# Properties

 Label 864.2.c.a Level $864$ Weight $2$ Character orbit 864.c Analytic conductor $6.899$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,2,Mod(863,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

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

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

chi := DirichletCharacter("864.863");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$864 = 2^{5} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 864.c (of order $$2$$, degree $$1$$, 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: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{5} - \beta_{7} q^{7}+O(q^{10})$$ q - b3 * q^5 - b7 * q^7 $$q - \beta_{3} q^{5} - \beta_{7} q^{7} + (\beta_{5} + \beta_1) q^{11} + ( - \beta_{6} - 2) q^{13} + (\beta_{4} + \beta_{3}) q^{17} + ( - 2 \beta_{7} + \beta_{2}) q^{19} - \beta_{5} q^{23} - \beta_{6} q^{25} + ( - 3 \beta_{4} + \beta_{3}) q^{29} + ( - \beta_{7} + 3 \beta_{2}) q^{31} + ( - 2 \beta_{5} - 3 \beta_1) q^{35} - \beta_{6} q^{37} + (3 \beta_{4} + \beta_{3}) q^{41} + ( - 2 \beta_{7} + 2 \beta_{2}) q^{43} + (2 \beta_{5} - 2 \beta_1) q^{47} - \beta_{6} q^{49} + ( - \beta_{4} - 4 \beta_{3}) q^{53} + ( - 3 \beta_{7} - 2 \beta_{2}) q^{55} + ( - 2 \beta_{5} + 2 \beta_1) q^{59} + 4 q^{61} + ( - \beta_{4} + 7 \beta_{3}) q^{65} + ( - 2 \beta_{7} - 4 \beta_{2}) q^{67} + (\beta_{5} + 6 \beta_1) q^{71} + ( - \beta_{6} - 3) q^{73} - 5 \beta_{3} q^{77} - \beta_{2} q^{79} + (5 \beta_{5} - \beta_1) q^{83} + (\beta_{6} + 6) q^{85} + (4 \beta_{4} - 2 \beta_{3}) q^{89} + (4 \beta_{7} + 5 \beta_{2}) q^{91} + ( - 3 \beta_{5} - 4 \beta_1) q^{95} - 5 q^{97}+O(q^{100})$$ q - b3 * q^5 - b7 * q^7 + (b5 + b1) * q^11 + (-b6 - 2) * q^13 + (b4 + b3) * q^17 + (-2*b7 + b2) * q^19 - b5 * q^23 - b6 * q^25 + (-3*b4 + b3) * q^29 + (-b7 + 3*b2) * q^31 + (-2*b5 - 3*b1) * q^35 - b6 * q^37 + (3*b4 + b3) * q^41 + (-2*b7 + 2*b2) * q^43 + (2*b5 - 2*b1) * q^47 - b6 * q^49 + (-b4 - 4*b3) * q^53 + (-3*b7 - 2*b2) * q^55 + (-2*b5 + 2*b1) * q^59 + 4 * q^61 + (-b4 + 7*b3) * q^65 + (-2*b7 - 4*b2) * q^67 + (b5 + 6*b1) * q^71 + (-b6 - 3) * q^73 - 5*b3 * q^77 - b2 * q^79 + (5*b5 - b1) * q^83 + (b6 + 6) * q^85 + (4*b4 - 2*b3) * q^89 + (4*b7 + 5*b2) * q^91 + (-3*b5 - 4*b1) * q^95 - 5 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q - 16 q^{13} + 32 q^{61} - 24 q^{73} + 48 q^{85} - 40 q^{97}+O(q^{100})$$ 8 * q - 16 * q^13 + 32 * q^61 - 24 * q^73 + 48 * q^85 - 40 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$-\zeta_{24}^{6} + 2\zeta_{24}^{2}$$ -v^6 + 2*v^2 $$\beta_{2}$$ $$=$$ $$2\zeta_{24}^{6}$$ 2*v^6 $$\beta_{3}$$ $$=$$ $$\zeta_{24}^{5} + 2\zeta_{24}^{4} + \zeta_{24}^{3} - \zeta_{24} - 1$$ v^5 + 2*v^4 + v^3 - v - 1 $$\beta_{4}$$ $$=$$ $$-\zeta_{24}^{5} + 2\zeta_{24}^{4} - \zeta_{24}^{3} + \zeta_{24} - 1$$ -v^5 + 2*v^4 - v^3 + v - 1 $$\beta_{5}$$ $$=$$ $$-2\zeta_{24}^{5} + 2\zeta_{24}^{3} + 2\zeta_{24}$$ -2*v^5 + 2*v^3 + 2*v $$\beta_{6}$$ $$=$$ $$-4\zeta_{24}^{7} + 2\zeta_{24}^{5} + 2\zeta_{24}^{3} + 2\zeta_{24}$$ -4*v^7 + 2*v^5 + 2*v^3 + 2*v $$\beta_{7}$$ $$=$$ $$2\zeta_{24}^{7} + \zeta_{24}^{6} + \zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24}$$ 2*v^7 + v^6 + v^5 - v^3 + v
 $$\zeta_{24}$$ $$=$$ $$( 2\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} ) / 8$$ (2*b7 + b6 + b5 + b4 - b3 - b2) / 8 $$\zeta_{24}^{2}$$ $$=$$ $$( \beta_{2} + 2\beta_1 ) / 4$$ (b2 + 2*b1) / 4 $$\zeta_{24}^{3}$$ $$=$$ $$( \beta_{5} - \beta_{4} + \beta_{3} ) / 4$$ (b5 - b4 + b3) / 4 $$\zeta_{24}^{4}$$ $$=$$ $$( \beta_{4} + \beta_{3} + 2 ) / 4$$ (b4 + b3 + 2) / 4 $$\zeta_{24}^{5}$$ $$=$$ $$( 2\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} ) / 8$$ (2*b7 + b6 - b5 - b4 + b3 - b2) / 8 $$\zeta_{24}^{6}$$ $$=$$ $$( \beta_{2} ) / 2$$ (b2) / 2 $$\zeta_{24}^{7}$$ $$=$$ $$( 2\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} ) / 8$$ (2*b7 - b6 + b5 - b4 + b3 - b2) / 8

