# Properties

 Label 1728.4.f.f Level $1728$ Weight $4$ Character orbit 1728.f Analytic conductor $101.955$ Analytic rank $0$ Dimension $8$ CM discriminant -24 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1728.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$101.955300490$$ 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: $$2^{10}\cdot 3^{8}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{5} + 2 \beta_{2}) q^{5} + (\beta_{4} - 8 \beta_1) q^{7}+O(q^{10})$$ q + (-b5 + 2*b2) * q^5 + (b4 - 8*b1) * q^7 $$q + ( - \beta_{5} + 2 \beta_{2}) q^{5} + (\beta_{4} - 8 \beta_1) q^{7} + ( - 2 \beta_{6} - 5 \beta_{3}) q^{11} + ( - 7 \beta_{7} + 196) q^{25} + ( - 7 \beta_{5} - 35 \beta_{2}) q^{29} + (23 \beta_{4} + 29 \beta_1) q^{31} + (7 \beta_{6} - 32 \beta_{3}) q^{35} + ( - 17 \beta_{7} - 108) q^{49} + ( - 26 \beta_{5} + 75 \beta_{2}) q^{53} + (47 \beta_{4} + 76 \beta_1) q^{55} + ( - 46 \beta_{6} - 92 \beta_{3}) q^{59} + ( - 41 \beta_{7} - 161) q^{73} + (59 \beta_{5} + 72 \beta_{2}) q^{77} + 685 \beta_1 q^{79} + ( - 44 \beta_{6} + 129 \beta_{3}) q^{83} + (62 \beta_{7} + 287) q^{97}+O(q^{100})$$ q + (-b5 + 2*b2) * q^5 + (b4 - 8*b1) * q^7 + (-2*b6 - 5*b3) * q^11 + (-7*b7 + 196) * q^25 + (-7*b5 - 35*b2) * q^29 + (23*b4 + 29*b1) * q^31 + (7*b6 - 32*b3) * q^35 + (-17*b7 - 108) * q^49 + (-26*b5 + 75*b2) * q^53 + (47*b4 + 76*b1) * q^55 + (-46*b6 - 92*b3) * q^59 + (-41*b7 - 161) * q^73 + (59*b5 + 72*b2) * q^77 + 685*b1 * q^79 + (-44*b6 + 129*b3) * q^83 + (62*b7 + 287) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q + 1568 q^{25} - 864 q^{49} - 1288 q^{73} + 2296 q^{97}+O(q^{100})$$ 8 * q + 1568 * q^25 - 864 * q^49 - 1288 * q^73 + 2296 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$-2\zeta_{24}^{6}$$ -2*v^6 $$\beta_{2}$$ $$=$$ $$2\zeta_{24}^{7} - 3\zeta_{24}^{6} - \zeta_{24}^{5} - \zeta_{24}^{3} + 6\zeta_{24}^{2} - \zeta_{24}$$ 2*v^7 - 3*v^6 - v^5 - v^3 + 6*v^2 - v $$\beta_{3}$$ $$=$$ $$4\zeta_{24}^{7} + 2\zeta_{24}^{5} + 6\zeta_{24}^{4} - 2\zeta_{24}^{3} + 2\zeta_{24} - 3$$ 4*v^7 + 2*v^5 + 6*v^4 - 2*v^3 + 2*v - 3 $$\beta_{4}$$ $$=$$ $$\zeta_{24}^{6} + 9\zeta_{24}^{5} + 9\zeta_{24}^{3} - 9\zeta_{24}$$ v^6 + 9*v^5 + 9*v^3 - 9*v $$\beta_{5}$$ $$=$$ $$-10\zeta_{24}^{7} - 3\zeta_{24}^{6} + 5\zeta_{24}^{5} + 5\zeta_{24}^{3} + 6\zeta_{24}^{2} + 5\zeta_{24}$$ -10*v^7 - 3*v^6 + 5*v^5 + 5*v^3 + 6*v^2 + 5*v $$\beta_{6}$$ $$=$$ $$-8\zeta_{24}^{7} - 4\zeta_{24}^{5} + 6\zeta_{24}^{4} + 4\zeta_{24}^{3} - 4\zeta_{24} - 3$$ -8*v^7 - 4*v^5 + 6*v^4 + 4*v^3 - 4*v - 3 $$\beta_{7}$$ $$=$$ $$-18\zeta_{24}^{5} + 18\zeta_{24}^{3} + 18\zeta_{24}$$ -18*v^5 + 18*v^3 + 18*v
 $$\zeta_{24}$$ $$=$$ $$( \beta_{7} - 3\beta_{6} + 3\beta_{5} - 2\beta_{4} + 3\beta_{3} - 3\beta_{2} - \beta_1 ) / 72$$ (b7 - 3*b6 + 3*b5 - 2*b4 + 3*b3 - 3*b2 - b1) / 72 $$\zeta_{24}^{2}$$ $$=$$ $$( \beta_{5} + 5\beta_{2} - 9\beta_1 ) / 36$$ (b5 + 5*b2 - 9*b1) / 36 $$\zeta_{24}^{3}$$ $$=$$ $$( \beta_{7} + 2\beta_{4} + \beta_1 ) / 36$$ (b7 + 2*b4 + b1) / 36 $$\zeta_{24}^{4}$$ $$=$$ $$( \beta_{6} + 2\beta_{3} + 9 ) / 18$$ (b6 + 2*b3 + 9) / 18 $$\zeta_{24}^{5}$$ $$=$$ $$( -\beta_{7} - 3\beta_{6} + 3\beta_{5} + 2\beta_{4} + 3\beta_{3} - 3\beta_{2} + \beta_1 ) / 72$$ (-b7 - 3*b6 + 3*b5 + 2*b4 + 3*b3 - 3*b2 + b1) / 72 $$\zeta_{24}^{6}$$ $$=$$ $$( -\beta_1 ) / 2$$ (-b1) / 2 $$\zeta_{24}^{7}$$ $$=$$ $$( \beta_{7} - 3\beta_{6} - 3\beta_{5} + 2\beta_{4} + 3\beta_{3} + 3\beta_{2} + \beta_1 ) / 72$$ (b7 - 3*b6 - 3*b5 + 2*b4 + 3*b3 + 3*b2 + b1) / 72

