Properties

Label 570.2.q.a
Level $570$
Weight $2$
Character orbit 570.q
Analytic conductor $4.551$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 570.q (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.55147291521\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{2} + ( \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{3} + ( 1 - \zeta_{24}^{4} ) q^{4} + ( -\zeta_{24} + \zeta_{24}^{2} + \zeta_{24}^{4} + 2 \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{5} + ( 1 - \zeta_{24}^{4} ) q^{6} + 3 \zeta_{24}^{6} q^{7} -\zeta_{24}^{6} q^{8} + ( 1 - \zeta_{24}^{4} ) q^{9} +O(q^{10})\) \( q + ( \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{2} + ( \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{3} + ( 1 - \zeta_{24}^{4} ) q^{4} + ( -\zeta_{24} + \zeta_{24}^{2} + \zeta_{24}^{4} + 2 \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{5} + ( 1 - \zeta_{24}^{4} ) q^{6} + 3 \zeta_{24}^{6} q^{7} -\zeta_{24}^{6} q^{8} + ( 1 - \zeta_{24}^{4} ) q^{9} + ( 1 + \zeta_{24}^{2} + \zeta_{24}^{3} - \zeta_{24}^{4} + \zeta_{24}^{7} ) q^{10} + 2 q^{11} -\zeta_{24}^{6} q^{12} + ( 2 \zeta_{24} - \zeta_{24}^{2} + \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{13} + 3 \zeta_{24}^{4} q^{14} + ( 1 + \zeta_{24}^{2} + \zeta_{24}^{3} - \zeta_{24}^{4} + \zeta_{24}^{7} ) q^{15} -\zeta_{24}^{4} q^{16} + ( -2 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{17} -\zeta_{24}^{6} q^{18} + ( 1 + 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{19} + ( 1 + \zeta_{24} + \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{20} + 3 \zeta_{24}^{4} q^{21} + ( 2 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{22} + ( 2 \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{23} -\zeta_{24}^{4} q^{24} + ( -4 \zeta_{24} - \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{25} + ( -1 + \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{26} -\zeta_{24}^{6} q^{27} + 3 \zeta_{24}^{2} q^{28} + ( -2 + 6 \zeta_{24} - 3 \zeta_{24}^{3} + 2 \zeta_{24}^{4} - 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{29} + ( 1 + \zeta_{24} + \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{30} + ( -3 + 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{31} -\zeta_{24}^{2} q^{32} + ( 2 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{33} + ( -4 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{34} + ( -3 \zeta_{24}^{2} - 6 \zeta_{24}^{3} + 3 \zeta_{24}^{4} + 3 \zeta_{24}^{6} + 3 \zeta_{24}^{7} ) q^{35} -\zeta_{24}^{4} q^{36} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 7 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{37} + ( -3 \zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{6} - 3 \zeta_{24}^{7} ) q^{38} + ( -1 + \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{39} + ( \zeta_{24}^{2} + 2 \zeta_{24}^{3} - \zeta_{24}^{4} - \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{40} + ( -\zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{41} + 3 \zeta_{24}^{2} q^{42} + ( -\zeta_{24} - 3 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 3 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{43} + ( 2 - 2 \zeta_{24}^{4} ) q^{44} + ( 1 + \zeta_{24} + \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{45} + ( \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{46} + ( -2 \zeta_{24} - 6 \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{47} -\zeta_{24}^{2} q^{48} -2 q^{49} + ( -1 + 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{50} + ( -4 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{51} + ( \zeta_{24} - \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{52} + ( -4 \zeta_{24} - 8 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{53} -\zeta_{24}^{4} q^{54} + ( -2 \zeta_{24} + 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} + 4 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{55} + 3 q^{56} + ( -3 \zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{6} - 