# Properties

 Label 81.2.c.b Level $81$ Weight $2$ Character orbit 81.c Analytic conductor $0.647$ Analytic rank $0$ Dimension $4$ CM no 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + (\beta_1 - 1) q^{4} + (\beta_{3} - \beta_{2}) q^{5} - 2 \beta_1 q^{7} - \beta_{3} q^{8}+O(q^{10})$$ q - b2 * q^2 + (b1 - 1) * q^4 + (b3 - b2) * q^5 - 2*b1 * q^7 - b3 * q^8 $$q - \beta_{2} q^{2} + (\beta_1 - 1) q^{4} + (\beta_{3} - \beta_{2}) q^{5} - 2 \beta_1 q^{7} - \beta_{3} q^{8} - 3 q^{10} + 2 \beta_{2} q^{11} + ( - \beta_1 + 1) q^{13} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{14} + 5 \beta_1 q^{16} + 3 \beta_{3} q^{17} + 2 q^{19} + \beta_{2} q^{20} + ( - 6 \beta_1 + 6) q^{22} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{23} + 2 \beta_1 q^{25} - \beta_{3} q^{26} + 2 q^{28} - \beta_{2} q^{29} + (8 \beta_1 - 8) q^{31} + (3 \beta_{3} - 3 \beta_{2}) q^{32} - 9 \beta_1 q^{34} - 2 \beta_{3} q^{35} - 7 q^{37} - 2 \beta_{2} q^{38} + (3 \beta_1 - 3) q^{40} + (4 \beta_{3} - 4 \beta_{2}) q^{41} - 2 \beta_1 q^{43} - 2 \beta_{3} q^{44} + 6 q^{46} - 4 \beta_{2} q^{47} + ( - 3 \beta_1 + 3) q^{49} + (2 \beta_{3} - 2 \beta_{2}) q^{50} + \beta_1 q^{52} + 6 q^{55} + 2 \beta_{2} q^{56} + (3 \beta_1 - 3) q^{58} + ( - 8 \beta_{3} + 8 \beta_{2}) q^{59} + 7 \beta_1 q^{61} + 8 \beta_{3} q^{62} + q^{64} - \beta_{2} q^{65} + ( - 10 \beta_1 + 10) q^{67} + ( - 3 \beta_{3} + 3 \beta_{2}) q^{68} + 6 \beta_1 q^{70} - 6 \beta_{3} q^{71} - 7 q^{73} + 7 \beta_{2} q^{74} + (2 \beta_1 - 2) q^{76} + (4 \beta_{3} - 4 \beta_{2}) q^{77} - 2 \beta_1 q^{79} + 5 \beta_{3} q^{80} - 12 q^{82} + 8 \beta_{2} q^{83} + ( - 9 \beta_1 + 9) q^{85} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{86} - 6 \beta_1 q^{88} - 3 \beta_{3} q^{89} - 2 q^{91} - 2 \beta_{2} q^{92} + (12 \beta_1 - 12) q^{94} + (2 \beta_{3} - 2 \beta_{2}) q^{95} - 2 \beta_1 q^{97} - 3 \beta_{3} q^{98}+O(q^{100})$$ q - b2 * q^2 + (b1 - 1) * q^4 + (b3 - b2) * q^5 - 2*b1 * q^7 - b3 * q^8 - 3 * q^10 + 2*b2 * q^11 + (-b1 + 1) * q^13 + (-2*b3 + 2*b2) * q^14 + 5*b1 * q^16 + 3*b3 * q^17 + 2 * q^19 + b2 * q^20 + (-6*b1 + 6) * q^22 + (-2*b3 + 2*b2) * q^23 + 2*b1 * q^25 - b3 * q^26 + 2 * q^28 - b2 * q^29 + (8*b1 - 8) * q^31 + (3*b3 - 3*b2) * q^32 - 9*b1 * q^34 - 2*b3 * q^35 - 7 * q^37 - 2*b2 * q^38 + (3*b1 - 3) * q^40 + (4*b3 - 4*b2) * q^41 - 2*b1 * q^43 - 2*b3 * q^44 + 6 * q^46 - 4*b2 * q^47 + (-3*b1 + 3) * q^49 + (2*b3 - 2*b2) * q^50 + b1 * q^52 + 6 * q^55 + 2*b2 * q^56 + (3*b1 - 3) * q^58 + (-8*b3 + 8*b2) * q^59 + 7*b1 * q^61 + 8*b3 * q^62 + q^64 - b2 * q^65 + (-10*b1 + 10) * q^67 + (-3*b3 + 3*b2) * q^68 + 6*b1 * q^70 - 6*b3 * q^71 - 7 * q^73 + 7*b2 * q^74 + (2*b1 - 2) * q^76 + (4*b3 - 4*b2) * q^77 - 2*b1 * q^79 + 5*b3 * q^80 - 12 * q^82 + 8*b2 * q^83 + (-9*b1 + 9) * q^85 + (-2*b3 + 2*b2) * q^86 - 6*b1 * q^88 - 3*b3 * q^89 - 2 * q^91 - 2*b2 * q^92 + (12*b1 - 12) * q^94 + (2*b3 - 2*b2) * q^95 - 2*b1 * q^97 - 3*b3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{4} - 4 q^{7}+O(q^{10})$$ 4 * q - 2 * q^4 - 4 * q^7 $$4 q - 2 q^{4} - 4 q^{7} - 12 q^{10} + 2 q^{13} + 10 q^{16} + 8 q^{19} + 12 q^{22} + 4 q^{25} + 8 q^{28} - 16 q^{31} - 18 q^{34} - 28 q^{37} - 6 q^{40} - 4 q^{43} + 24 q^{46} + 6 q^{49} + 2 q^{52} + 24 q^{55} - 6 q^{58} + 14 q^{61} + 4 q^{64} + 20 q^{67} + 12 q^{70} - 28 q^{73} - 4 q^{76} - 4 q^{79} - 48 q^{82} + 18 q^{85} - 12 q^{88} - 8 q^{91} - 24 q^{94} - 4 q^{97}+O(q^{100})$$ 4 * q - 2 * q^4 - 4 * q^7 - 12 * q^10 + 2 * q^13 + 10 * q^16 + 8 * q^19 + 12 * q^22 + 4 * q^25 + 8 * q^28 - 16 * q^31 - 18 * q^34 - 28 * q^37 - 6 * q^40 - 4 * q^43 + 24 * q^46 + 6 * q^49 + 2 * q^52 + 24 * q^55 - 6 * q^58 + 14 * q^61 + 4 * q^64 + 20 * q^67 + 12 * q^70 - 28 * q^73 - 4 * q^76 - 4 * q^79 - 48 * q^82 + 18 * q^85 - 12 * q^88 - 8 * q^91 - 24 * q^94 - 4 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{12}^{3} + \zeta_{12}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 3$$ (b3 + b2) / 3 $$\zeta_{12}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{12}^{3}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} ) / 3$$ (-b3 + 2*b2) / 3

## Character values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$-1 + \beta_{1}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
28.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
−0.866025 1.50000i 0 −0.500000 + 0.866025i 0.866025 1.50000i 0 −1.00000 1.73205i −1.73205 0 −3.00000
28.2 0.866025 + 1.50000i 0 −0.500000 + 0.866025i −0.866025 + 1.50000i 0 −1.00000 1.73205i 1.73205 0 −3.00000
55.1 −0.866025 + 1.50000i 0 −0.500000 0.866025i 0.866025 + 1.50000i 0 −1.00000 + 1.73205i −1.73205 0 −3.00000
55.2 0.866025 1.50000i 0 −0.500000 0.866025i −0.866025 1.50000i 0 −1.00000 + 1.73205i 1.73205 0 −3.00000
 $$n$$: 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
