# Properties

 Label 1008.6.a.j 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)

 $$f(q)$$ $$=$$ $$q - 26 q^{5} + 49 q^{7}+O(q^{10})$$ q - 26 * q^5 + 49 * q^7 $$q - 26 q^{5} + 49 q^{7} + 664 q^{11} + 318 q^{13} - 1582 q^{17} - 236 q^{19} + 2212 q^{23} - 2449 q^{25} + 4954 q^{29} + 7128 q^{31} - 1274 q^{35} + 4358 q^{37} - 10542 q^{41} + 8452 q^{43} + 5352 q^{47} + 2401 q^{49} + 33354 q^{53} - 17264 q^{55} - 15436 q^{59} - 36762 q^{61} - 8268 q^{65} - 40972 q^{67} - 9092 q^{71} - 73454 q^{73} + 32536 q^{77} - 89400 q^{79} - 6428 q^{83} + 41132 q^{85} + 122658 q^{89} + 15582 q^{91} + 6136 q^{95} + 21370 q^{97}+O(q^{100})$$ q - 26 * q^5 + 49 * q^7 + 664 * q^11 + 318 * q^13 - 1582 * q^17 - 236 * q^19 + 2212 * q^23 - 2449 * q^25 + 4954 * q^29 + 7128 * q^31 - 1274 * q^35 + 4358 * q^37 - 10542 * q^41 + 8452 * q^43 + 5352 * q^47 + 2401 * q^49 + 33354 * q^53 - 17264 * q^55 - 15436 * q^59 - 36762 * q^61 - 8268 * q^65 - 40972 * q^67 - 9092 * q^71 - 73454 * q^73 + 32536 * q^77 - 89400 * q^79 - 6428 * q^83 + 41132 * q^85 + 122658 * q^89 + 15582 * q^91 + 6136 * q^95 + 21370 * 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 −26.0000 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.j 1
3.b odd 2 1 336.6.a.h 1
4.b odd 2 1 126.6.a.i 1
12.b even 2 1 42.6.a.d 1
28.d even 2 1 882.6.a.s 1
60.h even 2 1 1050.6.a.k 1
60.l odd 4 2 1050.6.g.i 2
84.h odd 2 1 294.6.a.b 1
84.j odd 6 2 294.6.e.p 2
84.n even 6 2 294.6.e.i 2

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

 $$T_{5} + 26$$ T5 + 26 $$T_{11} - 664$$ T11 - 664

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 26$$
$7$ $$T - 49$$
$11$ $$T - 664$$
$13$ $$T - 318$$
$17$ $$T + 1582$$
$19$ $$T + 236$$
$23$ $$T - 2212$$
$29$ $$T - 4954$$
$31$ $$T - 7128$$
$37$ $$T - 4358$$
$41$ $$T + 10542$$
$43$ $$T - 8452$$
$47$ $$T - 5352$$
$53$ $$T - 33354$$
$59$ $$T + 15436$$
$61$ $$T + 36762$$
$67$ $$T + 40972$$
$71$ $$T + 9092$$
$73$ $$T + 73454$$
$79$ $$T + 89400$$
$83$ $$T + 6428$$
$89$ $$T - 122658$$
$97$ $$T - 21370$$