# Properties

 Label 1728.2.i.n Level $1728$ Weight $2$ Character orbit 1728.i Analytic conductor $13.798$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# 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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.170772624.1 Defining polynomial: $$x^{8} - 3 x^{7} + 5 x^{6} - 6 x^{5} + 6 x^{4} - 12 x^{3} + 20 x^{2} - 24 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}\cdot 3^{4}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{6} q^{5} + ( \beta_{1} - \beta_{7} ) q^{7} +O(q^{10})$$ $$q -\beta_{6} q^{5} + ( \beta_{1} - \beta_{7} ) q^{7} + \beta_{3} q^{11} + ( 2 \beta_{4} + \beta_{6} ) q^{13} + \beta_{5} q^{17} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{19} -\beta_{7} q^{23} + ( -4 + 3 \beta_{4} - \beta_{5} - \beta_{6} ) q^{25} + ( 3 - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{29} + ( -2 \beta_{2} - \beta_{7} ) q^{31} + ( 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{35} -4 q^{37} + \beta_{4} q^{41} + \beta_{3} q^{43} + ( -\beta_{1} + \beta_{7} ) q^{47} + ( -5 \beta_{4} + 3 \beta_{6} ) q^{49} -4 q^{53} + ( \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{55} + ( -\beta_{2} + 2 \beta_{7} ) q^{59} + ( 5 - 8 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} ) q^{61} + ( 7 - 8 \beta_{4} - \beta_{5} - \beta_{6} ) q^{65} + ( -\beta_{2} + 2 \beta_{7} ) q^{67} + 2 \beta_{1} q^{71} + ( 8 + \beta_{5} ) q^{73} + ( 6 \beta_{4} + 3 \beta_{6} ) q^{77} + ( \beta_{1} + 2 \beta_{3} - \beta_{7} ) q^{79} + ( -\beta_{1} + 2 \beta_{3} + \beta_{7} ) q^{83} + 8 \beta_{4} q^{85} + ( -8 - 2 \beta_{5} ) q^{89} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{91} + 4 \beta_{7} q^{95} + ( -9 + 9 \beta_{4} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 2q^{5} + O(q^{10})$$ $$8q + 2q^{5} + 6q^{13} - 4q^{17} - 14q^{25} + 10q^{29} - 32q^{37} + 4q^{41} - 26q^{49} - 32q^{53} + 26q^{61} + 30q^{65} + 60q^{73} + 18q^{77} + 32q^{85} - 56q^{89} - 36q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{7} + 5 x^{6} - 6 x^{5} + 6 x^{4} - 12 x^{3} + 20 x^{2} - 24 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{7} + 3 \nu^{4} + 6 \nu^{2} + 4 \nu$$$$)/4$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} + 3 \nu^{6} + 3 \nu^{5} - 6 \nu^{4} + 6 \nu^{3} + 4 \nu - 24$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} + 3 \nu^{6} - 9 \nu^{5} + 6 \nu^{4} - 6 \nu^{3} + 24 \nu^{2} - 44 \nu + 48$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$3 \nu^{7} - 5 \nu^{6} + 7 \nu^{5} - 6 \nu^{4} + 10 \nu^{3} - 24 \nu^{2} + 28 \nu - 24$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} + 4 \nu^{6} - 6 \nu^{5} + 5 \nu^{4} - 6 \nu^{3} + 14 \nu^{2} - 20 \nu + 20$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$-2 \nu^{7} + 3 \nu^{6} - 5 \nu^{5} + 4 \nu^{4} - 5 \nu^{3} + 13 \nu^{2} - 18 \nu + 20$$$$)/2$$ $$\beta_{7}$$ $$=$$ $$($$$$2 \nu^{7} - 3 \nu^{6} + 6 \nu^{5} - 3 \nu^{4} + 6 \nu^{3} - 15 \nu^{2} + 16 \nu - 24$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{1} + 2$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{7} - 3 \beta_{6} - 3 \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_{1}$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{6} + \beta_{5} + 10 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + 3 \beta_{1} - 4$$$$)/6$$ $$\nu^{4}$$ $$=$$ $$($$$$3 \beta_{7} + 3 \beta_{6} + 3 \beta_{5} + 3 \beta_{4} - \beta_{3} - 2 \beta_{2} + 3 \beta_{1}$$$$)/6$$ $$\nu^{5}$$ $$=$$ $$($$$$5 \beta_{7} - \beta_{6} - 2 \beta_{5} - 17 \beta_{4} + \beta_{2} + 32$$$$)/6$$ $$\nu^{6}$$ $$=$$ $$($$$$2 \beta_{7} + 9 \beta_{5} - 5 \beta_{3} + 5 \beta_{2} - \beta_{1} + 24$$$$)/6$$ $$\nu^{7}$$ $$=$$ $$($$$$-\beta_{7} - 13 \beta_{6} + 13 \beta_{5} - 5 \beta_{4} + 5 \beta_{3} + \beta_{1} + 8$$$$)/6$$

