# Properties

 Label 16.6.a.b Level $16$ Weight $6$ Character orbit 16.a Self dual yes Analytic conductor $2.566$ 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] = [16,6,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, 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("16.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$16 = 2^{4}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 16.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: $$2.56614111701$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 4) 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)

 $$f(q)$$ $$=$$ $$q + 12 q^{3} + 54 q^{5} + 88 q^{7} - 99 q^{9}+O(q^{10})$$ q + 12 * q^3 + 54 * q^5 + 88 * q^7 - 99 * q^9 $$q + 12 q^{3} + 54 q^{5} + 88 q^{7} - 99 q^{9} - 540 q^{11} - 418 q^{13} + 648 q^{15} + 594 q^{17} - 836 q^{19} + 1056 q^{21} + 4104 q^{23} - 209 q^{25} - 4104 q^{27} - 594 q^{29} - 4256 q^{31} - 6480 q^{33} + 4752 q^{35} - 298 q^{37} - 5016 q^{39} + 17226 q^{41} + 12100 q^{43} - 5346 q^{45} + 1296 q^{47} - 9063 q^{49} + 7128 q^{51} + 19494 q^{53} - 29160 q^{55} - 10032 q^{57} + 7668 q^{59} - 34738 q^{61} - 8712 q^{63} - 22572 q^{65} - 21812 q^{67} + 49248 q^{69} + 46872 q^{71} + 67562 q^{73} - 2508 q^{75} - 47520 q^{77} + 76912 q^{79} - 25191 q^{81} - 67716 q^{83} + 32076 q^{85} - 7128 q^{87} + 29754 q^{89} - 36784 q^{91} - 51072 q^{93} - 45144 q^{95} - 122398 q^{97} + 53460 q^{99}+O(q^{100})$$ q + 12 * q^3 + 54 * q^5 + 88 * q^7 - 99 * q^9 - 540 * q^11 - 418 * q^13 + 648 * q^15 + 594 * q^17 - 836 * q^19 + 1056 * q^21 + 4104 * q^23 - 209 * q^25 - 4104 * q^27 - 594 * q^29 - 4256 * q^31 - 6480 * q^33 + 4752 * q^35 - 298 * q^37 - 5016 * q^39 + 17226 * q^41 + 12100 * q^43 - 5346 * q^45 + 1296 * q^47 - 9063 * q^49 + 7128 * q^51 + 19494 * q^53 - 29160 * q^55 - 10032 * q^57 + 7668 * q^59 - 34738 * q^61 - 8712 * q^63 - 22572 * q^65 - 21812 * q^67 + 49248 * q^69 + 46872 * q^71 + 67562 * q^73 - 2508 * q^75 - 47520 * q^77 + 76912 * q^79 - 25191 * q^81 - 67716 * q^83 + 32076 * q^85 - 7128 * q^87 + 29754 * q^89 - 36784 * q^91 - 51072 * q^93 - 45144 * q^95 - 122398 * q^97 + 53460 * q^99

## Expression as an eta quotient

$$f(z) = \dfrac{\eta(4z)^{36}}{\eta(2z)^{12}\eta(8z)^{12}}=q\prod_{n=1}^\infty(1 - q^{2n})^{-12}(1 - q^{4n})^{36}(1 - q^{8n})^{-12}$$

## 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 12.0000 0 54.0000 0 88.0000 0 −99.0000 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 16.6.a.b 1
3.b odd 2 1 144.6.a.c 1
4.b odd 2 1 4.6.a.a 1
5.b even 2 1 400.6.a.d 1
5.c odd 4 2 400.6.c.f 2
7.b odd 2 1 784.6.a.d 1
8.b even 2 1 64.6.a.b 1
8.d odd 2 1 64.6.a.f 1
12.b even 2 1 36.6.a.a 1
16.e even 4 2 256.6.b.c 2
16.f odd 4 2 256.6.b.g 2
20.d odd 2 1 100.6.a.b 1
20.e even 4 2 100.6.c.b 2
24.f even 2 1 576.6.a.bc 1
24.h odd 2 1 576.6.a.bd 1
28.d even 2 1 196.6.a.e 1
28.f even 6 2 196.6.e.d 2
28.g odd 6 2 196.6.e.g 2
36.f odd 6 2 324.6.e.a 2
36.h even 6 2 324.6.e.d 2
44.c even 2 1 484.6.a.a 1
52.b odd 2 1 676.6.a.a 1
52.f even 4 2 676.6.d.a 2
60.h even 2 1 900.6.a.h 1
60.l odd 4 2 900.6.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.6.a.a 1 4.b odd 2 1
16.6.a.b 1 1.a even 1 1 trivial
36.6.a.a 1 12.b even 2 1
64.6.a.b 1 8.b even 2 1
64.6.a.f 1 8.d odd 2 1
100.6.a.b 1 20.d odd 2 1
100.6.c.b 2 20.e even 4 2
144.6.a.c 1 3.b odd 2 1
196.6.a.e 1 28.d even 2 1
196.6.e.d 2 28.f even 6 2
196.6.e.g 2 28.g odd 6 2
256.6.b.c 2 16.e even 4 2
256.6.b.g 2 16.f odd 4 2
324.6.e.a 2 36.f odd 6 2
324.6.e.d 2 36.h even 6 2
400.6.a.d 1 5.b even 2 1
400.6.c.f 2 5.c odd 4 2
484.6.a.a 1 44.c even 2 1
576.6.a.bc 1 24.f even 2 1
576.6.a.bd 1 24.h odd 2 1
676.6.a.a 1 52.b odd 2 1
676.6.d.a 2 52.f even 4 2
784.6.a.d 1 7.b odd 2 1
900.6.a.h 1 60.h even 2 1
900.6.d.a 2 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 12$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(16))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 12$$
$5$ $$T - 54$$
$7$ $$T - 88$$
$11$ $$T + 540$$
$13$ $$T + 418$$
$17$ $$T - 594$$
$19$ $$T + 836$$
$23$ $$T - 4104$$
$29$ $$T + 594$$
$31$ $$T + 4256$$
$37$ $$T + 298$$
$41$ $$T - 17226$$
$43$ $$T - 12100$$
$47$ $$T - 1296$$
$53$ $$T - 19494$$
$59$ $$T - 7668$$
$61$ $$T + 34738$$
$67$ $$T + 21812$$
$71$ $$T - 46872$$
$73$ $$T - 67562$$
$79$ $$T - 76912$$
$83$ $$T + 67716$$
$89$ $$T - 29754$$
$97$ $$T + 122398$$