# Properties

 Label 864.2.i.e Level $864$ Weight $2$ Character orbit 864.i Analytic conductor $6.899$ 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] = [864,2,Mod(289,864)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("864.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$864 = 2^{5} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 864.i (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: $$6.89907473464$$ 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: $$3$$ Twist minimal: no (minimal twist has level 288) 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{5} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{7}+O(q^{10})$$ q - b1 * q^5 + (-b3 + b2 - b1 + 1) * q^7 $$q - \beta_1 q^{5} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{7} + (\beta_{3} - \beta_{2} - \beta_1 + 1) q^{11} + (2 \beta_{2} - \beta_1) q^{13} + 2 \beta_{3} q^{17} - 4 q^{19} + ( - \beta_{2} + 3 \beta_1) q^{23} + ( - 4 \beta_1 + 4) q^{25} + (2 \beta_{3} - 2 \beta_{2} - 5 \beta_1 + 5) q^{29} + (\beta_{2} + 5 \beta_1) q^{31} + (\beta_{3} - 1) q^{35} + (2 \beta_{3} + 4) q^{37} + (2 \beta_{2} - 7 \beta_1) q^{41} + (3 \beta_{3} - 3 \beta_{2} - 5 \beta_1 + 5) q^{43} + ( - 3 \beta_{3} + 3 \beta_{2} + \beta_1 - 1) q^{47} + 2 \beta_{2} q^{49} + (2 \beta_{3} - 4) q^{53} + ( - \beta_{3} - 1) q^{55} + (3 \beta_{2} - 7 \beta_1) q^{59} + ( - 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 3) q^{61} + (2 \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{65} + ( - 3 \beta_{2} + 5 \beta_1) q^{67} + ( - 4 \beta_{3} - 2) q^{71} + 2 \beta_{3} q^{73} + 5 \beta_1 q^{77} + ( - \beta_{3} + \beta_{2} - 11 \beta_1 + 11) q^{79} + (\beta_{3} - \beta_{2} - 3 \beta_1 + 3) q^{83} - 2 \beta_{2} q^{85} + ( - 2 \beta_{3} + 8) q^{89} + (3 \beta_{3} - 13) q^{91} + 4 \beta_1 q^{95} + ( - 2 \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{97}+O(q^{100})$$ q - b1 * q^5 + (-b3 + b2 - b1 + 1) * q^7 + (b3 - b2 - b1 + 1) * q^11 + (2*b2 - b1) * q^13 + 2*b3 * q^17 - 4 * q^19 + (-b2 + 3*b1) * q^23 + (-4*b1 + 4) * q^25 + (2*b3 - 2*b2 - 5*b1 + 5) * q^29 + (b2 + 5*b1) * q^31 + (b3 - 1) * q^35 + (2*b3 + 4) * q^37 + (2*b2 - 7*b1) * q^41 + (3*b3 - 3*b2 - 5*b1 + 5) * q^43 + (-3*b3 + 3*b2 + b1 - 1) * q^47 + 2*b2 * q^49 + (2*b3 - 4) * q^53 + (-b3 - 1) * q^55 + (3*b2 - 7*b1) * q^59 + (-2*b3 + 2*b2 - 3*b1 + 3) * q^61 + (2*b3 - 2*b2 + b1 - 1) * q^65 + (-3*b2 + 5*b1) * q^67 + (-4*b3 - 2) * q^71 + 2*b3 * q^73 + 5*b1 * q^77 + (-b3 + b2 - 11*b1 + 11) * q^79 + (b3 - b2 - 3*b1 + 3) * q^83 - 2*b2 * q^85 + (-2*b3 + 8) * q^89 + (3*b3 - 13) * q^91 + 4*b1 * q^95 + (-2*b3 + 2*b2 + b1 - 1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{5} + 2 q^{7}+O(q^{10})$$ 4 * q - 2 * q^5 + 2 * q^7 $$4 q - 2 q^{5} + 2 q^{7} + 2 q^{11} - 2 q^{13} - 16 q^{19} + 6 q^{23} + 8 q^{25} + 10 q^{29} + 10 q^{31} - 4 q^{35} + 16 q^{37} - 14 q^{41} + 10 q^{43} - 2 q^{47} - 16 q^{53} - 4 q^{55} - 14 q^{59} + 6 q^{61} - 2 q^{65} + 10 q^{67} - 8 q^{71} + 10 q^{77} + 22 q^{79} + 6 q^{83} + 32 q^{89} - 52 q^{91} + 8 q^{95} - 2 q^{97}+O(q^{100})$$ 4 * q - 2 * q^5 + 2 * q^7 + 2 * q^11 - 2 * q^13 - 16 * q^19 + 6 * q^23 + 8 * q^25 + 10 * q^29 + 10 * q^31 - 4 * q^35 + 16 * q^37 - 14 * q^41 + 10 * q^43 - 2 * q^47 - 16 * q^53 - 4 * q^55 - 14 * q^59 + 6 * q^61 - 2 * q^65 + 10 * q^67 - 8 * q^71 + 10 * q^77 + 22 * q^79 + 6 * q^83 + 32 * q^89 - 52 * q^91 + 8 * q^95 - 2 * q^97

