Properties

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

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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: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 96)
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.

\(f(q)\) \(=\) \( q + ( 2 \zeta_{24} - 2 \zeta_{24}^{7} ) q^{5} -2 \zeta_{24}^{2} q^{7} +O(q^{10})\) \( q + ( 2 \zeta_{24} - 2 \zeta_{24}^{7} ) q^{5} -2 \zeta_{24}^{2} q^{7} + ( -2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{11} + 2 \zeta_{24}^{4} q^{13} -6 \zeta_{24}^{6} q^{19} + ( -4 \zeta_{24} - 4 \zeta_{24}^{7} ) q^{23} + ( 3 - 3 \zeta_{24}^{4} ) q^{25} + ( 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{29} + ( 2 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{31} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{35} + 6 q^{37} + ( 4 \zeta_{24} - 4 \zeta_{24}^{7} ) q^{41} -2 \zeta_{24}^{2} q^{43} + ( 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{47} -3 \zeta_{24}^{4} q^{49} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{53} + 8 \zeta_{24}^{6} q^{55} + ( -2 \zeta_{24} - 2 \zeta_{24}^{7} ) q^{59} + ( 2 - 2 \zeta_{24}^{4} ) q^{61} + ( 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{65} + ( 2 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{67} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{71} -6 q^{73} + ( 4 \zeta_{24} - 4 \zeta_{24}^{7} ) q^{77} + 14 \zeta_{24}^{2} q^{79} + ( 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{83} + ( 12 \zeta_{24} - 12 \zeta_{24}^{3} - 12 \zeta_{24}^{5} ) q^{89} -4 \zeta_{24}^{6} q^{91} + ( -12 \zeta_{24} - 12 \zeta_{24}^{7} ) q^{95} + ( -10 + 10 \zeta_{24}^{4} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + O(q^{10}) \) \( 8 q + 8 q^{13} + 12 q^{25} + 48 q^{37} - 12 q^{49} + 8 q^{61} - 48 q^{73} - 40 q^{97} + O(q^{100}) \)

