# Properties

 Label 1008.2.r.d Level $1008$ Weight $2$ Character orbit 1008.r 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(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: $$2$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 126) 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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{6} + 1) q^{3} + 3 \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7} - 3 q^{9}+O(q^{10})$$ q + (-2*z + 1) * q^3 + 3*z * q^5 + (-z + 1) * q^7 - 3 * q^9 $$q + ( - 2 \zeta_{6} + 1) q^{3} + 3 \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7} - 3 q^{9} + (6 \zeta_{6} - 6) q^{11} - 2 \zeta_{6} q^{13} + ( - 3 \zeta_{6} + 6) q^{15} + 6 q^{17} + 7 q^{19} + ( - \zeta_{6} - 1) q^{21} + 3 \zeta_{6} q^{23} + (4 \zeta_{6} - 4) q^{25} + (6 \zeta_{6} - 3) q^{27} + (6 \zeta_{6} - 6) q^{29} + 2 \zeta_{6} q^{31} + (6 \zeta_{6} + 6) q^{33} + 3 q^{35} + 2 q^{37} + (2 \zeta_{6} - 4) q^{39} + ( - 2 \zeta_{6} + 2) q^{43} - 9 \zeta_{6} q^{45} - \zeta_{6} q^{49} + ( - 12 \zeta_{6} + 6) q^{51} + 6 q^{53} - 18 q^{55} + ( - 14 \zeta_{6} + 7) q^{57} + (5 \zeta_{6} - 5) q^{61} + (3 \zeta_{6} - 3) q^{63} + ( - 6 \zeta_{6} + 6) q^{65} + 8 \zeta_{6} q^{67} + ( - 3 \zeta_{6} + 6) q^{69} - 3 q^{71} + 2 q^{73} + (4 \zeta_{6} + 4) q^{75} + 6 \zeta_{6} q^{77} + ( - 5 \zeta_{6} + 5) q^{79} + 9 q^{81} + ( - 12 \zeta_{6} + 12) q^{83} + 18 \zeta_{6} q^{85} + (6 \zeta_{6} + 6) q^{87} - 2 q^{91} + ( - 2 \zeta_{6} + 4) q^{93} + 21 \zeta_{6} q^{95} + (2 \zeta_{6} - 2) q^{97} + ( - 18 \zeta_{6} + 18) q^{99} +O(q^{100})$$ q + (-2*z + 1) * q^3 + 3*z * q^5 + (-z + 1) * q^7 - 3 * q^9 + (6*z - 6) * q^11 - 2*z * q^13 + (-3*z + 6) * q^15 + 6 * q^17 + 7 * q^19 + (-z - 1) * q^21 + 3*z * q^23 + (4*z - 4) * q^25 + (6*z - 3) * q^27 + (6*z - 6) * q^29 + 2*z * q^31 + (6*z + 6) * q^33 + 3 * q^35 + 2 * q^37 + (2*z - 4) * q^39 + (-2*z + 2) * q^43 - 9*z * q^45 - z * q^49 + (-12*z + 6) * q^51 + 6 * q^53 - 18 * q^55 + (-14*z + 7) * q^57 + (5*z - 5) * q^61 + (3*z - 3) * q^63 + (-6*z + 6) * q^65 + 8*z * q^67 + (-3*z + 6) * q^69 - 3 * q^71 + 2 * q^73 + (4*z + 4) * q^75 + 6*z * q^77 + (-5*z + 5) * q^79 + 9 * q^81 + (-12*z + 12) * q^83 + 18*z * q^85 + (6*z + 6) * q^87 - 2 * q^91 + (-2*z + 4) * q^93 + 21*z * q^95 + (2*z - 2) * q^97 + (-18*z + 18) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{5} + q^{7} - 6 q^{9}+O(q^{10})$$ 2 * q + 3 * q^5 + q^7 - 6 * q^9 $$2 q + 3 q^{5} + q^{7} - 6 q^{9} - 6 q^{11} - 2 q^{13} + 9 q^{15} + 12 q^{17} + 14 q^{19} - 3 q^{21} + 3 q^{23} - 4 q^{25} - 6 q^{29} + 2 q^{31} + 18 q^{33} + 6 q^{35} + 4 q^{37} - 6 q^{39} + 2 q^{43} - 9 q^{45} - q^{49} + 12 q^{53} - 36 q^{55} - 5 q^{61} - 3 q^{63} + 6 q^{65} + 8 q^{67} + 9 q^{69} - 6 q^{71} + 4 q^{73} + 12 q^{75} + 6 q^{77} + 5 q^{79} + 18 q^{81} + 12 q^{83} + 18 q^{85} + 18 q^{87} - 4 q^{91} + 6 q^{93} + 21 q^{95} - 2 q^{97} + 18 q^{99}+O(q^{100})$$ 2 * q + 3 * q^5 + q^7 - 6 * q^9 - 6 * q^11 - 2 * q^13 + 9 * q^15 + 12 * q^17 + 14 * q^19 - 3 * q^21 + 3 * q^23 - 4 * q^25 - 6 * q^29 + 2 * q^31 + 18 * q^33 + 6 * q^35 + 4 * q^37 - 6 * q^39 + 2 * q^43 - 9 * q^45 - q^49 + 12 * q^53 - 36 * q^55 - 5 * q^61 - 3 * q^63 + 6 * q^65 + 8 * q^67 + 9 * q^69 - 6 * q^71 + 4 * q^73 + 12 * q^75 + 6 * q^77 + 5 * q^79 + 18 * q^81 + 12 * q^83 + 18 * q^85 + 18 * q^87 - 4 * q^91 + 6 * q^93 + 21 * q^95 - 2 * q^97 + 18 * q^99

