# Properties

 Label 5586.2.a.bw Level $5586$ Weight $2$ Character orbit 5586.a Self dual yes Analytic conductor $44.604$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.29268.1 Defining polynomial: $$x^{4} - x^{3} - 9 x^{2} + 5 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} -\beta_{2} q^{5} + q^{6} - q^{8} + q^{9} +O(q^{10})$$ $$q - q^{2} - q^{3} + q^{4} -\beta_{2} q^{5} + q^{6} - q^{8} + q^{9} + \beta_{2} q^{10} + ( -\beta_{1} + \beta_{3} ) q^{11} - q^{12} + 2 \beta_{2} q^{13} + \beta_{2} q^{15} + q^{16} + ( 3 - \beta_{2} + 2 \beta_{3} ) q^{17} - q^{18} + q^{19} -\beta_{2} q^{20} + ( \beta_{1} - \beta_{3} ) q^{22} + ( -2 + 3 \beta_{1} - \beta_{2} ) q^{23} + q^{24} + ( 1 + \beta_{1} + \beta_{3} ) q^{25} -2 \beta_{2} q^{26} - q^{27} -3 \beta_{1} q^{29} -\beta_{2} q^{30} + ( 3 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{31} - q^{32} + ( \beta_{1} - \beta_{3} ) q^{33} + ( -3 + \beta_{2} - 2 \beta_{3} ) q^{34} + q^{36} + ( -3 - 2 \beta_{1} + \beta_{2} ) q^{37} - q^{38} -2 \beta_{2} q^{39} + \beta_{2} q^{40} + ( 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{41} + ( 5 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{43} + ( -\beta_{1} + \beta_{3} ) q^{44} -\beta_{2} q^{45} + ( 2 - 3 \beta_{1} + \beta_{2} ) q^{46} + ( 2 - 2 \beta_{2} + \beta_{3} ) q^{47} - q^{48} + ( -1 - \beta_{1} - \beta_{3} ) q^{50} + ( -3 + \beta_{2} - 2 \beta_{3} ) q^{51} + 2 \beta_{2} q^{52} + ( -1 + 3 \beta_{3} ) q^{53} + q^{54} + ( 1 - 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{55} - q^{57} + 3 \beta_{1} q^{58} + ( 3 - 5 \beta_{1} + \beta_{2} ) q^{59} + \beta_{2} q^{60} + ( 6 + \beta_{3} ) q^{61} + ( -3 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{62} + q^{64} + ( -12 - 2 \beta_{1} - 2 \beta_{3} ) q^{65} + ( -\beta_{1} + \beta_{3} ) q^{66} + ( 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{67} + ( 3 - \beta_{2} + 2 \beta_{3} ) q^{68} + ( 2 - 3 \beta_{1} + \beta_{2} ) q^{69} + ( -1 + 4 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{71} - q^{72} + ( -3 + 5 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{73} + ( 3 + 2 \beta_{1} - \beta_{2} ) q^{74} + ( -1 - \beta_{1} - \beta_{3} ) q^{75} + q^{76} + 2 \beta_{2} q^{78} + ( -2 - \beta_{1} + 2 \beta_{3} ) q^{79} -\beta_{2} q^{80} + q^{81} + ( -3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{82} + ( 5 - 4 \beta_{1} + 2 \beta_{3} ) q^{83} + ( 2 - 3 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{85} + ( -5 - 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{86} + 3 \beta_{1} q^{87} + ( \beta_{1} - \beta_{3} ) q^{88} + ( 2 - \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{89} + \beta_{2} q^{90} + ( -2 + 3 \beta_{1} - \beta_{2} ) q^{92} + ( -3 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{93} + ( -2 + 2 \beta_{2} - \beta_{3} ) q^{94} -\beta_{2} q^{95} + q^{96} + ( -1 + 3 \beta_{1} - \beta_{2} ) q^{97} + ( -\beta_{1} + \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{6} - 4 q^{8} + 4 q^{9} + O(q^{10})$$ $$4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{6} - 4 q^{8} + 4 q^{9} - 2 q^{11} - 4 q^{12} + 4 q^{16} + 10 q^{17} - 4 q^{18} + 4 q^{19} + 2 q^{22} - 5 q^{23} + 4 q^{24} + 4 q^{25} - 4 q^{27} - 3 q^{29} + 9 q^{31} - 4 q^{32} + 2 q^{33} - 10 q^{34} + 4 q^{36} - 14 q^{37} - 4 q^{38} + 4 q^{41} + 21 q^{43} - 2 q^{44} + 5 q^{46} + 7 q^{47} - 4 q^{48} - 4 q^{50} - 10 q^{51} - 7 q^{53} + 4 q^{54} - 4 q^{57} + 3 q^{58} + 7 q^{59} + 23 q^{61} - 9 q^{62} + 4 q^{64} - 48 q^{65} - 2 q^{66} + 6 q^{67} + 10 q^{68} + 5 q^{69} + 2 q^{71} - 4 q^{72} - 5 q^{73} + 14 q^{74} - 4 q^{75} + 4 q^{76} - 11 q^{79} + 4 q^{81} - 4 q^{82} + 14 q^{83} + 6 q^{85} - 21 q^{86} + 3 q^{87} + 2 q^{88} + 10 q^{89} - 5 q^{92} - 9 q^{93} - 7 q^{94} + 4 q^{96} - q^{97} - 2 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 9 x^{2} + 5 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - 5 \nu - 2$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} - 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 5$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{2} + 5 \beta_{1} + 2$$

