Properties

Label 570.2.f.c
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: \( x^{8} - 2x^{7} + x^{6} + 6x^{5} - 20x^{4} + 18x^{3} + 9x^{2} - 54x + 81 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - q^{2} + \beta_1 q^{3} + q^{4} + \beta_{2} q^{5} - \beta_1 q^{6} + ( - \beta_{7} - \beta_{6} - \beta_{4} + 1) q^{7} - q^{8} + (\beta_{7} - \beta_{5} + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta_1 q^{3} + q^{4} + \beta_{2} q^{5} - \beta_1 q^{6} + ( - \beta_{7} - \beta_{6} - \beta_{4} + 1) q^{7} - q^{8} + (\beta_{7} - \beta_{5} + \beta_{2}) q^{9} - \beta_{2} q^{10} + 2 \beta_{2} q^{11} + \beta_1 q^{12} + ( - \beta_{5} - 2 \beta_{2} + \beta_1) q^{13} + (\beta_{7} + \beta_{6} + \beta_{4} - 1) q^{14} + \beta_{6} q^{15} + q^{16} + (\beta_{6} - \beta_{4} - \beta_{3}) q^{17} + ( - \beta_{7} + \beta_{5} - \beta_{2}) q^{18} + ( - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_1 + 1) q^{19} + \beta_{2} q^{20} + ( - \beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{21} - 2 \beta_{2} q^{22} + (\beta_{5} - 2 \beta_{3} - \beta_1) q^{23} - \beta_1 q^{24} - q^{25} + (\beta_{5} + 2 \beta_{2} - \beta_1) q^{26} + (\beta_{7} + 2 \beta_{6} + 2 \beta_{5} + \beta_{2} - 3) q^{27} + ( - \beta_{7} - \beta_{6} - \beta_{4} + 1) q^{28} + (\beta_{5} + \beta_1) q^{29} - \beta_{6} q^{30} + ( - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 4 \beta_{2} + \beta_1) q^{31} - q^{32} + 2 \beta_{6} q^{33} + ( - \beta_{6} + \beta_{4} + \beta_{3}) q^{34} + ( - \beta_{5} + \beta_{3} + \beta_1) q^{35} + (\beta_{7} - \beta_{5} + \beta_{2}) q^{36} + ( - 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{37} + (\beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_1 - 1) q^{38} + (\beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{2} - 3) q^{39} - \beta_{2} q^{40} + ( - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_1 - 1) q^{41} + (\beta_{7} + \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{42} + (2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + 2 \beta_1 - 6) q^{43} + 2 \beta_{2} q^{44} + (\beta_{4} - \beta_{3} + \beta_{2} - 1) q^{45} + ( - \beta_{5} + 2 \beta_{3} + \beta_1) q^{46} + ( - 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{2} + 2 \beta_1) q^{47} + \beta_1 q^{48} + ( - 2 \beta_{7} - 4 \beta_{6} + \beta_{5} - 4 \beta_{4} + \beta_1 + 1) q^{49} + q^{50} + ( - \beta_{6} - \beta_{4} - 2 \beta_{3} + 5 \beta_{2} - \beta_1 - 2) q^{51} + ( - \beta_{5} - 2 \beta_{2} + \beta_1) q^{52} + (2 \beta_{6} + \beta_{5} + 2 \beta_{4} + \beta_1 - 4) q^{53} + ( - \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - \beta_{2} + 3) q^{54} - 2 q^{55} + (\beta_{7} + \beta_{6} + \beta_{4} - 1) q^{56} + (\beta_{7} - \beta_{6} - \beta_{5} - 3 \beta_{4} - 2 \beta_{2} + 3) q^{57} + ( - \beta_{5} - \beta_1) q^{58} + (3 \beta_{7} - \beta_{6} - \beta_{4} - 1) q^{59} + \beta_{6} q^{60} + (2 \beta_{7} + 4 \beta_{6} + 2 \beta_{5} + 4 \beta_{4} + 2 \beta_1) q^{61} + (\beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - 4 \beta_{2} - \beta_1) q^{62} + (\beta_{6} - 3 \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 4) q^{63} + q^{64} + (\beta_{6} + \beta_{4} + 2) q^{65} - 2 \beta_{6} q^{66} + ( - 2 \beta_{6} + 3 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} + 6 \beta_{2} - 3 \beta_1) q^{67} + (\beta_{6} - \beta_{4} - \beta_{3}) q^{68} + ( - \beta_{7} - 2 \beta_{6} + \beta_{5} - 4 \beta_{4} - 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{69} + (\beta_{5} - \beta_{3} - \beta_1) q^{70} + ( - 2 \beta_{7} - 4 \beta_{6} - 4 \beta_{4} - 2) q^{71} + ( - \beta_{7} + \beta_{5} - \beta_{2}) q^{72} + ( - 2 \beta_{7} + 2 \beta_{6} + \beta_{5} + 2 \beta_{4} + \beta_1) q^{73} + (2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_1) q^{74} - \beta_1 