# Properties

 Label 1008.2.t.d Level $1008$ Weight $2$ Character orbit 1008.t 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(193,1008)]

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, 2, 4]))

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

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

chi := DirichletCharacter("1008.193");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1008 = 2^{4} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1008.t (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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 2) q^{3} - q^{5} + ( - 3 \zeta_{6} + 1) q^{7} + ( - 3 \zeta_{6} + 3) q^{9}+O(q^{10})$$ q + (-z + 2) * q^3 - q^5 + (-3*z + 1) * q^7 + (-3*z + 3) * q^9 $$q + ( - \zeta_{6} + 2) q^{3} - q^{5} + ( - 3 \zeta_{6} + 1) q^{7} + ( - 3 \zeta_{6} + 3) q^{9} - 5 q^{11} + ( - 5 \zeta_{6} + 5) q^{13} + (\zeta_{6} - 2) q^{15} + (3 \zeta_{6} - 3) q^{17} + \zeta_{6} q^{19} + ( - 4 \zeta_{6} - 1) q^{21} - 3 q^{23} - 4 q^{25} + ( - 6 \zeta_{6} + 3) q^{27} + \zeta_{6} q^{29} + (5 \zeta_{6} - 10) q^{33} + (3 \zeta_{6} - 1) q^{35} - 3 \zeta_{6} q^{37} + ( - 10 \zeta_{6} + 5) q^{39} + ( - 5 \zeta_{6} + 5) q^{41} - \zeta_{6} q^{43} + (3 \zeta_{6} - 3) q^{45} + (3 \zeta_{6} - 8) q^{49} + (6 \zeta_{6} - 3) q^{51} + ( - 9 \zeta_{6} + 9) q^{53} + 5 q^{55} + (\zeta_{6} + 1) q^{57} + ( - 14 \zeta_{6} + 14) q^{61} + ( - 3 \zeta_{6} - 6) q^{63} + (5 \zeta_{6} - 5) q^{65} + 4 \zeta_{6} q^{67} + (3 \zeta_{6} - 6) q^{69} + 12 q^{71} + (3 \zeta_{6} - 3) q^{73} + (4 \zeta_{6} - 8) q^{75} + (15 \zeta_{6} - 5) q^{77} + ( - 8 \zeta_{6} + 8) q^{79} - 9 \zeta_{6} q^{81} - 9 \zeta_{6} q^{83} + ( - 3 \zeta_{6} + 3) q^{85} + (\zeta_{6} + 1) q^{87} + 13 \zeta_{6} q^{89} + ( - 5 \zeta_{6} - 10) q^{91} - \zeta_{6} q^{95} + 9 \zeta_{6} q^{97} + (15 \zeta_{6} - 15) q^{99} +O(q^{100})$$ q + (-z + 2) * q^3 - q^5 + (-3*z + 1) * q^7 + (-3*z + 3) * q^9 - 5 * q^11 + (-5*z + 5) * q^13 + (z - 2) * q^15 + (3*z - 3) * q^17 + z * q^19 + (-4*z - 1) * q^21 - 3 * q^23 - 4 * q^25 + (-6*z + 3) * q^27 + z * q^29 + (5*z - 10) * q^33 + (3*z - 1) * q^35 - 3*z * q^37 + (-10*z + 5) * q^39 + (-5*z + 5) * q^41 - z * q^43 + (3*z - 3) * q^45 + (3*z - 8) * q^49 + (6*z - 3) * q^51 + (-9*z + 9) * q^53 + 5 * q^55 + (z + 1) * q^57 + (-14*z + 14) * q^61 + (-3*z - 6) * q^63 + (5*z - 5) * q^65 + 4*z * q^67 + (3*z - 6) * q^69 + 12 * q^71 + (3*z - 3) * q^73 + (4*z - 8) * q^75 + (15*z - 5) * q^77 + (-8*z + 8) * q^79 - 9*z * q^81 - 9*z * q^83 + (-3*z + 3) * q^85 + (z + 1) * q^87 + 13*z * q^89 + (-5*z - 10) * q^91 - z * q^95 + 9*z * q^97 + (15*z - 15) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{3} - 2 q^{5} - q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q + 3 * q^3 - 2 * q^5 - q^7 + 3 * q^9 $$2 q + 3 q^{3} - 2 q^{5} - q^{7} + 3 q^{9} - 10 q^{11} + 5 q^{13} - 3 q^{15} - 3 q^{17} + q^{19} - 6 q^{21} - 6 q^{23} - 8 q^{25} + q^{29} - 15 q^{33} + q^{35} - 3 q^{37} + 5 q^{41} - q^{43} - 3 q^{45} - 13 q^{49} + 9 q^{53} + 10 q^{55} + 3 q^{57} + 14 q^{61} - 15 q^{63} - 5 q^{65} + 4 q^{67} - 9 q^{69} + 24 q^{71} - 3 q^{73} - 12 q^{75} + 5 q^{77} + 8 q^{79} - 9 q^{81} - 9 q^{83} + 3 q^{85} + 3 q^{87} + 13 q^{89} - 25 q^{91} - q^{95} + 9 q^{97} - 15 q^{99}+O(q^{100})$$ 2 * q + 3 * q^3 - 2 * q^5 - q^7 + 3 * q^9 - 10 * q^11 + 5 * q^13 - 3 * q^15 - 3 * q^17 + q^19 - 6 * q^21 - 6 * q^23 - 8 * q^25 + q^29 - 15 * q^33 + q^35 - 3 * q^37 + 5 * q^41 - q^43 - 3 * q^45 - 13 * q^49 + 9 * q^53 + 10 * q^55 + 3 * q^57 + 14 * q^61 - 15 * q^63 - 5 * q^65 + 4 * q^67 - 9 * q^69 + 24 * q^71 - 3 * q^73 - 12 * q^75 + 5 * q^77 + 8 * q^79 - 9 * q^81 - 9 * q^83 + 3 * q^85 + 3 * q^87 + 13 * q^89 - 25 * q^91 - q^95 + 9 * q^97 - 15 * q^99

