# Properties

 Label 243.2.a.f Level $243$ Weight $2$ Character orbit 243.a Self dual yes Analytic conductor $1.940$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# 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: $$3$$ Coefficient field: $$\Q(\zeta_{18})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 3x - 1$$ x^3 - 3*x - 1 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{2} + (\beta_{2} - 2 \beta_1 + 1) q^{4} + (\beta_1 + 2) q^{5} + ( - 2 \beta_{2} + \beta_1 - 1) q^{7} + (3 \beta_{2} - 2 \beta_1 + 2) q^{8}+O(q^{10})$$ q + (-b1 + 1) * q^2 + (b2 - 2*b1 + 1) * q^4 + (b1 + 2) * q^5 + (-2*b2 + b1 - 1) * q^7 + (3*b2 - 2*b1 + 2) * q^8 $$q + ( - \beta_1 + 1) q^{2} + (\beta_{2} - 2 \beta_1 + 1) q^{4} + (\beta_1 + 2) q^{5} + ( - 2 \beta_{2} + \beta_1 - 1) q^{7} + (3 \beta_{2} - 2 \beta_1 + 2) q^{8} + ( - \beta_{2} - \beta_1) q^{10} + ( - 3 \beta_{2} + 2 \beta_1 + 1) q^{11} + (\beta_{2} + \beta_1 - 1) q^{13} + ( - 3 \beta_{2} + 4 \beta_1 - 1) q^{14} + (3 \beta_{2} - 3 \beta_1 + 1) q^{16} + 3 q^{17} + (3 \beta_{2} - 1) q^{19} + ( - 2 \beta_1 - 1) q^{20} + ( - 5 \beta_{2} + 4 \beta_1) q^{22} + (3 \beta_{2} - 2 \beta_1 + 2) q^{23} + (\beta_{2} + 4 \beta_1 + 1) q^{25} + (\beta_1 - 4) q^{26} + ( - 3 \beta_{2} + 6 \beta_1 - 4) q^{28} + (3 \beta_{2} - \beta_1 + 4) q^{29} + ( - 2 \beta_{2} + \beta_1 - 4) q^{31} - 3 \beta_1 q^{32} + ( - 3 \beta_1 + 3) q^{34} + ( - 3 \beta_{2} - \beta_1 - 2) q^{35} + ( - 3 \beta_1 - 1) q^{37} + (3 \beta_{2} - 2 \beta_1 - 4) q^{38} + (4 \beta_{2} + \beta_1 + 3) q^{40} + (3 \beta_{2} - 5 \beta_1 - 1) q^{41} + (\beta_{2} - 2 \beta_1 - 4) q^{43} + ( - 3 \beta_{2} + 5 \beta_1 - 5) q^{44} + (5 \beta_{2} - 7 \beta_1 + 3) q^{46} + (5 \beta_1 - 2) q^{47} + (\beta_{2} - 2 \beta_1) q^{49} + ( - 3 \beta_{2} + 2 \beta_1 - 8) q^{50} + ( - 3 \beta_{2} + 3 \beta_1 - 4) q^{52} + ( - 3 \beta_{2} + 6) q^{53} + ( - 4 \beta_{2} + 2 \beta_1 + 3) q^{55} + ( - 3 \beta_{2} + 5 \beta_1 - 11) q^{56} + (4 \beta_{2} - 8 \beta_1 + 3) q^{58} + (\beta_1 - 7) q^{59} + ( - 5 \beta_{2} + \beta_1 + 2) q^{61} + ( - 3 \beta_{2} + 7 \beta_1 - 4) q^{62} + ( - 3 \beta_{2} + 3 \beta_1 + 4) q^{64} + (3 \beta_{2} + 2 \beta_1 + 1) q^{65} + (\beta_{2} - 5 \beta_1 + 2) q^{67} + (3 \beta_{2} - 6 \beta_1 + 3) q^{68} + ( - 2 \beta_{2} + 4 \beta_1 + 3) q^{70} + (3 \beta_{2} - 9 \beta_1 - 3) q^{71} + ( - 3 \beta_{2} + 6 \beta_1 + 2) q^{73} + (3 \beta_{2} - 2 \beta_1 + 5) q^{74} + ( - \beta_{2} - \beta_1 - 1) q^{76} + ( - 3 \beta_{2} - 2 \beta_1 + 8) q^{77} + (4 \beta_{2} - 5 \beta_1 + 2) q^{79} + (3 \beta_{2} - 2 \beta_1 - 1) q^{80} + (8 \beta_{2} - 7 \beta_1 + 6) q^{82} + ( - 3 \beta_{2} - \beta_1 - 2) q^{83} + (3 \beta_1 + 6) q^{85} + (3 \beta_{2} + \beta_1 - 1) q^{86} + (2 \beta_{2} + 5 \beta_1 - 12) q^{88} + ( - 6 \beta_{2} + 9 \beta_1) q^{89} + (4 \beta_{2} - 5 \beta_1 - 2) q^{91} + (6 \beta_{2} - 11 \beta_1 + 8) q^{92} + ( - 5 \beta_{2} + 7 \beta_1 - 12) q^{94} + (6 \beta_{2} + 2 \beta_1 + 1) q^{95} + (4 \beta_{2} - 8 \beta_1 + 5) q^{97} + (3 \beta_{2} - 3 \beta_1 + 3) q^{98}+O(q^{100})$$ q + (-b1 + 1) * q^2 + (b2 - 2*b1 + 1) * q^4 + (b1 + 2) * q^5 + (-2*b2 + b1 - 1) * q^7 + (3*b2 - 2*b1 + 2) * q^8 + (-b2 - b1) * q^10 + (-3*b2 + 2*b1 + 1) * q^11 + (b2 + b1 - 1) * q^13 + (-3*b2 + 4*b1 - 1) * q^14 + (3*b2 - 3*b1 + 1) * q^16 + 3 * q^17 + (3*b2 - 1) * q^19 + (-2*b1 - 1) * q^20 + (-5*b2 + 4*b1) * q^22 + (3*b2 - 2*b1 + 2) * q^23 + (b2 + 4*b1 + 1) * q^25 + (b1 - 4) * q^26 + (-3*b2 + 6*b1 - 4) * q^28 + (3*b2 - b1 + 4) * q^29 + (-2*b2 + b1 - 4) * q^31 - 3*b1 * q^32 + (-3*b1 + 3) * q^34 + (-3*b2 - b1 - 2) * q^35 + (-3*b1 - 1) * q^37 + (3*b2 - 2*b1 - 4) * q^38 + (4*b2 + b1 + 3) * q^40 + (3*b2 - 5*b1 - 1) * q^41 + (b2 - 2*b1 - 4) * q^43 + (-3*b2 + 5*b1 - 5) * q^44 + (5*b2 - 7*b1 + 3) * q^46 + (5*b1 - 2) * q^47 + (b2 - 2*b1) * q^49 + (-3*b2 + 2*b1 - 8) * q^50 + (-3*b2 + 3*b1 - 4) * q^52 + (-3*b2 + 6) * q^53 + (-4*b2 + 2*b1 + 3) * q^55 + (-3*b2 + 5*b1 - 11) * q^56 + (4*b2 - 8*b1 + 3) * q^58 + (b1 - 7) * q^59 + (-5*b2 + b1 + 2) * q^61 + (-3*b2 + 7*b1 - 4) * q^62 + (-3*b2 + 3*b1 + 4) * q^64 + (3*b2 + 2*b1 + 1) * q^65 + (b2 - 5*b1 + 2) * q^67 + (3*b2 - 6*b1 + 3) * q^68 + (-2*b2 + 4*b1 + 3) * q^70 + (3*b2 - 9*b1 - 3) * q^71 + (-3*b2 + 6*b1 + 2) * q^73 + (3*b2 - 2*b1 + 5) * q^74 + (-b2 - b1 - 1) * q^76 + (-3*b2 - 