Properties

Label 567.2.s.b
Level $567$
Weight $2$
Character orbit 567.s
Analytic conductor $4.528$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.s (of order \(6\), 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: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 189)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q -2 \zeta_{6} q^{4} + ( 3 - 2 \zeta_{6} ) q^{7} +O(q^{10})\) \( q -2 \zeta_{6} q^{4} + ( 3 - 2 \zeta_{6} ) q^{7} + ( -4 - 4 \zeta_{6} ) q^{13} + ( -4 + 4 \zeta_{6} ) q^{16} + ( 6 - 3 \zeta_{6} ) q^{19} -5 q^{25} + ( -4 - 2 \zeta_{6} ) q^{28} + ( 2 - \zeta_{6} ) q^{31} -10 \zeta_{6} q^{37} -13 \zeta_{6} q^{43} + ( 5 - 8 \zeta_{6} ) q^{49} + ( -8 + 16 \zeta_{6} ) q^{52} + ( 5 + 5 \zeta_{6} ) q^{61} + 8 q^{64} + 16 \zeta_{6} q^{67} + ( 9 + 9 \zeta_{6} ) q^{73} + ( -6 - 6 \zeta_{6} ) q^{76} + ( 4 - 4 \zeta_{6} ) q^{79} + ( -20 + 4 \zeta_{6} ) q^{91} + ( 22 - 11 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 4q^{7} + O(q^{10}) \) \( 2q - 2q^{4} + 4q^{7} - 12q^{13} - 4q^{16} + 9q^{19} - 10q^{25} - 10q^{28} + 3q^{31} - 10q^{37} - 13q^{43} + 2q^{49} + 15q^{61} + 16q^{64} + 16q^{67} + 27q^{73} - 18q^{76} + 4q^{79} - 36q^{91} + 33q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 −1.00000 + 1.73205i 0 0 2.00000 + 1.73205i 0 0 0
458.1 0 0 −1.00000 1.73205i 0 0 2.00000 1.73205i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
63.k odd 6 1 inner
63.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 567.2.s.b 2
3.b odd 2 1 CM 567.2.s.b 2
7.d odd 6 1 567.2.i.a 2
9.c even 3 1 189.2.p.a 2
9.c even 3 1 567.2.i.a 2
9.d odd 6 1 189.2.p.a 2
9.d odd 6 1 567.2.i.a 2
21.g even 6 1 567.2.i.a 2
63.g even 3 1 1323.2.c.a 2
63.i even 6 1 189.2.p.a 2
63.k odd 6 1 inner 567.2.s.b 2
63.k odd 6 1 1323.2.c.a 2
63.n odd 6 1 1323.2.c.a 2
63.s even 6 1 inner 567.2.s.b 2
63.s even 6 1 1323.2.c.a 2
63.t odd 6 1 189.2.p.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.p.a 2 9.c even 3 1
189.2.p.a 2 9.d odd 6 1
189.2.p.a 2 63.i even 6 1
189.2.p.a 2 63.t odd 6 1
567.2.i.a 2 7.d odd 6 1
567.2.i.a 2 9.c even 3 1
567.2.i.a 2 9.d odd 6 1
567.2.i.a 2 21.g even 6 1
567.2.s.b 2 1.a even 1 1 trivial
567.2.s.b 2 3.b odd 2 1 CM
567.2.s.b 2 63.k odd 6 1 inner
567.2.s.b 2 63.s even 6 1 inner
1323.2.c.a 2 63.g even 3 1
1323.2.c.a 2 63.k odd 6 1
1323.2.c.a 2 63.n odd 6 1
1323.2.c.a 2 63.s even 6 1

Hecke kernels

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

\( T_{2} \)
\( T_{11} \)
\( T_{13}^{2} + 12 T_{13} + 48 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 7 - 4 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 48 + 12 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 27 - 9 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 3 - 3 T + T^{2} \)
$37$ \( 100 + 10 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 169 + 13 T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 75 - 15 T + T^{2} \)
$67$ \( 256 - 16 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 243 - 27 T + T^{2} \)
$79$ \( 16 - 4 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 363 - 33 T + T^{2} \)
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