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.50000 0.866025i 0 −1.00000 0 −0.500000 2.59808i 0 1.50000 2.59808i 0
961.1 0 1.50000 + 0.866025i 0 −1.00000 0 −0.500000 + 2.59808i 0 1.50000 + 2.59808i 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
63.g 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.t.d 2
3.b odd 2 1 3024.2.t.d 2
4.b odd 2 1 63.2.g.a 2
7.c even 3 1 1008.2.q.c 2
9.c even 3 1 1008.2.q.c 2
9.d odd 6 1 3024.2.q.b 2
12.b even 2 1 189.2.g.a 2
21.h odd 6 1 3024.2.q.b 2
28.d even 2 1 441.2.g.a 2
28.f even 6 1 441.2.f.a 2
28.f even 6 1 441.2.h.a 2
28.g odd 6 1 63.2.h.a yes 2
28.g odd 6 1 441.2.f.b 2
36.f odd 6 1 63.2.h.a yes 2
36.f odd 6 1 567.2.e.a 2
36.h even 6 1 189.2.h.a 2
36.h even 6 1 567.2.e.b 2
63.g even 3 1 inner 1008.2.t.d 2
63.n odd 6 1 3024.2.t.d 2
84.h odd 2 1 1323.2.g.a 2
84.j odd 6 1 1323.2.f.b 2
84.j odd 6 1 1323.2.h.a 2
84.n even 6 1 189.2.h.a 2
84.n even 6 1 1323.2.f.a 2
252.n even 6 1 441.2.g.a 2
252.n even 6 1 3969.2.a.f 1
252.o even 6 1 189.2.g.a 2
252.o even 6 1 3969.2.a.c 1
252.r odd 6 1 1323.2.f.b 2
252.s odd 6 1 1323.2.h.a 2
252.u odd 6 1 441.2.f.b 2
252.u odd 6 1 567.2.e.a 2
252.bb even 6 1 567.2.e.b 2
252.bb even 6 1 1323.2.f.a 2
252.bi even 6 1 441.2.h.a 2
252.bj even 6 1 441.2.f.a 2
252.bl odd 6 1 63.2.g.a 2
252.bl odd 6 1 3969.2.a.d 1
252.bn odd 6 1 1323.2.g.a 2
252.bn odd 6 1 3969.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.a 2 4.b odd 2 1
63.2.g.a 2 252.bl odd 6 1
63.2.h.a yes 2 28.g odd 6 1
63.2.h.a yes 2 36.f odd 6 1
189.2.g.a 2 12.b even 2 1
189.2.g.a 2 252.o even 6 1
189.2.h.a 2 36.h even 6 1
189.2.h.a 2 84.n even 6 1
441.2.f.a 2 28.f even 6 1
441.2.f.a 2 252.bj even 6 1
441.2.f.b 2 28.g odd 6 1
441.2.f.b 2 252.u odd 6 1
441.2.g.a 2 28.d even 2 1
441.2.g.a 2 252.n even 6 1
441.2.h.a 2 28.f even 6 1
441.2.h.a 2 252.bi even 6 1
567.2.e.a 2 36.f odd 6 1
567.2.e.a 2 252.u odd 6 1
567.2.e.b 2 36.h even 6 1
567.2.e.b 2 252.bb even 6 1
1008.2.q.c 2 7.c even 3 1
1008.2.q.c 2 9.c even 3 1
1008.2.t.d 2 1.a even 1 1 trivial
1008.2.t.d 2 63.g even 3 1 inner
1323.2.f.a 2 84.n even 6 1
1323.2.f.a 2 252.bb even 6 1
1323.2.f.b 2 84.j odd 6 1
1323.2.f.b 2 252.r odd 6 1
1323.2.g.a 2 84.h odd 2 1
1323.2.g.a 2 252.bn odd 6 1
1323.2.h.a 2 84.j odd 6 1
1323.2.h.a 2 252.s odd 6 1
3024.2.q.b 2 9.d odd 6 1
3024.2.q.b 2 21.h odd 6 1
3024.2.t.d 2 3.b odd 2 1
3024.2.t.d 2 63.n odd 6 1
3969.2.a.a 1 252.bn odd 6 1
3969.2.a.c 1 252.o even 6 1
3969.2.a.d 1 252.bl odd 6 1
3969.2.a.f 1 252.n even 6 1

## Hecke kernels

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

 $$T_{5} + 1$$ T5 + 1 $$T_{11} + 5$$ T11 + 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 3T + 3$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} + T + 7$$
$11$ $$(T + 5)^{2}$$
$13$ $$T^{2} - 5T + 25$$
$17$ $$T^{2} + 3T + 9$$
$19$ $$T^{2} - T + 1$$
$23$ $$(T + 3)^{2}$$
$29$ $$T^{2} - T + 1$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 3T + 9$$
$41$ $$T^{2} - 5T + 25$$
$43$ $$T^{2} + T + 1$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 9T + 81$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 14T + 196$$
$67$ $$T^{2} - 4T + 16$$
$71$ $$(T - 12)^{2}$$
$73$ $$T^{2} + 3T + 9$$
$79$ $$T^{2} - 8T + 64$$
$83$ $$T^{2} + 9T + 81$$
$89$ $$T^{2} - 13T + 169$$
$97$ $$T^{2} - 9T + 81$$