# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1728.r (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.7981494693$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.12960000.1 Defining polynomial: $$x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1$$ x^8 - 3*x^6 + 8*x^4 - 3*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$2^{4}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 576) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{4} q^{5} + (\beta_{7} - \beta_{5}) q^{7}+O(q^{10})$$ q - b4 * q^5 + (b7 - b5) * q^7 $$q - \beta_{4} q^{5} + (\beta_{7} - \beta_{5}) q^{7} - 3 \beta_1 q^{11} - \beta_{4} q^{13} - 2 \beta_{3} q^{19} + \beta_{7} q^{23} + 10 \beta_{2} q^{25} - \beta_{6} q^{29} - \beta_{7} q^{31} + 15 \beta_{3} q^{35} + (2 \beta_{6} + 2 \beta_{4}) q^{37} + ( - 9 \beta_{2} + 9) q^{41} - 7 \beta_1 q^{43} + ( - \beta_{7} + \beta_{5}) q^{47} + (8 \beta_{2} - 8) q^{49} + (2 \beta_{6} + 2 \beta_{4}) q^{53} + 3 \beta_{5} q^{55} + ( - 3 \beta_{3} - 3 \beta_1) q^{59} - 3 \beta_{6} q^{61} + 15 \beta_{2} q^{65} + ( - 11 \beta_{3} - 11 \beta_1) q^{67} - 2 \beta_{5} q^{71} + 8 q^{73} + 3 \beta_{4} q^{77} + ( - \beta_{7} + \beta_{5}) q^{79} + 3 \beta_1 q^{83} + 12 q^{89} + 15 \beta_{3} q^{91} - 2 \beta_{7} q^{95} - \beta_{2} q^{97}+O(q^{100})$$ q - b4 * q^5 + (b7 - b5) * q^7 - 3*b1 * q^11 - b4 * q^13 - 2*b3 * q^19 + b7 * q^23 + 10*b2 * q^25 - b6 * q^29 - b7 * q^31 + 15*b3 * q^35 + (2*b6 + 2*b4) * q^37 + (-9*b2 + 9) * q^41 - 7*b1 * q^43 + (-b7 + b5) * q^47 + (8*b2 - 8) * q^49 + (2*b6 + 2*b4) * q^53 + 3*b5 * q^55 + (-3*b3 - 3*b1) * q^59 - 3*b6 * q^61 + 15*b2 * q^65 + (-11*b3 - 11*b1) * q^67 - 2*b5 * q^71 + 8 * q^73 + 3*b4 * q^77 + (-b7 + b5) * q^79 + 3*b1 * q^83 + 12 * q^89 + 15*b3 * q^91 - 2*b7 * q^95 - b2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q + 40 q^{25} + 36 q^{41} - 32 q^{49} + 60 q^{65} + 64 q^{73} + 96 q^{89} - 4 q^{97}+O(q^{100})$$ 8 * q + 40 * q^25 + 36 * q^41 - 32 * q^49 + 60 * q^65 + 64 * q^73 + 96 * q^89 - 4 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{7} + 13\nu ) / 8$$ (v^7 + 13*v) / 8 $$\beta_{2}$$ $$=$$ $$( -3\nu^{6} + 8\nu^{4} - 24\nu^{2} + 9 ) / 8$$ (-3*v^6 + 8*v^4 - 24*v^2 + 9) / 8 $$\beta_{3}$$ $$=$$ $$( -3\nu^{7} + 8\nu^{5} - 20\nu^{3} + \nu ) / 4$$ (-3*v^7 + 8*v^5 - 20*v^3 + v) / 4 $$\beta_{4}$$ $$=$$ $$( -5\nu^{6} + 24\nu^{4} - 56\nu^{2} + 39 ) / 8$$ (-5*v^6 + 24*v^4 - 56*v^2 + 39) / 8 $$\beta_{5}$$ $$=$$ $$( -5\nu^{7} + 16\nu^{5} - 44\nu^{3} + 31\nu ) / 4$$ (-5*v^7 + 16*v^5 - 44*v^3 + 31*v) / 4 $$\beta_{6}$$ $$=$$ $$( -11\nu^{6} + 24\nu^{4} - 56\nu^{2} - 15 ) / 8$$ (-11*v^6 + 24*v^4 - 56*v^2 - 15) / 8 $$\beta_{7}$$ $$=$$ $$( 13\nu^{7} - 32\nu^{5} + 88\nu^{3} + 25\nu ) / 8$$ (13*v^7 - 32*v^5 + 88*v^3 + 25*v) / 8
 $$\nu$$ $$=$$ $$( \beta_{7} + \beta_{5} - 3\beta_1 ) / 6$$ (b7 + b5 - 3*b1) / 6 $$\nu^{2}$$ $$=$$ $$( 2\beta_{6} + \beta_{4} - 9\beta_{2} + 9 ) / 6$$ (2*b6 + b4 - 9*b2 + 9) / 6 $$\nu^{3}$$ $$=$$ $$( 2\beta_{7} - \beta_{5} + 6\beta_{3} ) / 3$$ (2*b7 - b5 + 6*b3) / 3 $$\nu^{4}$$ $$=$$ $$( \beta_{6} + 2\beta_{4} - 7\beta_{2} ) / 2$$ (b6 + 2*b4 - 7*b2) / 2 $$\nu^{5}$$ $$=$$ $$( 5\beta_{7} - 10\beta_{5} + 33\beta_{3} + 33\beta_1 ) / 6$$ (5*b7 - 10*b5 + 33*b3 + 33*b1) / 6 $$\nu^{6}$$ $$=$$ $$( -4\beta_{6} + 4\beta_{4} - 27 ) / 3$$ (-4*b6 + 4*b4 - 27) / 3 $$\nu^{7}$$ $$=$$ $$( -13\beta_{7} - 13\beta_{5} + 87\beta_1 ) / 6$$ (-13*b7 - 13*b5 + 87*b1) / 6

