Properties

Label 2592.2.s.h
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: \(\Z[a_1, \ldots, a_{13}]\)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - \beta_{2} + 2) q^{5} + ( - \beta_{7} - \beta_{4} + \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - \beta_{2} + 2) q^{5} + ( - \beta_{7} - \beta_{4} + \beta_1) q^{7} + ( - 2 \beta_{7} + 2 \beta_{4} - 2 \beta_{3} + \beta_1) q^{11} + ( - 4 \beta_{6} + 2 \beta_{5} + 2 \beta_{2}) q^{13} + ( - 4 \beta_{2} + 2) q^{17} + ( - 2 \beta_{7} + 4 \beta_{4}) q^{19} + 2 \beta_{4} q^{23} + (2 \beta_{6} - 4 \beta_{5}) q^{25} + ( - 4 \beta_{6} - 2 \beta_{2} - 2) q^{29} + (2 \beta_{7} - \beta_{4} - 5 \beta_{3} + 5 \beta_1) q^{31} + ( - 4 \beta_{7} - 3 \beta_{3} + 6 \beta_1) q^{35} + (2 \beta_{6} + 2 \beta_{5}) q^{37} + ( - 2 \beta_{5} + 4 \beta_{2} - 8) q^{41} + ( - 2 \beta_{7} - 2 \beta_{4} - 2 \beta_1) q^{43} + ( - 4 \beta_{7} + 4 \beta_{4} + 4 \beta_{3} - 2 \beta_1) q^{47} + ( - 4 \beta_{6} + 2 \beta_{5}) q^{49} + ( - 3 \beta_{6} + 3 \beta_{5} + 10 \beta_{2} - 5) q^{53} + ( - 3 \beta_{7} + 6 \beta_{4} - 7 \beta_{3}) q^{55} + (4 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{59} + (4 \beta_{2} - 4) q^{61} + ( - 8 \beta_{6} + 6 \beta_{2} + 6) q^{65} + (4 \beta_{7} - 2 \beta_{4} + 10 \beta_{3} - 10 \beta_1) q^{67} + (2 \beta_{7} + 6 \beta_{3} - 12 \beta_1) q^{71} + (2 \beta_{6} + 2 \beta_{5} - 3) q^{73} + ( - 5 \beta_{5} - 5 \beta_{2} + 10) q^{77} + 2 \beta_1 q^{79} + ( - 10 \beta_{7} + 10 \beta_{4} + 2 \beta_{3} - \beta_1) q^{83} + (4 \beta_{6} - 2 \beta_{5} - 6 \beta_{2}) q^{85} + ( - 6 \beta_{6} + 6 \beta_{5} - 4 \beta_{2} + 2) q^{89} + (4 \beta_{7} - 8 \beta_{4} + 14 \beta_{3}) q^{91} + (6 \beta_{4} - 4 \beta_{3} - 4 \beta_1) q^{95} + ( - 5 \beta_{2} + 5) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{5} + 8 q^{13} - 24 q^{29} - 48 q^{41} - 16 q^{61} + 72 q^{65} - 24 q^{73} + 60 q^{77} - 24 q^{85} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(-1\)

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
−0.258819 0.965926i
−0.965926 + 0.258819i
0.965926 + 0.258819i
0.258819 0.965926i
−0.258819 + 0.965926i
−0.965926 0.258819i
0 0 0 0.275255 + 0.158919i 0 −1.25529 + 0.724745i 0 0 0
863.2 0 0 0 0.275255 + 0.158919i 0 1.25529 0.724745i 0 0 0
863.3 0 0 0 2.72474 + 1.57313i 0 −2.98735 + 1.72474i 0 0 0
863.4 0 0 0 2.72474 + 1.57313i 0 2.98735 1.72474i 0 0 0
1727.1 0 0 0 0.275255 0.158919i 0 −1.25529 0.724745i 0 0 0
1727.2 0 0 0 0.275255 0.158919i 0 1.25529 + 0.724745i 0 0 0
1727.3 0 0 0 2.72474 1.57313i 0 −2.98735 1.72474i 0 0 0
1727.4 0 0 0 2.72474 1.57313i 0 2.98735 + 1.72474i 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.h 8
3.b odd 2 1 2592.2.s.a 8
4.b odd 2 1 inner 2592.2.s.h 8
9.c even 3 1 864.2.c.a 8
9.c even 3 1 2592.2.s.a 8
9.d odd 6 1 864.2.c.a 8
9.d odd 6 1 inner 2592.2.s.h 8
12.b even 2 1 2592.2.s.a 8
36.f odd 6 1 864.2.c.a 8
36.f odd 6 1 2592.2.s.a 8
36.h even 6 1 864.2.c.a 8
36.h even 6 1 inner 2592.2.s.h 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 3.b odd 2 1
2592.2.s.a 8 9.c even 3 1
2592.2.s.a 8 12.b even 2 1
2592.2.s.a 8 36.f odd 6 1
2592.2.s.h 8 1.a even 1 1 trivial
2592.2.s.h 8 4.b odd 2 1 inner
2592.2.s.h 8 9.d odd 6 1 inner
2592.2.s.h 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 \(S_{2}^{\mathrm{new}}(2592, [\chi])\):

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