3 \zeta_{24}^{7} ) q^{57} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 2 \zeta_{24}^{6} - 6 \zeta_{24}^{7} ) q^{58} + ( 4 \zeta_{24} - 8 \zeta_{24}^{3} + 2 \zeta_{24}^{4} - 8 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{59} + ( \zeta_{24}^{2} + 2 \zeta_{24}^{3} - \zeta_{24}^{4} - \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{60} + ( 3 + 6 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{4} - 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{61} + ( 2 \zeta_{24} - 3 \zeta_{24}^{2} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 3 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{62} + 3 \zeta_{24}^{2} q^{63} - q^{64} + ( -4 + 2 \zeta_{24} + \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{65} + ( 2 - 2 \zeta_{24}^{4} ) q^{66} + ( 2 \zeta_{24} + \zeta_{24}^{2} + \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{67} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{68} + ( \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{69} + ( -3 - 6 \zeta_{24} + 3 \zeta_{24}^{2} + 3 \zeta_{24}^{4} + 3 \zeta_{24}^{5} ) q^{70} + ( -\zeta_{24} + 2 \zeta_{24}^{3} + 6 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{71} -\zeta_{24}^{2} q^{72} + ( -2 \zeta_{24} + 7 \zeta_{24}^{2} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 7 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{73} + ( \zeta_{24} - 2 \zeta_{24}^{3} - 7 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{74} + ( -1 + 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{75} + ( 1 - 3 \zeta_{24}^{3} - \zeta_{24}^{4} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{76} + 6 \zeta_{24}^{6} q^{77} + ( \zeta_{24} - \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{78} + ( 4 \zeta_{24} - 8 \zeta_{24}^{3} + 7 \zeta_{24}^{4} - 8 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{79} + ( 1 + 2 \zeta_{24} - \zeta_{24}^{2} - \zeta_{24}^{4} - \zeta_{24}^{5} ) q^{80} -\zeta_{24}^{4} q^{81} + ( 2 \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{82} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 6 \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{83} + 3 q^{84} + ( 6 - 8 \zeta_{24} - 6 \zeta_{24}^{2} - 6 \zeta_{24}^{4} + 4 \zeta_{24}^{5} ) q^{85} + ( -3 - 2 \zeta_{24} + \zeta_{24}^{3} + 3 \zeta_{24}^{4} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{86} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 2 \zeta_{24}^{6} - 6 \zeta_{24}^{7} ) q^{87} -2 \zeta_{24}^{6} q^{88} + ( -14 - 2 \zeta_{24} + \zeta_{24}^{3} + 14 \zeta_{24}^{4} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{89} + ( \zeta_{24}^{2} + 2 \zeta_{24}^{3} - \zeta_{24}^{4} - \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{90} + ( 3 - 6 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{4} + 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{91} + ( \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{92} + ( 2 \zeta_{24} - 3 \zeta_{24}^{2} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 3 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{93} + ( -6 - \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{94} + ( 6 - \zeta_{24} + 4 \zeta_{24}^{2} - 2 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} - 6 \zeta_{24}^{7} ) q^{95} - q^{96} + ( 6 \zeta_{24} - 4 \zeta_{24}^{2} + 12 \zeta_{24}^{3} - 12 \zeta_{24}^{5} + 4 \zeta_{24}^{6} - 6 \zeta_{24}^{7} ) q^{97} + ( -2 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{98} + ( 2 - 2 \zeta_{24}^{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{4} + 4q^{5} + 4q^{6} + 4q^{9} + O(q^{10}) \) \( 8q + 4q^{4} + 4q^{5} + 4q^{6} + 4q^{9} + 4q^{10} + 16q^{11} + 12q^{14} + 4q^{15} - 4q^{16} + 8q^{19} + 8q^{20} + 12q^{21} - 4q^{24} - 8q^{26} - 8q^{29} + 8q^{30} - 24q^{31} + 12q^{35} - 4q^{36} - 8q^{39} - 4q^{40} + 8q^{44} + 8q^{45} - 16q^{49} - 8q^{50} - 4q^{54} + 8q^{55} + 24q^{56} + 8q^{59} - 4q^{60} + 12q^{61} - 8q^{64} - 32q^{65} + 8q^{66} - 12q^{70} + 24q^{71} - 28q^{74} - 8q^{75} + 4q^{76} + 28q^{79} + 4q^{80} - 4q^{81} + 24q^{84} + 24q^{85} - 12q^{86} - 56q^{89} - 4q^{90} + 12q^{91} - 48q^{94} + 40q^{95} - 8q^{96} + 8q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/570\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(211\) \(457\)
