# Properties

 Label 1008.6.a.bm Level $1008$ Weight $6$ Character orbit 1008.a Self dual yes Analytic conductor $161.667$ Analytic rank $1$ Dimension $2$ Inner twists $2$

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{1099})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 1099$$ x^2 - 1099 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 252) 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{1099}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{5} + 49 q^{7}+O(q^{10})$$ q + b * q^5 + 49 * q^7 $$q + \beta q^{5} + 49 q^{7} - 5 \beta q^{11} - 54 q^{13} + 11 \beta q^{17} - 452 q^{19} - 11 \beta q^{23} + 1271 q^{25} - 20 \beta q^{29} - 3492 q^{31} + 49 \beta q^{35} + 3362 q^{37} - 159 \beta q^{41} - 9836 q^{43} + 30 \beta q^{47} + 2401 q^{49} - 66 \beta q^{53} - 21980 q^{55} - 622 \beta q^{59} + 31710 q^{61} - 54 \beta q^{65} - 39196 q^{67} - 977 \beta q^{71} + 37486 q^{73} - 245 \beta q^{77} - 62088 q^{79} + 568 \beta q^{83} + 48356 q^{85} - 1779 \beta q^{89} - 2646 q^{91} - 452 \beta q^{95} + 59038 q^{97} +O(q^{100})$$ q + b * q^5 + 49 * q^7 - 5*b * q^11 - 54 * q^13 + 11*b * q^17 - 452 * q^19 - 11*b * q^23 + 1271 * q^25 - 20*b * q^29 - 3492 * q^31 + 49*b * q^35 + 3362 * q^37 - 159*b * q^41 - 9836 * q^43 + 30*b * q^47 + 2401 * q^49 - 66*b * q^53 - 21980 * q^55 - 622*b * q^59 + 31710 * q^61 - 54*b * q^65 - 39196 * q^67 - 977*b * q^71 + 37486 * q^73 - 245*b * q^77 - 62088 * q^79 + 568*b * q^83 + 48356 * q^85 - 1779*b * q^89 - 2646 * q^91 - 452*b * q^95 + 59038 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 98 q^{7}+O(q^{10})$$ 2 * q + 98 * q^7 $$2 q + 98 q^{7} - 108 q^{13} - 904 q^{19} + 2542 q^{25} - 6984 q^{31} + 6724 q^{37} - 19672 q^{43} + 4802 q^{49} - 43960 q^{55} + 63420 q^{61} - 78392 q^{67} + 74972 q^{73} - 124176 q^{79} + 96712 q^{85} - 5292 q^{91} + 118076 q^{97}+O(q^{100})$$ 2 * q + 98 * q^7 - 108 * q^13 - 904 * q^19 + 2542 * q^25 - 6984 * q^31 + 6724 * q^37 - 19672 * q^43 + 4802 * q^49 - 43960 * q^55 + 63420 * q^61 - 78392 * q^67 + 74972 * q^73 - 124176 * q^79 + 96712 * q^85 - 5292 * q^91 + 118076 * 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
 −33.1512 33.1512
0 0 0 −66.3023 0 49.0000 0 0 0
1.2 0 0 0 66.3023 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.6.a.bm 2
3.b odd 2 1 inner 1008.6.a.bm 2
4.b odd 2 1 252.6.a.f 2
12.b even 2 1 252.6.a.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.6.a.f 2 4.b odd 2 1
252.6.a.f 2 12.b even 2 1
1008.6.a.bm 2 1.a even 1 1 trivial
1008.6.a.bm 2 3.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(1008))$$:

 $$T_{5}^{2} - 4396$$ T5^2 - 4396 $$T_{11}^{2} - 109900$$ T11^2 - 109900

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 4396$$
$7$ $$(T - 49)^{2}$$
$11$ $$T^{2} - 109900$$
$13$ $$(T + 54)^{2}$$
$17$ $$T^{2} - 531916$$
$19$ $$(T + 452)^{2}$$
$23$ $$T^{2} - 531916$$
$29$ $$T^{2} - 1758400$$
$31$ $$(T + 3492)^{2}$$
$37$ $$(T - 3362)^{2}$$
$41$ $$T^{2} - 111135276$$
$43$ $$(T + 9836)^{2}$$
$47$ $$T^{2} - 3956400$$
$53$ $$T^{2} - 19148976$$
$59$ $$T^{2} - 1700742064$$
$61$ $$(T - 31710)^{2}$$
$67$ $$(T + 39196)^{2}$$
$71$ $$T^{2} - 4196109484$$
$73$ $$(T - 37486)^{2}$$
$79$ $$(T + 62088)^{2}$$
$83$ $$T^{2} - 1418255104$$
$89$ $$T^{2} - 13912641036$$
$97$ $$(T - 59038)^{2}$$