# Properties

 Label 1008.2.r.h Level $1008$ Weight $2$ Character orbit 1008.r Analytic conductor $8.049$ Analytic rank $0$ Dimension $6$ 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: $$\Q(\zeta_{18})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{3} + 1$$ x^6 - x^3 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{4} + \beta_{2}) q^{3} + (\beta_{5} + \beta_1 - 1) q^{5} + \beta_1 q^{7} + ( - \beta_{5} - \beta_{4} + \beta_{2}) q^{9}+O(q^{10})$$ q + (b4 + b2) * q^3 + (b5 + b1 - 1) * q^5 + b1 * q^7 + (-b5 - b4 + b2) * q^9 $$q + (\beta_{4} + \beta_{2}) q^{3} + (\beta_{5} + \beta_1 - 1) q^{5} + \beta_1 q^{7} + ( - \beta_{5} - \beta_{4} + \beta_{2}) q^{9} + ( - \beta_{5} + \beta_{4} + \cdots + 2 \beta_1) q^{11}+ \cdots + ( - 3 \beta_{4} - 6 \beta_{3} + \cdots - 3) q^{99}+O(q^{100})$$ q + (b4 + b2) * q^3 + (b5 + b1 - 1) * q^5 + b1 * q^7 + (-b5 - b4 + b2) * q^9 + (-b5 + b4 + b3 + 2*b1) * q^11 + (4*b5 - 2*b3 + 2*b2 - b1 + 1) * q^13 + (-2*b5 + b4 + 2*b3 - 2*b2 - b1 + 2) * q^15 + (-b4 + 2) * q^17 + (-b4 - 2*b3 + 1) * q^19 + (-b5 + b4) * q^21 + (2*b5 - 3*b3 + 3*b2 - 4*b1 + 4) * q^23 + (-2*b5 + 2*b4 + 2*b3 - b2 + 2*b1) * q^25 + (3*b1 - 6) * q^27 + (-4*b5 + 4*b4 + 4*b3 + b2 - 3*b1) * q^29 + (3*b5 - 6*b3 + 6*b2 + b1 - 1) * q^31 + (-2*b5 + b4 - b3 + b2 + 2*b1 - 1) * q^33 + (b4 + b3 - 1) * q^35 + (3*b4 - 3*b3 - 1) * q^37 + (-3*b5 + 2*b4 + 6*b3 - b2 - 6*b1 + 6) * q^39 + (-b5 + 2*b3 - 2*b2) * q^41 + (b5 - b4 - b3 + 2*b2 - b1) * q^43 + (3*b5 - 3*b3 + 3*b2 + 3*b1) * q^45 + (-3*b5 + 3*b4 + 3*b3 - 5*b2 + b1) * q^47 + (b1 - 1) * q^49 + (3*b4 + b3 + b2 + b1 - 2) * q^51 + (2*b4 + 3*b3 + 2) * q^53 + (b4 + 2*b3) * q^55 + (2*b5 - 3*b3 + 2*b2 - 3*b1) * q^57 + (5*b3 - 5*b2 + b1 - 1) * q^59 + (3*b2 - 2*b1) * q^61 + (b5 - 2*b4 - 3*b3 + 2*b2) * q^63 + (-5*b5 + 5*b4 + 5*b3 - 6*b2 - 5*b1) * q^65 + (3*b5 - 3*b3 + 3*b2 + 4*b1 - 4) * q^67 + (2*b5 - b4 + b3 + 5*b2 - 5*b1 + 1) * q^69 + (6*b4 + 3*b3 - 3) * q^71 + (b4 - 4*b3 - 7) * q^73 + (-b5 - 3*b3 + 2*b2 + 3*b1) * q^75 + (-b5 + 2*b1 - 2) * q^77 + (-3*b5 + 3*b4 + 3*b3 - 3*b2 - 7*b1) * q^79 + (-3*b5 - 3*b4 - 6*b2) * q^81 + (b5 - b4 - b3 + 5*b2 - 6*b1) * q^83 + (4*b5 - 2*b3 + 2*b2 + 3*b1 - 3) * q^85 + (2*b5 - 7*b4 - 3*b3 + 4*b2 + 9*b1 - 6) * q^87 + (7*b4 + 3*b3 + 4) * q^89 + (4*b4 + 2*b3 + 1) * q^91 + (-4*b5 - 3*b4 + 2*b2 - 9*b1) * q^93 + (b5 + 4*b1 - 4) * q^95 + (8*b5 - 8*b4 - 8*b3 + b2 + b1) * q^97 + (-3*b4 - 6*b3 + 3*b2 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 3 q^{5} + 3 q^{7}+O(q^{10})$$ 6 * q - 3 * q^5 + 3 * q^7 $$6 q - 3 q^{5} + 3 q^{7} + 6 q^{11} + 3 q^{13} + 9 q^{15} + 12 q^{17} + 6 q^{19} + 12 q^{23} + 6 q^{25} - 27 q^{27} - 9 q^{29} - 3 q^{31} - 6 q^{35} - 6 q^{37} + 18 q^{39} - 3 q^{43} + 9 q^{45} + 3 q^{47} - 3 q^{49} - 9 q^{51} + 12 q^{53} - 9 q^{57} - 3 q^{59} - 6 q^{61} - 15 q^{65} - 12 q^{67} - 9 q^{69} - 18 q^{71} - 42 q^{73} + 9 q^{75} - 6 q^{77} - 21 q^{79} - 18 q^{83} - 9 q^{85} - 9 q^{87} + 24 q^{89} + 6 q^{91} - 27 q^{93} - 12 q^{95} + 3 q^{97} - 18 q^{99}+O(q^{100})$$ 6 * q - 3 * q^5 + 3 * q^7 + 6 * q^11 + 3 * q^13 + 9 * q^15 + 12 * q^17 + 6 * q^19 + 12 * q^23 + 6 * q^25 - 27 * q^27 - 9 * q^29 - 3 * q^31 - 6 * q^35 - 6 * q^37 + 18 * q^39 - 3 * q^43 + 9 * q^45 + 3 * q^47 - 3 * q^49 - 9 * q^51 + 12 * q^53 - 9 * q^57 - 3 * q^59 - 6 * q^61 - 15 * q^65 - 12 * q^67 - 9 * q^69 - 18 * q^71 - 42 * q^73 + 9 * q^75 - 6 * q^77 - 21 * q^79 - 18 * q^83 - 9 * q^85 - 9 * q^87 + 24 * q^89 + 6 * q^91 - 27 * q^93 - 12 * q^95 + 3 * q^97 - 18 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{18}^{3}$$ v^3 $$\beta_{2}$$ $$=$$ $$\zeta_{18}^{5} + \zeta_{18}$$ v^5 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18}$$ -v^4 + v^2 + v $$\beta_{4}$$ $$=$$ $$-\zeta_{18}^{5} + \zeta_{18}^{4}$$ -v^5 + v^4 $$\beta_{5}$$ $$=$$ $$-\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}$$ -v^5 - v^4 + v
 $$\zeta_{18}$$ $$=$$ $$( \beta_{5} + \beta_{4} + 2\beta_{2} ) / 3$$ (b5 + b4 + 2*b2) / 3 $$\zeta_{18}^{2}$$ $$=$$ $$( -2\beta_{5} + \beta_{4} + 3\beta_{3} - \beta_{2} ) / 3$$ (-2*b5 + b4 + 3*b3 - b2) / 3 $$\zeta_{18}^{3}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{18}^{4}$$ $$=$$ $$( -\beta_{5} + 2\beta_{4} + \beta_{2} ) / 3$$ (-b5 + 2*b4 + b2) / 3 $$\zeta_{18}^{5}$$ $$=$$ $$( -\beta_{5} - \beta_{4} + \beta_{2} ) / 3$$ (-b5 - b4 + b2) / 3

