# Properties

 Label 1008.2.r.e Level $1008$ Weight $2$ Character orbit 1008.r Analytic conductor $8.049$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{2} + 4$$ x^4 - 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} - 1) q^{3} + (\beta_{3} - \beta_{2} + \beta_1) q^{5} + (\beta_{2} - 1) q^{7} + (2 \beta_{3} - 1) q^{9}+O(q^{10})$$ q + (-b3 - 1) * q^3 + (b3 - b2 + b1) * q^5 + (b2 - 1) * q^7 + (2*b3 - 1) * q^9 $$q + ( - \beta_{3} - 1) q^{3} + (\beta_{3} - \beta_{2} + \beta_1) q^{5} + (\beta_{2} - 1) q^{7} + (2 \beta_{3} - 1) q^{9} + ( - 2 \beta_{2} + 2) q^{11} + ( - 2 \beta_{3} - 2 \beta_1) q^{13} + ( - \beta_{2} - 2 \beta_1 + 4) q^{15} + 2 q^{17} + ( - \beta_{3} + 2 \beta_1 - 5) q^{19} + ( - \beta_{2} + \beta_1 + 1) q^{21} - \beta_{2} q^{23} + ( - 4 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{25} + ( - \beta_{3} + 5) q^{27} + ( - 4 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{29} + 6 \beta_{2} q^{31} + (2 \beta_{2} - 2 \beta_1 - 2) q^{33} + (\beta_{3} - 2 \beta_1 + 1) q^{35} + ( - 4 \beta_{3} + 8 \beta_1 + 2) q^{37} + (2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 8) q^{39} + ( - 4 \beta_{3} - 4 \beta_1) q^{41} + ( - 4 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{43} + ( - 3 \beta_{3} + 5 \beta_{2} + \beta_1 - 8) q^{45} + ( - 8 \beta_{3} + 4 \beta_1) q^{47} - \beta_{2} q^{49} + ( - 2 \beta_{3} - 2) q^{51} + (2 \beta_{3} - 4 \beta_1 - 6) q^{53} + ( - 2 \beta_{3} + 4 \beta_1 - 2) q^{55} + (6 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 7) q^{57} - 2 \beta_{2} q^{59} + ( - 2 \beta_{3} + 9 \beta_{2} + \beta_1 - 9) q^{61} + ( - \beta_{2} - 2 \beta_1 + 1) q^{63} + (4 \beta_{3} - 12 \beta_{2} - 2 \beta_1 + 12) q^{65} + ( - 2 \beta_{3} - 8 \beta_{2} - 2 \beta_1) q^{67} + (\beta_{3} + \beta_{2} - \beta_1) q^{69} + ( - 2 \beta_{3} + 4 \beta_1 - 5) q^{71} + (2 \beta_{3} - 4 \beta_1 - 2) q^{73} + (4 \beta_{3} - 6 \beta_{2} - 2) q^{75} + 2 \beta_{2} q^{77} + (4 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 3) q^{79} + ( - 4 \beta_{3} - 7) q^{81} + ( - 2 \beta_{2} + 2) q^{83} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{85} + (4 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 6) q^{87} + (2 \beta_{3} - 4 \beta_1 - 12) q^{89} + ( - 2 \beta_{3} + 4 \beta_1) q^{91} + ( - 6 \beta_{3} - 6 \beta_{2} + 6 \beta_1) q^{93} + ( - 6 \beta_{3} + 11 \beta_{2} - 6 \beta_1) q^{95} + (4 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{97} + (2 \beta_{2} + 4 \beta_1 - 2) q^{99}+O(q^{100})$$ q + (-b3 - 1) * q^3 + (b3 - b2 + b1) * q^5 + (b2 - 1) * q^7 + (2*b3 - 1) * q^9 + (-2*b2 + 2) * q^11 + (-2*b3 - 2*b1) * q^13 + (-b2 - 2*b1 + 4) * q^15 + 2 * q^17 + (-b3 + 2*b1 - 5) * q^19 + (-b2 + b1 + 1) * q^21 - b2 * q^23 + (-4*b3 + 2*b2 + 2*b1 - 2) * q^25 + (-b3 + 5) * q^27 + (-4*b3 - 2*b2 + 2*b1 + 2) * q^29 + 6*b2 * q^31 + (2*b2 - 2*b1 - 2) * q^33 + (b3 - 2*b1 + 1) * q^35 + (-4*b3 + 8*b1 + 2) * q^37 + (2*b3 + 4*b2 + 2*b1 - 8) * q^39 + (-4*b3 - 4*b1) * q^41 + (-4*b3 - 2*b2 + 2*b1 + 2) * q^43 + (-3*b3 + 5*b2 + b1 - 8) * q^45 + (-8*b3 + 4*b1) * q^47 - b2 * q^49 + (-2*b3 - 2) * q^51 + (2*b3 - 4*b1 - 6) * q^53 + (-2*b3 + 4*b1 - 2) * q^55 + (6*b3 - 4*b2 - 2*b1 + 7) * q^57 - 2*b2 * q^59 + (-2*b3 + 9*b2 + b1 - 9) * q^61 + (-b2 - 2*b1 + 1) * q^63 + (4*b3 - 12*b2 - 2*b1 + 12) * q^65 + (-2*b3 - 8*b2 - 2*b1) * q^67 + (b3 + b2 - b1) * q^69 + (-2*b3 + 4*b1 - 5) * q^71 + (2*b3 - 4*b1 - 2) * q^73 + (4*b3 - 6*b2 - 2) * q^75 + 2*b2 * q^77 + (4*b3 - 3*b2 - 2*b1 + 3) * q^79 + (-4*b3 - 7) * q^81 + (-2*b2 + 2) * q^83 + (2*b3 - 2*b2 + 2*b1) * q^85 + (4*b3 - 2*b2 - 4*b1 - 6) * q^87 + (2*b3 - 4*b1 - 12) * q^89 + (-2*b3 + 4*b1) * q^91 + (-6*b3 - 6*b2 + 6*b1) * q^93 + (-6*b3 + 11*b2 - 6*b1) * q^95 + (4*b3 - 2*b2 - 2*b1 + 2) * q^97 + (2*b2 + 4*b1 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} - 2 q^{5} - 2 q^{7} - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^3 - 2 * q^5 - 2 * q^7 - 4 * q^9 $$4 q - 4 q^{3} - 2 q^{5} - 2 q^{7} - 4 q^{9} + 4 q^{11} + 14 q^{15} + 8 q^{17} - 20 q^{19} + 2 q^{21} - 2 q^{23} - 4 q^{25} + 20 q^{27} + 4 q^{29} + 12 q^{31} - 4 q^{33} + 4 q^{35} + 8 q^{37} - 24 q^{39} + 4 q^{43} - 22 q^{45} - 2 q^{49} - 8 q^{51} - 24 q^{53} - 8 q^{55} + 20 q^{57} - 4 q^{59} - 18 q^{61} + 2 q^{63} + 24 q^{65} - 16 q^{67} + 2 q^{69} - 20 q^{71} - 8 q^{73} - 20 q^{75} + 4 q^{77} + 6 q^{79} - 28 q^{81} + 4 q^{83} - 4 q^{85} - 28 q^{87} - 48 q^{89} - 12 q^{93} + 22 q^{95} + 4 q^{97} - 4 q^{99}+O(q^{100})$$ 4 * q - 4 * q^3 - 2 * q^5 - 2 * q^7 - 4 * q^9 + 4 * q^11 + 14 * q^15 + 8 * q^17 - 20 * q^19 + 2 * q^21 - 2 * q^23 - 4 * q^25 + 20 * q^27 + 4 * q^29 + 12 * q^31 - 4 * q^33 + 4 * q^35 + 8 * q^37 - 24 * q^39 + 4 * q^43 - 22 * q^45 - 2 * q^49 - 8 * q^51 - 24 * q^53 - 8 * q^55 + 20 * q^57 - 4 * q^59 - 18 * q^61 + 2 * q^63 + 24 * q^65 - 16 * q^67 + 2 * q^69 - 20 * q^71 - 8 * q^73 - 20 * q^75 + 4 * q^77 + 6 * q^79 - 28 * q^81 + 4 * q^83 - 4 * q^85 - 28 * q^87 - 48 * q^89 - 12 * q^93 + 22 * q^95 + 4 * q^97 - 4 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

