# Properties

 Label 5290.2.a.e 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: $$\Q(\sqrt{21})$$ Defining polynomial: $$x^{2} - x - 5$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 230) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{21})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 2.79129 −1.79129
−1.00000 −2.79129 1.00000 1.00000 2.79129 1.79129 −1.00000 4.79129 −1.00000
1.2 −1.00000 1.79129 1.00000 1.00000 −1.79129 −2.79129 −1.00000 0.208712 −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.e 2
23.b odd 2 1 230.2.a.a 2
69.c even 2 1 2070.2.a.x 2
92.b even 2 1 1840.2.a.n 2
115.c odd 2 1 1150.2.a.o 2
115.e even 4 2 1150.2.b.g 4
184.e odd 2 1 7360.2.a.bq 2
184.h even 2 1 7360.2.a.bk 2
460.g even 2 1 9200.2.a.bs 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.a 2 23.b odd 2 1
1150.2.a.o 2 115.c odd 2 1
1150.2.b.g 4 115.e even 4 2
1840.2.a.n 2 92.b even 2 1
2070.2.a.x 2 69.c even 2 1
5290.2.a.e 2 1.a even 1 1 trivial
7360.2.a.bk 2 184.h even 2 1
7360.2.a.bq 2 184.e odd 2 1
9200.2.a.bs 2 460.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5290))$$:

 $$T_{3}^{2} + T_{3} - 5$$ $$T_{7}^{2} + T_{7} - 5$$ $$T_{11}^{2} + 3 T_{11} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$-5 + T + T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$-5 + T + T^{2}$$
$11$ $$-3 + 3 T + T^{2}$$
$13$ $$7 - 7 T + T^{2}$$
$17$ $$-3 - 3 T + T^{2}$$
$19$ $$7 + 7 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$-12 - 6 T + T^{2}$$
$31$ $$-35 - 7 T + T^{2}$$
$37$ $$( -4 + T )^{2}$$
$41$ $$15 + 9 T + T^{2}$$
$43$ $$-80 + 4 T + T^{2}$$
$47$ $$60 + 18 T + T^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$60 + 18 T + T^{2}$$
$61$ $$-35 + 7 T + T^{2}$$
$67$ $$-80 + 4 T + T^{2}$$
$71$ $$-45 - 3 T + T^{2}$$
$73$ $$-188 + 2 T + T^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$( -6 + T )^{2}$$
$89$ $$-48 + 12 T + T^{2}$$
$97$ $$-119 + 7 T + T^{2}$$