# Properties

 Label 1728.2.i.g Level $1728$ Weight $2$ Character orbit 1728.i Analytic conductor $13.798$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1728.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.7981494693$$ 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 72) 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{5} + ( -3 + 3 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + \zeta_{6} q^{5} + ( -3 + 3 \zeta_{6} ) q^{7} + ( 5 - 5 \zeta_{6} ) q^{11} -5 \zeta_{6} q^{13} + 2 q^{17} -4 q^{19} + \zeta_{6} q^{23} + ( 4 - 4 \zeta_{6} ) q^{25} + ( 9 - 9 \zeta_{6} ) q^{29} -\zeta_{6} q^{31} -3 q^{35} + 6 q^{37} + 3 \zeta_{6} q^{41} + ( -1 + \zeta_{6} ) q^{43} + ( 3 - 3 \zeta_{6} ) q^{47} -2 \zeta_{6} q^{49} + 2 q^{53} + 5 q^{55} + 11 \zeta_{6} q^{59} + ( 7 - 7 \zeta_{6} ) q^{61} + ( 5 - 5 \zeta_{6} ) q^{65} + \zeta_{6} q^{67} + 4 q^{71} -2 q^{73} + 15 \zeta_{6} q^{77} + ( 1 - \zeta_{6} ) q^{79} + ( 1 - \zeta_{6} ) q^{83} + 2 \zeta_{6} q^{85} + 18 q^{89} + 15 q^{91} -4 \zeta_{6} q^{95} + ( 13 - 13 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{5} - 3q^{7} + O(q^{10})$$ $$2q + q^{5} - 3q^{7} + 5q^{11} - 5q^{13} + 4q^{17} - 8q^{19} + q^{23} + 4q^{25} + 9q^{29} - q^{31} - 6q^{35} + 12q^{37} + 3q^{41} - q^{43} + 3q^{47} - 2q^{49} + 4q^{53} + 10q^{55} + 11q^{59} + 7q^{61} + 5q^{65} + q^{67} + 8q^{71} - 4q^{73} + 15q^{77} + q^{79} + q^{83} + 2q^{85} + 36q^{89} + 30q^{91} - 4q^{95} + 13q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
577.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 0.500000 0.866025i 0 −1.50000 2.59808i 0 0 0
1153.1 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: 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 1728.2.i.g 2
3.b odd 2 1 576.2.i.c 2
4.b odd 2 1 1728.2.i.h 2
8.b even 2 1 432.2.i.a 2
8.d odd 2 1 216.2.i.a 2
9.c even 3 1 inner 1728.2.i.g 2
9.c even 3 1 5184.2.a.n 1
9.d odd 6 1 576.2.i.c 2
9.d odd 6 1 5184.2.a.x 1
12.b even 2 1 576.2.i.d 2
24.f even 2 1 72.2.i.a 2
24.h odd 2 1 144.2.i.b 2
36.f odd 6 1 1728.2.i.h 2
36.f odd 6 1 5184.2.a.i 1
36.h even 6 1 576.2.i.d 2
36.h even 6 1 5184.2.a.s 1
72.j odd 6 1 144.2.i.b 2
72.j odd 6 1 1296.2.a.e 1
72.l even 6 1 72.2.i.a 2
72.l even 6 1 648.2.a.a 1
72.n even 6 1 432.2.i.a 2
72.n even 6 1 1296.2.a.i 1
72.p odd 6 1 216.2.i.a 2
72.p odd 6 1 648.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.i.a 2 24.f even 2 1
72.2.i.a 2 72.l even 6 1
144.2.i.b 2 24.h odd 2 1
144.2.i.b 2 72.j odd 6 1
216.2.i.a 2 8.d odd 2 1
216.2.i.a 2 72.p odd 6 1
432.2.i.a 2 8.b even 2 1
432.2.i.a 2 72.n even 6 1
576.2.i.c 2 3.b odd 2 1
576.2.i.c 2 9.d odd 6 1
576.2.i.d 2 12.b even 2 1
576.2.i.d 2 36.h even 6 1
648.2.a.a 1 72.l even 6 1
648.2.a.c 1 72.p odd 6 1
1296.2.a.e 1 72.j odd 6 1
1296.2.a.i 1 72.n even 6 1
1728.2.i.g 2 1.a even 1 1 trivial
1728.2.i.g 2 9.c even 3 1 inner
1728.2.i.h 2 4.b odd 2 1
1728.2.i.h 2 36.f odd 6 1
5184.2.a.i 1 36.f odd 6 1
5184.2.a.n 1 9.c even 3 1
5184.2.a.s 1 36.h even 6 1
5184.2.a.x 1 9.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1728, [\chi])$$:

 $$T_{5}^{2} - T_{5} + 1$$ $$T_{7}^{2} + 3 T_{7} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$1 - T + T^{2}$$
$7$ $$9 + 3 T + T^{2}$$
$11$ $$25 - 5 T + T^{2}$$
$13$ $$25 + 5 T + T^{2}$$
$17$ $$( -2 + T )^{2}$$
$19$ $$( 4 + T )^{2}$$
$23$ $$1 - T + T^{2}$$
$29$ $$81 - 9 T + T^{2}$$
$31$ $$1 + T + T^{2}$$
$37$ $$( -6 + T )^{2}$$
$41$ $$9 - 3 T + T^{2}$$
$43$ $$1 + T + T^{2}$$
$47$ $$9 - 3 T + T^{2}$$
$53$ $$( -2 + T )^{2}$$
$59$ $$121 - 11 T + T^{2}$$
$61$ $$49 - 7 T + T^{2}$$
$67$ $$1 - T + T^{2}$$
$71$ $$( -4 + T )^{2}$$
$73$ $$( 2 + T )^{2}$$
$79$ $$1 - T + T^{2}$$
$83$ $$1 - T + T^{2}$$
$89$ $$( -18 + T )^{2}$$
$97$ $$169 - 13 T + T^{2}$$