# Properties

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

# Related objects

## 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 96) 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.

 $$f(q)$$ $$=$$ $$q -2 \zeta_{6} q^{5} + ( -4 + 4 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q -2 \zeta_{6} q^{5} + ( -4 + 4 \zeta_{6} ) q^{7} + ( 4 - 4 \zeta_{6} ) q^{11} + 2 \zeta_{6} q^{13} -6 q^{17} + 4 q^{19} + ( 1 - \zeta_{6} ) q^{25} + ( -2 + 2 \zeta_{6} ) q^{29} + 4 \zeta_{6} q^{31} + 8 q^{35} -2 q^{37} -2 \zeta_{6} q^{41} + ( 4 - 4 \zeta_{6} ) q^{43} + ( 8 - 8 \zeta_{6} ) q^{47} -9 \zeta_{6} q^{49} + 10 q^{53} -8 q^{55} -4 \zeta_{6} q^{59} + ( -6 + 6 \zeta_{6} ) q^{61} + ( 4 - 4 \zeta_{6} ) q^{65} + 4 \zeta_{6} q^{67} + 16 q^{71} -6 q^{73} + 16 \zeta_{6} q^{77} + ( 4 - 4 \zeta_{6} ) q^{79} + ( 12 - 12 \zeta_{6} ) q^{83} + 12 \zeta_{6} q^{85} + 10 q^{89} -8 q^{91} -8 \zeta_{6} q^{95} + ( 14 - 14 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{5} - 4q^{7} + O(q^{10})$$ $$2q - 2q^{5} - 4q^{7} + 4q^{11} + 2q^{13} - 12q^{17} + 8q^{19} + q^{25} - 2q^{29} + 4q^{31} + 16q^{35} - 4q^{37} - 2q^{41} + 4q^{43} + 8q^{47} - 9q^{49} + 20q^{53} - 16q^{55} - 4q^{59} - 6q^{61} + 4q^{65} + 4q^{67} + 32q^{71} - 12q^{73} + 16q^{77} + 4q^{79} + 12q^{83} + 12q^{85} + 20q^{89} - 16q^{91} - 8q^{95} + 14q^{97} + O(q^{100})$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 −1.00000 1.73205i 0 −2.00000 + 3.46410i 0 0 0
1729.1 0 0 0 −1.00000 + 1.73205i 0 −2.00000 3.46410i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.b 2
3.b odd 2 1 2592.2.i.q 2
4.b odd 2 1 2592.2.i.h 2
9.c even 3 1 96.2.a.a 1
9.c even 3 1 inner 2592.2.i.b 2
9.d odd 6 1 288.2.a.c 1
9.d odd 6 1 2592.2.i.q 2
12.b even 2 1 2592.2.i.w 2
36.f odd 6 1 96.2.a.b yes 1
36.f odd 6 1 2592.2.i.h 2
36.h even 6 1 288.2.a.b 1
36.h even 6 1 2592.2.i.w 2
45.h odd 6 1 7200.2.a.e 1
45.j even 6 1 2400.2.a.r 1
45.k odd 12 2 2400.2.f.a 2
45.l even 12 2 7200.2.f.x 2
63.l odd 6 1 4704.2.a.t 1
72.j odd 6 1 576.2.a.h 1
72.l even 6 1 576.2.a.g 1
72.n even 6 1 192.2.a.c 1
72.p odd 6 1 192.2.a.a 1
144.u even 12 2 2304.2.d.s 2
144.v odd 12 2 768.2.d.h 2
144.w odd 12 2 2304.2.d.c 2
144.x even 12 2 768.2.d.a 2
180.n even 6 1 7200.2.a.bx 1
180.p odd 6 1 2400.2.a.q 1
180.v odd 12 2 7200.2.f.f 2
180.x even 12 2 2400.2.f.r 2
252.bi even 6 1 4704.2.a.e 1
360.z odd 6 1 4800.2.a.co 1
360.bk even 6 1 4800.2.a.f 1
360.bo even 12 2 4800.2.f.e 2
360.bu odd 12 2 4800.2.f.bh 2
504.be even 6 1 9408.2.a.ct 1
504.bn odd 6 1 9408.2.a.bj 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.2.a.a 1 9.c even 3 1
96.2.a.b yes 1 36.f odd 6 1
192.2.a.a 1 72.p odd 6 1
192.2.a.c 1 72.n even 6 1
288.2.a.b 1 36.h even 6 1
288.2.a.c 1 9.d odd 6 1
576.2.a.g 1 72.l even 6 1
576.2.a.h 1 72.j odd 6 1
768.2.d.a 2 144.x even 12 2
768.2.d.h 2 144.v odd 12 2
2304.2.d.c 2 144.w odd 12 2
2304.2.d.s 2 144.u even 12 2
2400.2.a.q 1 180.p odd 6 1
2400.2.a.r 1 45.j even 6 1
2400.2.f.a 2 45.k odd 12 2
2400.2.f.r 2 180.x even 12 2
2592.2.i.b 2 1.a even 1 1 trivial
2592.2.i.b 2 9.c even 3 1 inner
2592.2.i.h 2 4.b odd 2 1
2592.2.i.h 2 36.f odd 6 1
2592.2.i.q 2 3.b odd 2 1
2592.2.i.q 2 9.d odd 6 1
2592.2.i.w 2 12.b even 2 1
2592.2.i.w 2 36.h even 6 1
4704.2.a.e 1 252.bi even 6 1
4704.2.a.t 1 63.l odd 6 1
4800.2.a.f 1 360.bk even 6 1
4800.2.a.co 1 360.z odd 6 1
4800.2.f.e 2 360.bo even 12 2
4800.2.f.bh 2 360.bu odd 12 2
7200.2.a.e 1 45.h odd 6 1
7200.2.a.bx 1 180.n even 6 1
7200.2.f.f 2 180.v odd 12 2
7200.2.f.x 2 45.l even 12 2
9408.2.a.bj 1 504.bn odd 6 1
9408.2.a.ct 1 504.be even 6 1

