# Properties

 Label 5586.2.a.bu Level $5586$ Weight $2$ Character orbit 5586.a Self dual yes Analytic conductor $44.604$ Analytic rank $0$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.404.1 Defining polynomial: $$x^{3} - x^{2} - 5 x - 1$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 5 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 3$$

## 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
 2.86620 −0.210756 −1.65544
1.00000 −1.00000 1.00000 −0.866198 −1.00000 0 1.00000 1.00000 −0.866198
1.2 1.00000 −1.00000 1.00000 2.21076 −1.00000 0 1.00000 1.00000 2.21076
1.3 1.00000 −1.00000 1.00000 3.65544 −1.00000 0 1.00000 1.00000 3.65544
 $$n$$: 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.bu 3
7.b odd 2 1 5586.2.a.bv 3
7.d odd 6 2 798.2.j.i 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.j.i 6 7.d odd 6 2
5586.2.a.bu 3 1.a even 1 1 trivial
5586.2.a.bv 3 7.b odd 2 1

## Hecke kernels

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

 $$T_{5}^{3} - 5 T_{5}^{2} + 3 T_{5} + 7$$ $$T_{11}^{3} - T_{11}^{2} - 17 T_{11} + 21$$ $$T_{13}^{3} - 6 T_{13}^{2} - 20 T_{13} + 88$$ $$T_{17}^{3} - 14 T_{17}^{2} + 60 T_{17} - 74$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{3}$$
$3$ $$( 1 + T )^{3}$$
$5$ $$7 + 3 T - 5 T^{2} + T^{3}$$
$7$ $$T^{3}$$
$11$ $$21 - 17 T - T^{2} + T^{3}$$
$13$ $$88 - 20 T - 6 T^{2} + T^{3}$$
$17$ $$-74 + 60 T - 14 T^{2} + T^{3}$$
$19$ $$( -1 + T )^{3}$$
$23$ $$-302 - 54 T + 6 T^{2} + T^{3}$$
$29$ $$43 - 41 T + T^{2} + T^{3}$$
$31$ $$7 + 19 T - 11 T^{2} + T^{3}$$
$37$ $$-194 - 44 T + 4 T^{2} + T^{3}$$
$41$ $$-42 + 46 T - 14 T^{2} + T^{3}$$
$43$ $$606 - 92 T - 6 T^{2} + T^{3}$$
$47$ $$144 - 96 T + 4 T^{2} + T^{3}$$
$53$ $$-101 - 5 T + 9 T^{2} + T^{3}$$
$59$ $$63 - 3 T - 7 T^{2} + T^{3}$$
$61$ $$-56 + 12 T + 10 T^{2} + T^{3}$$
$67$ $$-32 + 16 T + 12 T^{2} + T^{3}$$
$71$ $$206 - 92 T + 6 T^{2} + T^{3}$$
$73$ $$196 - 56 T - 2 T^{2} + T^{3}$$
$79$ $$-1067 - 81 T + 15 T^{2} + T^{3}$$
$83$ $$3279 - 157 T - 19 T^{2} + T^{3}$$
$89$ $$-42 + 46 T - 14 T^{2} + T^{3}$$
$97$ $$-873 + 301 T - 31 T^{2} + T^{3}$$