# Properties

 Label 640.2.f.f Level $640$ Weight $2$ Character orbit 640.f Analytic conductor $5.110$ Analytic rank $0$ Dimension $4$ CM discriminant -20 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [640,2,Mod(449,640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(640, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

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

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

chi := DirichletCharacter("640.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$640 = 2^{7} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 640.f (of order $$2$$, degree $$1$$, 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: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 4x^{2} + 9$$ x^4 - 4*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{3} - \beta_{2} q^{5} - 3 \beta_1 q^{7} + 7 q^{9}+O(q^{10})$$ q - b3 * q^3 - b2 * q^5 - 3*b1 * q^7 + 7 * q^9 $$q - \beta_{3} q^{3} - \beta_{2} q^{5} - 3 \beta_1 q^{7} + 7 q^{9} + 5 \beta_1 q^{15} + 6 \beta_{2} q^{21} - \beta_1 q^{23} - 5 q^{25} - 4 \beta_{3} q^{27} - 4 \beta_{2} q^{29} - 3 \beta_{3} q^{35} - 12 q^{41} - \beta_{3} q^{43} - 7 \beta_{2} q^{45} + 7 \beta_1 q^{47} - 11 q^{49} + 6 \beta_{2} q^{61} - 21 \beta_1 q^{63} + 5 \beta_{3} q^{67} + 2 \beta_{2} q^{69} + 5 \beta_{3} q^{75} + 19 q^{81} - 3 \beta_{3} q^{83} + 20 \beta_1 q^{87} - 6 q^{89}+O(q^{100})$$ q - b3 * q^3 - b2 * q^5 - 3*b1 * q^7 + 7 * q^9 + 5*b1 * q^15 + 6*b2 * q^21 - b1 * q^23 - 5 * q^25 - 4*b3 * q^27 - 4*b2 * q^29 - 3*b3 * q^35 - 12 * q^41 - b3 * q^43 - 7*b2 * q^45 + 7*b1 * q^47 - 11 * q^49 + 6*b2 * q^61 - 21*b1 * q^63 + 5*b3 * q^67 + 2*b2 * q^69 + 5*b3 * q^75 + 19 * q^81 - 3*b3 * q^83 + 20*b1 * q^87 - 6 * q^89 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 28 q^{9}+O(q^{10})$$ 4 * q + 28 * q^9 $$4 q + 28 q^{9} - 20 q^{25} - 48 q^{41} - 44 q^{49} + 76 q^{81} - 24 q^{89}+O(q^{100})$$ 4 * q + 28 * q^9 - 20 * q^25 - 48 * q^41 - 44 * q^49 + 76 * q^81 - 24 * q^89

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 4x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} - \nu ) / 3$$ (v^3 - v) / 3 $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 7\nu ) / 3$$ (-v^3 + 7*v) / 3
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2 $$\nu^{3}$$ $$=$$ $$( \beta_{3} + 7\beta_1 ) / 2$$ (b3 + 7*b1) / 2

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 1.58114 + 0.707107i 1.58114 − 0.707107i −1.58114 − 0.707107i −1.58114 + 0.707107i
0 −3.16228 0 2.23607i 0 4.24264i 0 7.00000 0
449.2 0 −3.16228 0 2.23607i 0 4.24264i 0 7.00000 0
449.3 0 3.16228 0 2.23607i 0 4.24264i 0 7.00000 0
449.4 0 3.16228 0 2.23607i 0 4.24264i 0 7.00000 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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
4.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
40.e odd 2 1 inner
40.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 640.2.f.f 4
4.b odd 2 1 inner 640.2.f.f 4
5.b even 2 1 inner 640.2.f.f 4
5.c odd 4 2 3200.2.d.i 4
8.b even 2 1 inner 640.2.f.f 4
8.d odd 2 1 inner 640.2.f.f 4
16.e even 4 2 1280.2.c.f 4
16.f odd 4 2 1280.2.c.f 4
20.d odd 2 1 CM 640.2.f.f 4
20.e even 4 2 3200.2.d.i 4
40.e odd 2 1 inner 640.2.f.f 4
40.f even 2 1 inner 640.2.f.f 4
40.i odd 4 2 3200.2.d.i 4
40.k even 4 2 3200.2.d.i 4
80.i odd 4 2 6400.2.a.cv 4
80.j even 4 2 6400.2.a.cv 4
80.k odd 4 2 1280.2.c.f 4
80.q even 4 2 1280.2.c.f 4
80.s even 4 2 6400.2.a.cv 4
80.t odd 4 2 6400.2.a.cv 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.f.f 4 1.a even 1 1 trivial
640.2.f.f 4 4.b odd 2 1 inner
640.2.f.f 4 5.b even 2 1 inner
640.2.f.f 4 8.b even 2 1 inner
640.2.f.f 4 8.d odd 2 1 inner
640.2.f.f 4 20.d odd 2 1 CM
640.2.f.f 4 40.e odd 2 1 inner
640.2.f.f 4 40.f even 2 1 inner
1280.2.c.f 4 16.e even 4 2
1280.2.c.f 4 16.f odd 4 2
1280.2.c.f 4 80.k odd 4 2
1280.2.c.f 4 80.q even 4 2
3200.2.d.i 4 5.c odd 4 2
3200.2.d.i 4 20.e even 4 2
3200.2.d.i 4 40.i odd 4 2
3200.2.d.i 4 40.k even 4 2
6400.2.a.cv 4 80.i odd 4 2
6400.2.a.cv 4 80.j even 4 2
6400.2.a.cv 4 80.s even 4 2
6400.2.a.cv 4 80.t odd 4 2

## Hecke kernels

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

 $$T_{3}^{2} - 10$$ T3^2 - 10 $$T_{7}^{2} + 18$$ T7^2 + 18 $$T_{13}$$ T13 $$T_{37}$$ T37

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - 10)^{2}$$
$5$ $$(T^{2} + 5)^{2}$$
$7$ $$(T^{2} + 18)^{2}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$(T^{2} + 2)^{2}$$
$29$ $$(T^{2} + 80)^{2}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$(T + 12)^{4}$$
$43$ $$(T^{2} - 10)^{2}$$
$47$ $$(T^{2} + 98)^{2}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$(T^{2} + 180)^{2}$$
$67$ $$(T^{2} - 250)^{2}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$(T^{2} - 90)^{2}$$
$89$ $$(T + 6)^{4}$$
$97$ $$T^{4}$$