# Properties

 Label 1008.6.a.d Level $1008$ Weight $6$ Character orbit 1008.a Self dual yes Analytic conductor $161.667$ 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] = [1008,6,Mod(1,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1008.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1008 = 2^{4} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 1008.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: $$161.666890371$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) 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)

 $$f(q)$$ $$=$$ $$q - 76 q^{5} + 49 q^{7}+O(q^{10})$$ q - 76 * q^5 + 49 * q^7 $$q - 76 q^{5} + 49 q^{7} + 650 q^{11} + 762 q^{13} + 556 q^{17} + 2452 q^{19} - 2950 q^{23} + 2651 q^{25} + 674 q^{29} + 3024 q^{31} - 3724 q^{35} + 7730 q^{37} + 17016 q^{41} - 21836 q^{43} - 23940 q^{47} + 2401 q^{49} - 15594 q^{53} - 49400 q^{55} + 5608 q^{59} + 150 q^{61} - 57912 q^{65} + 43784 q^{67} - 39178 q^{71} - 23570 q^{73} + 31850 q^{77} + 17892 q^{79} + 38972 q^{83} - 42256 q^{85} - 6024 q^{89} + 37338 q^{91} - 186352 q^{95} + 108430 q^{97}+O(q^{100})$$ q - 76 * q^5 + 49 * q^7 + 650 * q^11 + 762 * q^13 + 556 * q^17 + 2452 * q^19 - 2950 * q^23 + 2651 * q^25 + 674 * q^29 + 3024 * q^31 - 3724 * q^35 + 7730 * q^37 + 17016 * q^41 - 21836 * q^43 - 23940 * q^47 + 2401 * q^49 - 15594 * q^53 - 49400 * q^55 + 5608 * q^59 + 150 * q^61 - 57912 * q^65 + 43784 * q^67 - 39178 * q^71 - 23570 * q^73 + 31850 * q^77 + 17892 * q^79 + 38972 * q^83 - 42256 * q^85 - 6024 * q^89 + 37338 * q^91 - 186352 * q^95 + 108430 * q^97

## 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 0 0 −76.0000 0 49.0000 0 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$-1$$

## 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 1008.6.a.d 1
3.b odd 2 1 336.6.a.q 1
4.b odd 2 1 126.6.a.a 1
12.b even 2 1 42.6.a.e 1
28.d even 2 1 882.6.a.j 1
60.h even 2 1 1050.6.a.f 1
60.l odd 4 2 1050.6.g.h 2
84.h odd 2 1 294.6.a.k 1
84.j odd 6 2 294.6.e.c 2
84.n even 6 2 294.6.e.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.a.e 1 12.b even 2 1
126.6.a.a 1 4.b odd 2 1
294.6.a.k 1 84.h odd 2 1
294.6.e.c 2 84.j odd 6 2
294.6.e.d 2 84.n even 6 2
336.6.a.q 1 3.b odd 2 1
882.6.a.j 1 28.d even 2 1
1008.6.a.d 1 1.a even 1 1 trivial
1050.6.a.f 1 60.h even 2 1
1050.6.g.h 2 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(1008))$$:

 $$T_{5} + 76$$ T5 + 76 $$T_{11} - 650$$ T11 - 650

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 76$$
$7$ $$T - 49$$
$11$ $$T - 650$$
$13$ $$T - 762$$
$17$ $$T - 556$$
$19$ $$T - 2452$$
$23$ $$T + 2950$$
$29$ $$T - 674$$
$31$ $$T - 3024$$
$37$ $$T - 7730$$
$41$ $$T - 17016$$
$43$ $$T + 21836$$
$47$ $$T + 23940$$
$53$ $$T + 15594$$
$59$ $$T - 5608$$
$61$ $$T - 150$$
$67$ $$T - 43784$$
$71$ $$T + 39178$$
$73$ $$T + 23570$$
$79$ $$T - 17892$$
$83$ $$T - 38972$$
$89$ $$T + 6024$$
$97$ $$T - 108430$$