# Properties

 Label 640.2.a.k Level $640$ Weight $2$ Character orbit 640.a Self dual yes Analytic conductor $5.110$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$640 = 2^{7} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 640.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$5.11042572936$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{3} - q^{5} + ( 1 - \beta ) q^{7} + ( 3 + 2 \beta ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{3} - q^{5} + ( 1 - \beta ) q^{7} + ( 3 + 2 \beta ) q^{9} + 2 q^{11} + 2 \beta q^{13} + ( -1 - \beta ) q^{15} -2 \beta q^{17} -2 \beta q^{19} -4 q^{21} + ( 7 + \beta ) q^{23} + q^{25} + ( 10 + 2 \beta ) q^{27} -2 q^{29} + ( -2 + 2 \beta ) q^{31} + ( 2 + 2 \beta ) q^{33} + ( -1 + \beta ) q^{35} + ( -2 - 4 \beta ) q^{37} + ( 10 + 2 \beta ) q^{39} + ( 8 - 2 \beta ) q^{41} + ( 1 - 3 \beta ) q^{43} + ( -3 - 2 \beta ) q^{45} + ( 5 - \beta ) q^{47} + ( -1 - 2 \beta ) q^{49} + ( -10 - 2 \beta ) q^{51} + ( -4 - 2 \beta ) q^{53} -2 q^{55} + ( -10 - 2 \beta ) q^{57} + ( 4 - 2 \beta ) q^{59} -6 q^{61} + ( -7 - \beta ) q^{63} -2 \beta q^{65} + ( -1 + 3 \beta ) q^{67} + ( 12 + 8 \beta ) q^{69} + ( -2 - 2 \beta ) q^{71} -2 \beta q^{73} + ( 1 + \beta ) q^{75} + ( 2 - 2 \beta ) q^{77} + ( -4 + 4 \beta ) q^{79} + ( 11 + 6 \beta ) q^{81} + ( -3 - 3 \beta ) q^{83} + 2 \beta q^{85} + ( -2 - 2 \beta ) q^{87} + ( -6 + 4 \beta ) q^{89} + ( -10 + 2 \beta ) q^{91} + 8 q^{93} + 2 \beta q^{95} + ( -12 + 2 \beta ) q^{97} + ( 6 + 4 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 6 q^{9} + O(q^{10})$$ $$2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 6 q^{9} + 4 q^{11} - 2 q^{15} - 8 q^{21} + 14 q^{23} + 2 q^{25} + 20 q^{27} - 4 q^{29} - 4 q^{31} + 4 q^{33} - 2 q^{35} - 4 q^{37} + 20 q^{39} + 16 q^{41} + 2 q^{43} - 6 q^{45} + 10 q^{47} - 2 q^{49} - 20 q^{51} - 8 q^{53} - 4 q^{55} - 20 q^{57} + 8 q^{59} - 12 q^{61} - 14 q^{63} - 2 q^{67} + 24 q^{69} - 4 q^{71} + 2 q^{75} + 4 q^{77} - 8 q^{79} + 22 q^{81} - 6 q^{83} - 4 q^{87} - 12 q^{89} - 20 q^{91} + 16 q^{93} - 24 q^{97} + 12 q^{99} + O(q^{100})$$

## 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
 −0.618034 1.61803
0 −1.23607 0 −1.00000 0 3.23607 0 −1.47214 0
1.2 0 3.23607 0 −1.00000 0 −1.23607 0 7.47214 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$1$$

## 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 640.2.a.k yes 2
3.b odd 2 1 5760.2.a.ci 2
4.b odd 2 1 640.2.a.i 2
5.b even 2 1 3200.2.a.be 2
5.c odd 4 2 3200.2.c.w 4
8.b even 2 1 640.2.a.j yes 2
8.d odd 2 1 640.2.a.l yes 2
12.b even 2 1 5760.2.a.ch 2
16.e even 4 2 1280.2.d.k 4
16.f odd 4 2 1280.2.d.m 4
20.d odd 2 1 3200.2.a.bl 2
20.e even 4 2 3200.2.c.u 4
24.f even 2 1 5760.2.a.bw 2
24.h odd 2 1 5760.2.a.cd 2
40.e odd 2 1 3200.2.a.bf 2
40.f even 2 1 3200.2.a.bk 2
40.i odd 4 2 3200.2.c.v 4
40.k even 4 2 3200.2.c.x 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.a.i 2 4.b odd 2 1
640.2.a.j yes 2 8.b even 2 1
640.2.a.k yes 2 1.a even 1 1 trivial
640.2.a.l yes 2 8.d odd 2 1
1280.2.d.k 4 16.e even 4 2
1280.2.d.m 4 16.f odd 4 2
3200.2.a.be 2 5.b even 2 1
3200.2.a.bf 2 40.e odd 2 1
3200.2.a.bk 2 40.f even 2 1
3200.2.a.bl 2 20.d odd 2 1
3200.2.c.u 4 20.e even 4 2
3200.2.c.v 4 40.i odd 4 2
3200.2.c.w 4 5.c odd 4 2
3200.2.c.x 4 40.k even 4 2
5760.2.a.bw 2 24.f even 2 1
5760.2.a.cd 2 24.h odd 2 1
5760.2.a.ch 2 12.b even 2 1
5760.2.a.ci 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(640))$$:

 $$T_{3}^{2} - 2 T_{3} - 4$$ $$T_{7}^{2} - 2 T_{7} - 4$$ $$T_{11} - 2$$ $$T_{13}^{2} - 20$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-4 - 2 T + T^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$-4 - 2 T + T^{2}$$
$11$ $$( -2 + T )^{2}$$
$13$ $$-20 + T^{2}$$
$17$ $$-20 + T^{2}$$
$19$ $$-20 + T^{2}$$
$23$ $$44 - 14 T + T^{2}$$
$29$ $$( 2 + T )^{2}$$
$31$ $$-16 + 4 T + T^{2}$$
$37$ $$-76 + 4 T + T^{2}$$
$41$ $$44 - 16 T + T^{2}$$
$43$ $$-44 - 2 T + T^{2}$$
$47$ $$20 - 10 T + T^{2}$$
$53$ $$-4 + 8 T + T^{2}$$
$59$ $$-4 - 8 T + T^{2}$$
$61$ $$( 6 + T )^{2}$$
$67$ $$-44 + 2 T + T^{2}$$
$71$ $$-16 + 4 T + T^{2}$$
$73$ $$-20 + T^{2}$$
$79$ $$-64 + 8 T + T^{2}$$
$83$ $$-36 + 6 T + T^{2}$$
$89$ $$-44 + 12 T + T^{2}$$
$97$ $$124 + 24 T + T^{2}$$