# Properties

 Label 490.2.a.l Level $490$ Weight $2$ Character orbit 490.a Self dual yes Analytic conductor $3.913$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$490 = 2 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 490.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$3.91266969904$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + ( -2 + \beta ) q^{3} + q^{4} - q^{5} + ( 2 - \beta ) q^{6} - q^{8} + ( 3 - 4 \beta ) q^{9} +O(q^{10})$$ $$q - q^{2} + ( -2 + \beta ) q^{3} + q^{4} - q^{5} + ( 2 - \beta ) q^{6} - q^{8} + ( 3 - 4 \beta ) q^{9} + q^{10} + ( 2 + 2 \beta ) q^{11} + ( -2 + \beta ) q^{12} + ( 2 - 2 \beta ) q^{13} + ( 2 - \beta ) q^{15} + q^{16} + ( -4 - \beta ) q^{17} + ( -3 + 4 \beta ) q^{18} + ( -2 - \beta ) q^{19} - q^{20} + ( -2 - 2 \beta ) q^{22} + ( -4 - 2 \beta ) q^{23} + ( 2 - \beta ) q^{24} + q^{25} + ( -2 + 2 \beta ) q^{26} + ( -8 + 8 \beta ) q^{27} + ( -2 + 2 \beta ) q^{29} + ( -2 + \beta ) q^{30} -2 \beta q^{31} - q^{32} -2 \beta q^{33} + ( 4 + \beta ) q^{34} + ( 3 - 4 \beta ) q^{36} + ( -2 + 4 \beta ) q^{37} + ( 2 + \beta ) q^{38} + ( -8 + 6 \beta ) q^{39} + q^{40} + ( -4 - 5 \beta ) q^{41} + ( -6 + 2 \beta ) q^{43} + ( 2 + 2 \beta ) q^{44} + ( -3 + 4 \beta ) q^{45} + ( 4 + 2 \beta ) q^{46} + ( -8 - 2 \beta ) q^{47} + ( -2 + \beta ) q^{48} - q^{50} + ( 6 - 2 \beta ) q^{51} + ( 2 - 2 \beta ) q^{52} + ( -2 - 6 \beta ) q^{53} + ( 8 - 8 \beta ) q^{54} + ( -2 - 2 \beta ) q^{55} + 2 q^{57} + ( 2 - 2 \beta ) q^{58} + ( -10 - \beta ) q^{59} + ( 2 - \beta ) q^{60} + ( 2 + 8 \beta ) q^{61} + 2 \beta q^{62} + q^{64} + ( -2 + 2 \beta ) q^{65} + 2 \beta q^{66} + ( -4 - 4 \beta ) q^{67} + ( -4 - \beta ) q^{68} + 4 q^{69} + ( 4 + 6 \beta ) q^{71} + ( -3 + 4 \beta ) q^{72} + ( 8 - \beta ) q^{73} + ( 2 - 4 \beta ) q^{74} + ( -2 + \beta ) q^{75} + ( -2 - \beta ) q^{76} + ( 8 - 6 \beta ) q^{78} + ( -4 + 2 \beta ) q^{79} - q^{80} + ( 23 - 12 \beta ) q^{81} + ( 4 + 5 \beta ) q^{82} + ( -2 - 3 \beta ) q^{83} + ( 4 + \beta ) q^{85} + ( 6 - 2 \beta ) q^{86} + ( 8 - 6 \beta ) q^{87} + ( -2 - 2 \beta ) q^{88} + 9 \beta q^{89} + ( 3 - 4 \beta ) q^{90} + ( -4 - 2 \beta ) q^{92} + ( -4 + 4 \beta ) q^{93} + ( 8 + 2 \beta ) q^{94} + ( 2 + \beta ) q^{95} + ( 2 - \beta ) q^{96} + ( 12 + 3 \beta ) q^{97} + ( -10 - 2 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 4q^{3} + 2q^{4} - 2q^{5} + 4q^{6} - 2q^{8} + 6q^{9} + O(q^{10})$$ $$2q - 2q^{2} - 4q^{3} + 2q^{4} - 2q^{5} + 4q^{6} - 2q^{8} + 6q^{9} + 2q^{10} + 4q^{11} - 4q^{12} + 4q^{13} + 4q^{15} + 2q^{16} - 8q^{17} - 6q^{18} - 4q^{19} - 2q^{20} - 4q^{22} - 8q^{23} + 4q^{24} + 2q^{25} - 4q^{26} - 16q^{27} - 4q^{29} - 4q^{30} - 2q^{32} + 8q^{34} + 6q^{36} - 4q^{37} + 4q^{38} - 16q^{39} + 2q^{40} - 8q^{41} - 12q^{43} + 4q^{44} - 6q^{45} + 8q^{46} - 16q^{47} - 4q^{48} - 2q^{50} + 12q^{51} + 4q^{52} - 4q^{53} + 16q^{54} - 4q^{55} + 4q^{57} + 4q^{58} - 20q^{59} + 4q^{60} + 4q^{61} + 2q^{64} - 4q^{65} - 8q^{67} - 8q^{68} + 8q^{69} + 8q^{71} - 6q^{72} + 16q^{73} + 4q^{74} - 4q^{75} - 4q^{76} + 16q^{78} - 8q^{79} - 2q^{80} + 46q^{81} + 8q^{82} - 4q^{83} + 8q^{85} + 12q^{86} + 16q^{87} - 4q^{88} + 6q^{90} - 8q^{92} - 8q^{93} + 16q^{94} + 4q^{95} + 4q^{96} + 24q^{97} - 20q^{99} + O(q^{100})$$

