# Properties

 Label 5586.2.a.bq Level $5586$ Weight $2$ Character orbit 5586.a Self dual yes Analytic conductor $44.604$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −0.618034 1.61803
1.00000 1.00000 1.00000 −1.23607 1.00000 0 1.00000 1.00000 −1.23607
1.2 1.00000 1.00000 1.00000 3.23607 1.00000 0 1.00000 1.00000 3.23607
 $$n$$: 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.bq yes 2
7.b odd 2 1 5586.2.a.bh 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5586.2.a.bh 2 7.b odd 2 1
5586.2.a.bq yes 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5586))$$:

 $$T_{5}^{2} - 2 T_{5} - 4$$ $$T_{11}^{2} + 6 T_{11} + 4$$ $$T_{13}^{2} - 2 T_{13} - 4$$ $$T_{17}^{2} - 10 T_{17} + 20$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$-4 - 2 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$4 + 6 T + T^{2}$$
$13$ $$-4 - 2 T + T^{2}$$
$17$ $$20 - 10 T + T^{2}$$
$19$ $$( 1 + T )^{2}$$
$23$ $$4 - 6 T + T^{2}$$
$29$ $$-20 + T^{2}$$
$31$ $$-16 + 4 T + T^{2}$$
$37$ $$( -6 + T )^{2}$$
$41$ $$-80 + T^{2}$$
$43$ $$-16 - 4 T + T^{2}$$
$47$ $$-64 - 8 T + T^{2}$$
$53$ $$-180 + T^{2}$$
$59$ $$-80 + T^{2}$$
$61$ $$-16 + 4 T + T^{2}$$
$67$ $$44 - 14 T + T^{2}$$
$71$ $$-16 + 4 T + T^{2}$$
$73$ $$T^{2}$$
$79$ $$-20 - 10 T + T^{2}$$
$83$ $$80 + 20 T + T^{2}$$
$89$ $$-80 + T^{2}$$
$97$ $$-44 + 2 T + T^{2}$$