# Properties

 Label 1728.3.q.a Level $1728$ Weight $3$ Character orbit 1728.q Analytic conductor $47.085$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$47.0845896815$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 9) 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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 4 - 2 \zeta_{6} ) q^{5} + ( -2 + 2 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( 4 - 2 \zeta_{6} ) q^{5} + ( -2 + 2 \zeta_{6} ) q^{7} + ( -1 - \zeta_{6} ) q^{11} -4 \zeta_{6} q^{13} + ( -9 + 18 \zeta_{6} ) q^{17} -11 q^{19} + ( 32 - 16 \zeta_{6} ) q^{23} + ( -13 + 13 \zeta_{6} ) q^{25} + ( 26 + 26 \zeta_{6} ) q^{29} -32 \zeta_{6} q^{31} + ( -4 + 8 \zeta_{6} ) q^{35} + 34 q^{37} + ( 14 - 7 \zeta_{6} ) q^{41} + ( -61 + 61 \zeta_{6} ) q^{43} + ( 28 + 28 \zeta_{6} ) q^{47} + 45 \zeta_{6} q^{49} -6 q^{55} + ( 58 - 29 \zeta_{6} ) q^{59} + ( 56 - 56 \zeta_{6} ) q^{61} + ( -8 - 8 \zeta_{6} ) q^{65} -31 \zeta_{6} q^{67} + ( 18 - 36 \zeta_{6} ) q^{71} + 65 q^{73} + ( 4 - 2 \zeta_{6} ) q^{77} + ( -38 + 38 \zeta_{6} ) q^{79} + ( -28 - 28 \zeta_{6} ) q^{83} + 54 \zeta_{6} q^{85} + ( -72 + 144 \zeta_{6} ) q^{89} + 8 q^{91} + ( -44 + 22 \zeta_{6} ) q^{95} + ( 115 - 115 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{5} - 2 q^{7} + O(q^{10})$$ $$2 q + 6 q^{5} - 2 q^{7} - 3 q^{11} - 4 q^{13} - 22 q^{19} + 48 q^{23} - 13 q^{25} + 78 q^{29} - 32 q^{31} + 68 q^{37} + 21 q^{41} - 61 q^{43} + 84 q^{47} + 45 q^{49} - 12 q^{55} + 87 q^{59} + 56 q^{61} - 24 q^{65} - 31 q^{67} + 130 q^{73} + 6 q^{77} - 38 q^{79} - 84 q^{83} + 54 q^{85} + 16 q^{91} - 66 q^{95} + 115 q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$1$$ $$1$$ $$\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 3.00000 + 1.73205i 0 −1.00000 1.73205i 0 0 0
1601.1 0 0 0 3.00000 1.73205i 0 −1.00000 + 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
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.3.q.a 2
3.b odd 2 1 576.3.q.b 2
4.b odd 2 1 1728.3.q.b 2
8.b even 2 1 27.3.d.a 2
8.d odd 2 1 432.3.q.a 2
9.c even 3 1 576.3.q.b 2
9.d odd 6 1 inner 1728.3.q.a 2
12.b even 2 1 576.3.q.a 2
24.f even 2 1 144.3.q.a 2
24.h odd 2 1 9.3.d.a 2
36.f odd 6 1 576.3.q.a 2
36.h even 6 1 1728.3.q.b 2
40.f even 2 1 675.3.j.a 2
40.i odd 4 2 675.3.i.a 4
72.j odd 6 1 27.3.d.a 2
72.j odd 6 1 81.3.b.a 2
72.l even 6 1 432.3.q.a 2
72.l even 6 1 1296.3.e.a 2
72.n even 6 1 9.3.d.a 2
72.n even 6 1 81.3.b.a 2
72.p odd 6 1 144.3.q.a 2
72.p odd 6 1 1296.3.e.a 2
120.i odd 2 1 225.3.j.a 2
120.w even 4 2 225.3.i.a 4
168.i even 2 1 441.3.r.a 2
168.s odd 6 1 441.3.j.a 2
168.s odd 6 1 441.3.n.b 2
168.ba even 6 1 441.3.j.b 2
168.ba even 6 1 441.3.n.a 2
360.bh odd 6 1 675.3.j.a 2
360.bk even 6 1 225.3.j.a 2
360.br even 12 2 675.3.i.a 4
360.bu odd 12 2 225.3.i.a 4
504.w even 6 1 441.3.j.a 2
504.bn odd 6 1 441.3.r.a 2
504.bp odd 6 1 441.3.n.a 2
504.cq even 6 1 441.3.n.b 2
504.cw odd 6 1 441.3.j.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.3.d.a 2 24.h odd 2 1
9.3.d.a 2 72.n even 6 1
27.3.d.a 2 8.b even 2 1
27.3.d.a 2 72.j odd 6 1
81.3.b.a 2 72.j odd 6 1
81.3.b.a 2 72.n even 6 1
144.3.q.a 2 24.f even 2 1
144.3.q.a 2 72.p odd 6 1
225.3.i.a 4 120.w even 4 2
225.3.i.a 4 360.bu odd 12 2
225.3.j.a 2 120.i odd 2 1
225.3.j.a 2 360.bk even 6 1
432.3.q.a 2 8.d odd 2 1
432.3.q.a 2 72.l even 6 1
441.3.j.a 2 168.s odd 6 1
441.3.j.a 2 504.w even 6 1
441.3.j.b 2 168.ba even 6 1
441.3.j.b 2 504.cw odd 6 1
441.3.n.a 2 168.ba even 6 1
441.3.n.a 2 504.bp odd 6 1
441.3.n.b 2 168.s odd 6 1
441.3.n.b 2 504.cq even 6 1
441.3.r.a 2 168.i even 2 1
441.3.r.a 2 504.bn odd 6 1
576.3.q.a 2 12.b even 2 1
576.3.q.a 2 36.f odd 6 1
576.3.q.b 2 3.b odd 2 1
576.3.q.b 2 9.c even 3 1
675.3.i.a 4 40.i odd 4 2
675.3.i.a 4 360.br even 12 2
675.3.j.a 2 40.f even 2 1
675.3.j.a 2 360.bh odd 6 1
1296.3.e.a 2 72.l even 6 1
1296.3.e.a 2 72.p odd 6 1
1728.3.q.a 2 1.a even 1 1 trivial
1728.3.q.a 2 9.d odd 6 1 inner
1728.3.q.b 2 4.b odd 2 1
1728.3.q.b 2 36.h even 6 1

