Properties

Label 2592.2.s.g
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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{8} \)
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.

\(f(q)\) \(=\) \( q + ( 2 \zeta_{24} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{5} + ( 2 - 2 \zeta_{24}^{2} - \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{7} +O(q^{10})\) \( q + ( 2 \zeta_{24} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{5} + ( 2 - 2 \zeta_{24}^{2} - \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{7} + ( 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{11} + ( -1 + 2 \zeta_{24}^{2} + \zeta_{24}^{4} - 4 \zeta_{24}^{6} ) q^{13} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{17} + ( 1 - 2 \zeta_{24}^{4} ) q^{19} + ( 6 \zeta_{24} - 4 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{23} + ( -4 \zeta_{24}^{2} + 3 \zeta_{24}^{4} - 4 \zeta_{24}^{6} ) q^{25} + ( -4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{29} + ( 2 - 4 \zeta_{24}^{2} + 2 \zeta_{24}^{4} ) q^{31} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} - 8 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{35} + ( -3 - 4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{37} + ( 4 \zeta_{24} - 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{41} + ( 4 + 4 \zeta_{24}^{2} - 2 \zeta_{24}^{4} - 4 \zeta_{24}^{6} ) q^{43} + ( 4 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{47} + ( -4 \zeta_{24}^{2} + 8 \zeta_{24}^{6} ) q^{49} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} - 8 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{53} + 4 \zeta_{24}^{6} q^{55} + ( -2 \zeta_{24} + 8 \zeta_{24}^{3} - 6 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{59} + ( 6 \zeta_{24}^{2} - \zeta_{24}^{4} + 6 \zeta_{24}^{6} ) q^{61} + ( 4 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{65} + ( 1 + 8 \zeta_{24}^{2} + \zeta_{24}^{4} ) q^{67} + ( 8 \zeta_{24} + 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} ) q^{71} + ( 3 + 8 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{73} + ( -2 \zeta_{24} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{77} + ( -6 - 10 \zeta_{24}^{2} + 3 \zeta_{24}^{4} + 10 \zeta_{24}^{6} ) q^{79} + ( 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{83} + ( 4 \zeta_{24}^{2} - 8 \zeta_{24}^{6} ) q^{85} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{89} + ( -5 + 10 \zeta_{24}^{4} - 8 \zeta_{24}^{6} ) q^{91} + ( -2 \zeta_{24} + 4 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{95} -7 \zeta_{24}^{4} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{7} + O(q^{10}) \) \( 8 q + 12 q^{7} - 4 q^{13} + 12 q^{25} + 24 q^{31} - 24 q^{37} + 24 q^{43} - 4 q^{61} + 12 q^{67} + 24 q^{73} - 36 q^{79} - 28 q^{97} + O(q^{100}) \)

