# Properties

 Label 1008.2.r.k Level $1008$ Weight $2$ Character orbit 1008.r Analytic conductor $8.049$ Analytic rank $0$ Dimension $6$ 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(337,1008)]

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, 0]))

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

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

chi := DirichletCharacter("1008.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1008 = 2^{4} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1008.r (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: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.309123.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3$$ x^6 - 3*x^5 + 10*x^4 - 15*x^3 + 19*x^2 - 12*x + 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{5} + 1) q^{3} + ( - \beta_{5} + 2 \beta_{4} + \beta_{2} + 2) q^{5} + \beta_{4} q^{7} + ( - \beta_{4} - 2 \beta_{3} - \beta_{2} - 2) q^{9}+O(q^{10})$$ q + (-b5 + 1) * q^3 + (-b5 + 2*b4 + b2 + 2) * q^5 + b4 * q^7 + (-b4 - 2*b3 - b2 - 2) * q^9 $$q + ( - \beta_{5} + 1) q^{3} + ( - \beta_{5} + 2 \beta_{4} + \beta_{2} + 2) q^{5} + \beta_{4} q^{7} + ( - \beta_{4} - 2 \beta_{3} - \beta_{2} - 2) q^{9} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} + \cdots - 1) q^{11}+ \cdots + ( - \beta_{5} - 8 \beta_{4} + 2 \beta_{3} + \cdots + 4) q^{99}+O(q^{100})$$ q + (-b5 + 1) * q^3 + (-b5 + 2*b4 + b2 + 2) * q^5 + b4 * q^7 + (-b4 - 2*b3 - b2 - 2) * q^9 + (-b5 + 2*b4 - b3 + 3*b1 - 1) * q^11 + (-b4 - 1) * q^13 + (2*b4 - 2*b3 - 2*b2 - b1 - 1) * q^15 + (b4 - 2*b3 + b1 - 5) * q^17 + (b4 - 2*b3 + b1) * q^19 + (b5 - b2 - b1) * q^21 + (-3*b5 + b4 + b3 + 3*b2 + b1 + 1) * q^23 + (-3*b5 + b4 + 2*b3 - b1 + 2) * q^25 + (2*b5 + b4 - 4*b3 + 2*b1 - 2) * q^27 + (b5 - b1) * q^29 + (3*b5 - 2*b4 - 2*b3 - 3*b2 - 2*b1 - 2) * q^31 + (b5 - 2*b4 - b3 - 3*b2 + 2*b1 - 1) * q^33 + (-b4 + b3 - b2 - b1 - 1) * q^35 + (3*b4 - 3*b3 + 3*b2 + 3*b1 - 3) * q^37 + (b2 + b1 - 1) * q^39 + (b5 + 7*b4 - b2 + 7) * q^41 + (3*b5 - b4 + b3 - 5*b1 + 1) * q^43 + (6*b5 - b4 - 5*b3 - 4*b2 - 3*b1 - 2) * q^45 + (3*b3 - 6*b1 + 3) * q^47 + (-b4 - 1) * q^49 + (5*b5 + 3*b4 - 3*b3 - b2 + 2*b1 - 4) * q^51 + (2*b4 - b3 + 3*b2 + 2*b1 - 7) * q^53 + (-b4 - 4*b3 - 6*b2 - b1 + 1) * q^55 + (3*b4 - 3*b3 - b2 + 2*b1 + 1) * q^57 + (3*b5 - 4*b4 - b3 - 3*b2 - b1 - 4) * q^59 + (-3*b5 - 3*b4 + 2*b3 - b1 + 2) * q^61 + (b5 - b4 + b3 - 2*b1 + 2) * q^63 + (b5 - b4 - b3 + b1 - 1) * q^65 + (-6*b5 + 2*b4 + 7*b3 + 6*b2 + 7*b1 + 2) * q^67 + (-b5 + 3*b4 - 3*b3 - b2 + 2*b1 - 7) * q^69 + (2*b4 - b3 + 3*b2 + 2*b1 - 4) * q^71 + (-5*b4 + 4*b3 - 6*b2 - 5*b1 + 4) * q^73 + (3*b5 - 6*b4 - 3*b3 - 4*b2 - 4*b1 - 8) * q^75 + (b5 - b4 - 2*b3 - b2 - 2*b1 - 1) * q^77 + (-6*b5 - 4*b4 + b3 + 4*b1 + 1) * q^79 + (-b5 + 8*b4 - 2*b3 + b2 + 5*b1 + 6) * q^81 + (5*b4 - b3 + 2*b1 - 1) * q^83 + (6*b5 - 5*b4 - 2*b3 - 6*b2 - 2*b1 - 5) * q^85 + (2*b4 + b3 + b2 - b1 + 2) * q^87 + (-b4 + 7*b3 + 5*b2 - b1 + 2) * q^89 + q^91 + (2*b5 + 2*b2 - b1 + 5) * q^93 + (b5 + 5*b4 - 2*b3 - b2 - 2*b1 + 5) * q^95 + (-3*b5 + 4*b4 - 2*b3 + 7*b1 - 2) * q^97 + (-b5 - 8*b4 + 2*b3 + 2*b1 + 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 4 q^{3} + 5 q^{5} - 3 q^{7} - 4 q^{9}+O(q^{10})$$ 6 * q + 4 * q^3 + 5 * q^5 - 3 * q^7 - 4 * q^9 $$6 q + 4 q^{3} + 5 q^{5} - 3 q^{7} - 4 q^{9} - 2 q^{11} - 3 q^{13} - 11 q^{15} - 24 q^{17} + 6 q^{19} - 2 q^{21} - 6 q^{25} + 7 q^{27} - q^{29} - 3 q^{31} + 8 q^{33} - 10 q^{35} - 6 q^{37} - 2 q^{39} + 22 q^{41} - 3 q^{43} + 5 q^{45} - 9 q^{47} - 3 q^{49} - 9 q^{51} - 36 q^{53} + 12 q^{55} + 11 q^{57} - 9 q^{59} + 6 q^{61} + 8 q^{63} + 5 q^{65} - 39 q^{69} - 18 q^{71} + 6 q^{73} - 31 q^{75} - 2 q^{77} + 15 q^{79} + 32 q^{81} - 12 q^{83} - 9 q^{85} + q^{87} - 4 q^{89} + 6 q^{91} + 33 q^{93} + 16 q^{95} - 3 q^{97} + 46 q^{99}+O(q^{100})$$ 6 * q + 4 * q^3 + 5 * q^5 - 3 * q^7 - 4 * q^9 - 2 * q^11 - 3 * q^13 - 11 * q^15 - 24 * q^17 + 6 * q^19 - 2 * q^21 - 6 * q^25 + 7 * q^27 - q^29 - 3 * q^31 + 8 * q^33 - 10 * q^35 - 6 * q^37 - 2 * q^39 + 22 * q^41 - 3 * q^43 + 5 * q^45 - 9 * q^47 - 3 * q^49 - 9 * q^51 - 36 * q^53 + 12 * q^55 + 11 * q^57 - 9 * q^59 + 6 * q^61 + 8 * q^63 + 5 * q^65 - 39 * q^69 - 18 * q^71 + 6 * q^73 - 31 * q^75 - 2 * q^77 + 15 * q^79 + 32 * q^81 - 12 * q^83 - 9 * q^85 + q^87 - 4 * q^89 + 6 * q^91 + 33 * q^93 + 16 * q^95 - 3 * q^97 + 46 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{5} - \nu^{4} + 5\nu^{3} + \nu^{2} + 6 ) / 3$$ (v^5 - v^4 + 5*v^3 + v^2 + 6) / 3 $$\beta_{3}$$ $$=$$ $$( -\nu^{5} + \nu^{4} - 5\nu^{3} + 2\nu^{2} - 3\nu ) / 3$$ (-v^5 + v^4 - 5*v^3 + 2*v^2 - 3*v) / 3 $$\beta_{4}$$ $$=$$ $$( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 6 ) / 3$$ (-2*v^5 + 5*v^4 - 16*v^3 + 19*v^2 - 21*v + 6) / 3 $$\beta_{5}$$ $$=$$ $$( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 33\nu - 9 ) / 3$$ (2*v^5 - 5*v^4 + 19*v^3 - 22*v^2 + 33*v - 9) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta _1 - 2$$ b3 + b2 + b1 - 2 $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 3\beta _1 - 1$$ b5 + b4 + b3 + b2 - 3*b1 - 1 $$\nu^{4}$$ $$=$$ $$2\beta_{5} + 3\beta_{4} - 5\beta_{3} - 3\beta_{2} - 6\beta _1 + 6$$ 2*b5 + 3*b4 - 5*b3 - 3*b2 - 6*b1 + 6 $$\nu^{5}$$ $$=$$ $$-3\beta_{5} - 2\beta_{4} - 11\beta_{3} - 6\beta_{2} + 8\beta _1 + 7$$ -3*b5 - 2*b4 - 11*b3 - 6*b2 + 8*b1 + 7

