# Properties

 Label 1728.2.i.k 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: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 288) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} ) q^{5} + ( -\beta_{1} + \beta_{2} ) q^{7} +O(q^{10})$$ $$q + ( 1 - \beta_{1} ) q^{5} + ( -\beta_{1} + \beta_{2} ) q^{7} + ( \beta_{1} + \beta_{2} ) q^{11} + ( 1 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{13} + 2 \beta_{3} q^{17} -4 q^{19} + ( -3 + 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{23} + 4 \beta_{1} q^{25} + ( -5 \beta_{1} - 2 \beta_{2} ) q^{29} + ( -5 + 5 \beta_{1} + \beta_{2} - \beta_{3} ) q^{31} + ( -1 + \beta_{3} ) q^{35} + ( -4 - 2 \beta_{3} ) q^{37} + ( -7 + 7 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{41} + ( 5 \beta_{1} + 3 \beta_{2} ) q^{43} + ( \beta_{1} + 3 \beta_{2} ) q^{47} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{49} + ( 4 - 2 \beta_{3} ) q^{53} + ( 1 + \beta_{3} ) q^{55} + ( -7 + 7 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{59} + ( -3 \beta_{1} + 2 \beta_{2} ) q^{61} + ( -\beta_{1} + 2 \beta_{2} ) q^{65} + ( 5 - 5 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{67} + ( 2 + 4 \beta_{3} ) q^{71} + 2 \beta_{3} q^{73} + ( -5 + 5 \beta_{1} ) q^{77} + ( -11 \beta_{1} + \beta_{2} ) q^{79} + ( 3 \beta_{1} + \beta_{2} ) q^{83} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{85} + ( 8 - 2 \beta_{3} ) q^{89} + ( -13 + 3 \beta_{3} ) q^{91} + ( -4 + 4 \beta_{1} ) q^{95} + ( -\beta_{1} - 2 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{5} - 2q^{7} + O(q^{10})$$ $$4q + 2q^{5} - 2q^{7} + 2q^{11} + 2q^{13} - 16q^{19} - 6q^{23} + 8q^{25} - 10q^{29} - 10q^{31} - 4q^{35} - 16q^{37} - 14q^{41} + 10q^{43} + 2q^{47} + 16q^{53} + 4q^{55} - 14q^{59} - 6q^{61} - 2q^{65} + 10q^{67} + 8q^{71} - 10q^{77} - 22q^{79} + 6q^{83} + 32q^{89} - 52q^{91} - 8q^{95} - 2q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 4 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{3} + 4 \beta_{2}$$$$)/3$$