## Hecke kernels

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

 $$T_{5}^{2} + 2 T_{5} + 4$$ $$T_{7}^{2} + 4 T_{7} + 16$$ $$T_{11}^{2} - 4 T_{11} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4}$$
$7$ $$( 1 - T + 7 T^{2} )( 1 + 5 T + 7 T^{2} )$$
$11$ $$1 - 4 T + 5 T^{2} - 44 T^{3} + 121 T^{4}$$
$13$ $$( 1 - 7 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} )$$
$17$ $$( 1 + 6 T + 17 T^{2} )^{2}$$
$19$ $$( 1 - 4 T + 19 T^{2} )^{2}$$
$23$ $$1 - 23 T^{2} + 529 T^{4}$$
$29$ $$1 + 2 T - 25 T^{2} + 58 T^{3} + 841 T^{4}$$
$31$ $$( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} )$$
$37$ $$( 1 + 2 T + 37 T^{2} )^{2}$$
$41$ $$1 + 2 T - 37 T^{2} + 82 T^{3} + 1681 T^{4}$$
$43$ $$1 - 4 T - 27 T^{2} - 172 T^{3} + 1849 T^{4}$$
$47$ $$1 - 8 T + 17 T^{2} - 376 T^{3} + 2209 T^{4}$$
$53$ $$( 1 - 10 T + 53 T^{2} )^{2}$$
$59$ $$1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4}$$
$61$ $$1 + 6 T - 25 T^{2} + 366 T^{3} + 3721 T^{4}$$
$67$ $$1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4}$$
$71$ $$( 1 - 16 T + 71 T^{2} )^{2}$$
$73$ $$( 1 + 6 T + 73 T^{2} )^{2}$$
$79$ $$( 1 - 17 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} )$$
$83$ $$1 - 12 T + 61 T^{2} - 996 T^{3} + 6889 T^{4}$$
$89$ $$( 1 - 10 T + 89 T^{2} )^{2}$$
$97$ $$( 1 - 19 T + 97 T^{2} )( 1 + 5 T + 97 T^{2} )$$