## Character values

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

 $$n$$ $$127$$ $$577$$ $$757$$ $$785$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1 - \beta_{4}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 0.5 + 0.224437i 0.5 − 1.41036i 0.5 + 2.05195i 0.5 − 0.224437i 0.5 + 1.41036i 0.5 − 2.05195i
0 −0.349814 1.69636i 0 1.79418 3.10761i 0 −0.500000 0.866025i 0 −2.75526 + 1.18682i 0
337.2 0 0.619562 + 1.61745i 0 −0.590972 + 1.02359i 0 −0.500000 0.866025i 0 −2.23229 + 2.00422i 0
337.3 0 1.73025 + 0.0789082i 0 1.29679 2.24611i 0 −0.500000 0.866025i 0 2.98755 + 0.273062i 0
673.1 0 −0.349814 + 1.69636i 0 1.79418 + 3.10761i 0 −0.500000 + 0.866025i 0 −2.75526 1.18682i 0
673.2 0 0.619562 1.61745i 0 −0.590972 1.02359i 0 −0.500000 + 0.866025i 0 −2.23229 2.00422i 0
673.3 0 1.73025 0.0789082i 0 1.29679 + 2.24611i 0 −0.500000 + 0.866025i 0 2.98755 0.273062i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 337.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.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.r.k 6
3.b odd 2 1 3024.2.r.g 6
4.b odd 2 1 63.2.f.b 6
9.c even 3 1 inner 1008.2.r.k 6
9.c even 3 1 9072.2.a.bq 3
9.d odd 6 1 3024.2.r.g 6
9.d odd 6 1 9072.2.a.cd 3
12.b even 2 1 189.2.f.a 6
28.d even 2 1 441.2.f.d 6
28.f even 6 1 441.2.g.d 6
28.f even 6 1 441.2.h.b 6
28.g odd 6 1 441.2.g.e 6
28.g odd 6 1 441.2.h.c 6
36.f odd 6 1 63.2.f.b 6
36.f odd 6 1 567.2.a.d 3
36.h even 6 1 189.2.f.a 6
36.h even 6 1 567.2.a.g 3
84.h odd 2 1 1323.2.f.c 6
84.j odd 6 1 1323.2.g.b 6
84.j odd 6 1 1323.2.h.e 6
84.n even 6 1 1323.2.g.c 6
84.n even 6 1 1323.2.h.d 6
252.n even 6 1 441.2.h.b 6
252.o even 6 1 1323.2.h.d 6
252.r odd 6 1 1323.2.g.b 6
252.s odd 6 1 1323.2.f.c 6
252.s odd 6 1 3969.2.a.p 3
252.u odd 6 1 441.2.g.e 6
252.bb even 6 1 1323.2.g.c 6
252.bi even 6 1 441.2.f.d 6
252.bi even 6 1 3969.2.a.m 3
252.bj even 6 1 441.2.g.d 6
252.bl odd 6 1 441.2.h.c 6
252.bn odd 6 1 1323.2.h.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.f.b 6 4.b odd 2 1
63.2.f.b 6 36.f odd 6 1
189.2.f.a 6 12.b even 2 1
189.2.f.a 6 36.h even 6 1
441.2.f.d 6 28.d even 2 1
441.2.f.d 6 252.bi even 6 1
441.2.g.d 6 28.f even 6 1
441.2.g.d 6 252.bj even 6 1
441.2.g.e 6 28.g odd 6 1
441.2.g.e 6 252.u odd 6 1
441.2.h.b 6 28.f even 6 1
441.2.h.b 6 252.n even 6 1
441.2.h.c 6 28.g odd 6 1
441.2.h.c 6 252.bl odd 6 1
567.2.a.d 3 36.f odd 6 1
567.2.a.g 3 36.h even 6 1
1008.2.r.k 6 1.a even 1 1 trivial
1008.2.r.k 6 9.c even 3 1 inner
1323.2.f.c 6 84.h odd 2 1
1323.2.f.c 6 252.s odd 6 1
1323.2.g.b 6 84.j odd 6 1
1323.2.g.b 6 252.r odd 6 1
1323.2.g.c 6 84.n even 6 1
1323.2.g.c 6 252.bb even 6 1
1323.2.h.d 6 84.n even 6 1
1323.2.h.d 6 252.o even 6 1
1323.2.h.e 6 84.j odd 6 1
1323.2.h.e 6 252.bn odd 6 1
3024.2.r.g 6 3.b odd 2 1
3024.2.r.g 6 9.d odd 6 1
3969.2.a.m 3 252.bi even 6 1
3969.2.a.p 3 252.s odd 6 1
9072.2.a.bq 3 9.c even 3 1
9072.2.a.cd 3 9.d odd 6 1