2*b1 + 8) * q^77 + (4*b2 - 5*b1 + 2) * q^79 + (3*b2 - 2*b1 - 1) * q^80 + (8*b2 - 7*b1 + 6) * q^82 + (-3*b2 - b1 - 2) * q^83 + (3*b1 + 6) * q^85 + (3*b2 + b1 - 1) * q^86 + (2*b2 + 5*b1 - 12) * q^88 + (-6*b2 + 9*b1) * q^89 + (4*b2 - 5*b1 - 2) * q^91 + (6*b2 - 11*b1 + 8) * q^92 + (-5*b2 + 7*b1 - 12) * q^94 + (6*b2 + 2*b1 + 1) * q^95 + (4*b2 - 8*b1 + 5) * q^97 + (3*b2 - 3*b1 + 3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} - 3 q^{7} + 6 q^{8}+O(q^{10})$$ 3 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 - 3 * q^7 + 6 * q^8 $$3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} - 3 q^{7} + 6 q^{8} + 3 q^{11} - 3 q^{13} - 3 q^{14} + 3 q^{16} + 9 q^{17} - 3 q^{19} - 3 q^{20} + 6 q^{23} + 3 q^{25} - 12 q^{26} - 12 q^{28} + 12 q^{29} - 12 q^{31} + 9 q^{34} - 6 q^{35} - 3 q^{37} - 12 q^{38} + 9 q^{40} - 3 q^{41} - 12 q^{43} - 15 q^{44} + 9 q^{46} - 6 q^{47} - 24 q^{50} - 12 q^{52} + 18 q^{53} + 9 q^{55} - 33 q^{56} + 9 q^{58} - 21 q^{59} + 6 q^{61} - 12 q^{62} + 12 q^{64} + 3 q^{65} + 6 q^{67} + 9 q^{68} + 9 q^{70} - 9 q^{71} + 6 q^{73} + 15 q^{74} - 3 q^{76} + 24 q^{77} + 6 q^{79} - 3 q^{80} + 18 q^{82} - 6 q^{83} + 18 q^{85} - 3 q^{86} - 36 q^{88} - 6 q^{91} + 24 q^{92} - 36 q^{94} + 3 q^{95} + 15 q^{97} + 9 q^{98}+O(q^{100})$$ 3 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 - 3 * q^7 + 6 * q^8 + 3 * q^11 - 3 * q^13 - 3 * q^14 + 3 * q^16 + 9 * q^17 - 3 * q^19 - 3 * q^20 + 6 * q^23 + 3 * q^25 - 12 * q^26 - 12 * q^28 + 12 * q^29 - 12 * q^31 + 9 * q^34 - 6 * q^35 - 3 * q^37 - 12 * q^38 + 9 * q^40 - 3 * q^41 - 12 * q^43 - 15 * q^44 + 9 * q^46 - 6 * q^47 - 24 * q^50 - 12 * q^52 + 18 * q^53 + 9 * q^55 - 33 * q^56 + 9 * q^58 - 21 * q^59 + 6 * q^61 - 12 * q^62 + 12 * q^64 + 3 * q^65 + 6 * q^67 + 9 * q^68 + 9 * q^70 - 9 * q^71 + 6 * q^73 + 15 * q^74 - 3 * q^76 + 24 * q^77 + 6 * q^79 - 3 * q^80 + 18 * q^82 - 6 * q^83 + 18 * q^85 - 3 * q^86 - 36 * q^88 - 6 * q^91 + 24 * q^92 - 36 * q^94 + 3 * q^95 + 15 * q^97 + 9 * q^98

Basis of coefficient ring in terms of $$\nu = \zeta_{18} + \zeta_{18}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2

## 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
 1.87939 −0.347296 −1.53209
