Properties

 Label 5586.2.a.bt 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.

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

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 −1.65544 −0.210756
−1.00000 1.00000 1.00000 −2.34889 −1.00000 0 −1.00000 1.00000 2.34889
1.2 −1.00000 1.00000 1.00000 −1.39593 −1.00000 0 −1.00000 1.00000 1.39593
1.3 −1.00000 1.00000 1.00000 2.74483 −1.00000 0 −1.00000 1.00000 −2.74483
 $$n$$: 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.bt 3
7.b odd 2 1 5586.2.a.bs 3
7.d odd 6 2 798.2.j.j 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.j.j 6 7.d odd 6 2
5586.2.a.bs 3 7.b odd 2 1
5586.2.a.bt 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5586))$$:

 $$T_{5}^{3} + T_{5}^{2} - 7 T_{5} - 9$$ $$T_{11}^{3} - 7 T_{11}^{2} - T_{11} + 43$$ $$T_{13}^{3} + 2 T_{13}^{2} - 20 T_{13} + 8$$ $$T_{17}^{3} - 10 T_{17}^{2} + 26 T_{17} - 6$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{3}$$
$3$ $$( -1 + T )^{3}$$
$5$ $$-9 - 7 T + T^{2} + T^{3}$$
$7$ $$T^{3}$$
$11$ $$43 - T - 7 T^{2} + T^{3}$$
$13$ $$8 - 20 T + 2 T^{2} + T^{3}$$
$17$ $$-6 + 26 T - 10 T^{2} + T^{3}$$
$19$ $$( -1 + T )^{3}$$
$23$ $$82 + 8 T - 10 T^{2} + T^{3}$$
$29$ $$-27 - 25 T + 7 T^{2} + T^{3}$$
$31$ $$479 - 37 T - 11 T^{2} + T^{3}$$
$37$ $$-66 - 74 T + T^{3}$$
$41$ $$6 - 4 T - 2 T^{2} + T^{3}$$
$43$ $$18 - 26 T + 6 T^{2} + T^{3}$$
$47$ $$-32 - 32 T + T^{3}$$
$53$ $$29 - 21 T - T^{2} + T^{3}$$
$59$ $$-473 - 113 T + 3 T^{2} + T^{3}$$
$61$ $$-232 + 132 T - 22 T^{2} + T^{3}$$
$67$ $$48 + 104 T + 20 T^{2} + T^{3}$$
$71$ $$6 + 26 T + 10 T^{2} + T^{3}$$
$73$ $$-396 - 84 T + 10 T^{2} + T^{3}$$
$79$ $$301 - 197 T + 3 T^{2} + T^{3}$$
$83$ $$121 + 91 T + 19 T^{2} + T^{3}$$
$89$ $$178 - 68 T - 14 T^{2} + T^{3}$$
$97$ $$-839 - 77 T + 13 T^{2} + T^{3}$$