# Properties

 Label 1728.3.b.g Level $1728$ Weight $3$ Character orbit 1728.b Analytic conductor $47.085$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1728.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$47.0845896815$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{10}\cdot 3^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{5} - 7 \beta_1 q^{7}+O(q^{10})$$ q + b3 * q^5 - 7*b1 * q^7 $$q + \beta_{3} q^{5} - 7 \beta_1 q^{7} - \beta_{6} q^{11} + 11 \beta_{2} q^{13} - \beta_{5} q^{17} + 5 \beta_{4} q^{19} + \beta_{7} q^{23} - 47 q^{25} + 6 \beta_{3} q^{29} + 10 \beta_1 q^{31} - 7 \beta_{6} q^{35} - 11 \beta_{2} q^{37} - 4 \beta_{5} q^{41} - 24 \beta_{4} q^{43} + 5 \beta_{7} q^{47} - 2 \beta_{3} q^{53} + 72 \beta_1 q^{55} - 3 \beta_{6} q^{59} + 11 \beta_{2} q^{61} + 11 \beta_{5} q^{65} + 67 \beta_{4} q^{67} - 8 \beta_{7} q^{71} - 71 q^{73} - 7 \beta_{3} q^{77} + 151 \beta_1 q^{79} + 16 \beta_{6} q^{83} + 72 \beta_{2} q^{85} - 11 \beta_{5} q^{89} + 77 \beta_{4} q^{91} - 5 \beta_{7} q^{95} - 25 q^{97}+O(q^{100})$$ q + b3 * q^5 - 7*b1 * q^7 - b6 * q^11 + 11*b2 * q^13 - b5 * q^17 + 5*b4 * q^19 + b7 * q^23 - 47 * q^25 + 6*b3 * q^29 + 10*b1 * q^31 - 7*b6 * q^35 - 11*b2 * q^37 - 4*b5 * q^41 - 24*b4 * q^43 + 5*b7 * q^47 - 2*b3 * q^53 + 72*b1 * q^55 - 3*b6 * q^59 + 11*b2 * q^61 + 11*b5 * q^65 + 67*b4 * q^67 - 8*b7 * q^71 - 71 * q^73 - 7*b3 * q^77 + 151*b1 * q^79 + 16*b6 * q^83 + 72*b2 * q^85 - 11*b5 * q^89 + 77*b4 * q^91 - 5*b7 * q^95 - 25 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q - 376 q^{25} - 568 q^{73} - 200 q^{97}+O(q^{100})$$ 8 * q - 376 * q^25 - 568 * q^73 - 200 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{24}^{6}$$ v^6 $$\beta_{2}$$ $$=$$ $$2\zeta_{24}^{4} - 1$$ 2*v^4 - 1 $$\beta_{3}$$ $$=$$ $$-6\zeta_{24}^{5} - 6\zeta_{24}^{3} + 6\zeta_{24}$$ -6*v^5 - 6*v^3 + 6*v $$\beta_{4}$$ $$=$$ $$-\zeta_{24}^{6} + 2\zeta_{24}^{2}$$ -v^6 + 2*v^2 $$\beta_{5}$$ $$=$$ $$-12\zeta_{24}^{7} + 6\zeta_{24}^{5} + 6\zeta_{24}^{3} + 6\zeta_{24}$$ -12*v^7 + 6*v^5 + 6*v^3 + 6*v $$\beta_{6}$$ $$=$$ $$-6\zeta_{24}^{5} + 6\zeta_{24}^{3} + 6\zeta_{24}$$ -6*v^5 + 6*v^3 + 6*v $$\beta_{7}$$ $$=$$ $$12\zeta_{24}^{7} + 6\zeta_{24}^{5} - 6\zeta_{24}^{3} + 6\zeta_{24}$$ 12*v^7 + 6*v^5 - 6*v^3 + 6*v
 $$\zeta_{24}$$ $$=$$ $$( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} ) / 24$$ (b7 + b6 + b5 + b3) / 24 $$\zeta_{24}^{2}$$ $$=$$ $$( \beta_{4} + \beta_1 ) / 2$$ (b4 + b1) / 2 $$\zeta_{24}^{3}$$ $$=$$ $$( \beta_{6} - \beta_{3} ) / 12$$ (b6 - b3) / 12 $$\zeta_{24}^{4}$$ $$=$$ $$( \beta_{2} + 1 ) / 2$$ (b2 + 1) / 2 $$\zeta_{24}^{5}$$ $$=$$ $$( \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} ) / 24$$ (b7 - b6 + b5 - b3) / 24 $$\zeta_{24}^{6}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{24}^{7}$$ $$=$$ $$( \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} ) / 24$$ (b7 + b6 - b5 - b3) / 24

