Properties

Label 2592.1.bd.a
Level $2592$
Weight $1$
Character orbit 2592.bd
Analytic conductor $1.294$
Analytic rank $0$
Dimension $6$
Projective image $D_{9}$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2592 = 2^{5} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2592.bd (of order \(18\), degree \(6\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.29357651274\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \(x^{6} - x^{3} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 216)
Projective image: \(D_{9}\)
Projective field: Galois closure of 9.1.128536820158464.7

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q +O(q^{10})\) \( q + ( \zeta_{18}^{6} + \zeta_{18}^{8} ) q^{11} + ( \zeta_{18} - \zeta_{18}^{2} ) q^{17} + ( \zeta_{18} + \zeta_{18}^{5} ) q^{19} + \zeta_{18}^{4} q^{25} + ( \zeta_{18}^{3} + \zeta_{18}^{7} ) q^{41} + ( -\zeta_{18}^{2} + \zeta_{18}^{3} ) q^{43} + \zeta_{18}^{2} q^{49} + ( 1 + \zeta_{18}^{4} ) q^{59} + ( -\zeta_{18}^{4} - \zeta_{18}^{6} ) q^{67} + ( -\zeta_{18}^{7} + \zeta_{18}^{8} ) q^{73} -\zeta_{18}^{4} q^{83} -\zeta_{18}^{3} q^{89} + ( \zeta_{18}^{6} + \zeta_{18}^{8} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + O(q^{10}) \) \( 6q - 3q^{11} + 3q^{41} + 3q^{43} + 6q^{59} + 3q^{67} - 3q^{89} - 3q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(-1\) \(\zeta_{18}^{8}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
559.1
0.939693 0.342020i
−0.173648 0.984808i
−0.173648 + 0.984808i
0.939693 + 0.342020i
−0.766044 0.642788i
−0.766044 + 0.642788i
0 0 0 0 0 0 0 0 0
847.1 0 0 0 0 0 0 0 0 0
1423.1 0 0 0 0 0 0 0 0 0
1711.1 0 0 0 0 0 0 0 0 0
2287.1 0 0 0 0 0 0 0 0 0
2575.1 0 0 0 0 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2575.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
27.e even 9 1 inner
216.r odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2592.1.bd.a 6
3.b odd 2 1 864.1.bd.a 6
4.b odd 2 1 648.1.r.a 6
8.b even 2 1 648.1.r.a 6
8.d odd 2 1 CM 2592.1.bd.a 6
12.b even 2 1 216.1.r.a 6
24.f even 2 1 864.1.bd.a 6
24.h odd 2 1 216.1.r.a 6
27.e even 9 1 inner 2592.1.bd.a 6
27.f odd 18 1 864.1.bd.a 6
36.f odd 6 1 1944.1.r.c 6
36.f odd 6 1 1944.1.r.d 6
36.h even 6 1 1944.1.r.a 6
36.h even 6 1 1944.1.r.b 6
72.j odd 6 1 1944.1.r.a 6
72.j odd 6 1 1944.1.r.b 6
72.n even 6 1 1944.1.r.c 6
72.n even 6 1 1944.1.r.d 6
108.j odd 18 1 648.1.r.a 6
108.j odd 18 1 1944.1.r.c 6
108.j odd 18 1 1944.1.r.d 6
108.l even 18 1 216.1.r.a 6
108.l even 18 1 1944.1.r.a 6
108.l even 18 1 1944.1.r.b 6
216.r odd 18 1 inner 2592.1.bd.a 6
216.t even 18 1 648.1.r.a 6
216.t even 18 1 1944.1.r.c 6
216.t even 18 1 1944.1.r.d 6
216.v even 18 1 864.1.bd.a 6
216.x odd 18 1 216.1.r.a 6
216.x odd 18 1 1944.1.r.a 6
216.x odd 18 1 1944.1.r.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.1.r.a 6 12.b even 2 1
216.1.r.a 6 24.h odd 2 1
216.1.r.a 6 108.l even 18 1
216.1.r.a 6 216.x odd 18 1
648.1.r.a 6 4.b odd 2 1
648.1.r.a 6 8.b even 2 1
648.1.r.a 6 108.j odd 18 1
648.1.r.a 6 216.t even 18 1
864.1.bd.a 6 3.b odd 2 1
864.1.bd.a 6 24.f even 2 1
864.1.bd.a 6 27.f odd 18 1
864.1.bd.a 6 216.v even 18 1
1944.1.r.a 6 36.h even 6 1
1944.1.r.a 6 72.j odd 6 1
1944.1.r.a 6 108.l even 18 1
1944.1.r.a 6 216.x odd 18 1
1944.1.r.b 6 36.h even 6 1
1944.1.r.b 6 72.j odd 6 1
1944.1.r.b 6 108.l even 18 1
1944.1.r.b 6 216.x odd 18 1
1944.1.r.c 6 36.f odd 6 1
1944.1.r.c 6 72.n even 6 1
1944.1.r.c 6 108.j odd 18 1
1944.1.r.c 6 216.t even 18 1
1944.1.r.d 6 36.f odd 6 1
1944.1.r.d 6 72.n even 6 1
1944.1.r.d 6 108.j odd 18 1
1944.1.r.d 6 216.t even 18 1
2592.1.bd.a 6 1.a even 1 1 trivial
2592.1.bd.a 6 8.d odd 2 1 CM
2592.1.bd.a 6 27.e even 9 1 inner
2592.1.bd.a 6 216.r odd 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2592, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( T^{6} \)
$5$ \( T^{6} \)
$7$ \( T^{6} \)
$11$ \( 1 + 6 T + 12 T^{2} + 8 T^{3} + 6 T^{4} + 3 T^{5} + T^{6} \)
$13$ \( T^{6} \)
$17$ \( 1 - 3 T + 9 T^{2} - 2 T^{3} + 3 T^{4} + T^{6} \)
$19$ \( 1 - 3 T + 9 T^{2} - 2 T^{3} + 3 T^{4} + T^{6} \)
$23$ \( T^{6} \)
$29$ \( T^{6} \)
$31$ \( T^{6} \)
$37$ \( T^{6} \)
$41$ \( 1 - 6 T + 12 T^{2} - 8 T^{3} + 6 T^{4} - 3 T^{5} + T^{6} \)
$43$ \( 1 + 3 T + 3 T^{2} - 8 T^{3} + 6 T^{4} - 3 T^{5} + T^{6} \)
$47$ \( T^{6} \)
$53$ \( T^{6} \)
$59$ \( 1 - 3 T + 12 T^{2} - 19 T^{3} + 15 T^{4} - 6 T^{5} + T^{6} \)
$61$ \( T^{6} \)
$67$ \( 1 + 3 T + 3 T^{2} - 8 T^{3} + 6 T^{4} - 3 T^{5} + T^{6} \)
$71$ \( T^{6} \)
$73$ \( 1 + 3 T + 9 T^{2} + 2 T^{3} + 3 T^{4} + T^{6} \)
$79$ \( T^{6} \)
$83$ \( 1 - T^{3} + T^{6} \)
$89$ \( ( 1 + T + T^{2} )^{3} \)
$97$ \( 1 + 6 T + 12 T^{2} + 8 T^{3} + 6 T^{4} + 3 T^{5} + T^{6} \)
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