Properties

Label 2592.2.s.b
Level $2592$
Weight $2$
Character orbit 2592.s
Analytic conductor $20.697$
Analytic rank $1$
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: \(1\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( \zeta_{24} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{5} + ( -2 - \zeta_{24}^{2} + \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{7} +O(q^{10})\) \( q + ( \zeta_{24} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{5} + ( -2 - \zeta_{24}^{2} + \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{7} + ( -3 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{11} + ( -1 + 2 \zeta_{24}^{2} + \zeta_{24}^{4} - 4 \zeta_{24}^{6} ) q^{13} + ( \zeta_{24} - \zeta_{24}^{3} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{17} + ( -1 + 2 \zeta_{24}^{4} + 3 \zeta_{24}^{6} ) q^{19} + ( -\zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{23} + ( -\zeta_{24}^{2} - 3 \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{25} + ( -3 \zeta_{24} - 5 \zeta_{24}^{3} + 5 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{29} + ( -2 + 4 \zeta_{24}^{2} - 2 \zeta_{24}^{4} ) q^{31} + ( -\zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{35} + ( -6 + 2 \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{37} + ( 2 \zeta_{24} + 3 \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{41} + ( 2 - 7 \zeta_{24}^{2} - \zeta_{24}^{4} + 7 \zeta_{24}^{6} ) q^{43} + ( 4 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{47} + ( -3 + 2 \zeta_{24}^{2} + 3 \zeta_{24}^{4} - 4 \zeta_{24}^{6} ) q^{49} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{53} + ( 3 - 6 \zeta_{24}^{4} + 5 \zeta_{24}^{6} ) q^{55} + ( 10 \zeta_{24} - 4 \zeta_{24}^{3} - 6 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{59} + ( -3 \zeta_{24}^{2} - 4 \zeta_{24}^{4} - 3 \zeta_{24}^{6} ) q^{61} + ( 2 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{65} + ( -7 + \zeta_{24}^{2} - 7 \zeta_{24}^{4} ) q^{67} + ( -7 \zeta_{24} - 7 \zeta_{24}^{3} - 5 \zeta_{24}^{5} + 12 \zeta_{24}^{7} ) q^{71} + ( -12 + 2 \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{73} + ( 2 \zeta_{24} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{77} + ( -18 + \zeta_{24}^{2} + 9 \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{79} + ( -6 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{83} + ( \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{85} + ( -8 \zeta_{24} + 8 \zeta_{24}^{3} + \zeta_{24}^{5} - 7 \zeta_{24}^{7} ) q^{89} + ( -1 + 2 \zeta_{24}^{4} + 5 \zeta_{24}^{6} ) q^{91} + ( -2 \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{95} + ( -6 \zeta_{24}^{2} - 4 \zeta_{24}^{4} - 6 \zeta_{24}^{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{7} + O(q^{10}) \) \( 8 q - 12 q^{7} - 4 q^{13} - 12 q^{25} - 24 q^{31} - 48 q^{37} + 12 q^{43} - 12 q^{49} - 16 q^{61} - 84 q^{67} - 96 q^{73} - 108 q^{79} - 16 q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(1\) \(1 - \zeta_{24}^{4}\) \(-1\)

