Properties

Label 560.2.ci.a
Level 560
Weight 2
Character orbit 560.ci
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.ci (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.47162251319\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{3} + ( -\zeta_{12} - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} + ( \zeta_{12} + 2 \zeta_{12}^{3} ) q^{7} + ( -2 - \zeta_{12} + \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{9} +O(q^{10})\) \( q + ( \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{3} + ( -\zeta_{12} - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} + ( \zeta_{12} + 2 \zeta_{12}^{3} ) q^{7} + ( -2 - \zeta_{12} + \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{9} + ( -1 - \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{11} + ( 2 + 2 \zeta_{12}^{3} ) q^{13} + ( 2 - 3 \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{15} + ( 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{17} + ( \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{19} + ( 3 - 2 \zeta_{12} - \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{21} + ( 3 - \zeta_{12} - 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{23} + ( -3 + 4 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{25} + ( -1 - 3 \zeta_{12} + 3 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{27} -3 \zeta_{12}^{3} q^{29} + ( 2 + 4 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{31} + ( -\zeta_{12} + \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{33} + ( -1 + 4 \zeta_{12} - 2 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{35} + ( 4 - 2 \zeta_{12} - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{37} + ( 2 - 2 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{39} + ( -2 + 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{41} + ( -1 - 5 \zeta_{12} + 5 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{43} + ( 3 + 3 \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{45} + ( -4 - 5 \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{47} + ( -8 + 5 \zeta_{12}^{2} ) q^{49} + 2 \zeta_{12}^{2} q^{51} + ( 5 - 5 \zeta_{12} - 5 \zeta_{12}^{2} ) q^{53} + ( 3 + 4 \zeta_{12} - 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{55} + ( 3 - 2 \zeta_{12} - 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{57} + ( -3 - 3 \zeta_{12} + 3 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{59} + ( -4 + 5 \zeta_{12} + 2 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{61} + ( -1 - 4 \zeta_{12} - 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{63} + ( 2 \zeta_{12} - 6 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{65} + ( 6 + 6 \zeta_{12} - \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{67} + ( 7 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{69} + ( -3 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{71} + ( -8 - 8 \zeta_{12} + 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{73} + ( 1 + 3 \zeta_{12} - 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{75} + ( -4 - 3 \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{77} + ( -2 + 5 \zeta_{12} + \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{79} + ( -2 + 5 \zeta_{12} + 2 \zeta_{12}^{2} - 10 \zeta_{12}^{3} ) q^{81} + ( 2 - 3 \zeta_{12} - 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{83} + ( 4 + 2 \zeta_{12} - 6 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{85} + ( -3 + 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{87} + ( 5 \zeta_{12} - 8 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{89} + ( -6 + 2 \zeta_{12} + 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{91} + ( 2 + 2 \zeta_{12} ) q^{93} + ( -3 + 3 \zeta_{12} + \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{95} + ( 3 + 8 \zeta_{12} - 8 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{97} + ( 2 - 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} - 4q^{5} - 6q^{9} + O(q^{10}) \) \( 4q + 2q^{3} - 4q^{5} - 6q^{9} - 2q^{11} + 8q^{13} + 6q^{15} + 4q^{17} - 2q^{19} + 10q^{21} + 4q^{23} - 6q^{25} + 2q^{27} + 12q^{31} + 2q^{33} - 8q^{35} + 12q^{37} + 12q^{39} + 6q^{43} + 14q^{45} - 18q^{47} - 22q^{49} + 4q^{51} + 10q^{53} + 8q^{55} + 8q^{57} - 6q^{59} - 12q^{61} - 8q^{63} - 12q^{65} + 22q^{67} + 28q^{69} - 12q^{71} - 24q^{73} - 4q^{75} - 18q^{77} - 6q^{79} - 4q^{81} + 2q^{83} + 4q^{85} - 12q^{87} - 16q^{89} - 20q^{91} + 8q^{93} - 10q^{95} - 4q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(\zeta_{12}^{2}\) \(\zeta_{12}^{3}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
