Properties

Label 2592.2.s.f
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: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
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.

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

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.f 8
3.b odd 2 1 inner 2592.2.s.f 8
4.b odd 2 1 2592.2.s.b 8
9.c even 3 1 2592.2.c.a 8
9.c even 3 1 2592.2.s.b 8
9.d odd 6 1 2592.2.c.a 8
9.d odd 6 1 2592.2.s.b 8
12.b even 2 1 2592.2.s.b 8
36.f odd 6 1 2592.2.c.a 8
36.f odd 6 1 inner 2592.2.s.f 8
36.h even 6 1 2592.2.c.a 8
36.h even 6 1 inner 2592.2.s.f 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 4.b odd 2 1
2592.2.s.b 8 9.c even 3 1
2592.2.s.b 8 9.d odd 6 1
2592.2.s.b 8 12.b even 2 1
2592.2.s.f 8 1.a even 1 1 trivial
2592.2.s.f 8 3.b odd 2 1 inner
2592.2.s.f 8 36.f odd 6 1 inner
2592.2.s.f 8 36.h even 6 1 inner
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 \(S_{2}^{\mathrm{new}}(2592, [\chi])\):

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

Hecke characteristic polynomials

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