# Properties

 Label 1728.2.i.f Level $1728$ Weight $2$ Character orbit 1728.i 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.i (of order $$3$$, 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: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 18) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 2 - 2 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( 2 - 2 \zeta_{6} ) q^{7} + ( -3 + 3 \zeta_{6} ) q^{11} + 2 \zeta_{6} q^{13} + 3 q^{17} - q^{19} + 6 \zeta_{6} q^{23} + ( 5 - 5 \zeta_{6} ) q^{25} + ( -6 + 6 \zeta_{6} ) q^{29} -4 \zeta_{6} q^{31} + 4 q^{37} + 9 \zeta_{6} q^{41} + ( 1 - \zeta_{6} ) q^{43} + ( 6 - 6 \zeta_{6} ) q^{47} + 3 \zeta_{6} q^{49} + 12 q^{53} + 3 \zeta_{6} q^{59} + ( 8 - 8 \zeta_{6} ) q^{61} -5 \zeta_{6} q^{67} -12 q^{71} + 11 q^{73} + 6 \zeta_{6} q^{77} + ( -4 + 4 \zeta_{6} ) q^{79} + ( 12 - 12 \zeta_{6} ) q^{83} -6 q^{89} + 4 q^{91} + ( -5 + 5 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{7} + O(q^{10})$$ $$2q + 2q^{7} - 3q^{11} + 2q^{13} + 6q^{17} - 2q^{19} + 6q^{23} + 5q^{25} - 6q^{29} - 4q^{31} + 8q^{37} + 9q^{41} + q^{43} + 6q^{47} + 3q^{49} + 24q^{53} + 3q^{59} + 8q^{61} - 5q^{67} - 24q^{71} + 22q^{73} + 6q^{77} - 4q^{79} + 12q^{83} - 12q^{89} + 8q^{91} - 5q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
577.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 0 0 1.00000 + 1.73205i 0 0 0
1153.1 0 0 0 0 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.i.f 2
3.b odd 2 1 576.2.i.a 2
4.b odd 2 1 1728.2.i.e 2
8.b even 2 1 432.2.i.b 2
8.d odd 2 1 54.2.c.a 2
9.c even 3 1 inner 1728.2.i.f 2
9.c even 3 1 5184.2.a.p 1
9.d odd 6 1 576.2.i.a 2
9.d odd 6 1 5184.2.a.o 1
12.b even 2 1 576.2.i.g 2
24.f even 2 1 18.2.c.a 2
24.h odd 2 1 144.2.i.c 2
36.f odd 6 1 1728.2.i.e 2
36.f odd 6 1 5184.2.a.q 1
36.h even 6 1 576.2.i.g 2
36.h even 6 1 5184.2.a.r 1
40.e odd 2 1 1350.2.e.c 2
40.k even 4 2 1350.2.j.a 4
56.e even 2 1 2646.2.f.g 2
56.k odd 6 1 2646.2.e.b 2
56.k odd 6 1 2646.2.h.h 2
56.m even 6 1 2646.2.e.c 2
56.m even 6 1 2646.2.h.i 2
72.j odd 6 1 144.2.i.c 2
72.j odd 6 1 1296.2.a.g 1
72.l even 6 1 18.2.c.a 2
72.l even 6 1 162.2.a.c 1
72.n even 6 1 432.2.i.b 2
72.n even 6 1 1296.2.a.f 1
72.p odd 6 1 54.2.c.a 2
72.p odd 6 1 162.2.a.b 1
120.m even 2 1 450.2.e.i 2
120.q odd 4 2 450.2.j.e 4
168.e odd 2 1 882.2.f.d 2
168.v even 6 1 882.2.e.i 2
168.v even 6 1 882.2.h.c 2
168.be odd 6 1 882.2.e.g 2
168.be odd 6 1 882.2.h.b 2
360.z odd 6 1 1350.2.e.c 2
360.z odd 6 1 4050.2.a.v 1
360.bd even 6 1 450.2.e.i 2
360.bd even 6 1 4050.2.a.c 1
360.bo even 12 2 1350.2.j.a 4
360.bo even 12 2 4050.2.c.r 2
