Properties

 Label 5184.2.d.l 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: $$x^{4} + 9$$ 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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu^{2}$$$$/3$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + \nu^{2} + 3 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{3} + 6 \nu$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + 2 \beta_{2} - \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{1}$$$$/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{3} + 6 \beta_{2} - 3 \beta_{1}$$$$)/4$$

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.l 4
3.b odd 2 1 5184.2.d.e 4
4.b odd 2 1 inner 5184.2.d.l 4
8.b even 2 1 inner 5184.2.d.l 4
8.d odd 2 1 CM 5184.2.d.l 4
9.c even 3 2 576.2.r.c 8
9.d odd 6 2 1728.2.r.c 8
12.b even 2 1 5184.2.d.e 4
24.f even 2 1 5184.2.d.e 4
24.h odd 2 1 5184.2.d.e 4
36.f odd 6 2 576.2.r.c 8
36.h even 6 2 1728.2.r.c 8
72.j odd 6 2 1728.2.r.c 8
72.l even 6 2 1728.2.r.c 8
72.n even 6 2 576.2.r.c 8
72.p odd 6 2 576.2.r.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.2.r.c 8 9.c even 3 2
576.2.r.c 8 36.f odd 6 2
576.2.r.c 8 72.n even 6 2
576.2.r.c 8 72.p odd 6 2
1728.2.r.c 8 9.d odd 6 2
1728.2.r.c 8 36.h even 6 2
1728.2.r.c 8 72.j odd 6 2
1728.2.r.c 8 72.l even 6 2
5184.2.d.e 4 3.b odd 2 1
5184.2.d.e 4 12.b even 2 1
5184.2.d.e 4 24.f even 2 1
5184.2.d.e 4 24.h odd 2 1
5184.2.d.l 4 1.a even 1 1 trivial
5184.2.d.l 4 4.b odd 2 1 inner
5184.2.d.l 4 8.b even 2 1 inner
5184.2.d.l 4 8.d odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(5184, [\chi])$$:

 $$T_{5}$$ $$T_{7}$$ $$T_{17}^{2} - 6 T_{17} - 15$$ $$T_{23}$$ $$T_{41}^{2} - 6 T_{41} - 87$$

Hecke characteristic polynomials

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