# Properties

 Label 64.8.b.a Level $64$ Weight $8$ Character orbit 64.b Analytic conductor $19.993$ 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] = [64,8,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, 8, names="a")

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

chi := DirichletCharacter("64.33");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$64 = 2^{6}$$ Weight: $$k$$ $$=$$ $$8$$ 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: $$19.9926416310$$ 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.

 $$f(q)$$ $$=$$ $$q + 43 \beta q^{3} - 5209 q^{9}+O(q^{10})$$ q + 43*b * q^3 - 5209 * q^9 $$q + 43 \beta q^{3} - 5209 q^{9} + 4407 \beta q^{11} - 22182 q^{17} - 29861 \beta q^{19} + 78125 q^{25} - 129946 \beta q^{27} - 758004 q^{33} - 236886 q^{41} + 110255 \beta q^{43} - 823543 q^{49} - 953826 \beta q^{51} + 5136092 q^{57} + 515463 \beta q^{59} + 1925651 \beta q^{67} - 4865614 q^{73} + 3359375 \beta q^{75} + 10958629 q^{81} + 2404467 \beta q^{83} - 7073118 q^{89} + 9938890 q^{97} - 22956063 \beta q^{99} +O(q^{100})$$ q + 43*b * q^3 - 5209 * q^9 + 4407*b * q^11 - 22182 * q^17 - 29861*b * q^19 + 78125 * q^25 - 129946*b * q^27 - 758004 * q^33 - 236886 * q^41 + 110255*b * q^43 - 823543 * q^49 - 953826*b * q^51 + 5136092 * q^57 + 515463*b * q^59 + 1925651*b * q^67 - 4865614 * q^73 + 3359375*b * q^75 + 10958629 * q^81 + 2404467*b * q^83 - 7073118 * q^89 + 9938890 * q^97 - 22956063*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 10418 q^{9}+O(q^{10})$$ 2 * q - 10418 * q^9 $$2 q - 10418 q^{9} - 44364 q^{17} + 156250 q^{25} - 1516008 q^{33} - 473772 q^{41} - 1647086 q^{49} + 10272184 q^{57} - 9731228 q^{73} + 21917258 q^{81} - 14146236 q^{89} + 19877780 q^{97}+O(q^{100})$$ 2 * q - 10418 * q^9 - 44364 * q^17 + 156250 * q^25 - 1516008 * q^33 - 473772 * q^41 - 1647086 * q^49 + 10272184 * q^57 - 9731228 * q^73 + 21917258 * q^81 - 14146236 * q^89 + 19877780 * 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 86.0000i 0 0 0 0 0 −5209.00 0
33.2 0 86.0000i 0 0 0 0 0 −5209.00 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 64.8.b.a 2
3.b odd 2 1 576.8.d.a 2
4.b odd 2 1 inner 64.8.b.a 2
8.b even 2 1 inner 64.8.b.a 2
8.d odd 2 1 CM 64.8.b.a 2
12.b even 2 1 576.8.d.a 2
16.e even 4 1 256.8.a.a 1
16.e even 4 1 256.8.a.d 1
16.f odd 4 1 256.8.a.a 1
16.f odd 4 1 256.8.a.d 1
24.f even 2 1 576.8.d.a 2
24.h odd 2 1 576.8.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
64.8.b.a 2 1.a even 1 1 trivial
64.8.b.a 2 4.b odd 2 1 inner
64.8.b.a 2 8.b even 2 1 inner
64.8.b.a 2 8.d odd 2 1 CM
256.8.a.a 1 16.e even 4 1
256.8.a.a 1 16.f odd 4 1
256.8.a.d 1 16.e even 4 1
256.8.a.d 1 16.f odd 4 1
576.8.d.a 2 3.b odd 2 1
576.8.d.a 2 12.b even 2 1
576.8.d.a 2 24.f even 2 1
576.8.d.a 2 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 7396$$ acting on $$S_{8}^{\mathrm{new}}(64, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 7396$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 77686596$$
$13$ $$T^{2}$$
$17$ $$(T + 22182)^{2}$$
$19$ $$T^{2} + 3566717284$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$(T + 236886)^{2}$$
$43$ $$T^{2} + 48624660100$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2} + 1062808417476$$
$61$ $$T^{2}$$
$67$ $$T^{2} + 14832527095204$$
$71$ $$T^{2}$$
$73$ $$(T + 4865614)^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 23125846216356$$
$89$ $$(T + 7073118)^{2}$$
$97$ $$(T - 9938890)^{2}$$