Properties

Label 570.2.c.b
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: \(x^{4} + 1\)
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.

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

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.b yes 4
3.b odd 2 1 570.2.c.a 4
5.b even 2 1 inner 570.2.c.b yes 4
15.d odd 2 1 570.2.c.a 4
19.b odd 2 1 570.2.c.a 4
57.d even 2 1 inner 570.2.c.b yes 4
95.d odd 2 1 570.2.c.a 4
285.b even 2 1 inner 570.2.c.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.c.a 4 3.b odd 2 1
570.2.c.a 4 15.d odd 2 1
570.2.c.a 4 19.b odd 2 1
570.2.c.a 4 95.d odd 2 1
570.2.c.b yes 4 1.a even 1 1 trivial
570.2.c.b yes 4 5.b even 2 1 inner
570.2.c.b yes 4 57.d even 2 1 inner
570.2.c.b yes 4 285.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 \(S_{2}^{\mathrm{new}}(570, [\chi])\):

\( T_{7} \)
\( T_{11}^{2} + 8 \)
\( T_{29} - 6 \)
\( T_{37}^{2} - 18 \)

Hecke characteristic polynomials

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