# Properties

 Label 1728.2.i.i Level $1728$ Weight $2$ Character orbit 1728.i Analytic conductor $13.798$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 72) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{5} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} +O(q^{10})$$ $$q + \beta_{3} q^{5} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + ( -1 + \beta_{1} ) q^{11} + ( 2 \beta_{1} + \beta_{3} ) q^{13} + ( 2 - \beta_{2} ) q^{17} + ( -4 - \beta_{2} ) q^{19} + ( 2 \beta_{1} + \beta_{3} ) q^{23} + ( -4 + 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{25} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{29} + ( -4 \beta_{1} + \beta_{3} ) q^{31} + ( 7 - \beta_{2} ) q^{35} + ( -2 + 2 \beta_{2} ) q^{37} + ( -7 \beta_{1} + 2 \beta_{3} ) q^{41} + ( 5 - 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{43} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{47} + ( -5 \beta_{1} + 3 \beta_{3} ) q^{49} + ( -6 - 2 \beta_{2} ) q^{53} + ( -1 - \beta_{2} ) q^{55} -7 \beta_{1} q^{59} + ( -1 - \beta_{2} + \beta_{3} ) q^{61} + ( -11 + 8 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{65} + ( \beta_{1} + 2 \beta_{3} ) q^{67} + 4 q^{71} + ( -2 + 3 \beta_{2} ) q^{73} + ( -2 \beta_{1} + \beta_{3} ) q^{77} + ( 3 - 4 \beta_{1} - \beta_{2} + \beta_{3} ) q^{79} + ( -13 + 12 \beta_{1} - \beta_{2} + \beta_{3} ) q^{83} + ( -8 \beta_{1} + 2 \beta_{3} ) q^{85} -6 q^{89} + ( 5 + \beta_{2} ) q^{91} + ( -8 \beta_{1} - 4 \beta_{3} ) q^{95} + ( 5 - 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + q^{5} - 3q^{7} + O(q^{10})$$ $$4q + q^{5} - 3q^{7} - 2q^{11} + 5q^{13} + 10q^{17} - 14q^{19} + 5q^{23} - 7q^{25} + 3q^{29} - 7q^{31} + 30q^{35} - 12q^{37} - 12q^{41} + 8q^{43} + 3q^{47} - 7q^{49} - 20q^{53} - 2q^{55} - 14q^{59} - q^{61} - 19q^{65} + 4q^{67} + 16q^{71} - 14q^{73} - 3q^{77} + 7q^{79} - 25q^{83} - 14q^{85} - 24q^{89} + 18q^{91} - 20q^{95} + 8q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} - 2 \nu - 3$$$$)/6$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + \nu^{2} + 5 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{3} + \nu^{2} + 2 \nu - 9$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - 2 \beta_{1} + 2$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + 2 \beta_{2} + 8 \beta_{1} + 1$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$4 \beta_{3} - 2 \beta_{2} - 2 \beta_{1} + 11$$$$)/3$$

## Character values

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

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{1}$$

## 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
 −1.18614 + 1.26217i 1.68614 − 0.396143i −1.18614 − 1.26217i 1.68614 + 0.396143i
0 0 0 −1.18614 + 2.05446i 0 −2.18614 3.78651i 0 0 0
577.2 0 0 0 1.68614 2.92048i 0 0.686141 + 1.18843i 0 0 0
1153.1 0 0 0 −1.18614 2.05446i 0 −2.18614 + 3.78651i 0 0 0
1153.2 0 0 0 1.68614 + 2.92048i 0 0.686141 1.18843i 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.i 4
3.b odd 2 1 576.2.i.j 4
4.b odd 2 1 1728.2.i.j 4
8.b even 2 1 216.2.i.b 4
8.d odd 2 1 432.2.i.d 4
9.c even 3 1 inner 1728.2.i.i 4
9.c even 3 1 5184.2.a.bp 2
9.d odd 6 1 576.2.i.j 4
9.d odd 6 1 5184.2.a.bt 2
12.b even 2 1 576.2.i.l 4
24.f even 2 1 144.2.i.d 4
24.h odd 2 1 72.2.i.b 4
36.f odd 6 1 1728.2.i.j 4
36.f odd 6 1 5184.2.a.bo 2
36.h even 6 1 576.2.i.l 4
36.h even 6 1 5184.2.a.bs 2
72.j odd 6 1 72.2.i.b 4
72.j odd 6 1 648.2.a.f 2
72.l even 6 1 144.2.i.d 4
72.l even 6 1 1296.2.a.n 2
72.n even 6 1 216.2.i.b 4
72.n even 6 1 648.2.a.g 2
72.p odd 6 1 432.2.i.d 4
72.p odd 6 1 1296.2.a.p 2

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

## Hecke kernels

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

 $$T_{5}^{4} - T_{5}^{3} + 9 T_{5}^{2} + 8 T_{5} + 64$$ $$T_{7}^{4} + 3 T_{7}^{3} + 15 T_{7}^{2} - 18 T_{7} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$64 + 8 T + 9 T^{2} - T^{3} + T^{4}$$
$7$ $$36 - 18 T + 15 T^{2} + 3 T^{3} + T^{4}$$
$11$ $$( 1 + T + T^{2} )^{2}$$
$13$ $$4 + 10 T + 27 T^{2} - 5 T^{3} + T^{4}$$
$17$ $$( -2 - 5 T + T^{2} )^{2}$$
$19$ $$( 4 + 7 T + T^{2} )^{2}$$
$23$ $$4 + 10 T + 27 T^{2} - 5 T^{3} + T^{4}$$
$29$ $$36 + 18 T + 15 T^{2} - 3 T^{3} + T^{4}$$
$31$ $$16 + 28 T + 45 T^{2} + 7 T^{3} + T^{4}$$
$37$ $$( -24 + 6 T + T^{2} )^{2}$$
$41$ $$9 + 36 T + 141 T^{2} + 12 T^{3} + T^{4}$$
$43$ $$289 + 136 T + 81 T^{2} - 8 T^{3} + T^{4}$$
$47$ $$36 + 18 T + 15 T^{2} - 3 T^{3} + T^{4}$$
$53$ $$( -8 + 10 T + T^{2} )^{2}$$
$59$ $$( 49 + 7 T + T^{2} )^{2}$$
$61$ $$64 - 8 T + 9 T^{2} + T^{3} + T^{4}$$
$67$ $$841 + 116 T + 45 T^{2} - 4 T^{3} + T^{4}$$
$71$ $$( -4 + T )^{4}$$
$73$ $$( -62 + 7 T + T^{2} )^{2}$$
$79$ $$16 - 28 T + 45 T^{2} - 7 T^{3} + T^{4}$$
$83$ $$21904 + 3700 T + 477 T^{2} + 25 T^{3} + T^{4}$$
$89$ $$( 6 + T )^{4}$$
$97$ $$289 + 136 T + 81 T^{2} - 8 T^{3} + T^{4}$$