# Properties

 Label 1728.2.r.b Level $1728$ Weight $2$ Character orbit 1728.r 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.r (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.7981494693$$ 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: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 576) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 4 - 2 \zeta_{12}^{2} ) q^{5} +O(q^{10})$$ $$q + ( 4 - 2 \zeta_{12}^{2} ) q^{5} + 3 \zeta_{12} q^{11} + ( -4 + 2 \zeta_{12}^{2} ) q^{13} + 3 q^{17} + 7 \zeta_{12}^{3} q^{19} + ( 2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{23} + ( 7 - 7 \zeta_{12}^{2} ) q^{25} + ( 4 + 4 \zeta_{12}^{2} ) q^{29} + ( 4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{31} + ( 6 - 12 \zeta_{12}^{2} ) q^{37} + 3 \zeta_{12}^{2} q^{41} -5 \zeta_{12} q^{43} + ( 2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{47} + 7 \zeta_{12}^{2} q^{49} + ( 8 - 16 \zeta_{12}^{2} ) q^{53} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{55} + ( -9 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{59} + ( -4 - 4 \zeta_{12}^{2} ) q^{61} + ( -12 + 12 \zeta_{12}^{2} ) q^{65} + ( 5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{67} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{71} -7 q^{73} + ( 10 \zeta_{12} - 20 \zeta_{12}^{3} ) q^{79} -12 \zeta_{12} q^{83} + ( 12 - 6 \zeta_{12}^{2} ) q^{85} + 6 q^{89} + ( 14 \zeta_{12} + 14 \zeta_{12}^{3} ) q^{95} + ( -1 + \zeta_{12}^{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 12q^{5} + O(q^{10})$$ $$4q + 12q^{5} - 12q^{13} + 12q^{17} + 14q^{25} + 24q^{29} + 6q^{41} + 14q^{49} - 24q^{61} - 24q^{65} - 28q^{73} + 36q^{85} + 24q^{89} - 2q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
0 0 0 3.00000 1.73205i 0 0 0 0 0
289.2 0 0 0 3.00000 1.73205i 0 0 0 0 0
1441.1 0 0 0 3.00000 + 1.73205i 0 0 0 0 0
1441.2 0 0 0 3.00000 + 1.73205i 0 0 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
72.n even 6 1 inner
72.p odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.r.b 4
3.b odd 2 1 576.2.r.a 4
4.b odd 2 1 inner 1728.2.r.b 4
8.b even 2 1 1728.2.r.a 4
8.d odd 2 1 1728.2.r.a 4
9.c even 3 1 1728.2.r.a 4
9.c even 3 1 5184.2.d.j 4
9.d odd 6 1 576.2.r.b yes 4
9.d odd 6 1 5184.2.d.c 4
12.b even 2 1 576.2.r.a 4
24.f even 2 1 576.2.r.b yes 4
24.h odd 2 1 576.2.r.b yes 4
36.f odd 6 1 1728.2.r.a 4
36.f odd 6 1 5184.2.d.j 4
36.h even 6 1 576.2.r.b yes 4
36.h even 6 1 5184.2.d.c 4
72.j odd 6 1 576.2.r.a 4
72.j odd 6 1 5184.2.d.c 4
72.l even 6 1 576.2.r.a 4
72.l even 6 1 5184.2.d.c 4
72.n even 6 1 inner 1728.2.r.b 4
72.n even 6 1 5184.2.d.j 4
72.p odd 6 1 inner 1728.2.r.b 4
72.p odd 6 1 5184.2.d.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.2.r.a 4 3.b odd 2 1
576.2.r.a 4 12.b even 2 1
576.2.r.a 4 72.j odd 6 1
576.2.r.a 4 72.l even 6 1
576.2.r.b yes 4 9.d odd 6 1
576.2.r.b yes 4 24.f even 2 1
576.2.r.b yes 4 24.h odd 2 1
576.2.r.b yes 4 36.h even 6 1
1728.2.r.a 4 8.b even 2 1
1728.2.r.a 4 8.d odd 2 1
1728.2.r.a 4 9.c even 3 1
1728.2.r.a 4 36.f odd 6 1
1728.2.r.b 4 1.a even 1 1 trivial
1728.2.r.b 4 4.b odd 2 1 inner
1728.2.r.b 4 72.n even 6 1 inner
1728.2.r.b 4 72.p odd 6 1 inner
5184.2.d.c 4 9.d odd 6 1
5184.2.d.c 4 36.h even 6 1
5184.2.d.c 4 72.j odd 6 1
5184.2.d.c 4 72.l even 6 1
5184.2.d.j 4 9.c even 3 1
5184.2.d.j 4 36.f odd 6 1
5184.2.d.j 4 72.n even 6 1
5184.2.d.j 4 72.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 6 T_{5} + 12$$ acting on $$S_{2}^{\mathrm{new}}(1728, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 12 - 6 T + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$81 - 9 T^{2} + T^{4}$$
$13$ $$( 12 + 6 T + T^{2} )^{2}$$
$17$ $$( -3 + T )^{4}$$
$19$ $$( 49 + T^{2} )^{2}$$
$23$ $$144 + 12 T^{2} + T^{4}$$
$29$ $$( 48 - 12 T + T^{2} )^{2}$$
$31$ $$2304 + 48 T^{2} + T^{4}$$
$37$ $$( 108 + T^{2} )^{2}$$
$41$ $$( 9 - 3 T + T^{2} )^{2}$$
$43$ $$625 - 25 T^{2} + T^{4}$$
$47$ $$144 + 12 T^{2} + T^{4}$$
$53$ $$( 192 + T^{2} )^{2}$$
$59$ $$6561 - 81 T^{2} + T^{4}$$
$61$ $$( 48 + 12 T + T^{2} )^{2}$$
$67$ $$625 - 25 T^{2} + T^{4}$$
$71$ $$( -12 + T^{2} )^{2}$$
$73$ $$( 7 + T )^{4}$$
$79$ $$90000 + 300 T^{2} + T^{4}$$
$83$ $$20736 - 144 T^{2} + T^{4}$$
$89$ $$( -6 + T )^{4}$$
$97$ $$( 1 + T + T^{2} )^{2}$$