Properties

Label 570.2.i.f
Level $570$
Weight $2$
Character orbit 570.i
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.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.55147291521\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{2} + ( 1 - \beta_{1} ) q^{3} -\beta_{1} q^{4} + ( -1 + \beta_{1} ) q^{5} + \beta_{1} q^{6} + ( 1 - \beta_{3} ) q^{7} + q^{8} -\beta_{1} q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{2} + ( 1 - \beta_{1} ) q^{3} -\beta_{1} q^{4} + ( -1 + \beta_{1} ) q^{5} + \beta_{1} q^{6} + ( 1 - \beta_{3} ) q^{7} + q^{8} -\beta_{1} q^{9} -\beta_{1} q^{10} -3 q^{11} - q^{12} -2 \beta_{2} q^{13} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{14} + \beta_{1} q^{15} + ( -1 + \beta_{1} ) q^{16} + ( 4 - 4 \beta_{1} + \beta_{2} - \beta_{3} ) q^{17} + q^{18} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{19} + q^{20} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{21} + ( 3 - 3 \beta_{1} ) q^{22} + ( \beta_{1} + 2 \beta_{2} ) q^{23} + ( 1 - \beta_{1} ) q^{24} -\beta_{1} q^{25} + 2 \beta_{3} q^{26} - q^{27} + ( -\beta_{1} + \beta_{2} ) q^{28} + ( -2 \beta_{1} - 3 \beta_{2} ) q^{29} - q^{30} + ( 4 - 2 \beta_{3} ) q^{31} -\beta_{1} q^{32} + ( -3 + 3 \beta_{1} ) q^{33} + ( 4 \beta_{1} - \beta_{2} ) q^{34} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{35} + ( -1 + \beta_{1} ) q^{36} + ( -5 - 2 \beta_{3} ) q^{37} + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{38} -2 \beta_{3} q^{39} + ( -1 + \beta_{1} ) q^{40} + ( 3 - 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{41} + ( \beta_{1} - \beta_{2} ) q^{42} + ( 2 - 2 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{43} + 3 \beta_{1} q^{44} + q^{45} + ( -1 - 2 \beta_{3} ) q^{46} + ( 6 \beta_{1} - 2 \beta_{2} ) q^{47} + \beta_{1} q^{48} -2 \beta_{3} q^{49} + q^{50} + ( -4 \beta_{1} + \beta_{2} ) q^{51} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{52} + ( 5 \beta_{1} + \beta_{2} ) q^{53} + ( 1 - \beta_{1} ) q^{54} + ( 3 - 3 \beta_{1} ) q^{55} + ( 1 - \beta_{3} ) q^{56} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{57} + ( 2 + 3 \beta_{3} ) q^{58} + ( 6 - 6 \beta_{1} ) q^{59} + ( 1 - \beta_{1} ) q^{60} + ( -10 \beta_{1} - \beta_{2} ) q^{61} + ( -4 + 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{62} + ( -\beta_{1} + \beta_{2} ) q^{63} + q^{64} + 2 \beta_{3} q^{65} -3 \beta_{1} q^{66} + 3 \beta_{2} q^{67} + ( -4 + \beta_{3} ) q^{68} + ( 1 + 2 \beta_{3} ) q^{69} + ( -\beta_{1} + \beta_{2} ) q^{70} + ( -4 + 4 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{71} -\beta_{1} q^{72} + ( -8 + 8 \beta_{1} + \beta_{2} - \beta_{3} ) q^{73} + ( 5 - 5 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{74} - q^{75} + ( 1 + 2 \beta_{2} - \beta_{3} ) q^{76} + ( -3 + 3 \beta_{3} ) q^{77} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{78} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{79} -\beta_{1} q^{80} + ( -1 + \beta_{1} ) q^{81} + ( 3 \beta_{1} - 3 \beta_{2} ) q^{82} + ( 2 + 4 \beta_{3} ) q^{83} + ( -1 + \beta_{3} ) q^{84} + ( 4 \beta_{1} - \beta_{2} ) q^{85} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{86} + ( -2 - 3 \beta_{3} ) q^{87} -3 q^{88} + ( 3 \beta_{1} + \beta_{2} ) q^{89} + ( -1 + \beta_{1} ) q^{90} + ( 12 \beta_{1} - 2 \beta_{2} ) q^{91} + ( 1 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{92} + ( 4 - 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{93} + ( -6 + 2 \beta_{3} ) q^{94} + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{95} - q^{96} + ( 16 - 16 \beta_{1} - \beta_{2} + \beta_{3} ) q^{97} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{98} + 3 \beta_{1} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} + 2q^{3} - 2q^{4} - 2q^{5} + 2q^{6} + 4q^{7} + 4q^{8} - 2q^{9} + O(q^{10}) \) \( 4q - 2q^{2} + 2q^{3} - 2q^{4} - 2q^{5} + 2q^{6} + 4q^{7} + 4q^{8} - 2q^{9} - 2q^{10} - 12q^{11} - 4q^{12} - 2q^{14} + 2q^{15} - 2q^{16} + 8q^{17} + 4q^{18} - 2q^{19} + 4q^{20} + 2q^{21} + 6q^{22} + 2q^{23} + 2q^{24} - 2q^{25} - 4q^{27} - 2q^{28} - 4q^{29} - 4q^{30} + 16q^{31} - 2q^{32} - 6q^{33} + 8q^{34} - 2q^{35} - 2q^{36} - 20q^{37} - 2q^{38} - 2q^{40} + 6q^{41} + 2q^{42} + 4q^{43} + 6q^{44} + 4q^{45} - 4q^{46} + 12q^{47} + 2q^{48} + 4q^{50} - 8q^{51} + 10q^{53} + 2q^{54} + 6q^{55} + 4q^{56} + 2q^{57} + 8q^{58} + 12q^{59} + 2q^{60} - 20q^{61} - 8q^{62} - 2q^{63} + 4q^{64} - 6q^{66} - 16q^{68} + 4q^{69} - 2q^{70} - 8q^{71} - 2q^{72} - 16q^{73} + 10q^{74} - 4q^{75} + 4q^{76} - 12q^{77} - 4q^{79} - 2q^{80} - 2q^{81} + 6q^{82} + 8q^{83} - 4q^{84} + 8q^{85} + 4q^{86} - 8q^{87} - 12q^{88} + 6q^{89} - 2q^{90} + 24q^{91} + 2q^{92} + 8q^{93} - 24q^{94} - 2q^{95} - 4q^{96} + 32q^{97} + 6q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 2 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 4 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2}\)\()/3\)
\(\nu^{2}\)\(=\)\(2 \beta_{1}\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{3} + 4 \beta_{2}\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/570\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(211\) \(457\)
\(\chi(n)\) \(1\) \(-\beta_{1}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
1.22474 0.707107i
−1.22474 + 0.707107i
1.22474 + 0.707107i
−1.22474 0.707107i
−0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 0.500000 0.866025i −1.44949 1.00000 −0.500000 + 0.866025i −0.500000 + 0.866025i
121.2 −0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 0.500000 0.866025i 3.44949 1.00000 −0.500000 + 0.866025i −0.500000 + 0.866025i
391.1 −0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i −1.44949 1.00000 −0.500000 0.866025i −0.500000 0.866025i
391.2 −0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i 3.44949 1.00000 −0.500000 0.866025i −0.500000 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.i.f 4
3.b odd 2 1 1710.2.l.n 4
19.c even 3 1 inner 570.2.i.f 4
57.h odd 6 1 1710.2.l.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.i.f 4 1.a even 1 1 trivial
570.2.i.f 4 19.c even 3 1 inner
1710.2.l.n 4 3.b odd 2 1
1710.2.l.n 4 57.h odd 6 1

