Properties

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

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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: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
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.

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

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{24}^{4} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\zeta_{24}^{6} + 2\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -2\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 2\zeta_{24}^{7} + \zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{4} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{6} - \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} ) / 4 \) 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
0.965926 + 0.258819i
−0.965926 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
0.965926 0.258819i
−0.965926 + 0.258819i
−0.258819 + 0.965926i
0.258819 0.965926i
1.00000i −1.73205 −1.00000 −1.41421 1.73205i 1.73205i 2.44949i 1.00000i 3.00000 −1.73205 + 1.41421i
569.2 1.00000i −1.73205 −1.00000 1.41421 1.73205i 1.73205i 2.44949i 1.00000i 3.00000 −1.73205 1.41421i
569.3 1.00000i 1.73205 −1.00000 −1.41421 + 1.73205i 1.73205i 2.44949i 1.00000i 3.00000 1.73205 + 1.41421i
569.4 1.00000i 1.73205 −1.00000 1.41421 + 1.73205i 1.73205i 2.44949i 1.00000i 3.00000 1.73205 1.41421i
569.5 1.00000i −1.73205 −1.00000 −1.41421 + 1.73205i 1.73205i 2.44949i 1.00000i 3.00000 −1.73205 1.41421i
569.6 1.00000i −1.73205 −1.00000 1.41421 + 1.73205i 1.73205i 2.44949i 1.00000i 3.00000 −1.73205 + 1.41421i
569.7 1.00000i 1.73205 −1.00000 −1.41421 1.73205i 1.73205i 2.44949i 1.00000i 3.00000 1.73205 1.41421i
569.8 1.00000i 1.73205 −1.00000 1.41421 1.73205i 1.73205i 2.44949i 1.00000i 3.00000 1.73205 + 1.41421i
\(n\): 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
5.b even 2 1 inner
15.d odd 2 1 inner
19.b odd 2 1 inner
57.d even 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.d 8
3.b odd 2 1 inner 570.2.c.d 8
5.b even 2 1 inner 570.2.c.d 8
15.d odd 2 1 inner 570.2.c.d 8
19.b odd 2 1 inner 570.2.c.d 8
57.d even 2 1 inner 570.2.c.d 8
95.d odd 2 1 inner 570.2.c.d 8
285.b even 2 1 inner 570.2.c.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.c.d 8 1.a even 1 1 trivial
570.2.c.d 8 3.b odd 2 1 inner
570.2.c.d 8 5.b even 2 1 inner
570.2.c.d 8 15.d odd 2 1 inner
570.2.c.d 8 19.b odd 2 1 inner
570.2.c.d 8 57.d even 2 1 inner
570.2.c.d 8 95.d odd 2 1 inner
570.2.c.d 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 \(S_{2}^{\mathrm{new}}(570, [\chi])\):

\( T_{7}^{2} + 6 \) Copy content Toggle raw display
\( T_{11}^{2} + 12 \) Copy content Toggle raw display
\( T_{29}^{2} - 24 \) Copy content Toggle raw display
\( T_{37}^{2} - 48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} + 2 T^{2} + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 6)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 12)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{2} - 50)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 2 T + 19)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 32)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 24)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 18)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 48)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 6)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 96)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 8)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 144)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 54)^{4} \) Copy content Toggle raw display
$61$ \( (T + 10)^{8} \) Copy content Toggle raw display
$67$ \( (T^{2} - 192)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 24)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 96)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 18)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} - 150)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 12)^{4} \) Copy content Toggle raw display
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