# Properties

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

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## 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + \beta q^{3} + q^{4} + q^{5} -\beta q^{6} + ( 1 + \beta ) q^{7} - q^{8} +O(q^{10})$$ $$q - q^{2} + \beta q^{3} + q^{4} + q^{5} -\beta q^{6} + ( 1 + \beta ) q^{7} - q^{8} - q^{10} + ( -2 - 2 \beta ) q^{11} + \beta q^{12} + ( 2 + 2 \beta ) q^{13} + ( -1 - \beta ) q^{14} + \beta q^{15} + q^{16} + ( -3 - 2 \beta ) q^{17} + ( -2 - \beta ) q^{19} + q^{20} + ( 3 + \beta ) q^{21} + ( 2 + 2 \beta ) q^{22} -\beta q^{24} + q^{25} + ( -2 - 2 \beta ) q^{26} -3 \beta q^{27} + ( 1 + \beta ) q^{28} + ( -3 - \beta ) q^{29} -\beta q^{30} + ( -7 + \beta ) q^{31} - q^{32} + ( -6 - 2 \beta ) q^{33} + ( 3 + 2 \beta ) q^{34} + ( 1 + \beta ) q^{35} -2 \beta q^{37} + ( 2 + \beta ) q^{38} + ( 6 + 2 \beta ) q^{39} - q^{40} + ( 6 + 2 \beta ) q^{41} + ( -3 - \beta ) q^{42} + ( 1 - 4 \beta ) q^{43} + ( -2 - 2 \beta ) q^{44} + ( -1 - 7 \beta ) q^{47} + \beta q^{48} + ( -3 + 2 \beta ) q^{49} - q^{50} + ( -6 - 3 \beta ) q^{51} + ( 2 + 2 \beta ) q^{52} + ( 3 + \beta ) q^{53} + 3 \beta q^{54} + ( -2 - 2 \beta ) q^{55} + ( -1 - \beta ) q^{56} + ( -3 - 2 \beta ) q^{57} + ( 3 + \beta ) q^{58} -7 q^{59} + \beta q^{60} + ( -4 + 4 \beta ) q^{61} + ( 7 - \beta ) q^{62} + q^{64} + ( 2 + 2 \beta ) q^{65} + ( 6 + 2 \beta ) q^{66} + ( -1 - 4 \beta ) q^{67} + ( -3 - 2 \beta ) q^{68} + ( -1 - \beta ) q^{70} + ( 3 + 7 \beta ) q^{71} -\beta q^{73} + 2 \beta q^{74} + \beta q^{75} + ( -2 - \beta ) q^{76} + ( -8 - 4 \beta ) q^{77} + ( -6 - 2 \beta ) q^{78} + ( -3 + \beta ) q^{79} + q^{80} -9 q^{81} + ( -6 - 2 \beta ) q^{82} + ( -11 - 4 \beta ) q^{83} + ( 3 + \beta ) q^{84} + ( -3 - 2 \beta ) q^{85} + ( -1 + 4 \beta ) q^{86} + ( -3 - 3 \beta ) q^{87} + ( 2 + 2 \beta ) q^{88} + ( 8 - 2 \beta ) q^{89} + ( 8 + 4 \beta ) q^{91} + ( 3 - 7 \beta ) q^{93} + ( 1 + 7 \beta ) q^{94} + ( -2 - \beta ) q^{95} -\beta q^{96} + ( 8 + 4 \beta ) q^{97} + ( 3 - 2 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{4} + 2q^{5} + 2q^{7} - 2q^{8} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{4} + 2q^{5} + 2q^{7} - 2q^{8} - 2q^{10} - 4q^{11} + 4q^{13} - 2q^{14} + 2q^{16} - 6q^{17} - 4q^{19} + 2q^{20} + 6q^{21} + 4q^{22} + 2q^{25} - 4q^{26} + 2q^{28} - 6q^{29} - 14q^{31} - 2q^{32} - 12q^{33} + 6q^{34} + 2q^{35} + 4q^{38} + 12q^{39} - 2q^{40} + 12q^{41} - 6q^{42} + 2q^{43} - 4q^{44} - 2q^{47} - 6q^{49} - 2q^{50} - 12q^{51} + 4q^{52} + 6q^{53} - 4q^{55} - 2q^{56} - 6q^{57} + 6q^{58} - 14q^{59} - 8q^{61} + 14q^{62} + 2q^{64} + 4q^{65} + 12q^{66} - 2q^{67} - 6q^{68} - 2q^{70} + 6q^{71} - 4q^{76} - 16q^{77} - 12q^{78} - 6q^{79} + 2q^{80} - 18q^{81} - 12q^{82} - 22q^{83} + 6q^{84} - 6q^{85} - 2q^{86} - 6q^{87} + 4q^{88} + 16q^{89} + 16q^{91} + 6q^{93} + 2q^{94} - 4q^{95} + 16q^{97} + 6q^{98} + O(q^{100})$$

## 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 −0.732051 −1.00000 0 −1.00000
1.2 −1.00000 1.73205 1.00000 1.00000 −1.73205 2.73205 −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.g yes 2
23.b odd 2 1 5290.2.a.f 2

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

 $$T_{3}^{2} - 3$$ $$T_{7}^{2} - 2 T_{7} - 2$$ $$T_{11}^{2} + 4 T_{11} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$-3 + T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$-2 - 2 T + T^{2}$$
$11$ $$-8 + 4 T + T^{2}$$
$13$ $$-8 - 4 T + T^{2}$$
$17$ $$-3 + 6 T + T^{2}$$
$19$ $$1 + 4 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$6 + 6 T + T^{2}$$
$31$ $$46 + 14 T + T^{2}$$
$37$ $$-12 + T^{2}$$
$41$ $$24 - 12 T + T^{2}$$
$43$ $$-47 - 2 T + T^{2}$$
$47$ $$-146 + 2 T + T^{2}$$
$53$ $$6 - 6 T + T^{2}$$
$59$ $$( 7 + T )^{2}$$
$61$ $$-32 + 8 T + T^{2}$$
$67$ $$-47 + 2 T + T^{2}$$
$71$ $$-138 - 6 T + T^{2}$$
$73$ $$-3 + T^{2}$$
$79$ $$6 + 6 T + T^{2}$$
$83$ $$73 + 22 T + T^{2}$$
$89$ $$52 - 16 T + T^{2}$$
$97$ $$16 - 16 T + T^{2}$$
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