Properties

Label 2592.2.s.a
Level $2592$
Weight $2$
Character orbit 2592.s
Analytic conductor $20.697$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(20.6972242039\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 864)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 + \zeta_{24} + \zeta_{24}^{4} - \zeta_{24}^{7} ) q^{5} + ( -2 \zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{7} +O(q^{10})\) \( q + ( -2 + \zeta_{24} + \zeta_{24}^{4} - \zeta_{24}^{7} ) q^{5} + ( -2 \zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{7} + ( -\zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{11} + ( 2 \zeta_{24} - 4 \zeta_{24}^{3} + 2 \zeta_{24}^{4} - 4 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{13} + ( -2 + 4 \zeta_{24}^{4} ) q^{17} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{19} + ( -2 \zeta_{24} - 2 \zeta_{24}^{7} ) q^{23} + ( -4 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{25} + ( 2 + 4 \zeta_{24}^{3} + 2 \zeta_{24}^{4} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{29} + ( \zeta_{24} + 5 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 5 \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{31} + ( 4 \zeta_{24} - 6 \zeta_{24}^{2} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 3 \zeta_{24}^{6} ) q^{35} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{37} + ( 8 + 2 \zeta_{24} - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{7} ) q^{41} + ( -4 \zeta_{24} - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{43} + ( 2 \zeta_{24}^{2} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 4 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{47} + ( 2 \zeta_{24} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{49} + ( 5 - 3 \zeta_{24} + 3 \zeta_{24}^{3} - 10 \zeta_{24}^{4} + 3 \zeta_{24}^{5} ) q^{53} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 7 \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{55} + ( -4 \zeta_{24} - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{59} + ( -4 + 4 \zeta_{24}^{4} ) q^{61} + ( -6 + 8 \zeta_{24}^{3} - 6 \zeta_{24}^{4} + 8 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{65} + ( 2 \zeta_{24} - 10 \zeta_{24}^{2} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 10 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{67} + ( -2 \zeta_{24} + 12 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 6 \zeta_{24}^{6} ) q^{71} + ( -3 + 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{73} + ( -10 + 5 \zeta_{24} + 5 \zeta_{24}^{4} - 5 \zeta_{24}^{7} ) q^{77} + 2 \zeta_{24}^{2} q^{79} + ( \zeta_{24}^{2} + 10 \zeta_{24}^{3} - 10 \zeta_{24}^{5} - 2 \zeta_{24}^{6} - 10 \zeta_{24}^{7} ) q^{83} + ( -2 \zeta_{24} + 4 \zeta_{24}^{3} - 6 \zeta_{24}^{4} + 4 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{85} + ( -2 - 6 \zeta_{24} + 6 \zeta_{24}^{3} + 4 \zeta_{24}^{4} + 6 \zeta_{24}^{5} ) q^{89} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 14 \zeta_{24}^{6} - 8 \zeta_{24}^{7} ) q^{91} + ( -6 \zeta_{24} + 4 \zeta_{24}^{2} + 4 \zeta_{24}^{6} - 6 \zeta_{24}^{7} ) q^{95} + ( 5 - 5 \zeta_{24}^{4} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 12q^{5} + O(q^{10}) \) \( 8q - 12q^{5} + 8q^{13} + 24q^{29} + 48q^{41} - 16q^{61} - 72q^{65} - 24q^{73} - 60q^{77} - 24q^{85} + 20q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(1\) \(\zeta_{24}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
