# Properties

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

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

## 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 −1.93185 1.00000 1.00000 −1.93185 −0.482362 1.00000 0.732051 1.00000
1.2 1.00000 −0.517638 1.00000 1.00000 −0.517638 0.931852 1.00000 −2.73205 1.00000
1.3 1.00000 0.517638 1.00000 1.00000 0.517638 −2.93185 1.00000 −2.73205 1.00000
1.4 1.00000 1.93185 1.00000 1.00000 1.93185 −1.51764 1.00000 0.732051 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.z yes 4
23.b odd 2 1 5290.2.a.y 4

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{4}$$
$3$ $$1 - 4 T^{2} + T^{4}$$
$5$ $$( -1 + T )^{4}$$
$7$ $$-2 - 4 T + 2 T^{2} + 4 T^{3} + T^{4}$$
$11$ $$-32 - 32 T + 8 T^{2} + 8 T^{3} + T^{4}$$
$13$ $$-200 - 160 T - 28 T^{2} + 4 T^{3} + T^{4}$$
$17$ $$( -5 + 2 T + T^{2} )^{2}$$
$19$ $$-23 - 28 T - 4 T^{2} + 4 T^{3} + T^{4}$$
$23$ $$T^{4}$$
$29$ $$-50 + 100 T - 46 T^{2} - 4 T^{3} + T^{4}$$
$31$ $$-2 - 4 T + 6 T^{2} + 8 T^{3} + T^{4}$$
$37$ $$-3056 - 832 T + 8 T^{2} + 16 T^{3} + T^{4}$$
$41$ $$4 + 8 T - 16 T^{2} + 4 T^{3} + T^{4}$$
$43$ $$-239 - 276 T - 10 T^{2} + 12 T^{3} + T^{4}$$
$47$ $$-386 - 244 T - 18 T^{2} + 8 T^{3} + T^{4}$$
$53$ $$46 + 60 T - 34 T^{2} + T^{4}$$
$59$ $$-2087 - 1300 T - 166 T^{2} + 4 T^{3} + T^{4}$$
$61$ $$184 + 128 T - 76 T^{2} + 4 T^{3} + T^{4}$$
$67$ $$-743 + 576 T - 118 T^{2} + T^{4}$$
$71$ $$-6434 + 908 T + 162 T^{2} - 28 T^{3} + T^{4}$$
$73$ $$1225 - 1260 T - 124 T^{2} + 12 T^{3} + T^{4}$$
$79$ $$8878 + 4180 T + 674 T^{2} + 44 T^{3} + T^{4}$$
$83$ $$-7751 + 1628 T + 14 T^{2} - 20 T^{3} + T^{4}$$
$89$ $$-7664 + 800 T + 464 T^{2} + 40 T^{3} + T^{4}$$
$97$ $$10468 - 2936 T - 160 T^{2} + 20 T^{3} + T^{4}$$