Properties

Label 570.2.c.f
Level $570$
Weight $2$
Character orbit 570.c
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.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.55147291521\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1499238400.2
Defining polynomial: \(x^{8} - 4 x^{7} + 16 x^{6} - 34 x^{5} + 59 x^{4} - 66 x^{3} + 54 x^{2} - 26 x + 5\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} + 16 x^{6} - 34 x^{5} + 59 x^{4} - 66 x^{3} + 54 x^{2} - 26 x + 5\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 3 \nu^{7} - 8 \nu^{6} + 34 \nu^{5} - 50 \nu^{4} + 82 \nu^{3} - 62 \nu^{2} + 46 \nu - 10 \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{7} + 8 \nu^{6} - 34 \nu^{5} + 50 \nu^{4} - 82 \nu^{3} + 67 \nu^{2} - 51 \nu + 25 \)\()/5\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{7} - 7 \nu^{6} + 31 \nu^{5} - 60 \nu^{4} + 118 \nu^{3} - 118 \nu^{2} + 109 \nu - 30 \)\()/5\)
\(\beta_{5}\)\(=\)\((\)\( -4 \nu^{7} + 14 \nu^{6} - 57 \nu^{5} + 110 \nu^{4} - 186 \nu^{3} + 191 \nu^{2} - 138 \nu + 50 \)\()/5\)
\(\beta_{6}\)\(=\)\((\)\( -6 \nu^{7} + 21 \nu^{6} - 83 \nu^{5} + 155 \nu^{4} - 249 \nu^{3} + 229 \nu^{2} - 157 \nu + 45 \)\()/5\)
\(\beta_{7}\)\(=\)\((\)\( -7 \nu^{7} + 22 \nu^{6} - 96 \nu^{5} + 170 \nu^{4} - 313 \nu^{3} + 298 \nu^{2} - 249 \nu + 85 \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + \beta_{1} - 3\)
\(\nu^{3}\)\(=\)\(\beta_{7} + 2 \beta_{4} + \beta_{2} - 3 \beta_{1} - 3\)
\(\nu^{4}\)\(=\)\(\beta_{7} - \beta_{6} + 2 \beta_{5} + 3 \beta_{4} - 7 \beta_{3} - 6 \beta_{2} - 8 \beta_{1} + 13\)
\(\nu^{5}\)\(=\)\(-8 \beta_{7} - 2 \beta_{6} + 5 \beta_{5} - 12 \beta_{4} - 5 \beta_{3} - 13 \beta_{2} + 7 \beta_{1} + 31\)
\(\nu^{6}\)\(=\)\(-18 \beta_{7} + 6 \beta_{6} - 7 \beta_{5} - 35 \beta_{4} + 41 \beta_{3} + 25 \beta_{2} + 50 \beta_{1} - 43\)
\(\nu^{7}\)\(=\)\(32 \beta_{7} + 22 \beta_{6} - 42 \beta_{5} + 38 \beta_{4} + 70 \beta_{3} + 109 \beta_{2} + 8 \beta_{1} - 226\)