## Hecke kernels

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

 $$T_{5}^{2} - 6 T_{5} + 12$$ $$T_{7}^{2} + 2 T_{7} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$12 - 6 T + T^{2}$$
$7$ $$4 + 2 T + T^{2}$$
$11$ $$3 + 3 T + T^{2}$$
$13$ $$16 + 4 T + T^{2}$$
$17$ $$243 + T^{2}$$
$19$ $$( 11 + T )^{2}$$
$23$ $$768 - 48 T + T^{2}$$
$29$ $$2028 - 78 T + T^{2}$$
$31$ $$1024 + 32 T + T^{2}$$
$37$ $$( -34 + T )^{2}$$
$41$ $$147 - 21 T + T^{2}$$
$43$ $$3721 + 61 T + T^{2}$$
$47$ $$2352 - 84 T + T^{2}$$
$53$ $$T^{2}$$
$59$ $$2523 - 87 T + T^{2}$$
$61$ $$3136 - 56 T + T^{2}$$
$67$ $$961 + 31 T + T^{2}$$
$71$ $$972 + T^{2}$$
$73$ $$( -65 + T )^{2}$$
$79$ $$1444 + 38 T + T^{2}$$
$83$ $$2352 + 84 T + T^{2}$$
$89$ $$15552 + T^{2}$$
$97$ $$13225 - 115 T + T^{2}$$