# Properties

 Label 81.4.a.c Level $81$ Weight $4$ Character orbit 81.a Self dual yes Analytic conductor $4.779$ Analytic rank $1$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,4,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, 4, names="a")

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

chi := DirichletCharacter("81.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$81 = 3^{4}$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$4.77915471046$$ Analytic rank: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - 5 q^{4} - 7 \beta q^{5} - 22 q^{7} - 13 \beta q^{8} +O(q^{10})$$ q + b * q^2 - 5 * q^4 - 7*b * q^5 - 22 * q^7 - 13*b * q^8 $$q + \beta q^{2} - 5 q^{4} - 7 \beta q^{5} - 22 q^{7} - 13 \beta q^{8} - 21 q^{10} + 34 \beta q^{11} - 49 q^{13} - 22 \beta q^{14} + q^{16} + 21 \beta q^{17} - 70 q^{19} + 35 \beta q^{20} + 102 q^{22} + 14 \beta q^{23} + 22 q^{25} - 49 \beta q^{26} + 110 q^{28} - 161 \beta q^{29} - 112 q^{31} + 105 \beta q^{32} + 63 q^{34} + 154 \beta q^{35} + 281 q^{37} - 70 \beta q^{38} + 273 q^{40} - 28 \beta q^{41} + 50 q^{43} - 170 \beta q^{44} + 42 q^{46} - 140 \beta q^{47} + 141 q^{49} + 22 \beta q^{50} + 245 q^{52} + 216 \beta q^{53} - 714 q^{55} + 286 \beta q^{56} - 483 q^{58} + 56 \beta q^{59} - 679 q^{61} - 112 \beta q^{62} + 307 q^{64} + 343 \beta q^{65} - 274 q^{67} - 105 \beta q^{68} + 462 q^{70} - 258 \beta q^{71} - 511 q^{73} + 281 \beta q^{74} + 350 q^{76} - 748 \beta q^{77} - 526 q^{79} - 7 \beta q^{80} - 84 q^{82} - 224 \beta q^{83} - 441 q^{85} + 50 \beta q^{86} - 1326 q^{88} - 21 \beta q^{89} + 1078 q^{91} - 70 \beta q^{92} - 420 q^{94} + 490 \beta q^{95} + 1778 q^{97} + 141 \beta q^{98} +O(q^{100})$$ q + b * q^2 - 5 * q^4 - 7*b * q^5 - 22 * q^7 - 13*b * q^8 - 21 * q^10 + 34*b * q^11 - 49 * q^13 - 22*b * q^14 + q^16 + 21*b * q^17 - 70 * q^19 + 35*b * q^20 + 102 * q^22 + 14*b * q^23 + 22 * q^25 - 49*b * q^26 + 110 * q^28 - 161*b * q^29 - 112 * q^31 + 105*b * q^32 + 63 * q^34 + 154*b * q^35 + 281 * q^37 - 70*b * q^38 + 273 * q^40 - 28*b * q^41 + 50 * q^43 - 170*b * q^44 + 42 * q^46 - 140*b * q^47 + 141 * q^49 + 22*b * q^50 + 245 * q^52 + 216*b * q^53 - 714 * q^55 + 286*b * q^56 - 483 * q^58 + 56*b * q^59 - 679 * q^61 - 112*b * q^62 + 307 * q^64 + 343*b * q^65 - 274 * q^67 - 105*b * q^68 + 462 * q^70 - 258*b * q^71 - 511 * q^73 + 281*b * q^74 + 350 * q^76 - 748*b * q^77 - 526 * q^79 - 7*b * q^80 - 84 * q^82 - 224*b * q^83 - 441 * q^85 + 50*b * q^86 - 1326 * q^88 - 21*b * q^89 + 1078 * q^91 - 70*b * q^92 - 420 * q^94 + 490*b * q^95 + 1778 * q^97 + 141*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 10 q^{4} - 44 q^{7}+O(q^{10})$$ 2 * q - 10 * q^4 - 44 * q^7 $$2 q - 10 q^{4} - 44 q^{7} - 42 q^{10} - 98 q^{13} + 2 q^{16} - 140 q^{19} + 204 q^{22} + 44 q^{25} + 220 q^{28} - 224 q^{31} + 126 q^{34} + 562 q^{37} + 546 q^{40} + 100 q^{43} + 84 q^{46} + 282 q^{49} + 490 q^{52} - 1428 q^{55} - 966 q^{58} - 1358 q^{61} + 614 q^{64} - 548 q^{67} + 924 q^{70} - 1022 q^{73} + 700 q^{76} - 1052 q^{79} - 168 q^{82} - 882 q^{85} - 2652 q^{88} + 2156 q^{91} - 840 q^{94} + 3556 q^{97}+O(q^{100})$$ 2 * q - 10 * q^4 - 44 * q^7 - 42 * q^10 - 98 * q^13 + 2 * q^16 - 140 * q^19 + 204 * q^22 + 44 * q^25 + 220 * q^28 - 224 * q^31 + 126 * q^34 + 562 * q^37 + 546 * q^40 + 100 * q^43 + 84 * q^46 + 282 * q^49 + 490 * q^52 - 1428 * q^55 - 966 * q^58 - 1358 * q^61 + 614 * q^64 - 548 * q^67 + 924 * q^70 - 1022 * q^73 + 700 * q^76 - 1052 * q^79 - 168 * q^82 - 882 * q^85 - 2652 * q^88 + 2156 * q^91 - 840 * q^94 + 3556 * 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 −5.00000 12.1244 0 −22.0000 22.5167 0 −21.0000
1.2 1.73205 0 −5.00000 −12.1244 0 −22.0000 −22.5167 0 −21.0000
 $$n$$: 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 81.4.a.c 2
3.b odd 2 1 inner 81.4.a.c 2
4.b odd 2 1 1296.4.a.o 2
5.b even 2 1 2025.4.a.k 2
9.c even 3 2 81.4.c.f 4
9.d odd 6 2 81.4.c.f 4
12.b even 2 1 1296.4.a.o 2
15.d odd 2 1 2025.4.a.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
81.4.a.c 2 1.a even 1 1 trivial
81.4.a.c 2 3.b odd 2 1 inner
81.4.c.f 4 9.c even 3 2
81.4.c.f 4 9.d odd 6 2
1296.4.a.o 2 4.b odd 2 1
1296.4.a.o 2 12.b even 2 1
2025.4.a.k 2 5.b even 2 1
2025.4.a.k 2 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 3$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(81))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 147$$
$7$ $$(T + 22)^{2}$$
$11$ $$T^{2} - 3468$$
$13$ $$(T + 49)^{2}$$
$17$ $$T^{2} - 1323$$
$19$ $$(T + 70)^{2}$$
$23$ $$T^{2} - 588$$
$29$ $$T^{2} - 77763$$
$31$ $$(T + 112)^{2}$$
$37$ $$(T - 281)^{2}$$
$41$ $$T^{2} - 2352$$
$43$ $$(T - 50)^{2}$$
$47$ $$T^{2} - 58800$$
$53$ $$T^{2} - 139968$$
$59$ $$T^{2} - 9408$$
$61$ $$(T + 679)^{2}$$
$67$ $$(T + 274)^{2}$$
$71$ $$T^{2} - 199692$$
$73$ $$(T + 511)^{2}$$
$79$ $$(T + 526)^{2}$$
$83$ $$T^{2} - 150528$$
$89$ $$T^{2} - 1323$$
$97$ $$(T - 1778)^{2}$$