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
0.500000 + 0.0845405i
0.500000 1.08454i
0.500000 + 1.41267i
0.500000 2.41267i
0.500000 1.41267i
0.500000 + 2.41267i
0.500000 0.0845405i
0.500000 + 1.08454i
1.00000i 0.500000 1.65831i −1.00000 −0.584541 + 2.15831i −1.65831 0.500000i 4.86140i 1.00000i −2.50000 1.65831i 2.15831 + 0.584541i
569.2 1.00000i 0.500000 1.65831i −1.00000 0.584541 + 2.15831i −1.65831 0.500000i 4.86140i 1.00000i −2.50000 1.65831i 2.15831 0.584541i
569.3 1.00000i 0.500000 + 1.65831i −1.00000 −1.91267 1.15831i 1.65831 0.500000i 3.21974i 1.00000i −2.50000 + 1.65831i −1.15831 + 1.91267i
569.4 1.00000i 0.500000 + 1.65831i −1.00000 1.91267 1.15831i 1.65831 0.500000i 3.21974i 1.00000i −2.50000 + 1.65831i −1.15831 1.91267i
569.5 1.00000i 0.500000 1.65831i −1.00000 −1.91267 + 1.15831i 1.65831 + 0.500000i 3.21974i 1.00000i −2.50000 1.65831i −1.15831 1.91267i
569.6 1.00000i 0.500000 1.65831i −1.00000 1.91267 + 1.15831i 1.65831 + 0.500000i 3.21974i 1.00000i −2.50000 1.65831i −1.15831 + 1.91267i
569.7 1.00000i 0.500000 + 1.65831i −1.00000 −0.584541 2.15831i −1.65831 + 0.500000i 4.86140i 1.00000i −2.50000 + 1.65831i 2.15831 0.584541i
569.8 1.00000i 0.500000 + 1.65831i −1.00000 0.584541 2.15831i −1.65831 + 0.500000i 4.86140i 1.00000i −2.50000 + 1.65831i 2.15831 + 0.584541i
\(n\): 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
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.f yes 8
3.b odd 2 1 inner 570.2.c.f yes 8
5.b even 2 1 570.2.c.c 8
15.d odd 2 1 570.2.c.c 8
19.b odd 2 1 570.2.c.c 8
57.d even 2 1 570.2.c.c 8
95.d odd 2 1 inner 570.2.c.f yes 8
285.b even 2 1 inner 570.2.c.f yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.c.c 8 5.b even 2 1
570.2.c.c 8 15.d odd 2 1
570.2.c.c 8 19.b odd 2 1
570.2.c.c 8 57.d even 2 1
570.2.c.f yes 8 1.a even 1 1 trivial
570.2.c.f yes 8 3.b odd 2 1 inner
570.2.c.f yes 8 95.d odd 2 1 inner
570.2.c.f yes 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 \(S_{2}^{\mathrm{new}}(570, [\chi])\):

\( T_{7}^{4} + 34 T_{7}^{2} + 245 \)
\( T_{11}^{4} + 24 T_{11}^{2} + 100 \)
\( T_{29}^{4} - 70 T_{29}^{2} + 125 \)
\( T_{37} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{4} \)
$3$ \( ( 3 - T + T^{2} )^{4} \)
$5$ \( 625 + 100 T^{2} + 10 T^{4} + 4 T^{6} + T^{8} \)
$7$ \( ( 245 + 34 T^{2} + T^{4} )^{2} \)
$11$ \( ( 100 + 24 T^{2} + T^{4} )^{2} \)
$13$ \( ( -11 + T^{2} )^{4} \)
$17$ \( ( 245 - 34 T^{2} + T^{4} )^{2} \)
$19$ \( ( 361 + 76 T - 2 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$23$ \( ( 1805 - 86 T^{2} + T^{4} )^{2} \)
$29$ \( ( 125 - 70 T^{2} + T^{4} )^{2} \)
$31$ \( ( 500 + 80 T^{2} + T^{4} )^{2} \)
$37$ \( ( -2 + T )^{8} \)
$41$ \( ( 1620 - 144 T^{2} + T^{4} )^{2} \)
$43$ \( ( 20 + 16 T^{2} + T^{4} )^{2} \)
$47$ \( ( 2000 - 104 T^{2} + T^{4} )^{2} \)
$53$ \( ( 1 + T^{2} )^{4} \)
$59$ \( ( 3125 - 130 T^{2} + T^{4} )^{2} \)
$61$ \( ( -10 - 2 T + T^{2} )^{4} \)
$67$ \( ( -7 + T )^{8} \)
$71$ \( ( 12500 - 224 T^{2} + T^{4} )^{2} \)
$73$ \( ( 405 + 126 T^{2} + T^{4} )^{2} \)
$79$ \( ( 5120 + 256 T^{2} + T^{4} )^{2} \)
$83$ \( ( 3920 - 136 T^{2} + T^{4} )^{2} \)
$89$ \( ( 500 - 80 T^{2} + T^{4} )^{2} \)
$97$ \( ( 56 - 20 T + T^{2} )^{4} \)
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