# Properties

 Label 729.2.a.e Level $729$ Weight $2$ Character orbit 729.a Self dual yes Analytic conductor $5.821$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [729,2,Mod(1,729)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(729, 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("729.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$729 = 3^{6}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 729.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: $$5.82109430735$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.7459857.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 3x^{5} - 6x^{4} + 13x^{3} + 12x^{2} - 12x - 8$$ x^6 - 3*x^5 - 6*x^4 + 13*x^3 + 12*x^2 - 12*x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} + \beta_1) q^{2} + ( - \beta_{5} + \beta_{4} + \beta_{2} + \cdots + 1) q^{4}+ \cdots + (\beta_{4} + 3 \beta_{3} + \beta_1) q^{8}+O(q^{10})$$ q + (b3 + b1) * q^2 + (-b5 + b4 + b2 + b1 + 1) * q^4 + (-b4 + b3) * q^5 + (-b4 + b3 + b1 + 1) * q^7 + (b4 + 3*b3 + b1) * q^8 $$q + (\beta_{3} + \beta_1) q^{2} + ( - \beta_{5} + \beta_{4} + \beta_{2} + \cdots + 1) q^{4}+ \cdots + ( - 2 \beta_{5} - 4 \beta_{4} + \cdots + 5) q^{98}+O(q^{100})$$ q + (b3 + b1) * q^2 + (-b5 + b4 + b2 + b1 + 1) * q^4 + (-b4 + b3) * q^5 + (-b4 + b3 + b1 + 1) * q^7 + (b4 + 3*b3 + b1) * q^8 + (-b5 - b3 - b2 - b1 + 2) * q^10 + (b5 - b4 - b3 + b2 - 1) * q^11 + (b5 + b1) * q^13 + (-b5 - b2 + b1 + 4) * q^14 + (-b5 + 2*b4 + b3 + 3*b2 + 2) * q^16 + (-b5 + b4 - b3 + 2*b2 + b1 - 2) * q^17 + (b5 + b4 + b3 - b2 + 1) * q^19 + (b5 - b4 + 3*b3 - 5*b2 + b1 - 4) * q^20 + (b5 - 5*b3 + b2 - 2*b1 + 1) * q^22 + (-b5 - b2 + b1 - 2) * q^23 + (-b4 - b3 + b2 - 2*b1 + 3) * q^25 + (b4 - 3*b3 + 3*b2 + b1 + 3) * q^26 + (5*b3 - 4*b2 + 3*b1 - 1) * q^28 + (2*b5 - b4 - 2*b3 - b2 - 2*b1 + 4) * q^29 + (-b4 - 2*b3 - 2*b1 + 4) * q^31 + (-b5 + 3*b4 + b3 + 5*b2 + 2*b1 - 2) * q^32 + (b5 + b4 + 2*b3 - 1) * q^34 + (-b5 - b4 + b3 - b2 - 2*b1 + 7) * q^35 + (b5 - 2*b4 + 4*b3 - 4*b2 + 1) * q^37 + (-b5 + 2*b4 - b3 + 5*b2 + 2*b1 + 1) * q^38 + (-b5 - 2*b4 - 6*b3 + b2 - 2*b1 + 3) * q^40 + (2*b5 - b4 - b3 - 4*b2 - b1 - 2) * q^41 + (2*b5 - 2*b4 - 4*b3 - 2*b1 + 2) * q^43 + (3*b5 - b4 - 3*b2 - b1 - 6) * q^44 + (-2*b4 + b3 - 4*b2 - b1 + 1) * q^46 + (2*b4 + 3*b3 - b1 - 3) * q^47 + (-b5 - 2*b4 + 2*b3 - 3*b2 + b1 + 3) * q^49 + (b5 - b4 + 2*b3 - 2*b2 - 4) * q^50 + (b5 + b4 + 4*b3 + 2*b2 + 3*b1 - 2) * q^52 + (-2*b5 + b4 - 2*b2 - 1) * q^53 + (b4 - 2*b3 + 2*b2 - 4*b1 + 4) * q^55 + (-3*b5 + b4 - b3 + 3*b2 + 3) * q^56 + (2*b5 - 2*b4 - 3*b3 + b2 + b1 - 3) * q^58 + (-b4 + b3 - 3*b1 + 3) * q^59 + (-b5 + b4 - 3*b3 + 4*b2 - 2*b1 + 5) * q^61 + (2*b5 - 3*b4 + 3*b3 - 4*b2 + b1 - 5) * q^62 + (b5 + 4*b4 + 2*b3 + 3*b2 + 3*b1 - 3) * q^64 + (-2*b5 + b4 - 2*b3 + b2 - 2*b1 - 1) * q^65 + (2*b4 + 5*b3 - 2*b2 + b1 + 1) * q^67 + (2*b4 - b3 + 3*b2 - 2*b1 + 6) * q^68 + (-b5 - 2*b4 + 9*b3 - 5*b2 + 4*b1 - 3) * q^70 + (-2*b5 - 4*b3 + 4*b2 - 2*b1 + 2) * q^71 + (-3*b5 + 3*b3 + 3*b2 + 3*b1 + 2) * q^73 + (-4*b5 - b4 - 4*b3 - b2 - b1 + 7) * q^74 + (-b5 + 3*b4 + 4*b3 + 7*b2 + 5*b1 - 2) * q^76 + (3*b5 - 7*b3 + 3*b2 - 4*b1 + 3) * q^77 + (-2*b5 + 3*b4 - 3*b3 + b1 + 3) * q^79 + (4*b5 - 6*b4 - 2*b3 - 2*b2 - 3*b1 - 1) * q^80 + (b5 - 4*b4 - 9*b3 - b2 - 4*b1) * q^82 + (-2*b5 + b4 + b3 - 2*b2 + b1 - 1) * q^83 + (2*b5 + 2*b4 + b3 - 6*b2 - 5) * q^85 + (4*b5 - 4*b4 - 6*b3 - 2*b2 - 2*b1 - 4) * q^86 + (-2*b5 - b4 - 6*b3 + 2*b2 - 4*b1) * q^88 + (4*b4 - 2*b3 + 3*b2 - b1 - 3) * q^89 + (b5 - 6*b3 + 3*b2 + b1 + 2) * q^91 + (b5 - 5*b4 - b3 - 5*b2 - 4*b1 + 5) * q^92 + (-3*b5 + 5*b4 - b3 + 7*b2 - 2*b1 - 1) * q^94 + (-2*b5 - b4 - 4*b3 + 4*b2 - 4) * q^95 + (2*b5 - 2*b4 - b3 + b1 - 4) * q^97 + (-2*b5 - 4*b4 + 4*b3 - 8*b2 + 2*b1 + 5) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 3 q^{2} + 9 q^{4} - 3 q^{5} + 6 q^{7} + 6 q^{8}+O(q^{10})$$ 6 * q + 3 * q^2 + 9 * q^4 - 3 * q^5 + 6 * q^7 + 6 * q^8 $$6 q + 3 q^{2} + 9 q^{4} - 3 q^{5} + 6 q^{7} + 6 q^{8} + 6 q^{10} - 6 q^{11} + 6 q^{13} + 24 q^{14} + 15 q^{16} - 9 q^{17} + 12 q^{19} - 21 q^{20} + 3 q^{22} - 12 q^{23} + 9 q^{25} + 24 q^{26} + 3 q^{28} + 21 q^{29} + 15 q^{31} + 30 q^{35} + 3 q^{37} + 15 q^{38} + 3 q^{40} - 12 q^{41} + 6 q^{43} - 33 q^{44} - 3 q^{46} - 15 q^{47} + 12 q^{49} - 24 q^{50} + 3 q^{52} - 9 q^{53} + 15 q^{55} + 12 q^{56} - 15 q^{58} + 6 q^{59} + 24 q^{61} - 30 q^{62} + 6 q^{64} - 15 q^{65} + 15 q^{67} + 36 q^{68} - 15 q^{70} + 12 q^{73} + 24 q^{74} + 9 q^{76} + 15 