# Properties

 Label 1728.2.s.c 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: $$x^{2} - x + 1$$ 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.

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

## 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.c 2
3.b odd 2 1 576.2.s.b 2
4.b odd 2 1 1728.2.s.d 2
8.b even 2 1 432.2.s.a 2
8.d odd 2 1 432.2.s.b 2
9.c even 3 1 576.2.s.c 2
9.c even 3 1 5184.2.c.b 2
9.d odd 6 1 1728.2.s.d 2
9.d odd 6 1 5184.2.c.d 2
12.b even 2 1 576.2.s.c 2
24.f even 2 1 144.2.s.c yes 2
24.h odd 2 1 144.2.s.b 2
36.f odd 6 1 576.2.s.b 2
36.f odd 6 1 5184.2.c.d 2
36.h even 6 1 inner 1728.2.s.c 2
36.h even 6 1 5184.2.c.b 2
72.j odd 6 1 432.2.s.b 2
72.j odd 6 1 1296.2.c.a 2
72.l even 6 1 432.2.s.a 2
72.l even 6 1 1296.2.c.c 2
72.n even 6 1 144.2.s.c yes 2
72.n even 6 1 1296.2.c.c 2
72.p odd 6 1 144.2.s.b 2
72.p odd 6 1 1296.2.c.a 2

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

## Hecke kernels

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

 $$T_{5}^{2} - 3T_{5} + 3$$ T5^2 - 3*T5 + 3 $$T_{7}^{2} + 3T_{7} + 3$$ T7^2 + 3*T7 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 3T + 3$$
$7$ $$T^{2} + 3T + 3$$
$11$ $$T^{2} - 3T + 9$$
$13$ $$T^{2} + 5T + 25$$
$17$ $$T^{2} + 48$$
$19$ $$T^{2} + 12$$
$23$ $$T^{2} + 9T + 81$$
$29$ $$T^{2} - 3T + 3$$
$31$ $$T^{2} + 9T + 27$$
$37$ $$(T + 2)^{2}$$
$41$ $$T^{2} - 9T + 27$$
$43$ $$T^{2} + 9T + 27$$
$47$ $$T^{2} + 3T + 9$$
$53$ $$T^{2}$$
$59$ $$T^{2} + 3T + 9$$
$61$ $$T^{2} + T + 1$$
$67$ $$T^{2} + 15T + 75$$
$71$ $$(T - 12)^{2}$$
$73$ $$(T + 2)^{2}$$
$79$ $$T^{2} + 15T + 75$$
$83$ $$T^{2} - 15T + 225$$
$89$ $$T^{2} + 48$$
$97$ $$T^{2} - 5T + 25$$