# Properties

 Label 640.1.m.b Level $640$ Weight $1$ Character orbit 640.m Analytic conductor $0.319$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -4 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [640,1,Mod(193,640)]

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([0, 2, 3]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("640.193");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$640 = 2^{7} \cdot 5$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 640.m (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: $$0.319401608085$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ 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 Projective image: $$D_{4}$$ Projective field: Galois closure of 4.0.32000.1 Artin image: $C_4\wr C_2$ Artin field: Galois closure of 8.0.131072000.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + i q^{5} - i q^{9} +O(q^{10})$$ q + z * q^5 - z * q^9 $$q + i q^{5} - i q^{9} + (i + 1) q^{13} + (i + 1) q^{17} - q^{25} - 2 q^{29} + ( - i + 1) q^{37} + q^{45} - i q^{49} + ( - i - 1) q^{53} - 2 i q^{61} + (i - 1) q^{65} + (i - 1) q^{73} - q^{81} + (i - 1) q^{85} + ( - i - 1) q^{97} +O(q^{100})$$ q + z * q^5 - z * q^9 + (z + 1) * q^13 + (z + 1) * q^17 - q^25 - 2 * q^29 + (-z + 1) * q^37 + q^45 - z * q^49 + (-z - 1) * q^53 - 2*z * q^61 + (z - 1) * q^65 + (z - 1) * q^73 - q^81 + (z - 1) * q^85 + (-z - 1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q + 2 q^{13} + 2 q^{17} - 2 q^{25} - 4 q^{29} + 2 q^{37} + 2 q^{45} - 2 q^{53} - 2 q^{65} - 2 q^{73} - 2 q^{81} - 2 q^{85} - 2 q^{97}+O(q^{100})$$ 2 * q + 2 * q^13 + 2 * q^17 - 2 * q^25 - 4 * q^29 + 2 * q^37 + 2 * q^45 - 2 * q^53 - 2 * q^65 - 2 * q^73 - 2 * q^81 - 2 * q^85 - 2 * 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 − 1.00000i 1.00000i
0 0 0 1.00000i 0 0 0 1.00000i 0
577.1 0 0 0 1.00000i 0 0 0 1.00000i 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
40.i odd 4 1 inner
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 640.1.m.b yes 2
4.b odd 2 1 CM 640.1.m.b yes 2
5.b even 2 1 3200.1.m.a 2
5.c odd 4 1 640.1.m.a 2
5.c odd 4 1 3200.1.m.b 2
8.b even 2 1 640.1.m.a 2
8.d odd 2 1 640.1.m.a 2
16.e even 4 1 1280.1.p.a 2
16.e even 4 1 1280.1.p.b 2
16.f odd 4 1 1280.1.p.a 2
16.f odd 4 1 1280.1.p.b 2
20.d odd 2 1 3200.1.m.a 2
20.e even 4 1 640.1.m.a 2
20.e even 4 1 3200.1.m.b 2
40.e odd 2 1 3200.1.m.b 2
40.f even 2 1 3200.1.m.b 2
40.i odd 4 1 inner 640.1.m.b yes 2
40.i odd 4 1 3200.1.m.a 2
40.k even 4 1 inner 640.1.m.b yes 2
40.k even 4 1 3200.1.m.a 2
80.i odd 4 1 1280.1.p.b 2
80.j even 4 1 1280.1.p.a 2
80.s even 4 1 1280.1.p.b 2
80.t odd 4 1 1280.1.p.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.1.m.a 2 5.c odd 4 1
640.1.m.a 2 8.b even 2 1
640.1.m.a 2 8.d odd 2 1
640.1.m.a 2 20.e even 4 1
640.1.m.b yes 2 1.a even 1 1 trivial
640.1.m.b yes 2 4.b odd 2 1 CM
640.1.m.b yes 2 40.i odd 4 1 inner
640.1.m.b yes 2 40.k even 4 1 inner
1280.1.p.a 2 16.e even 4 1
1280.1.p.a 2 16.f odd 4 1
1280.1.p.a 2 80.j even 4 1
1280.1.p.a 2 80.t odd 4 1
1280.1.p.b 2 16.e even 4 1
1280.1.p.b 2 16.f odd 4 1
1280.1.p.b 2 80.i odd 4 1
1280.1.p.b 2 80.s even 4 1
3200.1.m.a 2 5.b even 2 1
3200.1.m.a 2 20.d odd 2 1
3200.1.m.a 2 40.i odd 4 1
3200.1.m.a 2 40.k even 4 1
3200.1.m.b 2 5.c odd 4 1
3200.1.m.b 2 20.e even 4 1
3200.1.m.b 2 40.e odd 2 1
3200.1.m.b 2 40.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13}^{2} - 2T_{13} + 2$$ acting on $$S_{1}^{\mathrm{new}}(640, [\chi])$$.

## Hecke characteristic polynomials

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