Properties

Label 570.2.i.g
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: \(\Q(\sqrt{-3}, \sqrt{73})\)
Defining polynomial: \(x^{4} - x^{3} + 19 x^{2} + 18 x + 324\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 19 x^{2} + 18 x + 324\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 19 \nu^{2} - 19 \nu + 324 \)\()/342\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 37 \)\()/19\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 18 \beta_{2} + \beta_{1} - 19\)
\(\nu^{3}\)\(=\)\(19 \beta_{3} - 37\)

Character values

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

\(n\) \(191\) \(211\) \(457\)
\(\chi(n)\) \(1\) \(-\beta_{2}\) \(1\)

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
2.38600 4.13267i
−1.88600 + 3.26665i
2.38600 + 4.13267i
−1.88600 3.26665i
0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 0.500000 0.866025i −1.00000 −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 −1.00000 −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 −1.00000 −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 −1.00000 −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.g 4
3.b odd 2 1 1710.2.l.l 4
19.c even 3 1 inner 570.2.i.g 4
57.h odd 6 1 1710.2.l.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.i.g 4 1.a even 1 1 trivial
570.2.i.g 4 19.c even 3 1 inner
1710.2.l.l 4 3.b odd 2 1
1710.2.l.l 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 \(S_{2}^{\mathrm{new}}(570, [\chi])\):

\( T_{7} + 1 \)
\( T_{11}^{2} + T_{11} - 18 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{2} \)
$3$ \( ( 1 + T + T^{2} )^{2} \)
$5$ \( ( 1 + T + T^{2} )^{2} \)
$7$ \( ( 1 + T )^{4} \)
$11$ \( ( -18 + T + T^{2} )^{2} \)
$13$ \( 324 + 18 T + 19 T^{2} - T^{3} + T^{4} \)
$17$ \( T^{4} \)
$19$ \( 361 - 57 T + 22 T^{2} - 3 T^{3} + T^{4} \)
$23$ \( 324 - 18 T + 19 T^{2} + T^{3} + T^{4} \)
$29$ \( ( 36 - 6 T + T^{2} )^{2} \)
$31$ \( ( -16 + 3 T + T^{2} )^{2} \)
$37$ \( ( 1 + T )^{4} \)
$41$ \( 324 - 18 T + 19 T^{2} + T^{3} + T^{4} \)
$43$ \( 4 + 18 T + 79 T^{2} + 9 T^{3} + T^{4} \)
$47$ \( T^{4} \)
$53$ \( 144 + 132 T + 109 T^{2} + 11 T^{3} + T^{4} \)
$59$ \( 2304 + 480 T + 148 T^{2} - 10 T^{3} + T^{4} \)
$61$ \( 256 + 48 T + 25 T^{2} - 3 T^{3} + T^{4} \)
$67$ \( 576 - 312 T + 145 T^{2} - 13 T^{3} + T^{4} \)
$71$ \( 5184 - 144 T + 76 T^{2} + 2 T^{3} + T^{4} \)
$73$ \( 1444 - 570 T + 187 T^{2} - 15 T^{3} + T^{4} \)
$79$ \( 24964 + 790 T + 183 T^{2} - 5 T^{3} + T^{4} \)
$83$ \( ( -6 + T )^{4} \)
$89$ \( 20736 - 1296 T + 225 T^{2} + 9 T^{3} + T^{4} \)
$97$ \( ( 100 - 10 T + T^{2} )^{2} \)
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