# Properties

 Label 81.2.c.a Level $81$ Weight $2$ Character orbit 81.c Analytic conductor $0.647$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,2,Mod(28,81)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(81, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2]))

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

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

chi := DirichletCharacter("81.28");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$81 = 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 81.c (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: $$0.646788256372$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 27) Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{4} + ( - \zeta_{6} + 1) q^{7}+O(q^{10})$$ q + 2*z * q^4 + (-z + 1) * q^7 $$q + 2 \zeta_{6} q^{4} + ( - \zeta_{6} + 1) q^{7} - 5 \zeta_{6} q^{13} + (4 \zeta_{6} - 4) q^{16} - 7 q^{19} + ( - 5 \zeta_{6} + 5) q^{25} + 2 q^{28} + 4 \zeta_{6} q^{31} + 11 q^{37} + (8 \zeta_{6} - 8) q^{43} + 6 \zeta_{6} q^{49} + ( - 10 \zeta_{6} + 10) q^{52} + ( - \zeta_{6} + 1) q^{61} - 8 q^{64} - 5 \zeta_{6} q^{67} - 7 q^{73} - 14 \zeta_{6} q^{76} + (17 \zeta_{6} - 17) q^{79} - 5 q^{91} + ( - 19 \zeta_{6} + 19) q^{97} +O(q^{100})$$ q + 2*z * q^4 + (-z + 1) * q^7 - 5*z * q^13 + (4*z - 4) * q^16 - 7 * q^19 + (-5*z + 5) * q^25 + 2 * q^28 + 4*z * q^31 + 11 * q^37 + (8*z - 8) * q^43 + 6*z * q^49 + (-10*z + 10) * q^52 + (-z + 1) * q^61 - 8 * q^64 - 5*z * q^67 - 7 * q^73 - 14*z * q^76 + (17*z - 17) * q^79 - 5 * q^91 + (-19*z + 19) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} + q^{7}+O(q^{10})$$ 2 * q + 2 * q^4 + q^7 $$2 q + 2 q^{4} + q^{7} - 5 q^{13} - 4 q^{16} - 14 q^{19} + 5 q^{25} + 4 q^{28} + 4 q^{31} + 22 q^{37} - 8 q^{43} + 6 q^{49} + 10 q^{52} + q^{61} - 16 q^{64} - 5 q^{67} - 14 q^{73} - 14 q^{76} - 17 q^{79} - 10 q^{91} + 19 q^{97}+O(q^{100})$$ 2 * q + 2 * q^4 + q^7 - 5 * q^13 - 4 * q^16 - 14 * q^19 + 5 * q^25 + 4 * q^28 + 4 * q^31 + 22 * q^37 - 8 * q^43 + 6 * q^49 + 10 * q^52 + q^61 - 16 * q^64 - 5 * q^67 - 14 * q^73 - 14 * q^76 - 17 * q^79 - 10 * q^91 + 19 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/81\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
28.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 1.00000 1.73205i 0 0 0.500000 + 0.866025i 0 0 0
55.1 0 0 1.00000 + 1.73205i 0 0 0.500000 0.866025i 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 CM by $$\Q(\sqrt{-3})$$
9.c even 3 1 inner
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 81.2.c.a 2
3.b odd 2 1 CM 81.2.c.a 2
4.b odd 2 1 1296.2.i.i 2
9.c even 3 1 27.2.a.a 1
9.c even 3 1 inner 81.2.c.a 2
9.d odd 6 1 27.2.a.a 1
9.d odd 6 1 inner 81.2.c.a 2
12.b even 2 1 1296.2.i.i 2
27.e even 9 6 729.2.e.f 6
27.f odd 18 6 729.2.e.f 6
36.f odd 6 1 432.2.a.e 1
36.f odd 6 1 1296.2.i.i 2
36.h even 6 1 432.2.a.e 1
36.h even 6 1 1296.2.i.i 2
45.h odd 6 1 675.2.a.e 1
45.j even 6 1 675.2.a.e 1
45.k odd 12 2 675.2.b.f 2
45.l even 12 2 675.2.b.f 2
63.l odd 6 1 1323.2.a.i 1
63.o even 6 1 1323.2.a.i 1
72.j odd 6 1 1728.2.a.n 1
72.l even 6 1 1728.2.a.o 1
72.n even 6 1 1728.2.a.n 1
72.p odd 6 1 1728.2.a.o 1
99.g even 6 1 3267.2.a.f 1
99.h odd 6 1 3267.2.a.f 1
117.n odd 6 1 4563.2.a.e 1
117.t even 6 1 4563.2.a.e 1
153.h even 6 1 7803.2.a.k 1
153.i odd 6 1 7803.2.a.k 1
171.l even 6 1 9747.2.a.f 1
171.o odd 6 1 9747.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.2.a.a 1 9.c even 3 1
27.2.a.a 1 9.d odd 6 1
81.2.c.a 2 1.a even 1 1 trivial
81.2.c.a 2 3.b odd 2 1 CM
81.2.c.a 2 9.c even 3 1 inner
81.2.c.a 2 9.d odd 6 1 inner
432.2.a.e 1 36.f odd 6 1
432.2.a.e 1 36.h even 6 1
675.2.a.e 1 45.h odd 6 1
675.2.a.e 1 45.j even 6 1
675.2.b.f 2 45.k odd 12 2
675.2.b.f 2 45.l even 12 2
729.2.e.f 6 27.e even 9 6
729.2.e.f 6 27.f odd 18 6
1296.2.i.i 2 4.b odd 2 1
1296.2.i.i 2 12.b even 2 1
1296.2.i.i 2 36.f odd 6 1
1296.2.i.i 2 36.h even 6 1
1323.2.a.i 1 63.l odd 6 1
1323.2.a.i 1 63.o even 6 1
1728.2.a.n 1 72.j odd 6 1
1728.2.a.n 1 72.n even 6 1
1728.2.a.o 1 72.l even 6 1
1728.2.a.o 1 72.p odd 6 1
3267.2.a.f 1 99.g even 6 1
3267.2.a.f 1 99.h odd 6 1
4563.2.a.e 1 117.n odd 6 1
4563.2.a.e 1 117.t even 6 1
7803.2.a.k 1 153.h even 6 1
7803.2.a.k 1 153.i odd 6 1
9747.2.a.f 1 171.l even 6 1
9747.2.a.f 1 171.o odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{2}^{\mathrm{new}}(81, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - T + 1$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 5T + 25$$
$17$ $$T^{2}$$
$19$ $$(T + 7)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} - 4T + 16$$
$37$ $$(T - 11)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 8T + 64$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} - T + 1$$
$67$ $$T^{2} + 5T + 25$$
$71$ $$T^{2}$$
$73$ $$(T + 7)^{2}$$
$79$ $$T^{2} + 17T + 289$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} - 19T + 361$$