# Properties

 Label 490.2.a.m Level $490$ Weight $2$ Character orbit 490.a Self dual yes Analytic conductor $3.913$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ 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.

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

## 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 0.585786 1.00000 1.00000 −0.585786 0 −1.00000 −2.65685 −1.00000
1.2 −1.00000 3.41421 1.00000 1.00000 −3.41421 0 −1.00000 8.65685 −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.m yes 2
3.b odd 2 1 4410.2.a.bt 2
4.b odd 2 1 3920.2.a.bm 2
5.b even 2 1 2450.2.a.bn 2
5.c odd 4 2 2450.2.c.t 4
7.b odd 2 1 490.2.a.l 2
7.c even 3 2 490.2.e.i 4
7.d odd 6 2 490.2.e.j 4
21.c even 2 1 4410.2.a.by 2
28.d even 2 1 3920.2.a.ca 2
35.c odd 2 1 2450.2.a.bs 2
35.f even 4 2 2450.2.c.w 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.2.a.l 2 7.b odd 2 1
490.2.a.m yes 2 1.a even 1 1 trivial
490.2.e.i 4 7.c even 3 2
490.2.e.j 4 7.d odd 6 2
2450.2.a.bn 2 5.b even 2 1
2450.2.a.bs 2 35.c odd 2 1
2450.2.c.t 4 5.c odd 4 2
2450.2.c.w 4 35.f even 4 2
3920.2.a.bm 2 4.b odd 2 1
3920.2.a.ca 2 28.d even 2 1
4410.2.a.bt 2 3.b odd 2 1
4410.2.a.by 2 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(490))$$:

 $$T_{3}^{2} - 4T_{3} + 2$$ T3^2 - 4*T3 + 2 $$T_{11}^{2} - 4T_{11} - 4$$ T11^2 - 4*T11 - 4 $$T_{13}^{2} + 4T_{13} - 4$$ T13^2 + 4*T13 - 4

## Hecke characteristic polynomials

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