# Properties

 Label 1728.2.s.a Level $1728$ Weight $2$ Character orbit 1728.s 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.s (of order $$6$$, 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: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 144) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -4 + 2 \zeta_{6} ) q^{5} + ( -2 - 2 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( -4 + 2 \zeta_{6} ) q^{5} + ( -2 - 2 \zeta_{6} ) q^{7} + ( -3 + 3 \zeta_{6} ) q^{11} + 4 \zeta_{6} q^{13} + ( -1 + 2 \zeta_{6} ) q^{17} + ( 1 - 2 \zeta_{6} ) q^{19} + ( 7 - 7 \zeta_{6} ) q^{25} + ( -2 - 2 \zeta_{6} ) q^{29} + 12 q^{35} -2 q^{37} + ( 6 - 3 \zeta_{6} ) q^{41} + ( -3 - 3 \zeta_{6} ) q^{43} + ( 12 - 12 \zeta_{6} ) q^{47} + 5 \zeta_{6} q^{49} + ( 6 - 12 \zeta_{6} ) q^{55} -15 \zeta_{6} q^{59} + ( 8 - 8 \zeta_{6} ) q^{61} + ( -8 - 8 \zeta_{6} ) q^{65} + ( 10 - 5 \zeta_{6} ) q^{67} + 6 q^{71} -11 q^{73} + ( 12 - 6 \zeta_{6} ) q^{77} + ( 2 + 2 \zeta_{6} ) q^{79} + ( 12 - 12 \zeta_{6} ) q^{83} -6 \zeta_{6} q^{85} + ( -8 + 16 \zeta_{6} ) q^{89} + ( 8 - 16 \zeta_{6} ) q^{91} + 6 \zeta_{6} q^{95} + ( -13 + 13 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 6q^{5} - 6q^{7} + O(q^{10})$$ $$2q - 6q^{5} - 6q^{7} - 3q^{11} + 4q^{13} + 7q^{25} - 6q^{29} + 24q^{35} - 4q^{37} + 9q^{41} - 9q^{43} + 12q^{47} + 5q^{49} - 15q^{59} + 8q^{61} - 24q^{65} + 15q^{67} + 12q^{71} - 22q^{73} + 18q^{77} + 6q^{79} + 12q^{83} - 6q^{85} + 6q^{95} - 13q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
575.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 −3.00000 + 1.73205i 0 −3.00000 1.73205i 0 0 0
1151.1 0 0 0 −3.00000 1.73205i 0 −3.00000 + 1.73205i 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
36.h even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.s.a 2
3.b odd 2 1 576.2.s.d 2
4.b odd 2 1 1728.2.s.b 2
8.b even 2 1 432.2.s.c 2
8.d odd 2 1 432.2.s.d 2
9.c even 3 1 576.2.s.a 2
9.c even 3 1 5184.2.c.c 2
9.d odd 6 1 1728.2.s.b 2
9.d odd 6 1 5184.2.c.a 2
12.b even 2 1 576.2.s.a 2
24.f even 2 1 144.2.s.d yes 2
24.h odd 2 1 144.2.s.a 2
36.f odd 6 1 576.2.s.d 2
36.f odd 6 1 5184.2.c.a 2
36.h even 6 1 inner 1728.2.s.a 2
36.h even 6 1 5184.2.c.c 2
72.j odd 6 1 432.2.s.d 2
72.j odd 6 1 1296.2.c.d 2
72.l even 6 1 432.2.s.c 2
72.l even 6 1 1296.2.c.b 2
72.n even 6 1 144.2.s.d yes 2
72.n even 6 1 1296.2.c.b 2
72.p odd 6 1 144.2.s.a 2
72.p odd 6 1 1296.2.c.d 2

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

## Hecke kernels

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

 $$T_{5}^{2} + 6 T_{5} + 12$$ $$T_{7}^{2} + 6 T_{7} + 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$12 + 6 T + T^{2}$$
$7$ $$12 + 6 T + T^{2}$$
$11$ $$9 + 3 T + T^{2}$$
$13$ $$16 - 4 T + T^{2}$$
$17$ $$3 + T^{2}$$
$19$ $$3 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$12 + 6 T + T^{2}$$
$31$ $$T^{2}$$
$37$ $$( 2 + T )^{2}$$
$41$ $$27 - 9 T + T^{2}$$
$43$ $$27 + 9 T + T^{2}$$
$47$ $$144 - 12 T + T^{2}$$
$53$ $$T^{2}$$
$59$ $$225 + 15 T + T^{2}$$
$61$ $$64 - 8 T + T^{2}$$
$67$ $$75 - 15 T + T^{2}$$
$71$ $$( -6 + T )^{2}$$
$73$ $$( 11 + T )^{2}$$
$79$ $$12 - 6 T + T^{2}$$
$83$ $$144 - 12 T + T^{2}$$
$89$ $$192 + T^{2}$$
$97$ $$169 + 13 T + T^{2}$$