Properties

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

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(4.55147291521\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.7278137344.1
Defining polynomial: \(x^{8} - 2 x^{7} + x^{6} + 6 x^{5} - 20 x^{4} + 18 x^{3} + 9 x^{2} - 54 x + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} + x^{6} + 6 x^{5} - 20 x^{4} + 18 x^{3} + 9 x^{2} - 54 x + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} + \nu^{6} - 5 \nu^{5} + 9 \nu^{4} - 2 \nu^{3} - 15 \nu^{2} + 36 \nu - 27 \)\()/54\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{5} - \nu^{4} + 2 \nu^{3} - 3 \nu^{2} - \nu + 12 \)\()/6\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} - \nu^{6} - 4 \nu^{5} + 9 \nu^{4} - 7 \nu^{3} - 12 \nu^{2} + 36 \nu - 54 \)\()/54\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} + 2 \nu^{6} - \nu^{5} - 6 \nu^{4} + 20 \nu^{3} - 18 \nu^{2} - 9 \nu + 54 \)\()/27\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} - 2 \nu^{6} + \nu^{5} + 6 \nu^{4} - 11 \nu^{3} + 9 \nu^{2} + 9 \nu - 27 \)\()/18\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{7} + \nu^{6} + \nu^{5} - 7 \nu^{4} + 14 \nu^{3} + 11 \nu^{2} - 18 \nu + 45 \)\()/18\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} - \beta_{5} + \beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{7} + 2 \beta_{6} + 2 \beta_{5} + \beta_{2} - 3\)
\(\nu^{4}\)\(=\)\(\beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} - 3 \beta_{1} + 4\)
\(\nu^{5}\)\(=\)\(-2 \beta_{7} + 2 \beta_{6} + 5 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} + 2 \beta_{1} + 2\)
\(\nu^{6}\)\(=\)\(-10 \beta_{6} - 6 \beta_{5} - 6 \beta_{4} - 6 \beta_{3} + 12 \beta_{2} - 2 \beta_{1} + 9\)
\(\nu^{7}\)\(=\)\(-2 \beta_{7} + 6 \beta_{6} + 2 \beta_{5} - 22 \beta_{4} + 4 \beta_{3} + 12 \beta_{2} + 3 \beta_{1} - 14\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
341.1
1.67936 + 0.423958i
1.67936 0.423958i
0.828750 + 1.52091i
0.828750 1.52091i
0.209196 + 1.71937i
0.209196 1.71937i
−1.71731 + 0.225499i
−1.71731 0.225499i
1.00000 −1.67936 0.423958i 1.00000 1.00000i −1.67936 0.423958i −2.47197 1.00000 2.64052 + 1.42396i 1.00000i
341.2 1.00000 −1.67936 + 0.423958i 1.00000 1.00000i −1.67936 + 0.423958i −2.47197 1.00000 2.64052 1.42396i 1.00000i
341.3 1.00000 −0.828750 1.52091i 1.00000 1.00000i −0.828750 1.52091i 4.83942 1.00000 −1.62635 + 2.52091i 1.00000i
341.4 1.00000 −0.828750 + 1.52091i 1.00000 1.00000i −0.828750 + 1.52091i 4.83942 1.00000 −1.62635 2.52091i 1.00000i
341.5 1.00000 −0.209196 1.71937i 1.00000 1.00000i −0.209196 1.71937i 0.264536 1.00000 −2.91247 + 0.719371i 1.00000i
341.6 1.00000 −0.209196 + 1.71937i 1.00000 1.00000i −0.209196 + 1.71937i 0.264536 1.00000 −2.91247 0.719371i 1.00000i
341.7 1.00000 1.71731 0.225499i 1.00000 1.00000i 1.71731 0.225499i −0.631989 1.00000 2.89830 0.774501i 1.00000i
341.8 1.00000 1.71731 + 0.225499i 1.00000 1.00000i 1.71731 + 0.225499i −0.631989 1.00000 2.89830 + 0.774501i 1.00000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 341.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.d 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.f.d yes 8
3.b odd 2 1 570.2.f.c 8
19.b odd 2 1 570.2.f.c 8
57.d even 2 1 inner 570.2.f.d yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.f.c 8 3.b odd 2 1
570.2.f.c 8 19.b odd 2 1
570.2.f.d yes 8 1.a even 1 1 trivial
570.2.f.d yes 8 57.d 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 \(S_{2}^{\mathrm{new}}(570, [\chi])\):

\( T_{7}^{4} - 2 T_{7}^{3} - 13 T_{7}^{2} - 4 T_{7} + 2 \)
\( T_{29}^{4} + 2 T_{29}^{3} - 11 T_{29}^{2} - 24 T_{29} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{8} \)
$3$ \( 81 + 54 T + 9 T^{2} - 18 T^{3} - 20 T^{4} - 6 T^{5} + T^{6} + 2 T^{7} + T^{8} \)
$5$ \( ( 1 + T^{2} )^{4} \)
$7$ \( ( 2 - 4 T - 13 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$11$ \( ( 4 + T^{2} )^{4} \)
$13$ \( 256 + 480 T^{2} + 265 T^{4} + 38 T^{6} + T^{8} \)
$17$ \( 100 + 5116 T^{2} + 1069 T^{4} + 66 T^{6} + T^{8} \)
$19$ \( 130321 - 82308 T + 28880 T^{2} - 7524 T^{3} + 1678 T^{4} - 396 T^{5} + 80 T^{6} - 12 T^{7} + T^{8} \)
$23$ \( 15376 + 25048 T^{2} + 2761 T^{4} + 94 T^{6} + T^{8} \)
$29$ \( ( -8 - 24 T - 11 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$31$ \( 193600 + 42464 T^{2} + 3252 T^{4} + 100 T^{6} + T^{8} \)
$37$ \( 16384 + 174080 T^{2} + 11072 T^{4} + 208 T^{6} + T^{8} \)
$41$ \( ( 16 + 24 T - 2 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$43$ \( ( -176 - 112 T + 88 T^{2} + 20 T^{3} + T^{4} )^{2} \)
$47$ \( 430336 + 86016 T^{2} + 5344 T^{4} + 128 T^{6} + T^{8} \)
$53$ \( ( -424 + 168 T + 33 T^{2} - 14 T^{3} + T^{4} )^{2} \)
$59$ \( ( 118 - 280 T - 161 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$61$ \( ( 3104 + 848 T - 140 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$67$ \( 1849600 + 4202464 T^{2} + 82801 T^{4} + 510 T^{6} + T^{8} \)
$71$ \( ( -5440 + 1696 T - 96 T^{2} - 12 T^{3} + T^{4} )^{2} \)
$73$ \( ( -1228 - 764 T - 119 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$79$ \( 4326400 + 459904 T^{2} + 16132 T^{4} + 220 T^{6} + T^{8} \)
$83$ \( 121000000 + 6119584 T^{2} + 93076 T^{4} + 540 T^{6} + T^{8} \)
$89$ \( ( 11384 + 4720 T + 698 T^{2} + 44 T^{3} + T^{4} )^{2} \)
$97$ \( 65536 + 38912 T^{2} + 4240 T^{4} + 120 T^{6} + T^{8} \)
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