# Properties

 Label 1728.3.b.a Level $1728$ Weight $3$ Character orbit 1728.b Analytic conductor $47.085$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1728.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$47.0845896815$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 6 \zeta_{12}^{3} q^{5} -5 \zeta_{12}^{3} q^{7} +O(q^{10})$$ $$q + 6 \zeta_{12}^{3} q^{5} -5 \zeta_{12}^{3} q^{7} -6 q^{11} + ( 3 - 6 \zeta_{12}^{2} ) q^{13} + ( 36 \zeta_{12} - 18 \zeta_{12}^{3} ) q^{17} + ( -30 \zeta_{12} + 15 \zeta_{12}^{3} ) q^{19} + ( -18 + 36 \zeta_{12}^{2} ) q^{23} -11 q^{25} -36 \zeta_{12}^{3} q^{29} + 26 \zeta_{12}^{3} q^{31} + 30 q^{35} + ( -15 + 30 \zeta_{12}^{2} ) q^{37} + ( -24 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{43} + ( -18 + 36 \zeta_{12}^{2} ) q^{47} + 24 q^{49} + 60 \zeta_{12}^{3} q^{53} -36 \zeta_{12}^{3} q^{55} -18 q^{59} + ( -45 + 90 \zeta_{12}^{2} ) q^{61} + ( 36 \zeta_{12} - 18 \zeta_{12}^{3} ) q^{65} + ( -90 \zeta_{12} + 45 \zeta_{12}^{3} ) q^{67} + 25 q^{73} + 30 \zeta_{12}^{3} q^{77} -31 \zeta_{12}^{3} q^{79} -120 q^{83} + ( -108 + 216 \zeta_{12}^{2} ) q^{85} + ( -180 \zeta_{12} + 90 \zeta_{12}^{3} ) q^{89} + ( -30 \zeta_{12} + 15 \zeta_{12}^{3} ) q^{91} + ( 90 - 180 \zeta_{12}^{2} ) q^{95} -85 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + O(q^{10})$$ $$4 q - 24 q^{11} - 44 q^{25} + 120 q^{35} + 96 q^{49} - 72 q^{59} + 100 q^{73} - 480 q^{83} - 340 q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1567.1
 −0.866025 − 0.500000i 0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i
0 0 0 6.00000i 0 5.00000i 0 0 0
1567.2 0 0 0 6.00000i 0 5.00000i 0 0 0
1567.3 0 0 0 6.00000i 0 5.00000i 0 0 0
1567.4 0 0 0 6.00000i 0 5.00000i 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
8.d odd 2 1 inner
12.b even 2 1 inner
24.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.3.b.a 4
3.b odd 2 1 1728.3.b.f yes 4
4.b odd 2 1 1728.3.b.f yes 4
8.b even 2 1 1728.3.b.f yes 4
8.d odd 2 1 inner 1728.3.b.a 4
12.b even 2 1 inner 1728.3.b.a 4
24.f even 2 1 1728.3.b.f yes 4
24.h odd 2 1 inner 1728.3.b.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1728.3.b.a 4 1.a even 1 1 trivial
1728.3.b.a 4 8.d odd 2 1 inner
1728.3.b.a 4 12.b even 2 1 inner
1728.3.b.a 4 24.h odd 2 1 inner
1728.3.b.f yes 4 3.b odd 2 1
1728.3.b.f yes 4 4.b odd 2 1
1728.3.b.f yes 4 8.b even 2 1
1728.3.b.f yes 4 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1728, [\chi])$$:

 $$T_{5}^{2} + 36$$ $$T_{7}^{2} + 25$$ $$T_{11} + 6$$ $$T_{17}^{2} - 972$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 36 + T^{2} )^{2}$$
$7$ $$( 25 + T^{2} )^{2}$$
$11$ $$( 6 + T )^{4}$$
$13$ $$( 27 + T^{2} )^{2}$$
$17$ $$( -972 + T^{2} )^{2}$$
$19$ $$( -675 + T^{2} )^{2}$$
$23$ $$( 972 + T^{2} )^{2}$$
$29$ $$( 1296 + T^{2} )^{2}$$
$31$ $$( 676 + T^{2} )^{2}$$
$37$ $$( 675 + T^{2} )^{2}$$
$41$ $$T^{4}$$
$43$ $$( -432 + T^{2} )^{2}$$
$47$ $$( 972 + T^{2} )^{2}$$
$53$ $$( 3600 + T^{2} )^{2}$$
$59$ $$( 18 + T )^{4}$$
$61$ $$( 6075 + T^{2} )^{2}$$
$67$ $$( -6075 + T^{2} )^{2}$$
$71$ $$T^{4}$$
$73$ $$( -25 + T )^{4}$$
$79$ $$( 961 + T^{2} )^{2}$$
$83$ $$( 120 + T )^{4}$$
$89$ $$( -24300 + T^{2} )^{2}$$
$97$ $$( 85 + T )^{4}$$