# Properties

 Label 1728.4.d.g 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: 8.0.1731891456.1 Defining polynomial: $$x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256$$ x^8 - 9*x^6 + 65*x^4 - 144*x^2 + 256 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{12}\cdot 3^{6}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{5} + \beta_{6} q^{7}+O(q^{10})$$ q + b2 * q^5 + b6 * q^7 $$q + \beta_{2} q^{5} + \beta_{6} q^{7} - \beta_{7} q^{11} + \beta_{5} q^{13} + 5 \beta_{4} q^{17} - 7 \beta_1 q^{19} - 11 \beta_{3} q^{23} + 17 q^{25} - 26 \beta_{2} q^{29} + 10 \beta_{6} q^{31} + 9 \beta_{7} q^{35} + 11 \beta_{5} q^{37} - 16 \beta_{4} q^{41} - 92 \beta_1 q^{43} - 11 \beta_{3} q^{47} + 116 q^{49} + 2 \beta_{2} q^{53} + 12 \beta_{6} q^{55} - 7 \beta_{7} q^{59} + 21 \beta_{5} q^{61} + 9 \beta_{4} q^{65} - 353 \beta_1 q^{67} - 76 \beta_{3} q^{71} + 425 q^{73} - 51 \beta_{2} q^{77} - 61 \beta_{6} q^{79} - 24 \beta_{7} q^{83} - 60 \beta_{5} q^{85} + 3 \beta_{4} q^{89} - 459 \beta_1 q^{91} + 7 \beta_{3} q^{95} + 799 q^{97}+O(q^{100})$$ q + b2 * q^5 + b6 * q^7 - b7 * q^11 + b5 * q^13 + 5*b4 * q^17 - 7*b1 * q^19 - 11*b3 * q^23 + 17 * q^25 - 26*b2 * q^29 + 10*b6 * q^31 + 9*b7 * q^35 + 11*b5 * q^37 - 16*b4 * q^41 - 92*b1 * q^43 - 11*b3 * q^47 + 116 * q^49 + 2*b2 * q^53 + 12*b6 * q^55 - 7*b7 * q^59 + 21*b5 * q^61 + 9*b4 * q^65 - 353*b1 * q^67 - 76*b3 * q^71 + 425 * q^73 - 51*b2 * q^77 - 61*b6 * q^79 - 24*b7 * q^83 - 60*b5 * q^85 + 3*b4 * q^89 - 459*b1 * q^91 + 7*b3 * q^95 + 799 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q + 136 q^{25} + 928 q^{49} + 3400 q^{73} + 6392 q^{97}+O(q^{100})$$ 8 * q + 136 * q^25 + 928 * q^49 + 3400 * q^73 + 6392 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256$$ :

