# Properties

 Label 1728.3.b.b 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: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q - 2 \beta_{2} q^{5} + \beta_1 q^{7}+O(q^{10})$$ q - 2*b2 * q^5 + b1 * q^7 $$q - 2 \beta_{2} q^{5} + \beta_1 q^{7} - 10 \beta_{3} q^{11} - \beta_{2} q^{13} - 6 q^{17} + \beta_{3} q^{19} + 30 \beta_1 q^{23} + 13 q^{25} + 12 \beta_{2} q^{29} + 14 \beta_1 q^{31} + 2 \beta_{3} q^{35} - 11 \beta_{2} q^{37} + 48 q^{41} + 12 \beta_{3} q^{43} - 66 \beta_1 q^{47} + 48 q^{49} + 28 \beta_{2} q^{53} + 60 \beta_1 q^{55} + 18 \beta_{3} q^{59} - 25 \beta_{2} q^{61} - 6 q^{65} + 35 \beta_{3} q^{67} + 48 \beta_1 q^{71} + 49 q^{73} - 10 \beta_{2} q^{77} + 83 \beta_1 q^{79} - 8 \beta_{3} q^{83} + 12 \beta_{2} q^{85} - 66 q^{89} + \beta_{3} q^{91} - 6 \beta_1 q^{95} + 107 q^{97}+O(q^{100})$$ q - 2*b2 * q^5 + b1 * q^7 - 10*b3 * q^11 - b2 * q^13 - 6 * q^17 + b3 * q^19 + 30*b1 * q^23 + 13 * q^25 + 12*b2 * q^29 + 14*b1 * q^31 + 2*b3 * q^35 - 11*b2 * q^37 + 48 * q^41 + 12*b3 * q^43 - 66*b1 * q^47 + 48 * q^49 + 28*b2 * q^53 + 60*b1 * q^55 + 18*b3 * q^59 - 25*b2 * q^61 - 6 * q^65 + 35*b3 * q^67 + 48*b1 * q^71 + 49 * q^73 - 10*b2 * q^77 + 83*b1 * q^79 - 8*b3 * q^83 + 12*b2 * q^85 - 66 * q^89 + b3 * q^91 - 6*b1 * q^95 + 107 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 24 q^{17} + 52 q^{25} + 192 q^{41} + 192 q^{49} - 24 q^{65} + 196 q^{73} - 264 q^{89} + 428 q^{97}+O(q^{100})$$ 4 * q - 24 * q^17 + 52 * q^25 + 192 * q^41 + 192 * q^49 - 24 * q^65 + 196 * q^73 - 264 * q^89 + 428 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{3}$$ v^3 $$\beta_{2}$$ $$=$$ $$2\zeta_{12}^{2} - 1$$ 2*v^2 - 1 $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\zeta_{12}^{2}$$ $$=$$ $$( \beta_{2} + 1 ) / 2$$ (b2 + 1) / 2 $$\zeta_{12}^{3}$$ $$=$$ $$\beta_1$$ 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 3.46410i 0 1.00000i 0 0 0
1567.2 0 0 0 3.46410i 0 1.00000i 0 0 0
1567.3 0 0 0 3.46410i 0 1.00000i 0 0 0
1567.4 0 0 0 3.46410i 0 1.00000i 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
4.b odd 2 1 inner
8.b even 2 1 inner
8.d 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.b 4
3.b odd 2 1 1728.3.b.e yes 4
4.b odd 2 1 inner 1728.3.b.b 4
8.b even 2 1 inner 1728.3.b.b 4
8.d odd 2 1 inner 1728.3.b.b 4
12.b even 2 1 1728.3.b.e yes 4
24.f even 2 1 1728.3.b.e yes 4
24.h odd 2 1 1728.3.b.e yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1728.3.b.b 4 1.a even 1 1 trivial
1728.3.b.b 4 4.b odd 2 1 inner
1728.3.b.b 4 8.b even 2 1 inner
1728.3.b.b 4 8.d odd 2 1 inner
1728.3.b.e yes 4 3.b odd 2 1
1728.3.b.e yes 4 12.b even 2 1
1728.3.b.e yes 4 24.f even 2 1
1728.3.b.e yes 4 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1728, [\chi])$$:

 $$T_{5}^{2} + 12$$ T5^2 + 12 $$T_{7}^{2} + 1$$ T7^2 + 1 $$T_{11}^{2} - 300$$ T11^2 - 300 $$T_{17} + 6$$ T17 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 12)^{2}$$
$7$ $$(T^{2} + 1)^{2}$$
$11$ $$(T^{2} - 300)^{2}$$
$13$ $$(T^{2} + 3)^{2}$$
$17$ $$(T + 6)^{4}$$
$19$ $$(T^{2} - 3)^{2}$$
$23$ $$(T^{2} + 900)^{2}$$
$29$ $$(T^{2} + 432)^{2}$$
$31$ $$(T^{2} + 196)^{2}$$
$37$ $$(T^{2} + 363)^{2}$$
$41$ $$(T - 48)^{4}$$
$43$ $$(T^{2} - 432)^{2}$$
$47$ $$(T^{2} + 4356)^{2}$$
$53$ $$(T^{2} + 2352)^{2}$$
$59$ $$(T^{2} - 972)^{2}$$
$61$ $$(T^{2} + 1875)^{2}$$
$67$ $$(T^{2} - 3675)^{2}$$
$71$ $$(T^{2} + 2304)^{2}$$
$73$ $$(T - 49)^{4}$$
$79$ $$(T^{2} + 6889)^{2}$$
$83$ $$(T^{2} - 192)^{2}$$
$89$ $$(T + 66)^{4}$$
$97$ $$(T - 107)^{4}$$