## Character values

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

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1567.1
 −0.258819 − 0.965926i 0.965926 + 0.258819i −0.965926 + 0.258819i 0.258819 − 0.965926i 0.258819 + 0.965926i −0.965926 − 0.258819i 0.965926 − 0.258819i −0.258819 + 0.965926i
0 0 0 8.48528i 0 7.00000i 0 0 0
1567.2 0 0 0 8.48528i 0 7.00000i 0 0 0
1567.3 0 0 0 8.48528i 0 7.00000i 0 0 0
1567.4 0 0 0 8.48528i 0 7.00000i 0 0 0
1567.5 0 0 0 8.48528i 0 7.00000i 0 0 0
1567.6 0 0 0 8.48528i 0 7.00000i 0 0 0
1567.7 0 0 0 8.48528i 0 7.00000i 0 0 0
1567.8 0 0 0 8.48528i 0 7.00000i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1567.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.3.b.g 8
3.b odd 2 1 inner 1728.3.b.g 8
4.b odd 2 1 inner 1728.3.b.g 8
8.b even 2 1 inner 1728.3.b.g 8
8.d odd 2 1 inner 1728.3.b.g 8
12.b even 2 1 inner 1728.3.b.g 8
24.f even 2 1 inner 1728.3.b.g 8
24.h odd 2 1 inner 1728.3.b.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1728.3.b.g 8 1.a even 1 1 trivial
1728.3.b.g 8 3.b odd 2 1 inner
1728.3.b.g 8 4.b odd 2 1 inner
1728.3.b.g 8 8.b even 2 1 inner
1728.3.b.g 8 8.d odd 2 1 inner
1728.3.b.g 8 12.b even 2 1 inner
1728.3.b.g 8 24.f even 2 1 inner
1728.3.b.g 8 24.h odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1728, [\chi])$$:

 $$T_{5}^{2} + 72$$ T5^2 + 72 $$T_{7}^{2} + 49$$ T7^2 + 49 $$T_{11}^{2} - 72$$ T11^2 - 72 $$T_{17}^{2} - 216$$ T17^2 - 216

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$(T^{2} + 72)^{4}$$
$7$ $$(T^{2} + 49)^{4}$$
$11$ $$(T^{2} - 72)^{4}$$
$13$ $$(T^{2} + 363)^{4}$$
$17$ $$(T^{2} - 216)^{4}$$
$19$ $$(T^{2} - 75)^{4}$$
$23$ $$(T^{2} + 216)^{4}$$
$29$ $$(T^{2} + 2592)^{4}$$
$31$ $$(T^{2} + 100)^{4}$$
$37$ $$(T^{2} + 363)^{4}$$
$41$ $$(T^{2} - 3456)^{4}$$
$43$ $$(T^{2} - 1728)^{4}$$
$47$ $$(T^{2} + 5400)^{4}$$
$53$ $$(T^{2} + 288)^{4}$$
$59$ $$(T^{2} - 648)^{4}$$
$61$ $$(T^{2} + 363)^{4}$$
$67$ $$(T^{2} - 13467)^{4}$$
$71$ $$(T^{2} + 13824)^{4}$$
$73$ $$(T + 71)^{8}$$
$79$ $$(T^{2} + 22801)^{4}$$
$83$ $$(T^{2} - 18432)^{4}$$
$89$ $$(T^{2} - 26136)^{4}$$
$97$ $$(T + 25)^{8}$$