Properties

Label 570.2.u.c
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.

\(f(q)\) \(=\) \( q + (\zeta_{18}^{4} - \zeta_{18}) q^{2} - \zeta_{18} q^{3} - \zeta_{18}^{5} q^{4} + \zeta_{18}^{4} q^{5} + ( - \zeta_{18}^{5} + \zeta_{18}^{2}) q^{6} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{7} + \zeta_{18}^{3} q^{8} + \zeta_{18}^{2} q^{9} +O(q^{10}) \) 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} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{7} + \zeta_{18}^{3} q^{8} + \zeta_{18}^{2} q^{9} - \zeta_{18}^{2} q^{10} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} - \zeta_{18}) q^{11} + (\zeta_{18}^{3} - 1) q^{12} + ( - 2 \zeta_{18}^{5} + \zeta_{18}^{3} + 2 \zeta_{18}^{2} - \zeta_{18} - 1) q^{13} + ( - \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{14} - \zeta_{18}^{5} q^{15} - \zeta_{18} q^{16} + ( - \zeta_{18}^{5} + 2 \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 2 \zeta_{18} + 1) q^{17} - q^{18} + ( - 4 \zeta_{18}^{5} + 2 \zeta_{18}^{4} - \zeta_{18}^{3} + 2 \zeta_{18}^{2} + 2) q^{19} + q^{20} + (\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18} - 1) q^{21} + ( - \zeta_{18}^{5} + \zeta_{18}^{3} + \zeta_{18} - 1) q^{22} + (\zeta_{18}^{5} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - \zeta_{18} - 2) q^{23} - \zeta_{18}^{4} q^{24} + (\zeta_{18}^{5} - \zeta_{18}^{2}) q^{25} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + 2 \zeta_{18}^{3} + \zeta_{18}^{2} - 2) q^{26} - \zeta_{18}^{3} q^{27} + ( - \zeta_{18}^{4} + \zeta_{18}^{2} - 1) q^{28} + (2 \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18}^{2} + \zeta_{18} + 2) q^{29} + \zeta_{18}^{3} q^{30} + ( - 2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 4 \zeta_{18}^{3} - \zeta_{18}^{2} + 3 \zeta_{18} + 4) q^{31} + ( - \zeta_{18}^{5} + \zeta_{18}^{2}) q^{32} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{2} - 1) q^{33} + ( - 2 \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{3} + 2 \zeta_{18} - 3) q^{34} + ( - \zeta_{18}^{5} + \zeta_{18}^{3} + \zeta_{18}^{2} - \zeta_{18}) q^{35} + ( - \zeta_{18}^{4} + \zeta_{18}) q^{36} + ( - 3 \zeta_{18}^{5} - \zeta_{18}^{4} + 4 \zeta_{18}^{2} + 4 \zeta_{18} - 3) q^{37} + (2 \zeta_{18}^{4} + 4 \zeta_{18}^{3} - 2 \zeta_{18}^{2} - \zeta_{18} - 2) q^{38} + ( - \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} - 2) q^{39} + (\zeta_{18}^{4} - \zeta_{18}) q^{40} + (3 \zeta_{18}^{2} + 5 \zeta_{18} + 3) q^{41} + (\zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}) q^{42} + (5 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 2) q^{43} + (\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}^{2}) q^{44} + (\zeta_{18}^{3} - 1) q^{45} + ( - \zeta_{18}^{5} - 2 \zeta_{18}^{4} - \zeta_{18}^{3} - 2 \zeta_{18}^{2} - \zeta_{18}) q^{46} + (5 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + \zeta_{18}^{2} - 2 \zeta_{18} + 5) q^{47} + \zeta_{18}^{2} q^{48} + (3 \zeta_{18}^{5} - \zeta_{18}^{4} + 4 \zeta_{18}^{3} - \zeta_{18}^{2} + 3 \zeta_{18}) q^{49} + ( - \zeta_{18}^{3} + 1) q^{50} + ( - 2 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - \zeta_{18} - 1) q^{51} + ( - 2 \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{52} + (8 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 3 \zeta_{18} + 1) q^{53} + \zeta_{18} q^{54} + ( - \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}^{2} + \zeta_{18}) q^{55} + ( - \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} - 