# Properties

 Label 5184.2.d.k Level $5184$ Weight $2$ Character orbit 5184.d Analytic conductor $41.394$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5184 = 2^{6} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5184.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$41.3944484078$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{4}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{5} - \beta_{3} q^{7}+O(q^{10})$$ q + b2 * q^5 - b3 * q^7 $$q + \beta_{2} q^{5} - \beta_{3} q^{7} + \beta_{2} q^{13} + 3 q^{17} - \beta_1 q^{19} - 2 \beta_{3} q^{23} + 2 q^{25} - 5 \beta_{2} q^{29} - \beta_{3} q^{31} - 3 \beta_1 q^{35} + 5 \beta_{2} q^{37} - 6 q^{41} - 2 \beta_1 q^{43} + \beta_{3} q^{47} + 5 q^{49} - 4 \beta_{2} q^{53} + 6 \beta_1 q^{59} - 3 \beta_{2} q^{61} - 3 q^{65} - 5 \beta_1 q^{67} - \beta_{3} q^{71} + 5 q^{73} - 2 \beta_{3} q^{79} + 3 \beta_1 q^{83} + 3 \beta_{2} q^{85} + 15 q^{89} - 3 \beta_1 q^{91} + \beta_{3} q^{95} + 10 q^{97}+O(q^{100})$$ q + b2 * q^5 - b3 * q^7 + b2 * q^13 + 3 * q^17 - b1 * q^19 - 2*b3 * q^23 + 2 * q^25 - 5*b2 * q^29 - b3 * q^31 - 3*b1 * q^35 + 5*b2 * q^37 - 6 * q^41 - 2*b1 * q^43 + b3 * q^47 + 5 * q^49 - 4*b2 * q^53 + 6*b1 * q^59 - 3*b2 * q^61 - 3 * q^65 - 5*b1 * q^67 - b3 * q^71 + 5 * q^73 - 2*b3 * q^79 + 3*b1 * q^83 + 3*b2 * q^85 + 15 * q^89 - 3*b1 * q^91 + b3 * q^95 + 10 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 12 q^{17} + 8 q^{25} - 24 q^{41} + 20 q^{49} - 12 q^{65} + 20 q^{73} + 60 q^{89} + 40 q^{97}+O(q^{100})$$ 4 * q + 12 * q^17 + 8 * q^25 - 24 * q^41 + 20 * q^49 - 12 * q^65 + 20 * q^73 + 60 * q^89 + 40 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$2\zeta_{12}^{3}$$ 2*v^3 $$\beta_{2}$$ $$=$$ $$2\zeta_{12}^{2} - 1$$ 2*v^2 - 1 $$\beta_{3}$$ $$=$$ $$-2\zeta_{12}^{3} + 4\zeta_{12}$$ -2*v^3 + 4*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 4$$ (b3 + b1) / 4 $$\zeta_{12}^{2}$$ $$=$$ $$( \beta_{2} + 1 ) / 2$$ (b2 + 1) / 2 $$\zeta_{12}^{3}$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5184\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$1217$$ $$2431$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2593.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
0 0 0 1.73205i 0 −3.46410 0 0 0
2593.2 0 0 0 1.73205i 0 3.46410 0 0 0
2593.3 0 0 0 1.73205i 0 −3.46410 0 0 0
2593.4 0 0 0 1.73205i 0 3.46410 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 5184.2.d.k yes 4
3.b odd 2 1 5184.2.d.d 4
4.b odd 2 1 inner 5184.2.d.k yes 4
8.b even 2 1 inner 5184.2.d.k yes 4
8.d odd 2 1 inner 5184.2.d.k yes 4
12.b even 2 1 5184.2.d.d 4
24.f even 2 1 5184.2.d.d 4
24.h odd 2 1 5184.2.d.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5184.2.d.d 4 3.b odd 2 1
5184.2.d.d 4 12.b even 2 1
5184.2.d.d 4 24.f even 2 1
5184.2.d.d 4 24.h odd 2 1
5184.2.d.k yes 4 1.a even 1 1 trivial
5184.2.d.k yes 4 4.b odd 2 1 inner
5184.2.d.k yes 4 8.b even 2 1 inner
5184.2.d.k yes 4 8.d 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 $$S_{2}^{\mathrm{new}}(5184, [\chi])$$:

 $$T_{5}^{2} + 3$$ T5^2 + 3 $$T_{7}^{2} - 12$$ T7^2 - 12 $$T_{17} - 3$$ T17 - 3 $$T_{23}^{2} - 48$$ T23^2 - 48 $$T_{41} + 6$$ T41 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 3)^{2}$$
$7$ $$(T^{2} - 12)^{2}$$
$11$ $$T^{4}$$
$13$ $$(T^{2} + 3)^{2}$$
$17$ $$(T - 3)^{4}$$
$19$ $$(T^{2} + 4)^{2}$$
$23$ $$(T^{2} - 48)^{2}$$
$29$ $$(T^{2} + 75)^{2}$$
$31$ $$(T^{2} - 12)^{2}$$
$37$ $$(T^{2} + 75)^{2}$$
$41$ $$(T + 6)^{4}$$
$43$ $$(T^{2} + 16)^{2}$$
$47$ $$(T^{2} - 12)^{2}$$
$53$ $$(T^{2} + 48)^{2}$$
$59$ $$(T^{2} + 144)^{2}$$
$61$ $$(T^{2} + 27)^{2}$$
$67$ $$(T^{2} + 100)^{2}$$
$71$ $$(T^{2} - 12)^{2}$$
$73$ $$(T - 5)^{4}$$
$79$ $$(T^{2} - 48)^{2}$$
$83$ $$(T^{2} + 36)^{2}$$
$89$ $$(T - 15)^{4}$$
$97$ $$(T - 10)^{4}$$