Properties

Label 567.2.f.d
Level $567$
Weight $2$
Character orbit 567.f
Analytic conductor $4.528$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.f (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.52751779461\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 189)
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.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{4} -3 \zeta_{6} q^{5} + ( -1 + \zeta_{6} ) q^{7} +O(q^{10})\) \( q + 2 \zeta_{6} q^{4} -3 \zeta_{6} q^{5} + ( -1 + \zeta_{6} ) q^{7} + ( -6 + 6 \zeta_{6} ) q^{11} + 4 \zeta_{6} q^{13} + ( -4 + 4 \zeta_{6} ) q^{16} + 3 q^{17} + 2 q^{19} + ( 6 - 6 \zeta_{6} ) q^{20} + 6 \zeta_{6} q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} -2 q^{28} + ( 6 - 6 \zeta_{6} ) q^{29} + 4 \zeta_{6} q^{31} + 3 q^{35} -7 q^{37} + 3 \zeta_{6} q^{41} + ( 1 - \zeta_{6} ) q^{43} -12 q^{44} + ( -9 + 9 \zeta_{6} ) q^{47} -\zeta_{6} q^{49} + ( -8 + 8 \zeta_{6} ) q^{52} -6 q^{53} + 18 q^{55} -9 \zeta_{6} q^{59} + ( 10 - 10 \zeta_{6} ) q^{61} -8 q^{64} + ( 12 - 12 \zeta_{6} ) q^{65} + 4 \zeta_{6} q^{67} + 6 \zeta_{6} q^{68} + 2 q^{73} + 4 \zeta_{6} q^{76} -6 \zeta_{6} q^{77} + ( 1 - \zeta_{6} ) q^{79} + 12 q^{80} + ( -3 + 3 \zeta_{6} ) q^{83} -9 \zeta_{6} q^{85} + 6 q^{89} -4 q^{91} + ( -12 + 12 \zeta_{6} ) q^{92} -6 \zeta_{6} q^{95} + ( 10 - 10 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{4} - 3q^{5} - q^{7} + O(q^{10}) \) \( 2q + 2q^{4} - 3q^{5} - q^{7} - 6q^{11} + 4q^{13} - 4q^{16} + 6q^{17} + 4q^{19} + 6q^{20} + 6q^{23} - 4q^{25} - 4q^{28} + 6q^{29} + 4q^{31} + 6q^{35} - 14q^{37} + 3q^{41} + q^{43} - 24q^{44} - 9q^{47} - q^{49} - 8q^{52} - 12q^{53} + 36q^{55} - 9q^{59} + 10q^{61} - 16q^{64} + 12q^{65} + 4q^{67} + 6q^{68} + 4q^{73} + 4q^{76} - 6q^{77} + q^{79} + 24q^{80} - 3q^{83} - 9q^{85} + 12q^{89} - 8q^{91} - 12q^{92} - 6q^{95} + 10q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
190.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 1.00000 1.73205i −1.50000 + 2.59808i 0 −0.500000 0.866025i 0 0 0
379.1 0 0 1.00000 + 1.73205i −1.50000 2.59808i 0 −0.500000 + 0.866025i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 567.2.f.d 2
3.b odd 2 1 567.2.f.e 2
9.c even 3 1 189.2.a.c yes 1
9.c even 3 1 inner 567.2.f.d 2
9.d odd 6 1 189.2.a.b 1
9.d odd 6 1 567.2.f.e 2
36.f odd 6 1 3024.2.a.y 1
36.h even 6 1 3024.2.a.f 1
45.h odd 6 1 4725.2.a.i 1
45.j even 6 1 4725.2.a.k 1
63.l odd 6 1 1323.2.a.h 1
63.o even 6 1 1323.2.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.a.b 1 9.d odd 6 1
189.2.a.c yes 1 9.c even 3 1
567.2.f.d 2 1.a even 1 1 trivial
567.2.f.d 2 9.c even 3 1 inner
567.2.f.e 2 3.b odd 2 1
567.2.f.e 2 9.d odd 6 1
1323.2.a.h 1 63.l odd 6 1
1323.2.a.l 1 63.o even 6 1
3024.2.a.f 1 36.h even 6 1
3024.2.a.y 1 36.f odd 6 1
4725.2.a.i 1 45.h odd 6 1
4725.2.a.k 1 45.j even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(567, [\chi])\):

\( T_{2} \)
\( T_{5}^{2} + 3 T_{5} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 9 + 3 T + T^{2} \)
$7$ \( 1 + T + T^{2} \)
$11$ \( 36 + 6 T + T^{2} \)
$13$ \( 16 - 4 T + T^{2} \)
$17$ \( ( -3 + T )^{2} \)
$19$ \( ( -2 + T )^{2} \)
$23$ \( 36 - 6 T + T^{2} \)
$29$ \( 36 - 6 T + T^{2} \)
$31$ \( 16 - 4 T + T^{2} \)
$37$ \( ( 7 + T )^{2} \)
$41$ \( 9 - 3 T + T^{2} \)
$43$ \( 1 - T + T^{2} \)
$47$ \( 81 + 9 T + T^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( 81 + 9 T + T^{2} \)
$61$ \( 100 - 10 T + T^{2} \)
$67$ \( 16 - 4 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( -2 + T )^{2} \)
$79$ \( 1 - T + T^{2} \)
$83$ \( 9 + 3 T + T^{2} \)
$89$ \( ( -6 + T )^{2} \)
$97$ \( 100 - 10 T + T^{2} \)
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