Properties

Label 570.2.i.j
Level $570$
Weight $2$
Character orbit 570.i
Analytic conductor $4.551$
Analytic rank $0$
Dimension $6$
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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.29654208.1
Defining polynomial: \( x^{6} + 14x^{4} + 49x^{2} + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_1 - 1) q^{3} - \beta_1 q^{4} + (\beta_1 - 1) q^{5} - \beta_1 q^{6} - \beta_{3} q^{7} + q^{8} - \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + (\beta_1 - 1) q^{3} - \beta_1 q^{4} + (\beta_1 - 1) q^{5} - \beta_1 q^{6} - \beta_{3} q^{7} + q^{8} - \beta_1 q^{9} - \beta_1 q^{10} + (\beta_{2} + 1) q^{11} + q^{12} + ( - \beta_{4} - \beta_{2}) q^{13} - \beta_{5} q^{14} - \beta_1 q^{15} + (\beta_1 - 1) q^{16} + ( - \beta_{5} - 2 \beta_{4} - \beta_1 + 1) q^{17} + q^{18} + ( - \beta_{5} - \beta_{4} - \beta_{2}) q^{19} + q^{20} - \beta_{5} q^{21} + (\beta_{4} + \beta_1 - 1) q^{22} + (\beta_{4} + \beta_{2} + \beta_1) q^{23} + (\beta_1 - 1) q^{24} - \beta_1 q^{25} + \beta_{2} q^{26} + q^{27} + (\beta_{5} + \beta_{3}) q^{28} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} + 2 \beta_{2} - 3 \beta_1) q^{29} + q^{30} + \beta_{2} q^{31} - \beta_1 q^{32} + (\beta_{4} + \beta_1 - 1) q^{33} + (\beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1) q^{34} - \beta_{5} q^{35} + (\beta_1 - 1) q^{36} - q^{37} + (\beta_{5} + \beta_{3} + \beta_{2}) q^{38} + \beta_{2} q^{39} + (\beta_1 - 1) q^{40} + ( - 3 \beta_{5} - \beta_{4} + 2 \beta_1 - 2) q^{41} + (\beta_{5} + \beta_{3}) q^{42} + (2 \beta_{5} + \beta_{4} + 4 \beta_1 - 4) q^{43} + ( - \beta_{4} - \beta_{2} - \beta_1) q^{44} + q^{45} + ( - \beta_{2} - 1) q^{46} + ( - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2}) q^{47} - \beta_1 q^{48} + (2 \beta_{3} + 2 \beta_{2} + 6) q^{49} + q^{50} + (\beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1) q^{51} + \beta_{4} q^{52} + (\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 6 \beta_1) q^{53} + (\beta_1 - 1) q^{54} + (\beta_{4} + \beta_1 - 1) q^{55} - \beta_{3} q^{56} + (\beta_{5} + \beta_{3} + \beta_{2}) q^{57} + (\beta_{3} - 2 \beta_{2} + 3) q^{58} + (6 \beta_1 - 6) q^{59} + (\beta_1 - 1) q^{60} + ( - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + \beta_1) q^{61} + \beta_{4} q^{62} + (\beta_{5} + \beta_{3}) q^{63} + q^{64} + \beta_{2} q^{65} + ( - \beta_{4} - \beta_{2} - \beta_1) q^{66} + ( - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - 5 \beta_1) q^{67} + ( - \beta_{3} - 2 \beta_{2} - 1) q^{68} + ( - \beta_{2} - 1) q^{69} + (\beta_{5} + \beta_{3}) q^{70} + ( - \beta_{5} - 2 \beta_{4} - \beta_1 + 1) q^{71} - \beta_1 q^{72} + (\beta_{5} + \beta_{4} - 7 \beta_1 + 7) q^{73} + ( - \beta_1 + 1) q^{74} + q^{75} + (\beta_{4} - \beta_{3}) q^{76} + ( - \beta_{3} + 3 \beta_{2} + 4) q^{77} + \beta_{4} q^{78} + ( - \beta_{4} - 2 \beta_1 + 2) q^{79} - \beta_1 q^{80} + (\beta_1 - 1) q^{81} + (3 \beta_{5} + \beta_{4} + 3 \beta_{3} + \beta_{2} - 2 \beta_1) q^{82} + ( - 4 \beta_{2} + 2) q^{83} - \beta_{3} q^{84} + (\beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1) q^{85} + ( - 2 \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} - 4 \beta_1) q^{86} + (\beta_{3} - 2 \beta_{2} + 3) q^{87} + (\beta_{2} + 1) q^{88} + ( - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + 6 \beta_1) q^{89} + (\beta_1 - 1) q^{90} + ( - 3 \beta_{4} - 3 \beta_{2} - 4 \beta_1) q^{91} + ( - \beta_{4} - \beta_1 + 1) q^{92} + \beta_{4} q^{93} + (2 \beta_{3} + 2 \beta_{2}) q^{94} + (\beta_{5} + \beta_{3} + \beta_{2}) q^{95} + q^{96} + (\beta_{5} + 13 \beta_1 - 13) q^{97} + (2 \beta_{5} + 2 \beta_{4} + 6 \beta_1 - 6) q^{98} + ( - \beta_{4} - \beta_{2} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} - 3 q^{3} - 3 q^{4} - 3 q^{5} - 3 q^{6} + 2 q^{7} + 6 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{2} - 3 q^{3} - 3 q^{4} - 3 q^{5} - 3 q^{6} + 2 q^{7} + 6 q^{8} - 3 q^{9} - 3 q^{10} + 8 q^{11} + 6 q^{12} - q^{13} - q^{14} - 3 q^{15} - 3 q^{16} + 4 q^{17} + 6 q^{18} - 2 q^{19} + 6 q^{20} - q^{21} - 4 q^{22} + 4 q^{23} - 3 q^{24} - 3 q^{25} + 2 q^{26} + 6 q^{27} - q^{28} - 6 q^{29} + 6 q^{30} + 2 q^{31} - 3 q^{32} - 4 q^{33} + 4 q^{34} - q^{35} - 3 q^{36} - 6 q^{37} + q^{38} + 2 q^{39} - 3 q^{40} - 8 q^{41} - q^{42} - 11 q^{43} - 4 q^{44} + 6 q^{45} - 8 q^{46} - 3 q^{48} + 36 q^{49} + 6 q^{50} + 4 q^{51} - q^{52} - 18 q^{53} - 3 q^{54} - 4 q^{55} + 2 q^{56} + q^{57} + 12 q^{58} - 18 q^{59} - 3 q^{60} + 3 q^{61} - q^{62} - q^{63} + 6 q^{64} + 2 q^{65} - 4 q^{66} - 15 q^{67} - 8 q^{68} - 8 q^{69} - q^{70} + 4 q^{71} - 3 q^{72} + 21 q^{73} + 3 q^{74} + 6 q^{75} + q^{76} + 32 q^{77} - q^{78} + 7 q^{79} - 3 q^{80} - 3 q^{81} - 8 q^{82} + 4 q^{83} + 2 q^{84} + 4 q^{85} - 11 q^{86} + 12 q^{87} + 8 q^{88} + 18 q^{89} - 3 q^{90} - 15 q^{91} + 4 q^{92} - q^{93} + q^{95} + 6 q^{96} - 38 q^{97} - 18 q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 14x^{4} + 49x^{2} + 12 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 7\nu + 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{4} - 9\nu^{2} - 10 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + 12\nu^{3} - 2\nu^{2} + 35\nu - 10 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 12\nu^{3} + 9\nu^{2} - 29\nu + 10 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{5} + 2\beta_{4} + \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -14\beta_{5} - 14\beta_{4} - 7\beta_{3} - 7\beta_{2} + 12\beta _1 - 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{3} - 9\beta_{2} + 35 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 98\beta_{5} + 110\beta_{4} + 49\beta_{3} + 55\beta_{2} - 144\beta _1 + 72 ) / 3 \) Copy content Toggle raw display