q^{77} + 24 q^{79} - 21 q^{80} - 21 q^{82} - 6 q^{83} - 18 q^{85} - 30 q^{86} - 21 q^{88} - 9 q^{89} + 18 q^{91} + 6 q^{92} - 6 q^{94} - 33 q^{95} - 21 q^{97} + 18 q^{98}+O(q^{100})$$ 6 * q + 3 * q^2 + 9 * q^4 - 3 * q^5 + 6 * q^7 + 6 * q^8 + 6 * q^10 - 6 * q^11 + 6 * q^13 + 24 * q^14 + 15 * q^16 - 9 * q^17 + 12 * q^19 - 21 * q^20 + 3 * q^22 - 12 * q^23 + 9 * q^25 + 24 * q^26 + 3 * q^28 + 21 * q^29 + 15 * q^31 + 30 * q^35 + 3 * q^37 + 15 * q^38 + 3 * q^40 - 12 * q^41 + 6 * q^43 - 33 * q^44 - 3 * q^46 - 15 * q^47 + 12 * q^49 - 24 * q^50 + 3 * q^52 - 9 * q^53 + 15 * q^55 + 12 * q^56 - 15 * q^58 + 6 * q^59 + 24 * q^61 - 30 * q^62 + 6 * q^64 - 15 * q^65 + 15 * q^67 + 36 * q^68 - 15 * q^70 + 12 * q^73 + 24 * q^74 + 9 * q^76 + 15 * q^77 + 24 * q^79 - 21 * q^80 - 21 * q^82 - 6 * q^83 - 18 * q^85 - 30 * q^86 - 21 * q^88 - 9 * q^89 + 18 * q^91 + 6 * q^92 - 6 * q^94 - 33 * q^95 - 21 * q^97 + 18 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3x^{5} - 6x^{4} + 13x^{3} + 12x^{2} - 12x - 8$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{4} - 3\nu^{3} - 4\nu^{2} + 7\nu + 4 ) / 2$$ (v^4 - 3*v^3 - 4*v^2 + 7*v + 4) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{5} - 3\nu^{4} - 6\nu^{3} + 13\nu^{2} + 8\nu - 8 ) / 4$$ (v^5 - 3*v^4 - 6*v^3 + 13*v^2 + 8*v - 8) / 4 $$\beta_{4}$$ $$=$$ $$( \nu^{5} - 3\nu^{4} - 4\nu^{3} + 7\nu^{2} + 4\nu + 2 ) / 2$$ (v^5 - 3*v^4 - 4*v^3 + 7*v^2 + 4*v + 2) / 2 $$\beta_{5}$$ $$=$$ $$( 3\nu^{5} - 9\nu^{4} - 14\nu^{3} + 31\nu^{2} + 12\nu - 16 ) / 4$$ (3*v^5 - 9*v^4 - 14*v^3 + 31*v^2 + 12*v - 16) / 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} - \beta_{4} - \beta_{3} + \beta _1 + 3$$ b5 - b4 - b3 + b1 + 3 $$\nu^{3}$$ $$=$$ $$3\beta_{5} - 2\beta_{4} - 5\beta_{3} + 5\beta _1 + 4$$ 3*b5 - 2*b4 - 5*b3 + 5*b1 + 4 $$\nu^{4}$$ $$=$$ $$13\beta_{5} - 10\beta_{4} - 19\beta_{3} + 2\beta_{2} + 12\beta _1 + 20$$ 13*b5 - 10*b4 - 19*b3 + 2*b2 + 12*b1 + 20 $$\nu^{5}$$ $$=$$ $$44\beta_{5} - 29\beta_{4} - 70\beta_{3} + 6\beta_{2} + 45\beta _1 + 53$$ 44*b5 - 29*b4 - 70*b3 + 6*b2 + 45*b1 + 53

## 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
 −0.578404 −1.12503 −1.70506 3.45779 1.77773 1.17298
−2.45779 0 4.04073 −3.08026 0 −2.65867 −5.01568 0 7.57064
