Properties

Label 560.2.g.f
Level 560
Weight 2
Character orbit 560.g
Analytic conductor 4.472
Analytic rank 0
Dimension 6
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: \(6\)
Coefficient field: 6.0.5161984.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{5} q^{3} + ( \beta_{2} + \beta_{5} ) q^{5} + \beta_{4} q^{7} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q -\beta_{5} q^{3} + ( \beta_{2} + \beta_{5} ) q^{5} + \beta_{4} q^{7} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{9} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{11} + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{13} + ( 3 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} ) q^{15} + ( -\beta_{1} + \beta_{2} - \beta_{5} ) q^{17} + ( 2 + \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{19} + \beta_{3} q^{21} + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} ) q^{23} + ( -2 - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{25} + ( \beta_{1} - \beta_{2} - 6 \beta_{4} + \beta_{5} ) q^{27} + ( 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{29} + ( -4 - 2 \beta_{3} ) q^{31} + ( \beta_{1} - \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{33} + ( -\beta_{1} - \beta_{3} ) q^{35} -6 \beta_{4} q^{37} + ( \beta_{1} + \beta_{2} + 5 \beta_{3} ) q^{39} + ( 6 + 2 \beta_{1} + 2 \beta_{2} ) q^{41} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{43} + ( -4 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{45} + ( -3 \beta_{1} + 3 \beta_{2} - 2 \beta_{4} + 3 \beta_{5} ) q^{47} - q^{49} + ( -6 - \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{51} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 4 \beta_{5} ) q^{53} + ( 2 - \beta_{1} - 3 \beta_{2} + 4 \beta_{4} - \beta_{5} ) q^{55} + ( 2 \beta_{1} - 2 \beta_{2} - 6 \beta_{4} - 2 \beta_{5} ) q^{57} + ( 2 - 3 \beta_{1} - 3 \beta_{2} ) q^{59} + ( 8 - \beta_{1} - \beta_{2} ) q^{61} + ( \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} ) q^{63} + ( -5 + \beta_{1} - \beta_{2} - 4 \beta_{3} + 5 \beta_{4} - \beta_{5} ) q^{65} + ( -2 \beta_{1} + 2 \beta_{2} - 4 \beta_{4} - 2 \beta_{5} ) q^{67} + ( -4 + 6 \beta_{3} ) q^{69} + ( -2 - 2 \beta_{3} ) q^{71} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} ) q^{73} + ( 6 + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} ) q^{75} + ( -\beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{77} + ( 4 - \beta_{1} - \beta_{2} - 5 \beta_{3} ) q^{79} + ( 3 - 2 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} ) q^{81} + ( -\beta_{1} + \beta_{2} + 6 \beta_{4} + 6 \beta_{5} ) q^{83} + ( 2 + 3 \beta_{1} + \beta_{2} - \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{85} + ( 3 \beta_{1} - 3 \beta_{2} - 6 \beta_{4} - 3 \beta_{5} ) q^{87} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} ) q^{89} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{91} + ( -2 \beta_{1} + 2 \beta_{2} + 8 \beta_{4} + 2 \beta_{5} ) q^{93} + ( 1 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + 7 \beta_{4} + 3 \beta_{5} ) q^{95} + ( -\beta_{1} + \beta_{2} + 4 \beta_{4} + 3 \beta_{5} ) q^{97} + ( 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 8q^{9} + O(q^{10}) \) \( 6q - 8q^{9} - 14q^{11} + 18q^{15} + 8q^{19} - 2q^{21} - 10q^{25} - 6q^{29} - 20q^{31} + 2q^{35} - 10q^{39} + 36q^{41} - 28q^{45} - 6q^{49} - 42q^{51} + 12q^{55} + 12q^{59} + 48q^{61} - 22q^{65} - 36q^{69} - 8q^{71} + 40q^{75} + 34q^{79} + 30q^{81} + 14q^{85} - 10q^{91} + 4q^{95} + 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 4 x^{3} + 25 x^{2} - 20 x + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -5 \nu^{5} - 2 \nu^{4} - 25 \nu^{3} + 10 \nu^{2} - 121 \nu + 100 \)\()/121\)
\(\beta_{3}\)\(=\)\((\)\( 7 \nu^{5} + 27 \nu^{4} + 35 \nu^{3} - 14 \nu^{2} + 223 \)\()/121\)
\(\beta_{4}\)\(=\)\((\)\( -25 \nu^{5} - 10 \nu^{4} - 4 \nu^{3} + 50 \nu^{2} - 605 \nu + 258 \)\()/242\)
\(\beta_{5}\)\(=\)\((\)\( -65 \nu^{5} - 26 \nu^{4} + 38 \nu^{3} + 372 \nu^{2} - 1331 \nu + 574 \)\()/242\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - 3 \beta_{4} + \beta_{2} - \beta_{1}\)
\(\nu^{3}\)\(=\)\(2 \beta_{4} - 5 \beta_{2} + 2\)
\(\nu^{4}\)\(=\)\(5 \beta_{3} + 7 \beta_{2} + 7 \beta_{1} - 15\)
\(\nu^{5}\)\(=\)\(2 \beta_{5} - 16 \beta_{4} - 2 \beta_{3} - 29 \beta_{1} + 16\)