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

 $$n$$ $$127$$ $$577$$ $$757$$ $$785$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-\beta_{2}$$

## 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
 −1.22474 + 0.707107i 1.22474 − 0.707107i 1.22474 + 0.707107i −1.22474 − 0.707107i
0 −1.00000 1.41421i 0 −1.72474 + 2.98735i 0 −0.500000 0.866025i 0 −1.00000 + 2.82843i 0
337.2 0 −1.00000 + 1.41421i 0 0.724745 1.25529i 0 −0.500000 0.866025i 0 −1.00000 2.82843i 0
673.1 0 −1.00000 1.41421i 0 0.724745 + 1.25529i 0 −0.500000 + 0.866025i 0 −1.00000 + 2.82843i 0
673.2 0 −1.00000 + 1.41421i 0 −1.72474 2.98735i 0 −0.500000 + 0.866025i 0 −1.00000 2.82843i 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.e 4
3.b odd 2 1 3024.2.r.e 4
4.b odd 2 1 126.2.f.c 4
9.c even 3 1 inner 1008.2.r.e 4
9.c even 3 1 9072.2.a.bk 2
9.d odd 6 1 3024.2.r.e 4
9.d odd 6 1 9072.2.a.bd 2
12.b even 2 1 378.2.f.d 4
28.d even 2 1 882.2.f.j 4
28.f even 6 1 882.2.e.n 4
28.f even 6 1 882.2.h.l 4
28.g odd 6 1 882.2.e.m 4
28.g odd 6 1 882.2.h.k 4
36.f odd 6 1 126.2.f.c 4
36.f odd 6 1 1134.2.a.p 2
36.h even 6 1 378.2.f.d 4
36.h even 6 1 1134.2.a.i 2
84.h odd 2 1 2646.2.f.k 4
84.j odd 6 1 2646.2.e.k 4
84.j odd 6 1 2646.2.h.n 4
84.n even 6 1 2646.2.e.l 4
84.n even 6 1 2646.2.h.m 4
252.n even 6 1 882.2.e.n 4
252.o even 6 1 2646.2.e.l 4
252.r odd 6 1 2646.2.h.n 4
252.s odd 6 1 2646.2.f.k 4
252.s odd 6 1 7938.2.a.bm 2
252.u odd 6 1 882.2.h.k 4
252.bb even 6 1 2646.2.h.m 4
252.bi even 6 1 882.2.f.j 4
252.bi even 6 1 7938.2.a.bn 2
252.bj even 6 1 882.2.h.l 4
252.bl odd 6 1 882.2.e.m 4
252.bn odd 6 1 2646.2.e.k 4

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

 $$T_{5}^{4} + 2T_{5}^{3} + 9T_{5}^{2} - 10T_{5} + 25$$ T5^4 + 2*T5^3 + 9*T5^2 - 10*T5 + 25 $$T_{11}^{2} - 2T_{11} + 4$$ T11^2 - 2*T11 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 2 T + 3)^{2}$$
$5$ $$T^{4} + 2 T^{3} + 9 T^{2} - 10 T + 25$$
$7$ $$(T^{2} + T + 1)^{2}$$
$11$ $$(T^{2} - 2 T + 4)^{2}$$
$13$ $$T^{4} + 24T^{2} + 576$$
$17$ $$(T - 2)^{4}$$
$19$ $$(T^{2} + 10 T + 19)^{2}$$
$23$ $$(T^{2} + T + 1)^{2}$$
$29$ $$T^{4} - 4 T^{3} + 36 T^{2} + 80 T + 400$$
$31$ $$(T^{2} - 6 T + 36)^{2}$$
$37$ $$(T^{2} - 4 T - 92)^{2}$$
$41$ $$T^{4} + 96T^{2} + 9216$$
$43$ $$T^{4} - 4 T^{3} + 36 T^{2} + 80 T + 400$$
$47$ $$T^{4} + 96T^{2} + 9216$$
$53$ $$(T^{2} + 12 T + 12)^{2}$$
$59$ $$(T^{2} + 2 T + 4)^{2}$$
$61$ $$T^{4} + 18 T^{3} + 249 T^{2} + \cdots + 5625$$
$67$ $$T^{4} + 16 T^{3} + 216 T^{2} + \cdots + 1600$$
$71$ $$(T^{2} + 10 T + 1)^{2}$$
$73$ $$(T^{2} + 4 T - 20)^{2}$$
$79$ $$T^{4} - 6 T^{3} + 51 T^{2} + 90 T + 225$$
$83$ $$(T^{2} - 2 T + 4)^{2}$$
$89$ $$(T^{2} + 24 T + 120)^{2}$$
$97$ $$T^{4} - 4 T^{3} + 36 T^{2} + 80 T + 400$$