Properties

Label 570.2.c.a
Level $570$
Weight $2$
Character orbit 570.c
Analytic conductor $4.551$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) 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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{8}^{2} q^{2} + ( - \zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}) q^{3} - q^{4} + (2 \zeta_{8}^{3} - \zeta_{8}) q^{5} + (\zeta_{8}^{3} + \zeta_{8} - 1) q^{6} - \zeta_{8}^{2} q^{8} + (2 \zeta_{8}^{3} + 2 \zeta_{8} + 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{8}^{2} q^{2} + ( - \zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}) q^{3} - q^{4} + (2 \zeta_{8}^{3} - \zeta_{8}) q^{5} + (\zeta_{8}^{3} + \zeta_{8} - 1) q^{6} - \zeta_{8}^{2} q^{8} + (2 \zeta_{8}^{3} + 2 \zeta_{8} + 1) q^{9} + ( - \zeta_{8}^{3} - 2 \zeta_{8}) q^{10} + (2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{11} + (\zeta_{8}^{3} - \zeta_{8}^{2} - \zeta_{8}) q^{12} + (3 \zeta_{8}^{3} - 3 \zeta_{8}) q^{13} + ( - \zeta_{8}^{3} + \zeta_{8}^{2} - 2 \zeta_{8} - 3) q^{15} + q^{16} + (2 \zeta_{8}^{3} + \zeta_{8}^{2} - 2 \zeta_{8}) q^{18} + (3 \zeta_{8}^{3} + 3 \zeta_{8} + 1) q^{19} + ( - 2 \zeta_{8}^{3} + \zeta_{8}) q^{20} + (2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{22} + (3 \zeta_{8}^{3} - 3 \zeta_{8}) q^{23} + ( - \zeta_{8}^{3} - \zeta_{8} + 1) q^{24} + ( - 3 \zeta_{8}^{2} + 4) q^{25} + ( - 3 \zeta_{8}^{3} - 3 \zeta_{8}) q^{26} + (\zeta_{8}^{3} + 5 \zeta_{8}^{2} - \zeta_{8}) q^{27} - 6 q^{29} + ( - 2 \zeta_{8}^{3} - 3 \zeta_{8}^{2} + \zeta_{8} - 1) q^{30} + ( - 3 \zeta_{8}^{3} - 3 \zeta_{8}) q^{31} + \zeta_{8}^{2} q^{32} + (2 \zeta_{8}^{3} + 4 \zeta_{8}^{2} - 2 \zeta_{8}) q^{33} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8} - 1) q^{36} + (3 \zeta_{8}^{3} - 3 \zeta_{8}) q^{37} + (3 \zeta_{8}^{3} + \zeta_{8}^{2} - 3 \zeta_{8}) q^{38} + ( - 3 \zeta_{8}^{3} - 3 \zeta_{8} - 6) q^{39} + (\zeta_{8}^{3} + 2 \zeta_{8}) q^{40} + 6 q^{41} + 12 \zeta_{8}^{2} q^{43} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{44} + (2 \zeta_{8}^{3} - 6 \zeta_{8}^{2} - \zeta_{8} - 2) q^{45} + ( - 3 \zeta_{8}^{3} - 3 \zeta_{8}) q^{46} + ( - 9 \zeta_{8}^{3} + 9 \zeta_{8}) q^{47} + ( - \zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}) q^{48} + 7 q^{49} + (4 \zeta_{8}^{2} + 3) q^{50} + ( - 3 \zeta_{8}^{3} + 3 \zeta_{8}) q^{52} - 6 \zeta_{8}^{2} q^{53} + ( - \zeta_{8}^{3} - \zeta_{8} - 5) q^{54} + ( - 6 \zeta_{8}^{2} - 2) q^{55} + (2 \zeta_{8}^{3} + 7 \zeta_{8}^{2} - 2 \zeta_{8}) q^{57} - 6 \zeta_{8}^{2} q^{58} - 6 q^{59} + (\zeta_{8}^{3} - \zeta_{8}^{2} + 2 \zeta_{8} + 3) q^{60} + 10 q^{61} + ( - 3 \zeta_{8}^{3} + 3 \zeta_{8}) q^{62} - q^{64} + ( - 3 \zeta_{8}^{2} + 9) q^{65} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8} - 4) q^{66} + ( - 3 \zeta_{8}^{3} - 3 \zeta_{8} - 6) q^{69} + 12 q^{71} + ( - 2 \zeta_{8}^{3} - \zeta_{8}^{2} + 2 \zeta_{8}) q^{72} + 6 \zeta_{8}^{2} q^{73} + ( - 3 \zeta_{8}^{3} - 3 \zeta_{8}) q^{74} + ( - 7 \zeta_{8}^{3} + 4 \zeta_{8}^{2} + \zeta_{8} + 3) q^{75} + ( - 3 \zeta_{8}^{3} - 3 \zeta_{8} - 1) q^{76} + ( - 3 \zeta_{8}^{3} - 6 \zeta_{8}^{2} + 3 \zeta_{8}) q^{78} + (9 \zeta_{8}^{3} + 9 \zeta_{8}) q^{79} + (2 \zeta_{8}^{3} - \zeta_{8}) q^{80} + (4 \zeta_{8}^{3} + 4 \zeta_{8} - 7) q^{81} + 6 \zeta_{8}^{2} q^{82} - 