Properties

Label 2592.2.i.t
Level $2592$
Weight $2$
Character orbit 2592.i
Analytic conductor $20.697$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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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: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 32)
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{5} +O(q^{10})\) \( q + 2 \zeta_{6} q^{5} -6 \zeta_{6} q^{13} + 2 q^{17} + ( 1 - \zeta_{6} ) q^{25} + ( 10 - 10 \zeta_{6} ) q^{29} -2 q^{37} -10 \zeta_{6} q^{41} + 7 \zeta_{6} q^{49} + 14 q^{53} + ( 10 - 10 \zeta_{6} ) q^{61} + ( 12 - 12 \zeta_{6} ) q^{65} -6 q^{73} + 4 \zeta_{6} q^{85} + 10 q^{89} + ( -18 + 18 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} + O(q^{10}) \) \( 2q + 2q^{5} - 6q^{13} + 4q^{17} + q^{25} + 10q^{29} - 4q^{37} - 10q^{41} + 7q^{49} + 28q^{53} + 10q^{61} + 12q^{65} - 12q^{73} + 4q^{85} + 20q^{89} - 18q^{97} + O(q^{100}) \)

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 1.00000 + 1.73205i 0 0 0 0 0
1729.1 0 0 0 1.00000 1.73205i 0 0 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
9.c even 3 1 inner
36.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2592.2.i.t 2
3.b odd 2 1 2592.2.i.e 2
4.b odd 2 1 CM 2592.2.i.t 2
9.c even 3 1 32.2.a.a 1
9.c even 3 1 inner 2592.2.i.t 2
9.d odd 6 1 288.2.a.d 1
9.d odd 6 1 2592.2.i.e 2
12.b even 2 1 2592.2.i.e 2
36.f odd 6 1 32.2.a.a 1
36.f odd 6 1 inner 2592.2.i.t 2
36.h even 6 1 288.2.a.d 1
36.h even 6 1 2592.2.i.e 2
45.h odd 6 1 7200.2.a.v 1
45.j even 6 1 800.2.a.d 1
45.k odd 12 2 800.2.c.e 2
45.l even 12 2 7200.2.f.m 2
63.g even 3 1 1568.2.i.g 2
63.h even 3 1 1568.2.i.g 2
63.k odd 6 1 1568.2.i.f 2
63.l odd 6 1 1568.2.a.e 1
63.t odd 6 1 1568.2.i.f 2
72.j odd 6 1 576.2.a.c 1
72.l even 6 1 576.2.a.c 1
72.n even 6 1 64.2.a.a 1
72.p odd 6 1 64.2.a.a 1
99.h odd 6 1 3872.2.a.f 1
117.t even 6 1 5408.2.a.g 1
144.u even 12 2 2304.2.d.j 2
144.v odd 12 2 256.2.b.b 2
144.w odd 12 2 2304.2.d.j 2
144.x even 12 2 256.2.b.b 2
153.h even 6 1 9248.2.a.f 1
180.n even 6 1 7200.2.a.v 1
180.p odd 6 1 800.2.a.d 1
180.v odd 12 2 7200.2.f.m 2
180.x even 12 2 800.2.c.e 2
252.n even 6 1 1568.2.i.f 2
252.u odd 6 1 1568.2.i.g 2
252.bi even 6 1 1568.2.a.e 1
252.bj even 6 1 1568.2.i.f 2
252.bl odd 6 1 1568.2.i.g 2
288.bc even 24 4 1024.2.e.j 4
288.bd odd 24 4 1024.2.e.j 4
360.z odd 6 1 1600.2.a.n 1
360.bk even 6 1 1600.2.a.n 1
360.bo even 12 2 1600.2.c.l 2
360.bu odd 12 2 1600.2.c.l 2
396.k even 6 1 3872.2.a.f 1
468.bg odd 6 1 5408.2.a.g 1
504.be even 6 1 3136.2.a.m 1
504.bn odd 6 1 3136.2.a.m 1
612.q odd 6 1 9248.2.a.f 1
792.z even 6 1 7744.2.a.v 1
792.be odd 6 1 7744.2.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.a.a 1 9.c even 3 1
32.2.a.a 1 36.f odd 6 1
64.2.a.a 1 72.n even 6 1
64.2.a.a 1 72.p odd 6 1
256.2.b.b 2 144.v odd 12 2
256.2.b.b 2 144.x even 12 2
288.2.a.d 1 9.d odd 6 1
288.2.a.d 1 36.h even 6 1
576.2.a.c 1 72.j odd 6 1
576.2.a.c 1 72.l even 6 1
800.2.a.d 1 45.j even 6 1
800.2.a.d 1 180.p odd 6 1
800.2.c.e 2 45.k odd 12 2
800.2.c.e 2 180.x even 12 2
1024.2.e.j 4 288.bc even 24 4
1024.2.e.j 4 288.bd odd 24 4
1568.2.a.e 1 63.l odd 6 1
1568.2.a.e 1 252.bi even 6 1
1568.2.i.f 2 63.k odd 6 1
1568.2.i.f 2 63.t odd 6 1
1568.2.i.f 2 252.n even 6 1
1568.2.i.f 2 252.bj even 6 1
1568.2.i.g 2 63.g even 3 1
1568.2.i.g 2 63.h even 3 1
1568.2.i.g 2 252.u odd 6 1
1568.2.i.g 2 252.bl odd 6 1
1600.2.a.n 1 360.z odd 6 1
1600.2.a.n 1 360.bk even 6 1
1600.2.c.l 2 360.bo even 12 2
1600.2.c.l 2 360.bu odd 12 2
2304.2.d.j 2 144.u even 12 2
2304.2.d.j 2 144.w odd 12 2
2592.2.i.e 2 3.b odd 2 1
2592.2.i.e 2 9.d odd 6 1
2592.2.i.e 2 12.b even 2 1
2592.2.i.e 2 36.h even 6 1
2592.2.i.t 2 1.a even 1 1 trivial
2592.2.i.t 2 4.b odd 2 1 CM
2592.2.i.t 2 9.c even 3 1 inner
2592.2.i.t 2 36.f odd 6 1 inner
3136.2.a.m 1 504.be even 6 1
3136.2.a.m 1 504.bn odd 6 1
3872.2.a.f 1 99.h odd 6 1
3872.2.a.f 1 396.k even 6 1
5408.2.a.g 1 117.t even 6 1
5408.2.a.g 1 468.bg odd 6 1
7200.2.a.v 1 45.h odd 6 1
7200.2.a.v 1 180.n even 6 1
7200.2.f.m 2 45.l even 12 2
7200.2.f.m 2 180.v odd 12 2
7744.2.a.v 1 792.z even 6 1
7744.2.a.v 1 792.be odd 6 1
9248.2.a.f 1 153.h even 6 1
9248.2.a.f 1 612.q odd 6 1

Hecke kernels

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

\( T_{5}^{2} - 2 T_{5} + 4 \)
\( T_{7} \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 4 - 2 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( 36 + 6 T + T^{2} \)
$17$ \( ( -2 + T )^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( 100 - 10 T + T^{2} \)
$31$ \( T^{2} \)
$37$ \( ( 2 + T )^{2} \)
$41$ \( 100 + 10 T + T^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( ( -14 + T )^{2} \)
$59$ \( T^{2} \)
$61$ \( 100 - 10 T + T^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( 6 + T )^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( ( -10 + T )^{2} \)
$97$ \( 324 + 18 T + T^{2} \)
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