# Properties

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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$13.7981494693$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 144) 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.

 $$f(q)$$ $$=$$ $$q + ( 2 - \zeta_{6} ) q^{5} + ( 1 + \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( 2 - \zeta_{6} ) q^{5} + ( 1 + \zeta_{6} ) q^{7} + ( -3 + 3 \zeta_{6} ) q^{11} -5 \zeta_{6} q^{13} + ( -4 + 8 \zeta_{6} ) q^{17} + ( -2 + 4 \zeta_{6} ) q^{19} + 9 \zeta_{6} q^{23} + ( -2 + 2 \zeta_{6} ) q^{25} + ( 1 + \zeta_{6} ) q^{29} + ( 6 - 3 \zeta_{6} ) q^{31} + 3 q^{35} -2 q^{37} + ( 6 - 3 \zeta_{6} ) q^{41} + ( 3 + 3 \zeta_{6} ) q^{43} + ( 3 - 3 \zeta_{6} ) q^{47} -4 \zeta_{6} q^{49} + ( -3 + 6 \zeta_{6} ) q^{55} + 3 \zeta_{6} q^{59} + ( -1 + \zeta_{6} ) q^{61} + ( -5 - 5 \zeta_{6} ) q^{65} + ( 10 - 5 \zeta_{6} ) q^{67} -12 q^{71} -2 q^{73} + ( -6 + 3 \zeta_{6} ) q^{77} + ( 5 + 5 \zeta_{6} ) q^{79} + ( -15 + 15 \zeta_{6} ) q^{83} + 12 \zeta_{6} q^{85} + ( 4 - 8 \zeta_{6} ) q^{89} + ( 5 - 10 \zeta_{6} ) q^{91} + 6 \zeta_{6} q^{95} + ( 5 - 5 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{5} + 3q^{7} + O(q^{10})$$ $$2q + 3q^{5} + 3q^{7} - 3q^{11} - 5q^{13} + 9q^{23} - 2q^{25} + 3q^{29} + 9q^{31} + 6q^{35} - 4q^{37} + 9q^{41} + 9q^{43} + 3q^{47} - 4q^{49} + 3q^{59} - q^{61} - 15q^{65} + 15q^{67} - 24q^{71} - 4q^{73} - 9q^{77} + 15q^{79} - 15q^{83} + 12q^{85} + 6q^{95} + 5q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
575.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 1.50000 0.866025i 0 1.50000 + 0.866025i 0 0 0
1151.1 0 0 0 1.50000 + 0.866025i 0 1.50000 0.866025i 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
36.h even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.s.d 2
3.b odd 2 1 576.2.s.c 2
4.b odd 2 1 1728.2.s.c 2
8.b even 2 1 432.2.s.b 2
8.d odd 2 1 432.2.s.a 2
9.c even 3 1 576.2.s.b 2
9.c even 3 1 5184.2.c.d 2
9.d odd 6 1 1728.2.s.c 2
9.d odd 6 1 5184.2.c.b 2
12.b even 2 1 576.2.s.b 2
24.f even 2 1 144.2.s.b 2
24.h odd 2 1 144.2.s.c yes 2
36.f odd 6 1 576.2.s.c 2
36.f odd 6 1 5184.2.c.b 2
36.h even 6 1 inner 1728.2.s.d 2
36.h even 6 1 5184.2.c.d 2
72.j odd 6 1 432.2.s.a 2
72.j odd 6 1 1296.2.c.c 2
72.l even 6 1 432.2.s.b 2
72.l even 6 1 1296.2.c.a 2
72.n even 6 1 144.2.s.b 2
72.n even 6 1 1296.2.c.a 2
72.p odd 6 1 144.2.s.c yes 2
72.p odd 6 1 1296.2.c.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.s.b 2 24.f even 2 1
144.2.s.b 2 72.n even 6 1
144.2.s.c yes 2 24.h odd 2 1
144.2.s.c yes 2 72.p odd 6 1
432.2.s.a 2 8.d odd 2 1
432.2.s.a 2 72.j odd 6 1
432.2.s.b 2 8.b even 2 1
432.2.s.b 2 72.l even 6 1
576.2.s.b 2 9.c even 3 1
576.2.s.b 2 12.b even 2 1
576.2.s.c 2 3.b odd 2 1
576.2.s.c 2 36.f odd 6 1
1296.2.c.a 2 72.l even 6 1
1296.2.c.a 2 72.n even 6 1
1296.2.c.c 2 72.j odd 6 1
1296.2.c.c 2 72.p odd 6 1
1728.2.s.c 2 4.b odd 2 1
1728.2.s.c 2 9.d odd 6 1
1728.2.s.d 2 1.a even 1 1 trivial
1728.2.s.d 2 36.h even 6 1 inner
5184.2.c.b 2 9.d odd 6 1
5184.2.c.b 2 36.f odd 6 1
5184.2.c.d 2 9.c even 3 1
5184.2.c.d 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 $$S_{2}^{\mathrm{new}}(1728, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4}$$
$7$ $$( 1 - 4 T + 7 T^{2} )( 1 + T + 7 T^{2} )$$
$11$ $$1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4}$$
$13$ $$( 1 - 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$
$17$ $$1 + 14 T^{2} + 289 T^{4}$$
$19$ $$( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$
$23$ $$1 - 9 T + 58 T^{2} - 207 T^{3} + 529 T^{4}$$
$29$ $$1 - 3 T + 32 T^{2} - 87 T^{3} + 841 T^{4}$$
$31$ $$1 - 9 T + 58 T^{2} - 279 T^{3} + 961 T^{4}$$
$37$ $$( 1 + 2 T + 37 T^{2} )^{2}$$
$41$ $$1 - 9 T + 68 T^{2} - 369 T^{3} + 1681 T^{4}$$
$43$ $$1 - 9 T + 70 T^{2} - 387 T^{3} + 1849 T^{4}$$
$47$ $$1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4}$$
$53$ $$( 1 - 53 T^{2} )^{2}$$
$59$ $$1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4}$$
$61$ $$( 1 - 13 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} )$$
$67$ $$1 - 15 T + 142 T^{2} - 1005 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 12 T + 71 T^{2} )^{2}$$
$73$ $$( 1 + 2 T + 73 T^{2} )^{2}$$
$79$ $$1 - 15 T + 154 T^{2} - 1185 T^{3} + 6241 T^{4}$$
$83$ $$1 + 15 T + 142 T^{2} + 1245 T^{3} + 6889 T^{4}$$
$89$ $$1 - 130 T^{2} + 7921 T^{4}$$
$97$ $$( 1 - 19 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} )$$