# Properties

 Label 1728.1.g.a Level $1728$ Weight $1$ Character orbit 1728.g Analytic conductor $0.862$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1728.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.862384341830$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 432) Projective image: $$D_{6}$$ Projective field: Galois closure of 6.0.186624.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{6} + \zeta_{6}^{2} ) q^{7} +O(q^{10})$$ $$q + ( \zeta_{6} + \zeta_{6}^{2} ) q^{7} - q^{13} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{19} - q^{25} + q^{37} + ( -1 - \zeta_{6} + \zeta_{6}^{2} ) q^{49} + q^{61} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{67} + q^{73} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{79} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{91} + q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + O(q^{10})$$ $$2 q - 2 q^{13} - 2 q^{25} + 2 q^{37} - 4 q^{49} + 2 q^{61} + 2 q^{73} + 2 q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
703.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 0 0 1.73205i 0 0 0
703.2 0 0 0 0 0 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
4.b odd 2 1 inner
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.1.g.a 2
3.b odd 2 1 CM 1728.1.g.a 2
4.b odd 2 1 inner 1728.1.g.a 2
8.b even 2 1 432.1.g.a 2
8.d odd 2 1 432.1.g.a 2
12.b even 2 1 inner 1728.1.g.a 2
24.f even 2 1 432.1.g.a 2
24.h odd 2 1 432.1.g.a 2
72.j odd 6 1 1296.1.o.a 2
72.j odd 6 1 1296.1.o.c 2
72.l even 6 1 1296.1.o.a 2
72.l even 6 1 1296.1.o.c 2
72.n even 6 1 1296.1.o.a 2
72.n even 6 1 1296.1.o.c 2
72.p odd 6 1 1296.1.o.a 2
72.p odd 6 1 1296.1.o.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.1.g.a 2 8.b even 2 1
432.1.g.a 2 8.d odd 2 1
432.1.g.a 2 24.f even 2 1
432.1.g.a 2 24.h odd 2 1
1296.1.o.a 2 72.j odd 6 1
1296.1.o.a 2 72.l even 6 1
1296.1.o.a 2 72.n even 6 1
1296.1.o.a 2 72.p odd 6 1
1296.1.o.c 2 72.j odd 6 1
1296.1.o.c 2 72.l even 6 1
1296.1.o.c 2 72.n even 6 1
1296.1.o.c 2 72.p odd 6 1
1728.1.g.a 2 1.a even 1 1 trivial
1728.1.g.a 2 3.b odd 2 1 CM
1728.1.g.a 2 4.b odd 2 1 inner
1728.1.g.a 2 12.b even 2 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1728, [\chi])$$.

## Hecke characteristic polynomials

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