# Properties

 Label 1728.4.d.f Level $1728$ Weight $4$ Character orbit 1728.d Analytic conductor $101.955$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$101.955300490$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.592240896.1 Defining polynomial: $$x^{8} - 7x^{6} + 40x^{4} - 63x^{2} + 81$$ x^8 - 7*x^6 + 40*x^4 - 63*x^2 + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{12}\cdot 3^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{6} q^{5} + 7 \beta_1 q^{7}+O(q^{10})$$ q - b6 * q^5 + 7*b1 * q^7 $$q - \beta_{6} q^{5} + 7 \beta_1 q^{7} - \beta_{7} q^{11} - 9 \beta_{2} q^{13} - 3 \beta_{4} q^{17} + 49 \beta_{3} q^{19} + 5 \beta_{5} q^{23} - 31 q^{25} + 2 \beta_{6} q^{29} + 14 \beta_1 q^{31} - 7 \beta_{7} q^{35} - 59 \beta_{2} q^{37} - 16 \beta_{4} q^{41} + 260 \beta_{3} q^{43} + 29 \beta_{5} q^{47} - 196 q^{49} + 46 \beta_{6} q^{53} - 156 \beta_1 q^{55} - 15 \beta_{7} q^{59} - 101 \beta_{2} q^{61} + 9 \beta_{4} q^{65} - 241 \beta_{3} q^{67} - 20 \beta_{5} q^{71} + 353 q^{73} - 21 \beta_{6} q^{77} - 3 \beta_1 q^{79} - 48 \beta_{7} q^{83} - 468 \beta_{2} q^{85} - 37 \beta_{4} q^{89} + 189 \beta_{3} q^{91} - 49 \beta_{5} q^{95} + 1111 q^{97}+O(q^{100})$$ q - b6 * q^5 + 7*b1 * q^7 - b7 * q^11 - 9*b2 * q^13 - 3*b4 * q^17 + 49*b3 * q^19 + 5*b5 * q^23 - 31 * q^25 + 2*b6 * q^29 + 14*b1 * q^31 - 7*b7 * q^35 - 59*b2 * q^37 - 16*b4 * q^41 + 260*b3 * q^43 + 29*b5 * q^47 - 196 * q^49 + 46*b6 * q^53 - 156*b1 * q^55 - 15*b7 * q^59 - 101*b2 * q^61 + 9*b4 * q^65 - 241*b3 * q^67 - 20*b5 * q^71 + 353 * q^73 - 21*b6 * q^77 - 3*b1 * q^79 - 48*b7 * q^83 - 468*b2 * q^85 - 37*b4 * q^89 + 189*b3 * q^91 - 49*b5 * q^95 + 1111 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q - 248 q^{25} - 1568 q^{49} + 2824 q^{73} + 8888 q^{97}+O(q^{100})$$ 8 * q - 248 * q^25 - 1568 * q^49 + 2824 * q^73 + 8888 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 7x^{6} + 40x^{4} - 63x^{2} + 81$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{7} - 16\nu^{5} + 76\nu^{3} - 207\nu ) / 108$$ (v^7 - 16*v^5 + 76*v^3 - 207*v) / 108 $$\beta_{2}$$ $$=$$ $$( 7\nu^{6} - 40\nu^{4} + 280\nu^{2} - 261 ) / 180$$ (7*v^6 - 40*v^4 + 280*v^2 - 261) / 180 $$\beta_{3}$$ $$=$$ $$( 7\nu^{7} - 40\nu^{5} + 190\nu^{3} - 81\nu ) / 270$$ (7*v^7 - 40*v^5 + 190*v^3 - 81*v) / 270 $$\beta_{4}$$ $$=$$ $$( 3\nu^{6} + 231 ) / 10$$ (3*v^6 + 231) / 10 $$\beta_{5}$$ $$=$$ $$( -13\nu^{7} + 100\nu^{5} - 610\nu^{3} + 1719\nu ) / 135$$ (-13*v^7 + 100*v^5 - 610*v^3 + 1719*v) / 135 $$\beta_{6}$$ $$=$$ $$( -4\nu^{6} + 28\nu^{4} - 124\nu^{2} + 126 ) / 9$$ (-4*v^6 + 28*v^4 - 124*v^2 + 126) / 9 $$\beta_{7}$$ $$=$$ $$( 7\nu^{7} - 40\nu^{5} + 244\nu^{3} - 81\nu ) / 18$$ (7*v^7 - 40*v^5 + 244*v^3 - 81*v) / 18
 $$\nu$$ $$=$$ $$( \beta_{7} + 3\beta_{5} - 6\beta_{3} + 6\beta_1 ) / 24$$ (b7 + 3*b5 - 6*b3 + 6*b1) / 24 $$\nu^{2}$$ $$=$$ $$( 3\beta_{6} - \beta_{4} + 42\beta_{2} + 42 ) / 24$$ (3*b6 - b4 + 42*b2 + 42) / 24 $$\nu^{3}$$ $$=$$ $$( \beta_{7} - 15\beta_{3} ) / 3$$ (b7 - 15*b3) / 3 $$\nu^{4}$$ $$=$$ $$( 21\beta_{6} + 7\beta_{4} + 186\beta_{2} - 186 ) / 24$$ (21*b6 + 7*b4 + 186*b2 - 186) / 24 $$\nu^{5}$$ $$=$$ $$( 19\beta_{7} - 57\beta_{5} - 366\beta_{3} - 366\beta_1 ) / 24$$ (19*b7 - 57*b5 - 366*b3 - 366*b1) / 24 $$\nu^{6}$$ $$=$$ $$( 10\beta_{4} - 231 ) / 3$$ (10*b4 - 231) / 3 $$\nu^{7}$$ $$=$$ $$( -97\beta_{7} - 291\beta_{5} + 2022\beta_{3} - 2022\beta_1 ) / 24$$ (-97*b7 - 291*b5 + 2022*b3 - 2022*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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
865.1
 −1.99426 − 1.15139i 1.12824 − 0.651388i 1.99426 + 1.15139i −1.12824 + 0.651388i 1.12824 + 0.651388i −1.99426 + 1.15139i −1.12824 − 0.651388i 1.99426 − 1.15139i
0 0 0 12.4900i 0 −12.1244 0 0 0
865.2 0 0 0 12.4900i 0 −12.1244 0 0 0
865.3 0 0 0 12.4900i 0 12.1244 0 0 0
865.4 0 0 0 12.4900i 0 12.1244 0 0 0
865.5 0 0 0 12.4900i 0 −12.1244 0 0 0
865.6 0 0 0 12.4900i 0 −12.1244 0 0 0
865.7 0 0 0 12.4900i 0 12.1244 0 0 0
865.8 0 0 0 12.4900i 0 12.1244 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 865.8 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 1728.4.d.f 8
3.b odd 2 1 inner 1728.4.d.f 8
4.b odd 2 1 inner 1728.4.d.f 8
8.b even 2 1 inner 1728.4.d.f 8
8.d odd 2 1 inner 1728.4.d.f 8
12.b even 2 1 inner 1728.4.d.f 8
24.f even 2 1 inner 1728.4.d.f 8
24.h odd 2 1 inner 1728.4.d.f 8