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 $$\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$353$$ $$703$$ $$\chi(n)$$ $$1$$ $$-\beta_{1}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 1.22474 + 0.707107i −1.22474 − 0.707107i 1.22474 − 0.707107i −1.22474 + 0.707107i
0 0 0 −0.500000 0.866025i 0 −0.724745 + 1.25529i 0 0 0
289.2 0 0 0 −0.500000 0.866025i 0 1.72474 2.98735i 0 0 0
577.1 0 0 0 −0.500000 + 0.866025i 0 −0.724745 1.25529i 0 0 0
577.2 0 0 0 −0.500000 + 0.866025i 0 1.72474 + 2.98735i 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
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 864.2.i.e 4
3.b odd 2 1 288.2.i.c 4
4.b odd 2 1 864.2.i.c 4
8.b even 2 1 1728.2.i.m 4
8.d odd 2 1 1728.2.i.k 4
9.c even 3 1 inner 864.2.i.e 4
9.c even 3 1 2592.2.a.o 2
9.d odd 6 1 288.2.i.c 4
9.d odd 6 1 2592.2.a.j 2
12.b even 2 1 288.2.i.e yes 4
24.f even 2 1 576.2.i.i 4
24.h odd 2 1 576.2.i.m 4
36.f odd 6 1 864.2.i.c 4
36.f odd 6 1 2592.2.a.s 2
36.h even 6 1 288.2.i.e yes 4
36.h even 6 1 2592.2.a.n 2
72.j odd 6 1 576.2.i.m 4
72.j odd 6 1 5184.2.a.bu 2
72.l even 6 1 576.2.i.i 4
72.l even 6 1 5184.2.a.by 2
72.n even 6 1 1728.2.i.m 4
72.n even 6 1 5184.2.a.bj 2
72.p odd 6 1 1728.2.i.k 4
72.p odd 6 1 5184.2.a.bn 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.i.c 4 3.b odd 2 1
288.2.i.c 4 9.d odd 6 1
288.2.i.e yes 4 12.b even 2 1
288.2.i.e yes 4 36.h even 6 1
576.2.i.i 4 24.f even 2 1
576.2.i.i 4 72.l even 6 1
576.2.i.m 4 24.h odd 2 1
576.2.i.m 4 72.j odd 6 1
864.2.i.c 4 4.b odd 2 1
864.2.i.c 4 36.f odd 6 1
864.2.i.e 4 1.a even 1 1 trivial
864.2.i.e 4 9.c even 3 1 inner
1728.2.i.k 4 8.d odd 2 1
1728.2.i.k 4 72.p odd 6 1
1728.2.i.m 4 8.b even 2 1
1728.2.i.m 4 72.n even 6 1
2592.2.a.j 2 9.d odd 6 1
2592.2.a.n 2 36.h even 6 1
2592.2.a.o 2 9.c even 3 1
2592.2.a.s 2 36.f odd 6 1
5184.2.a.bj 2 72.n even 6 1
5184.2.a.bn 2 72.p odd 6 1
5184.2.a.bu 2 72.j odd 6 1
5184.2.a.by 2 72.l even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(864, [\chi])$$:

 $$T_{5}^{2} + T_{5} + 1$$ T5^2 + T5 + 1 $$T_{7}^{4} - 2T_{7}^{3} + 9T_{7}^{2} + 10T_{7} + 25$$ T7^4 - 2*T7^3 + 9*T7^2 + 10*T7 + 25

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + T + 1)^{2}$$
$7$ $$T^{4} - 2 T^{3} + 9 T^{2} + 10 T + 25$$
$11$ $$T^{4} - 2 T^{3} + 9 T^{2} + 10 T + 25$$
$13$ $$T^{4} + 2 T^{3} + 27 T^{2} - 46 T + 529$$
$17$ $$(T^{2} - 24)^{2}$$
$19$ $$(T + 4)^{4}$$
$23$ $$T^{4} - 6 T^{3} + 33 T^{2} - 18 T + 9$$
$29$ $$T^{4} - 10 T^{3} + 99 T^{2} - 10 T + 1$$
$31$ $$T^{4} - 10 T^{3} + 81 T^{2} + \cdots + 361$$
$37$ $$(T^{2} - 8 T - 8)^{2}$$
$41$ $$T^{4} + 14 T^{3} + 171 T^{2} + \cdots + 625$$
$43$ $$T^{4} - 10 T^{3} + 129 T^{2} + \cdots + 841$$
$47$ $$T^{4} + 2 T^{3} + 57 T^{2} + \cdots + 2809$$
$53$ $$(T^{2} + 8 T - 8)^{2}$$
$59$ $$T^{4} + 14 T^{3} + 201 T^{2} + \cdots + 25$$
$61$ $$T^{4} - 6 T^{3} + 51 T^{2} + 90 T + 225$$
$67$ $$T^{4} - 10 T^{3} + 129 T^{2} + \cdots + 841$$
$71$ $$(T^{2} + 4 T - 92)^{2}$$
$73$ $$(T^{2} - 24)^{2}$$
$79$ $$T^{4} - 22 T^{3} + 369 T^{2} + \cdots + 13225$$
$83$ $$T^{4} - 6 T^{3} + 33 T^{2} - 18 T + 9$$
$89$ $$(T^{2} - 16 T + 40)^{2}$$
$97$ $$T^{4} + 2 T^{3} + 27 T^{2} - 46 T + 529$$