Hecke kernels

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

\( T_{7}^{2} - 2 T_{7} - 5 \)
\( T_{11} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( ( 1 - T + T^{2} )^{2} \)
$5$ \( ( 1 + T + T^{2} )^{2} \)
$7$ \( ( -5 - 2 T + T^{2} )^{2} \)
$11$ \( ( 3 + T )^{4} \)
$13$ \( 576 + 24 T^{2} + T^{4} \)
$17$ \( 100 - 80 T + 54 T^{2} - 8 T^{3} + T^{4} \)
$19$ \( 361 + 38 T - 15 T^{2} + 2 T^{3} + T^{4} \)
$23$ \( 529 + 46 T + 27 T^{2} - 2 T^{3} + T^{4} \)
$29$ \( 2500 - 200 T + 66 T^{2} + 4 T^{3} + T^{4} \)
$31$ \( ( -8 - 8 T + T^{2} )^{2} \)
$37$ \( ( 1 + 10 T + T^{2} )^{2} \)
$41$ \( 2025 + 270 T + 81 T^{2} - 6 T^{3} + T^{4} \)
$43$ \( 8464 + 368 T + 108 T^{2} - 4 T^{3} + T^{4} \)
$47$ \( 144 - 144 T + 132 T^{2} - 12 T^{3} + T^{4} \)
$53$ \( 361 - 190 T + 81 T^{2} - 10 T^{3} + T^{4} \)
$59$ \( ( 36 - 6 T + T^{2} )^{2} \)
$61$ \( 8836 + 1880 T + 306 T^{2} + 20 T^{3} + T^{4} \)
$67$ \( 2916 + 54 T^{2} + T^{4} \)
$71$ \( 1444 - 304 T + 102 T^{2} + 8 T^{3} + T^{4} \)
$73$ \( 3364 + 928 T + 198 T^{2} + 16 T^{3} + T^{4} \)
$79$ \( 400 - 80 T + 36 T^{2} + 4 T^{3} + T^{4} \)
$83$ \( ( -92 - 4 T + T^{2} )^{2} \)
$89$ \( 9 - 18 T + 33 T^{2} - 6 T^{3} + T^{4} \)
$97$ \( 62500 - 8000 T + 774 T^{2} - 32 T^{3} + T^{4} \)
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