Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 570.c (of order \(2\), degree \(1\), minimal)

Newform invariants

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

$q$-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.

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

Character values

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

\(n\) \(191\) \(211\) \(457\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
569.1
0.965926 0.258819i
−0.258819 + 0.965926i
−0.965926 + 0.258819i
0.258819 0.965926i
0.965926 + 0.258819i
−0.258819 0.965926i
−0.965926 0.258819i
0.258819 + 0.965926i
1.00000i −1.41421 + 1.00000i −1.00000 −1.73205 + 1.41421i 1.00000 + 1.41421i 2.44949i 1.00000i 1.00000 2.82843i 1.41421 + 1.73205i
569.2 1.00000i −1.41421 + 1.00000i −1.00000 1.73205 + 1.41421i 1.00000 + 1.41421i 2.44949i 1.00000i 1.00000 2.82843i 1.41421 1.73205i
569.3 1.00000i 1.41421 + 1.00000i −1.00000 −1.73205 1.41421i 1.00000 1.41421i 2.44949i 1.00000i 1.00000 + 2.82843i −1.41421 + 1.73205i
569.4 1.00000i 1.41421 + 1.00000i −1.00000 1.73205 1.41421i 1.00000 1.41421i 2.44949i 1.00000i 1.00000 + 2.82843i −1.41421 1.73205i
569.5 1.00000i −1.41421 1.00000i −1.00000 −1.73205 1.41421i 1.00000 1.41421i 2.44949i 1.00000i 1.00000 + 2.82843i 1.41421 1.73205i
569.6 1.00000i −1.41421 1.00000i −1.00000 1.73205 1.41421i 1.00000 1.41421i 2.44949i 1.00000i 1.00000 + 2.82843i 1.41421 + 1.73205i
569.7 1.00000i 1.41421 1.00000i −1.00000 −1.73205 + 1.41421i 1.00000 + 1.41421i 2.44949i 1.00000i 1.00000 2.82843i −1.41421 1.73205i
569.8 1.00000i 1.41421 1.00000i −1.00000 1.73205 + 1.41421i 1.00000 + 1.41421i 2.44949i 1.00000i 1.00000 2.82843i −1.41421 + 1.73205i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 569.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner
19.b odd 2 1 inner
57.d even 2 1 inner
95.d odd 2 1 inner
285.b even 2 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(570, [\chi])\):

\( T_{7}^{2} + 6 \)
\( T_{11}^{2} + 2 \)
\( T_{29}^{2} - 6 \)
\( T_{37}^{2} - 18 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{4} \)
$3$ \( ( 9 - 2 T^{2} + T^{4} )^{2} \)
$5$ \( ( 25 - 2 T^{2} + T^{4} )^{2} \)
$7$ \( ( 6 + T^{2} )^{4} \)
$11$ \( ( 2 + T^{2} )^{4} \)
$13$ \( ( -18 + T^{2} )^{4} \)
$17$ \( ( -48 + T^{2} )^{4} \)
$19$ \( ( 19 - 8 T + T^{2} )^{4} \)
$23$ \( ( -12 + T^{2} )^{4} \)
$29$ \( ( -6 + T^{2} )^{4} \)
$31$ \( ( 48 + T^{2} )^{4} \)
$37$ \( ( -18 + T^{2} )^{4} \)
$41$ \( ( -150 + T^{2} )^{4} \)
$43$ \( ( 54 + T^{2} )^{4} \)
$47$ \( ( -12 + T^{2} )^{4} \)
$53$ \( ( 36 + T^{2} )^{4} \)
$59$ \( ( -24 + T^{2} )^{4} \)
$61$ \( ( 8 + T )^{8} \)
$67$ \( T^{8} \)
$71$ \( ( -24 + T^{2} )^{4} \)
$73$ \( ( 216 + T^{2} )^{4} \)
$79$ \( ( 108 + T^{2} )^{4} \)
$83$ \( ( -108 + T^{2} )^{4} \)
$89$ \( ( -6 + T^{2} )^{4} \)
$97$ \( ( -162 + T^{2} )^{4} \)
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