# Properties

 Label 5290.2.a.r Level $5290$ Weight $2$ Character orbit 5290.a Self dual yes Analytic conductor $42.241$ Analytic rank $0$ 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: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.1101.1 Defining polynomial: $$x^{3} - x^{2} - 9 x + 12$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 9 x + 12$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu - 7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - \beta_{1} + 7$$

## 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
 −3.11903 1.43163 2.68740
1.00000 −3.11903 1.00000 1.00000 −3.11903 −4.50973 1.00000 6.72833 1.00000
1.2 1.00000 1.43163 1.00000 1.00000 1.43163 −3.08719 1.00000 −0.950444 1.00000
1.3 1.00000 2.68740 1.00000 1.00000 2.68740 4.59692 1.00000 4.22212 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.r 3
23.b odd 2 1 230.2.a.d 3
69.c even 2 1 2070.2.a.z 3
92.b even 2 1 1840.2.a.r 3
115.c odd 2 1 1150.2.a.q 3
115.e even 4 2 1150.2.b.j 6
184.e odd 2 1 7360.2.a.bz 3
184.h even 2 1 7360.2.a.ce 3
460.g even 2 1 9200.2.a.cf 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.d 3 23.b odd 2 1
1150.2.a.q 3 115.c odd 2 1
1150.2.b.j 6 115.e even 4 2
1840.2.a.r 3 92.b even 2 1
2070.2.a.z 3 69.c even 2 1
5290.2.a.r 3 1.a even 1 1 trivial
7360.2.a.bz 3 184.e odd 2 1
7360.2.a.ce 3 184.h even 2 1
9200.2.a.cf 3 460.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5290))$$:

 $$T_{3}^{3} - T_{3}^{2} - 9 T_{3} + 12$$ $$T_{7}^{3} + 3 T_{7}^{2} - 21 T_{7} - 64$$ $$T_{11}^{3} + 3 T_{11}^{2} - 39 T_{11} - 144$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{3}$$
$3$ $$12 - 9 T - T^{2} + T^{3}$$
$5$ $$( -1 + T )^{3}$$
$7$ $$-64 - 21 T + 3 T^{2} + T^{3}$$
$11$ $$-144 - 39 T + 3 T^{2} + T^{3}$$
$13$ $$-18 - 15 T + T^{2} + T^{3}$$
$17$ $$18 + 7 T - 7 T^{2} + T^{3}$$
$19$ $$-64 - 21 T + 3 T^{2} + T^{3}$$
$23$ $$T^{3}$$
$29$ $$24 - 32 T + 4 T^{2} + T^{3}$$
$31$ $$-8 - 7 T + 5 T^{2} + T^{3}$$
$37$ $$32 - 40 T - 2 T^{2} + T^{3}$$
$41$ $$186 - 59 T - T^{2} + T^{3}$$
$43$ $$( 8 + T )^{3}$$
$47$ $$-288 + 4 T + 14 T^{2} + T^{3}$$
$53$ $$( -6 + T )^{3}$$
$59$ $$144 + 28 T - 14 T^{2} + T^{3}$$
$61$ $$-526 - 157 T + T^{2} + T^{3}$$
$67$ $$-384 - 144 T + 8 T^{2} + T^{3}$$
$71$ $$-24 + 31 T - 11 T^{2} + T^{3}$$
$73$ $$-248 - 40 T + 8 T^{2} + T^{3}$$
$79$ $$1152 - 240 T - 4 T^{2} + T^{3}$$
$83$ $$-96 - 20 T + 8 T^{2} + T^{3}$$
$89$ $$-1152 - 48 T + 18 T^{2} + T^{3}$$
$97$ $$-166 + 279 T - 33 T^{2} + T^{3}$$