# Properties

 Label 640.2.o.b Level $640$ Weight $2$ Character orbit 640.o Analytic conductor $5.110$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [640,2,Mod(63,640)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(640, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 2, 3]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("640.63");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$640 = 2^{7} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 640.o (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$5.11042572936$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (2 i - 2) q^{3} + ( - i + 2) q^{5} + ( - 2 i + 2) q^{7} - 5 i q^{9} +O(q^{10})$$ q + (2*i - 2) * q^3 + (-i + 2) * q^5 + (-2*i + 2) * q^7 - 5*i * q^9 $$q + (2 i - 2) q^{3} + ( - i + 2) q^{5} + ( - 2 i + 2) q^{7} - 5 i q^{9} - 4 q^{11} + ( - 3 i - 3) q^{13} + (6 i - 2) q^{15} + ( - 3 i - 3) q^{17} + 8 i q^{21} + ( - 6 i - 6) q^{23} + ( - 4 i + 3) q^{25} + (4 i + 4) q^{27} + 2 q^{29} + 4 i q^{31} + ( - 8 i + 8) q^{33} + ( - 6 i + 2) q^{35} + (3 i - 3) q^{37} + 12 q^{39} + ( - 6 i + 6) q^{43} + ( - 10 i - 5) q^{45} + ( - 6 i + 6) q^{47} - i q^{49} + 12 q^{51} + (3 i + 3) q^{53} + (4 i - 8) q^{55} - 8 i q^{59} - 6 i q^{61} + ( - 10 i - 10) q^{63} + ( - 3 i - 9) q^{65} + ( - 6 i - 6) q^{67} + 24 q^{69} + 12 i q^{71} + (5 i - 5) q^{73} + (14 i + 2) q^{75} + (8 i - 8) q^{77} + 8 q^{79} - q^{81} + (6 i - 6) q^{83} + ( - 3 i - 9) q^{85} + (4 i - 4) q^{87} - 12 q^{91} + ( - 8 i - 8) q^{93} + (11 i + 11) q^{97} + 20 i q^{99} +O(q^{100})$$ q + (2*i - 2) * q^3 + (-i + 2) * q^5 + (-2*i + 2) * q^7 - 5*i * q^9 - 4 * q^11 + (-3*i - 3) * q^13 + (6*i - 2) * q^15 + (-3*i - 3) * q^17 + 8*i * q^21 + (-6*i - 6) * q^23 + (-4*i + 3) * q^25 + (4*i + 4) * q^27 + 2 * q^29 + 4*i * q^31 + (-8*i + 8) * q^33 + (-6*i + 2) * q^35 + (3*i - 3) * q^37 + 12 * q^39 + (-6*i + 6) * q^43 + (-10*i - 5) * q^45 + (-6*i + 6) * q^47 - i * q^49 + 12 * q^51 + (3*i + 3) * q^53 + (4*i - 8) * q^55 - 8*i * q^59 - 6*i * q^61 + (-10*i - 10) * q^63 + (-3*i - 9) * q^65 + (-6*i - 6) * q^67 + 24 * q^69 + 12*i * q^71 + (5*i - 5) * q^73 + (14*i + 2) * q^75 + (8*i - 8) * q^77 + 8 * q^79 - q^81 + (6*i - 6) * q^83 + (-3*i - 9) * q^85 + (4*i - 4) * q^87 - 12 * q^91 + (-8*i - 8) * q^93 + (11*i + 11) * q^97 + 20*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{3} + 4 q^{5} + 4 q^{7}+O(q^{10})$$ 2 * q - 4 * q^3 + 4 * q^5 + 4 * q^7 $$2 q - 4 q^{3} + 4 q^{5} + 4 q^{7} - 8 q^{11} - 6 q^{13} - 4 q^{15} - 6 q^{17} - 12 q^{23} + 6 q^{25} + 8 q^{27} + 4 q^{29} + 16 q^{33} + 4 q^{35} - 6 q^{37} + 24 q^{39} + 12 q^{43} - 10 q^{45} + 12 q^{47} + 24 q^{51} + 6 q^{53} - 16 q^{55} - 20 q^{63} - 18 q^{65} - 12 q^{67} + 48 q^{69} - 10 q^{73} + 4 q^{75} - 16 q^{77} + 16 q^{79} - 2 q^{81} - 12 q^{83} - 18 q^{85} - 8 q^{87} - 24 q^{91} - 16 q^{93} + 22 q^{97}+O(q^{100})$$ 2 * q - 4 * q^3 + 4 * q^5 + 4 * q^7 - 8 * q^11 - 6 * q^13 - 4 * q^15 - 6 * q^17 - 12 * q^23 + 6 * q^25 + 8 * q^27 + 4 * q^29 + 16 * q^33 + 4 * q^35 - 6 * q^37 + 24 * q^39 + 12 * q^43 - 10 * q^45 + 12 * q^47 + 24 * q^51 + 6 * q^53 - 16 * q^55 - 20 * q^63 - 18 * q^65 - 12 * q^67 + 48 * q^69 - 10 * q^73 + 4 * q^75 - 16 * q^77 + 16 * q^79 - 2 * q^81 - 12 * q^83 - 18 * q^85 - 8 * q^87 - 24 * q^91 - 16 * q^93 + 22 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/640\mathbb{Z}\right)^\times$$.

