# Properties

 Label 1008.2.bt.a Level $1008$ Weight $2$ Character orbit 1008.bt Analytic conductor $8.049$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,2,Mod(17,1008)]

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, 3, 1]))

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

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

chi := DirichletCharacter("1008.17");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1008 = 2^{4} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1008.bt (of order $$6$$, 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 $$\Q(\zeta_{6})$$ 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 252) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{5} + (3 \beta_1 - 2) q^{7}+O(q^{10})$$ q - b2 * q^5 + (3*b1 - 2) * q^7 $$q - \beta_{2} q^{5} + (3 \beta_1 - 2) q^{7} + (\beta_{3} + \beta_{2}) q^{11} + (2 \beta_1 - 1) q^{13} + (2 \beta_{3} - 2 \beta_{2}) q^{17} + (3 \beta_1 - 6) q^{19} + (\beta_1 - 1) q^{25} + ( - 2 \beta_{3} + 4 \beta_{2}) q^{29} + (\beta_1 + 1) q^{31} + (3 \beta_{3} - \beta_{2}) q^{35} + 5 \beta_1 q^{37} + 5 \beta_{3} q^{41} + 11 q^{43} + \beta_{2} q^{47} + ( - 3 \beta_1 - 5) q^{49} + (2 \beta_{3} + 2 \beta_{2}) q^{53} + ( - 12 \beta_1 + 6) q^{55} + (2 \beta_{3} - 2 \beta_{2}) q^{59} + (2 \beta_1 - 4) q^{61} + (2 \beta_{3} - \beta_{2}) q^{65} + (7 \beta_1 - 7) q^{67} + (3 \beta_{3} - 6 \beta_{2}) q^{71} + (9 \beta_1 + 9) q^{73} + ( - 5 \beta_{3} + 4 \beta_{2}) q^{77} + 11 \beta_1 q^{79} - 5 \beta_{3} q^{83} - 12 q^{85} - 6 \beta_{2} q^{89} + ( - \beta_1 - 4) q^{91} + (3 \beta_{3} + 3 \beta_{2}) q^{95} + (4 \beta_1 - 2) q^{97}+O(q^{100})$$ q - b2 * q^5 + (3*b1 - 2) * q^7 + (b3 + b2) * q^11 + (2*b1 - 1) * q^13 + (2*b3 - 2*b2) * q^17 + (3*b1 - 6) * q^19 + (b1 - 1) * q^25 + (-2*b3 + 4*b2) * q^29 + (b1 + 1) * q^31 + (3*b3 - b2) * q^35 + 5*b1 * q^37 + 5*b3 * q^41 + 11 * q^43 + b2 * q^47 + (-3*b1 - 5) * q^49 + (2*b3 + 2*b2) * q^53 + (-12*b1 + 6) * q^55 + (2*b3 - 2*b2) * q^59 + (2*b1 - 4) * q^61 + (2*b3 - b2) * q^65 + (7*b1 - 7) * q^67 + (3*b3 - 6*b2) * q^71 + (9*b1 + 9) * q^73 + (-5*b3 + 4*b2) * q^77 + 11*b1 * q^79 - 5*b3 * q^83 - 12 * q^85 - 6*b2 * q^89 + (-b1 - 4) * q^91 + (3*b3 + 3*b2) * q^95 + (4*b1 - 2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{7}+O(q^{10})$$ 4 * q - 2 * q^7 $$4 q - 2 q^{7} - 18 q^{19} - 2 q^{25} + 6 q^{31} + 10 q^{37} + 44 q^{43} - 26 q^{49} - 12 q^{61} - 14 q^{67} + 54 q^{73} + 22 q^{79} - 48 q^{85} - 18 q^{91}+O(q^{100})$$ 4 * q - 2 * q^7 - 18 * q^19 - 2 * q^25 + 6 * q^31 + 10 * q^37 + 44 * q^43 - 26 * q^49 - 12 * q^61 - 14 * q^67 + 54 * q^73 + 22 * q^79 - 48 * q^85 - 18 * q^91

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

 $$\beta_{1}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 2\nu ) / 2$$ (v^3 + 2*v) / 2 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 4\nu ) / 2$$ (-v^3 + 4*v) / 2
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 3$$ (b3 + b2) / 3 $$\nu^{2}$$ $$=$$ $$2\beta_1$$ 2*b1 $$\nu^{3}$$ $$=$$ $$( -2\beta_{3} + 4\beta_{2} ) / 3$$ (-2*b3 + 4*b2) / 3

