Properties

 Label 5586.2.a.bp 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: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ 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 $$\beta = 2\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

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

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
 −1.41421 1.41421
1.00000 1.00000 1.00000 −2.82843 1.00000 0 1.00000 1.00000 −2.82843
1.2 1.00000 1.00000 1.00000 2.82843 1.00000 0 1.00000 1.00000 2.82843
 $$n$$: 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.bp 2
7.b odd 2 1 798.2.a.l 2
21.c even 2 1 2394.2.a.r 2
28.d even 2 1 6384.2.a.bq 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.a.l 2 7.b odd 2 1
2394.2.a.r 2 21.c even 2 1
5586.2.a.bp 2 1.a even 1 1 trivial
6384.2.a.bq 2 28.d even 2 1

Hecke kernels

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

 $$T_{5}^{2} - 8$$ $$T_{11} - 2$$ $$T_{13}^{2} - 8$$ $$T_{17}$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$-8 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( -2 + T )^{2}$$
$13$ $$-8 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$( 1 + T )^{2}$$
$23$ $$-4 - 4 T + T^{2}$$
$29$ $$-28 - 4 T + T^{2}$$
$31$ $$8 + 8 T + T^{2}$$
$37$ $$-68 - 4 T + T^{2}$$
$41$ $$-28 - 4 T + T^{2}$$
$43$ $$( -8 + T )^{2}$$
$47$ $$-8 + T^{2}$$
$53$ $$-28 - 4 T + T^{2}$$
$59$ $$-16 - 8 T + T^{2}$$
$61$ $$-124 - 4 T + T^{2}$$
$67$ $$( -10 + T )^{2}$$
$71$ $$-16 + 8 T + T^{2}$$
$73$ $$4 - 12 T + T^{2}$$
$79$ $$28 + 12 T + T^{2}$$
$83$ $$-128 + T^{2}$$
$89$ $$( -10 + T )^{2}$$
$97$ $$-112 - 8 T + T^{2}$$