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 1.73205i 0 1.50000 2.59808i 0 0.500000 + 0.866025i 0 −3.00000 0
673.1 0 1.73205i 0 1.50000 + 2.59808i 0 0.500000 0.866025i 0 −3.00000 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
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.d 2
3.b odd 2 1 3024.2.r.a 2
4.b odd 2 1 126.2.f.a 2
9.c even 3 1 inner 1008.2.r.d 2
9.c even 3 1 9072.2.a.c 1
9.d odd 6 1 3024.2.r.a 2
9.d odd 6 1 9072.2.a.w 1
12.b even 2 1 378.2.f.a 2
28.d even 2 1 882.2.f.h 2
28.f even 6 1 882.2.e.d 2
28.f even 6 1 882.2.h.f 2
28.g odd 6 1 882.2.e.b 2
28.g odd 6 1 882.2.h.j 2
36.f odd 6 1 126.2.f.a 2
36.f odd 6 1 1134.2.a.a 1
36.h even 6 1 378.2.f.a 2
36.h even 6 1 1134.2.a.h 1
84.h odd 2 1 2646.2.f.c 2
84.j odd 6 1 2646.2.e.j 2
84.j odd 6 1 2646.2.h.a 2
84.n even 6 1 2646.2.e.f 2
84.n even 6 1 2646.2.h.e 2
252.n even 6 1 882.2.e.d 2
252.o even 6 1 2646.2.e.f 2
252.r odd 6 1 2646.2.h.a 2
252.s odd 6 1 2646.2.f.c 2
252.s odd 6 1 7938.2.a.u 1
252.u odd 6 1 882.2.h.j 2
252.bb even 6 1 2646.2.h.e 2
252.bi even 6 1 882.2.f.h 2
252.bi even 6 1 7938.2.a.l 1
252.bj even 6 1 882.2.h.f 2
252.bl odd 6 1 882.2.e.b 2
252.bn odd 6 1 2646.2.e.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.a 2 4.b odd 2 1
126.2.f.a 2 36.f odd 6 1
378.2.f.a 2 12.b even 2 1
378.2.f.a 2 36.h even 6 1
882.2.e.b 2 28.g odd 6 1
882.2.e.b 2 252.bl odd 6 1
882.2.e.d 2 28.f even 6 1
882.2.e.d 2 252.n even 6 1
882.2.f.h 2 28.d even 2 1
882.2.f.h 2 252.bi even 6 1
882.2.h.f 2 28.f even 6 1
882.2.h.f 2 252.bj even 6 1
882.2.h.j 2 28.g odd 6 1
882.2.h.j 2 252.u odd 6 1
1008.2.r.d 2 1.a even 1 1 trivial
1008.2.r.d 2 9.c even 3 1 inner
1134.2.a.a 1 36.f odd 6 1
1134.2.a.h 1 36.h even 6 1
2646.2.e.f 2 84.n even 6 1
2646.2.e.f 2 252.o even 6 1
2646.2.e.j 2 84.j odd 6 1
2646.2.e.j 2 252.bn odd 6 1
2646.2.f.c 2 84.h odd 2 1
2646.2.f.c 2 252.s odd 6 1
2646.2.h.a 2 84.j odd 6 1
2646.2.h.a 2 252.r odd 6 1
2646.2.h.e 2 84.n even 6 1
2646.2.h.e 2 252.bb even 6 1
3024.2.r.a 2 3.b odd 2 1
3024.2.r.a 2 9.d odd 6 1
7938.2.a.l 1 252.bi even 6 1
7938.2.a.u 1 252.s odd 6 1
9072.2.a.c 1 9.c even 3 1
9072.2.a.w 1 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 $$S_{2}^{\mathrm{new}}(1008, [\chi])$$:

 $$T_{5}^{2} - 3T_{5} + 9$$ T5^2 - 3*T5 + 9 $$T_{11}^{2} + 6T_{11} + 36$$ T11^2 + 6*T11 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 3$$
$5$ $$T^{2} - 3T + 9$$
$7$ $$T^{2} - T + 1$$
$11$ $$T^{2} + 6T + 36$$
$13$ $$T^{2} + 2T + 4$$
$17$ $$(T - 6)^{2}$$
$19$ $$(T - 7)^{2}$$
$23$ $$T^{2} - 3T + 9$$
$29$ $$T^{2} + 6T + 36$$
$31$ $$T^{2} - 2T + 4$$
$37$ $$(T - 2)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} - 2T + 4$$
$47$ $$T^{2}$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 5T + 25$$
$67$ $$T^{2} - 8T + 64$$
$71$ $$(T + 3)^{2}$$
$73$ $$(T - 2)^{2}$$
$79$ $$T^{2} - 5T + 25$$
$83$ $$T^{2} - 12T + 144$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 2T + 4$$