Properties

 Label 5586.2.a.bd Level $5586$ Weight $2$ Character orbit 5586.a Self dual yes Analytic conductor $44.604$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ 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 $$\beta = 2\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

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

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
 −1.73205 1.73205
−1.00000 1.00000 1.00000 −3.46410 −1.00000 0 −1.00000 1.00000 3.46410
1.2 −1.00000 1.00000 1.00000 3.46410 −1.00000 0 −1.00000 1.00000 −3.46410
 $$n$$: 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.bd 2
7.b odd 2 1 798.2.a.k 2
21.c even 2 1 2394.2.a.x 2
28.d even 2 1 6384.2.a.br 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.a.k 2 7.b odd 2 1
2394.2.a.x 2 21.c even 2 1
5586.2.a.bd 2 1.a even 1 1 trivial
6384.2.a.br 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5586))$$:

 $$T_{5}^{2} - 12$$ $$T_{11}^{2} + 4 T_{11} - 8$$ $$T_{13}^{2} - 12$$ $$T_{17}^{2} + 8 T_{17} + 4$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$-12 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-8 + 4 T + T^{2}$$
$13$ $$-12 + T^{2}$$
$17$ $$4 + 8 T + T^{2}$$
$19$ $$( 1 + T )^{2}$$
$23$ $$-8 + 4 T + T^{2}$$
$29$ $$( 6 + T )^{2}$$
$31$ $$-48 + T^{2}$$
$37$ $$( -10 + T )^{2}$$
$41$ $$( -2 + T )^{2}$$
$43$ $$-48 + T^{2}$$
$47$ $$-32 + 8 T + T^{2}$$
$53$ $$( -2 + T )^{2}$$
$59$ $$( 4 + T )^{2}$$
$61$ $$-44 + 4 T + T^{2}$$
$67$ $$24 - 12 T + T^{2}$$
$71$ $$-32 + 8 T + T^{2}$$
$73$ $$( 10 + T )^{2}$$
$79$ $$-104 + 4 T + T^{2}$$
$83$ $$( 8 + T )^{2}$$
$89$ $$( -2 + T )^{2}$$
$97$ $$52 + 16 T + T^{2}$$