 $$\beta_{1}$$ $$=$$ $$( -9\nu^{7} + 65\nu^{5} - 377\nu^{3} + 256\nu ) / 832$$ (-9*v^7 + 65*v^5 - 377*v^3 + 256*v) / 832 $$\beta_{2}$$ $$=$$ $$( -27\nu^{6} + 195\nu^{4} - 1755\nu^{2} + 2328 ) / 260$$ (-27*v^6 + 195*v^4 - 1755*v^2 + 2328) / 260 $$\beta_{3}$$ $$=$$ $$( -3\nu^{7} + 75\nu^{5} - 435\nu^{3} + 1632\nu ) / 160$$ (-3*v^7 + 75*v^5 - 435*v^3 + 1632*v) / 160 $$\beta_{4}$$ $$=$$ $$( -12\nu^{6} - 1782 ) / 65$$ (-12*v^6 - 1782) / 65 $$\beta_{5}$$ $$=$$ $$( -3\nu^{6} + 27\nu^{4} - 147\nu^{2} + 216 ) / 8$$ (-3*v^6 + 27*v^4 - 147*v^2 + 216) / 8 $$\beta_{6}$$ $$=$$ $$( 51\nu^{7} - 507\nu^{5} + 3939\nu^{3} - 15456\nu ) / 832$$ (51*v^7 - 507*v^5 + 3939*v^3 - 15456*v) / 832 $$\beta_{7}$$ $$=$$ $$( 27\nu^{7} - 195\nu^{5} + 1515\nu^{3} - 768\nu ) / 160$$ (27*v^7 - 195*v^5 + 1515*v^3 - 768*v) / 160
 $$\nu$$ $$=$$ $$( \beta_{7} - 2\beta_{6} - \beta_{3} + 6\beta_1 ) / 24$$ (b7 - 2*b6 - b3 + 6*b1) / 24 $$\nu^{2}$$ $$=$$ $$( 2\beta_{5} + \beta_{4} - 9\beta_{2} + 54 ) / 24$$ (2*b5 + b4 - 9*b2 + 54) / 24 $$\nu^{3}$$ $$=$$ $$( 5\beta_{7} + 78\beta_1 ) / 12$$ (5*b7 + 78*b1) / 12 $$\nu^{4}$$ $$=$$ $$( 18\beta_{5} - 9\beta_{4} - 49\beta_{2} - 294 ) / 24$$ (18*b5 - 9*b4 - 49*b2 - 294) / 24 $$\nu^{5}$$ $$=$$ $$( 29\beta_{7} + 58\beta_{6} + 101\beta_{3} + 606\beta_1 ) / 24$$ (29*b7 + 58*b6 + 101*b3 + 606*b1) / 24 $$\nu^{6}$$ $$=$$ $$( -65\beta_{4} - 1782 ) / 12$$ (-65*b4 - 1782) / 12 $$\nu^{7}$$ $$=$$ $$( -181\beta_{7} + 362\beta_{6} + 701\beta_{3} - 4206\beta_1 ) / 24$$ (-181*b7 + 362*b6 + 701*b3 - 4206*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
 1.35234 + 0.780776i 2.21837 + 1.28078i −1.35234 − 0.780776i −2.21837 − 1.28078i 2.21837 − 1.28078i 1.35234 − 0.780776i −2.21837 + 1.28078i −1.35234 + 0.780776i
0 0 0 10.3923i 0 −21.4243 0 0 0
865.2 0 0 0 10.3923i 0 −21.4243 0 0 0
865.3 0 0 0 10.3923i 0 21.4243 0 0 0
865.4 0 0 0 10.3923i 0 21.4243 0 0 0
865.5 0 0 0 10.3923i 0 −21.4243 0 0 0
865.6 0 0 0 10.3923i 0 −21.4243 0 0 0
865.7 0 0 0 10.3923i 0 21.4243 0 0 0
865.8 0 0 0 10.3923i 0 21.4243 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.g 8
3.b odd 2 1 inner 1728.4.d.g 8
4.b odd 2 1 inner 1728.4.d.g 8
8.b even 2 1 inner 1728.4.d.g 8
8.d odd 2 1 inner 1728.4.d.g 8
12.b even 2 1 inner 1728.4.d.g 8
24.f even 2 1 inner 1728.4.d.g 8
24.h odd 2 1 inner 1728.4.d.g 8

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

 $$T_{5}^{2} + 108$$ T5^2 + 108 $$T_{7}^{2} - 459$$ T7^2 - 459 $$T_{17}^{2} - 15300$$ T17^2 - 15300 $$T_{23}^{2} - 13068$$ T23^2 - 13068

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$(T^{2} + 108)^{4}$$
$7$ $$(T^{2} - 459)^{4}$$
$11$ $$(T^{2} + 612)^{4}$$
$13$ $$(T^{2} + 459)^{4}$$
$17$ $$(T^{2} - 15300)^{4}$$
$19$ $$(T^{2} + 49)^{4}$$
$23$ $$(T^{2} - 13068)^{4}$$
$29$ $$(T^{2} + 73008)^{4}$$
$31$ $$(T^{2} - 45900)^{4}$$
$37$ $$(T^{2} + 55539)^{4}$$
$41$ $$(T^{2} - 156672)^{4}$$
$43$ $$(T^{2} + 8464)^{4}$$
$47$ $$(T^{2} - 13068)^{4}$$
$53$ $$(T^{2} + 432)^{4}$$
$59$ $$(T^{2} + 29988)^{4}$$
$61$ $$(T^{2} + 202419)^{4}$$
$67$ $$(T^{2} + 124609)^{4}$$
$71$ $$(T^{2} - 623808)^{4}$$
$73$ $$(T - 425)^{8}$$
$79$ $$(T^{2} - 1707939)^{4}$$
$83$ $$(T^{2} + 352512)^{4}$$
$89$ $$(T^{2} - 5508)^{4}$$
$97$ $$(T - 799)^{8}$$