Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 2592 = 2^{5} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2592.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(20.6972242039\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 864)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{5} + ( 3 - 3 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + \zeta_{6} q^{5} + ( 3 - 3 \zeta_{6} ) q^{7} + ( -3 + 3 \zeta_{6} ) q^{11} + 4 q^{17} -6 q^{19} + 6 \zeta_{6} q^{23} + ( 4 - 4 \zeta_{6} ) q^{25} + ( 2 - 2 \zeta_{6} ) q^{29} + 9 \zeta_{6} q^{31} + 3 q^{35} -2 q^{37} + 10 \zeta_{6} q^{41} + ( 6 - 6 \zeta_{6} ) q^{43} + ( 6 - 6 \zeta_{6} ) q^{47} -2 \zeta_{6} q^{49} + 13 q^{53} -3 q^{55} -12 \zeta_{6} q^{59} + ( -8 + 8 \zeta_{6} ) q^{61} + 6 \zeta_{6} q^{67} -12 q^{71} + 9 q^{73} + 9 \zeta_{6} q^{77} + ( -3 + 3 \zeta_{6} ) q^{83} + 4 \zeta_{6} q^{85} + 14 q^{89} -6 \zeta_{6} q^{95} + ( 9 - 9 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{5} + 3q^{7} + O(q^{10}) \) \( 2q + q^{5} + 3q^{7} - 3q^{11} + 8q^{17} - 12q^{19} + 6q^{23} + 4q^{25} + 2q^{29} + 9q^{31} + 6q^{35} - 4q^{37} + 10q^{41} + 6q^{43} + 6q^{47} - 2q^{49} + 26q^{53} - 6q^{55} - 12q^{59} - 8q^{61} + 6q^{67} - 24q^{71} + 18q^{73} + 9q^{77} - 3q^{83} + 4q^{85} + 28q^{89} - 6q^{95} + 9q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
865.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 0.500000 + 0.866025i 0 1.50000 2.59808i 0 0 0
1729.1 0 0 0 0.500000 0.866025i 0 1.50000 + 2.59808i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2592.2.i.p 2
3.b odd 2 1 2592.2.i.l 2
4.b odd 2 1 2592.2.i.m 2
9.c even 3 1 864.2.a.e 1
9.c even 3 1 inner 2592.2.i.p 2
9.d odd 6 1 864.2.a.g yes 1
9.d odd 6 1 2592.2.i.l 2
12.b even 2 1 2592.2.i.i 2
36.f odd 6 1 864.2.a.f yes 1
36.f odd 6 1 2592.2.i.m 2
36.h even 6 1 864.2.a.h yes 1
36.h even 6 1 2592.2.i.i 2
72.j odd 6 1 1728.2.a.j 1
72.l even 6 1 1728.2.a.k 1
72.n even 6 1 1728.2.a.q 1
72.p odd 6 1 1728.2.a.t 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.a.e 1 9.c even 3 1
864.2.a.f yes 1 36.f odd 6 1
864.2.a.g yes 1 9.d odd 6 1
864.2.a.h yes 1 36.h even 6 1
1728.2.a.j 1 72.j odd 6 1
1728.2.a.k 1 72.l even 6 1
1728.2.a.q 1 72.n even 6 1
1728.2.a.t 1 72.p odd 6 1
2592.2.i.i 2 12.b even 2 1
2592.2.i.i 2 36.h even 6 1
2592.2.i.l 2 3.b odd 2 1
2592.2.i.l 2 9.d odd 6 1
2592.2.i.m 2 4.b odd 2 1
2592.2.i.m 2 36.f odd 6 1
2592.2.i.p 2 1.a even 1 1 trivial
2592.2.i.p 2 9.c even 3 1 inner

Hecke kernels

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

\( T_{5}^{2} - T_{5} + 1 \)
\( T_{7}^{2} - 3 T_{7} + 9 \)
\( T_{11}^{2} + 3 T_{11} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 1 - T + T^{2} \)
$7$ \( 9 - 3 T + T^{2} \)
$11$ \( 9 + 3 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( ( -4 + T )^{2} \)
$19$ \( ( 6 + T )^{2} \)
$23$ \( 36 - 6 T + T^{2} \)
$29$ \( 4 - 2 T + T^{2} \)
$31$ \( 81 - 9 T + T^{2} \)
$37$ \( ( 2 + T )^{2} \)
$41$ \( 100 - 10 T + T^{2} \)
$43$ \( 36 - 6 T + T^{2} \)
$47$ \( 36 - 6 T + T^{2} \)
$53$ \( ( -13 + T )^{2} \)
$59$ \( 144 + 12 T + T^{2} \)
$61$ \( 64 + 8 T + T^{2} \)
$67$ \( 36 - 6 T + T^{2} \)
$71$ \( ( 12 + T )^{2} \)
$73$ \( ( -9 + T )^{2} \)
$79$ \( T^{2} \)
$83$ \( 9 + 3 T + T^{2} \)
$89$ \( ( -14 + T )^{2} \)
$97$ \( 81 - 9 T + T^{2} \)
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