# Properties

 Label 1008.6.a.a 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 21) 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)

 $$f(q)$$ $$=$$ $$q - 94 q^{5} + 49 q^{7}+O(q^{10})$$ q - 94 * q^5 + 49 * q^7 $$q - 94 q^{5} + 49 q^{7} + 52 q^{11} - 770 q^{13} + 2022 q^{17} - 1732 q^{19} - 576 q^{23} + 5711 q^{25} - 5518 q^{29} - 6336 q^{31} - 4606 q^{35} - 7338 q^{37} + 3262 q^{41} - 5420 q^{43} + 864 q^{47} + 2401 q^{49} - 4182 q^{53} - 4888 q^{55} - 11220 q^{59} - 45602 q^{61} + 72380 q^{65} - 1396 q^{67} + 18720 q^{71} + 46362 q^{73} + 2548 q^{77} - 97424 q^{79} - 81228 q^{83} - 190068 q^{85} + 3182 q^{89} - 37730 q^{91} + 162808 q^{95} + 4914 q^{97}+O(q^{100})$$ q - 94 * q^5 + 49 * q^7 + 52 * q^11 - 770 * q^13 + 2022 * q^17 - 1732 * q^19 - 576 * q^23 + 5711 * q^25 - 5518 * q^29 - 6336 * q^31 - 4606 * q^35 - 7338 * q^37 + 3262 * q^41 - 5420 * q^43 + 864 * q^47 + 2401 * q^49 - 4182 * q^53 - 4888 * q^55 - 11220 * q^59 - 45602 * q^61 + 72380 * q^65 - 1396 * q^67 + 18720 * q^71 + 46362 * q^73 + 2548 * q^77 - 97424 * q^79 - 81228 * q^83 - 190068 * q^85 + 3182 * q^89 - 37730 * q^91 + 162808 * q^95 + 4914 * 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 −94.0000 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.a 1
3.b odd 2 1 336.6.a.i 1
4.b odd 2 1 63.6.a.b 1
12.b even 2 1 21.6.a.c 1
28.d even 2 1 441.6.a.c 1
60.h even 2 1 525.6.a.b 1
60.l odd 4 2 525.6.d.c 2
84.h odd 2 1 147.6.a.f 1
84.j odd 6 2 147.6.e.d 2
84.n even 6 2 147.6.e.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.c 1 12.b even 2 1
63.6.a.b 1 4.b odd 2 1
147.6.a.f 1 84.h odd 2 1
147.6.e.c 2 84.n even 6 2
147.6.e.d 2 84.j odd 6 2
336.6.a.i 1 3.b odd 2 1
441.6.a.c 1 28.d even 2 1
525.6.a.b 1 60.h even 2 1
525.6.d.c 2 60.l odd 4 2
1008.6.a.a 1 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{5} + 94$$ T5 + 94 $$T_{11} - 52$$ T11 - 52

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 94$$
$7$ $$T - 49$$
$11$ $$T - 52$$
$13$ $$T + 770$$
$17$ $$T - 2022$$
$19$ $$T + 1732$$
$23$ $$T + 576$$
$29$ $$T + 5518$$
$31$ $$T + 6336$$
$37$ $$T + 7338$$
$41$ $$T - 3262$$
$43$ $$T + 5420$$
$47$ $$T - 864$$
$53$ $$T + 4182$$
$59$ $$T + 11220$$
$61$ $$T + 45602$$
$67$ $$T + 1396$$
$71$ $$T - 18720$$
$73$ $$T - 46362$$
$79$ $$T + 97424$$
$83$ $$T + 81228$$
$89$ $$T - 3182$$
$97$ $$T - 4914$$