# Properties

 Label 5586.2.a.bm Level $5586$ Weight $2$ Character orbit 5586.a Self dual yes Analytic conductor $44.604$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5586 = 2 \cdot 3 \cdot 7^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5586.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$44.6044345691$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 798) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + ( -2 + \beta ) q^{5} + q^{6} + q^{8} + q^{9} +O(q^{10})$$ $$q + q^{2} + q^{3} + q^{4} + ( -2 + \beta ) q^{5} + q^{6} + q^{8} + q^{9} + ( -2 + \beta ) q^{10} - q^{11} + q^{12} + ( -2 - 2 \beta ) q^{13} + ( -2 + \beta ) q^{15} + q^{16} + ( -3 - \beta ) q^{17} + q^{18} + q^{19} + ( -2 + \beta ) q^{20} - q^{22} + ( -1 - \beta ) q^{23} + q^{24} + ( 2 - 4 \beta ) q^{25} + ( -2 - 2 \beta ) q^{26} + q^{27} + q^{29} + ( -2 + \beta ) q^{30} + ( -1 + 2 \beta ) q^{31} + q^{32} - q^{33} + ( -3 - \beta ) q^{34} + q^{36} + ( 1 + 3 \beta ) q^{37} + q^{38} + ( -2 - 2 \beta ) q^{39} + ( -2 + \beta ) q^{40} + ( -7 + \beta ) q^{41} + ( -5 + 3 \beta ) q^{43} - q^{44} + ( -2 + \beta ) q^{45} + ( -1 - \beta ) q^{46} + ( -8 - 2 \beta ) q^{47} + q^{48} + ( 2 - 4 \beta ) q^{50} + ( -3 - \beta ) q^{51} + ( -2 - 2 \beta ) q^{52} + ( 5 - 2 \beta ) q^{53} + q^{54} + ( 2 - \beta ) q^{55} + q^{57} + q^{58} + \beta q^{59} + ( -2 + \beta ) q^{60} + ( -8 + 2 \beta ) q^{61} + ( -1 + 2 \beta ) q^{62} + q^{64} + ( -2 + 2 \beta ) q^{65} - q^{66} + ( 6 - 4 \beta ) q^{67} + ( -3 - \beta ) q^{68} + ( -1 - \beta ) q^{69} + ( -7 - 3 \beta ) q^{71} + q^{72} -4 \beta q^{73} + ( 1 + 3 \beta ) q^{74} + ( 2 - 4 \beta ) q^{75} + q^{76} + ( -2 - 2 \beta ) q^{78} + ( -1 - 6 \beta ) q^{79} + ( -2 + \beta ) q^{80} + q^{81} + ( -7 + \beta ) q^{82} + 3 q^{83} + ( 3 - \beta ) q^{85} + ( -5 + 3 \beta ) q^{86} + q^{87} - q^{88} + ( -9 + 5 \beta ) q^{89} + ( -2 + \beta ) q^{90} + ( -1 - \beta ) q^{92} + ( -1 + 2 \beta ) q^{93} + ( -8 - 2 \beta ) q^{94} + ( -2 + \beta ) q^{95} + q^{96} + ( -4 - \beta ) q^{97} - q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 2q^{3} + 2q^{4} - 4q^{5} + 2q^{6} + 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{2} + 2q^{3} + 2q^{4} - 4q^{5} + 2q^{6} + 2q^{8} + 2q^{9} - 4q^{10} - 2q^{11} + 2q^{12} - 4q^{13} - 4q^{15} + 2q^{16} - 6q^{17} + 2q^{18} + 2q^{19} - 4q^{20} - 2q^{22} - 2q^{23} + 2q^{24} + 4q^{25} - 4q^{26} + 2q^{27} + 2q^{29} - 4q^{30} - 2q^{31} + 2q^{32} - 2q^{33} - 6q^{34} + 2q^{36} + 2q^{37} + 2q^{38} - 4q^{39} - 4q^{40} - 14q^{41} - 10q^{43} - 2q^{44} - 4q^{45} - 2q^{46} - 16q^{47} + 2q^{48} + 4q^{50} - 6q^{51} - 4q^{52} + 10q^{53} + 2q^{54} + 4q^{55} + 2q^{57} + 2q^{58} - 4q^{60} - 16q^{61} - 2q^{62} + 2q^{64} - 4q^{65} - 2q^{66} + 12q^{67} - 6q^{68} - 2q^{69} - 14q^{71} + 2q^{72} + 2q^{74} + 4q^{75} + 2q^{76} - 4q^{78} - 2q^{79} - 4q^{80} + 2q^{81} - 14q^{82} + 6q^{83} + 6q^{85} - 10q^{86} + 2q^{87} - 2q^{88} - 18q^{89} - 4q^{90} - 2q^{92} - 2q^{93} - 16q^{94} - 4q^{95} + 2q^{96} - 8q^{97} - 2q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.73205 1.73205
1.00000 1.00000 1.00000 −3.73205 1.00000 0 1.00000 1.00000 −3.73205
1.2 1.00000 1.00000 1.00000 −0.267949 1.00000 0 1.00000 1.00000 −0.267949
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$-1$$
$$19$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5586.2.a.bm 2
7.b odd 2 1 5586.2.a.bl 2
7.d odd 6 2 798.2.j.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.j.g 4 7.d odd 6 2
5586.2.a.bl 2 7.b odd 2 1
5586.2.a.bm 2 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5586))$$:

 $$T_{5}^{2} + 4 T_{5} + 1$$ $$T_{11} + 1$$ $$T_{13}^{2} + 4 T_{13} - 8$$ $$T_{17}^{2} + 6 T_{17} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$1 + 4 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$-8 + 4 T + T^{2}$$
$17$ $$6 + 6 T + T^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$-2 + 2 T + T^{2}$$
$29$ $$( -1 + T )^{2}$$
$31$ $$-11 + 2 T + T^{2}$$
$37$ $$-26 - 2 T + T^{2}$$
$41$ $$46 + 14 T + T^{2}$$
$43$ $$-2 + 10 T + T^{2}$$
$47$ $$52 + 16 T + T^{2}$$
$53$ $$13 - 10 T + T^{2}$$
$59$ $$-3 + T^{2}$$
$61$ $$52 + 16 T + T^{2}$$
$67$ $$-12 - 12 T + T^{2}$$
$71$ $$22 + 14 T + T^{2}$$
$73$ $$-48 + T^{2}$$
$79$ $$-107 + 2 T + T^{2}$$
$83$ $$( -3 + T )^{2}$$
$89$ $$6 + 18 T + T^{2}$$
$97$ $$13 + 8 T + T^{2}$$