## Character values

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

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1 + \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 0.535233 + 0.309017i −0.535233 − 0.309017i 1.40126 + 0.809017i −1.40126 − 0.809017i 0.535233 − 0.309017i −0.535233 + 0.309017i 1.40126 − 0.809017i −1.40126 + 0.809017i
0 0 0 −3.35410 + 1.93649i 0 −1.93649 + 3.35410i 0 0 0
289.2 0 0 0 −3.35410 + 1.93649i 0 1.93649 3.35410i 0 0 0
289.3 0 0 0 3.35410 1.93649i 0 −1.93649 + 3.35410i 0 0 0
289.4 0 0 0 3.35410 1.93649i 0 1.93649 3.35410i 0 0 0
1441.1 0 0 0 −3.35410 1.93649i 0 −1.93649 3.35410i 0 0 0
1441.2 0 0 0 −3.35410 1.93649i 0 1.93649 + 3.35410i 0 0 0
1441.3 0 0 0 3.35410 + 1.93649i 0 −1.93649 3.35410i 0 0 0
1441.4 0 0 0 3.35410 + 1.93649i 0 1.93649 + 3.35410i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1441.4 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
8.b even 2 1 inner
8.d odd 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner
72.n even 6 1 inner
72.p 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.r.d 8
3.b odd 2 1 576.2.r.d 8
4.b odd 2 1 inner 1728.2.r.d 8
8.b even 2 1 inner 1728.2.r.d 8
8.d odd 2 1 inner 1728.2.r.d 8
9.c even 3 1 inner 1728.2.r.d 8
9.c even 3 1 5184.2.d.g 4
9.d odd 6 1 576.2.r.d 8
9.d odd 6 1 5184.2.d.h 4
12.b even 2 1 576.2.r.d 8
24.f even 2 1 576.2.r.d 8
24.h odd 2 1 576.2.r.d 8
36.f odd 6 1 inner 1728.2.r.d 8
36.f odd 6 1 5184.2.d.g 4
36.h even 6 1 576.2.r.d 8
36.h even 6 1 5184.2.d.h 4
72.j odd 6 1 576.2.r.d 8
72.j odd 6 1 5184.2.d.h 4
72.l even 6 1 576.2.r.d 8
72.l even 6 1 5184.2.d.h 4
72.n even 6 1 inner 1728.2.r.d 8
72.n even 6 1 5184.2.d.g 4
72.p odd 6 1 inner 1728.2.r.d 8
72.p odd 6 1 5184.2.d.g 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 15T_{5}^{2} + 225$$ acting on $$S_{2}^{\mathrm{new}}(1728, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} - 15 T^{2} + 225)^{2}$$
$7$ $$(T^{4} + 15 T^{2} + 225)^{2}$$
$11$ $$(T^{4} - 9 T^{2} + 81)^{2}$$
$13$ $$(T^{4} - 15 T^{2} + 225)^{2}$$
$17$ $$T^{8}$$
$19$ $$(T^{2} + 4)^{4}$$
$23$ $$(T^{4} + 15 T^{2} + 225)^{2}$$
$29$ $$(T^{4} - 15 T^{2} + 225)^{2}$$
$31$ $$(T^{4} + 15 T^{2} + 225)^{2}$$
$37$ $$(T^{2} + 60)^{4}$$
$41$ $$(T^{2} - 9 T + 81)^{4}$$
$43$ $$(T^{4} - 49 T^{2} + 2401)^{2}$$
$47$ $$(T^{4} + 15 T^{2} + 225)^{2}$$
$53$ $$(T^{2} + 60)^{4}$$
$59$ $$(T^{4} - 9 T^{2} + 81)^{2}$$
$61$ $$(T^{4} - 135 T^{2} + 18225)^{2}$$
$67$ $$(T^{4} - 121 T^{2} + 14641)^{2}$$
$71$ $$(T^{2} - 60)^{4}$$
$73$ $$(T - 8)^{8}$$
$79$ $$(T^{4} + 15 T^{2} + 225)^{2}$$
$83$ $$(T^{4} - 9 T^{2} + 81)^{2}$$
$89$ $$(T - 12)^{8}$$
$97$ $$(T^{2} + T + 1)^{4}$$