# Properties

 Label 1728.1.o.a Level $1728$ Weight $1$ Character orbit 1728.o Analytic conductor $0.862$ Analytic rank $0$ Dimension $4$ Projective image $A_{4}$ CM/RM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.862384341830$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 288) Projective image: $$A_{4}$$ Projective field: Galois closure of 4.0.5184.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{12}^{4} q^{5} + \zeta_{12}^{5} q^{7} +O(q^{10})$$ $$q -\zeta_{12}^{4} q^{5} + \zeta_{12}^{5} q^{7} -\zeta_{12}^{5} q^{11} -\zeta_{12}^{4} q^{13} -\zeta_{12} q^{23} -\zeta_{12}^{2} q^{29} + \zeta_{12} q^{31} + \zeta_{12}^{3} q^{35} -\zeta_{12}^{4} q^{41} -\zeta_{12}^{5} q^{43} -\zeta_{12}^{5} q^{47} -\zeta_{12}^{3} q^{55} -\zeta_{12} q^{59} + \zeta_{12}^{2} q^{61} -\zeta_{12}^{2} q^{65} -\zeta_{12} q^{67} + 2 \zeta_{12}^{3} q^{71} + \zeta_{12}^{4} q^{77} -\zeta_{12}^{5} q^{79} + \zeta_{12}^{5} q^{83} + \zeta_{12}^{3} q^{91} -\zeta_{12}^{2} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{5} + O(q^{10})$$ $$4 q + 2 q^{5} + 2 q^{13} - 2 q^{29} + 2 q^{41} + 2 q^{61} - 2 q^{65} - 2 q^{77} - 2 q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$1$$ $$-1$$ $$\zeta_{12}^{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
0 0 0 0.500000 + 0.866025i 0 −0.866025 0.500000i 0 0 0
127.2 0 0 0 0.500000 + 0.866025i 0 0.866025 + 0.500000i 0 0 0
1279.1 0 0 0 0.500000 0.866025i 0 −0.866025 + 0.500000i 0 0 0
1279.2 0 0 0 0.500000 0.866025i 0 0.866025 0.500000i 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 inner
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 1728.1.o.a 4
3.b odd 2 1 576.1.o.a 4
4.b odd 2 1 inner 1728.1.o.a 4
8.b even 2 1 864.1.o.a 4
8.d odd 2 1 864.1.o.a 4
9.c even 3 1 inner 1728.1.o.a 4
9.d odd 6 1 576.1.o.a 4
12.b even 2 1 576.1.o.a 4
24.f even 2 1 288.1.o.a 4
24.h odd 2 1 288.1.o.a 4
36.f odd 6 1 inner 1728.1.o.a 4
36.h even 6 1 576.1.o.a 4
48.i odd 4 1 2304.1.t.a 4
48.i odd 4 1 2304.1.t.b 4
48.k even 4 1 2304.1.t.a 4
48.k even 4 1 2304.1.t.b 4
72.j odd 6 1 288.1.o.a 4
72.j odd 6 1 2592.1.g.a 2
72.l even 6 1 288.1.o.a 4
72.l even 6 1 2592.1.g.a 2
72.n even 6 1 864.1.o.a 4
72.n even 6 1 2592.1.g.b 2
72.p odd 6 1 864.1.o.a 4
72.p odd 6 1 2592.1.g.b 2
144.u even 12 1 2304.1.t.a 4
144.u even 12 1 2304.1.t.b 4
144.w odd 12 1 2304.1.t.a 4
144.w odd 12 1 2304.1.t.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.1.o.a 4 24.f even 2 1
288.1.o.a 4 24.h odd 2 1
288.1.o.a 4 72.j odd 6 1
288.1.o.a 4 72.l even 6 1
576.1.o.a 4 3.b odd 2 1
576.1.o.a 4 9.d odd 6 1
576.1.o.a 4 12.b even 2 1
576.1.o.a 4 36.h even 6 1
864.1.o.a 4 8.b even 2 1
864.1.o.a 4 8.d odd 2 1
864.1.o.a 4 72.n even 6 1
864.1.o.a 4 72.p odd 6 1
1728.1.o.a 4 1.a even 1 1 trivial
1728.1.o.a 4 4.b odd 2 1 inner
1728.1.o.a 4 9.c even 3 1 inner
1728.1.o.a 4 36.f odd 6 1 inner
2304.1.t.a 4 48.i odd 4 1
2304.1.t.a 4 48.k even 4 1
2304.1.t.a 4 144.u even 12 1
2304.1.t.a 4 144.w odd 12 1
2304.1.t.b 4 48.i odd 4 1
2304.1.t.b 4 48.k even 4 1
2304.1.t.b 4 144.u even 12 1
2304.1.t.b 4 144.w odd 12 1
2592.1.g.a 2 72.j odd 6 1
2592.1.g.a 2 72.l even 6 1
2592.1.g.b 2 72.n even 6 1
2592.1.g.b 2 72.p odd 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

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