# Properties

 Label 1008.6.a.f 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 126) 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 $$q - 54 q^{5} - 49 q^{7} + 594 q^{11} + 26 q^{13} + 534 q^{17} + 3004 q^{19} + 3510 q^{23} - 209 q^{25} - 4296 q^{29} - 8036 q^{31} + 2646 q^{35} - 502 q^{37} - 9870 q^{41} - 9068 q^{43} + 1140 q^{47} + 2401 q^{49} - 28356 q^{53} - 32076 q^{55} - 8196 q^{59} + 29822 q^{61} - 1404 q^{65} + 62884 q^{67} - 34398 q^{71} + 56990 q^{73} - 29106 q^{77} - 49496 q^{79} - 52512 q^{83} - 28836 q^{85} + 48282 q^{89} - 1274 q^{91} - 162216 q^{95} - 83938 q^{97}+O(q^{100})$$ q - 54 * q^5 - 49 * q^7 + 594 * q^11 + 26 * q^13 + 534 * q^17 + 3004 * q^19 + 3510 * q^23 - 209 * q^25 - 4296 * q^29 - 8036 * q^31 + 2646 * q^35 - 502 * q^37 - 9870 * q^41 - 9068 * q^43 + 1140 * q^47 + 2401 * q^49 - 28356 * q^53 - 32076 * q^55 - 8196 * q^59 + 29822 * q^61 - 1404 * q^65 + 62884 * q^67 - 34398 * q^71 + 56990 * q^73 - 29106 * q^77 - 49496 * q^79 - 52512 * q^83 - 28836 * q^85 + 48282 * q^89 - 1274 * q^91 - 162216 * q^95 - 83938 * 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.f 1
3.b odd 2 1 1008.6.a.w 1
4.b odd 2 1 126.6.a.g yes 1
12.b even 2 1 126.6.a.e 1
28.d even 2 1 882.6.a.w 1
84.h odd 2 1 882.6.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.6.a.e 1 12.b even 2 1
126.6.a.g yes 1 4.b odd 2 1
882.6.a.b 1 84.h odd 2 1
882.6.a.w 1 28.d even 2 1
1008.6.a.f 1 1.a even 1 1 trivial
1008.6.a.w 1 3.b odd 2 1

## 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 $$T_{11} - 594$$ T11 - 594

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 54$$
$7$ $$T + 49$$
$11$ $$T - 594$$
$13$ $$T - 26$$
$17$ $$T - 534$$
$19$ $$T - 3004$$
$23$ $$T - 3510$$
$29$ $$T + 4296$$
$31$ $$T + 8036$$
$37$ $$T + 502$$
$41$ $$T + 9870$$
$43$ $$T + 9068$$
$47$ $$T - 1140$$
$53$ $$T + 28356$$
$59$ $$T + 8196$$
$61$ $$T - 29822$$
$67$ $$T - 62884$$
$71$ $$T + 34398$$
$73$ $$T - 56990$$
$79$ $$T + 49496$$
$83$ $$T + 52512$$
$89$ $$T - 48282$$
$97$ $$T + 83938$$