# Properties

 Label 1728.2.f.g Level $1728$ Weight $2$ Character orbit 1728.f Analytic conductor $13.798$ Analytic rank $0$ Dimension $8$ CM discriminant -24 Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$13.7981494693$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{10}\cdot 3^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{5} + \beta_{3} q^{7}+O(q^{10})$$ q - b1 * q^5 + b3 * q^7 $$q - \beta_1 q^{5} + \beta_{3} q^{7} - \beta_{6} q^{11} + ( - \beta_{5} + 4) q^{25} + ( - \beta_{7} - 2 \beta_1) q^{29} + ( - 3 \beta_{4} - \beta_{3}) q^{31} + (2 \beta_{6} - 3 \beta_{2}) q^{35} + (\beta_{5} - 12) q^{49} + ( - \beta_{7} + 3 \beta_1) q^{53} + (4 \beta_{4} - \beta_{3}) q^{55} - 2 \beta_{2} q^{59} + (\beta_{5} + 7) q^{73} + ( - 2 \beta_{7} - 3 \beta_1) q^{77} + 5 \beta_{4} q^{79} + (\beta_{6} - 2 \beta_{2}) q^{83} + (2 \beta_{5} - 1) q^{97}+O(q^{100})$$ q - b1 * q^5 + b3 * q^7 - b6 * q^11 + (-b5 + 4) * q^25 + (-b7 - 2*b1) * q^29 + (-3*b4 - b3) * q^31 + (2*b6 - 3*b2) * q^35 + (b5 - 12) * q^49 + (-b7 + 3*b1) * q^53 + (4*b4 - b3) * q^55 - 2*b2 * q^59 + (b5 + 7) * q^73 + (-2*b7 - 3*b1) * q^77 + 5*b4 * q^79 + (b6 - 2*b2) * q^83 + (2*b5 - 1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q + 32 q^{25} - 96 q^{49} + 56 q^{73} - 8 q^{97}+O(q^{100})$$ 8 * q + 32 * q^25 - 96 * q^49 + 56 * q^73 - 8 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$2\zeta_{24}^{7} - \zeta_{24}^{6} - \zeta_{24}^{5} - \zeta_{24}^{3} + 2\zeta_{24}^{2} - \zeta_{24}$$ 2*v^7 - v^6 - v^5 - v^3 + 2*v^2 - v $$\beta_{2}$$ $$=$$ $$6\zeta_{24}^{4} - 3$$ 6*v^4 - 3 $$\beta_{3}$$ $$=$$ $$\zeta_{24}^{6} - 3\zeta_{24}^{5} - 3\zeta_{24}^{3} + 3\zeta_{24}$$ v^6 - 3*v^5 - 3*v^3 + 3*v $$\beta_{4}$$ $$=$$ $$-2\zeta_{24}^{6}$$ -2*v^6 $$\beta_{5}$$ $$=$$ $$-6\zeta_{24}^{5} + 6\zeta_{24}^{3} + 6\zeta_{24}$$ -6*v^5 + 6*v^3 + 6*v $$\beta_{6}$$ $$=$$ $$4\zeta_{24}^{7} + 2\zeta_{24}^{5} + 2\zeta_{24}^{4} - 2\zeta_{24}^{3} + 2\zeta_{24} - 1$$ 4*v^7 + 2*v^5 + 2*v^4 - 2*v^3 + 2*v - 1 $$\beta_{7}$$ $$=$$ $$-4\zeta_{24}^{7} - 4\zeta_{24}^{6} + 2\zeta_{24}^{5} + 2\zeta_{24}^{3} + 8\zeta_{24}^{2} + 2\zeta_{24}$$ -4*v^7 - 4*v^6 + 2*v^5 + 2*v^3 + 8*v^2 + 2*v
 $$\zeta_{24}$$ $$=$$ $$( \beta_{7} + 3\beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{3} - \beta_{2} - 4\beta_1 ) / 24$$ (b7 + 3*b6 + b5 + b4 + 2*b3 - b2 - 4*b1) / 24 $$\zeta_{24}^{2}$$ $$=$$ $$( \beta_{7} - 3\beta_{4} + 2\beta_1 ) / 12$$ (b7 - 3*b4 + 2*b1) / 12 $$\zeta_{24}^{3}$$ $$=$$ $$( \beta_{5} - \beta_{4} - 2\beta_{3} ) / 12$$ (b5 - b4 - 2*b3) / 12 $$\zeta_{24}^{4}$$ $$=$$ $$( \beta_{2} + 3 ) / 6$$ (b2 + 3) / 6 $$\zeta_{24}^{5}$$ $$=$$ $$( \beta_{7} + 3\beta_{6} - \beta_{5} - \beta_{4} - 2\beta_{3} - \beta_{2} - 4\beta_1 ) / 24$$ (b7 + 3*b6 - b5 - b4 - 2*b3 - b2 - 4*b1) / 24 $$\zeta_{24}^{6}$$ $$=$$ $$( -\beta_{4} ) / 2$$ (-b4) / 2 $$\zeta_{24}^{7}$$ $$=$$ $$( -\beta_{7} + 3\beta_{6} + \beta_{5} - \beta_{4} - 2\beta_{3} - \beta_{2} + 4\beta_1 ) / 24$$ (-b7 + 3*b6 + b5 - b4 - 2*b3 - b2 + 4*b1) / 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
863.1
 −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 0.258819 + 0.965926i
0 0 0 −4.18154 0 5.24264i 0 0 0
863.2 0 0 0 −4.18154 0 5.24264i 0 0 0
863.3 0 0 0 −0.717439 0 3.24264i 0 0 0
863.4 0 0 0 −0.717439 0 3.24264i 0 0 0
863.5 0 0 0 0.717439 0 3.24264i 0 0 0
863.6 0 0 0 0.717439 0 3.24264i 0 0 0
863.7 0 0 0 4.18154 0 5.24264i 0 0 0
863.8 0 0 0 4.18154 0 5.24264i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 863.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by $$\Q(\sqrt{-6})$$
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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.f.g 8
3.b odd 2 1 inner 1728.2.f.g 8
4.b odd 2 1 inner 1728.2.f.g 8
8.b even 2 1 inner 1728.2.f.g 8
8.d odd 2 1 inner 1728.2.f.g 8
12.b even 2 1 inner 1728.2.f.g 8
24.f even 2 1 inner 1728.2.f.g 8
24.h odd 2 1 CM 1728.2.f.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1728.2.f.g 8 1.a even 1 1 trivial
1728.2.f.g 8 3.b odd 2 1 inner
1728.2.f.g 8 4.b odd 2 1 inner
1728.2.f.g 8 8.b even 2 1 inner
1728.2.f.g 8 8.d odd 2 1 inner
1728.2.f.g 8 12.b even 2 1 inner
1728.2.f.g 8 24.f even 2 1 inner
1728.2.f.g 8 24.h odd 2 1 CM

