# Properties

 Label 729.2.a.a Level $729$ Weight $2$ Character orbit 729.a Self dual yes Analytic conductor $5.821$ Analytic rank $1$ 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: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.1397493.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 3x^{5} - 3x^{4} + 10x^{3} + 3x^{2} - 6x + 1$$ x^6 - 3*x^5 - 3*x^4 + 10*x^3 + 3*x^2 - 6*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 27) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{5} - \beta_1) q^{2} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{4} + ( - \beta_{4} - 1) q^{5} + (\beta_{5} + \beta_{4} + \beta_{2}) q^{7} + (\beta_{5} + 2 \beta_{3} + \beta_{2} - 1) q^{8}+O(q^{10})$$ q + (-b5 - b1) * q^2 + (-b3 - b2 + b1) * q^4 + (-b4 - 1) * q^5 + (b5 + b4 + b2) * q^7 + (b5 + 2*b3 + b2 - 1) * q^8 $$q + ( - \beta_{5} - \beta_1) q^{2} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{4} + ( - \beta_{4} - 1) q^{5} + (\beta_{5} + \beta_{4} + \beta_{2}) q^{7} + (\beta_{5} + 2 \beta_{3} + \beta_{2} - 1) q^{8} + (2 \beta_{5} + \beta_{4} + \cdots + \beta_1) q^{10}+ \cdots + ( - \beta_{5} - \beta_{4} - 2 \beta_{3} + \cdots + 6) q^{98}+O(q^{100})$$ q + (-b5 - b1) * q^2 + (-b3 - b2 + b1) * q^4 + (-b4 - 1) * q^5 + (b5 + b4 + b2) * q^7 + (b5 + 2*b3 + b2 - 1) * q^8 + (2*b5 + b4 + b3 + b1) * q^10 + (b4 - 2) * q^11 + (-b4 + 2*b1 - 1) * q^13 + (b5 - b3 + b2 - 1) * q^14 + (b5 - b4 + b2 + b1 - 1) * q^16 + (b3 - b2 + b1 - 2) * q^17 + (b5 + b4 + b3 - b2 - b1 + 1) * q^19 + (-2*b5 - b3 + b2 - 1) * q^20 + (b5 - b4 - b3 + 2*b1) * q^22 + (2*b5 + b4 - b1 - 2) * q^23 + (2*b5 + b4 + 2*b3 + b2 - 1) * q^25 + (2*b5 + b4 + 3*b3 - b1 - 2) * q^26 + (2*b5 + b3 - 1) * q^28 + (-b5 - 3*b3 + 2*b1 - 3) * q^29 + (-2*b5 - 2*b4 + b2) * q^31 + (2*b5 + 2*b4 - 2*b3 - b2) * q^32 + (-b5 - 2*b4 - b2 + 2*b1 - 1) * q^34 + (-2*b5 - b4 - b3 - 2*b2 - 2) * q^35 + (-2*b5 - 2*b4 - 2*b3 - b2 - b1 + 1) * q^37 + (-5*b5 - 3*b4 - 3*b3 + b1) * q^38 + (-b3 - b2 - 2*b1 + 2) * q^40 + (b5 - 3*b2 + b1 - 3) * q^41 + (-b5 + 3*b4 - b2 - 2*b1 + 1) * q^43 + (2*b5 + 4*b3 + 2*b2 - 3*b1 + 1) * q^44 + (b5 - b4 - 2*b3 + 2*b2 + 3*b1 - 