12 q^{86} + (6 \zeta_{8}^{3} - 6 \zeta_{8}^{2} - 6 \zeta_{8}) q^{87} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{88} - 12 q^{89} + ( - \zeta_{8}^{3} - 2 \zeta_{8}^{2} - 2 \zeta_{8} + 6) q^{90} + ( - 3 \zeta_{8}^{3} + 3 \zeta_{8}) q^{92} + ( - 3 \zeta_{8}^{3} - 6 \zeta_{8}^{2} + 3 \zeta_{8}) q^{93} + (9 \zeta_{8}^{3} + 9 \zeta_{8}) q^{94} + (2 \zeta_{8}^{3} - 9 \zeta_{8}^{2} - \zeta_{8} - 3) q^{95} + (\zeta_{8}^{3} + \zeta_{8} - 1) q^{96} + 7 \zeta_{8}^{2} q^{98} + (2 \zeta_{8}^{3} + 2 \zeta_{8} - 8) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 4 q^{6} + 4 q^{9} - 12 q^{15} + 4 q^{16} + 4 q^{19} + 4 q^{24} + 16 q^{25} - 24 q^{29} - 4 q^{30} - 4 q^{36} - 24 q^{39} + 24 q^{41} - 8 q^{45} + 28 q^{49} + 12 q^{50} - 20 q^{54} - 8 q^{55} - 24 q^{59} + 12 q^{60} + 40 q^{61} - 4 q^{64} + 36 q^{65} - 16 q^{66} - 24 q^{69} + 48 q^{71} + 12 q^{75} - 4 q^{76} - 28 q^{81} - 48 q^{86} - 48 q^{89} + 24 q^{90} - 12 q^{95} - 4 q^{96} - 32 q^{99}+O(q^{100}) \) 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
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
1.00000i −1.41421 1.00000i −1.00000 2.12132 + 0.707107i −1.00000 + 1.41421i 0 1.00000i 1.00000 + 2.82843i 0.707107 2.12132i
569.2 1.00000i 1.41421 1.00000i −1.00000 −2.12132 0.707107i −1.00000 1.41421i 0 1.00000i 1.00000 2.82843i −0.707107 + 2.12132i
569.3 1.00000i −1.41421 + 1.00000i −1.00000 2.12132 0.707107i −1.00000 1.41421i 0 1.00000i 1.00000 2.82843i 0.707107 + 2.12132i
569.4 1.00000i 1.41421 + 1.00000i −1.00000 −2.12132 + 0.707107i −1.00000 + 1.41421i 0 1.00000i 1.00000 + 2.82843i −0.707107 2.12132i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
57.d even 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.a 4
3.b odd 2 1 570.2.c.b yes 4
5.b even 2 1 inner 570.2.c.a 4
15.d odd 2 1 570.2.c.b yes 4
19.b odd 2 1 570.2.c.b yes 4
57.d even 2 1 inner 570.2.c.a 4
95.d odd 2 1 570.2.c.b yes 4
285.b even 2 1 inner 570.2.c.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.c.a 4 1.a even 1 1 trivial
570.2.c.a 4 5.b even 2 1 inner
570.2.c.a 4 57.d even 2 1 inner
570.2.c.a 4 285.b even 2 1 inner
570.2.c.b yes 4 3.b odd 2 1
570.2.c.b yes 4 15.d odd 2 1
570.2.c.b yes 4 19.b odd 2 1
570.2.c.b yes 4 95.d odd 2 1

Hecke kernels

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

\( T_{7} \) Copy content Toggle raw display
\( T_{11}^{2} + 8 \) Copy content Toggle raw display
\( T_{29} + 6 \) Copy content Toggle raw display
\( T_{37}^{2} - 18 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} - 2T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} - 8T^{2} + 25 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 2 T + 19)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$29$ \( (T + 6)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$41$ \( (T - 6)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 162)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$59$ \( (T + 6)^{4} \) Copy content Toggle raw display
$61$ \( (T - 10)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( (T - 12)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 162)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T + 12)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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