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: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
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.

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

Character values

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

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

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$3$ \( T^{6} + T^{3} + 1 \) Copy content Toggle raw display
$5$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$7$ \( T^{6} + 6 T^{5} + 33 T^{4} + 56 T^{3} + \cdots + 361 \) Copy content Toggle raw display
$11$ \( T^{6} - 12 T^{5} + 99 T^{4} + \cdots + 2601 \) Copy content Toggle raw display
$13$ \( T^{6} + 125 T^{3} + 15625 \) Copy content Toggle raw display
$17$ \( T^{6} - 6 T^{5} + 240 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$19$ \( T^{6} + 18 T^{5} + 162 T^{4} + \cdots + 6859 \) Copy content Toggle raw display
$23$ \( T^{6} - 21 T^{5} + 252 T^{4} + \cdots + 45369 \) Copy content Toggle raw display
$29$ \( T^{6} - 6 T^{5} + 36 T^{4} - 192 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$31$ \( T^{6} + 84 T^{4} - 592 T^{3} + \cdots + 87616 \) Copy content Toggle raw display
$37$ \( (T^{3} - 12 T^{2} + 9 T + 73)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 3 T^{5} + 54 T^{4} + \cdots + 288369 \) Copy content Toggle raw display
$43$ \( T^{6} + 80 T^{3} + 576 T^{2} + \cdots + 18496 \) Copy content Toggle raw display
$47$ \( T^{6} - 24 T^{5} + 306 T^{4} + \cdots + 47961 \) Copy content Toggle raw display
$53$ \( T^{6} - 24 T^{5} + 333 T^{4} + \cdots + 106929 \) Copy content Toggle raw display
$59$ \( T^{6} - 12 T^{5} + 162 T^{4} + \cdots + 751689 \) Copy content Toggle raw display
$61$ \( T^{6} - 640 T^{3} + 9216 T^{2} + \cdots + 1183744 \) Copy content Toggle raw display
$67$ \( T^{6} + 216 T^{4} + 1160 T^{3} + \cdots + 87616 \) Copy content Toggle raw display
$71$ \( T^{6} + 36 T^{5} + 540 T^{4} + \cdots + 46656 \) Copy content Toggle raw display
$73$ \( T^{6} - 24 T^{5} + 240 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$79$ \( T^{6} + 36 T^{5} + 720 T^{4} + \cdots + 1183744 \) Copy content Toggle raw display
$83$ \( T^{6} + 36 T^{4} + 144 T^{3} + \cdots + 5184 \) Copy content Toggle raw display
$89$ \( T^{6} - 48 T^{5} + 1035 T^{4} + \cdots + 1962801 \) Copy content Toggle raw display
$97$ \( T^{6} - 42 T^{5} + 924 T^{4} + \cdots + 2050624 \) Copy content Toggle raw display
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