# Properties

 Label 1008.2.s.d Level $1008$ Weight $2$ Character orbit 1008.s Analytic conductor $8.049$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,2,Mod(289,1008)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1008.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1008 = 2^{4} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1008.s (of order $$3$$, degree $$2$$, not minimal)

## 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: no Analytic conductor: $$8.04892052375$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 2 \zeta_{6} q^{5} + (\zeta_{6} + 2) q^{7} +O(q^{10})$$ q - 2*z * q^5 + (z + 2) * q^7 $$q - 2 \zeta_{6} q^{5} + (\zeta_{6} + 2) q^{7} + ( - 2 \zeta_{6} + 2) q^{11} + q^{13} + \zeta_{6} q^{19} + ( - \zeta_{6} + 1) q^{25} - 4 q^{29} + ( - 9 \zeta_{6} + 9) q^{31} + ( - 6 \zeta_{6} + 2) q^{35} - 3 \zeta_{6} q^{37} + 10 q^{41} - 5 q^{43} + 6 \zeta_{6} q^{47} + (5 \zeta_{6} + 3) q^{49} + ( - 12 \zeta_{6} + 12) q^{53} - 4 q^{55} + ( - 12 \zeta_{6} + 12) q^{59} - 10 \zeta_{6} q^{61} - 2 \zeta_{6} q^{65} + (5 \zeta_{6} - 5) q^{67} - 6 q^{71} + ( - 3 \zeta_{6} + 3) q^{73} + ( - 4 \zeta_{6} + 6) q^{77} - \zeta_{6} q^{79} + 6 q^{83} + 16 \zeta_{6} q^{89} + (\zeta_{6} + 2) q^{91} + ( - 2 \zeta_{6} + 2) q^{95} - 6 q^{97} +O(q^{100})$$ q - 2*z * q^5 + (z + 2) * q^7 + (-2*z + 2) * q^11 + q^13 + z * q^19 + (-z + 1) * q^25 - 4 * q^29 + (-9*z + 9) * q^31 + (-6*z + 2) * q^35 - 3*z * q^37 + 10 * q^41 - 5 * q^43 + 6*z * q^47 + (5*z + 3) * q^49 + (-12*z + 12) * q^53 - 4 * q^55 + (-12*z + 12) * q^59 - 10*z * q^61 - 2*z * q^65 + (5*z - 5) * q^67 - 6 * q^71 + (-3*z + 3) * q^73 + (-4*z + 6) * q^77 - z * q^79 + 6 * q^83 + 16*z * q^89 + (z + 2) * q^91 + (-2*z + 2) * q^95 - 6 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} + 5 q^{7}+O(q^{10})$$ 2 * q - 2 * q^5 + 5 * q^7 $$2 q - 2 q^{5} + 5 q^{7} + 2 q^{11} + 2 q^{13} + q^{19} + q^{25} - 8 q^{29} + 9 q^{31} - 2 q^{35} - 3 q^{37} + 20 q^{41} - 10 q^{43} + 6 q^{47} + 11 q^{49} + 12 q^{53} - 8 q^{55} + 12 q^{59} - 10 q^{61} - 2 q^{65} - 5 q^{67} - 12 q^{71} + 3 q^{73} + 8 q^{77} - q^{79} + 12 q^{83} + 16 q^{89} + 5 q^{91} + 2 q^{95} - 12 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 + 5 * q^7 + 2 * q^11 + 2 * q^13 + q^19 + q^25 - 8 * q^29 + 9 * q^31 - 2 * q^35 - 3 * q^37 + 20 * q^41 - 10 * q^43 + 6 * q^47 + 11 * q^49 + 12 * q^53 - 8 * q^55 + 12 * q^59 - 10 * q^61 - 2 * q^65 - 5 * q^67 - 12 * q^71 + 3 * q^73 + 8 * q^77 - q^79 + 12 * q^83 + 16 * q^89 + 5 * q^91 + 2 * q^95 - 12 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$577$$ $$757$$ $$785$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 −1.00000 + 1.73205i 0 2.50000 0.866025i 0 0 0
865.1 0 0 0 −1.00000 1.73205i 0 2.50000 + 0.866025i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.s.d 2
3.b odd 2 1 336.2.q.f 2
4.b odd 2 1 63.2.e.b 2
7.c even 3 1 inner 1008.2.s.d 2
7.c even 3 1 7056.2.a.bp 1
7.d odd 6 1 7056.2.a.m 1
12.b even 2 1 21.2.e.a 2
21.c even 2 1 2352.2.q.c 2
