# Properties

 Label 5290.2.a.l Level $5290$ Weight $2$ Character orbit 5290.a Self dual yes Analytic conductor $42.241$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + ( -1 + \beta ) q^{3} + q^{4} + q^{5} + ( -1 + \beta ) q^{6} + ( 2 - \beta ) q^{7} + q^{8} -2 \beta q^{9} +O(q^{10})$$ $$q + q^{2} + ( -1 + \beta ) q^{3} + q^{4} + q^{5} + ( -1 + \beta ) q^{6} + ( 2 - \beta ) q^{7} + q^{8} -2 \beta q^{9} + q^{10} + 2 q^{11} + ( -1 + \beta ) q^{12} + 4 \beta q^{13} + ( 2 - \beta ) q^{14} + ( -1 + \beta ) q^{15} + q^{16} + ( -1 + 5 \beta ) q^{17} -2 \beta q^{18} -3 q^{19} + q^{20} + ( -4 + 3 \beta ) q^{21} + 2 q^{22} + ( -1 + \beta ) q^{24} + q^{25} + 4 \beta q^{26} + ( -1 - \beta ) q^{27} + ( 2 - \beta ) q^{28} + ( 2 - 5 \beta ) q^{29} + ( -1 + \beta ) q^{30} + ( 8 - \beta ) q^{31} + q^{32} + ( -2 + 2 \beta ) q^{33} + ( -1 + 5 \beta ) q^{34} + ( 2 - \beta ) q^{35} -2 \beta q^{36} + ( 2 - 2 \beta ) q^{37} -3 q^{38} + ( 8 - 4 \beta ) q^{39} + q^{40} + 4 \beta q^{41} + ( -4 + 3 \beta ) q^{42} + ( 3 + 3 \beta ) q^{43} + 2 q^{44} -2 \beta q^{45} + ( -6 - 3 \beta ) q^{47} + ( -1 + \beta ) q^{48} + ( -1 - 4 \beta ) q^{49} + q^{50} + ( 11 - 6 \beta ) q^{51} + 4 \beta q^{52} + ( 8 + \beta ) q^{53} + ( -1 - \beta ) q^{54} + 2 q^{55} + ( 2 - \beta ) q^{56} + ( 3 - 3 \beta ) q^{57} + ( 2 - 5 \beta ) q^{58} + ( 1 - 6 \beta ) q^{59} + ( -1 + \beta ) q^{60} -4 \beta q^{61} + ( 8 - \beta ) q^{62} + ( 4 - 4 \beta ) q^{63} + q^{64} + 4 \beta q^{65} + ( -2 + 2 \beta ) q^{66} + ( -5 - 5 \beta ) q^{67} + ( -1 + 5 \beta ) q^{68} + ( 2 - \beta ) q^{70} -5 \beta q^{71} -2 \beta q^{72} + ( -5 + 5 \beta ) q^{73} + ( 2 - 2 \beta ) q^{74} + ( -1 + \beta ) q^{75} -3 q^{76} + ( 4 - 2 \beta ) q^{77} + ( 8 - 4 \beta ) q^{78} + ( 6 - \beta ) q^{79} + q^{80} + ( -1 + 6 \beta ) q^{81} + 4 \beta q^{82} + ( 5 + 7 \beta ) q^{83} + ( -4 + 3 \beta ) q^{84} + ( -1 + 5 \beta ) q^{85} + ( 3 + 3 \beta ) q^{86} + ( -12 + 7 \beta ) q^{87} + 2 q^{88} + ( 12 + 2 \beta ) q^{89} -2 \beta q^{90} + ( -8 + 8 \beta ) q^{91} + ( -10 + 9 \beta ) q^{93} + ( -6 - 3 \beta ) q^{94} -3 q^{95} + ( -1 + \beta ) q^{96} + ( 6 + 2 \beta ) q^{97} + ( -1 - 4 \beta ) q^{98} -4 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 4q^{7} + 2q^{8} + O(q^{10})$$ $$2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 4q^{7} + 2q^{8} + 2q^{10} + 4q^{11} - 2q^{12} + 4q^{14} - 2q^{15} + 2q^{16} - 2q^{17} - 6q^{19} + 2q^{20} - 8q^{21} + 4q^{22} - 2q^{24} + 2q^{25} - 2q^{27} + 4q^{28} + 4q^{29} - 2q^{30} + 16q^{31} + 2q^{32} - 4q^{33} - 2q^{34} + 4q^{35} + 4q^{37} - 6q^{38} + 16q^{39} + 2q^{40} - 8q^{42} + 6q^{43} + 4q^{44} - 12q^{47} - 2q^{48} - 2q^{49} + 2q^{50} + 22q^{51} + 16q^{53} - 2q^{54} + 4q^{55} + 4q^{56} + 6q^{57} + 4q^{58} + 2q^{59} - 2q^{60} + 16q^{62} + 8q^{63} + 2q^{64} - 4q^{66} - 10q^{67} - 2q^{68} + 4q^{70} - 10q^{73} + 4q^{74} - 2q^{75} - 6q^{76} + 8q^{77} + 16q^{78} + 12q^{79} + 2q^{80} - 2q^{81} + 10q^{83} - 8q^{84} - 2q^{85} + 6q^{86} - 24q^{87} + 4q^{88} + 24q^{89} - 16q^{91} - 20q^{93} - 12q^{94} - 6q^{95} - 2q^{96} + 12q^{97} - 2q^{98} + O(q^{100})$$

## 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
 −1.41421 1.41421
1.00000 −2.41421 1.00000 1.00000 −2.41421 3.41421 1.00000 2.82843 1.00000
1.2 1.00000 0.414214 1.00000 1.00000 0.414214 0.585786 1.00000 −2.82843 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.l yes 2
23.b odd 2 1 5290.2.a.k 2

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

 $$T_{3}^{2} + 2 T_{3} - 1$$ $$T_{7}^{2} - 4 T_{7} + 2$$ $$T_{11} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$-1 + 2 T + T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$2 - 4 T + T^{2}$$
$11$ $$( -2 + T )^{2}$$
$13$ $$-32 + T^{2}$$
$17$ $$-49 + 2 T + T^{2}$$
$19$ $$( 3 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$-46 - 4 T + T^{2}$$
$31$ $$62 - 16 T + T^{2}$$
$37$ $$-4 - 4 T + T^{2}$$
$41$ $$-32 + T^{2}$$
$43$ $$-9 - 6 T + T^{2}$$
$47$ $$18 + 12 T + T^{2}$$
$53$ $$62 - 16 T + T^{2}$$
$59$ $$-71 - 2 T + T^{2}$$
$61$ $$-32 + T^{2}$$
$67$ $$-25 + 10 T + T^{2}$$
$71$ $$-50 + T^{2}$$
$73$ $$-25 + 10 T + T^{2}$$
$79$ $$34 - 12 T + T^{2}$$
$83$ $$-73 - 10 T + T^{2}$$
$89$ $$136 - 24 T + T^{2}$$
$97$ $$28 - 12 T + T^{2}$$