## Hecke kernels

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

 $$T_{5}^{6} - 5T_{5}^{5} + 23T_{5}^{4} - 32T_{5}^{3} + 59T_{5}^{2} + 22T_{5} + 121$$ T5^6 - 5*T5^5 + 23*T5^4 - 32*T5^3 + 59*T5^2 + 22*T5 + 121 $$T_{11}^{6} + 2T_{11}^{5} + 23T_{11}^{4} + 56T_{11}^{3} + 455T_{11}^{2} + 893T_{11} + 2209$$ T11^6 + 2*T11^5 + 23*T11^4 + 56*T11^3 + 455*T11^2 + 893*T11 + 2209

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} - 4 T^{5} + \cdots + 27$$
$5$ $$T^{6} - 5 T^{5} + \cdots + 121$$
$7$ $$(T^{2} + T + 1)^{3}$$
$11$ $$T^{6} + 2 T^{5} + \cdots + 2209$$
$13$ $$(T^{2} + T + 1)^{3}$$
$17$ $$(T^{3} + 12 T^{2} + \cdots + 27)^{2}$$
$19$ $$(T^{3} - 3 T^{2} - 6 T + 7)^{2}$$
$23$ $$T^{6} + 33 T^{4} + \cdots + 81$$
$29$ $$T^{6} + T^{5} + 5 T^{4} + \cdots + 1$$
$31$ $$T^{6} + 3 T^{5} + \cdots + 729$$
$37$ $$(T^{3} + 3 T^{2} - 54 T + 81)^{2}$$
$41$ $$T^{6} - 22 T^{5} + \cdots + 124609$$
$43$ $$T^{6} + 3 T^{5} + \cdots + 14641$$
$47$ $$T^{6} + 9 T^{5} + \cdots + 35721$$
$53$ $$(T^{3} + 18 T^{2} + 75 T + 9)^{2}$$
$59$ $$T^{6} + 9 T^{5} + \cdots + 3969$$
$61$ $$T^{6} - 6 T^{5} + \cdots + 4489$$
$67$ $$T^{6} + 207 T^{4} + \cdots + 466489$$
$71$ $$(T^{3} + 9 T^{2} - 6 T - 81)^{2}$$
$73$ $$(T^{3} - 3 T^{2} + \cdots - 243)^{2}$$
$79$ $$T^{6} - 15 T^{5} + \cdots + 591361$$
$83$ $$T^{6} + 12 T^{5} + \cdots + 729$$
$89$ $$(T^{3} + 2 T^{2} + \cdots + 379)^{2}$$
$97$ $$T^{6} + 3 T^{5} + \cdots + 363609$$