Character values

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

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

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 −3.34607 1.93185i 0 3.23205 1.86603i 0 0 0
863.2 0 0 0 −0.896575 0.517638i 0 −0.232051 + 0.133975i 0 0 0
863.3 0 0 0 0.896575 + 0.517638i 0 −0.232051 + 0.133975i 0 0 0
863.4 0 0 0 3.34607 + 1.93185i 0 3.23205 1.86603i 0 0 0
1727.1 0 0 0 −3.34607 + 1.93185i 0 3.23205 + 1.86603i 0 0 0
1727.2 0 0 0 −0.896575 + 0.517638i 0 −0.232051 0.133975i 0 0 0
1727.3 0 0 0 0.896575 0.517638i 0 −0.232051 0.133975i 0 0 0
1727.4 0 0 0 3.34607 1.93185i 0 3.23205 + 1.86603i 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
3.b odd 2 1 inner
36.f 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.g 8
3.b odd 2 1 inner 2592.2.s.g 8
4.b odd 2 1 2592.2.s.c 8
9.c even 3 1 864.2.c.b 8
9.c even 3 1 2592.2.s.c 8
9.d odd 6 1 864.2.c.b 8
9.d odd 6 1 2592.2.s.c 8
12.b even 2 1 2592.2.s.c 8
36.f odd 6 1 864.2.c.b 8
36.f odd 6 1 inner 2592.2.s.g 8
36.h even 6 1 864.2.c.b 8
36.h even 6 1 inner 2592.2.s.g 8
72.j odd 6 1 1728.2.c.f 8
72.l even 6 1 1728.2.c.f 8
72.n even 6 1 1728.2.c.f 8
72.p odd 6 1 1728.2.c.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.c.b 8 9.c even 3 1
864.2.c.b 8 9.d odd 6 1
864.2.c.b 8 36.f odd 6 1
864.2.c.b 8 36.h even 6 1
1728.2.c.f 8 72.j odd 6 1
1728.2.c.f 8 72.l even 6 1
1728.2.c.f 8 72.n even 6 1
1728.2.c.f 8 72.p odd 6 1
2592.2.s.c 8 4.b odd 2 1
2592.2.s.c 8 9.c even 3 1
2592.2.s.c 8 9.d odd 6 1
2592.2.s.c 8 12.b even 2 1
2592.2.s.g 8 1.a even 1 1 trivial
2592.2.s.g 8 3.b odd 2 1 inner
2592.2.s.g 8 36.f odd 6 1 inner
2592.2.s.g 8 36.h even 6 1 inner

Hecke kernels

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

\( T_{5}^{8} - 16 T_{5}^{6} + 240 T_{5}^{4} - 256 T_{5}^{2} + 256 \)
\( T_{7}^{4} - 6 T_{7}^{3} + 11 T_{7}^{2} + 6 T_{7} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( 256 - 256 T^{2} + 240 T^{4} - 16 T^{6} + T^{8} \)
$7$ \( ( 1 + 6 T + 11 T^{2} - 6 T^{3} + T^{4} )^{2} \)
$11$ \( 256 + 256 T^{2} + 240 T^{4} + 16 T^{6} + T^{8} \)
$13$ \( ( 121 - 22 T + 15 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$17$ \( ( 144 + 48 T^{2} + T^{4} )^{2} \)
$19$ \( ( 3 + T^{2} )^{4} \)
$23$ \( 7311616 + 302848 T^{2} + 9840 T^{4} + 112 T^{6} + T^{8} \)
$29$ \( 65536 - 16384 T^{2} + 3840 T^{4} - 64 T^{6} + T^{8} \)
$31$ \( ( 16 + 48 T + 44 T^{2} - 12 T^{3} + T^{4} )^{2} \)
$37$ \( ( -3 + 6 T + T^{2} )^{4} \)
$41$ \( 65536 - 16384 T^{2} + 3840 T^{4} - 64 T^{6} + T^{8} \)
$43$ \( ( 16 + 48 T + 44 T^{2} - 12 T^{3} + T^{4} )^{2} \)
$47$ \( 3748096 + 216832 T^{2} + 10608 T^{4} + 112 T^{6} + T^{8} \)
$53$ \( ( 2304 + 192 T^{2} + T^{4} )^{2} \)
$59$ \( 71639296 + 1760512 T^{2} + 34800 T^{4} + 208 T^{6} + T^{8} \)
$61$ \( ( 11449 - 214 T + 111 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$67$ \( ( 3721 + 366 T - 49 T^{2} - 6 T^{3} + T^{4} )^{2} \)
$71$ \( ( -128 + T^{2} )^{4} \)
$73$ \( ( -39 - 6 T + T^{2} )^{4} \)
$79$ \( ( 5329 - 1314 T + 35 T^{2} + 18 T^{3} + T^{4} )^{2} \)
$83$ \( 65536 + 16384 T^{2} + 3840 T^{4} + 64 T^{6} + T^{8} \)
$89$ \( ( 144 + 48 T^{2} + T^{4} )^{2} \)
$97$ \( ( 49 + 7 T + T^{2} )^{4} \)
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