# Properties

 Label 64.4.b.a Level $64$ Weight $4$ Character orbit 64.b Analytic conductor $3.776$ Analytic rank $0$ Dimension $2$ CM discriminant -8 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [64,4,Mod(33,64)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(64, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("64.33");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$64 = 2^{6}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 64.b (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: $$3.77612224037$$ 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: $$\Z[a_1, \ldots, a_{11}]$$ 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 $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 5 \beta q^{3} - 73 q^{9}+O(q^{10})$$ q + 5*b * q^3 - 73 * q^9 $$q + 5 \beta q^{3} - 73 q^{9} + 9 \beta q^{11} + 90 q^{17} + 53 \beta q^{19} + 125 q^{25} - 230 \beta q^{27} - 180 q^{33} + 522 q^{41} + 145 \beta q^{43} - 343 q^{49} + 450 \beta q^{51} - 1060 q^{57} - 423 \beta q^{59} - 35 \beta q^{67} - 430 q^{73} + 625 \beta q^{75} + 2629 q^{81} - 675 \beta q^{83} + 1026 q^{89} - 1910 q^{97} - 657 \beta q^{99} +O(q^{100})$$ q + 5*b * q^3 - 73 * q^9 + 9*b * q^11 + 90 * q^17 + 53*b * q^19 + 125 * q^25 - 230*b * q^27 - 180 * q^33 + 522 * q^41 + 145*b * q^43 - 343 * q^49 + 450*b * q^51 - 1060 * q^57 - 423*b * q^59 - 35*b * q^67 - 430 * q^73 + 625*b * q^75 + 2629 * q^81 - 675*b * q^83 + 1026 * q^89 - 1910 * q^97 - 657*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 146 q^{9}+O(q^{10})$$ 2 * q - 146 * q^9 $$2 q - 146 q^{9} + 180 q^{17} + 250 q^{25} - 360 q^{33} + 1044 q^{41} - 686 q^{49} - 2120 q^{57} - 860 q^{73} + 5258 q^{81} + 2052 q^{89} - 3820 q^{97}+O(q^{100})$$ 2 * q - 146 * q^9 + 180 * q^17 + 250 * q^25 - 360 * q^33 + 1044 * q^41 - 686 * q^49 - 2120 * q^57 - 860 * q^73 + 5258 * q^81 + 2052 * q^89 - 3820 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/64\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$63$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
33.1
 − 1.00000i 1.00000i
0 10.0000i 0 0 0 0 0 −73.0000 0
33.2 0 10.0000i 0 0 0 0 0 −73.0000 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
4.b odd 2 1 inner
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 64.4.b.a 2
3.b odd 2 1 576.4.d.a 2
4.b odd 2 1 inner 64.4.b.a 2
8.b even 2 1 inner 64.4.b.a 2
8.d odd 2 1 CM 64.4.b.a 2
12.b even 2 1 576.4.d.a 2
16.e even 4 1 256.4.a.a 1
16.e even 4 1 256.4.a.h 1
16.f odd 4 1 256.4.a.a 1
16.f odd 4 1 256.4.a.h 1
24.f even 2 1 576.4.d.a 2
24.h odd 2 1 576.4.d.a 2
48.i odd 4 1 2304.4.a.h 1
48.i odd 4 1 2304.4.a.i 1
48.k even 4 1 2304.4.a.h 1
48.k even 4 1 2304.4.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
64.4.b.a 2 1.a even 1 1 trivial
64.4.b.a 2 4.b odd 2 1 inner
64.4.b.a 2 8.b even 2 1 inner
64.4.b.a 2 8.d odd 2 1 CM
256.4.a.a 1 16.e even 4 1
256.4.a.a 1 16.f odd 4 1
256.4.a.h 1 16.e even 4 1
256.4.a.h 1 16.f odd 4 1
576.4.d.a 2 3.b odd 2 1
576.4.d.a 2 12.b even 2 1
576.4.d.a 2 24.f even 2 1
576.4.d.a 2 24.h odd 2 1
2304.4.a.h 1 48.i odd 4 1
2304.4.a.h 1 48.k even 4 1
2304.4.a.i 1 48.i odd 4 1
2304.4.a.i 1 48.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 100$$ acting on $$S_{4}^{\mathrm{new}}(64, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 100$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 324$$
$13$ $$T^{2}$$
$17$ $$(T - 90)^{2}$$
$19$ $$T^{2} + 11236$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$(T - 522)^{2}$$
$43$ $$T^{2} + 84100$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2} + 715716$$
$61$ $$T^{2}$$
$67$ $$T^{2} + 4900$$
$71$ $$T^{2}$$
$73$ $$(T + 430)^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 1822500$$
$89$ $$(T - 1026)^{2}$$
$97$ $$(T + 1910)^{2}$$