# Properties

 Label 5290.2.a.m Level $5290$ Weight $2$ Character orbit 5290.a Self dual yes Analytic conductor $42.241$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5290 = 2 \cdot 5 \cdot 23^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5290.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$42.2408626693$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + \beta q^{3} + q^{4} - q^{5} + \beta q^{6} + ( -1 + \beta ) q^{7} + q^{8} +O(q^{10})$$ $$q + q^{2} + \beta q^{3} + q^{4} - q^{5} + \beta q^{6} + ( -1 + \beta ) q^{7} + q^{8} - q^{10} -2 \beta q^{11} + \beta q^{12} + ( -1 + \beta ) q^{14} -\beta q^{15} + q^{16} + ( -2 - \beta ) q^{17} + ( -1 - 2 \beta ) q^{19} - q^{20} + ( 3 - \beta ) q^{21} -2 \beta q^{22} + \beta q^{24} + q^{25} -3 \beta q^{27} + ( -1 + \beta ) q^{28} + ( 1 - \beta ) q^{29} -\beta q^{30} + ( -5 + 3 \beta ) q^{31} + q^{32} -6 q^{33} + ( -2 - \beta ) q^{34} + ( 1 - \beta ) q^{35} + ( -8 + 2 \beta ) q^{37} + ( -1 - 2 \beta ) q^{38} - q^{40} + ( 2 - 6 \beta ) q^{41} + ( 3 - \beta ) q^{42} + ( -4 - 3 \beta ) q^{43} -2 \beta q^{44} + ( 5 + 3 \beta ) q^{47} + \beta q^{48} + ( -3 - 2 \beta ) q^{49} + q^{50} + ( -3 - 2 \beta ) q^{51} + ( 3 + 3 \beta ) q^{53} -3 \beta q^{54} + 2 \beta q^{55} + ( -1 + \beta ) q^{56} + ( -6 - \beta ) q^{57} + ( 1 - \beta ) q^{58} + ( 1 - 4 \beta ) q^{59} -\beta q^{60} + 4 q^{61} + ( -5 + 3 \beta ) q^{62} + q^{64} -6 q^{66} + \beta q^{67} + ( -2 - \beta ) q^{68} + ( 1 - \beta ) q^{70} + ( 7 - \beta ) q^{71} + ( 4 - 3 \beta ) q^{73} + ( -8 + 2 \beta ) q^{74} + \beta q^{75} + ( -1 - 2 \beta ) q^{76} + ( -6 + 2 \beta ) q^{77} + ( -3 + 3 \beta ) q^{79} - q^{80} -9 q^{81} + ( 2 - 6 \beta ) q^{82} + ( -8 - 5 \beta ) q^{83} + ( 3 - \beta ) q^{84} + ( 2 + \beta ) q^{85} + ( -4 - 3 \beta ) q^{86} + ( -3 + \beta ) q^{87} -2 \beta q^{88} -4 \beta q^{89} + ( 9 - 5 \beta ) q^{93} + ( 5 + 3 \beta ) q^{94} + ( 1 + 2 \beta ) q^{95} + \beta q^{96} + ( -4 + 6 \beta ) q^{97} + ( -3 - 2 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 2q^{4} - 2q^{5} - 2q^{7} + 2q^{8} + O(q^{10})$$ $$2q + 2q^{2} + 2q^{4} - 2q^{5} - 2q^{7} + 2q^{8} - 2q^{10} - 2q^{14} + 2q^{16} - 4q^{17} - 2q^{19} - 2q^{20} + 6q^{21} + 2q^{25} - 2q^{28} + 2q^{29} - 10q^{31} + 2q^{32} - 12q^{33} - 4q^{34} + 2q^{35} - 16q^{37} - 2q^{38} - 2q^{40} + 4q^{41} + 6q^{42} - 8q^{43} + 10q^{47} - 6q^{49} + 2q^{50} - 6q^{51} + 6q^{53} - 2q^{56} - 12q^{57} + 2q^{58} + 2q^{59} + 8q^{61} - 10q^{62} + 2q^{64} - 12q^{66} - 4q^{68} + 2q^{70} + 14q^{71} + 8q^{73} - 16q^{74} - 2q^{76} - 12q^{77} - 6q^{79} - 2q^{80} - 18q^{81} + 4q^{82} - 16q^{83} + 6q^{84} + 4q^{85} - 8q^{86} - 6q^{87} + 18q^{93} + 10q^{94} + 2q^{95} - 8q^{97} - 6q^{98} + O(q^{100})$$

## 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 −1.73205 1.00000 −1.00000 −1.73205 −2.73205 1.00000 0 −1.00000
1.2 1.00000 1.73205 1.00000 −1.00000 1.73205 0.732051 1.00000 0 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$1$$
$$23$$ $$-1$$

## 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 5290.2.a.m 2
23.b odd 2 1 5290.2.a.n yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5290.2.a.m 2 1.a even 1 1 trivial
5290.2.a.n yes 2 23.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5290))$$:

 $$T_{3}^{2} - 3$$ $$T_{7}^{2} + 2 T_{7} - 2$$ $$T_{11}^{2} - 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$-3 + T^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$-2 + 2 T + T^{2}$$
$11$ $$-12 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$1 + 4 T + T^{2}$$
$19$ $$-11 + 2 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$-2 - 2 T + T^{2}$$
$31$ $$-2 + 10 T + T^{2}$$
$37$ $$52 + 16 T + T^{2}$$
$41$ $$-104 - 4 T + T^{2}$$
$43$ $$-11 + 8 T + T^{2}$$
$47$ $$-2 - 10 T + T^{2}$$
$53$ $$-18 - 6 T + T^{2}$$
$59$ $$-47 - 2 T + T^{2}$$
$61$ $$( -4 + T )^{2}$$
$67$ $$-3 + T^{2}$$
$71$ $$46 - 14 T + T^{2}$$
$73$ $$-11 - 8 T + T^{2}$$
$79$ $$-18 + 6 T + T^{2}$$
$83$ $$-11 + 16 T + T^{2}$$
$89$ $$-48 + T^{2}$$
$97$ $$-92 + 8 T + T^{2}$$