## Character values

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

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

## 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
 −1.22474 − 0.707107i 1.22474 + 0.707107i −1.22474 + 0.707107i 1.22474 − 0.707107i
0 0 0 0.500000 0.866025i 0 −1.72474 2.98735i 0 0 0
577.2 0 0 0 0.500000 0.866025i 0 0.724745 + 1.25529i 0 0 0
1153.1 0 0 0 0.500000 + 0.866025i 0 −1.72474 + 2.98735i 0 0 0
1153.2 0 0 0 0.500000 + 0.866025i 0 0.724745 1.25529i 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.k 4
3.b odd 2 1 576.2.i.i 4
4.b odd 2 1 1728.2.i.m 4
8.b even 2 1 864.2.i.c 4
8.d odd 2 1 864.2.i.e 4
9.c even 3 1 inner 1728.2.i.k 4
9.c even 3 1 5184.2.a.bn 2
9.d odd 6 1 576.2.i.i 4
9.d odd 6 1 5184.2.a.by 2
12.b even 2 1 576.2.i.m 4
24.f even 2 1 288.2.i.c 4
24.h odd 2 1 288.2.i.e yes 4
36.f odd 6 1 1728.2.i.m 4
36.f odd 6 1 5184.2.a.bj 2
36.h even 6 1 576.2.i.m 4
36.h even 6 1 5184.2.a.bu 2
72.j odd 6 1 288.2.i.e yes 4
72.j odd 6 1 2592.2.a.n 2
72.l even 6 1 288.2.i.c 4
72.l even 6 1 2592.2.a.j 2
72.n even 6 1 864.2.i.c 4
72.n even 6 1 2592.2.a.s 2
72.p odd 6 1 864.2.i.e 4
72.p odd 6 1 2592.2.a.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.i.c 4 24.f even 2 1
288.2.i.c 4 72.l even 6 1
288.2.i.e yes 4 24.h odd 2 1
288.2.i.e yes 4 72.j odd 6 1
576.2.i.i 4 3.b odd 2 1
576.2.i.i 4 9.d odd 6 1
576.2.i.m 4 12.b even 2 1
576.2.i.m 4 36.h even 6 1
864.2.i.c 4 8.b even 2 1
864.2.i.c 4 72.n even 6 1
864.2.i.e 4 8.d odd 2 1
864.2.i.e 4 72.p odd 6 1
1728.2.i.k 4 1.a even 1 1 trivial
1728.2.i.k 4 9.c even 3 1 inner
1728.2.i.m 4 4.b odd 2 1
1728.2.i.m 4 36.f odd 6 1
2592.2.a.j 2 72.l even 6 1
2592.2.a.n 2 72.j odd 6 1
2592.2.a.o 2 72.p odd 6 1
2592.2.a.s 2 72.n even 6 1
5184.2.a.bj 2 36.f odd 6 1
5184.2.a.bn 2 9.c even 3 1
5184.2.a.bu 2 36.h even 6 1
5184.2.a.by 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 $$S_{2}^{\mathrm{new}}(1728, [\chi])$$:

 $$T_{5}^{2} - T_{5} + 1$$ $$T_{7}^{4} + 2 T_{7}^{3} + 9 T_{7}^{2} - 10 T_{7} + 25$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 1 - T + T^{2} )^{2}$$
$7$ $$25 - 10 T + 9 T^{2} + 2 T^{3} + T^{4}$$
$11$ $$25 + 10 T + 9 T^{2} - 2 T^{3} + T^{4}$$
$13$ $$529 + 46 T + 27 T^{2} - 2 T^{3} + T^{4}$$
$17$ $$( -24 + T^{2} )^{2}$$
$19$ $$( 4 + T )^{4}$$
$23$ $$9 + 18 T + 33 T^{2} + 6 T^{3} + T^{4}$$
$29$ $$1 + 10 T + 99 T^{2} + 10 T^{3} + T^{4}$$
$31$ $$361 + 190 T + 81 T^{2} + 10 T^{3} + T^{4}$$
$37$ $$( -8 + 8 T + T^{2} )^{2}$$
$41$ $$625 + 350 T + 171 T^{2} + 14 T^{3} + T^{4}$$
$43$ $$841 + 290 T + 129 T^{2} - 10 T^{3} + T^{4}$$
$47$ $$2809 + 106 T + 57 T^{2} - 2 T^{3} + T^{4}$$
$53$ $$( -8 - 8 T + T^{2} )^{2}$$
$59$ $$25 - 70 T + 201 T^{2} + 14 T^{3} + T^{4}$$
$61$ $$225 - 90 T + 51 T^{2} + 6 T^{3} + T^{4}$$
$67$ $$841 + 290 T + 129 T^{2} - 10 T^{3} + T^{4}$$
$71$ $$( -92 - 4 T + T^{2} )^{2}$$
$73$ $$( -24 + T^{2} )^{2}$$
$79$ $$13225 + 2530 T + 369 T^{2} + 22 T^{3} + T^{4}$$
$83$ $$9 - 18 T + 33 T^{2} - 6 T^{3} + T^{4}$$
$89$ $$( 40 - 16 T + T^{2} )^{2}$$
$97$ $$529 - 46 T + 27 T^{2} + 2 T^{3} + T^{4}$$