# Properties

 Label 2592.2.i.e Level $2592$ Weight $2$ Character orbit 2592.i Analytic conductor $20.697$ Analytic rank $1$ Dimension $2$ CM discriminant -4 Inner twists $4$

# 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: $$1$$ 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
 0.5 + 0.866025i 0.5 − 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.e 2
3.b odd 2 1 2592.2.i.t 2
4.b odd 2 1 CM 2592.2.i.e 2
9.c even 3 1 288.2.a.d 1
9.c even 3 1 inner 2592.2.i.e 2
9.d odd 6 1 32.2.a.a 1
9.d odd 6 1 2592.2.i.t 2
12.b even 2 1 2592.2.i.t 2
36.f odd 6 1 288.2.a.d 1
36.f odd 6 1 inner 2592.2.i.e 2
36.h even 6 1 32.2.a.a 1
36.h even 6 1 2592.2.i.t 2
45.h odd 6 1 800.2.a.d 1
45.j even 6 1 7200.2.a.v 1
45.k odd 12 2 7200.2.f.m 2
45.l even 12 2 800.2.c.e 2
63.i even 6 1 1568.2.i.f 2
63.j odd 6 1 1568.2.i.g 2
63.n odd 6 1 1568.2.i.g 2
63.o even 6 1 1568.2.a.e 1
63.s even 6 1 1568.2.i.f 2
72.j odd 6 1 64.2.a.a 1
72.l even 6 1 64.2.a.a 1
72.n even 6 1 576.2.a.c 1
72.p odd 6 1 576.2.a.c 1
99.g even 6 1 3872.2.a.f 1
117.n odd 6 1 5408.2.a.g 1
144.u even 12 2 256.2.b.b 2
144.v odd 12 2 2304.2.d.j 2
144.w odd 12 2 256.2.b.b 2
144.x even 12 2 2304.2.d.j 2
153.i odd 6 1 9248.2.a.f 1
180.n even 6 1 800.2.a.d 1
180.p odd 6 1 7200.2.a.v 1
180.v odd 12 2 800.2.c.e 2
180.x even 12 2 7200.2.f.m 2
252.o even 6 1 1568.2.i.g 2
252.r odd 6 1 1568.2.i.f 2
252.s odd 6 1 1568.2.a.e 1
252.bb even 6 1 1568.2.i.g 2
252.bn odd 6 1 1568.2.i.f 2
288.be odd 24 4 1024.2.e.j 4
288.bf even 24 4 1024.2.e.j 4
360.bd even 6 1 1600.2.a.n 1
360.bh odd 6 1 1600.2.a.n 1
360.br even 12 2 1600.2.c.l 2
360.bt odd 12 2 1600.2.c.l 2
396.o odd 6 1 3872.2.a.f 1
468.x even 6 1 5408.2.a.g 1
504.cc even 6 1 3136.2.a.m 1
504.co odd 6 1 3136.2.a.m 1
612.n even 6 1 9248.2.a.f 1
792.s odd 6 1 7744.2.a.v 1
792.w even 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.d odd 6 1
32.2.a.a 1 36.h even 6 1
64.2.a.a 1 72.j odd 6 1
64.2.a.a 1 72.l even 6 1
256.2.b.b 2 144.u even 12 2
256.2.b.b 2 144.w odd 12 2
288.2.a.d 1 9.c even 3 1
288.2.a.d 1 36.f odd 6 1
576.2.a.c 1 72.n even 6 1
576.2.a.c 1 72.p odd 6 1
800.2.a.d 1 45.h odd 6 1
800.2.a.d 1 180.n even 6 1
800.2.c.e 2 45.l even 12 2
800.2.c.e 2 180.v odd 12 2
1024.2.e.j 4 288.be odd 24 4
1024.2.e.j 4 288.bf even 24 4
1568.2.a.e 1 63.o even 6 1
1568.2.a.e 1 252.s odd 6 1
1568.2.i.f 2 63.i even 6 1
1568.2.i.f 2 63.s even 6 1
1568.2.i.f 2 252.r odd 6 1
1568.2.i.f 2 252.bn odd 6 1
1568.2.i.g 2 63.j odd 6 1
1568.2.i.g 2 63.n odd 6 1
1568.2.i.g 2 252.o even 6 1
1568.2.i.g 2 252.bb even 6 1
1600.2.a.n 1 360.bd even 6 1
1600.2.a.n 1 360.bh odd 6 1
1600.2.c.l 2 360.br even 12 2
1600.2.c.l 2 360.bt odd 12 2
2304.2.d.j 2 144.v odd 12 2
2304.2.d.j 2 144.x even 12 2
2592.2.i.e 2 1.a even 1 1 trivial
2592.2.i.e 2 4.b odd 2 1 CM
2592.2.i.e 2 9.c even 3 1 inner
2592.2.i.e 2 36.f odd 6 1 inner
2592.2.i.t 2 3.b odd 2 1
2592.2.i.t 2 9.d odd 6 1
2592.2.i.t 2 12.b even 2 1
2592.2.i.t 2 36.h even 6 1
3136.2.a.m 1 504.cc even 6 1
3136.2.a.m 1 504.co odd 6 1
3872.2.a.f 1 99.g even 6 1
3872.2.a.f 1 396.o odd 6 1
5408.2.a.g 1 117.n odd 6 1
5408.2.a.g 1 468.x even 6 1
7200.2.a.v 1 45.j even 6 1
7200.2.a.v 1 180.p odd 6 1
7200.2.f.m 2 45.k odd 12 2
7200.2.f.m 2 180.x even 12 2
7744.2.a.v 1 792.s odd 6 1
7744.2.a.v 1 792.w even 6 1
9248.2.a.f 1 153.i odd 6 1
9248.2.a.f 1 612.n even 6 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

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