863.1
−0.258819 0.965926i
−0.965926 + 0.258819i
0.965926 0.258819i
0.258819 + 0.965926i
−0.258819 + 0.965926i
−0.965926 0.258819i
0.965926 + 0.258819i
0.258819 0.965926i
0 0 0 −2.72474 1.57313i 0 −2.98735 + 1.72474i 0 0 0
863.2 0 0 0 −2.72474 1.57313i 0 2.98735 1.72474i 0 0 0
863.3 0 0 0 −0.275255 0.158919i 0 −1.25529 + 0.724745i 0 0 0
863.4 0 0 0 −0.275255 0.158919i 0 1.25529 0.724745i 0 0 0
1727.1 0 0 0 −2.72474 + 1.57313i 0 −2.98735 1.72474i 0 0 0
1727.2 0 0 0 −2.72474 + 1.57313i 0 2.98735 + 1.72474i 0 0 0
1727.3 0 0 0 −0.275255 + 0.158919i 0 −1.25529 0.724745i 0 0 0
1727.4 0 0 0 −0.275255 + 0.158919i 0 1.25529 + 0.724745i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1727.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.d odd 6 1 inner
36.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2592.2.s.a 8
3.b odd 2 1 2592.2.s.h 8
4.b odd 2 1 inner 2592.2.s.a 8
9.c even 3 1 864.2.c.a 8
9.c even 3 1 2592.2.s.h 8
9.d odd 6 1 864.2.c.a 8
9.d odd 6 1 inner 2592.2.s.a 8
12.b even 2 1 2592.2.s.h 8
36.f odd 6 1 864.2.c.a 8
36.f odd 6 1 2592.2.s.h 8
36.h even 6 1 864.2.c.a 8
36.h even 6 1 inner 2592.2.s.a 8
72.j odd 6 1 1728.2.c.g 8
72.l even 6 1 1728.2.c.g 8
72.n even 6 1 1728.2.c.g 8
72.p odd 6 1 1728.2.c.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.c.a 8 9.c even 3 1
864.2.c.a 8 9.d odd 6 1
864.2.c.a 8 36.f odd 6 1
864.2.c.a 8 36.h even 6 1
1728.2.c.g 8 72.j odd 6 1
1728.2.c.g 8 72.l even 6 1
1728.2.c.g 8 72.n even 6 1
1728.2.c.g 8 72.p odd 6 1
2592.2.s.a 8 1.a even 1 1 trivial
2592.2.s.a 8 4.b odd 2 1 inner
2592.2.s.a 8 9.d odd 6 1 inner
2592.2.s.a 8 36.h even 6 1 inner
2592.2.s.h 8 3.b odd 2 1
2592.2.s.h 8 9.c even 3 1
2592.2.s.h 8 12.b even 2 1
2592.2.s.h 8 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2592, [\chi])\):

\( T_{5}^{4} + 6 T_{5}^{3} + 13 T_{5}^{2} + 6 T_{5} + 1 \)
\( T_{7}^{8} - 14 T_{7}^{6} + 171 T_{7}^{4} - 350 T_{7}^{2} + 625 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 1 + 6 T + 13 T^{2} + 6 T^{3} + T^{4} )^{2} \)
$7$ \( 625 - 350 T^{2} + 171 T^{4} - 14 T^{6} + T^{8} \)
$11$ \( 625 + 550 T^{2} + 459 T^{4} + 22 T^{6} + T^{8} \)
$13$ \( ( 400 + 80 T + 36 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$17$ \( ( 12 + T^{2} )^{4} \)
$19$ \( ( 24 + T^{2} )^{4} \)
$23$ \( ( 64 + 8 T^{2} + T^{4} )^{2} \)
$29$ \( ( 400 + 240 T + 28 T^{2} - 12 T^{3} + T^{4} )^{2} \)
$31$ \( 130321 - 22382 T^{2} + 3483 T^{4} - 62 T^{6} + T^{8} \)
$37$ \( ( -24 + T^{2} )^{4} \)
$41$ \( ( 1600 - 960 T + 232 T^{2} - 24 T^{3} + T^{4} )^{2} \)
$43$ \( 160000 - 22400 T^{2} + 2736 T^{4} - 56 T^{6} + T^{8} \)
$47$ \( 160000 + 35200 T^{2} + 7344 T^{4} + 88 T^{6} + T^{8} \)
$53$ \( ( 3249 + 186 T^{2} + T^{4} )^{2} \)
$59$ \( 160000 + 35200 T^{2} + 7344 T^{4} + 88 T^{6} + T^{8} \)
$61$ \( ( 16 + 4 T + T^{2} )^{4} \)
$67$ \( 33362176 - 1432448 T^{2} + 55728 T^{4} - 248 T^{6} + T^{8} \)
$71$ \( ( 10000 - 232 T^{2} + T^{4} )^{2} \)
$73$ \( ( -15 + 6 T + T^{2} )^{4} \)
$79$ \( ( 16 - 4 T^{2} + T^{4} )^{2} \)
$83$ \( 1506138481 + 15756454 T^{2} + 126027 T^{4} + 406 T^{6} + T^{8} \)
$89$ \( ( 3600 + 168 T^{2} + T^{4} )^{2} \)
$97$ \( ( 25 - 5 T + T^{2} )^{4} \)
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