Properties

Label 570.2.f.a
Level $570$
Weight $2$
Character orbit 570.f
Analytic conductor $4.551$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 570.f (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: \( 2 \)
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.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
341.1
−0.707107 + 0.707107i
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
−1.00000 −1.00000 1.41421i 1.00000 1.00000i 1.00000 + 1.41421i 0.585786 −1.00000 −1.00000 + 2.82843i 1.00000i
341.2 −1.00000 −1.00000 1.41421i 1.00000 1.00000i 1.00000 + 1.41421i 3.41421 −1.00000 −1.00000 + 2.82843i 1.00000i
341.3 −1.00000 −1.00000 + 1.41421i 1.00000 1.00000i 1.00000 1.41421i 3.41421 −1.00000 −1.00000 2.82843i 1.00000i
341.4 −1.00000 −1.00000 + 1.41421i 1.00000 1.00000i 1.00000 1.41421i 0.585786 −1.00000 −1.00000 2.82843i 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

\( T_{7}^{2} - 4 T_{7} + 2 \)
\( T_{29}^{2} - 4 T_{29} - 46 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{4} \)
$3$ \( ( 3 + 2 T + T^{2} )^{2} \)
$5$ \( ( 1 + T^{2} )^{2} \)
$7$ \( ( 2 - 4 T + T^{2} )^{2} \)
$11$ \( 196 + 36 T^{2} + T^{4} \)
$13$ \( 196 + 44 T^{2} + T^{4} \)
$17$ \( ( 8 + T^{2} )^{2} \)
$19$ \( 361 - 34 T^{2} + T^{4} \)
$23$ \( ( 36 + T^{2} )^{2} \)
$29$ \( ( -46 - 4 T + T^{2} )^{2} \)
$31$ \( 64 + 48 T^{2} + T^{4} \)
$37$ \( 196 + 44 T^{2} + T^{4} \)
$41$ \( ( -34 - 8 T + T^{2} )^{2} \)
$43$ \( ( 62 - 16 T + T^{2} )^{2} \)
$47$ \( 784 + 88 T^{2} + T^{4} \)
$53$ \( ( 28 - 12 T + T^{2} )^{2} \)
$59$ \( ( 92 - 20 T + T^{2} )^{2} \)
$61$ \( ( -56 - 8 T + T^{2} )^{2} \)
$67$ \( T^{4} \)
$71$ \( ( -8 + T^{2} )^{2} \)
$73$ \( ( -36 + 12 T + T^{2} )^{2} \)
$79$ \( 15376 + 264 T^{2} + T^{4} \)
$83$ \( 8464 + 216 T^{2} + T^{4} \)
$89$ \( ( 142 + 24 T + T^{2} )^{2} \)
$97$ \( 3844 + 132 T^{2} + T^{4} \)
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