# Properties

 Label 243.2.a.c Level $243$ Weight $2$ Character orbit 243.a Self dual yes Analytic conductor $1.940$ 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] = [243,2,Mod(1,243)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(243, 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("243.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$243 = 3^{5}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 243.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: $$1.94036476912$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + q^{4} + 2 \beta q^{5} - q^{7} - \beta q^{8} +O(q^{10})$$ q + b * q^2 + q^4 + 2*b * q^5 - q^7 - b * q^8 $$q + \beta q^{2} + q^{4} + 2 \beta q^{5} - q^{7} - \beta q^{8} + 6 q^{10} - 2 \beta q^{11} + 5 q^{13} - \beta q^{14} - 5 q^{16} - q^{19} + 2 \beta q^{20} - 6 q^{22} - 4 \beta q^{23} + 7 q^{25} + 5 \beta q^{26} - q^{28} - 2 \beta q^{29} + 5 q^{31} - 3 \beta q^{32} - 2 \beta q^{35} - q^{37} - \beta q^{38} - 6 q^{40} + 2 \beta q^{41} - q^{43} - 2 \beta q^{44} - 12 q^{46} - 2 \beta q^{47} - 6 q^{49} + 7 \beta q^{50} + 5 q^{52} - 6 \beta q^{53} - 12 q^{55} + \beta q^{56} - 6 q^{58} + 2 \beta q^{59} + 2 q^{61} + 5 \beta q^{62} + q^{64} + 10 \beta q^{65} + 8 q^{67} - 6 q^{70} + 6 \beta q^{71} + 2 q^{73} - \beta q^{74} - q^{76} + 2 \beta q^{77} - q^{79} - 10 \beta q^{80} + 6 q^{82} + 4 \beta q^{83} - \beta q^{86} + 6 q^{88} + 6 \beta q^{89} - 5 q^{91} - 4 \beta q^{92} - 6 q^{94} - 2 \beta q^{95} + 17 q^{97} - 6 \beta q^{98} +O(q^{100})$$ q + b * q^2 + q^4 + 2*b * q^5 - q^7 - b * q^8 + 6 * q^10 - 2*b * q^11 + 5 * q^13 - b * q^14 - 5 * q^16 - q^19 + 2*b * q^20 - 6 * q^22 - 4*b * q^23 + 7 * q^25 + 5*b * q^26 - q^28 - 2*b * q^29 + 5 * q^31 - 3*b * q^32 - 2*b * q^35 - q^37 - b * q^38 - 6 * q^40 + 2*b * q^41 - q^43 - 2*b * q^44 - 12 * q^46 - 2*b * q^47 - 6 * q^49 + 7*b * q^50 + 5 * q^52 - 6*b * q^53 - 12 * q^55 + b * q^56 - 6 * q^58 + 2*b * q^59 + 2 * q^61 + 5*b * q^62 + q^64 + 10*b * q^65 + 8 * q^67 - 6 * q^70 + 6*b * q^71 + 2 * q^73 - b * q^74 - q^76 + 2*b * q^77 - q^79 - 10*b * q^80 + 6 * q^82 + 4*b * q^83 - b * q^86 + 6 * q^88 + 6*b * q^89 - 5 * q^91 - 4*b * q^92 - 6 * q^94 - 2*b * q^95 + 17 * q^97 - 6*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} - 2 q^{7}+O(q^{10})$$ 2 * q + 2 * q^4 - 2 * q^7 $$2 q + 2 q^{4} - 2 q^{7} + 12 q^{10} + 10 q^{13} - 10 q^{16} - 2 q^{19} - 12 q^{22} + 14 q^{25} - 2 q^{28} + 10 q^{31} - 2 q^{37} - 12 q^{40} - 2 q^{43} - 24 q^{46} - 12 q^{49} + 10 q^{52} - 24 q^{55} - 12 q^{58} + 4 q^{61} + 2 q^{64} + 16 q^{67} - 12 q^{70} + 4 q^{73} - 2 q^{76} - 2 q^{79} + 12 q^{82} + 12 q^{88} - 10 q^{91} - 12 q^{94} + 34 q^{97}+O(q^{100})$$ 2 * q + 2 * q^4 - 2 * q^7 + 12 * q^10 + 10 * q^13 - 10 * q^16 - 2 * q^19 - 12 * q^22 + 14 * q^25 - 2 * q^28 + 10 * q^31 - 2 * q^37 - 12 * q^40 - 2 * q^43 - 24 * q^46 - 12 * q^49 + 10 * q^52 - 24 * q^55 - 12 * q^58 + 4 * q^61 + 2 * q^64 + 16 * q^67 - 12 * q^70 + 4 * q^73 - 2 * q^76 - 2 * q^79 + 12 * q^82 + 12 * q^88 - 10 * q^91 - 12 * q^94 + 34 * 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 −3.46410 0 −1.00000 1.73205 0 6.00000
1.2 1.73205 0 1.00000 3.46410 0 −1.00000 −1.73205 0 6.00000
 $$n$$: 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 243.2.a.c 2
3.b odd 2 1 inner 243.2.a.c 2
4.b odd 2 1 3888.2.a.ba 2
5.b even 2 1 6075.2.a.bm 2
9.c even 3 2 243.2.c.d 4
9.d odd 6 2 243.2.c.d 4
12.b even 2 1 3888.2.a.ba 2
15.d odd 2 1 6075.2.a.bm 2
27.e even 9 6 729.2.e.n 12
27.f odd 18 6 729.2.e.n 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
243.2.a.c 2 1.a even 1 1 trivial
243.2.a.c 2 3.b odd 2 1 inner
243.2.c.d 4 9.c even 3 2
243.2.c.d 4 9.d odd 6 2
729.2.e.n 12 27.e even 9 6
729.2.e.n 12 27.f odd 18 6
3888.2.a.ba 2 4.b odd 2 1
3888.2.a.ba 2 12.b even 2 1
6075.2.a.bm 2 5.b even 2 1
6075.2.a.bm 2 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(243))$$:

 $$T_{2}^{2} - 3$$ T2^2 - 3 $$T_{7} + 1$$ T7 + 1

## Hecke characteristic polynomials

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