# Properties

 Label 5290.2.a.v Level $5290$ Weight $2$ Character orbit 5290.a Self dual yes Analytic conductor $42.241$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(\zeta_{24})^+$$ Defining polynomial: $$x^{4} - 4 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.93185 0.517638 −0.517638 1.93185
−1.00000 −0.414214 1.00000 1.00000 0.414214 −1.71744 −1.00000 −2.82843 −1.00000
1.2 −1.00000 −0.414214 1.00000 1.00000 0.414214 −0.282561 −1.00000 −2.82843 −1.00000
1.3 −1.00000 2.41421 1.00000 1.00000 −2.41421 −5.18154 −1.00000 2.82843 −1.00000
1.4 −1.00000 2.41421 1.00000 1.00000 −2.41421 3.18154 −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.v yes 4
23.b odd 2 1 5290.2.a.u 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5290.2.a.u 4 23.b odd 2 1
5290.2.a.v yes 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5290))$$:

 $$T_{3}^{2} - 2 T_{3} - 1$$ $$T_{7}^{4} + 4 T_{7}^{3} - 12 T_{7}^{2} - 32 T_{7} - 8$$ $$T_{11}^{4} - 28 T_{11}^{2} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{4}$$
$3$ $$( -1 - 2 T + T^{2} )^{2}$$
$5$ $$( -1 + T )^{4}$$
$7$ $$-8 - 32 T - 12 T^{2} + 4 T^{3} + T^{4}$$
$11$ $$4 - 28 T^{2} + T^{4}$$
$13$ $$-32 - 32 T + 8 T^{2} + 8 T^{3} + T^{4}$$
$17$ $$-47 - 36 T + 44 T^{2} - 12 T^{3} + T^{4}$$
$19$ $$49 - 56 T - 4 T^{2} + 8 T^{3} + T^{4}$$
$23$ $$T^{4}$$
$29$ $$-8 + 20 T^{2} + 12 T^{3} + T^{4}$$
$31$ $$1912 - 1232 T + 284 T^{2} - 28 T^{3} + T^{4}$$
$37$ $$( -4 - 4 T + T^{2} )^{2}$$
$41$ $$-1472 - 704 T - 64 T^{2} + 8 T^{3} + T^{4}$$
$43$ $$-47 + 36 T + 44 T^{2} + 12 T^{3} + T^{4}$$
$47$ $$-8 + 32 T - 28 T^{2} + 4 T^{3} + T^{4}$$
$53$ $$-968 + 176 T + 92 T^{2} - 20 T^{3} + T^{4}$$
$59$ $$( -3 + T )^{4}$$
$61$ $$1600 - 112 T^{2} + T^{4}$$
$67$ $$-5903 + 2796 T - 196 T^{2} - 12 T^{3} + T^{4}$$
$71$ $$-2312 + 1088 T + 20 T^{2} - 20 T^{3} + T^{4}$$
$73$ $$1081 - 496 T - 34 T^{2} + 16 T^{3} + T^{4}$$
$79$ $$13432 - 1360 T - 340 T^{2} + 4 T^{3} + T^{4}$$
$83$ $$81 + 108 T - 36 T^{2} - 12 T^{3} + T^{4}$$
$89$ $$4036 - 2800 T + 540 T^{2} - 40 T^{3} + T^{4}$$
$97$ $$2692 + 2272 T - 148 T^{2} - 16 T^{3} + T^{4}$$