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$353$$ $$703$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
863.1
 0.965926 + 0.258819i 0.258819 − 0.965926i −0.258819 + 0.965926i −0.965926 − 0.258819i −0.965926 + 0.258819i −0.258819 − 0.965926i 0.258819 + 0.965926i 0.965926 − 0.258819i
0 0 0 3.14626i 0 3.44949i 0 0 0
863.2 0 0 0 3.14626i 0 3.44949i 0 0 0
863.3 0 0 0 0.317837i 0 1.44949i 0 0 0
863.4 0 0 0 0.317837i 0 1.44949i 0 0 0
863.5 0 0 0 0.317837i 0 1.44949i 0 0 0
863.6 0 0 0 0.317837i 0 1.44949i 0 0 0
863.7 0 0 0 3.14626i 0 3.44949i 0 0 0
863.8 0 0 0 3.14626i 0 3.44949i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 863.8 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
4.b odd 2 1 inner
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.2.c.a 8
3.b odd 2 1 inner 864.2.c.a 8
4.b odd 2 1 inner 864.2.c.a 8
8.b even 2 1 1728.2.c.g 8
8.d odd 2 1 1728.2.c.g 8
9.c even 3 1 2592.2.s.a 8
9.c even 3 1 2592.2.s.h 8
9.d odd 6 1 2592.2.s.a 8
9.d odd 6 1 2592.2.s.h 8
12.b even 2 1 inner 864.2.c.a 8
24.f even 2 1 1728.2.c.g 8
24.h odd 2 1 1728.2.c.g 8
36.f odd 6 1 2592.2.s.a 8
36.f odd 6 1 2592.2.s.h 8
36.h even 6 1 2592.2.s.a 8
36.h even 6 1 2592.2.s.h 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.c.a 8 1.a even 1 1 trivial
864.2.c.a 8 3.b odd 2 1 inner
864.2.c.a 8 4.b odd 2 1 inner
864.2.c.a 8 12.b even 2 1 inner
1728.2.c.g 8 8.b even 2 1
1728.2.c.g 8 8.d odd 2 1
1728.2.c.g 8 24.f even 2 1
1728.2.c.g 8 24.h odd 2 1
2592.2.s.a 8 9.c even 3 1
2592.2.s.a 8 9.d odd 6 1
2592.2.s.a 8 36.f odd 6 1
2592.2.s.a 8 36.h even 6 1
2592.2.s.h 8 9.c even 3 1
2592.2.s.h 8 9.d odd 6 1
2592.2.s.h 8 36.f odd 6 1
2592.2.s.h 8 36.h even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 10T_{5}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(864, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} + 10 T^{2} + 1)^{2}$$
$7$ $$(T^{4} + 14 T^{2} + 25)^{2}$$
$11$ $$(T^{4} - 22 T^{2} + 25)^{2}$$
$13$ $$(T^{2} + 4 T - 20)^{4}$$
$17$ $$(T^{2} + 12)^{4}$$
$19$ $$(T^{2} + 24)^{4}$$
$23$ $$(T^{2} - 8)^{4}$$
$29$ $$(T^{4} + 88 T^{2} + 400)^{2}$$
$31$ $$(T^{4} + 62 T^{2} + 361)^{2}$$
$37$ $$(T^{2} - 24)^{4}$$
$41$ $$(T^{4} + 112 T^{2} + 1600)^{2}$$
$43$ $$(T^{4} + 56 T^{2} + 400)^{2}$$
$47$ $$(T^{4} - 88 T^{2} + 400)^{2}$$
$53$ $$(T^{4} + 186 T^{2} + 3249)^{2}$$
$59$ $$(T^{4} - 88 T^{2} + 400)^{2}$$
$61$ $$(T - 4)^{8}$$
$67$ $$(T^{4} + 248 T^{2} + 5776)^{2}$$
$71$ $$(T^{4} - 232 T^{2} + 10000)^{2}$$
$73$ $$(T^{2} + 6 T - 15)^{4}$$
$79$ $$(T^{2} + 4)^{4}$$
$83$ $$(T^{4} - 406 T^{2} + 38809)^{2}$$
$89$ $$(T^{4} + 168 T^{2} + 3600)^{2}$$
$97$ $$(T + 5)^{8}$$