Properties

Label 570.2.u.b
Level $570$
Weight $2$
Character orbit 570.u
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.u (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.55147291521\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \(x^{6} - x^{3} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(191\) \(211\) \(457\)
\(\chi(n)\) \(1\) \(-\zeta_{18}^{5}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1
−0.173648 0.984808i
−0.173648 + 0.984808i
0.939693 0.342020i
0.939693 + 0.342020i
−0.766044 0.642788i
−0.766044 + 0.642788i
0.939693 + 0.342020i 0.173648 + 0.984808i 0.766044 + 0.642788i −0.766044 + 0.642788i −0.173648 + 0.984808i −2.11334 + 3.66041i 0.500000 + 0.866025i −0.939693 + 0.342020i −0.939693 + 0.342020i
271.1 0.939693 0.342020i 0.173648 0.984808i 0.766044 0.642788i −0.766044 0.642788i −0.173648 0.984808i −2.11334 3.66041i 0.500000 0.866025i −0.939693 0.342020i −0.939693 0.342020i
301.1 −0.766044 0.642788i −0.939693 + 0.342020i 0.173648 + 0.984808i −0.173648 + 0.984808i 0.939693 + 0.342020i 0.705737 + 1.22237i 0.500000 0.866025i 0.766044 0.642788i 0.766044 0.642788i
481.1 −0.766044 + 0.642788i −0.939693 0.342020i 0.173648 0.984808i −0.173648 0.984808i 0.939693 0.342020i 0.705737 1.22237i 0.500000 + 0.866025i 0.766044 + 0.642788i 0.766044 + 0.642788i
511.1 −0.173648 + 0.984808i 0.766044 + 0.642788i −0.939693 0.342020i 0.939693 0.342020i −0.766044 + 0.642788i −1.59240 2.75811i 0.500000 0.866025i 0.173648 + 0.984808i 0.173648 + 0.984808i
541.1 −0.173648 0.984808i 0.766044 0.642788i −0.939693 + 0.342020i 0.939693 + 0.342020i −0.766044 0.642788i −1.59240 + 2.75811i 0.500000 + 0.866025i 0.173648 0.984808i 0.173648 0.984808i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 541.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.u.b 6
19.e even 9 1 inner 570.2.u.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.u.b 6 1.a even 1 1 trivial
570.2.u.b 6 19.e even 9 1 inner

Hecke kernels

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

\( T_{7}^{6} + 6 T_{7}^{5} + 33 T_{7}^{4} + 56 T_{7}^{3} + 123 T_{7}^{2} - 57 T_{7} + 361 \)
\( T_{11}^{6} - 12 T_{11}^{5} + 99 T_{11}^{4} - 438 T_{11}^{3} + 1413 T_{11}^{2} - 2295 T_{11} + 2601 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{3} + T^{6} \)
$3$ \( 1 + T^{3} + T^{6} \)
$5$ \( 1 - T^{3} + T^{6} \)
$7$ \( 361 - 57 T + 123 T^{2} + 56 T^{3} + 33 T^{4} + 6 T^{5} + T^{6} \)
$11$ \( 2601 - 2295 T + 1413 T^{2} - 438 T^{3} + 99 T^{4} - 12 T^{5} + T^{6} \)
$13$ \( 15625 + 125 T^{3} + T^{6} \)
$17$ \( 576 + 576 T^{2} + 240 T^{3} - 6 T^{5} + T^{6} \)
$19$ \( 6859 + 6498 T + 3078 T^{2} + 883 T^{3} + 162 T^{4} + 18 T^{5} + T^{6} \)
$23$ \( 45369 - 32589 T + 10971 T^{2} - 2076 T^{3} + 252 T^{4} - 21 T^{5} + T^{6} \)
$29$ \( 576 - 864 T + 576 T^{2} - 192 T^{3} + 36 T^{4} - 6 T^{5} + T^{6} \)
$31$ \( 87616 - 24864 T + 7056 T^{2} - 592 T^{3} + 84 T^{4} + T^{6} \)
$37$ \( ( 73 + 9 T - 12 T^{2} + T^{3} )^{2} \)
$41$ \( 288369 - 72495 T + 3357 T^{2} + 24 T^{3} + 54 T^{4} + 3 T^{5} + T^{6} \)
$43$ \( 18496 + 4896 T + 576 T^{2} + 80 T^{3} + T^{6} \)
$47$ \( 47961 - 39420 T + 15102 T^{2} - 2703 T^{3} + 306 T^{4} - 24 T^{5} + T^{6} \)
$53$ \( 106929 - 67689 T + 18720 T^{2} - 3027 T^{3} + 333 T^{4} - 24 T^{5} + T^{6} \)
$59$ \( 751689 - 5202 T^{2} - 969 T^{3} + 162 T^{4} - 12 T^{5} + T^{6} \)
$61$ \( 1183744 - 156672 T + 9216 T^{2} - 640 T^{3} + T^{6} \)
$67$ \( 87616 - 42624 T + 1440 T^{2} + 1160 T^{3} + 216 T^{4} + T^{6} \)
$71$ \( 46656 + 23328 T + 15552 T^{2} + 4104 T^{3} + 540 T^{4} + 36 T^{5} + T^{6} \)
$73$ \( 64 + 96 T + 192 T^{2} + 368 T^{3} + 240 T^{4} - 24 T^{5} + T^{6} \)
$79$ \( 1183744 + 470016 T + 87552 T^{2} + 9728 T^{3} + 720 T^{4} + 36 T^{5} + T^{6} \)
$83$ \( 5184 + 2592 T + 1296 T^{2} + 144 T^{3} + 36 T^{4} + T^{6} \)
$89$ \( 1962801 - 592623 T + 107172 T^{2} - 13011 T^{3} + 1035 T^{4} - 48 T^{5} + T^{6} \)
$97$ \( 2050624 - 738912 T + 122592 T^{2} - 12736 T^{3} + 924 T^{4} - 42 T^{5} + T^{6} \)
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