0 0.500000 0.133975i 0 −1.86603 1.23205i 0 0.866025 + 2.50000i 0 −2.36603 + 1.36603i 0
33.1 0 0.500000 + 0.133975i 0 −1.86603 + 1.23205i 0 0.866025 2.50000i 0 −2.36603 1.36603i 0
257.1 0 0.500000 1.86603i 0 −0.133975 + 2.23205i 0 −0.866025 + 2.50000i 0 −0.633975 0.366025i 0
353.1 0 0.500000 + 1.86603i 0 −0.133975 2.23205i 0 −0.866025 2.50000i 0 −0.633975 + 0.366025i 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
35.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.ci.a 4
4.b odd 2 1 35.2.k.a 4
5.c odd 4 1 560.2.ci.b 4
7.d odd 6 1 560.2.ci.b 4
12.b even 2 1 315.2.bz.b 4
20.d odd 2 1 175.2.o.b 4
20.e even 4 1 35.2.k.b yes 4
20.e even 4 1 175.2.o.a 4
28.d even 2 1 245.2.l.a 4
28.f even 6 1 35.2.k.b yes 4
28.f even 6 1 245.2.f.b 4
28.g odd 6 1 245.2.f.a 4
28.g odd 6 1 245.2.l.b 4
35.k even 12 1 inner 560.2.ci.a 4
60.l odd 4 1 315.2.bz.a 4
84.j odd 6 1 315.2.bz.a 4
140.j odd 4 1 245.2.l.b 4
140.s even 6 1 175.2.o.a 4
140.w even 12 1 245.2.f.b 4
140.w even 12 1 245.2.l.a 4
140.x odd 12 1 35.2.k.a 4
140.x odd 12 1 175.2.o.b 4
140.x odd 12 1 245.2.f.a 4
420.br even 12 1 315.2.bz.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.k.a 4 4.b odd 2 1
35.2.k.a 4 140.x odd 12 1
35.2.k.b yes 4 20.e even 4 1
35.2.k.b yes 4 28.f even 6 1
175.2.o.a 4 20.e even 4 1
175.2.o.a 4 140.s even 6 1
175.2.o.b 4 20.d odd 2 1
175.2.o.b 4 140.x odd 12 1
245.2.f.a 4 28.g odd 6 1
245.2.f.a 4 140.x odd 12 1
245.2.f.b 4 28.f even 6 1
245.2.f.b 4 140.w even 12 1
245.2.l.a 4 28.d even 2 1
245.2.l.a 4 140.w even 12 1
245.2.l.b 4 28.g odd 6 1
245.2.l.b 4 140.j odd 4 1
315.2.bz.a 4 60.l odd 4 1
315.2.bz.a 4 84.j odd 6 1
315.2.bz.b 4 12.b even 2 1
315.2.bz.b 4 420.br even 12 1
560.2.ci.a 4 1.a even 1 1 trivial
560.2.ci.a 4 35.k even 12 1 inner
560.2.ci.b 4 5.c odd 4 1
560.2.ci.b 4 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 2 T_{3}^{3} + 5 T_{3}^{2} - 4 T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(560, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 2 T + 5 T^{2} - 10 T^{3} + 16 T^{4} - 30 T^{5} + 45 T^{6} - 54 T^{7} + 81 T^{8} \)
$5$ \( 1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4} \)
$7$ \( 1 + 11 T^{2} + 49 T^{4} \)
$11$ \( 1 + 2 T - 16 T^{2} - 4 T^{3} + 235 T^{4} - 44 T^{5} - 1936 T^{6} + 2662 T^{7} + 14641 T^{8} \)
$13$ \( ( 1 - 4 T + 8 T^{2} - 52 T^{3} + 169 T^{4} )^{2} \)
$17$ \( 1 - 4 T + 20 T^{2} - 100 T^{3} + 271 T^{4} - 1700 T^{5} + 5780 T^{6} - 19652 T^{7} + 83521 T^{8} \)
$19$ \( 1 + 2 T - 32 T^{2} - 4 T^{3} + 859 T^{4} - 76 T^{5} - 11552 T^{6} + 13718 T^{7} + 130321 T^{8} \)
$23$ \( 1 - 4 T + 53 T^{2} - 244 T^{3} + 1588 T^{4} - 5612 T^{5} + 28037 T^{6} - 48668 T^{7} + 279841 T^{8} \)
$29$ \( ( 1 - 49 T^{2} + 841 T^{4} )^{2} \)
$31$ \( 1 - 12 T + 106 T^{2} - 696 T^{3} + 3891 T^{4} - 21576 T^{5} + 101866 T^{6} - 357492 T^{7} + 923521 T^{8} \)
$37$ \( 1 - 12 T + 72 T^{2} - 288 T^{3} + 983 T^{4} - 10656 T^{5} + 98568 T^{6} - 607836 T^{7} + 1874161 T^{8} \)
$41$ \( 1 - 122 T^{2} + 6651 T^{4} - 205082 T^{6} + 2825761 T^{8} \)
$43$ \( 1 - 6 T + 18 T^{2} - 60 T^{3} - 889 T^{4} - 2580 T^{5} + 33282 T^{6} - 477042 T^{7} + 3418801 T^{8} \)
$47$ \( 1 + 18 T + 90 T^{2} - 528 T^{3} - 8377 T^{4} - 24816 T^{5} + 198810 T^{6} + 1868814 T^{7} + 4879681 T^{8} \)
$53$ \( ( 1 - 14 T + 143 T^{2} - 742 T^{3} + 2809 T^{4} )( 1 + 4 T - 37 T^{2} + 212 T^{3} + 2809 T^{4} ) \)
$59$ \( 1 + 6 T - 64 T^{2} - 108 T^{3} + 4395 T^{4} - 6372 T^{5} - 222784 T^{6} + 1232274 T^{7} + 12117361 T^{8} \)
$61$ \( 1 + 12 T + 157 T^{2} + 1308 T^{3} + 11088 T^{4} + 79788 T^{5} + 584197 T^{6} + 2723772 T^{7} + 13845841 T^{8} \)
$67$ \( 1 - 22 T + 137 T^{2} + 834 T^{3} - 16648 T^{4} + 55878 T^{5} + 614993 T^{6} - 6616786 T^{7} + 20151121 T^{8} \)
$71$ \( ( 1 + 6 T + 148 T^{2} + 426 T^{3} + 5041 T^{4} )^{2} \)
$73$ \( 1 + 24 T + 144 T^{2} - 1752 T^{3} - 31057 T^{4} - 127896 T^{5} + 767376 T^{6} + 9336408 T^{7} + 28398241 T^{8} \)
$79$ \( 1 + 6 T + 148 T^{2} + 816 T^{3} + 13203 T^{4} + 64464 T^{5} + 923668 T^{6} + 2958234 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 2 T + 2 T^{2} - 140 T^{3} + 9631 T^{4} - 11620 T^{5} + 13778 T^{6} - 1143574 T^{7} + 47458321 T^{8} \)
$89$ \( ( 1 + 16 T + 89 T^{2} )^{2}( 1 - 16 T + 167 T^{2} - 1424 T^{3} + 7921 T^{4} ) \)
$97$ \( 1 + 4 T + 8 T^{2} + 12 T^{3} - 8818 T^{4} + 1164 T^{5} + 75272 T^{6} + 3650692 T^{7} + 88529281 T^{8} \)
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