−0.879385 0 −1.22668 3.87939 0 −2.18479 2.83750 0 −3.41147
1.2 1.34730 0 −0.184793 1.65270 0 2.41147 −2.94356 0 2.22668
1.3 2.53209 0 4.41147 0.467911 0 −3.22668 6.10607 0 1.18479
 $$n$$: 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 243.2.a.f yes 3
3.b odd 2 1 243.2.a.e 3
4.b odd 2 1 3888.2.a.bk 3
5.b even 2 1 6075.2.a.bq 3
9.c even 3 2 243.2.c.e 6
9.d odd 6 2 243.2.c.f 6
12.b even 2 1 3888.2.a.bd 3
15.d odd 2 1 6075.2.a.bv 3
27.e even 9 2 729.2.e.b 6
27.e even 9 2 729.2.e.c 6
27.e even 9 2 729.2.e.i 6
27.f odd 18 2 729.2.e.a 6
27.f odd 18 2 729.2.e.g 6
27.f odd 18 2 729.2.e.h 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
243.2.a.e 3 3.b odd 2 1
243.2.a.f yes 3 1.a even 1 1 trivial
243.2.c.e 6 9.c even 3 2
243.2.c.f 6 9.d odd 6 2
729.2.e.a 6 27.f odd 18 2
729.2.e.b 6 27.e even 9 2
729.2.e.c 6 27.e even 9 2
729.2.e.g 6 27.f odd 18 2
729.2.e.h 6 27.f odd 18 2
729.2.e.i 6 27.e even 9 2
3888.2.a.bd 3 12.b even 2 1
3888.2.a.bk 3 4.b odd 2 1
6075.2.a.bq 3 5.b even 2 1
6075.2.a.bv 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(243))$$:

 $$T_{2}^{3} - 3T_{2}^{2} + 3$$ T2^3 - 3*T2^2 + 3 $$T_{7}^{3} + 3T_{7}^{2} - 6T_{7} - 17$$ T7^3 + 3*T7^2 - 6*T7 - 17

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 3T^{2} + 3$$
$3$ $$T^{3}$$
$5$ $$T^{3} - 6 T^{2} + 9 T - 3$$
$7$ $$T^{3} + 3 T^{2} - 6 T - 17$$
$11$ $$T^{3} - 3 T^{2} - 18 T + 3$$
$13$ $$T^{3} + 3 T^{2} - 6 T - 17$$
$17$ $$(T - 3)^{3}$$
$19$ $$T^{3} + 3 T^{2} - 24 T + 1$$
$23$ $$T^{3} - 6 T^{2} - 9 T + 51$$
$29$ $$T^{3} - 12 T^{2} + 27 T + 57$$
$31$ $$T^{3} + 12 T^{2} + 39 T + 19$$
$37$ $$T^{3} + 3 T^{2} - 24 T + 1$$
$41$ $$T^{3} + 3 T^{2} - 54 T - 219$$
$43$ $$T^{3} + 12 T^{2} + 39 T + 19$$
$47$ $$T^{3} + 6 T^{2} - 63 T - 267$$
$53$ $$T^{3} - 18 T^{2} + 81 T - 81$$
$59$ $$T^{3} + 21 T^{2} + 144 T + 321$$
$61$ $$T^{3} - 6 T^{2} - 51 T - 53$$
$67$ $$T^{3} - 6 T^{2} - 51 T + 109$$
$71$ $$T^{3} + 9 T^{2} - 162 T - 999$$
$73$ $$T^{3} - 6 T^{2} - 69 T + 397$$
$79$ $$T^{3} - 6 T^{2} - 51 T - 53$$
$83$ $$T^{3} + 6 T^{2} - 27 T - 51$$
$89$ $$T^{3} - 189T + 999$$
$97$ $$T^{3} - 15 T^{2} - 69 T + 19$$