Properties

Label 567.2.o.a
Level 567
Weight 2
Character orbit 567.o
Analytic conductor 4.528
Analytic rank 1
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.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.52751779461\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q -2 \zeta_{6} q^{4} + ( -2 - \zeta_{6} ) q^{7} +O(q^{10})\) \( q -2 \zeta_{6} q^{4} + ( -2 - \zeta_{6} ) q^{7} + ( -6 + 3 \zeta_{6} ) q^{13} + ( -4 + 4 \zeta_{6} ) q^{16} + ( -3 + 6 \zeta_{6} ) q^{19} + ( 5 - 5 \zeta_{6} ) q^{25} + ( -2 + 6 \zeta_{6} ) q^{28} + ( -12 + 6 \zeta_{6} ) q^{31} -11 q^{37} + ( 8 - 8 \zeta_{6} ) q^{43} + ( 3 + 5 \zeta_{6} ) q^{49} + ( 6 + 6 \zeta_{6} ) q^{52} + ( -9 - 9 \zeta_{6} ) q^{61} + 8 q^{64} -5 \zeta_{6} q^{67} + ( 9 - 18 \zeta_{6} ) q^{73} + ( 12 - 6 \zeta_{6} ) q^{76} + ( -17 + 17 \zeta_{6} ) q^{79} + ( 15 - 3 \zeta_{6} ) q^{91} + ( 3 + 3 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 5q^{7} + O(q^{10}) \) \( 2q - 2q^{4} - 5q^{7} - 9q^{13} - 4q^{16} + 5q^{25} + 2q^{28} - 18q^{31} - 22q^{37} + 8q^{43} + 11q^{49} + 18q^{52} - 27q^{61} + 16q^{64} - 5q^{67} + 18q^{76} - 17q^{79} + 27q^{91} + 9q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
188.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 −1.00000 1.73205i 0 0 −2.50000 0.866025i 0 0 0
377.1 0 0 −1.00000 + 1.73205i 0 0 −2.50000 + 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
63.l odd 6 1 inner
63.o 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.o.a 2
3.b odd 2 1 CM 567.2.o.a 2
7.b odd 2 1 567.2.o.b 2
9.c even 3 1 189.2.c.a 2
9.c even 3 1 567.2.o.b 2
9.d odd 6 1 189.2.c.a 2
9.d odd 6 1 567.2.o.b 2
21.c even 2 1 567.2.o.b 2
36.f odd 6 1 3024.2.k.b 2
36.h even 6 1 3024.2.k.b 2
63.l odd 6 1 189.2.c.a 2
63.l odd 6 1 inner 567.2.o.a 2
63.o even 6 1 189.2.c.a 2
63.o even 6 1 inner 567.2.o.a 2
252.s odd 6 1 3024.2.k.b 2
252.bi even 6 1 3024.2.k.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.c.a 2 9.c even 3 1
189.2.c.a 2 9.d odd 6 1
189.2.c.a 2 63.l odd 6 1
189.2.c.a 2 63.o even 6 1
567.2.o.a 2 1.a even 1 1 trivial
567.2.o.a 2 3.b odd 2 1 CM
567.2.o.a 2 63.l odd 6 1 inner
567.2.o.a 2 63.o even 6 1 inner
567.2.o.b 2 7.b odd 2 1
567.2.o.b 2 9.c even 3 1
567.2.o.b 2 9.d odd 6 1
567.2.o.b 2 21.c even 2 1
3024.2.k.b 2 36.f odd 6 1
3024.2.k.b 2 36.h even 6 1
3024.2.k.b 2 252.s odd 6 1
3024.2.k.b 2 252.bi even 6 1

Hecke kernels

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

\( T_{2} \)
\( T_{13}^{2} + 9 T_{13} + 27 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T^{2} + 4 T^{4} \)
$3$ 1
$5$ \( 1 - 5 T^{2} + 25 T^{4} \)
$7$ \( 1 + 5 T + 7 T^{2} \)
$11$ \( 1 + 11 T^{2} + 121 T^{4} \)
$13$ \( ( 1 + 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \)
$17$ \( ( 1 + 17 T^{2} )^{2} \)
$19$ \( ( 1 - 7 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \)
$23$ \( 1 + 23 T^{2} + 529 T^{4} \)
$29$ \( 1 + 29 T^{2} + 841 T^{4} \)
$31$ \( ( 1 + 7 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \)
$37$ \( ( 1 + 11 T + 37 T^{2} )^{2} \)
$41$ \( 1 - 41 T^{2} + 1681 T^{4} \)
$43$ \( ( 1 - 13 T + 43 T^{2} )( 1 + 5 T + 43 T^{2} ) \)
$47$ \( 1 - 47 T^{2} + 2209 T^{4} \)
$53$ \( ( 1 - 53 T^{2} )^{2} \)
$59$ \( 1 - 59 T^{2} + 3481 T^{4} \)
$61$ \( ( 1 + 13 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \)
$67$ \( ( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \)
$71$ \( ( 1 - 71 T^{2} )^{2} \)
$73$ \( ( 1 - 7 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \)
$79$ \( ( 1 + 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \)
$83$ \( 1 - 83 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 + 89 T^{2} )^{2} \)
$97$ \( ( 1 - 14 T + 97 T^{2} )( 1 + 5 T + 97 T^{2} ) \)
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