q^{75} + ( - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_1 + 1) q^{76} + ( - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_1) q^{77} + ( - \beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{2} + 3) q^{78} + (\beta_{6} - \beta_{5} - \beta_{4} + 3 \beta_{3} + \beta_1) q^{79} + \beta_{2} q^{80} + (\beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 4) q^{81} + (\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_1 + 1) q^{82} + ( - 3 \beta_{6} - 5 \beta_{5} + 3 \beta_{4} + \beta_{3} - 2 \beta_{2} + 5 \beta_1) q^{83} + ( - \beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{84} + ( - \beta_{7} - \beta_{5} - \beta_1 + 1) q^{85} + ( - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_1 + 6) q^{86} + (\beta_{7} - \beta_{5} + \beta_{2} + 3) q^{87} - 2 \beta_{2} q^{88} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_1 + 11) q^{89} + ( - \beta_{4} + \beta_{3} - \beta_{2} + 1) q^{90} + ( - \beta_{5} + 2 \beta_{2} + \beta_1) q^{91} + (\beta_{5} - 2 \beta_{3} - \beta_1) q^{92} + (\beta_{7} + 3 \beta_{6} - \beta_{5} - 3 \beta_{4} - 2 \beta_{2} - \beta_1 - 3) q^{93} + (2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{2} - 2 \beta_1) q^{94} + ( - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} + \beta_1 + 1) q^{95} - \beta_1 q^{96} + (2 \beta_{5} - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{97} + (2 \beta_{7} + 4 \beta_{6} - \beta_{5} + 4 \beta_{4} - \beta_1 - 1) q^{98} + (2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 2 q^{3} + 8 q^{4} - 2 q^{6} + 4 q^{7} - 8 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} + 2 q^{3} + 8 q^{4} - 2 q^{6} + 4 q^{7} - 8 q^{8} + 2 q^{9} + 2 q^{12} - 4 q^{14} + 8 q^{16} - 2 q^{18} + 12 q^{19} + 2 q^{21} - 2 q^{24} - 8 q^{25} - 16 q^{27} + 4 q^{28} + 4 q^{29} - 8 q^{32} + 2 q^{36} - 12 q^{38} - 22 q^{39} - 16 q^{41} - 2 q^{42} - 40 q^{43} - 8 q^{45} + 2 q^{48} + 4 q^{49} + 8 q^{50} - 18 q^{51} - 28 q^{53} + 16 q^{54} - 16 q^{55} - 4 q^{56} + 26 q^{57} - 4 q^{58} + 4 q^{59} + 16 q^{61} - 34 q^{63} + 8 q^{64} + 16 q^{65} + 2 q^{69} - 24 q^{71} - 2 q^{72} - 4 q^{73} - 2 q^{75} + 12 q^{76} + 22 q^{78} + 34 q^{81} + 16 q^{82} + 2 q^{84} + 40 q^{86} + 26 q^{87} + 88 q^{89} + 8 q^{90} - 24 q^{93} + 8 q^{95} - 2 q^{96} - 4 q^{98} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + \nu^{6} - 5\nu^{5} + 9\nu^{4} - 2\nu^{3} - 15\nu^{2} + 36\nu - 27 ) / 54 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} - \nu^{4} + 2\nu^{3} - 3\nu^{2} - \nu + 12 ) / 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - \nu^{6} - 4\nu^{5} + 9\nu^{4} - 7\nu^{3} - 12\nu^{2} + 36\nu - 54 ) / 54 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} - \nu^{5} - 6\nu^{4} + 20\nu^{3} - 18\nu^{2} - 9\nu + 54 ) / 27 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + \nu^{5} + 6\nu^{4} - 11\nu^{3} + 9\nu^{2} + 9\nu - 27 ) / 18 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} + \nu^{6} + \nu^{5} - 7\nu^{4} + 14\nu^{3} + 11\nu^{2} - 18\nu + 45 ) / 18 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{5} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 2\beta_{6} + 2\beta_{5} + \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + 2\beta_{6} + 2\beta_{5} + 2\beta_{4} - 2\beta_{3} + 3\beta_{2} - 3\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{7} + 2\beta_{6} + 5\beta_{5} - 2\beta_{4} - 4\beta_{3} - 4\beta_{2} + 2\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -10\beta_{6} - 6\beta_{5} - 6\beta_{4} - 6\beta_{3} + 12\beta_{2} - 2\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -2\beta_{7} + 6\beta_{6} + 2\beta_{5} - 22\beta_{4} + 4\beta_{3} + 12\beta_{2} + 3\beta _1 - 14 \) Copy content Toggle raw display