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
1.32001 + 1.32001i
0.432320 0.432320i
−1.75233 + 1.75233i
−1.75233 1.75233i
0.432320 + 0.432320i
1.32001 1.32001i
0 3.12489i 0 1.32001 + 1.80487i 0 1.00000i 0 −6.76491 0
449.2 0 1.76156i 0 0.432320 + 2.19388i 0 1.00000i 0 −0.103084 0
449.3 0 0.363328i 0 −1.75233 1.38900i 0 1.00000i 0 2.86799 0
449.4 0 0.363328i 0 −1.75233 + 1.38900i 0 1.00000i 0 2.86799 0
449.5 0 1.76156i 0 0.432320 2.19388i 0 1.00000i 0 −0.103084 0
449.6 0 3.12489i 0 1.32001 1.80487i 0 1.00000i 0 −6.76491 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.6
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.f 6
3.b odd 2 1 5040.2.t.y 6
4.b odd 2 1 280.2.g.b 6
5.b even 2 1 inner 560.2.g.f 6
5.c odd 4 1 2800.2.a.bq 3
5.c odd 4 1 2800.2.a.br 3
8.b even 2 1 2240.2.g.m 6
8.d odd 2 1 2240.2.g.l 6
12.b even 2 1 2520.2.t.g 6
15.d odd 2 1 5040.2.t.y 6
20.d odd 2 1 280.2.g.b 6
20.e even 4 1 1400.2.a.s 3
20.e even 4 1 1400.2.a.t 3
28.d even 2 1 1960.2.g.c 6
40.e odd 2 1 2240.2.g.l 6
40.f even 2 1 2240.2.g.m 6
60.h even 2 1 2520.2.t.g 6
140.c even 2 1 1960.2.g.c 6
140.j odd 4 1 9800.2.a.cd 3
140.j odd 4 1 9800.2.a.cg 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.g.b 6 4.b odd 2 1
280.2.g.b 6 20.d odd 2 1
560.2.g.f 6 1.a even 1 1 trivial
560.2.g.f 6 5.b even 2 1 inner
1400.2.a.s 3 20.e even 4 1
1400.2.a.t 3 20.e even 4 1
1960.2.g.c 6 28.d even 2 1
1960.2.g.c 6 140.c even 2 1
2240.2.g.l 6 8.d odd 2 1
2240.2.g.l 6 40.e odd 2 1
2240.2.g.m 6 8.b even 2 1
2240.2.g.m 6 40.f even 2 1
2520.2.t.g 6 12.b even 2 1
2520.2.t.g 6 60.h even 2 1
2800.2.a.bq 3 5.c odd 4 1
2800.2.a.br 3 5.c odd 4 1
5040.2.t.y 6 3.b odd 2 1
5040.2.t.y 6 15.d odd 2 1
9800.2.a.cd 3 140.j odd 4 1
9800.2.a.cg 3 140.j odd 4 1