## Character values

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

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{4}$$

## 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
 −1.02187 − 0.977642i 0.335728 + 1.37379i 0.774115 − 1.18353i 1.41203 − 0.0786378i −1.02187 + 0.977642i 0.335728 − 1.37379i 0.774115 + 1.18353i 1.41203 + 0.0786378i
0 0 0 −1.18614 + 2.05446i 0 −1.10489 1.91373i 0 0 0
577.2 0 0 0 −1.18614 + 2.05446i 0 1.10489 + 1.91373i 0 0 0
577.3 0 0 0 1.68614 2.92048i 0 −2.35143 4.07279i 0 0 0
577.4 0 0 0 1.68614 2.92048i 0 2.35143 + 4.07279i 0 0 0
1153.1 0 0 0 −1.18614 2.05446i 0 −1.10489 + 1.91373i 0 0 0
1153.2 0 0 0 −1.18614 2.05446i 0 1.10489 1.91373i 0 0 0
1153.3 0 0 0 1.68614 + 2.92048i 0 −2.35143 + 4.07279i 0 0 0
1153.4 0 0 0 1.68614 + 2.92048i 0 2.35143 4.07279i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1153.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.i.n 8
3.b odd 2 1 576.2.i.n 8
4.b odd 2 1 inner 1728.2.i.n 8
8.b even 2 1 864.2.i.f 8
8.d odd 2 1 864.2.i.f 8
9.c even 3 1 inner 1728.2.i.n 8
9.c even 3 1 5184.2.a.cc 4
9.d odd 6 1 576.2.i.n 8
9.d odd 6 1 5184.2.a.cf 4
12.b even 2 1 576.2.i.n 8
24.f even 2 1 288.2.i.f 8
24.h odd 2 1 288.2.i.f 8
36.f odd 6 1 inner 1728.2.i.n 8
36.f odd 6 1 5184.2.a.cc 4
36.h even 6 1 576.2.i.n 8
36.h even 6 1 5184.2.a.cf 4
72.j odd 6 1 288.2.i.f 8
72.j odd 6 1 2592.2.a.u 4
72.l even 6 1 288.2.i.f 8
72.l even 6 1 2592.2.a.u 4
72.n even 6 1 864.2.i.f 8
72.n even 6 1 2592.2.a.x 4
72.p odd 6 1 864.2.i.f 8
72.p odd 6 1 2592.2.a.x 4

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

 $$T_{5}^{4} - T_{5}^{3} + 9 T_{5}^{2} + 8 T_{5} + 64$$ $$T_{7}^{8} + 27 T_{7}^{6} + 621 T_{7}^{4} + 2916 T_{7}^{2} + 11664$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 64 + 8 T + 9 T^{2} - T^{3} + T^{4} )^{2}$$
$7$ $$11664 + 2916 T^{2} + 621 T^{4} + 27 T^{6} + T^{8}$$
$11$ $$729 + 972 T^{2} + 1269 T^{4} + 36 T^{6} + T^{8}$$
$13$ $$( 36 + 18 T + 15 T^{2} - 3 T^{3} + T^{4} )^{2}$$
$17$ $$( -8 + T + T^{2} )^{4}$$
$19$ $$( 432 - 45 T^{2} + T^{4} )^{2}$$
$23$ $$11664 + 2916 T^{2} + 621 T^{4} + 27 T^{6} + T^{8}$$
$29$ $$( 4 + 10 T + 27 T^{2} - 5 T^{3} + T^{4} )^{2}$$
$31$ $$15116544 + 524880 T^{2} + 14337 T^{4} + 135 T^{6} + T^{8}$$
$37$ $$( 4 + T )^{8}$$
$41$ $$( 1 - T + T^{2} )^{4}$$
$43$ $$729 + 972 T^{2} + 1269 T^{4} + 36 T^{6} + T^{8}$$
$47$ $$11664 + 2916 T^{2} + 621 T^{4} + 27 T^{6} + T^{8}$$
$53$ $$( 4 + T )^{8}$$
$59$ $$60886809 + 1404540 T^{2} + 24597 T^{4} + 180 T^{6} + T^{8}$$
$61$ $$( 1024 + 416 T + 201 T^{2} - 13 T^{3} + T^{4} )^{2}$$
$67$ $$60886809 + 1404540 T^{2} + 24597 T^{4} + 180 T^{6} + T^{8}$$
$71$ $$( 1728 - 108 T^{2} + T^{4} )^{2}$$
$73$ $$( 48 - 15 T + T^{2} )^{4}$$
$79$ $$15116544 + 524880 T^{2} + 14337 T^{4} + 135 T^{6} + T^{8}$$
$83$ $$2985984 + 357696 T^{2} + 41121 T^{4} + 207 T^{6} + T^{8}$$
$89$ $$( 16 + 14 T + T^{2} )^{4}$$
$97$ $$( 81 + 9 T + T^{2} )^{4}$$