# Properties

 Label 1728.2.c.e Level $1728$ Weight $2$ Character orbit 1728.c 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.c (of order $$2$$, degree $$1$$, not 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 432) 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.

 $$f(q)$$ $$=$$ $$q + 3 \zeta_{12}^{3} q^{5} + ( 1 - 2 \zeta_{12}^{2} ) q^{7} +O(q^{10})$$ $$q + 3 \zeta_{12}^{3} q^{5} + ( 1 - 2 \zeta_{12}^{2} ) q^{7} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{11} + 2 q^{13} -6 \zeta_{12}^{3} q^{17} + ( 4 - 8 \zeta_{12}^{2} ) q^{19} -4 q^{25} + 6 \zeta_{12}^{3} q^{29} + ( 3 - 6 \zeta_{12}^{2} ) q^{31} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{35} -8 q^{37} + ( -6 + 12 \zeta_{12}^{2} ) q^{43} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{47} + 4 q^{49} -9 \zeta_{12}^{3} q^{53} + ( -9 + 18 \zeta_{12}^{2} ) q^{55} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{59} + 4 q^{61} + 6 \zeta_{12}^{3} q^{65} + ( -2 + 4 \zeta_{12}^{2} ) q^{67} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{71} + q^{73} -9 \zeta_{12}^{3} q^{77} + ( 2 - 4 \zeta_{12}^{2} ) q^{79} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{83} + 18 q^{85} + 6 \zeta_{12}^{3} q^{89} + ( 2 - 4 \zeta_{12}^{2} ) q^{91} + ( 24 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{95} -5 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 8q^{13} - 16q^{25} - 32q^{37} + 16q^{49} + 16q^{61} + 4q^{73} + 72q^{85} - 20q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1727.1
 −0.866025 − 0.500000i 0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i
0 0 0 3.00000i 0 1.73205i 0 0 0
1727.2 0 0 0 3.00000i 0 1.73205i 0 0 0
1727.3 0 0 0 3.00000i 0 1.73205i 0 0 0
1727.4 0 0 0 3.00000i 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 inner
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 1728.2.c.e 4
3.b odd 2 1 inner 1728.2.c.e 4
4.b odd 2 1 inner 1728.2.c.e 4
8.b even 2 1 432.2.c.c 4
8.d odd 2 1 432.2.c.c 4
12.b even 2 1 inner 1728.2.c.e 4
24.f even 2 1 432.2.c.c 4
24.h odd 2 1 432.2.c.c 4
72.j odd 6 1 1296.2.s.g 4
72.j odd 6 1 1296.2.s.i 4
72.l even 6 1 1296.2.s.g 4
72.l even 6 1 1296.2.s.i 4
72.n even 6 1 1296.2.s.g 4
72.n even 6 1 1296.2.s.i 4
72.p odd 6 1 1296.2.s.g 4
72.p odd 6 1 1296.2.s.i 4

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

## Hecke kernels

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

 $$T_{5}^{2} + 9$$ $$T_{7}^{2} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 9 + T^{2} )^{2}$$
$7$ $$( 3 + T^{2} )^{2}$$
$11$ $$( -27 + T^{2} )^{2}$$
$13$ $$( -2 + T )^{4}$$
$17$ $$( 36 + T^{2} )^{2}$$
$19$ $$( 48 + T^{2} )^{2}$$
$23$ $$T^{4}$$
$29$ $$( 36 + T^{2} )^{2}$$
$31$ $$( 27 + T^{2} )^{2}$$
$37$ $$( 8 + T )^{4}$$
$41$ $$T^{4}$$
$43$ $$( 108 + T^{2} )^{2}$$
$47$ $$( -108 + T^{2} )^{2}$$
$53$ $$( 81 + T^{2} )^{2}$$
$59$ $$( -108 + T^{2} )^{2}$$
$61$ $$( -4 + T )^{4}$$
$67$ $$( 12 + T^{2} )^{2}$$
$71$ $$( -108 + T^{2} )^{2}$$
$73$ $$( -1 + T )^{4}$$
$79$ $$( 12 + T^{2} )^{2}$$
$83$ $$( -27 + T^{2} )^{2}$$
$89$ $$( 36 + T^{2} )^{2}$$
$97$ $$( 5 + T )^{4}$$