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 −1.67303 0.965926i 0 −0.633975 + 0.366025i 0 0 0
863.2 0 0 0 −0.448288 0.258819i 0 −2.36603 + 1.36603i 0 0 0
863.3 0 0 0 0.448288 + 0.258819i 0 −2.36603 + 1.36603i 0 0 0
863.4 0 0 0 1.67303 + 0.965926i 0 −0.633975 + 0.366025i 0 0 0
1727.1 0 0 0 −1.67303 + 0.965926i 0 −0.633975 0.366025i 0 0 0
1727.2 0 0 0 −0.448288 + 0.258819i 0 −2.36603 1.36603i 0 0 0
1727.3 0 0 0 0.448288 0.258819i 0 −2.36603 1.36603i 0 0 0
1727.4 0 0 0 1.67303 0.965926i 0 −0.633975 0.366025i 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.b 8
3.b odd 2 1 inner 2592.2.s.b 8
4.b odd 2 1 2592.2.s.f 8
9.c even 3 1 2592.2.c.a 8
9.c even 3 1 2592.2.s.f 8
9.d odd 6 1 2592.2.c.a 8
9.d odd 6 1 2592.2.s.f 8
12.b even 2 1 2592.2.s.f 8
36.f odd 6 1 2592.2.c.a 8
36.f odd 6 1 inner 2592.2.s.b 8
36.h even 6 1 2592.2.c.a 8
36.h even 6 1 inner 2592.2.s.b 8
72.j odd 6 1 5184.2.c.i 8
72.l even 6 1 5184.2.c.i 8
72.n even 6 1 5184.2.c.i 8
72.p odd 6 1 5184.2.c.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2592.2.c.a 8 9.c even 3 1
2592.2.c.a 8 9.d odd 6 1
2592.2.c.a 8 36.f odd 6 1
2592.2.c.a 8 36.h even 6 1
2592.2.s.b 8 1.a even 1 1 trivial
2592.2.s.b 8 3.b odd 2 1 inner
2592.2.s.b 8 36.f odd 6 1 inner
2592.2.s.b 8 36.h even 6 1 inner
2592.2.s.f 8 4.b odd 2 1
2592.2.s.f 8 9.c even 3 1
2592.2.s.f 8 9.d odd 6 1
2592.2.s.f 8 12.b even 2 1
5184.2.c.i 8 72.j odd 6 1
5184.2.c.i 8 72.l even 6 1
5184.2.c.i 8 72.n even 6 1
5184.2.c.i 8 72.p odd 6 1

Hecke kernels

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

\( T_{5}^{8} - 4 T_{5}^{6} + 15 T_{5}^{4} - 4 T_{5}^{2} + 1 \)
\( T_{7}^{4} + 6 T_{7}^{3} + 14 T_{7}^{2} + 12 T_{7} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( 1 - 4 T^{2} + 15 T^{4} - 4 T^{6} + T^{8} \)
$7$ \( ( 4 + 12 T + 14 T^{2} + 6 T^{3} + T^{4} )^{2} \)
$11$ \( 16 + 112 T^{2} + 780 T^{4} + 28 T^{6} + T^{8} \)
$13$ \( ( 121 - 22 T + 15 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$17$ \( ( 9 + 12 T^{2} + T^{4} )^{2} \)
$19$ \( ( 36 + 24 T^{2} + T^{4} )^{2} \)
$23$ \( ( 4 + 2 T^{2} + T^{4} )^{2} \)
$29$ \( 1874161 - 104044 T^{2} + 4407 T^{4} - 76 T^{6} + T^{8} \)
$31$ \( ( 16 - 48 T + 44 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$37$ \( ( 33 + 12 T + T^{2} )^{4} \)
$41$ \( 16 - 112 T^{2} + 780 T^{4} - 28 T^{6} + T^{8} \)
$43$ \( ( 2116 + 276 T - 34 T^{2} - 6 T^{3} + T^{4} )^{2} \)
$47$ \( 3748096 + 216832 T^{2} + 10608 T^{4} + 112 T^{6} + T^{8} \)
$53$ \( ( 6 + T^{2} )^{4} \)
$59$ \( 479785216 + 6658816 T^{2} + 70512 T^{4} + 304 T^{6} + T^{8} \)
$61$ \( ( 121 - 88 T + 75 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$67$ \( ( 21316 + 6132 T + 734 T^{2} + 42 T^{3} + T^{4} )^{2} \)
$71$ \( ( 45796 - 436 T^{2} + T^{4} )^{2} \)
$73$ \( ( 141 + 24 T + T^{2} )^{4} \)
$79$ \( ( 58564 + 13068 T + 1214 T^{2} + 54 T^{3} + T^{4} )^{2} \)
$83$ \( 4096 + 7168 T^{2} + 12480 T^{4} + 112 T^{6} + T^{8} \)
$89$ \( ( 1089 + 228 T^{2} + T^{4} )^{2} \)
$97$ \( ( 8464 - 736 T + 156 T^{2} + 8 T^{3} + T^{4} )^{2} \)
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