# Properties

 Label 1008.6.a.x Level $1008$ Weight $6$ Character orbit 1008.a Self dual yes Analytic conductor $161.667$ Analytic rank $1$ 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: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q + 54 q^{5} - 49 q^{7}+O(q^{10})$$ q + 54 * q^5 - 49 * q^7 $$q + 54 q^{5} - 49 q^{7} + 216 q^{11} + 998 q^{13} - 1302 q^{17} - 884 q^{19} - 2268 q^{23} - 209 q^{25} + 1482 q^{29} - 8360 q^{31} - 2646 q^{35} - 4714 q^{37} + 9786 q^{41} - 19436 q^{43} + 22200 q^{47} + 2401 q^{49} - 26790 q^{53} + 11664 q^{55} + 28092 q^{59} - 38866 q^{61} + 53892 q^{65} - 23948 q^{67} - 20628 q^{71} + 290 q^{73} - 10584 q^{77} + 99544 q^{79} + 19308 q^{83} - 70308 q^{85} - 36390 q^{89} - 48902 q^{91} - 47736 q^{95} - 79078 q^{97}+O(q^{100})$$ q + 54 * q^5 - 49 * q^7 + 216 * q^11 + 998 * q^13 - 1302 * q^17 - 884 * q^19 - 2268 * q^23 - 209 * q^25 + 1482 * q^29 - 8360 * q^31 - 2646 * q^35 - 4714 * q^37 + 9786 * q^41 - 19436 * q^43 + 22200 * q^47 + 2401 * q^49 - 26790 * q^53 + 11664 * q^55 + 28092 * q^59 - 38866 * q^61 + 53892 * q^65 - 23948 * q^67 - 20628 * q^71 + 290 * q^73 - 10584 * q^77 + 99544 * q^79 + 19308 * q^83 - 70308 * q^85 - 36390 * q^89 - 48902 * q^91 - 47736 * q^95 - 79078 * 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 54.0000 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.x 1
3.b odd 2 1 336.6.a.j 1
4.b odd 2 1 126.6.a.k 1
12.b even 2 1 42.6.a.a 1
28.d even 2 1 882.6.a.o 1
60.h even 2 1 1050.6.a.n 1
60.l odd 4 2 1050.6.g.o 2
84.h odd 2 1 294.6.a.h 1
84.j odd 6 2 294.6.e.h 2
84.n even 6 2 294.6.e.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.a.a 1 12.b even 2 1
126.6.a.k 1 4.b odd 2 1
294.6.a.h 1 84.h odd 2 1
294.6.e.h 2 84.j odd 6 2
294.6.e.r 2 84.n even 6 2
336.6.a.j 1 3.b odd 2 1
882.6.a.o 1 28.d even 2 1
1008.6.a.x 1 1.a even 1 1 trivial
1050.6.a.n 1 60.h even 2 1
1050.6.g.o 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 $$S_{6}^{\mathrm{new}}(\Gamma_0(1008))$$:

 $$T_{5} - 54$$ T5 - 54 $$T_{11} - 216$$ T11 - 216

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 54$$
$7$ $$T + 49$$
$11$ $$T - 216$$
$13$ $$T - 998$$
$17$ $$T + 1302$$
$19$ $$T + 884$$
$23$ $$T + 2268$$
$29$ $$T - 1482$$
$31$ $$T + 8360$$
$37$ $$T + 4714$$
$41$ $$T - 9786$$
$43$ $$T + 19436$$
$47$ $$T - 22200$$
$53$ $$T + 26790$$
$59$ $$T - 28092$$
$61$ $$T + 38866$$
$67$ $$T + 23948$$
$71$ $$T + 20628$$
$73$ $$T - 290$$
$79$ $$T - 99544$$
$83$ $$T - 19308$$
$89$ $$T + 36390$$
$97$ $$T + 79078$$