# Properties

 Label 5290.2.a.j 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: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.30278 2.30278
−1.00000 −0.302776 1.00000 −1.00000 0.302776 −3.30278 −1.00000 −2.90833 1.00000
1.2 −1.00000 3.30278 1.00000 −1.00000 −3.30278 0.302776 −1.00000 7.90833 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.j 2
23.b odd 2 1 230.2.a.b 2
69.c even 2 1 2070.2.a.w 2
92.b even 2 1 1840.2.a.j 2
115.c odd 2 1 1150.2.a.m 2
115.e even 4 2 1150.2.b.f 4
184.e odd 2 1 7360.2.a.bc 2
184.h even 2 1 7360.2.a.bu 2
460.g even 2 1 9200.2.a.ca 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.b 2 23.b odd 2 1
1150.2.a.m 2 115.c odd 2 1
1150.2.b.f 4 115.e even 4 2
1840.2.a.j 2 92.b even 2 1
2070.2.a.w 2 69.c even 2 1
5290.2.a.j 2 1.a even 1 1 trivial
7360.2.a.bc 2 184.e odd 2 1
7360.2.a.bu 2 184.h even 2 1
9200.2.a.ca 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5290))$$:

 $$T_{3}^{2} - 3 T_{3} - 1$$ $$T_{7}^{2} + 3 T_{7} - 1$$ $$T_{11}^{2} - 7 T_{11} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$-1 - 3 T + T^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$-1 + 3 T + T^{2}$$
$11$ $$9 - 7 T + T^{2}$$
$13$ $$-1 - 3 T + T^{2}$$
$17$ $$-27 + 3 T + T^{2}$$
$19$ $$-29 + T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$-12 - 2 T + T^{2}$$
$31$ $$-23 + 5 T + T^{2}$$
$37$ $$( 8 + T )^{2}$$
$41$ $$-9 + 9 T + T^{2}$$
$43$ $$-48 - 4 T + T^{2}$$
$47$ $$-12 - 2 T + T^{2}$$
$53$ $$-36 - 8 T + T^{2}$$
$59$ $$36 + 14 T + T^{2}$$
$61$ $$-75 + 5 T + T^{2}$$
$67$ $$( -4 + T )^{2}$$
$71$ $$207 + 29 T + T^{2}$$
$73$ $$-92 - 10 T + T^{2}$$
$79$ $$-208 + T^{2}$$
$83$ $$-36 + 8 T + T^{2}$$
$89$ $$T^{2}$$
$97$ $$17 + 9 T + T^{2}$$