# Properties

 Label 1728.3.b.f Level $1728$ Weight $3$ Character orbit 1728.b Analytic conductor $47.085$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# 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: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 6 \beta_1 q^{5} + 5 \beta_1 q^{7}+O(q^{10})$$ q + 6*b1 * q^5 + 5*b1 * q^7 $$q + 6 \beta_1 q^{5} + 5 \beta_1 q^{7} + 6 q^{11} + \beta_{2} q^{13} - 6 \beta_{3} q^{17} - 5 \beta_{3} q^{19} + 6 \beta_{2} q^{23} - 11 q^{25} - 36 \beta_1 q^{29} - 26 \beta_1 q^{31} - 30 q^{35} - 5 \beta_{2} q^{37} - 4 \beta_{3} q^{43} + 6 \beta_{2} q^{47} + 24 q^{49} + 60 \beta_1 q^{53} + 36 \beta_1 q^{55} + 18 q^{59} - 15 \beta_{2} q^{61} - 6 \beta_{3} q^{65} - 15 \beta_{3} q^{67} + 25 q^{73} + 30 \beta_1 q^{77} + 31 \beta_1 q^{79} + 120 q^{83} - 36 \beta_{2} q^{85} + 30 \beta_{3} q^{89} - 5 \beta_{3} q^{91} - 30 \beta_{2} q^{95} - 85 q^{97}+O(q^{100})$$ q + 6*b1 * q^5 + 5*b1 * q^7 + 6 * q^11 + b2 * q^13 - 6*b3 * q^17 - 5*b3 * q^19 + 6*b2 * q^23 - 11 * q^25 - 36*b1 * q^29 - 26*b1 * q^31 - 30 * q^35 - 5*b2 * q^37 - 4*b3 * q^43 + 6*b2 * q^47 + 24 * q^49 + 60*b1 * q^53 + 36*b1 * q^55 + 18 * q^59 - 15*b2 * q^61 - 6*b3 * q^65 - 15*b3 * q^67 + 25 * q^73 + 30*b1 * q^77 + 31*b1 * q^79 + 120 * q^83 - 36*b2 * q^85 + 30*b3 * q^89 - 5*b3 * q^91 - 30*b2 * q^95 - 85 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 24 q^{11} - 44 q^{25} - 120 q^{35} + 96 q^{49} + 72 q^{59} + 100 q^{73} + 480 q^{83} - 340 q^{97}+O(q^{100})$$ 4 * q + 24 * q^11 - 44 * q^25 - 120 * q^35 + 96 * q^49 + 72 * q^59 + 100 * q^73 + 480 * q^83 - 340 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{3}$$ v^3 $$\beta_{2}$$ $$=$$ $$6\zeta_{12}^{2} - 3$$ 6*v^2 - 3 $$\beta_{3}$$ $$=$$ $$-3\zeta_{12}^{3} + 6\zeta_{12}$$ -3*v^3 + 6*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + 3\beta_1 ) / 6$$ (b3 + 3*b1) / 6 $$\zeta_{12}^{2}$$ $$=$$ $$( \beta_{2} + 3 ) / 6$$ (b2 + 3) / 6 $$\zeta_{12}^{3}$$ $$=$$ $$\beta_1$$ b1

## 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
 0.866025 − 0.500000i −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i
0 0 0 6.00000i 0 5.00000i 0 0 0
1567.2 0 0 0 6.00000i 0 5.00000i 0 0 0
1567.3 0 0 0 6.00000i 0 5.00000i 0 0 0
1567.4 0 0 0 6.00000i 0 5.00000i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
12.b 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.f yes 4
3.b odd 2 1 1728.3.b.a 4
4.b odd 2 1 1728.3.b.a 4
8.b even 2 1 1728.3.b.a 4
8.d odd 2 1 inner 1728.3.b.f yes 4
12.b even 2 1 inner 1728.3.b.f yes 4
24.f even 2 1 1728.3.b.a 4
24.h odd 2 1 inner 1728.3.b.f yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1728.3.b.a 4 3.b odd 2 1
1728.3.b.a 4 4.b odd 2 1
1728.3.b.a 4 8.b even 2 1
1728.3.b.a 4 24.f even 2 1
1728.3.b.f yes 4 1.a even 1 1 trivial
1728.3.b.f yes 4 8.d odd 2 1 inner
1728.3.b.f yes 4 12.b even 2 1 inner
1728.3.b.f yes 4 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 $$S_{3}^{\mathrm{new}}(1728, [\chi])$$:

 $$T_{5}^{2} + 36$$ T5^2 + 36 $$T_{7}^{2} + 25$$ T7^2 + 25 $$T_{11} - 6$$ T11 - 6 $$T_{17}^{2} - 972$$ T17^2 - 972

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 36)^{2}$$
$7$ $$(T^{2} + 25)^{2}$$
$11$ $$(T - 6)^{4}$$
$13$ $$(T^{2} + 27)^{2}$$
$17$ $$(T^{2} - 972)^{2}$$
$19$ $$(T^{2} - 675)^{2}$$
$23$ $$(T^{2} + 972)^{2}$$
$29$ $$(T^{2} + 1296)^{2}$$
$31$ $$(T^{2} + 676)^{2}$$
$37$ $$(T^{2} + 675)^{2}$$
$41$ $$T^{4}$$
$43$ $$(T^{2} - 432)^{2}$$
$47$ $$(T^{2} + 972)^{2}$$
$53$ $$(T^{2} + 3600)^{2}$$
$59$ $$(T - 18)^{4}$$
$61$ $$(T^{2} + 6075)^{2}$$
$67$ $$(T^{2} - 6075)^{2}$$
$71$ $$T^{4}$$
$73$ $$(T - 25)^{4}$$
$79$ $$(T^{2} + 961)^{2}$$
$83$ $$(T - 120)^{4}$$
$89$ $$(T^{2} - 24300)^{2}$$
$97$ $$(T + 85)^{4}$$