Hecke kernels

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

\( T_{3}^{6} + 13 T_{3}^{4} + 32 T_{3}^{2} + 4 \)
\( T_{11}^{3} + 7 T_{11}^{2} + 8 T_{11} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 5 T^{2} + 11 T^{4} - 26 T^{6} + 99 T^{8} - 405 T^{10} + 729 T^{12} \)
$5$ \( 1 + 5 T^{2} + 8 T^{3} + 25 T^{4} + 125 T^{6} \)
$7$ \( ( 1 + T^{2} )^{3} \)
$11$ \( ( 1 + 7 T + 41 T^{2} + 146 T^{3} + 451 T^{4} + 847 T^{5} + 1331 T^{6} )^{2} \)
$13$ \( 1 - 9 T^{2} + 491 T^{4} - 2882 T^{6} + 82979 T^{8} - 257049 T^{10} + 4826809 T^{12} \)
$17$ \( 1 - 53 T^{2} + 1539 T^{4} - 31118 T^{6} + 444771 T^{8} - 4426613 T^{10} + 24137569 T^{12} \)
$19$ \( ( 1 - 4 T + 43 T^{2} - 160 T^{3} + 817 T^{4} - 1444 T^{5} + 6859 T^{6} )^{2} \)
$23$ \( 1 - 46 T^{2} + 1887 T^{4} - 43972 T^{6} + 998223 T^{8} - 12872686 T^{10} + 148035889 T^{12} \)
$29$ \( ( 1 + 3 T + 15 T^{2} + 66 T^{3} + 435 T^{4} + 2523 T^{5} + 24389 T^{6} )^{2} \)
$31$ \( ( 1 + 10 T + 101 T^{2} + 540 T^{3} + 3131 T^{4} + 9610 T^{5} + 29791 T^{6} )^{2} \)
$37$ \( ( 1 - 38 T^{2} + 1369 T^{4} )^{3} \)
$41$ \( ( 1 - 18 T + 191 T^{2} - 1388 T^{3} + 7831 T^{4} - 30258 T^{5} + 68921 T^{6} )^{2} \)
$43$ \( 1 - 178 T^{2} + 15575 T^{4} - 836124 T^{6} + 28798175 T^{8} - 608546578 T^{10} + 6321363049 T^{12} \)
$47$ \( 1 - 105 T^{2} + 6339 T^{4} - 285798 T^{6} + 14002851 T^{8} - 512366505 T^{10} + 10779215329 T^{12} \)
$53$ \( 1 - 130 T^{2} + 13655 T^{4} - 792060 T^{6} + 38356895 T^{8} - 1025762530 T^{10} + 22164361129 T^{12} \)
$59$ \( ( 1 - 6 T + 99 T^{2} - 752 T^{3} + 5841 T^{4} - 20886 T^{5} + 205379 T^{6} )^{2} \)
$61$ \( ( 1 - 24 T + 365 T^{2} - 3368 T^{3} + 22265 T^{4} - 89304 T^{5} + 226981 T^{6} )^{2} \)
$67$ \( 1 - 174 T^{2} + 20567 T^{4} - 1533188 T^{6} + 92325263 T^{8} - 3506295054 T^{10} + 90458382169 T^{12} \)
$71$ \( ( 1 + 4 T + 193 T^{2} + 504 T^{3} + 13703 T^{4} + 20164 T^{5} + 357911 T^{6} )^{2} \)
$73$ \( 1 - 346 T^{2} + 55487 T^{4} - 5172972 T^{6} + 295690223 T^{8} - 9825791386 T^{10} + 151334226289 T^{12} \)
$79$ \( ( 1 - 17 T + 205 T^{2} - 2138 T^{3} + 16195 T^{4} - 106097 T^{5} + 493039 T^{6} )^{2} \)
$83$ \( 1 - 70 T^{2} + 1179 T^{4} + 304148 T^{6} + 8122131 T^{8} - 3322082470 T^{10} + 326940373369 T^{12} \)
$89$ \( ( 1 + 95 T^{2} + 464 T^{3} + 8455 T^{4} + 704969 T^{6} )^{2} \)
$97$ \( 1 - 469 T^{2} + 98339 T^{4} - 12075534 T^{6} + 925271651 T^{8} - 41520232789 T^{10} + 832972004929 T^{12} \)
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