## 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
 −1.41421 1.41421
−1.00000 −3.41421 1.00000 −1.00000 3.41421 0 −1.00000 8.65685 1.00000
1.2 −1.00000 −0.585786 1.00000 −1.00000 0.585786 0 −1.00000 −2.65685 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
$$2$$ $$1$$
$$5$$ $$1$$
$$7$$ $$1$$

## 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 490.2.a.l 2
3.b odd 2 1 4410.2.a.by 2
4.b odd 2 1 3920.2.a.ca 2
5.b even 2 1 2450.2.a.bs 2
5.c odd 4 2 2450.2.c.w 4
7.b odd 2 1 490.2.a.m yes 2
7.c even 3 2 490.2.e.j 4
7.d odd 6 2 490.2.e.i 4
21.c even 2 1 4410.2.a.bt 2
28.d even 2 1 3920.2.a.bm 2
35.c odd 2 1 2450.2.a.bn 2
35.f even 4 2 2450.2.c.t 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.2.a.l 2 1.a even 1 1 trivial
490.2.a.m yes 2 7.b odd 2 1
490.2.e.i 4 7.d odd 6 2
490.2.e.j 4 7.c even 3 2
2450.2.a.bn 2 35.c odd 2 1
2450.2.a.bs 2 5.b even 2 1
2450.2.c.t 4 35.f even 4 2
2450.2.c.w 4 5.c odd 4 2
3920.2.a.bm 2 28.d even 2 1
3920.2.a.ca 2 4.b odd 2 1
4410.2.a.bt 2 21.c even 2 1
4410.2.a.by 2 3.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(490))$$:

 $$T_{3}^{2} + 4 T_{3} + 2$$ $$T_{11}^{2} - 4 T_{11} - 4$$ $$T_{13}^{2} - 4 T_{13} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$2 + 4 T + T^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$T^{2}$$
$11$ $$-4 - 4 T + T^{2}$$
$13$ $$-4 - 4 T + T^{2}$$
$17$ $$14 + 8 T + T^{2}$$
$19$ $$2 + 4 T + T^{2}$$
$23$ $$8 + 8 T + T^{2}$$
$29$ $$-4 + 4 T + T^{2}$$
$31$ $$-8 + T^{2}$$
$37$ $$-28 + 4 T + T^{2}$$
$41$ $$-34 + 8 T + T^{2}$$
$43$ $$28 + 12 T + T^{2}$$
$47$ $$56 + 16 T + T^{2}$$
$53$ $$-68 + 4 T + T^{2}$$
$59$ $$98 + 20 T + T^{2}$$
$61$ $$-124 - 4 T + T^{2}$$
$67$ $$-16 + 8 T + T^{2}$$
$71$ $$-56 - 8 T + T^{2}$$
$73$ $$62 - 16 T + T^{2}$$
$79$ $$8 + 8 T + T^{2}$$
$83$ $$-14 + 4 T + T^{2}$$
$89$ $$-162 + T^{2}$$
$97$ $$126 - 24 T + T^{2}$$
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