# Properties

 Label 1008.6.a.bl 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: $$\Q(\sqrt{7})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 7$$ x^2 - 7 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 63) 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 $$\beta = 2\sqrt{7}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 7 \beta q^{5} + 49 q^{7}+O(q^{10})$$ q + 7*b * q^5 + 49 * q^7 $$q + 7 \beta q^{5} + 49 q^{7} - 43 \beta q^{11} - 518 q^{13} - 147 \beta q^{17} + 1484 q^{19} + 723 \beta q^{23} - 1753 q^{25} - 20 \beta q^{29} + 2604 q^{31} + 343 \beta q^{35} + 402 q^{37} + 119 \beta q^{41} - 6956 q^{43} - 5166 \beta q^{47} + 2401 q^{49} - 5766 \beta q^{53} - 8428 q^{55} + 8526 \beta q^{59} - 22610 q^{61} - 3626 \beta q^{65} + 13124 q^{67} - 9111 \beta q^{71} - 82866 q^{73} - 2107 \beta q^{77} + 81112 q^{79} - 12600 \beta q^{83} - 28812 q^{85} + 23947 \beta q^{89} - 25382 q^{91} + 10388 \beta q^{95} - 10626 q^{97} +O(q^{100})$$ q + 7*b * q^5 + 49 * q^7 - 43*b * q^11 - 518 * q^13 - 147*b * q^17 + 1484 * q^19 + 723*b * q^23 - 1753 * q^25 - 20*b * q^29 + 2604 * q^31 + 343*b * q^35 + 402 * q^37 + 119*b * q^41 - 6956 * q^43 - 5166*b * q^47 + 2401 * q^49 - 5766*b * q^53 - 8428 * q^55 + 8526*b * q^59 - 22610 * q^61 - 3626*b * q^65 + 13124 * q^67 - 9111*b * q^71 - 82866 * q^73 - 2107*b * q^77 + 81112 * q^79 - 12600*b * q^83 - 28812 * q^85 + 23947*b * q^89 - 25382 * q^91 + 10388*b * q^95 - 10626 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 98 q^{7}+O(q^{10})$$ 2 * q + 98 * q^7 $$2 q + 98 q^{7} - 1036 q^{13} + 2968 q^{19} - 3506 q^{25} + 5208 q^{31} + 804 q^{37} - 13912 q^{43} + 4802 q^{49} - 16856 q^{55} - 45220 q^{61} + 26248 q^{67} - 165732 q^{73} + 162224 q^{79} - 57624 q^{85} - 50764 q^{91} - 21252 q^{97}+O(q^{100})$$ 2 * q + 98 * q^7 - 1036 * q^13 + 2968 * q^19 - 3506 * q^25 + 5208 * q^31 + 804 * q^37 - 13912 * q^43 + 4802 * q^49 - 16856 * q^55 - 45220 * q^61 + 26248 * q^67 - 165732 * q^73 + 162224 * q^79 - 57624 * q^85 - 50764 * q^91 - 21252 * 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
 −2.64575 2.64575
0 0 0 −37.0405 0 49.0000 0 0 0
1.2 0 0 0 37.0405 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.bl 2
3.b odd 2 1 inner 1008.6.a.bl 2
4.b odd 2 1 63.6.a.g 2
12.b even 2 1 63.6.a.g 2
28.d even 2 1 441.6.a.p 2
84.h odd 2 1 441.6.a.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.6.a.g 2 4.b odd 2 1
63.6.a.g 2 12.b even 2 1
441.6.a.p 2 28.d even 2 1
441.6.a.p 2 84.h odd 2 1
1008.6.a.bl 2 1.a even 1 1 trivial
1008.6.a.bl 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 $$S_{6}^{\mathrm{new}}(\Gamma_0(1008))$$:

 $$T_{5}^{2} - 1372$$ T5^2 - 1372 $$T_{11}^{2} - 51772$$ T11^2 - 51772

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 1372$$
$7$ $$(T - 49)^{2}$$
$11$ $$T^{2} - 51772$$
$13$ $$(T + 518)^{2}$$
$17$ $$T^{2} - 605052$$
$19$ $$(T - 1484)^{2}$$
$23$ $$T^{2} - 14636412$$
$29$ $$T^{2} - 11200$$
$31$ $$(T - 2604)^{2}$$
$37$ $$(T - 402)^{2}$$
$41$ $$T^{2} - 396508$$
$43$ $$(T + 6956)^{2}$$
$47$ $$T^{2} - 747251568$$
$53$ $$T^{2} - 930909168$$
$59$ $$T^{2} - 2035394928$$
$61$ $$(T + 22610)^{2}$$
$67$ $$(T - 13124)^{2}$$
$71$ $$T^{2} - 2324288988$$
$73$ $$(T + 82866)^{2}$$
$79$ $$(T - 81112)^{2}$$
$83$ $$T^{2} - 4445280000$$
$89$ $$T^{2} - 16056846652$$
$97$ $$(T + 10626)^{2}$$