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 1.22474 + 0.707107i −1.22474 − 0.707107i 1.22474 − 0.707107i −1.22474 + 0.707107i
0 0 0 −1.22474 2.12132i 0 −0.500000 + 2.59808i 0 0 0
17.2 0 0 0 1.22474 + 2.12132i 0 −0.500000 + 2.59808i 0 0 0
593.1 0 0 0 −1.22474 + 2.12132i 0 −0.500000 2.59808i 0 0 0
593.2 0 0 0 1.22474 2.12132i 0 −0.500000 2.59808i 0 0 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
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.bt.a 4
3.b odd 2 1 inner 1008.2.bt.a 4
4.b odd 2 1 252.2.t.a 4
7.c even 3 1 7056.2.k.a 4
7.d odd 6 1 inner 1008.2.bt.a 4
7.d odd 6 1 7056.2.k.a 4
12.b even 2 1 252.2.t.a 4
20.d odd 2 1 6300.2.ch.a 4
20.e even 4 2 6300.2.dd.a 8
21.g even 6 1 inner 1008.2.bt.a 4
21.g even 6 1 7056.2.k.a 4
21.h odd 6 1 7056.2.k.a 4
28.d even 2 1 1764.2.t.a 4
28.f even 6 1 252.2.t.a 4
28.f even 6 1 1764.2.f.a 4
28.g odd 6 1 1764.2.f.a 4
28.g odd 6 1 1764.2.t.a 4
36.f odd 6 1 2268.2.w.h 4
36.f odd 6 1 2268.2.bm.g 4
36.h even 6 1 2268.2.w.h 4
36.h even 6 1 2268.2.bm.g 4
60.h even 2 1 6300.2.ch.a 4
60.l odd 4 2 6300.2.dd.a 8
84.h odd 2 1 1764.2.t.a 4
84.j odd 6 1 252.2.t.a 4
84.j odd 6 1 1764.2.f.a 4
84.n even 6 1 1764.2.f.a 4
84.n even 6 1 1764.2.t.a 4
140.s even 6 1 6300.2.ch.a 4
140.x odd 12 2 6300.2.dd.a 8
252.n even 6 1 2268.2.w.h 4
252.r odd 6 1 2268.2.bm.g 4
252.bj even 6 1 2268.2.bm.g 4
252.bn odd 6 1 2268.2.w.h 4
420.be odd 6 1 6300.2.ch.a 4
420.br even 12 2 6300.2.dd.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.t.a 4 4.b odd 2 1
252.2.t.a 4 12.b even 2 1
252.2.t.a 4 28.f even 6 1
252.2.t.a 4 84.j odd 6 1
1008.2.bt.a 4 1.a even 1 1 trivial
1008.2.bt.a 4 3.b odd 2 1 inner
1008.2.bt.a 4 7.d odd 6 1 inner
1008.2.bt.a 4 21.g even 6 1 inner
1764.2.f.a 4 28.f even 6 1
1764.2.f.a 4 28.g odd 6 1
1764.2.f.a 4 84.j odd 6 1
1764.2.f.a 4 84.n even 6 1
1764.2.t.a 4 28.d even 2 1
1764.2.t.a 4 28.g odd 6 1
1764.2.t.a 4 84.h odd 2 1
1764.2.t.a 4 84.n even 6 1
2268.2.w.h 4 36.f odd 6 1
2268.2.w.h 4 36.h even 6 1
2268.2.w.h 4 252.n even 6 1
2268.2.w.h 4 252.bn odd 6 1
2268.2.bm.g 4 36.f odd 6 1
2268.2.bm.g 4 36.h even 6 1
2268.2.bm.g 4 252.r odd 6 1
2268.2.bm.g 4 252.bj even 6 1
6300.2.ch.a 4 20.d odd 2 1
6300.2.ch.a 4 60.h even 2 1
6300.2.ch.a 4 140.s even 6 1
6300.2.ch.a 4 420.be odd 6 1
6300.2.dd.a 8 20.e even 4 2
6300.2.dd.a 8 60.l odd 4 2
6300.2.dd.a 8 140.x odd 12 2
6300.2.dd.a 8 420.br even 12 2
7056.2.k.a 4 7.c even 3 1
7056.2.k.a 4 7.d odd 6 1
7056.2.k.a 4 21.g even 6 1
7056.2.k.a 4 21.h odd 6 1

## Hecke kernels

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

 $$T_{5}^{4} + 6T_{5}^{2} + 36$$ T5^4 + 6*T5^2 + 36 $$T_{11}^{4} - 18T_{11}^{2} + 324$$ T11^4 - 18*T11^2 + 324

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 6T^{2} + 36$$
$7$ $$(T^{2} + T + 7)^{2}$$
$11$ $$T^{4} - 18T^{2} + 324$$
$13$ $$(T^{2} + 3)^{2}$$
$17$ $$T^{4} + 24T^{2} + 576$$
$19$ $$(T^{2} + 9 T + 27)^{2}$$
$23$ $$T^{4}$$
$29$ $$(T^{2} + 72)^{2}$$
$31$ $$(T^{2} - 3 T + 3)^{2}$$
$37$ $$(T^{2} - 5 T + 25)^{2}$$
$41$ $$(T^{2} - 150)^{2}$$
$43$ $$(T - 11)^{4}$$
$47$ $$T^{4} + 6T^{2} + 36$$
$53$ $$T^{4} - 72T^{2} + 5184$$
$59$ $$T^{4} + 24T^{2} + 576$$
$61$ $$(T^{2} + 6 T + 12)^{2}$$
$67$ $$(T^{2} + 7 T + 49)^{2}$$
$71$ $$(T^{2} + 162)^{2}$$
$73$ $$(T^{2} - 27 T + 243)^{2}$$
$79$ $$(T^{2} - 11 T + 121)^{2}$$
$83$ $$(T^{2} - 150)^{2}$$
$89$ $$T^{4} + 216 T^{2} + 46656$$
$97$ $$(T^{2} + 12)^{2}$$