Properties

 Label 64.2.b.a Level $64$ Weight $2$ Character orbit 64.b Analytic conductor $0.511$ 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] = [64,2,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, 2, names="a")

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

chi := DirichletCharacter("64.33");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$64 = 2^{6}$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$0.511042572936$$ 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: $$\Z[a_1, a_2, a_3]$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{3} - q^{9}+O(q^{10})$$ q + i * q^3 - q^9 $$q + i q^{3} - q^{9} - 3 i q^{11} - 6 q^{17} + i q^{19} + 5 q^{25} + 2 i q^{27} + 12 q^{33} - 6 q^{41} + 5 i q^{43} - 7 q^{49} - 6 i q^{51} - 4 q^{57} - 3 i q^{59} - 7 i q^{67} + 2 q^{73} + 5 i q^{75} - 11 q^{81} + 9 i q^{83} + 18 q^{89} + 10 q^{97} + 3 i q^{99} +O(q^{100})$$ q + i * q^3 - q^9 - 3*i * q^11 - 6 * q^17 + i * q^19 + 5 * q^25 + 2*i * q^27 + 12 * q^33 - 6 * q^41 + 5*i * q^43 - 7 * q^49 - 6*i * q^51 - 4 * q^57 - 3*i * q^59 - 7*i * q^67 + 2 * q^73 + 5*i * q^75 - 11 * q^81 + 9*i * q^83 + 18 * q^89 + 10 * q^97 + 3*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} - 12 q^{17} + 10 q^{25} + 24 q^{33} - 12 q^{41} - 14 q^{49} - 8 q^{57} + 4 q^{73} - 22 q^{81} + 36 q^{89} + 20 q^{97}+O(q^{100})$$ 2 * q - 2 * q^9 - 12 * q^17 + 10 * q^25 + 24 * q^33 - 12 * q^41 - 14 * q^49 - 8 * q^57 + 4 * q^73 - 22 * q^81 + 36 * q^89 + 20 * 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 2.00000i 0 0 0 0 0 −1.00000 0
33.2 0 2.00000i 0 0 0 0 0 −1.00000 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
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 64.2.b.a 2
3.b odd 2 1 576.2.d.a 2
4.b odd 2 1 inner 64.2.b.a 2
5.b even 2 1 1600.2.d.a 2
5.c odd 4 1 1600.2.f.a 2
5.c odd 4 1 1600.2.f.b 2
7.b odd 2 1 3136.2.b.b 2
8.b even 2 1 inner 64.2.b.a 2
8.d odd 2 1 CM 64.2.b.a 2
12.b even 2 1 576.2.d.a 2
16.e even 4 1 256.2.a.a 1
16.e even 4 1 256.2.a.d 1
16.f odd 4 1 256.2.a.a 1
16.f odd 4 1 256.2.a.d 1
20.d odd 2 1 1600.2.d.a 2
20.e even 4 1 1600.2.f.a 2
20.e even 4 1 1600.2.f.b 2
24.f even 2 1 576.2.d.a 2
24.h odd 2 1 576.2.d.a 2
28.d even 2 1 3136.2.b.b 2
32.g even 8 4 1024.2.e.l 4
32.h odd 8 4 1024.2.e.l 4
40.e odd 2 1 1600.2.d.a 2
40.f even 2 1 1600.2.d.a 2
40.i odd 4 1 1600.2.f.a 2
40.i odd 4 1 1600.2.f.b 2
40.k even 4 1 1600.2.f.a 2
40.k even 4 1 1600.2.f.b 2
48.i odd 4 1 2304.2.a.h 1
48.i odd 4 1 2304.2.a.i 1
48.k even 4 1 2304.2.a.h 1
48.k even 4 1 2304.2.a.i 1
56.e even 2 1 3136.2.b.b 2
56.h odd 2 1 3136.2.b.b 2
80.k odd 4 1 6400.2.a.a 1
80.k odd 4 1 6400.2.a.x 1
80.q even 4 1 6400.2.a.a 1
80.q even 4 1 6400.2.a.x 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
64.2.b.a 2 1.a even 1 1 trivial
64.2.b.a 2 4.b odd 2 1 inner
64.2.b.a 2 8.b even 2 1 inner
64.2.b.a 2 8.d odd 2 1 CM
256.2.a.a 1 16.e even 4 1
256.2.a.a 1 16.f odd 4 1
256.2.a.d 1 16.e even 4 1
256.2.a.d 1 16.f odd 4 1
576.2.d.a 2 3.b odd 2 1
576.2.d.a 2 12.b even 2 1
576.2.d.a 2 24.f even 2 1
576.2.d.a 2 24.h odd 2 1
1024.2.e.l 4 32.g even 8 4
1024.2.e.l 4 32.h odd 8 4
1600.2.d.a 2 5.b even 2 1
1600.2.d.a 2 20.d odd 2 1
1600.2.d.a 2 40.e odd 2 1
1600.2.d.a 2 40.f even 2 1
1600.2.f.a 2 5.c odd 4 1
1600.2.f.a 2 20.e even 4 1
1600.2.f.a 2 40.i odd 4 1
1600.2.f.a 2 40.k even 4 1
1600.2.f.b 2 5.c odd 4 1
1600.2.f.b 2 20.e even 4 1
1600.2.f.b 2 40.i odd 4 1
1600.2.f.b 2 40.k even 4 1
2304.2.a.h 1 48.i odd 4 1
2304.2.a.h 1 48.k even 4 1
2304.2.a.i 1 48.i odd 4 1
2304.2.a.i 1 48.k even 4 1
3136.2.b.b 2 7.b odd 2 1
3136.2.b.b 2 28.d even 2 1
3136.2.b.b 2 56.e even 2 1
3136.2.b.b 2 56.h odd 2 1
6400.2.a.a 1 80.k odd 4 1
6400.2.a.a 1 80.q even 4 1
6400.2.a.x 1 80.k odd 4 1
6400.2.a.x 1 80.q even 4 1

Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(64, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 36$$
$13$ $$T^{2}$$
$17$ $$(T + 6)^{2}$$
$19$ $$T^{2} + 4$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$(T + 6)^{2}$$
$43$ $$T^{2} + 100$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2} + 36$$
$61$ $$T^{2}$$
$67$ $$T^{2} + 196$$
$71$ $$T^{2}$$
$73$ $$(T - 2)^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 324$$
$89$ $$(T - 18)^{2}$$
$97$ $$(T - 10)^{2}$$