 $$n$$ $$257$$ $$261$$ $$511$$ $$\chi(n)$$ $$i$$ $$-1$$ $$-1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
63.1
 − 1.00000i 1.00000i
0 −2.00000 2.00000i 0 2.00000 + 1.00000i 0 2.00000 + 2.00000i 0 5.00000i 0
447.1 0 −2.00000 + 2.00000i 0 2.00000 1.00000i 0 2.00000 2.00000i 0 5.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 640.2.o.b yes 2
4.b odd 2 1 640.2.o.h yes 2
5.c odd 4 1 640.2.o.a 2
8.b even 2 1 640.2.o.g yes 2
8.d odd 2 1 640.2.o.a 2
16.e even 4 1 1280.2.n.a 2
16.e even 4 1 1280.2.n.l 2
16.f odd 4 1 1280.2.n.b 2
16.f odd 4 1 1280.2.n.k 2
20.e even 4 1 640.2.o.g yes 2
40.i odd 4 1 640.2.o.h yes 2
40.k even 4 1 inner 640.2.o.b yes 2
80.i odd 4 1 1280.2.n.k 2
80.j even 4 1 1280.2.n.l 2
80.s even 4 1 1280.2.n.a 2
80.t odd 4 1 1280.2.n.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.o.a 2 5.c odd 4 1
640.2.o.a 2 8.d odd 2 1
640.2.o.b yes 2 1.a even 1 1 trivial
640.2.o.b yes 2 40.k even 4 1 inner
640.2.o.g yes 2 8.b even 2 1
640.2.o.g yes 2 20.e even 4 1
640.2.o.h yes 2 4.b odd 2 1
640.2.o.h yes 2 40.i odd 4 1
1280.2.n.a 2 16.e even 4 1
1280.2.n.a 2 80.s even 4 1
1280.2.n.b 2 16.f odd 4 1
1280.2.n.b 2 80.t odd 4 1
1280.2.n.k 2 16.f odd 4 1
1280.2.n.k 2 80.i odd 4 1
1280.2.n.l 2 16.e even 4 1
1280.2.n.l 2 80.j even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(640, [\chi])$$:

 $$T_{3}^{2} + 4T_{3} + 8$$ T3^2 + 4*T3 + 8 $$T_{7}^{2} - 4T_{7} + 8$$ T7^2 - 4*T7 + 8 $$T_{13}^{2} + 6T_{13} + 18$$ T13^2 + 6*T13 + 18

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 4T + 8$$
$5$ $$T^{2} - 4T + 5$$
$7$ $$T^{2} - 4T + 8$$
$11$ $$(T + 4)^{2}$$
$13$ $$T^{2} + 6T + 18$$
$17$ $$T^{2} + 6T + 18$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 12T + 72$$
$29$ $$(T - 2)^{2}$$
$31$ $$T^{2} + 16$$
$37$ $$T^{2} + 6T + 18$$
$41$ $$T^{2}$$
$43$ $$T^{2} - 12T + 72$$
$47$ $$T^{2} - 12T + 72$$
$53$ $$T^{2} - 6T + 18$$
$59$ $$T^{2} + 64$$
$61$ $$T^{2} + 36$$
$67$ $$T^{2} + 12T + 72$$
$71$ $$T^{2} + 144$$
$73$ $$T^{2} + 10T + 50$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 12T + 72$$
$89$ $$T^{2}$$
$97$ $$T^{2} - 22T + 242$$