Properties

Label 570.2.u.f
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.

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

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 1.43969 2.49362i 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 1.43969 + 2.49362i 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 −0.266044 0.460802i 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 −0.266044 + 0.460802i 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 0.326352 + 0.565258i 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 0.326352 0.565258i 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.f 6
19.e even 9 1 inner 570.2.u.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.u.f 6 1.a even 1 1 trivial
570.2.u.f 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 \(S_{2}^{\mathrm{new}}(570, [\chi])\):

\( T_{7}^{6} - 3 T_{7}^{5} + 9 T_{7}^{4} - 2 T_{7}^{3} + 3 T_{7}^{2} + 1 \)
\( T_{11}^{6} + 3 T_{11}^{5} + 27 T_{11}^{4} + 60 T_{11}^{3} + 495 T_{11}^{2} + 1026 T_{11} + 3249 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{3} + T^{6} \)
$3$ \( 1 + T^{3} + T^{6} \)
$5$ \( 1 - T^{3} + T^{6} \)
$7$ \( 1 + 3 T^{2} - 2 T^{3} + 9 T^{4} - 3 T^{5} + T^{6} \)
$11$ \( 3249 + 1026 T + 495 T^{2} + 60 T^{3} + 27 T^{4} + 3 T^{5} + T^{6} \)
$13$ \( 5041 + 4899 T + 2211 T^{2} + 620 T^{3} + 120 T^{4} + 15 T^{5} + T^{6} \)
$17$ \( 1 + 18 T^{2} + 28 T^{3} + 18 T^{4} - 9 T^{5} + T^{6} \)
$19$ \( 6859 - 3249 T + 179 T^{3} - 9 T^{5} + T^{6} \)
$23$ \( 9 + 243 T + 2790 T^{2} - 408 T^{3} + 81 T^{4} - 15 T^{5} + T^{6} \)
$29$ \( 546121 - 64293 T + 2622 T^{2} - 296 T^{3} + 15 T^{4} - 3 T^{5} + T^{6} \)
$31$ \( 29241 + 16929 T + 6723 T^{2} + 1440 T^{3} + 225 T^{4} + 18 T^{5} + T^{6} \)
$37$ \( ( 127 + 120 T + 21 T^{2} + T^{3} )^{2} \)
$41$ \( 292681 - 1623 T + 1110 T^{2} + 224 T^{3} + 42 T^{4} + 12 T^{5} + T^{6} \)
$43$ \( 54289 - 15378 T + 3993 T^{2} - 271 T^{3} + 12 T^{4} + 15 T^{5} + T^{6} \)
$47$ \( 47961 + 61101 T + 26496 T^{2} + 3918 T^{3} + 396 T^{4} + 24 T^{5} + T^{6} \)
$53$ \( 729 + 729 T + 972 T^{2} + 513 T^{3} + 135 T^{4} + 18 T^{5} + T^{6} \)
$59$ \( 104329 + 7752 T - 1551 T^{2} + 251 T^{3} + 48 T^{4} - 3 T^{5} + T^{6} \)
$61$ \( 2601 - 14229 T + 23121 T^{2} - 3912 T^{3} + 396 T^{4} - 21 T^{5} + T^{6} \)
$67$ \( 751689 - 304317 T + 58518 T^{2} - 5424 T^{3} + 432 T^{4} - 30 T^{5} + T^{6} \)
$71$ \( 239121 - 92421 T + 35406 T^{2} - 7104 T^{3} + 756 T^{4} - 42 T^{5} + T^{6} \)
$73$ \( 2809 + 3498 T + 312 T^{2} - 656 T^{3} + 246 T^{4} + 3 T^{5} + T^{6} \)
$79$ \( 3249 - 1539 T + 15534 T^{2} + 4776 T^{3} + 621 T^{4} + 39 T^{5} + T^{6} \)
$83$ \( 24117921 - 1591164 T + 178641 T^{2} - 4962 T^{3} + 549 T^{4} - 15 T^{5} + T^{6} \)
$89$ \( 2809 + 1272 T + 831 T^{2} + 181 T^{3} - 12 T^{4} - 3 T^{5} + T^{6} \)
$97$ \( 289 - 306 T + 1062 T^{2} - 415 T^{3} + 54 T^{4} + 18 T^{5} + T^{6} \)
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