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 −0.766044 + 0.642788i −0.173648 − 0.984808i 0.939693 + 0.342020i −0.766044 − 0.642788i −0.173648 + 0.984808i 0.939693 − 0.342020i
0 −1.70574 + 0.300767i 0 −1.26604 + 2.19285i 0 0.500000 + 0.866025i 0 2.81908 1.02606i 0
337.2 0 0.592396 1.62760i 0 −0.673648 + 1.16679i 0 0.500000 + 0.866025i 0 −2.29813 1.92836i 0
337.3 0 1.11334 + 1.32683i 0 0.439693 0.761570i 0 0.500000 + 0.866025i 0 −0.520945 + 2.95442i 0
673.1 0 −1.70574 0.300767i 0 −1.26604 2.19285i 0 0.500000 0.866025i 0 2.81908 + 1.02606i 0
673.2 0 0.592396 + 1.62760i 0 −0.673648 1.16679i 0 0.500000 0.866025i 0 −2.29813 + 1.92836i 0
673.3 0 1.11334 1.32683i 0 0.439693 + 0.761570i 0 0.500000 0.866025i 0 −0.520945 2.95442i 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.h 6
3.b odd 2 1 3024.2.r.k 6
4.b odd 2 1 63.2.f.a 6
9.c even 3 1 inner 1008.2.r.h 6
9.c even 3 1 9072.2.a.ca 3
9.d odd 6 1 3024.2.r.k 6
9.d odd 6 1 9072.2.a.bs 3
12.b even 2 1 189.2.f.b 6
28.d even 2 1 441.2.f.c 6
28.f even 6 1 441.2.g.b 6
28.f even 6 1 441.2.h.e 6
28.g odd 6 1 441.2.g.c 6
28.g odd 6 1 441.2.h.d 6
36.f odd 6 1 63.2.f.a 6
36.f odd 6 1 567.2.a.h 3
36.h even 6 1 189.2.f.b 6
36.h even 6 1 567.2.a.c 3
84.h odd 2 1 1323.2.f.d 6
84.j odd 6 1 1323.2.g.e 6
84.j odd 6 1 1323.2.h.b 6
84.n even 6 1 1323.2.g.d 6
84.n even 6 1 1323.2.h.c 6
252.n even 6 1 441.2.h.e 6
252.o even 6 1 1323.2.h.c 6
252.r odd 6 1 1323.2.g.e 6
252.s odd 6 1 1323.2.f.d 6
252.s odd 6 1 3969.2.a.l 3
252.u odd 6 1 441.2.g.c 6
252.bb even 6 1 1323.2.g.d 6
252.bi even 6 1 441.2.f.c 6
252.bi even 6 1 3969.2.a.q 3
252.bj even 6 1 441.2.g.b 6
252.bl odd 6 1 441.2.h.d 6
252.bn odd 6 1 1323.2.h.b 6