1) q^{56} + ( - 2 \zeta_{18}^{5} + \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 2 \zeta_{18} - 4) q^{57} + (\zeta_{18}^{5} + 2 \zeta_{18}^{4} - 3 \zeta_{18}^{2} - 3 \zeta_{18} - 1) q^{58} + ( - 6 \zeta_{18}^{5} + \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - \zeta_{18} + 6) q^{59} - \zeta_{18} q^{60} + (2 \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} + 8 \zeta_{18} - 7) q^{61} + (3 \zeta_{18}^{5} + 4 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - \zeta_{18}^{2} + 1) q^{62} + (\zeta_{18}^{5} - \zeta_{18}^{3} - \zeta_{18}^{2} + \zeta_{18} + 1) q^{63} + (\zeta_{18}^{3} - 1) q^{64} + ( - \zeta_{18}^{5} + 2 \zeta_{18}^{3} - \zeta_{18}) q^{65} + ( - \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}^{2} + \zeta_{18} - 1) q^{66} + ( - \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 5 \zeta_{18}^{2} - 2 \zeta_{18} - 1) q^{67} + (2 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 2 \zeta_{18}) q^{68} + ( - 3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} + 2 \zeta_{18} + 1) q^{69} + ( - \zeta_{18}^{5} + \zeta_{18}^{3} + \zeta_{18}^{2} - \zeta_{18} - 1) q^{70} + (\zeta_{18}^{5} - \zeta_{18}^{4} - 4 \zeta_{18}^{3} - 5 \zeta_{18}^{2} + 5) q^{71} + \zeta_{18}^{5} q^{72} + (5 \zeta_{18}^{5} - 5 \zeta_{18}^{3} - 2 \zeta_{18} + 5) q^{73} + (4 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 3 \zeta_{18} - 4) q^{74} + q^{75} + ( - \zeta_{18}^{5} - 2 \zeta_{18}^{4} - \zeta_{18}^{2} - 2 \zeta_{18} + 2) q^{76} + (\zeta_{18}^{5} - \zeta_{18}^{4}) q^{77} + (\zeta_{18}^{5} - 2 \zeta_{18}^{4} + 2 \zeta_{18} - 1) q^{78} + ( - \zeta_{18}^{5} + \zeta_{18}^{3} + 3 \zeta_{18} - 1) q^{79} - \zeta_{18}^{5} q^{80} + \zeta_{18}^{4} q^{81} + (5 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - 5 \zeta_{18}^{2} - 3 \zeta_{18} - 3) q^{82} + (\zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 3 \zeta_{18} - 1) q^{83} + (\zeta_{18}^{5} - \zeta_{18}^{3} + \zeta_{18}) q^{84} + ( - 2 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 3 \zeta_{18} - 2) q^{85} + (2 \zeta_{18}^{4} - 5 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 5 \zeta_{18} + 2) q^{86} + ( - 2 \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} - \zeta_{18}^{2} - 2 \zeta_{18}) q^{87} + ( - \zeta_{18}^{3} + \zeta_{18}^{2} - \zeta_{18} + 1) q^{88} + ( - 3 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + 7 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 5 \zeta_{18} - 5) q^{89} - \zeta_{18}^{4} q^{90} + (4 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 4 \zeta_{18} - 1) q^{91} + ( - \zeta_{18}^{5} + \zeta_{18}^{3} + 3 \zeta_{18}^{2} + \zeta_{18} + 2) q^{92} + (2 \zeta_{18}^{5} + 4 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{2} - 4 \zeta_{18} - 2) q^{93} + ( - 2 \zeta_{18}^{5} + 5 \zeta_{18}^{4} - 3 \zeta_{18}^{2} - 3 \zeta_{18} - 1) q^{94} + (2 \zeta_{18}^{5} + \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + \zeta_{18} + 2) q^{95} - q^{96} + ( - \zeta_{18}^{4} - 8 \zeta_{18}^{3} + 8 \zeta_{18}^{2} + \zeta_{18}) q^{97} + (3 \zeta_{18}^{5} - 3 \zeta_{18}^{3} - 2 \zeta_{18}^{2} - 4 \zeta_{18} + 1) q^{98} + ( - \zeta_{18}^{5} + \zeta_{18} - 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{7} + 3 q^{8} - 3 q^{11} - 3 q^{12} - 3 q^{13} - 3 q^{14} - 3 