# Properties

 Label 5290.2.a.p Level $5290$ Weight $2$ Character orbit 5290.a Self dual yes Analytic conductor $42.241$ Analytic rank $1$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.1509.1 Defining polynomial: $$x^{3} - x^{2} - 7 x + 4$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 7 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 5$$

## 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
 2.92542 0.551929 −2.47735
1.00000 −2.92542 1.00000 −1.00000 −2.92542 −4.55810 1.00000 5.55810 −1.00000
1.2 1.00000 −0.551929 1.00000 −1.00000 −0.551929 3.69537 1.00000 −2.69537 −1.00000
1.3 1.00000 2.47735 1.00000 −1.00000 2.47735 −2.13727 1.00000 3.13727 −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.p 3
23.b odd 2 1 5290.2.a.q yes 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5290.2.a.p 3 1.a even 1 1 trivial
5290.2.a.q yes 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5290))$$:

 $$T_{3}^{3} + T_{3}^{2} - 7 T_{3} - 4$$ $$T_{7}^{3} + 3 T_{7}^{2} - 15 T_{7} - 36$$ $$T_{11}^{3} + 5 T_{11}^{2} + T_{11} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{3}$$
$3$ $$-4 - 7 T + T^{2} + T^{3}$$
$5$ $$( 1 + T )^{3}$$
$7$ $$-36 - 15 T + 3 T^{2} + T^{3}$$
$11$ $$-6 + T + 5 T^{2} + T^{3}$$
$13$ $$2 - 15 T + 3 T^{2} + T^{3}$$
$17$ $$48 + 49 T + 13 T^{2} + T^{3}$$
$19$ $$114 - 29 T - 5 T^{2} + T^{3}$$
$23$ $$T^{3}$$
$29$ $$-96 - 80 T - 2 T^{2} + T^{3}$$
$31$ $$-76 - 49 T - 5 T^{2} + T^{3}$$
$37$ $$-24 - 116 T - 2 T^{2} + T^{3}$$
$41$ $$6 + T - 5 T^{2} + T^{3}$$
$43$ $$96 - 80 T + 2 T^{2} + T^{3}$$
$47$ $$-528 - 116 T + 4 T^{2} + T^{3}$$
$53$ $$-72 - 24 T + 12 T^{2} + T^{3}$$
$59$ $$-432 + 4 T + 16 T^{2} + T^{3}$$
$61$ $$-1278 - 141 T + 9 T^{2} + T^{3}$$
$67$ $$72 - 8 T - 8 T^{2} + T^{3}$$
$71$ $$48 + 49 T + 13 T^{2} + T^{3}$$
$73$ $$-352 + 80 T + 22 T^{2} + T^{3}$$
$79$ $$-528 - 116 T + 4 T^{2} + T^{3}$$
$83$ $$-72 - 128 T - 8 T^{2} + T^{3}$$
$89$ $$-288 - 60 T + 6 T^{2} + T^{3}$$
$97$ $$-648 + 297 T - 33 T^{2} + T^{3}$$