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.b 4
3.b odd 2 1 inner 81.2.c.b 4
4.b odd 2 1 1296.2.i.s 4
9.c even 3 1 81.2.a.a 2
9.c even 3 1 inner 81.2.c.b 4
9.d odd 6 1 81.2.a.a 2
9.d odd 6 1 inner 81.2.c.b 4
12.b even 2 1 1296.2.i.s 4
27.e even 9 6 729.2.e.o 12
27.f odd 18 6 729.2.e.o 12
36.f odd 6 1 1296.2.a.o 2
36.f odd 6 1 1296.2.i.s 4
36.h even 6 1 1296.2.a.o 2
36.h even 6 1 1296.2.i.s 4
45.h odd 6 1 2025.2.a.j 2
45.j even 6 1 2025.2.a.j 2
45.k odd 12 2 2025.2.b.k 4
45.l even 12 2 2025.2.b.k 4
63.l odd 6 1 3969.2.a.i 2
63.o even 6 1 3969.2.a.i 2
72.j odd 6 1 5184.2.a.br 2
72.l even 6 1 5184.2.a.bq 2
72.n even 6 1 5184.2.a.br 2
72.p odd 6 1 5184.2.a.bq 2
99.g even 6 1 9801.2.a.v 2
99.h odd 6 1 9801.2.a.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
81.2.a.a 2 9.c even 3 1
81.2.a.a 2 9.d odd 6 1
81.2.c.b 4 1.a even 1 1 trivial
81.2.c.b 4 3.b odd 2 1 inner
81.2.c.b 4 9.c even 3 1 inner
81.2.c.b 4 9.d odd 6 1 inner
729.2.e.o 12 27.e even 9 6
729.2.e.o 12 27.f odd 18 6
1296.2.a.o 2 36.f odd 6 1
1296.2.a.o 2 36.h even 6 1
1296.2.i.s 4 4.b odd 2 1
1296.2.i.s 4 12.b even 2 1
1296.2.i.s 4 36.f odd 6 1
1296.2.i.s 4 36.h even 6 1
2025.2.a.j 2 45.h odd 6 1
2025.2.a.j 2 45.j even 6 1
2025.2.b.k 4 45.k odd 12 2
2025.2.b.k 4 45.l even 12 2
3969.2.a.i 2 63.l odd 6 1
3969.2.a.i 2 63.o even 6 1
5184.2.a.bq 2 72.l even 6 1
5184.2.a.bq 2 72.p odd 6 1
5184.2.a.br 2 72.j odd 6 1
5184.2.a.br 2 72.n even 6 1
9801.2.a.v 2 99.g even 6 1
9801.2.a.v 2 99.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 3T_{2}^{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(81, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 3T^{2} + 9$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 3T^{2} + 9$$
$7$ $$(T^{2} + 2 T + 4)^{2}$$
$11$ $$T^{4} + 12T^{2} + 144$$
$13$ $$(T^{2} - T + 1)^{2}$$
$17$ $$(T^{2} - 27)^{2}$$
$19$ $$(T - 2)^{4}$$
$23$ $$T^{4} + 12T^{2} + 144$$
$29$ $$T^{4} + 3T^{2} + 9$$
$31$ $$(T^{2} + 8 T + 64)^{2}$$
$37$ $$(T + 7)^{4}$$
$41$ $$T^{4} + 48T^{2} + 2304$$
$43$ $$(T^{2} + 2 T + 4)^{2}$$
$47$ $$T^{4} + 48T^{2} + 2304$$
$53$ $$T^{4}$$
$59$ $$T^{4} + 192 T^{2} + 36864$$
$61$ $$(T^{2} - 7 T + 49)^{2}$$
$67$ $$(T^{2} - 10 T + 100)^{2}$$
$71$ $$(T^{2} - 108)^{2}$$
$73$ $$(T + 7)^{4}$$
$79$ $$(T^{2} + 2 T + 4)^{2}$$
$83$ $$T^{4} + 192 T^{2} + 36864$$
$89$ $$(T^{2} - 27)^{2}$$
$97$ $$(T^{2} + 2 T + 4)^{2}$$