# Properties

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

# 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: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ 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 primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 \zeta_{12}^{2} ) q^{5} + ( 3 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{7} +O(q^{10})$$ $$q + ( 1 - 2 \zeta_{12}^{2} ) q^{5} + ( 3 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{7} + ( -1 + 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{11} + 3 \zeta_{12}^{3} q^{13} + ( 6 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{17} + ( 3 - 6 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{19} + ( -3 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{23} + 2 q^{25} + ( 1 - 2 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{29} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{31} + ( 3 - 6 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{35} + ( 2 - 4 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{37} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{41} + ( 3 - 6 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{43} + ( 6 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{47} + ( 5 + 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{49} + ( -4 + 8 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{53} + ( 3 + 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{55} + ( 2 - 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{59} + ( 2 - 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{61} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{65} + ( -3 + 6 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{67} + ( -3 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{71} + 5 q^{73} + ( -6 + 12 \zeta_{12}^{2} + 12 \zeta_{12}^{3} ) q^{77} + ( -9 + 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{79} + ( -6 + 12 \zeta_{12}^{2} ) q^{83} + ( 6 - 12 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{85} + ( -6 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{89} + ( -3 + 6 \zeta_{12}^{2} + 9 \zeta_{12}^{3} ) q^{91} + ( -9 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{95} + 4 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 12 q^{7} + O(q^{10})$$ $$4 q + 12 q^{7} + 24 q^{17} - 12 q^{23} + 8 q^{25} + 24 q^{47} + 20 q^{49} + 12 q^{55} - 12 q^{71} + 20 q^{73} - 36 q^{79} - 24 q^{89} - 36 q^{95} + 16 q^{97} + O(q^{100})$$

## 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 1.26795 0 0 0
2593.2 0 0 0 1.73205i 0 4.73205 0 0 0
2593.3 0 0 0 1.73205i 0 1.26795 0 0 0
2593.4 0 0 0 1.73205i 0 4.73205 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.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.n yes 4
3.b odd 2 1 5184.2.d.m yes 4
4.b odd 2 1 5184.2.d.b yes 4
8.b even 2 1 inner 5184.2.d.n yes 4
8.d odd 2 1 5184.2.d.b yes 4
12.b even 2 1 5184.2.d.a 4
24.f even 2 1 5184.2.d.a 4
24.h odd 2 1 5184.2.d.m yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5184.2.d.a 4 12.b even 2 1
5184.2.d.a 4 24.f even 2 1
5184.2.d.b yes 4 4.b odd 2 1
5184.2.d.b yes 4 8.d odd 2 1
5184.2.d.m yes 4 3.b odd 2 1
5184.2.d.m yes 4 24.h odd 2 1
5184.2.d.n yes 4 1.a even 1 1 trivial
5184.2.d.n yes 4 8.b even 2 1 inner

## Hecke kernels

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

 $$T_{5}^{2} + 3$$ $$T_{7}^{2} - 6 T_{7} + 6$$ $$T_{17}^{2} - 12 T_{17} + 33$$ $$T_{23}^{2} + 6 T_{23} + 6$$ $$T_{41}^{2} - 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 3 + T^{2} )^{2}$$
$7$ $$( 6 - 6 T + T^{2} )^{2}$$
$11$ $$36 + 24 T^{2} + T^{4}$$
$13$ $$( 9 + T^{2} )^{2}$$
$17$ $$( 33 - 12 T + T^{2} )^{2}$$
$19$ $$676 + 56 T^{2} + T^{4}$$
$23$ $$( 6 + 6 T + T^{2} )^{2}$$
$29$ $$1089 + 78 T^{2} + T^{4}$$
$31$ $$( -12 + T^{2} )^{2}$$
$37$ $$9 + 42 T^{2} + T^{4}$$
$41$ $$( -12 + T^{2} )^{2}$$
$43$ $$676 + 56 T^{2} + T^{4}$$
$47$ $$( -12 - 12 T + T^{2} )^{2}$$
$53$ $$144 + 168 T^{2} + T^{4}$$
$59$ $$576 + 96 T^{2} + T^{4}$$
$61$ $$9 + 42 T^{2} + T^{4}$$
$67$ $$676 + 56 T^{2} + T^{4}$$
$71$ $$( 6 + 6 T + T^{2} )^{2}$$
$73$ $$( -5 + T )^{4}$$
$79$ $$( 54 + 18 T + T^{2} )^{2}$$
$83$ $$( 108 + T^{2} )^{2}$$
$89$ $$( 9 + 12 T + T^{2} )^{2}$$
$97$ $$( -4 + T )^{4}$$