\(\chi(n)\) \(1\) \(-1 + \zeta_{24}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.258819 0.965926i
0.258819 + 0.965926i
0.965926 0.258819i
−0.965926 + 0.258819i
−0.258819 + 0.965926i
0.258819 0.965926i
0.965926 + 0.258819i
−0.965926 0.258819i
−0.866025 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i −2.03906 0.917738i 0.500000 + 0.866025i 3.00000i 1.00000i 0.500000 + 0.866025i 1.30701 + 1.81431i
49.2 −0.866025 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i 1.30701 1.81431i 0.500000 + 0.866025i 3.00000i 1.00000i 0.500000 + 0.866025i −2.03906 + 0.917738i
49.3 0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 0.917738 2.03906i 0.500000 + 0.866025i 3.00000i 1.00000i 0.500000 + 0.866025i 1.81431 1.30701i
49.4 0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 1.81431 + 1.30701i 0.500000 + 0.866025i 3.00000i 1.00000i 0.500000 + 0.866025i 0.917738 + 2.03906i
349.1 −0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i −2.03906 + 0.917738i 0.500000 0.866025i 3.00000i 1.00000i 0.500000 0.866025i 1.30701 1.81431i
349.2 −0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i 1.30701 + 1.81431i 0.500000 0.866025i 3.00000i 1.00000i 0.500000 0.866025i −2.03906 0.917738i
349.3 0.866025 0.500000i 0.866025 0.500000i 0.500000 0.866025i 0.917738 + 2.03906i 0.500000 0.866025i 3.00000i 1.00000i 0.500000 0.866025i 1.81431 + 1.30701i
349.4 0.866025 0.500000i 0.866025 0.500000i 0.500000 0.866025i 1.81431 1.30701i 0.500000 0.866025i 3.00000i 1.00000i 0.500000 0.866025i 0.917738 2.03906i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 349.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.q.a 8
3.b odd 2 1 1710.2.t.a 8
5.b even 2 1 inner 570.2.q.a 8
15.d odd 2 1 1710.2.t.a 8
19.c even 3 1 inner 570.2.q.a 8
57.h odd 6 1 1710.2.t.a 8
95.i even 6 1 inner 570.2.q.a 8
285.n odd 6 1 1710.2.t.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.q.a 8 1.a even 1 1 trivial
570.2.q.a 8 5.b even 2 1 inner
570.2.q.a 8 19.c even 3 1 inner
570.2.q.a 8 95.i even 6 1 inner
1710.2.t.a 8 3.b odd 2 1
1710.2.t.a 8 15.d odd 2 1
1710.2.t.a 8 57.h odd 6 1
1710.2.t.a 8 285.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(570, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$3$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$5$ \( 625 - 500 T + 200 T^{2} + 40 T^{3} - 41 T^{4} + 8 T^{5} + 8 T^{6} - 4 T^{7} + T^{8} \)
$7$ \( ( 9 + T^{2} )^{4} \)
$11$ \( ( -2 + T )^{8} \)
$13$ \( 625 - 350 T^{2} + 171 T^{4} - 14 T^{6} + T^{8} \)
$17$ \( ( 576 - 24 T^{2} + T^{4} )^{2} \)
$19$ \( ( 19 - 2 T + T^{2} )^{4} \)
$23$ \( ( 36 - 6 T^{2} + T^{4} )^{2} \)
$29$ \( ( 2500 - 200 T + 66 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$31$ \( ( -15 + 6 T + T^{2} )^{4} \)
$37$ \( ( 1849 + 110 T^{2} + T^{4} )^{2} \)
$41$ \( ( 36 + 6 T^{2} + T^{4} )^{2} \)
$43$ \( 81 - 270 T^{2} + 891 T^{4} - 30 T^{6} + T^{8} \)
$47$ \( 810000 - 75600 T^{2} + 6156 T^{4} - 84 T^{6} + T^{8} \)
$53$ \( 2560000 - 281600 T^{2} + 29376 T^{4} - 176 T^{6} + T^{8} \)
$59$ \( ( 8464 + 368 T + 108 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$61$ \( ( 2025 + 270 T + 81 T^{2} - 6 T^{3} + T^{4} )^{2} \)
$67$ \( 625 - 350 T^{2} + 171 T^{4} - 14 T^{6} + T^{8} \)
$71$ \( ( 900 - 360 T + 114 T^{2} - 12 T^{3} + T^{4} )^{2} \)
$73$ \( 390625 - 91250 T^{2} + 20691 T^{4} - 146 T^{6} + T^{8} \)
$79$ \( ( 2209 + 658 T + 243 T^{2} - 14 T^{3} + T^{4} )^{2} \)
$83$ \( ( 324 + 180 T^{2} + T^{4} )^{2} \)
$89$ \( ( 36100 + 5320 T + 594 T^{2} + 28 T^{3} + T^{4} )^{2} \)
$97$ \( 1600000000 - 18560000 T^{2} + 175296 T^{4} - 464 T^{6} + T^{8} \)
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