## 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
 0.258819 − 0.965926i 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 0 0 −22.3426 0 4.27208i 0 0 0
863.2 0 0 0 −22.3426 0 4.27208i 0 0 0
863.3 0 0 0 −11.9503 0 29.7279i 0 0 0
863.4 0 0 0 −11.9503 0 29.7279i 0 0 0
863.5 0 0 0 11.9503 0 29.7279i 0 0 0
863.6 0 0 0 11.9503 0 29.7279i 0 0 0
863.7 0 0 0 22.3426 0 4.27208i 0 0 0
863.8 0 0 0 22.3426 0 4.27208i 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 1728.4.f.f 8
3.b odd 2 1 inner 1728.4.f.f 8
4.b odd 2 1 inner 1728.4.f.f 8
8.b even 2 1 inner 1728.4.f.f 8
8.d odd 2 1 inner 1728.4.f.f 8
12.b even 2 1 inner 1728.4.f.f 8
24.f even 2 1 inner 1728.4.f.f 8
24.h odd 2 1 CM 1728.4.f.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1728.4.f.f 8 1.a even 1 1 trivial
1728.4.f.f 8 3.b odd 2 1 inner
1728.4.f.f 8 4.b odd 2 1 inner
1728.4.f.f 8 8.b even 2 1 inner
1728.4.f.f 8 8.d odd 2 1 inner
1728.4.f.f 8 12.b even 2 1 inner
1728.4.f.f 8 24.f even 2 1 inner
1728.4.f.f 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 $$S_{4}^{\mathrm{new}}(1728, [\chi])$$:

 $$T_{5}^{4} - 642T_{5}^{2} + 71289$$ T5^4 - 642*T5^2 + 71289 $$T_{7}^{4} + 902T_{7}^{2} + 16129$$ T7^4 + 902*T7^2 + 16129 $$T_{23}$$ T23

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} - 642 T^{2} + 71289)^{2}$$
$7$ $$(T^{4} + 902 T^{2} + 16129)^{2}$$
$11$ $$(T^{4} + 2694 T^{2} + 1687401)^{2}$$
$13$ $$T^{8}$$
$17$ $$T^{8}$$
$19$ $$T^{8}$$
$23$ $$T^{8}$$
$29$ $$(T^{2} - 47628)^{4}$$
$31$ $$(T^{4} + 173846 T^{2} + \cdots + 7135687729)^{2}$$
$37$ $$T^{8}$$
$41$ $$T^{8}$$
$43$ $$T^{8}$$
$47$ $$T^{8}$$
$53$ $$(T^{4} - 633954 T^{2} + \cdots + 35089906329)^{2}$$
$59$ $$(T^{2} + 514188)^{4}$$
$61$ $$T^{8}$$
$67$ $$T^{8}$$
$71$ $$T^{8}$$
$73$ $$(T^{2} + 322 T - 1063367)^{4}$$
$79$ $$(T^{2} + 1876900)^{4}$$
$83$ $$(T^{4} + 2650422 T^{2} + \cdots + 874339073721)^{2}$$
$89$ $$T^{8}$$
$97$ $$(T^{2} - 574 T - 2408543)^{4}$$