## 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
 2.85121 −1.25548 −2.43292 1.83719
−1.00000 −1.00000 1.00000 −3.46130 1.00000 0 −1.00000 1.00000 3.46130
1.2 −1.00000 −1.00000 1.00000 −1.14924 1.00000 0 −1.00000 1.00000 1.14924
1.3 −1.00000 −1.00000 1.00000 2.11806 1.00000 0 −1.00000 1.00000 −2.11806
1.4 −1.00000 −1.00000 1.00000 2.49248 1.00000 0 −1.00000 1.00000 −2.49248
 $$n$$: 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.bw 4
7.b odd 2 1 5586.2.a.bz 4
7.c even 3 2 798.2.j.l 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.j.l 8 7.c even 3 2
5586.2.a.bw 4 1.a even 1 1 trivial
5586.2.a.bz 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5586))$$:

 $$T_{5}^{4} - 12 T_{5}^{2} + 6 T_{5} + 21$$ $$T_{11}^{4} + 2 T_{11}^{3} - 12 T_{11}^{2} - 22 T_{11} + 7$$ $$T_{13}^{4} - 48 T_{13}^{2} - 48 T_{13} + 336$$ $$T_{17}^{4} - 10 T_{17}^{3} - 6 T_{17}^{2} + 254 T_{17} - 452$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{4}$$
$3$ $$( 1 + T )^{4}$$
$5$ $$21 + 6 T - 12 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$7 - 22 T - 12 T^{2} + 2 T^{3} + T^{4}$$
$13$ $$336 - 48 T - 48 T^{2} + T^{4}$$
$17$ $$-452 + 254 T - 6 T^{2} - 10 T^{3} + T^{4}$$
$19$ $$( -1 + T )^{4}$$
$23$ $$922 - 190 T - 60 T^{2} + 5 T^{3} + T^{4}$$
$29$ $$1296 - 135 T - 81 T^{2} + 3 T^{3} + T^{4}$$
$31$ $$-1068 + 477 T - 33 T^{2} - 9 T^{3} + T^{4}$$
$37$ $$-8 - 22 T + 42 T^{2} + 14 T^{3} + T^{4}$$
$41$ $$-344 + 506 T - 96 T^{2} - 4 T^{3} + T^{4}$$
$43$ $$-654 + 24 T + 114 T^{2} - 21 T^{3} + T^{4}$$
$47$ $$256 + 128 T - 24 T^{2} - 7 T^{3} + T^{4}$$
$53$ $$976 - 419 T - 93 T^{2} + 7 T^{3} + T^{4}$$
$59$ $$9202 + 653 T - 183 T^{2} - 7 T^{3} + T^{4}$$
$61$ $$712 - 620 T + 186 T^{2} - 23 T^{3} + T^{4}$$
$67$ $$7968 + 1008 T - 240 T^{2} - 6 T^{3} + T^{4}$$
$71$ $$28 + 46 T - 126 T^{2} - 2 T^{3} + T^{4}$$
$73$ $$-1676 - 1240 T - 186 T^{2} + 5 T^{3} + T^{4}$$
$79$ $$172 - 151 T + 3 T^{2} + 11 T^{3} + T^{4}$$
$83$ $$43 + 286 T - 60 T^{2} - 14 T^{3} + T^{4}$$
$89$ $$-4844 + 1958 T - 156 T^{2} - 10 T^{3} + T^{4}$$
$97$ $$1048 - 59 T - 69 T^{2} + T^{3} + T^{4}$$