# Properties

 Label 1008.6.a.bc 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)

 $$f(q)$$ $$=$$ $$q + 106 q^{5} + 49 q^{7}+O(q^{10})$$ q + 106 * q^5 + 49 * q^7 $$q + 106 q^{5} + 49 q^{7} + 92 q^{11} + 670 q^{13} + 222 q^{17} + 908 q^{19} - 1176 q^{23} + 8111 q^{25} - 1118 q^{29} - 3696 q^{31} + 5194 q^{35} + 4182 q^{37} + 6662 q^{41} + 3700 q^{43} - 7056 q^{47} + 2401 q^{49} + 37578 q^{53} + 9752 q^{55} + 32700 q^{59} - 10802 q^{61} + 71020 q^{65} - 64996 q^{67} - 61320 q^{71} + 38922 q^{73} + 4508 q^{77} + 88096 q^{79} + 71892 q^{83} + 23532 q^{85} - 111818 q^{89} + 32830 q^{91} + 96248 q^{95} - 150846 q^{97}+O(q^{100})$$ q + 106 * q^5 + 49 * q^7 + 92 * q^11 + 670 * q^13 + 222 * q^17 + 908 * q^19 - 1176 * q^23 + 8111 * q^25 - 1118 * q^29 - 3696 * q^31 + 5194 * q^35 + 4182 * q^37 + 6662 * q^41 + 3700 * q^43 - 7056 * q^47 + 2401 * q^49 + 37578 * q^53 + 9752 * q^55 + 32700 * q^59 - 10802 * q^61 + 71020 * q^65 - 64996 * q^67 - 61320 * q^71 + 38922 * q^73 + 4508 * q^77 + 88096 * q^79 + 71892 * q^83 + 23532 * q^85 - 111818 * q^89 + 32830 * q^91 + 96248 * q^95 - 150846 * 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 0 106.000 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.bc 1
3.b odd 2 1 336.6.a.a 1
4.b odd 2 1 63.6.a.a 1
12.b even 2 1 21.6.a.d 1
28.d even 2 1 441.6.a.b 1
60.h even 2 1 525.6.a.a 1
60.l odd 4 2 525.6.d.a 2
84.h odd 2 1 147.6.a.g 1
84.j odd 6 2 147.6.e.b 2
84.n even 6 2 147.6.e.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.d 1 12.b even 2 1
63.6.a.a 1 4.b odd 2 1
147.6.a.g 1 84.h odd 2 1
147.6.e.a 2 84.n even 6 2
147.6.e.b 2 84.j odd 6 2
336.6.a.a 1 3.b odd 2 1
441.6.a.b 1 28.d even 2 1
525.6.a.a 1 60.h even 2 1
525.6.d.a 2 60.l odd 4 2
1008.6.a.bc 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 $$S_{6}^{\mathrm{new}}(\Gamma_0(1008))$$:

 $$T_{5} - 106$$ T5 - 106 $$T_{11} - 92$$ T11 - 92

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 106$$
$7$ $$T - 49$$
$11$ $$T - 92$$
$13$ $$T - 670$$
$17$ $$T - 222$$
$19$ $$T - 908$$
$23$ $$T + 1176$$
$29$ $$T + 1118$$
$31$ $$T + 3696$$
$37$ $$T - 4182$$
$41$ $$T - 6662$$
$43$ $$T - 3700$$
$47$ $$T + 7056$$
$53$ $$T - 37578$$
$59$ $$T - 32700$$
$61$ $$T + 10802$$
$67$ $$T + 64996$$
$71$ $$T + 61320$$
$73$ $$T - 38922$$
$79$ $$T - 88096$$
$83$ $$T - 71892$$
$89$ $$T + 111818$$
$97$ $$T + 150846$$