# Properties

 Label 81.4.c.b Level $81$ Weight $4$ Character orbit 81.c Analytic conductor $4.779$ 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] = [81,4,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, 4, names="a")

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

chi := DirichletCharacter("81.28");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$81 = 3^{4}$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$4.77915471046$$ 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: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 9) 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.

 $$f(q)$$ $$=$$ $$q + 8 \zeta_{6} q^{4} + (20 \zeta_{6} - 20) q^{7}+O(q^{10})$$ q + 8*z * q^4 + (20*z - 20) * q^7 $$q + 8 \zeta_{6} q^{4} + (20 \zeta_{6} - 20) q^{7} + 70 \zeta_{6} q^{13} + (64 \zeta_{6} - 64) q^{16} + 56 q^{19} + ( - 125 \zeta_{6} + 125) q^{25} - 160 q^{28} - 308 \zeta_{6} q^{31} + 110 q^{37} + ( - 520 \zeta_{6} + 520) q^{43} - 57 \zeta_{6} q^{49} + (560 \zeta_{6} - 560) q^{52} + (182 \zeta_{6} - 182) q^{61} - 512 q^{64} + 880 \zeta_{6} q^{67} + 1190 q^{73} + 448 \zeta_{6} q^{76} + (884 \zeta_{6} - 884) q^{79} - 1400 q^{91} + ( - 1330 \zeta_{6} + 1330) q^{97} +O(q^{100})$$ q + 8*z * q^4 + (20*z - 20) * q^7 + 70*z * q^13 + (64*z - 64) * q^16 + 56 * q^19 + (-125*z + 125) * q^25 - 160 * q^28 - 308*z * q^31 + 110 * q^37 + (-520*z + 520) * q^43 - 57*z * q^49 + (560*z - 560) * q^52 + (182*z - 182) * q^61 - 512 * q^64 + 880*z * q^67 + 1190 * q^73 + 448*z * q^76 + (884*z - 884) * q^79 - 1400 * q^91 + (-1330*z + 1330) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 8 q^{4} - 20 q^{7}+O(q^{10})$$ 2 * q + 8 * q^4 - 20 * q^7 $$2 q + 8 q^{4} - 20 q^{7} + 70 q^{13} - 64 q^{16} + 112 q^{19} + 125 q^{25} - 320 q^{28} - 308 q^{31} + 220 q^{37} + 520 q^{43} - 57 q^{49} - 560 q^{52} - 182 q^{61} - 1024 q^{64} + 880 q^{67} + 2380 q^{73} + 448 q^{76} - 884 q^{79} - 2800 q^{91} + 1330 q^{97}+O(q^{100})$$ 2 * q + 8 * q^4 - 20 * q^7 + 70 * q^13 - 64 * q^16 + 112 * q^19 + 125 * q^25 - 320 * q^28 - 308 * q^31 + 220 * q^37 + 520 * q^43 - 57 * q^49 - 560 * q^52 - 182 * q^61 - 1024 * q^64 + 880 * q^67 + 2380 * q^73 + 448 * q^76 - 884 * q^79 - 2800 * q^91 + 1330 * 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 4.00000 6.92820i 0 0 −10.0000 17.3205i 0 0 0
55.1 0 0 4.00000 + 6.92820i 0 0 −10.0000 + 17.3205i 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 81.4.c.b 2
3.b odd 2 1 CM 81.4.c.b 2
9.c even 3 1 9.4.a.a 1
9.c even 3 1 inner 81.4.c.b 2
9.d odd 6 1 9.4.a.a 1
9.d odd 6 1 inner 81.4.c.b 2
36.f odd 6 1 144.4.a.d 1
36.h even 6 1 144.4.a.d 1
45.h odd 6 1 225.4.a.d 1
45.j even 6 1 225.4.a.d 1
45.k odd 12 2 225.4.b.g 2
45.l even 12 2 225.4.b.g 2
63.g even 3 1 441.4.e.i 2
63.h even 3 1 441.4.e.i 2
63.i even 6 1 441.4.e.j 2
63.j odd 6 1 441.4.e.i 2
63.k odd 6 1 441.4.e.j 2
63.l odd 6 1 441.4.a.f 1
63.n odd 6 1 441.4.e.i 2
63.o even 6 1 441.4.a.f 1
63.s even 6 1 441.4.e.j 2
63.t odd 6 1 441.4.e.j 2
72.j odd 6 1 576.4.a.m 1
72.l even 6 1 576.4.a.l 1
72.n even 6 1 576.4.a.m 1
72.p odd 6 1 576.4.a.l 1
99.g even 6 1 1089.4.a.g 1
99.h odd 6 1 1089.4.a.g 1
117.n odd 6 1 1521.4.a.g 1
117.t even 6 1 1521.4.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.a.a 1 9.c even 3 1
9.4.a.a 1 9.d odd 6 1
81.4.c.b 2 1.a even 1 1 trivial
81.4.c.b 2 3.b odd 2 1 CM
81.4.c.b 2 9.c even 3 1 inner
81.4.c.b 2 9.d odd 6 1 inner
144.4.a.d 1 36.f odd 6 1
144.4.a.d 1 36.h even 6 1
225.4.a.d 1 45.h odd 6 1
225.4.a.d 1 45.j even 6 1
225.4.b.g 2 45.k odd 12 2
225.4.b.g 2 45.l even 12 2
441.4.a.f 1 63.l odd 6 1
441.4.a.f 1 63.o even 6 1
441.4.e.i 2 63.g even 3 1
441.4.e.i 2 63.h even 3 1
441.4.e.i 2 63.j odd 6 1
441.4.e.i 2 63.n odd 6 1
441.4.e.j 2 63.i even 6 1
441.4.e.j 2 63.k odd 6 1
441.4.e.j 2 63.s even 6 1
441.4.e.j 2 63.t odd 6 1
576.4.a.l 1 72.l even 6 1
576.4.a.l 1 72.p odd 6 1
576.4.a.m 1 72.j odd 6 1
576.4.a.m 1 72.n even 6 1
1089.4.a.g 1 99.g even 6 1
1089.4.a.g 1 99.h odd 6 1
1521.4.a.g 1 117.n odd 6 1
1521.4.a.g 1 117.t even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{4}^{\mathrm{new}}(81, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 20T + 400$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 70T + 4900$$
$17$ $$T^{2}$$
$19$ $$(T - 56)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 308T + 94864$$
$37$ $$(T - 110)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} - 520T + 270400$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 182T + 33124$$
$67$ $$T^{2} - 880T + 774400$$
$71$ $$T^{2}$$
$73$ $$(T - 1190)^{2}$$
$79$ $$T^{2} + 884T + 781456$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} - 1330 T + 1768900$$