# Properties

 Label 1008.6.a.b 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 14) 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)

 $$f(q)$$ $$=$$ $$q - 84 q^{5} - 49 q^{7}+O(q^{10})$$ q - 84 * q^5 - 49 * q^7 $$q - 84 q^{5} - 49 q^{7} - 336 q^{11} + 584 q^{13} + 1458 q^{17} - 470 q^{19} - 4200 q^{23} + 3931 q^{25} - 4866 q^{29} + 7372 q^{31} + 4116 q^{35} + 14330 q^{37} - 6222 q^{41} - 3704 q^{43} - 1812 q^{47} + 2401 q^{49} + 37242 q^{53} + 28224 q^{55} + 34302 q^{59} + 24476 q^{61} - 49056 q^{65} + 17452 q^{67} + 28224 q^{71} + 3602 q^{73} + 16464 q^{77} - 42872 q^{79} - 35202 q^{83} - 122472 q^{85} - 26730 q^{89} - 28616 q^{91} + 39480 q^{95} - 16978 q^{97}+O(q^{100})$$ q - 84 * q^5 - 49 * q^7 - 336 * q^11 + 584 * q^13 + 1458 * q^17 - 470 * q^19 - 4200 * q^23 + 3931 * q^25 - 4866 * q^29 + 7372 * q^31 + 4116 * q^35 + 14330 * q^37 - 6222 * q^41 - 3704 * q^43 - 1812 * q^47 + 2401 * q^49 + 37242 * q^53 + 28224 * q^55 + 34302 * q^59 + 24476 * q^61 - 49056 * q^65 + 17452 * q^67 + 28224 * q^71 + 3602 * q^73 + 16464 * q^77 - 42872 * q^79 - 35202 * q^83 - 122472 * q^85 - 26730 * q^89 - 28616 * q^91 + 39480 * q^95 - 16978 * 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 −84.0000 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.b 1
3.b odd 2 1 112.6.a.c 1
4.b odd 2 1 126.6.a.f 1
12.b even 2 1 14.6.a.a 1
21.c even 2 1 784.6.a.i 1
24.f even 2 1 448.6.a.e 1
24.h odd 2 1 448.6.a.l 1
28.d even 2 1 882.6.a.x 1
60.h even 2 1 350.6.a.i 1
60.l odd 4 2 350.6.c.d 2
84.h odd 2 1 98.6.a.a 1
84.j odd 6 2 98.6.c.d 2
84.n even 6 2 98.6.c.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.6.a.a 1 12.b even 2 1
98.6.a.a 1 84.h odd 2 1
98.6.c.c 2 84.n even 6 2
98.6.c.d 2 84.j odd 6 2
112.6.a.c 1 3.b odd 2 1
126.6.a.f 1 4.b odd 2 1
350.6.a.i 1 60.h even 2 1
350.6.c.d 2 60.l odd 4 2
448.6.a.e 1 24.f even 2 1
448.6.a.l 1 24.h odd 2 1
784.6.a.i 1 21.c even 2 1
882.6.a.x 1 28.d even 2 1
1008.6.a.b 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 $$S_{6}^{\mathrm{new}}(\Gamma_0(1008))$$:

 $$T_{5} + 84$$ T5 + 84 $$T_{11} + 336$$ T11 + 336

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 84$$
$7$ $$T + 49$$
$11$ $$T + 336$$
$13$ $$T - 584$$
$17$ $$T - 1458$$
$19$ $$T + 470$$
$23$ $$T + 4200$$
$29$ $$T + 4866$$
$31$ $$T - 7372$$
$37$ $$T - 14330$$
$41$ $$T + 6222$$
$43$ $$T + 3704$$
$47$ $$T + 1812$$
$53$ $$T - 37242$$
$59$ $$T - 34302$$
$61$ $$T - 24476$$
$67$ $$T - 17452$$
$71$ $$T - 28224$$
$73$ $$T - 3602$$
$79$ $$T + 42872$$
$83$ $$T + 35202$$
$89$ $$T + 26730$$
$97$ $$T + 16978$$