## Hecke kernels

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

 $$T_{5}^{4} - 18T_{5}^{2} + 9$$ T5^4 - 18*T5^2 + 9 $$T_{7}^{4} + 38T_{7}^{2} + 289$$ T7^4 + 38*T7^2 + 289 $$T_{23}$$ T23

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} - 18 T^{2} + 9)^{2}$$
$7$ $$(T^{4} + 38 T^{2} + 289)^{2}$$
$11$ $$(T^{4} + 54 T^{2} + 441)^{2}$$
$13$ $$T^{8}$$
$17$ $$T^{8}$$
$19$ $$T^{8}$$
$23$ $$T^{8}$$
$29$ $$(T^{2} - 108)^{4}$$
$31$ $$(T^{4} + 86 T^{2} + 49)^{2}$$
$37$ $$T^{8}$$
$41$ $$T^{8}$$
$43$ $$T^{8}$$
$47$ $$T^{8}$$
$53$ $$(T^{4} - 306 T^{2} + 21609)^{2}$$
$59$ $$(T^{2} + 108)^{4}$$
$61$ $$T^{8}$$
$67$ $$T^{8}$$
$71$ $$T^{8}$$
$73$ $$(T^{2} - 14 T - 23)^{4}$$
$79$ $$(T^{2} + 100)^{4}$$
$83$ $$(T^{4} + 198 T^{2} + 2601)^{2}$$
$89$ $$T^{8}$$
$97$ $$(T^{2} + 2 T - 287)^{4}$$