Properties

 Label 5184.2.d.e Level $5184$ Weight $2$ Character orbit 5184.d Analytic conductor $41.394$ Analytic rank $0$ Dimension $4$ CM discriminant -8 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: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 576) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q+O(q^{10})$$ q $$q + (\beta_{2} + \beta_1) q^{11} + (\beta_{3} - 3) q^{17} + (3 \beta_{2} - \beta_1) q^{19} + 5 q^{25} + ( - 2 \beta_{3} - 3) q^{41} + ( - 3 \beta_{2} + 4 \beta_1) q^{43} - 7 q^{49} + ( - 5 \beta_{2} + 4 \beta_1) q^{59} + ( - 3 \beta_{2} - 2 \beta_1) q^{67} + (3 \beta_{3} - 1) q^{73} + 9 \beta_1 q^{83} - 18 q^{89} + ( - 3 \beta_{3} - 5) q^{97}+O(q^{100})$$ q + (b2 + b1) * q^11 + (b3 - 3) * q^17 + (3*b2 - b1) * q^19 + 5 * q^25 + (-2*b3 - 3) * q^41 + (-3*b2 + 4*b1) * q^43 - 7 * q^49 + (-5*b2 + 4*b1) * q^59 + (-3*b2 - 2*b1) * q^67 + (3*b3 - 1) * q^73 + 9*b1 * q^83 - 18 * q^89 + (-3*b3 - 5) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 12 q^{17} + 20 q^{25} - 12 q^{41} - 28 q^{49} - 4 q^{73} - 72 q^{89} - 20 q^{97}+O(q^{100})$$ 4 * q - 12 * q^17 + 20 * q^25 - 12 * q^41 - 28 * q^49 - 4 * q^73 - 72 * q^89 - 20 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( 2\nu^{2} ) / 3$$ (2*v^2) / 3 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + \nu^{2} + 3\nu ) / 3$$ (v^3 + v^2 + 3*v) / 3 $$\beta_{3}$$ $$=$$ $$( -2\nu^{3} + 6\nu ) / 3$$ (-2*v^3 + 6*v) / 3
 $$\nu$$ $$=$$ $$( \beta_{3} + 2\beta_{2} - \beta_1 ) / 4$$ (b3 + 2*b2 - b1) / 4 $$\nu^{2}$$ $$=$$ $$( 3\beta_1 ) / 2$$ (3*b1) / 2 $$\nu^{3}$$ $$=$$ $$( -3\beta_{3} + 6\beta_{2} - 3\beta_1 ) / 4$$ (-3*b3 + 6*b2 - 3*b1) / 4

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
 1.22474 − 1.22474i −1.22474 + 1.22474i −1.22474 − 1.22474i 1.22474 + 1.22474i
0 0 0 0 0 0 0 0 0
2593.2 0 0 0 0 0 0 0 0 0
2593.3 0 0 0 0 0 0 0 0 0
2593.4 0 0 0 0 0 0 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 CM by $$\Q(\sqrt{-2})$$
4.b odd 2 1 inner
8.b even 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.e 4
3.b odd 2 1 5184.2.d.l 4
4.b odd 2 1 inner 5184.2.d.e 4
8.b even 2 1 inner 5184.2.d.e 4
8.d odd 2 1 CM 5184.2.d.e 4
9.c even 3 2 1728.2.r.c 8
9.d odd 6 2 576.2.r.c 8
12.b even 2 1 5184.2.d.l 4
24.f even 2 1 5184.2.d.l 4
24.h odd 2 1 5184.2.d.l 4
36.f odd 6 2 1728.2.r.c 8
36.h even 6 2 576.2.r.c 8
72.j odd 6 2 576.2.r.c 8
72.l even 6 2 576.2.r.c 8
72.n even 6 2 1728.2.r.c 8
72.p odd 6 2 1728.2.r.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.2.r.c 8 9.d odd 6 2
576.2.r.c 8 36.h even 6 2
576.2.r.c 8 72.j odd 6 2
576.2.r.c 8 72.l even 6 2
1728.2.r.c 8 9.c even 3 2
1728.2.r.c 8 36.f odd 6 2
1728.2.r.c 8 72.n even 6 2
1728.2.r.c 8 72.p odd 6 2
5184.2.d.e 4 1.a even 1 1 trivial
5184.2.d.e 4 4.b odd 2 1 inner
5184.2.d.e 4 8.b even 2 1 inner
5184.2.d.e 4 8.d odd 2 1 CM
5184.2.d.l 4 3.b odd 2 1
5184.2.d.l 4 12.b even 2 1
5184.2.d.l 4 24.f even 2 1
5184.2.d.l 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 $$S_{2}^{\mathrm{new}}(5184, [\chi])$$:

 $$T_{5}$$ T5 $$T_{7}$$ T7 $$T_{17}^{2} + 6T_{17} - 15$$ T17^2 + 6*T17 - 15 $$T_{23}$$ T23 $$T_{41}^{2} + 6T_{41} - 87$$ T41^2 + 6*T41 - 87

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4} + 30T^{2} + 9$$
$13$ $$T^{4}$$
$17$ $$(T^{2} + 6 T - 15)^{2}$$
$19$ $$T^{4} + 110T^{2} + 2809$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$(T^{2} + 6 T - 87)^{2}$$
$43$ $$T^{4} + 158T^{2} + 841$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4} + 318 T^{2} + 19881$$
$61$ $$T^{4}$$
$67$ $$T^{4} + 206T^{2} + 25$$
$71$ $$T^{4}$$
$73$ $$(T^{2} + 2 T - 215)^{2}$$
$79$ $$T^{4}$$
$83$ $$(T^{2} + 324)^{2}$$
$89$ $$(T + 18)^{4}$$
$97$ $$(T^{2} + 10 T - 191)^{2}$$