Character values

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

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

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
−0.965926 + 0.258819i
−0.258819 0.965926i
0.965926 0.258819i
0.258819 + 0.965926i
−0.965926 0.258819i
−0.258819 + 0.965926i
0.965926 + 0.258819i
0.258819 0.965926i
0 0 0 −2.44949 1.41421i 0 −1.73205 + 1.00000i 0 0 0
863.2 0 0 0 −2.44949 1.41421i 0 1.73205 1.00000i 0 0 0
863.3 0 0 0 2.44949 + 1.41421i 0 −1.73205 + 1.00000i 0 0 0
863.4 0 0 0 2.44949 + 1.41421i 0 1.73205 1.00000i 0 0 0
1727.1 0 0 0 −2.44949 + 1.41421i 0 −1.73205 1.00000i 0 0 0
1727.2 0 0 0 −2.44949 + 1.41421i 0 1.73205 + 1.00000i 0 0 0
1727.3 0 0 0 2.44949 1.41421i 0 −1.73205 1.00000i 0 0 0
1727.4 0 0 0 2.44949 1.41421i 0 1.73205 + 1.00000i 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
4.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
12.b even 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.e 8
3.b odd 2 1 inner 2592.2.s.e 8
4.b odd 2 1 inner 2592.2.s.e 8
9.c even 3 1 96.2.c.a 4
9.c even 3 1 inner 2592.2.s.e 8
9.d odd 6 1 96.2.c.a 4
9.d odd 6 1 inner 2592.2.s.e 8
12.b even 2 1 inner 2592.2.s.e 8
36.f odd 6 1 96.2.c.a 4
36.f odd 6 1 inner 2592.2.s.e 8
36.h even 6 1 96.2.c.a 4
36.h even 6 1 inner 2592.2.s.e 8
45.h odd 6 1 2400.2.h.c 4
45.j even 6 1 2400.2.h.c 4
45.k odd 12 1 2400.2.o.a 4
45.k odd 12 1 2400.2.o.h 4
45.l even 12 1 2400.2.o.a 4
45.l even 12 1 2400.2.o.h 4
72.j odd 6 1 192.2.c.b 4
72.l even 6 1 192.2.c.b 4
72.n even 6 1 192.2.c.b 4
72.p odd 6 1 192.2.c.b 4
144.u even 12 1 768.2.f.a 4
144.u even 12 1 768.2.f.g 4
144.v odd 12 1 768.2.f.a 4
144.v odd 12 1 768.2.f.g 4
144.w odd 12 1 768.2.f.a 4
144.w odd 12 1 768.2.f.g 4
144.x even 12 1 768.2.f.a 4
144.x even 12 1 768.2.f.g 4
180.n even 6 1 2400.2.h.c 4
180.p odd 6 1 2400.2.h.c 4
180.v odd 12 1 2400.2.o.a 4
180.v odd 12 1 2400.2.o.h 4
180.x even 12 1 2400.2.o.a 4
180.x even 12 1 2400.2.o.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.2.c.a 4 9.c even 3 1
96.2.c.a 4 9.d odd 6 1
96.2.c.a 4 36.f odd 6 1
96.2.c.a 4 36.h even 6 1
192.2.c.b 4 72.j odd 6 1
192.2.c.b 4 72.l even 6 1
192.2.c.b 4 72.n even 6 1
192.2.c.b 4 72.p odd 6 1
768.2.f.a 4 144.u even 12 1
768.2.f.a 4 144.v odd 12 1
768.2.f.a 4 144.w odd 12 1
768.2.f.a 4 144.x even 12 1
768.2.f.g 4 144.u even 12 1
768.2.f.g 4 144.v odd 12 1
768.2.f.g 4 144.w odd 12 1
768.2.f.g 4 144.x even 12 1
2400.2.h.c 4 45.h odd 6 1
2400.2.h.c 4 45.j even 6 1
2400.2.h.c 4 180.n even 6 1
2400.2.h.c 4 180.p odd 6 1
2400.2.o.a 4 45.k odd 12 1
2400.2.o.a 4 45.l even 12 1
2400.2.o.a 4 180.v odd 12 1
2400.2.o.a 4 180.x even 12 1
2400.2.o.h 4 45.k odd 12 1
2400.2.o.h 4 45.l even 12 1
2400.2.o.h 4 180.v odd 12 1
2400.2.o.h 4 180.x even 12 1
2592.2.s.e 8 1.a even 1 1 trivial
2592.2.s.e 8 3.b odd 2 1 inner
2592.2.s.e 8 4.b odd 2 1 inner
2592.2.s.e 8 9.c even 3 1 inner
2592.2.s.e 8 9.d odd 6 1 inner
2592.2.s.e 8 12.b even 2 1 inner
2592.2.s.e 8 36.f odd 6 1 inner
2592.2.s.e 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 \(S_{2}^{\mathrm{new}}(2592, [\chi])\):

\( T_{5}^{4} - 8 T_{5}^{2} + 64 \)
\( T_{7}^{4} - 4 T_{7}^{2} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 64 - 8 T^{2} + T^{4} )^{2} \)
$7$ \( ( 16 - 4 T^{2} + T^{4} )^{2} \)
$11$ \( ( 64 + 8 T^{2} + T^{4} )^{2} \)
$13$ \( ( 4 - 2 T + T^{2} )^{4} \)
$17$ \( T^{8} \)
$19$ \( ( 36 + T^{2} )^{4} \)
$23$ \( ( 1024 + 32 T^{2} + T^{4} )^{2} \)
$29$ \( ( 64 - 8 T^{2} + T^{4} )^{2} \)
$31$ \( ( 16 - 4 T^{2} + T^{4} )^{2} \)
$37$ \( ( -6 + T )^{8} \)
$41$ \( ( 1024 - 32 T^{2} + T^{4} )^{2} \)
$43$ \( ( 16 - 4 T^{2} + T^{4} )^{2} \)
$47$ \( ( 16384 + 128 T^{2} + T^{4} )^{2} \)
$53$ \( ( 72 + T^{2} )^{4} \)
$59$ \( ( 64 + 8 T^{2} + T^{4} )^{2} \)
$61$ \( ( 4 - 2 T + T^{2} )^{4} \)
$67$ \( ( 16 - 4 T^{2} + T^{4} )^{2} \)
$71$ \( ( -32 + T^{2} )^{4} \)
$73$ \( ( 6 + T )^{8} \)
$79$ \( ( 38416 - 196 T^{2} + T^{4} )^{2} \)
$83$ \( ( 64 + 8 T^{2} + T^{4} )^{2} \)
$89$ \( ( 288 + T^{2} )^{4} \)
$97$ \( ( 100 + 10 T + T^{2} )^{4} \)
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