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
Defining polynomial: \( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} + (\beta_{5} + \beta_{2}) q^{5} + \beta_{4} q^{7} + (\beta_{3} - \beta_{2} - \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{3} + (\beta_{5} + \beta_{2}) q^{5} + \beta_{4} q^{7} + (\beta_{3} - \beta_{2} - \beta_1 - 1) q^{9} + (\beta_{3} + \beta_{2} + \beta_1 - 2) q^{11} + (\beta_{5} + 2 \beta_{4} + 2 \beta_{2} - 2 \beta_1) q^{13} + ( - \beta_{5} + \beta_{4} + \beta_{2} + \beta_1 + 3) q^{15} + ( - \beta_{5} + \beta_{2} - \beta_1) q^{17} + (2 \beta_{3} + \beta_{2} + \beta_1 + 2) q^{19} + \beta_{3} q^{21} + (2 \beta_{4} + 2 \beta_{2} - 2 \beta_1) q^{23} + (\beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_{2} - 2) q^{25} + (\beta_{5} - 6 \beta_{4} - \beta_{2} + \beta_1) q^{27} + (3 \beta_{3} + 3 \beta_{2} + 3 \beta_1) q^{29} + ( - 2 \beta_{3} - 4) q^{31} + (\beta_{5} - 2 \beta_{4} - \beta_{2} + \beta_1) q^{33} + ( - \beta_{3} - \beta_1) q^{35} - 6 \beta_{4} q^{37} + (5 \beta_{3} + \beta_{2} + \beta_1) q^{39} + (2 \beta_{2} + 2 \beta_1 + 6) q^{41} + ( - 2 \beta_{2} + 2 \beta_1) q^{43} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - \beta_1 - 4) q^{45} + (3 \beta_{5} - 2 \beta_{4} + 3 \beta_{2} - 3 \beta_1) q^{47} - q^{49} + (3 \beta_{3} - \beta_{2} - \beta_1 - 6) q^{51} + ( - 4 \beta_{5} + 2 \beta_{4} - 2 \beta_{2} + 2 \beta_1) q^{53} + ( - \beta_{5} + 4 \beta_{4} - 3 \beta_{2} - \beta_1 + 2) q^{55} + ( - 2 \beta_{5} - 6 \beta_{4} - 2 \beta_{2} + 2 \beta_1) q^{57} + ( - 3 \beta_{2} - 3 \beta_1 + 2) q^{59} + ( - \beta_{2} - \beta_1 + 8) q^{61} + (\beta_{5} - \beta_{4} - \beta_{2} + \beta_1) q^{63} + ( - \beta_{5} + 5 \beta_{4} - 4 \beta_{3} - \beta_{2} + \beta_1 - 5) q^{65} + ( - 2 \beta_{5} - 4 \beta_{4} + 2 \beta_{2} - 2 \beta_1) q^{67} + (6 \beta_{3} - 4) q^{69} + ( - 2 \beta_{3} - 2) q^{71} + ( - 2 \beta_{4} - 2 \beta_{2} + 2 \beta_1) q^{73} + (3 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 6) q^{75} + (\beta_{5} - 2 \beta_{4} + \beta_{2} - \beta_1) q^{77} + ( - 5 \beta_{3} - \beta_{2} - \beta_1 + 4) q^{79} + ( - 6 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 3) q^{81} + (6 \beta_{5} + 6 \beta_{4} + \beta_{2} - \beta_1) q^{83} + ( - 2 \beta_{5} + 4 \beta_{4} - \beta_{3} + \beta_{2} + 3 \beta_1 + 2) q^{85} + ( - 3 \beta_{5} - 6 \beta_{4} - 3 \beta_{2} + 3 \beta_1) q^{87} + ( - 6 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{89} + ( - \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{91} + (2 \beta_{5} + 8 \beta_{4} + 2 \beta_{2} - 2 \beta_1) q^{93} + (3 \beta_{5} + 7 \beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 1) q^{95} + (3 \beta_{5} + 4 \beta_{4} + \beta_{2} - \beta_1) q^{97} + ( - 2 \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 8 q^{9} - 14 q^{11} + 18 q^{15} + 8 q^{19} - 2 q^{21} - 10 q^{25} - 6 q^{29} - 20 q^{31} + 2 q^{35} - 10 q^{39} + 36 q^{41} - 28 q^{45} - 6 q^{49} - 42 q^{51} + 12 q^{55} + 12 q^{59} + 48 q^{61} - 22 q^{65} - 36 q^{69} - 8 q^{71} + 40 q^{75} + 34 q^{79} + 30 q^{81} + 14 q^{85} - 10 q^{91} + 4 q^{95} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{5} - 2\nu^{4} - 25\nu^{3} + 10\nu^{2} - 121\nu + 100 ) / 121 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{5} + 27\nu^{4} + 35\nu^{3} - 14\nu^{2} + 223 ) / 121 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -25\nu^{5} - 10\nu^{4} - 4\nu^{3} + 50\nu^{2} - 605\nu + 258 ) / 242 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -65\nu^{5} - 26\nu^{4} + 38\nu^{3} + 372\nu^{2} - 1331\nu + 574 ) / 242 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - 3\beta_{4} + \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{4} - 5\beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{3} + 7\beta_{2} + 7\beta _1 - 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{5} - 16\beta_{4} - 2\beta_{3} - 29\beta _1 + 16 \) Copy content Toggle raw display

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 \(S_{2}^{\mathrm{new}}(560, [\chi])\):

\( T_{3}^{6} + 13T_{3}^{4} + 32T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{3} + 7T_{11}^{2} + 8T_{11} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 13 T^{4} + 32 T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{6} + 5 T^{4} + 8 T^{3} + 25 T^{2} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$11$ \( (T^{3} + 7 T^{2} + 8 T - 8)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 69 T^{4} + 1544 T^{2} + \cdots + 11236 \) Copy content Toggle raw display
$17$ \( T^{6} + 49 T^{4} + 536 T^{2} + \cdots + 400 \) Copy content Toggle raw display
$19$ \( (T^{3} - 4 T^{2} - 14 T - 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 92 T^{4} + 2416 T^{2} + \cdots + 18496 \) Copy content Toggle raw display
$29$ \( (T^{3} + 3 T^{2} - 72 T - 108)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 10 T^{2} + 8 T - 80)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 36)^{3} \) Copy content Toggle raw display
$41$ \( (T^{3} - 18 T^{2} + 68 T + 88)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 80 T^{4} + 1600 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$47$ \( T^{6} + 177 T^{4} + 6480 T^{2} + \cdots + 53824 \) Copy content Toggle raw display
$53$ \( T^{6} + 188 T^{4} + 11376 T^{2} + \cdots + 222784 \) Copy content Toggle raw display
$59$ \( (T^{3} - 6 T^{2} - 78 T - 44)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 24 T^{2} + 182 T - 440)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 228 T^{4} + 14336 T^{2} + \cdots + 262144 \) Copy content Toggle raw display
$71$ \( (T^{3} + 4 T^{2} - 20 T - 64)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 92 T^{4} + 2416 T^{2} + \cdots + 18496 \) Copy content Toggle raw display
$79$ \( (T^{3} - 17 T^{2} - 32 T + 548)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 428 T^{4} + 39940 T^{2} + \cdots + 678976 \) Copy content Toggle raw display
$89$ \( (T^{3} - 172 T + 464)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 113 T^{4} + 1048 T^{2} + \cdots + 1936 \) Copy content Toggle raw display
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