# Properties

 Label 490.2.a.g Level $490$ Weight $2$ Character orbit 490.a Self dual yes Analytic conductor $3.913$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} - 2q^{3} + q^{4} - q^{5} - 2q^{6} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} - 2q^{3} + q^{4} - q^{5} - 2q^{6} + q^{8} + q^{9} - q^{10} + 3q^{11} - 2q^{12} + 5q^{13} + 2q^{15} + q^{16} + 6q^{17} + q^{18} - q^{19} - q^{20} + 3q^{22} + 3q^{23} - 2q^{24} + q^{25} + 5q^{26} + 4q^{27} - 6q^{29} + 2q^{30} - 4q^{31} + q^{32} - 6q^{33} + 6q^{34} + q^{36} + 11q^{37} - q^{38} - 10q^{39} - q^{40} + 3q^{41} - 10q^{43} + 3q^{44} - q^{45} + 3q^{46} + 3q^{47} - 2q^{48} + q^{50} - 12q^{51} + 5q^{52} + 3q^{53} + 4q^{54} - 3q^{55} + 2q^{57} - 6q^{58} + 2q^{60} - 4q^{61} - 4q^{62} + q^{64} - 5q^{65} - 6q^{66} - 4q^{67} + 6q^{68} - 6q^{69} + 12q^{71} + q^{72} - 4q^{73} + 11q^{74} - 2q^{75} - q^{76} - 10q^{78} - 10q^{79} - q^{80} - 11q^{81} + 3q^{82} - 12q^{83} - 6q^{85} - 10q^{86} + 12q^{87} + 3q^{88} + 6q^{89} - q^{90} + 3q^{92} + 8q^{93} + 3q^{94} + q^{95} - 2q^{96} + 14q^{97} + 3q^{99} + O(q^{100})$$

## 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
 0
1.00000 −2.00000 1.00000 −1.00000 −2.00000 0 1.00000 1.00000 −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.g 1
3.b odd 2 1 4410.2.a.m 1
4.b odd 2 1 3920.2.a.be 1
5.b even 2 1 2450.2.a.p 1
5.c odd 4 2 2450.2.c.f 2
7.b odd 2 1 490.2.a.j 1
7.c even 3 2 70.2.e.b 2
7.d odd 6 2 490.2.e.a 2
21.c even 2 1 4410.2.a.c 1
21.h odd 6 2 630.2.k.e 2
28.d even 2 1 3920.2.a.g 1
28.g odd 6 2 560.2.q.d 2
35.c odd 2 1 2450.2.a.f 1
35.f even 4 2 2450.2.c.p 2
35.j even 6 2 350.2.e.h 2
35.l odd 12 4 350.2.j.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.b 2 7.c even 3 2
350.2.e.h 2 35.j even 6 2
350.2.j.a 4 35.l odd 12 4
490.2.a.g 1 1.a even 1 1 trivial
490.2.a.j 1 7.b odd 2 1
490.2.e.a 2 7.d odd 6 2
560.2.q.d 2 28.g odd 6 2
630.2.k.e 2 21.h odd 6 2
2450.2.a.f 1 35.c odd 2 1
2450.2.a.p 1 5.b even 2 1
2450.2.c.f 2 5.c odd 4 2
2450.2.c.p 2 35.f even 4 2
3920.2.a.g 1 28.d even 2 1
3920.2.a.be 1 4.b odd 2 1
4410.2.a.c 1 21.c even 2 1
4410.2.a.m 1 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(490))$$:

 $$T_{3} + 2$$ $$T_{11} - 3$$ $$T_{13} - 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$2 + T$$
$5$ $$1 + T$$
$7$ $$T$$
$11$ $$-3 + T$$
$13$ $$-5 + T$$
$17$ $$-6 + T$$
$19$ $$1 + T$$
$23$ $$-3 + T$$
$29$ $$6 + T$$
$31$ $$4 + T$$
$37$ $$-11 + T$$
$41$ $$-3 + T$$
$43$ $$10 + T$$
$47$ $$-3 + T$$
$53$ $$-3 + T$$
$59$ $$T$$
$61$ $$4 + T$$
$67$ $$4 + T$$
$71$ $$-12 + T$$
$73$ $$4 + T$$
$79$ $$10 + T$$
$83$ $$12 + T$$
$89$ $$-6 + T$$
$97$ $$-14 + T$$