21.g even 6 1 2352.2.a.w 1
21.g even 6 1 2352.2.q.c 2
21.h odd 6 1 336.2.q.f 2
21.h odd 6 1 2352.2.a.d 1
24.f even 2 1 1344.2.q.m 2
24.h odd 2 1 1344.2.q.c 2
28.d even 2 1 441.2.e.e 2
28.f even 6 1 441.2.a.a 1
28.f even 6 1 441.2.e.e 2
28.g odd 6 1 63.2.e.b 2
28.g odd 6 1 441.2.a.b 1
36.f odd 6 1 567.2.g.f 2
36.f odd 6 1 567.2.h.a 2
36.h even 6 1 567.2.g.a 2
36.h even 6 1 567.2.h.f 2
60.h even 2 1 525.2.i.e 2
60.l odd 4 2 525.2.r.e 4
84.h odd 2 1 147.2.e.a 2
84.j odd 6 1 147.2.a.b 1
84.j odd 6 1 147.2.e.a 2
84.n even 6 1 21.2.e.a 2
84.n even 6 1 147.2.a.c 1
168.s odd 6 1 1344.2.q.c 2
168.s odd 6 1 9408.2.a.cv 1
168.v even 6 1 1344.2.q.m 2
168.v even 6 1 9408.2.a.bg 1
168.ba even 6 1 9408.2.a.k 1
168.be odd 6 1 9408.2.a.bz 1
252.o even 6 1 567.2.h.f 2
252.u odd 6 1 567.2.g.f 2
252.bb even 6 1 567.2.g.a 2
252.bl odd 6 1 567.2.h.a 2
420.ba even 6 1 525.2.i.e 2
420.ba even 6 1 3675.2.a.a 1
420.be odd 6 1 3675.2.a.c 1
420.bp odd 12 2 525.2.r.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.e.a 2 12.b even 2 1
21.2.e.a 2 84.n even 6 1
63.2.e.b 2 4.b odd 2 1
63.2.e.b 2 28.g odd 6 1
147.2.a.b 1 84.j odd 6 1
147.2.a.c 1 84.n even 6 1
147.2.e.a 2 84.h odd 2 1
147.2.e.a 2 84.j odd 6 1
336.2.q.f 2 3.b odd 2 1
336.2.q.f 2 21.h odd 6 1
441.2.a.a 1 28.f even 6 1
441.2.a.b 1 28.g odd 6 1
441.2.e.e 2 28.d even 2 1
441.2.e.e 2 28.f even 6 1
525.2.i.e 2 60.h even 2 1
525.2.i.e 2 420.ba even 6 1
525.2.r.e 4 60.l odd 4 2
525.2.r.e 4 420.bp odd 12 2
567.2.g.a 2 36.h even 6 1
567.2.g.a 2 252.bb even 6 1
567.2.g.f 2 36.f odd 6 1
567.2.g.f 2 252.u odd 6 1
567.2.h.a 2 36.f odd 6 1
567.2.h.a 2 252.bl odd 6 1
567.2.h.f 2 36.h even 6 1
567.2.h.f 2 252.o even 6 1
1008.2.s.d 2 1.a even 1 1 trivial
1008.2.s.d 2 7.c even 3 1 inner
1344.2.q.c 2 24.h odd 2 1
1344.2.q.c 2 168.s odd 6 1
1344.2.q.m 2 24.f even 2 1
1344.2.q.m 2 168.v even 6 1
2352.2.a.d 1 21.h odd 6 1
2352.2.a.w 1 21.g even 6 1
2352.2.q.c 2 21.c even 2 1
2352.2.q.c 2 21.g even 6 1
3675.2.a.a 1 420.ba even 6 1
3675.2.a.c 1 420.be odd 6 1
7056.2.a.m 1 7.d odd 6 1
7056.2.a.bp 1 7.c even 3 1
9408.2.a.k 1 168.ba even 6 1
9408.2.a.bg 1 168.v even 6 1
9408.2.a.bz 1 168.be odd 6 1
9408.2.a.cv 1 168.s odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1008, [\chi])$$:

 $$T_{5}^{2} + 2T_{5} + 4$$ T5^2 + 2*T5 + 4 $$T_{11}^{2} - 2T_{11} + 4$$ T11^2 - 2*T11 + 4 $$T_{13} - 1$$ T13 - 1 $$T_{17}$$ T17

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 2T + 4$$
$7$ $$T^{2} - 5T + 7$$
$11$ $$T^{2} - 2T + 4$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} - T + 1$$
$23$ $$T^{2}$$
$29$ $$(T + 4)^{2}$$
$31$ $$T^{2} - 9T + 81$$
$37$ $$T^{2} + 3T + 9$$
$41$ $$(T - 10)^{2}$$
$43$ $$(T + 5)^{2}$$
$47$ $$T^{2} - 6T + 36$$
$53$ $$T^{2} - 12T + 144$$
$59$ $$T^{2} - 12T + 144$$
$61$ $$T^{2} + 10T + 100$$
$67$ $$T^{2} + 5T + 25$$
$71$ $$(T + 6)^{2}$$
$73$ $$T^{2} - 3T + 9$$
$79$ $$T^{2} + T + 1$$
$83$ $$(T - 6)^{2}$$
$89$ $$T^{2} - 16T + 256$$
$97$ $$(T + 6)^{2}$$