# Properties

 Label 1008.6.a.k 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)

 $$f(q)$$ $$=$$ $$q - 24 q^{5} - 49 q^{7}+O(q^{10})$$ q - 24 * q^5 - 49 * q^7 $$q - 24 q^{5} - 49 q^{7} + 66 q^{11} + 98 q^{13} + 216 q^{17} + 340 q^{19} - 1038 q^{23} - 2549 q^{25} + 2490 q^{29} + 7048 q^{31} + 1176 q^{35} - 12238 q^{37} - 6468 q^{41} + 15412 q^{43} + 20604 q^{47} + 2401 q^{49} - 32490 q^{53} - 1584 q^{55} + 34224 q^{59} + 35654 q^{61} - 2352 q^{65} - 12680 q^{67} - 42642 q^{71} + 33734 q^{73} - 3234 q^{77} + 85108 q^{79} - 106764 q^{83} - 5184 q^{85} - 34884 q^{89} - 4802 q^{91} - 8160 q^{95} + 18662 q^{97}+O(q^{100})$$ q - 24 * q^5 - 49 * q^7 + 66 * q^11 + 98 * q^13 + 216 * q^17 + 340 * q^19 - 1038 * q^23 - 2549 * q^25 + 2490 * q^29 + 7048 * q^31 + 1176 * q^35 - 12238 * q^37 - 6468 * q^41 + 15412 * q^43 + 20604 * q^47 + 2401 * q^49 - 32490 * q^53 - 1584 * q^55 + 34224 * q^59 + 35654 * q^61 - 2352 * q^65 - 12680 * q^67 - 42642 * q^71 + 33734 * q^73 - 3234 * q^77 + 85108 * q^79 - 106764 * q^83 - 5184 * q^85 - 34884 * q^89 - 4802 * q^91 - 8160 * q^95 + 18662 * 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 −24.0000 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.k 1
3.b odd 2 1 336.6.a.g 1
4.b odd 2 1 126.6.a.b 1
12.b even 2 1 42.6.a.f 1
28.d even 2 1 882.6.a.i 1
60.h even 2 1 1050.6.a.a 1
60.l odd 4 2 1050.6.g.m 2
84.h odd 2 1 294.6.a.i 1
84.j odd 6 2 294.6.e.f 2
84.n even 6 2 294.6.e.b 2

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

 $$T_{5} + 24$$ T5 + 24 $$T_{11} - 66$$ T11 - 66

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 24$$
$7$ $$T + 49$$
$11$ $$T - 66$$
$13$ $$T - 98$$
$17$ $$T - 216$$
$19$ $$T - 340$$
$23$ $$T + 1038$$
$29$ $$T - 2490$$
$31$ $$T - 7048$$
$37$ $$T + 12238$$
$41$ $$T + 6468$$
$43$ $$T - 15412$$
$47$ $$T - 20604$$
$53$ $$T + 32490$$
$59$ $$T - 34224$$
$61$ $$T - 35654$$
$67$ $$T + 12680$$
$71$ $$T + 42642$$
$73$ $$T - 33734$$
$79$ $$T - 85108$$
$83$ $$T + 106764$$
$89$ $$T + 34884$$
$97$ $$T - 18662$$