Properties

Label 570.2.i.b
Level $570$
Weight $2$
Character orbit 570.i
Analytic conductor $4.551$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(191\) \(211\) \(457\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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

\( T_{7} + 1 \) Copy content Toggle raw display
\( T_{11} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T - 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$17$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$19$ \( T^{2} - 8T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$31$ \( (T - 1)^{2} \) Copy content Toggle raw display
$37$ \( (T + 5)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$43$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$59$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$61$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$67$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$79$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$83$ \( (T - 2)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
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