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
−1.71731 0.225499i
−1.71731 + 0.225499i
0.209196 1.71937i
0.209196 + 1.71937i
0.828750 1.52091i
0.828750 + 1.52091i
1.67936 0.423958i
1.67936 + 0.423958i
−1.00000 −1.71731 0.225499i 1.00000 1.00000i 1.71731 + 0.225499i −0.631989 −1.00000 2.89830 + 0.774501i 1.00000i
341.2 −1.00000 −1.71731 + 0.225499i 1.00000 1.00000i 1.71731 0.225499i −0.631989 −1.00000 2.89830 0.774501i 1.00000i
341.3 −1.00000 0.209196 1.71937i 1.00000 1.00000i −0.209196 + 1.71937i 0.264536 −1.00000 −2.91247 0.719371i 1.00000i
341.4 −1.00000 0.209196 + 1.71937i 1.00000 1.00000i −0.209196 1.71937i 0.264536 −1.00000 −2.91247 + 0.719371i 1.00000i
341.5 −1.00000 0.828750 1.52091i 1.00000 1.00000i −0.828750 + 1.52091i 4.83942 −1.00000 −1.62635 2.52091i 1.00000i
341.6 −1.00000 0.828750 + 1.52091i 1.00000 1.00000i −0.828750 1.52091i 4.83942 −1.00000 −1.62635 + 2.52091i 1.00000i
341.7 −1.00000 1.67936 0.423958i 1.00000 1.00000i −1.67936 + 0.423958i −2.47197 −1.00000 2.64052 1.42396i 1.00000i
341.8 −1.00000 1.67936 + 0.423958i 1.00000 1.00000i −1.67936 0.423958i −2.47197 −1.00000 2.64052 + 1.42396i 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.c 8
3.b odd 2 1 570.2.f.d yes 8
19.b odd 2 1 570.2.f.d yes 8
57.d even 2 1 inner 570.2.f.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.f.c 8 1.a even 1 1 trivial
570.2.f.c 8 57.d even 2 1 inner
570.2.f.d yes 8 3.b odd 2 1
570.2.f.d yes 8 19.b odd 2 1

Hecke kernels

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

\( T_{7}^{4} - 2T_{7}^{3} - 13T_{7}^{2} - 4T_{7} + 2 \) Copy content Toggle raw display
\( T_{29}^{4} - 2T_{29}^{3} - 11T_{29}^{2} + 24T_{29} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{7} + T^{6} + 6 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} - 2 T^{3} - 13 T^{2} - 4 T + 2)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} + 38 T^{6} + 265 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{8} + 66 T^{6} + 1069 T^{4} + \cdots + 100 \) Copy content Toggle raw display
$19$ \( T^{8} - 12 T^{7} + 80 T^{6} + \cdots + 130321 \) Copy content Toggle raw display
$23$ \( T^{8} + 94 T^{6} + 2761 T^{4} + \cdots + 15376 \) Copy content Toggle raw display
$29$ \( (T^{4} - 2 T^{3} - 11 T^{2} + 24 T - 8)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 100 T^{6} + 3252 T^{4} + \cdots + 193600 \) Copy content Toggle raw display
$37$ \( T^{8} + 208 T^{6} + 11072 T^{4} + \cdots + 16384 \) Copy content Toggle raw display
$41$ \( (T^{4} + 8 T^{3} - 2 T^{2} - 24 T + 16)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 20 T^{3} + 88 T^{2} - 112 T - 176)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 128 T^{6} + 5344 T^{4} + \cdots + 430336 \) Copy content Toggle raw display
$53$ \( (T^{4} + 14 T^{3} + 33 T^{2} - 168 T - 424)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 2 T^{3} - 161 T^{2} + 280 T + 118)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 8 T^{3} - 140 T^{2} + 848 T + 3104)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 510 T^{6} + 82801 T^{4} + \cdots + 1849600 \) Copy content Toggle raw display
$71$ \( (T^{4} + 12 T^{3} - 96 T^{2} - 1696 T - 5440)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 2 T^{3} - 119 T^{2} - 764 T - 1228)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 220 T^{6} + 16132 T^{4} + \cdots + 4326400 \) Copy content Toggle raw display
$83$ \( T^{8} + 540 T^{6} + \cdots + 121000000 \) Copy content Toggle raw display
$89$ \( (T^{4} - 44 T^{3} + 698 T^{2} + \cdots + 11384)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 120 T^{6} + 4240 T^{4} + \cdots + 65536 \) Copy content Toggle raw display
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