1.2 −0.777732 0 −1.39513 −2.37635 0 −2.50138 2.64050 0 1.84816
1.3 −0.172976 0 −1.97008 3.73656 0 3.03150 0.686728 0 −0.646335
1.4 1.57840 0 0.491360 −1.67851 0 2.77928 −2.38124 0 −2.64936
1.5 2.12503 0 2.51575 2.07094 0 4.84867 1.09598 0 4.40081
1.6 2.70506 0 5.31738 −1.67238 0 0.500591 8.97372 0 −4.52391
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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 729.2.a.e yes 6
3.b odd 2 1 729.2.a.b 6
9.c even 3 2 729.2.c.a 12
9.d odd 6 2 729.2.c.d 12
27.e even 9 2 729.2.e.k 12
27.e even 9 2 729.2.e.l 12
27.e even 9 2 729.2.e.u 12
27.f odd 18 2 729.2.e.j 12
27.f odd 18 2 729.2.e.s 12
27.f odd 18 2 729.2.e.t 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
729.2.a.b 6 3.b odd 2 1
729.2.a.e yes 6 1.a even 1 1 trivial
729.2.c.a 12 9.c even 3 2
729.2.c.d 12 9.d odd 6 2
729.2.e.j 12 27.f odd 18 2
729.2.e.k 12 27.e even 9 2
729.2.e.l 12 27.e even 9 2
729.2.e.s 12 27.f odd 18 2
729.2.e.t 12 27.f odd 18 2
729.2.e.u 12 27.e even 9 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} - 3T_{2}^{5} - 6T_{2}^{4} + 21T_{2}^{3} - 18T_{2} - 3$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(729))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 3 T^{5} + \cdots - 3$$
$3$ $$T^{6}$$
$5$ $$T^{6} + 3 T^{5} + \cdots + 159$$
$7$ $$T^{6} - 6 T^{5} + \cdots + 136$$
$11$ $$T^{6} + 6 T^{5} + \cdots + 888$$
$13$ $$T^{6} - 6 T^{5} + \cdots - 89$$
$17$ $$T^{6} + 9 T^{5} + \cdots + 459$$
$19$ $$T^{6} - 12 T^{5} + \cdots - 296$$
$23$ $$T^{6} + 12 T^{5} + \cdots + 456$$
$29$ $$T^{6} - 21 T^{5} + \cdots + 9879$$
$31$ $$T^{6} - 15 T^{5} + \cdots - 1016$$
$37$ $$T^{6} - 3 T^{5} + \cdots - 16109$$
$41$ $$T^{6} + 12 T^{5} + \cdots + 23109$$
$43$ $$T^{6} - 6 T^{5} + \cdots + 17344$$
$47$ $$T^{6} + 15 T^{5} + \cdots + 17736$$
$53$ $$T^{6} + 9 T^{5} + \cdots + 1944$$
$59$ $$T^{6} - 6 T^{5} + \cdots - 408$$
$61$ $$T^{6} - 24 T^{5} + \cdots + 1576$$
$67$ $$T^{6} - 15 T^{5} + \cdots - 17288$$
$71$ $$T^{6} - 180 T^{4} + \cdots + 29376$$
$73$ $$T^{6} - 12 T^{5} + \cdots + 57601$$
$79$ $$T^{6} - 24 T^{5} + \cdots - 93176$$
$83$ $$T^{6} + 6 T^{5} + \cdots + 4344$$
$89$ $$T^{6} + 9 T^{5} + \cdots - 123957$$
$97$ $$T^{6} + 21 T^{5} + \cdots - 18251$$