# Properties

 Label 81.2.a.a Level $81$ Weight $2$ Character orbit 81.a Self dual yes Analytic conductor $0.647$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,2,Mod(1,81)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(81, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("81.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$81 = 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 81.a (trivial)

## 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: yes 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: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.73205 1.73205
−1.73205 0 1.00000 1.73205 0 2.00000 1.73205 0 −3.00000
1.2 1.73205 0 1.00000 −1.73205 0 2.00000 −1.73205 0 −3.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 81.2.a.a 2
3.b odd 2 1 inner 81.2.a.a 2
4.b odd 2 1 1296.2.a.o 2
5.b even 2 1 2025.2.a.j 2
5.c odd 4 2 2025.2.b.k 4
7.b odd 2 1 3969.2.a.i 2
8.b even 2 1 5184.2.a.br 2
8.d odd 2 1 5184.2.a.bq 2
9.c even 3 2 81.2.c.b 4
9.d odd 6 2 81.2.c.b 4
11.b odd 2 1 9801.2.a.v 2
12.b even 2 1 1296.2.a.o 2
15.d odd 2 1 2025.2.a.j 2
15.e even 4 2 2025.2.b.k 4
21.c even 2 1 3969.2.a.i 2
24.f even 2 1 5184.2.a.bq 2
24.h odd 2 1 5184.2.a.br 2
27.e even 9 6 729.2.e.o 12
27.f odd 18 6 729.2.e.o 12
33.d even 2 1 9801.2.a.v 2
36.f odd 6 2 1296.2.i.s 4
36.h even 6 2 1296.2.i.s 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
81.2.a.a 2 1.a even 1 1 trivial
81.2.a.a 2 3.b odd 2 1 inner
81.2.c.b 4 9.c even 3 2
81.2.c.b 4 9.d odd 6 2
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 4.b odd 2 1
1296.2.a.o 2 12.b even 2 1
1296.2.i.s 4 36.f odd 6 2
1296.2.i.s 4 36.h even 6 2
2025.2.a.j 2 5.b even 2 1
2025.2.a.j 2 15.d odd 2 1
2025.2.b.k 4 5.c odd 4 2
2025.2.b.k 4 15.e even 4 2
3969.2.a.i 2 7.b odd 2 1
3969.2.a.i 2 21.c even 2 1
5184.2.a.bq 2 8.d odd 2 1
5184.2.a.bq 2 24.f even 2 1
5184.2.a.br 2 8.b even 2 1
5184.2.a.br 2 24.h odd 2 1
9801.2.a.v 2 11.b odd 2 1
9801.2.a.v 2 33.d even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(\Gamma_0(81))$$.

## Hecke characteristic polynomials

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