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

 $$T_{5}^{2} + 156$$ T5^2 + 156 $$T_{7}^{2} - 147$$ T7^2 - 147 $$T_{17}^{2} - 4212$$ T17^2 - 4212 $$T_{23}^{2} - 3900$$ T23^2 - 3900

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$(T^{2} + 156)^{4}$$
$7$ $$(T^{2} - 147)^{4}$$
$11$ $$(T^{2} + 468)^{4}$$
$13$ $$(T^{2} + 243)^{4}$$
$17$ $$(T^{2} - 4212)^{4}$$
$19$ $$(T^{2} + 2401)^{4}$$
$23$ $$(T^{2} - 3900)^{4}$$
$29$ $$(T^{2} + 624)^{4}$$
$31$ $$(T^{2} - 588)^{4}$$
$37$ $$(T^{2} + 10443)^{4}$$
$41$ $$(T^{2} - 119808)^{4}$$
$43$ $$(T^{2} + 67600)^{4}$$
$47$ $$(T^{2} - 131196)^{4}$$
$53$ $$(T^{2} + 330096)^{4}$$
$59$ $$(T^{2} + 105300)^{4}$$
$61$ $$(T^{2} + 30603)^{4}$$
$67$ $$(T^{2} + 58081)^{4}$$
$71$ $$(T^{2} - 62400)^{4}$$
$73$ $$(T - 353)^{8}$$
$79$ $$(T^{2} - 27)^{4}$$
$83$ $$(T^{2} + 1078272)^{4}$$
$89$ $$(T^{2} - 640692)^{4}$$
$97$ $$(T - 1111)^{8}$$