1 + 6 T - 9 T^{2} - 534 T^{3} - 4876 T^{4} - 35778 T^{5} - 40401 T^{6} + 1804578 T^{7} + 20151121 T^{8} \)
$71$ \( ( 1 - 12 T + 71 T^{2} )^{4} \)
$73$ \( 1 - 4 T - 102 T^{2} + 112 T^{3} + 7427 T^{4} + 8176 T^{5} - 543558 T^{6} - 1556068 T^{7} + 28398241 T^{8} \)
$79$ \( ( 1 - 13 T + 79 T^{2} )^{2}( 1 + 17 T + 79 T^{2} )^{2} \)
$83$ \( ( 1 - 18 T + 229 T^{2} - 1494 T^{3} + 6889 T^{4} )^{2} \)
$89$ \( 1 - 22 T + 217 T^{2} - 1958 T^{3} + 20292 T^{4} - 174262 T^{5} + 1718857 T^{6} - 15509318 T^{7} + 62742241 T^{8} \)
$97$ \( ( 1 - 6 T + 97 T^{2} )^{4} \)
show more
show less