# Properties

 Label 1728.2.d.e Level $1728$ Weight $2$ Character orbit 1728.d Analytic conductor $13.798$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$13.7981494693$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 2 - 4 \zeta_{12}^{2} ) q^{5} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{7} +O(q^{10})$$ $$q + ( 2 - 4 \zeta_{12}^{2} ) q^{5} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{7} -6 \zeta_{12}^{3} q^{11} + ( -3 + 6 \zeta_{12}^{2} ) q^{13} -6 q^{17} -5 \zeta_{12}^{3} q^{19} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{23} -7 q^{25} + ( -4 + 8 \zeta_{12}^{2} ) q^{29} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{31} + 6 \zeta_{12}^{3} q^{35} + ( -1 + 2 \zeta_{12}^{2} ) q^{37} -4 \zeta_{12}^{3} q^{43} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{47} -4 q^{49} + ( 4 - 8 \zeta_{12}^{2} ) q^{53} + ( -24 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{55} + 6 \zeta_{12}^{3} q^{59} + ( -7 + 14 \zeta_{12}^{2} ) q^{61} + 18 q^{65} + 5 \zeta_{12}^{3} q^{67} + ( -16 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{71} -7 q^{73} + ( -6 + 12 \zeta_{12}^{2} ) q^{77} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{79} + ( -12 + 24 \zeta_{12}^{2} ) q^{85} + 6 q^{89} -9 \zeta_{12}^{3} q^{91} + ( -20 \zeta_{12} + 10 \zeta_{12}^{3} ) q^{95} + 7 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 24q^{17} - 28q^{25} - 16q^{49} + 72q^{65} - 28q^{73} + 24q^{89} + 28q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
865.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
0 0 0 3.46410i 0 −1.73205 0 0 0
865.2 0 0 0 3.46410i 0 1.73205 0 0 0
865.3 0 0 0 3.46410i 0 −1.73205 0 0 0
865.4 0 0 0 3.46410i 0 1.73205 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
8.b even 2 1 inner
8.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.d.e 4
3.b odd 2 1 1728.2.d.j yes 4
4.b odd 2 1 inner 1728.2.d.e 4
8.b even 2 1 inner 1728.2.d.e 4
8.d odd 2 1 inner 1728.2.d.e 4
12.b even 2 1 1728.2.d.j yes 4
16.e even 4 1 6912.2.a.bg 2
16.e even 4 1 6912.2.a.bq 2
16.f odd 4 1 6912.2.a.bg 2
16.f odd 4 1 6912.2.a.bq 2
24.f even 2 1 1728.2.d.j yes 4
24.h odd 2 1 1728.2.d.j yes 4
48.i odd 4 1 6912.2.a.bh 2
48.i odd 4 1 6912.2.a.br 2
48.k even 4 1 6912.2.a.bh 2
48.k even 4 1 6912.2.a.br 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1728.2.d.e 4 1.a even 1 1 trivial
1728.2.d.e 4 4.b odd 2 1 inner
1728.2.d.e 4 8.b even 2 1 inner
1728.2.d.e 4 8.d odd 2 1 inner
1728.2.d.j yes 4 3.b odd 2 1
1728.2.d.j yes 4 12.b even 2 1
1728.2.d.j yes 4 24.f even 2 1
1728.2.d.j yes 4 24.h odd 2 1
6912.2.a.bg 2 16.e even 4 1
6912.2.a.bg 2 16.f odd 4 1
6912.2.a.bh 2 48.i odd 4 1
6912.2.a.bh 2 48.k even 4 1
6912.2.a.bq 2 16.e even 4 1
6912.2.a.bq 2 16.f odd 4 1
6912.2.a.br 2 48.i odd 4 1
6912.2.a.br 2 48.k even 4 1

## Hecke kernels

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

 $$T_{5}^{2} + 12$$ $$T_{7}^{2} - 3$$ $$T_{17} + 6$$ $$T_{41}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 12 + T^{2} )^{2}$$
$7$ $$( -3 + T^{2} )^{2}$$
$11$ $$( 36 + T^{2} )^{2}$$
$13$ $$( 27 + T^{2} )^{2}$$
$17$ $$( 6 + T )^{4}$$
$19$ $$( 25 + T^{2} )^{2}$$
$23$ $$( -12 + T^{2} )^{2}$$
$29$ $$( 48 + T^{2} )^{2}$$
$31$ $$( -12 + T^{2} )^{2}$$
$37$ $$( 3 + T^{2} )^{2}$$
$41$ $$T^{4}$$
$43$ $$( 16 + T^{2} )^{2}$$
$47$ $$( -12 + T^{2} )^{2}$$
$53$ $$( 48 + T^{2} )^{2}$$
$59$ $$( 36 + T^{2} )^{2}$$
$61$ $$( 147 + T^{2} )^{2}$$
$67$ $$( 25 + T^{2} )^{2}$$
$71$ $$( -192 + T^{2} )^{2}$$
$73$ $$( 7 + T )^{4}$$
$79$ $$( -27 + T^{2} )^{2}$$
$83$ $$T^{4}$$
$89$ $$( -6 + T )^{4}$$
$97$ $$( -7 + T )^{4}$$