Character values

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

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

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
2.35084i
2.86514i
0.514306i
2.35084i
2.86514i
0.514306i
−0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −4.59821 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 1.75353 1.00000 −0.500000 + 0.866025i −0.500000 + 0.866025i
121.3 −0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 3.84469 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 −4.59821 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 1.75353 1.00000 −0.500000 0.866025i −0.500000 0.866025i
391.3 −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 3.84469 1.00000 −0.500000 0.866025i −0.500000 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 391.3
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.j 6
3.b odd 2 1 1710.2.l.q 6
19.c even 3 1 inner 570.2.i.j 6
57.h odd 6 1 1710.2.l.q 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.i.j 6 1.a even 1 1 trivial
570.2.i.j 6 19.c even 3 1 inner
1710.2.l.q 6 3.b odd 2 1
1710.2.l.q 6 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 \(S_{2}^{\mathrm{new}}(570, [\chi])\):

\( T_{7}^{3} - T_{7}^{2} - 19T_{7} + 31 \) Copy content Toggle raw display
\( T_{11}^{3} - 4T_{11}^{2} - 11T_{11} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$7$ \( (T^{3} - T^{2} - 19 T + 31)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} - 4 T^{2} - 11 T + 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + T^{5} + 17 T^{4} - 32 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$17$ \( T^{6} - 4 T^{5} + 66 T^{4} + \cdots + 46656 \) Copy content Toggle raw display
$19$ \( T^{6} + 2 T^{5} + 8 T^{4} + 140 T^{3} + \cdots + 6859 \) Copy content Toggle raw display
$23$ \( T^{6} - 4 T^{5} + 27 T^{4} + 32 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$29$ \( T^{6} + 6 T^{5} + 138 T^{4} + \cdots + 467856 \) Copy content Toggle raw display
$31$ \( (T^{3} - T^{2} - 16 T - 8)^{2} \) Copy content Toggle raw display
$37$ \( (T + 1)^{6} \) Copy content Toggle raw display
$41$ \( T^{6} + 8 T^{5} + 189 T^{4} + \cdots + 1077444 \) Copy content Toggle raw display
$43$ \( T^{6} + 11 T^{5} + 145 T^{4} + \cdots + 1296 \) Copy content Toggle raw display
$47$ \( T^{6} + 84 T^{4} + 288 T^{3} + \cdots + 20736 \) Copy content Toggle raw display
$53$ \( T^{6} + 18 T^{5} + 237 T^{4} + \cdots + 11664 \) Copy content Toggle raw display
$59$ \( (T^{2} + 6 T + 36)^{3} \) Copy content Toggle raw display
$61$ \( T^{6} - 3 T^{5} + 27 T^{4} + 50 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$67$ \( T^{6} + 15 T^{5} + 171 T^{4} + 806 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$71$ \( T^{6} - 4 T^{5} + 66 T^{4} + \cdots + 46656 \) Copy content Toggle raw display
$73$ \( T^{6} - 21 T^{5} + 315 T^{4} + \cdots + 45796 \) Copy content Toggle raw display
$79$ \( T^{6} - 7 T^{5} + 49 T^{4} - 24 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$83$ \( (T^{3} - 2 T^{2} - 260 T + 1032)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 18 T^{5} + 237 T^{4} + \cdots + 11664 \) Copy content Toggle raw display
$97$ \( T^{6} + 38 T^{5} + 982 T^{4} + \cdots + 3283344 \) Copy content Toggle raw display
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