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: \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 59x^{4} - 66x^{3} + 54x^{2} - 26x + 5 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + (\beta_{7} + \beta_{6} + \beta_{4} - \beta_{3}) q^{3} - q^{4} + ( - \beta_{7} - 2 \beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} + 1) q^{5} + (\beta_{6} - \beta_{3}) q^{6} + (\beta_{7} + 2 \beta_{6} - \beta_{4} + \beta_{2} - 3 \beta_1 + 1) q^{7} - \beta_{6} q^{8} + (\beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} + (\beta_{7} + \beta_{6} + \beta_{4} - \beta_{3}) q^{3} - q^{4} + ( - \beta_{7} - 2 \beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} + 1) q^{5} + (\beta_{6} - \beta_{3}) q^{6} + (\beta_{7} + 2 \beta_{6} - \beta_{4} + \beta_{2} - 3 \beta_1 + 1) q^{7} - \beta_{6} q^{8} + (\beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} - 3) q^{9} + ( - \beta_{6} + \beta_{3} + \beta_1) q^{10} + (2 \beta_{7} + \beta_{6} + 2 \beta_{4} - 2 \beta_{3} - 1) q^{11} + ( - \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3}) q^{12} + ( - \beta_{6} + 2 \beta_{3}) q^{13} + (\beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{4} - 3 \beta_{2} + \beta_1) q^{14} + ( - \beta_{7} - \beta_{6} + \beta_{3} - \beta_{2} - \beta_1 + 4) q^{15} + q^{16} + (\beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{4} - 3 \beta_{2} + \beta_1) q^{17} + ( - 2 \beta_{6} - \beta_{3}) q^{18} + ( - \beta_{7} - 2 \beta_{6} + \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{19} + (\beta_{7} + 2 \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{20} + (3 \beta_{7} + 2 \beta_{6} - 5 \beta_{5} + 2 \beta_{4} + \beta_1 - 3) q^{21} + (\beta_{6} - 2 \beta_{3} + 1) q^{22} + (\beta_{7} - 2 \beta_{6} - 2 \beta_{5} + \beta_{4} - 5 \beta_{2} + \beta_1 + 1) q^{23} + ( - \beta_{6} + \beta_{3}) q^{24} + (\beta_{7} + \beta_{6} - \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 - 2) q^{25} + (2 \beta_{7} + 2 \beta_{6} + 2 \beta_{4} - 2 \beta_{3} - 1) q^{26} + ( - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{4} + 2 \beta_{3} - 3) q^{27} + ( - \beta_{7} - 2 \beta_{6} + \beta_{4} - \beta_{2} + 3 \beta_1 - 1) q^{28} + (\beta_{7} + 2 \beta_{6} - 2 \beta_{5} + \beta_{4} + 3 \beta_{2} + \beta_1 - 3) q^{29} + (2 \beta_{6} + \beta_{5} - \beta_{2} + 1) q^{30} + (2 \beta_{7} + 3 \beta_{6} - 2 \beta_{4} + 2 \beta_{2} - 4 \beta_1 + 1) q^{31} + \beta_{6} q^{32} + (\beta_{7} + \beta_{4} - 6) q^{33} + ( - \beta_{7} - 2 \beta_{6} + \beta_{4} - \beta_{2} + 3 \beta_1 - 1) q^{34} + ( - 2 \beta_{7} - 4 \beta_{6} + 6 \beta_{5} - 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + \cdots + 3) q^{35}+ \cdots + ( - 5 \beta_{7} - 3 \beta_{6} - 5 \beta_{4} + 6 \beta_{3} - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} - 8 q^{4} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} - 8 q^{4} - 20 q^{9} + 4 q^{10} - 4 q^{12} + 22 q^{15} + 8 q^{16} - 8 q^{19} + 8 q^{22} - 8 q^{25} - 32 q^{27} + 2 q^{30} - 44 q^{33} + 20 q^{36} + 16 q^{37} - 4 q^{40} + 22 q^{45} + 4 q^{48} - 80 q^{49} + 40 q^{55} - 4 q^{57} - 22 q^{60} + 8 q^{61} - 8 q^{64} + 4 q^{66} + 56 q^{67} + 36 q^{70} - 4 q^{75} + 8 q^{76} - 44 q^{78} + 28 q^{81} + 36 q^{85} - 8 q^{88} - 10 q^{90} + 80 q^{97} - 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{7} - 8\nu^{6} + 34\nu^{5} - 50\nu^{4} + 82\nu^{3} - 62\nu^{2} + 46\nu - 10 ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{7} + 8\nu^{6} - 34\nu^{5} + 50\nu^{4} - 82\nu^{3} + 67\nu^{2} - 51\nu + 25 ) / 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{7} - 7\nu^{6} + 31\nu^{5} - 60\nu^{4} + 118\nu^{3} - 118\nu^{2} + 109\nu - 30 ) / 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4\nu^{7} + 14\nu^{6} - 57\nu^{5} + 110\nu^{4} - 186\nu^{3} + 191\nu^{2} - 138\nu + 50 ) / 5 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -6\nu^{7} + 21\nu^{6} - 83\nu^{5} + 155\nu^{4} - 249\nu^{3} + 229\nu^{2} - 157\nu + 45 ) / 5 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -7\nu^{7} + 22\nu^{6} - 96\nu^{5} + 170\nu^{4} - 313\nu^{3} + 298\nu^{2} - 249\nu + 85 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 2\beta_{4} + \beta_{2} - 3\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - \beta_{6} + 2\beta_{5} + 3\beta_{4} - 7\beta_{3} - 6\beta_{2} - 8\beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{7} - 2\beta_{6} + 5\beta_{5} - 12\beta_{4} - 5\beta_{3} - 13\beta_{2} + 7\beta _1 + 31 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -18\beta_{7} + 6\beta_{6} - 7\beta_{5} - 35\beta_{4} + 41\beta_{3} + 25\beta_{2} + 50\beta _1 - 43 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 32\beta_{7} + 22\beta_{6} - 42\beta_{5} + 38\beta_{4} + 70\beta_{3} + 109\beta_{2} + 8\beta _1 - 226 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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 \(S_{2}^{\mathrm{new}}(570, [\chi])\):

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

Hecke characteristic polynomials

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