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

## Hecke kernels

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

 $$T_{5}^{6} + 3T_{5}^{5} + 9T_{5}^{4} + 6T_{5}^{3} + 9T_{5}^{2} + 9$$ T5^6 + 3*T5^5 + 9*T5^4 + 6*T5^3 + 9*T5^2 + 9 $$T_{11}^{6} - 6T_{11}^{5} + 27T_{11}^{4} - 48T_{11}^{3} + 63T_{11}^{2} - 27T_{11} + 9$$ T11^6 - 6*T11^5 + 27*T11^4 - 48*T11^3 + 63*T11^2 - 27*T11 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} + 9T^{3} + 27$$
$5$ $$T^{6} + 3 T^{5} + \cdots + 9$$
$7$ $$(T^{2} - T + 1)^{3}$$
$11$ $$T^{6} - 6 T^{5} + \cdots + 9$$
$13$ $$T^{6} - 3 T^{5} + \cdots + 11449$$
$17$ $$(T^{3} - 6 T^{2} + 9 T - 3)^{2}$$
$19$ $$(T^{3} - 3 T^{2} - 6 T + 17)^{2}$$
$23$ $$T^{6} - 12 T^{5} + \cdots + 9$$
$29$ $$T^{6} + 9 T^{5} + \cdots + 110889$$
$31$ $$T^{6} + 3 T^{5} + \cdots + 104329$$
$37$ $$(T^{3} + 3 T^{2} + \cdots - 323)^{2}$$
$41$ $$T^{6} + 9 T^{4} + \cdots + 81$$
$43$ $$T^{6} + 3 T^{5} + \cdots + 1$$
$47$ $$T^{6} - 3 T^{5} + \cdots + 2601$$
$53$ $$(T^{3} - 6 T^{2} - 9 T - 3)^{2}$$
$59$ $$T^{6} + 3 T^{5} + \cdots + 2601$$
$61$ $$T^{6} + 6 T^{5} + \cdots + 361$$
$67$ $$T^{6} + 12 T^{5} + \cdots + 289$$
$71$ $$(T^{3} + 9 T^{2} - 54 T + 27)^{2}$$
$73$ $$(T^{3} + 21 T^{2} + \cdots - 269)^{2}$$
$79$ $$T^{6} + 21 T^{5} + \cdots + 32761$$
$83$ $$T^{6} + 18 T^{5} + \cdots + 81$$
$89$ $$(T^{3} - 12 T^{2} + \cdots + 813)^{2}$$
$97$ $$T^{6} - 3 T^{5} + \cdots + 104329$$