360.bt odd 12 2 450.2.j.e 4
360.bt odd 12 2 4050.2.c.c 2
504.u odd 6 1 882.2.e.g 2
504.ba odd 6 1 2646.2.e.b 2
504.be even 6 1 2646.2.f.g 2
504.be even 6 1 7938.2.a.i 1
504.bf even 6 1 2646.2.h.i 2
504.bt even 6 1 882.2.h.c 2
504.ce odd 6 1 2646.2.h.h 2
504.cm odd 6 1 882.2.h.b 2
504.co odd 6 1 882.2.f.d 2
504.co odd 6 1 7938.2.a.x 1
504.cy even 6 1 882.2.e.i 2
504.cz even 6 1 2646.2.e.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.2.c.a 2 24.f even 2 1
18.2.c.a 2 72.l even 6 1
54.2.c.a 2 8.d odd 2 1
54.2.c.a 2 72.p odd 6 1
144.2.i.c 2 24.h odd 2 1
144.2.i.c 2 72.j odd 6 1
162.2.a.b 1 72.p odd 6 1
162.2.a.c 1 72.l even 6 1
432.2.i.b 2 8.b even 2 1
432.2.i.b 2 72.n even 6 1
450.2.e.i 2 120.m even 2 1
450.2.e.i 2 360.bd even 6 1
450.2.j.e 4 120.q odd 4 2
450.2.j.e 4 360.bt odd 12 2
576.2.i.a 2 3.b odd 2 1
576.2.i.a 2 9.d odd 6 1
576.2.i.g 2 12.b even 2 1
576.2.i.g 2 36.h even 6 1
882.2.e.g 2 168.be odd 6 1
882.2.e.g 2 504.u odd 6 1
882.2.e.i 2 168.v even 6 1
882.2.e.i 2 504.cy even 6 1
882.2.f.d 2 168.e odd 2 1
882.2.f.d 2 504.co odd 6 1
882.2.h.b 2 168.be odd 6 1
882.2.h.b 2 504.cm odd 6 1
882.2.h.c 2 168.v even 6 1
882.2.h.c 2 504.bt even 6 1
1296.2.a.f 1 72.n even 6 1
1296.2.a.g 1 72.j odd 6 1
1350.2.e.c 2 40.e odd 2 1
1350.2.e.c 2 360.z odd 6 1
1350.2.j.a 4 40.k even 4 2
1350.2.j.a 4 360.bo even 12 2
1728.2.i.e 2 4.b odd 2 1
1728.2.i.e 2 36.f odd 6 1
1728.2.i.f 2 1.a even 1 1 trivial
1728.2.i.f 2 9.c even 3 1 inner
2646.2.e.b 2 56.k odd 6 1
2646.2.e.b 2 504.ba odd 6 1
2646.2.e.c 2 56.m even 6 1
2646.2.e.c 2 504.cz even 6 1
2646.2.f.g 2 56.e even 2 1
2646.2.f.g 2 504.be even 6 1
2646.2.h.h 2 56.k odd 6 1
2646.2.h.h 2 504.ce odd 6 1
2646.2.h.i 2 56.m even 6 1
2646.2.h.i 2 504.bf even 6 1
4050.2.a.c 1 360.bd even 6 1
4050.2.a.v 1 360.z odd 6 1
4050.2.c.c 2 360.bt odd 12 2
4050.2.c.r 2 360.bo even 12 2
5184.2.a.o 1 9.d odd 6 1
5184.2.a.p 1 9.c even 3 1
5184.2.a.q 1 36.f odd 6 1
5184.2.a.r 1 36.h even 6 1
7938.2.a.i 1 504.be even 6 1
7938.2.a.x 1 504.co odd 6 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$4 - 2 T + T^{2}$$
$11$ $$9 + 3 T + T^{2}$$
$13$ $$4 - 2 T + T^{2}$$
$17$ $$( -3 + T )^{2}$$
$19$ $$( 1 + T )^{2}$$
$23$ $$36 - 6 T + T^{2}$$
$29$ $$36 + 6 T + T^{2}$$
$31$ $$16 + 4 T + T^{2}$$
$37$ $$( -4 + T )^{2}$$
$41$ $$81 - 9 T + T^{2}$$
$43$ $$1 - T + T^{2}$$
$47$ $$36 - 6 T + T^{2}$$
$53$ $$( -12 + T )^{2}$$
$59$ $$9 - 3 T + T^{2}$$
$61$ $$64 - 8 T + T^{2}$$
$67$ $$25 + 5 T + T^{2}$$
$71$ $$( 12 + T )^{2}$$
$73$ $$( -11 + T )^{2}$$
$79$ $$16 + 4 T + T^{2}$$
$83$ $$144 - 12 T + T^{2}$$
$89$ $$( 6 + T )^{2}$$
$97$ $$25 + 5 T + T^{2}$$