Properties

 Label 8.6.a.a Level $8$ Weight $6$ Character orbit 8.a Self dual yes Analytic conductor $1.283$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8,6,Mod(1,8)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("8.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8 = 2^{3}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 8.a (trivial)

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: yes Analytic conductor: $$1.28307055850$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + 20 q^{3} - 74 q^{5} - 24 q^{7} + 157 q^{9}+O(q^{10})$$ q + 20 * q^3 - 74 * q^5 - 24 * q^7 + 157 * q^9 $$q + 20 q^{3} - 74 q^{5} - 24 q^{7} + 157 q^{9} + 124 q^{11} + 478 q^{13} - 1480 q^{15} - 1198 q^{17} + 3044 q^{19} - 480 q^{21} + 184 q^{23} + 2351 q^{25} - 1720 q^{27} - 3282 q^{29} - 5728 q^{31} + 2480 q^{33} + 1776 q^{35} + 10326 q^{37} + 9560 q^{39} - 8886 q^{41} - 9188 q^{43} - 11618 q^{45} + 23664 q^{47} - 16231 q^{49} - 23960 q^{51} + 11686 q^{53} - 9176 q^{55} + 60880 q^{57} + 16876 q^{59} - 18482 q^{61} - 3768 q^{63} - 35372 q^{65} - 15532 q^{67} + 3680 q^{69} - 31960 q^{71} - 4886 q^{73} + 47020 q^{75} - 2976 q^{77} + 44560 q^{79} - 72551 q^{81} + 67364 q^{83} + 88652 q^{85} - 65640 q^{87} + 71994 q^{89} - 11472 q^{91} - 114560 q^{93} - 225256 q^{95} + 48866 q^{97} + 19468 q^{99}+O(q^{100})$$ q + 20 * q^3 - 74 * q^5 - 24 * q^7 + 157 * q^9 + 124 * q^11 + 478 * q^13 - 1480 * q^15 - 1198 * q^17 + 3044 * q^19 - 480 * q^21 + 184 * q^23 + 2351 * q^25 - 1720 * q^27 - 3282 * q^29 - 5728 * q^31 + 2480 * q^33 + 1776 * q^35 + 10326 * q^37 + 9560 * q^39 - 8886 * q^41 - 9188 * q^43 - 11618 * q^45 + 23664 * q^47 - 16231 * q^49 - 23960 * q^51 + 11686 * q^53 - 9176 * q^55 + 60880 * q^57 + 16876 * q^59 - 18482 * q^61 - 3768 * q^63 - 35372 * q^65 - 15532 * q^67 + 3680 * q^69 - 31960 * q^71 - 4886 * q^73 + 47020 * q^75 - 2976 * q^77 + 44560 * q^79 - 72551 * q^81 + 67364 * q^83 + 88652 * q^85 - 65640 * q^87 + 71994 * q^89 - 11472 * q^91 - 114560 * q^93 - 225256 * q^95 + 48866 * q^97 + 19468 * q^99

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 20.0000 0 −74.0000 0 −24.0000 0 157.000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.6.a.a 1
3.b odd 2 1 72.6.a.f 1
4.b odd 2 1 16.6.a.a 1
5.b even 2 1 200.6.a.a 1
5.c odd 4 2 200.6.c.a 2
7.b odd 2 1 392.6.a.b 1
7.c even 3 2 392.6.i.b 2
7.d odd 6 2 392.6.i.e 2
8.b even 2 1 64.6.a.a 1
8.d odd 2 1 64.6.a.g 1
11.b odd 2 1 968.6.a.a 1
12.b even 2 1 144.6.a.k 1
16.e even 4 2 256.6.b.f 2
16.f odd 4 2 256.6.b.d 2
20.d odd 2 1 400.6.a.l 1
20.e even 4 2 400.6.c.d 2
24.f even 2 1 576.6.a.h 1
24.h odd 2 1 576.6.a.g 1
28.d even 2 1 784.6.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.6.a.a 1 1.a even 1 1 trivial
16.6.a.a 1 4.b odd 2 1
64.6.a.a 1 8.b even 2 1
64.6.a.g 1 8.d odd 2 1
72.6.a.f 1 3.b odd 2 1
144.6.a.k 1 12.b even 2 1
200.6.a.a 1 5.b even 2 1
200.6.c.a 2 5.c odd 4 2
256.6.b.d 2 16.f odd 4 2
256.6.b.f 2 16.e even 4 2
392.6.a.b 1 7.b odd 2 1
392.6.i.b 2 7.c even 3 2
392.6.i.e 2 7.d odd 6 2
400.6.a.l 1 20.d odd 2 1
400.6.c.d 2 20.e even 4 2
576.6.a.g 1 24.h odd 2 1
576.6.a.h 1 24.f even 2 1
784.6.a.l 1 28.d even 2 1
968.6.a.a 1 11.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(\Gamma_0(8))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 20$$
$5$ $$T + 74$$
$7$ $$T + 24$$
$11$ $$T - 124$$
$13$ $$T - 478$$
$17$ $$T + 1198$$
$19$ $$T - 3044$$
$23$ $$T - 184$$
$29$ $$T + 3282$$
$31$ $$T + 5728$$
$37$ $$T - 10326$$
$41$ $$T + 8886$$
$43$ $$T + 9188$$
$47$ $$T - 23664$$
$53$ $$T - 11686$$
$59$ $$T - 16876$$
$61$ $$T + 18482$$
$67$ $$T + 15532$$
$71$ $$T + 31960$$
$73$ $$T + 4886$$
$79$ $$T - 44560$$
$83$ $$T - 67364$$
$89$ $$T - 71994$$
$97$ $$T - 48866$$