# Properties

 Label 5586.2.a.bf 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: $$\Q(\sqrt{7})$$ Defining polynomial: $$x^{2} - 7$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{7}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −2.64575 2.64575
−1.00000 1.00000 1.00000 −1.64575 −1.00000 0 −1.00000 1.00000 1.64575
1.2 −1.00000 1.00000 1.00000 3.64575 −1.00000 0 −1.00000 1.00000 −3.64575
 $$n$$: 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.bf yes 2
7.b odd 2 1 5586.2.a.bc 2

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

 $$T_{5}^{2} - 2 T_{5} - 6$$ $$T_{11} - 2$$ $$T_{13}^{2} - 6 T_{13} + 2$$ $$T_{17}^{2} - 4 T_{17} - 24$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$-6 - 2 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( -2 + T )^{2}$$
$13$ $$2 - 6 T + T^{2}$$
$17$ $$-24 - 4 T + T^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$-6 + 2 T + T^{2}$$
$29$ $$-12 - 8 T + T^{2}$$
$31$ $$-6 + 2 T + T^{2}$$
$37$ $$-6 + 2 T + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$-12 + 8 T + T^{2}$$
$47$ $$-6 - 2 T + T^{2}$$
$53$ $$-12 - 8 T + T^{2}$$
$59$ $$( -2 + T )^{2}$$
$61$ $$-12 + 8 T + T^{2}$$
$67$ $$( -10 + T )^{2}$$
$71$ $$-24 + 4 T + T^{2}$$
$73$ $$-76 + 12 T + T^{2}$$
$79$ $$-14 - 14 T + T^{2}$$
$83$ $$( -12 + T )^{2}$$
$89$ $$( 8 + T )^{2}$$
$97$ $$72 + 20 T + T^{2}$$