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: \( x^{4} - x^{3} + 19x^{2} + 18x + 324 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 1) q^{2} + (\beta_{2} - 1) q^{3} - \beta_{2} q^{4} + (\beta_{2} - 1) q^{5} + \beta_{2} q^{6} - q^{7} - q^{8} - \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 1) q^{2} + (\beta_{2} - 1) q^{3} - \beta_{2} q^{4} + (\beta_{2} - 1) q^{5} + \beta_{2} q^{6} - q^{7} - q^{8} - \beta_{2} q^{9} + \beta_{2} q^{10} + (\beta_{3} - 1) q^{11} + q^{12} + (\beta_{2} - \beta_1) q^{13} + (\beta_{2} - 1) q^{14} - \beta_{2} q^{15} + (\beta_{2} - 1) q^{16} - q^{18} + ( - \beta_1 + 1) q^{19} + q^{20} + ( - \beta_{2} + 1) q^{21} + (\beta_{3} + \beta_1 - 1) q^{22} - \beta_1 q^{23} + ( - \beta_{2} + 1) q^{24} - \beta_{2} q^{25} + \beta_{3} q^{26} + q^{27} + \beta_{2} q^{28} + 6 \beta_{2} q^{29} - q^{30} + (\beta_{3} - 2) q^{31} + \beta_{2} q^{32} + ( - \beta_{3} - \beta_1 + 1) q^{33} + ( - \beta_{2} + 1) q^{35} + (\beta_{2} - 1) q^{36} - q^{37} + (\beta_{3} - \beta_{2}) q^{38} - \beta_{3} q^{39} + ( - \beta_{2} + 1) q^{40} + (\beta_{3} + \beta_1 - 1) q^{41} - \beta_{2} q^{42} + ( - \beta_{3} + 5 \beta_{2} - \beta_1 - 4) q^{43} + \beta_1 q^{44} + q^{45} + (\beta_{3} - 1) q^{46} - \beta_{2} q^{48} - 6 q^{49} - q^{50} + (\beta_{3} - \beta_{2} + \beta_1) q^{52} + ( - 6 \beta_{2} + \beta_1) q^{53} + ( - \beta_{2} + 1) q^{54} + ( - \beta_{3} - \beta_1 + 1) q^{55} + q^{56} + ( - \beta_{3} + \beta_{2}) q^{57} + 6 q^{58} + (2 \beta_{3} - 6 \beta_{2} + 2 \beta_1 + 4) q^{59} + (\beta_{2} - 1) q^{60} + (\beta_{2} + \beta_1) q^{61} + (\beta_{3} + \beta_{2} + \beta_1 - 2) q^{62} + \beta_{2} q^{63} + q^{64} - \beta_{3} q^{65} - \beta_1 q^{66} + (7 \beta_{2} - \beta_1) q^{67} + ( - \beta_{3} + 1) q^{69} - \beta_{2} q^{70} + (2 \beta_{3} + 2 \beta_1 - 2) q^{71} + \beta_{2} q^{72} + ( - \beta_{3} - 7 \beta_{2} - \beta_1 + 8) q^{73} + (\beta_{2} - 1) q^{74} + q^{75} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{76} + ( - \beta_{3} + 1) q^{77} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{78} + ( - 3 \beta_{3} - \beta_{2} - 3 \beta_1 + 4) q^{79} - \beta_{2} q^{80} + (\beta_{2} - 1) q^{81} + \beta_1 q^{82} + 6 q^{83} - q^{84} + (5 \beta_{2} - \beta_1) q^{86} - 6 q^{87} + ( - \beta_{3} + 1) q^{88} + ( - 6 \beta_{2} + 3 \beta_1) q^{89} + ( - \beta_{2} + 1) q^{90} + ( - \beta_{2} + \beta_1) q^{91} + (\beta_{3} + \beta_1 - 1) q^{92} + ( - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{93} + ( - \beta_{3} + \beta_{2}) q^{95} - q^{96} + ( - 10 \beta_{2} + 10) q^{97} + (6 \beta_{2} - 6) q^{98} + \beta_1 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{3} - 2 q^{4} - 2 q^{5} + 2 q^{6} - 4 q^{7} - 4 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{3} - 2 q^{4} - 2 q^{5} + 2 q^{6} - 4 q^{7} - 4 q^{8} - 2 q^{9} + 2 q^{10} - 2 q^{11} + 4 q^{12} + q^{13} - 2 q^{14} - 2 q^{15} - 2 q^{16} - 4 q^{18} + 3 q^{19} + 4 q^{20} + 2 q^{21} - q^{22} - q^{23} + 2 q^{24} - 2 q^{25} + 2 q^{26} + 4 q^{27} + 2 q^{28} + 12 q^{29} - 4 q^{30} - 6 q^{31} + 2 q^{32} + q^{33} + 2 q^{35} - 2 q^{36} - 4 q^{37} - 2 q^{39} + 2 q^{40} - q^{41} - 2 q^{42} - 9 q^{43} + q^{44} + 4 q^{45} - 2 q^{46} - 2 q^{48} - 24 q^{49} - 4 q^{50} + q^{52} - 11 q^{53} + 2 q^{54} + q^{55} + 4 q^{56} + 24 q^{58} + 10 q^{59} - 2 q^{60} + 3 q^{61} - 3 q^{62} + 2 q^{63} + 4 q^{64} - 2 q^{65} - q^{66} + 13 q^{67} + 2 q^{69} - 2 q^{70} - 2 q^{71} + 2 q^{72} + 15 q^{73} - 2 q^{74} + 4 q^{75} - 3 q^{76} + 2 q^{77} - q^{78} + 5 q^{79} - 2 q^{80} - 2 q^{81} + q^{82} + 24 q^{83} - 4 q^{84} + 9 q^{86} - 24 q^{87} + 2 q^{88} - 9 q^{89} + 2 q^{90} - q^{91} - q^{92} + 3 q^{93} - 4 q^{96} + 20 q^{97} - 12 q^{98} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 19\nu^{2} - 19\nu + 324 ) / 342 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 37 ) / 19 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 18\beta_{2} + \beta _1 - 19 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 19\beta_{3} - 37 \) Copy content Toggle raw display

Character values

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

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

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 \(S_{2}^{\mathrm{new}}(570, [\chi])\):

\( T_{7} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} + T_{11} - 18 \) Copy content Toggle raw display

Hecke characteristic polynomials

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