# Properties

 Label 8.4.a.a Level $8$ Weight $4$ Character orbit 8.a Self dual yes Analytic conductor $0.472$ 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] = [8,4,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, 4, names="a")

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

chi := DirichletCharacter("8.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8 = 2^{3}$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$0.472015280046$$ 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)

 $$f(q)$$ $$=$$ $$q - 4 q^{3} - 2 q^{5} + 24 q^{7} - 11 q^{9}+O(q^{10})$$ q - 4 * q^3 - 2 * q^5 + 24 * q^7 - 11 * q^9 $$q - 4 q^{3} - 2 q^{5} + 24 q^{7} - 11 q^{9} - 44 q^{11} + 22 q^{13} + 8 q^{15} + 50 q^{17} + 44 q^{19} - 96 q^{21} - 56 q^{23} - 121 q^{25} + 152 q^{27} + 198 q^{29} - 160 q^{31} + 176 q^{33} - 48 q^{35} - 162 q^{37} - 88 q^{39} - 198 q^{41} + 52 q^{43} + 22 q^{45} + 528 q^{47} + 233 q^{49} - 200 q^{51} - 242 q^{53} + 88 q^{55} - 176 q^{57} - 668 q^{59} + 550 q^{61} - 264 q^{63} - 44 q^{65} + 188 q^{67} + 224 q^{69} + 728 q^{71} + 154 q^{73} + 484 q^{75} - 1056 q^{77} - 656 q^{79} - 311 q^{81} + 236 q^{83} - 100 q^{85} - 792 q^{87} + 714 q^{89} + 528 q^{91} + 640 q^{93} - 88 q^{95} - 478 q^{97} + 484 q^{99}+O(q^{100})$$ q - 4 * q^3 - 2 * q^5 + 24 * q^7 - 11 * q^9 - 44 * q^11 + 22 * q^13 + 8 * q^15 + 50 * q^17 + 44 * q^19 - 96 * q^21 - 56 * q^23 - 121 * q^25 + 152 * q^27 + 198 * q^29 - 160 * q^31 + 176 * q^33 - 48 * q^35 - 162 * q^37 - 88 * q^39 - 198 * q^41 + 52 * q^43 + 22 * q^45 + 528 * q^47 + 233 * q^49 - 200 * q^51 - 242 * q^53 + 88 * q^55 - 176 * q^57 - 668 * q^59 + 550 * q^61 - 264 * q^63 - 44 * q^65 + 188 * q^67 + 224 * q^69 + 728 * q^71 + 154 * q^73 + 484 * q^75 - 1056 * q^77 - 656 * q^79 - 311 * q^81 + 236 * q^83 - 100 * q^85 - 792 * q^87 + 714 * q^89 + 528 * q^91 + 640 * q^93 - 88 * q^95 - 478 * q^97 + 484 * q^99

## Expression as an eta quotient

$$f(z) = \eta(2z)^{4}\eta(4z)^{4}=q\prod_{n=1}^\infty(1 - q^{2n})^{4}(1 - q^{4n})^{4}$$

## 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 −4.00000 0 −2.00000 0 24.0000 0 −11.0000 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 8.4.a.a 1
3.b odd 2 1 72.4.a.c 1
4.b odd 2 1 16.4.a.a 1
5.b even 2 1 200.4.a.g 1
5.c odd 4 2 200.4.c.e 2
7.b odd 2 1 392.4.a.e 1
7.c even 3 2 392.4.i.g 2
7.d odd 6 2 392.4.i.b 2
8.b even 2 1 64.4.a.d 1
8.d odd 2 1 64.4.a.b 1
9.c even 3 2 648.4.i.h 2
9.d odd 6 2 648.4.i.e 2
11.b odd 2 1 968.4.a.a 1
12.b even 2 1 144.4.a.e 1
13.b even 2 1 1352.4.a.a 1
15.d odd 2 1 1800.4.a.d 1
15.e even 4 2 1800.4.f.u 2
16.e even 4 2 256.4.b.a 2
16.f odd 4 2 256.4.b.g 2
17.b even 2 1 2312.4.a.a 1
20.d odd 2 1 400.4.a.g 1
20.e even 4 2 400.4.c.i 2
24.f even 2 1 576.4.a.j 1
24.h odd 2 1 576.4.a.k 1
28.d even 2 1 784.4.a.e 1
40.e odd 2 1 1600.4.a.bm 1
40.f even 2 1 1600.4.a.o 1
44.c even 2 1 1936.4.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.4.a.a 1 1.a even 1 1 trivial
16.4.a.a 1 4.b odd 2 1
64.4.a.b 1 8.d odd 2 1
64.4.a.d 1 8.b even 2 1
72.4.a.c 1 3.b odd 2 1
144.4.a.e 1 12.b even 2 1
200.4.a.g 1 5.b even 2 1
200.4.c.e 2 5.c odd 4 2
256.4.b.a 2 16.e even 4 2
256.4.b.g 2 16.f odd 4 2
392.4.a.e 1 7.b odd 2 1
392.4.i.b 2 7.d odd 6 2
392.4.i.g 2 7.c even 3 2
400.4.a.g 1 20.d odd 2 1
400.4.c.i 2 20.e even 4 2
576.4.a.j 1 24.f even 2 1
576.4.a.k 1 24.h odd 2 1
648.4.i.e 2 9.d odd 6 2
648.4.i.h 2 9.c even 3 2
784.4.a.e 1 28.d even 2 1
968.4.a.a 1 11.b odd 2 1
1352.4.a.a 1 13.b even 2 1
1600.4.a.o 1 40.f even 2 1
1600.4.a.bm 1 40.e odd 2 1
1800.4.a.d 1 15.d odd 2 1
1800.4.f.u 2 15.e even 4 2
1936.4.a.l 1 44.c even 2 1
2312.4.a.a 1 17.b even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(\Gamma_0(8))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 4$$
$5$ $$T + 2$$
$7$ $$T - 24$$
$11$ $$T + 44$$
$13$ $$T - 22$$
$17$ $$T - 50$$
$19$ $$T - 44$$
$23$ $$T + 56$$
$29$ $$T - 198$$
$31$ $$T + 160$$
$37$ $$T + 162$$
$41$ $$T + 198$$
$43$ $$T - 52$$
$47$ $$T - 528$$
$53$ $$T + 242$$
$59$ $$T + 668$$
$61$ $$T - 550$$
$67$ $$T - 188$$
$71$ $$T - 728$$
$73$ $$T - 154$$
$79$ $$T + 656$$
$83$ $$T - 236$$
$89$ $$T - 714$$
$97$ $$T + 478$$