q^{17} - 6 q^{18} + 9 q^{19} + 6 q^{20} - 6 q^{21} - 3 q^{22} - 3 q^{23} - 6 q^{26} - 3 q^{27} - 6 q^{28} + 15 q^{29} + 3 q^{30} + 12 q^{31} - 6 q^{33} - 15 q^{34} + 3 q^{35} - 18 q^{37} - 12 q^{39} + 18 q^{41} - 3 q^{42} + 21 q^{43} + 3 q^{44} - 3 q^{45} - 3 q^{46} + 24 q^{47} + 12 q^{49} + 3 q^{50} - 12 q^{51} - 3 q^{52} + 12 q^{53} + 3 q^{55} - 6 q^{56} - 18 q^{57} - 6 q^{58} + 27 q^{59} - 45 q^{61} + 12 q^{62} + 3 q^{63} - 3 q^{64} + 6 q^{65} - 3 q^{66} - 12 q^{67} + 6 q^{68} + 3 q^{69} - 3 q^{70} + 18 q^{71} + 15 q^{73} - 15 q^{74} + 6 q^{75} + 12 q^{76} - 6 q^{78} - 3 q^{79} - 18 q^{82} - 3 q^{83} - 3 q^{84} - 3 q^{85} - 3 q^{86} - 3 q^{87} + 3 q^{88} - 9 q^{89} - 15 q^{91} + 15 q^{92} - 3 q^{93} - 6 q^{94} + 18 q^{95} - 6 q^{96} - 24 q^{97} - 3 q^{98} - 6 q^{99}+O(q^{100}) \) 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 −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.c 6
19.e even 9 1 inner 570.2.u.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.u.c 6 1.a even 1 1 trivial
570.2.u.c 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 \(S_{2}^{\mathrm{new}}(570, [\chi])\):

\( T_{7}^{6} + 3T_{7}^{5} + 9T_{7}^{4} + 2T_{7}^{3} + 3T_{7}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{6} + 3T_{11}^{5} + 9T_{11}^{4} + 2T_{11}^{3} + 3T_{11}^{2} + 1 \) 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
$7$ \( T^{6} + 3 T^{5} + 9 T^{4} + 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{6} + 3 T^{5} + 9 T^{4} + 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{6} + 3 T^{5} + 18 T^{4} + 24 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$17$ \( T^{6} + 3 T^{5} + 54 T^{4} + 24 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$19$ \( T^{6} - 9 T^{5} + 179 T^{3} + \cdots + 6859 \) Copy content Toggle raw display
$23$ \( T^{6} + 3 T^{5} + 39 T^{4} + 28 T^{3} + \cdots + 289 \) Copy content Toggle raw display
$29$ \( T^{6} - 15 T^{5} + 99 T^{4} - 300 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$31$ \( T^{6} - 12 T^{5} + 117 T^{4} + \cdots + 1369 \) Copy content Toggle raw display
$37$ \( (T^{3} + 9 T^{2} - 12 T - 179)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 18 T^{5} + 90 T^{4} + \cdots + 289 \) Copy content Toggle raw display
$43$ \( T^{6} - 21 T^{5} + 132 T^{4} + \cdots + 1369 \) Copy content Toggle raw display
$47$ \( T^{6} - 24 T^{5} + 270 T^{4} + \cdots + 47961 \) Copy content Toggle raw display
$53$ \( T^{6} - 12 T^{5} + 93 T^{4} + \cdots + 94249 \) Copy content Toggle raw display
$59$ \( T^{6} - 27 T^{5} + 324 T^{4} + \cdots + 104329 \) Copy content Toggle raw display
$61$ \( T^{6} + 45 T^{5} + 882 T^{4} + \cdots + 237169 \) Copy content Toggle raw display
$67$ \( T^{6} + 12 T^{5} + 54 T^{4} + \cdots + 3249 \) Copy content Toggle raw display
$71$ \( T^{6} - 18 T^{5} + 180 T^{4} + \cdots + 54289 \) Copy content Toggle raw display
$73$ \( T^{6} - 15 T^{5} + 120 T^{4} + \cdots + 289 \) Copy content Toggle raw display
$79$ \( T^{6} + 3 T^{5} - 3 T^{4} - 64 T^{3} + \cdots + 361 \) Copy content Toggle raw display
$83$ \( T^{6} + 3 T^{5} + 27 T^{4} + \cdots + 3249 \) Copy content Toggle raw display
$89$ \( T^{6} + 9 T^{5} + 252 T^{4} + \cdots + 23409 \) Copy content Toggle raw display
$97$ \( T^{6} + 24 T^{5} + 336 T^{4} + \cdots + 494209 \) Copy content Toggle raw display
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