Properties

Label 570.2.v.b
Level $570$
Weight $2$
Character orbit 570.v
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.v (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.55147291521\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(191\) \(211\) \(457\)
\(\chi(n)\) \(-1\) \(-1 + \zeta_{24}^{4}\) \(-\zeta_{24}^{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
83.1
0.965926 + 0.258819i
−0.965926 0.258819i
0.258819 0.965926i
−0.258819 + 0.965926i
0.258819 + 0.965926i
−0.258819 0.965926i
0.965926 0.258819i
−0.965926 + 0.258819i
−0.258819 + 0.965926i 1.72474 + 0.158919i −0.866025 0.500000i −1.67303 1.48356i −0.599900 + 1.62484i −2.12132 + 2.12132i 0.707107 0.707107i 2.94949 + 0.548188i 1.86603 1.23205i
83.2 0.258819 0.965926i −0.724745 + 1.57313i −0.866025 0.500000i 1.67303 + 1.48356i 1.33195 + 1.10721i 2.12132 2.12132i −0.707107 + 0.707107i −1.94949 2.28024i 1.86603 1.23205i
197.1 −0.965926 0.258819i 1.72474 + 0.158919i 0.866025 + 0.500000i 0.448288 + 2.19067i −1.62484 0.599900i 2.12132 + 2.12132i −0.707107 0.707107i 2.94949 + 0.548188i 0.133975 2.23205i
197.2 0.965926 + 0.258819i −0.724745 + 1.57313i 0.866025 + 0.500000i −0.448288 2.19067i −1.10721 + 1.33195i −2.12132 2.12132i 0.707107 + 0.707107i −1.94949 2.28024i 0.133975 2.23205i
353.1 −0.965926 + 0.258819i 1.72474 0.158919i 0.866025 0.500000i 0.448288 2.19067i −1.62484 + 0.599900i 2.12132 2.12132i −0.707107 + 0.707107i 2.94949 0.548188i 0.133975 + 2.23205i
353.2 0.965926 0.258819i −0.724745 1.57313i 0.866025 0.500000i −0.448288 + 2.19067i −1.10721 1.33195i −2.12132 + 2.12132i 0.707107 0.707107i −1.94949 + 2.28024i 0.133975 + 2.23205i
467.1 −0.258819 0.965926i 1.72474 0.158919i −0.866025 + 0.500000i −1.67303 + 1.48356i −0.599900 1.62484i −2.12132 2.12132i 0.707107 + 0.707107i 2.94949 0.548188i 1.86603 + 1.23205i
467.2 0.258819 + 0.965926i −0.724745 1.57313i −0.866025 + 0.500000i 1.67303 1.48356i 1.33195 1.10721i 2.12132 + 2.12132i −0.707107 0.707107i −1.94949 + 2.28024i 1.86603 + 1.23205i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 467.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.e even 4 1 inner
19.c even 3 1 inner
285.v even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.v.b yes 8
3.b odd 2 1 570.2.v.a 8
5.c odd 4 1 570.2.v.a 8
15.e even 4 1 inner 570.2.v.b yes 8
19.c even 3 1 inner 570.2.v.b yes 8
57.h odd 6 1 570.2.v.a 8
95.m odd 12 1 570.2.v.a 8
285.v even 12 1 inner 570.2.v.b yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.v.a 8 3.b odd 2 1
570.2.v.a 8 5.c odd 4 1
570.2.v.a 8 57.h odd 6 1
570.2.v.a 8 95.m odd 12 1
570.2.v.b yes 8 1.a even 1 1 trivial
570.2.v.b yes 8 15.e even 4 1 inner
570.2.v.b yes 8 19.c even 3 1 inner
570.2.v.b yes 8 285.v even 12 1 inner

Hecke kernels

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

\( T_{7}^{4} + 81 \)
\(T_{17}^{8} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{4} + T^{8} \)
$3$ \( ( 9 - 6 T + T^{2} - 2 T^{3} + T^{4} )^{2} \)
$5$ \( 625 + 200 T^{2} + 39 T^{4} + 8 T^{6} + T^{8} \)
$7$ \( ( 81 + T^{4} )^{2} \)
$11$ \( ( 1 + 34 T^{2} + T^{4} )^{2} \)
$13$ \( ( 64 - 32 T + 8 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$17$ \( 38416 - 32928 T + 14112 T^{2} - 7392 T^{3} + 2972 T^{4} - 528 T^{5} + 72 T^{6} - 12 T^{7} + T^{8} \)
$19$ \( ( 361 - 37 T^{2} + T^{4} )^{2} \)
$23$ \( 6561 - 81 T^{4} + T^{8} \)
$29$ \( ( 784 + 336 T + 116 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$31$ \( ( -14 - 4 T + T^{2} )^{4} \)
$37$ \( ( 81 + T^{4} )^{2} \)
$41$ \( ( 81 - 9 T^{2} + T^{4} )^{2} \)
$43$ \( ( 324 + 108 T + 18 T^{2} + 6 T^{3} + T^{4} )^{2} \)
$47$ \( 21381376 - 7546368 T + 1331712 T^{2} - 248064 T^{3} + 39152 T^{4} - 3648 T^{5} + 288 T^{6} - 24 T^{7} + T^{8} \)
$53$ \( 25411681 - 8589864 T + 1451808 T^{2} - 248784 T^{3} + 37007 T^{4} - 3504 T^{5} + 288 T^{6} - 24 T^{7} + T^{8} \)
$59$ \( ( 1024 + 32 T^{2} + T^{4} )^{2} \)
$61$ \( ( 196 + 56 T + 30 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$67$ \( ( 1024 - 256 T + 32 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$71$ \( 406586896 - 5887888 T^{2} + 65100 T^{4} - 292 T^{6} + T^{8} \)
$73$ \( 252047376 - 24004512 T + 1143072 T^{2} - 489888 T^{3} + 7452 T^{4} + 3888 T^{5} + 72 T^{6} + 12 T^{7} + T^{8} \)
$79$ \( 21381376 - 702848 T^{2} + 18480 T^{4} - 152 T^{6} + T^{8} \)
$83$ \( ( 18 - 6 T + T^{2} )^{4} \)
$89$ \( ( 5329 + 1314 T + 251 T^{2} + 18 T^{3} + T^{4} )^{2} \)
$97$ \( 1679616 + 1119744 T + 373248 T^{2} + 186624 T^{3} + 60912 T^{4} + 5184 T^{5} + 288 T^{6} + 24 T^{7} + T^{8} \)
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