Properties

Label 570.2.i.c
Level $570$
Weight $2$
Character orbit 570.i
Analytic conductor $4.551$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{2} + ( - \zeta_{6} + 1) q^{3} - \zeta_{6} q^{4} + ( - \zeta_{6} + 1) q^{5} + \zeta_{6} q^{6} + 3 q^{7} + q^{8} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{2} + ( - \zeta_{6} + 1) q^{3} - \zeta_{6} q^{4} + ( - \zeta_{6} + 1) q^{5} + \zeta_{6} q^{6} + 3 q^{7} + q^{8} - \zeta_{6} q^{9} + \zeta_{6} q^{10} + q^{11} - q^{12} - 2 \zeta_{6} q^{13} + (3 \zeta_{6} - 3) q^{14} - \zeta_{6} q^{15} + (\zeta_{6} - 1) q^{16} + (2 \zeta_{6} - 2) q^{17} + q^{18} + ( - 5 \zeta_{6} + 3) q^{19} - q^{20} + ( - 3 \zeta_{6} + 3) q^{21} + (\zeta_{6} - 1) q^{22} - \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{24} - \zeta_{6} q^{25} + 2 q^{26} - q^{27} - 3 \zeta_{6} q^{28} + q^{30} + 4 q^{31} - \zeta_{6} q^{32} + ( - \zeta_{6} + 1) q^{33} - 2 \zeta_{6} q^{34} + ( - 3 \zeta_{6} + 3) q^{35} + (\zeta_{6} - 1) q^{36} + 9 q^{37} + (3 \zeta_{6} + 2) q^{38} - 2 q^{39} + ( - \zeta_{6} + 1) q^{40} + (\zeta_{6} - 1) q^{41} + 3 \zeta_{6} q^{42} + ( - 10 \zeta_{6} + 10) q^{43} - \zeta_{6} q^{44} - q^{45} + q^{46} + \zeta_{6} q^{48} + 2 q^{49} + q^{50} + 2 \zeta_{6} q^{51} + (2 \zeta_{6} - 2) q^{52} - 3 \zeta_{6} q^{53} + ( - \zeta_{6} + 1) q^{54} + ( - \zeta_{6} + 1) q^{55} + 3 q^{56} + ( - 3 \zeta_{6} - 2) q^{57} + (12 \zeta_{6} - 12) q^{59} + (\zeta_{6} - 1) q^{60} + 2 \zeta_{6} q^{61} + (4 \zeta_{6} - 4) q^{62} - 3 \zeta_{6} q^{63} + q^{64} - 2 q^{65} + \zeta_{6} q^{66} + 2 \zeta_{6} q^{67} + 2 q^{68} - q^{69} + 3 \zeta_{6} q^{70} + (8 \zeta_{6} - 8) q^{71} - \zeta_{6} q^{72} + ( - 12 \zeta_{6} + 12) q^{73} + (9 \zeta_{6} - 9) q^{74} - q^{75} + (2 \zeta_{6} - 5) q^{76} + 3 q^{77} + ( - 2 \zeta_{6} + 2) q^{78} + (14 \zeta_{6} - 14) q^{79} + \zeta_{6} q^{80} + (\zeta_{6} - 1) q^{81} - \zeta_{6} q^{82} + 6 q^{83} - 3 q^{84} + 2 \zeta_{6} q^{85} + 10 \zeta_{6} q^{86} + q^{88} + 5 \zeta_{6} q^{89} + ( - \zeta_{6} + 1) q^{90} - 6 \zeta_{6} q^{91} + (\zeta_{6} - 1) q^{92} + ( - 4 \zeta_{6} + 4) q^{93} + ( - 3 \zeta_{6} - 2) q^{95} - q^{96} + (8 \zeta_{6} - 8) q^{97} + (2 \zeta_{6} - 2) q^{98} - \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + q^{3} - q^{4} + q^{5} + q^{6} + 6 q^{7} + 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + q^{3} - q^{4} + q^{5} + q^{6} + 6 q^{7} + 2 q^{8} - q^{9} + q^{10} + 2 q^{11} - 2 q^{12} - 2 q^{13} - 3 q^{14} - q^{15} - q^{16} - 2 q^{17} + 2 q^{18} + q^{19} - 2 q^{20} + 3 q^{21} - q^{22} - q^{23} + q^{24} - q^{25} + 4 q^{26} - 2 q^{27} - 3 q^{28} + 2 q^{30} + 8 q^{31} - q^{32} + q^{33} - 2 q^{34} + 3 q^{35} - q^{36} + 18 q^{37} + 7 q^{38} - 4 q^{39} + q^{40} - q^{41} + 3 q^{42} + 10 q^{43} - q^{44} - 2 q^{45} + 2 q^{46} + q^{48} + 4 q^{49} + 2 q^{50} + 2 q^{51} - 2 q^{52} - 3 q^{53} + q^{54} + q^{55} + 6 q^{56} - 7 q^{57} - 12 q^{59} - q^{60} + 2 q^{61} - 4 q^{62} - 3 q^{63} + 2 q^{64} - 4 q^{65} + q^{66} + 2 q^{67} + 4 q^{68} - 2 q^{69} + 3 q^{70} - 8 q^{71} - q^{72} + 12 q^{73} - 9 q^{74} - 2 q^{75} - 8 q^{76} + 6 q^{77} + 2 q^{78} - 14 q^{79} + q^{80} - q^{81} - q^{82} + 12 q^{83} - 6 q^{84} + 2 q^{85} + 10 q^{86} + 2 q^{88} + 5 q^{89} + q^{90} - 6 q^{91} - q^{92} + 4 q^{93} - 7 q^{95} - 2 q^{96} - 8 q^{97} - 2 q^{98} - q^{99}+O(q^{100}) \) 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 3.00000 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 3.00000 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.c 2
3.b odd 2 1 1710.2.l.i 2
19.c even 3 1 inner 570.2.i.c 2
57.h odd 6 1 1710.2.l.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.i.c 2 1.a even 1 1 trivial
570.2.i.c 2 19.c even 3 1 inner
1710.2.l.i 2 3.b odd 2 1
1710.2.l.i 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 \(S_{2}^{\mathrm{new}}(570, [\chi])\):

\( T_{7} - 3 \) Copy content Toggle raw display
\( T_{11} - 1 \) 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
$7$ \( (T - 3)^{2} \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} - T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( (T - 9)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$43$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$59$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$61$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$71$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$73$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$79$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$97$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
show more
show less