1) * q^46 + (-2*b5 - b4 - 3*b3 - 3*b2 + b1 - 4) * q^47 + (-4*b5 - b2 - 2*b1 - 1) * q^49 + (-2*b4 - 3*b3 + 3*b1 - 2) * q^50 + (-2*b5 - 2*b4 - 5*b3 - b2 + 2*b1 + 1) * q^52 + (-3*b5 - 3*b4 + 3*b1 - 3) * q^53 + (-2*b5 + 2*b4 - 2*b3 - b2 - 1) * q^55 + (-2*b5 - b4 + b3 - b2 + 2*b1) * q^56 + (6*b5 + 3*b4 + 5*b3 + 2*b2 - 2*b1 - 1) * q^58 + (3*b5 + 2*b4 + 3*b3 + 3*b2 - 4) * q^59 + (2*b3 - b2 - 5*b1 + 1) * q^61 + (4*b5 + 3*b4 + 2*b3 - 2*b2 + 2) * q^62 + (-4*b5 + b4 + 2*b2 - 4*b1) * q^64 + (b4 - 2*b3 - b2 - 2*b1 + 2) * q^65 + (-b5 + 4*b4 - b3 + b2 - 3*b1) * q^67 + (b5 + b4 + 2*b3 + b2 - 3*b1 + 3) * q^68 + (2*b3 - b2 + b1 + 2) * q^70 + (-2*b3 + 2*b2 + b1 - 5) * q^71 + (-6*b5 - 3*b4 - 1) * q^73 + (b5 + 3*b4 + 3*b3 - 2*b1 + 3) * q^74 + (4*b5 + 4*b4 + 5*b3 - 2*b1 + 2) * q^76 + (-b5 - 2*b4 + b3 - b2 + 2) * q^77 + (-6*b5 - b4 - 4*b1 + 2) * q^79 + (b5 + b3 - b2 - b1 + 4) * q^80 + (-3*b5 - 3*b4 + b3 + b2 + 2*b1 - 2) * q^82 + (5*b5 + 3*b4 + 3*b3 + 3*b2 - 4*b1) * q^83 + (-2*b5 + 2*b4 - 3*b3 + b2 - 2*b1 + 1) * q^85 + (-6*b5 - 4*b4 - 5*b3 - b2 + b1 + 3) * q^86 + (-3*b5 - 5*b3 - 2*b2 + 2*b1 + 1) * q^88 + (-3*b5 - b3 + 4*b2 - b1 - 1) * q^89 + (-b4 + 3*b3 + 2*b1 - 2) * q^91 + (4*b5 + 3*b4 + 6*b3 + 3*b2 - 2*b1) * q^92 + (2*b5 + b4 + 5*b3 + b2 + 1) * q^94 + (-3*b5 - 2*b4 + 3*b2 - 2) * q^95 + (-b5 - 3*b4 - 3*b3 - b2 - 2*b1 + 1) * q^97 + (-b5 - b4 - 2*b3 - 4*b2 + 3*b1 + 6) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 3 q^{2} + 3 q^{4} - 6 q^{5} - 6 q^{8}+O(q^{10})$$ 6 * q - 3 * q^2 + 3 * q^4 - 6 * q^5 - 6 * q^8 $$6 q - 3 q^{2} + 3 q^{4} - 6 q^{5} - 6 q^{8} + 3 q^{10} - 12 q^{11} - 6 q^{14} - 3 q^{16} - 9 q^{17} + 3 q^{19} - 6 q^{20} + 6 q^{22} - 15 q^{23} - 6 q^{25} - 15 q^{26} - 6 q^{28} - 12 q^{29} - 12 q^{35} + 3 q^{37} + 3 q^{38} + 6 q^{40} - 15 q^{41} - 3 q^{44} + 3 q^{46} - 21 q^{47} - 12 q^{49} - 3 q^{50} + 12 q^{52} - 9 q^{53} - 6 q^{55} + 6 q^{56} - 12 q^{58} - 24 q^{59} - 9 q^{61} + 12 q^{62} - 12 q^{64} + 6 q^{65} - 9 q^{67} + 9 q^{68} + 15 q^{70} - 27 q^{71} - 6 q^{73} + 12 q^{74} + 6 q^{76} + 12 q^{77} + 21 q^{80} - 6 q^{82} - 12 q^{83} + 21 q^{86} + 12 q^{88} - 9 q^{89} - 6 q^{91} - 6 q^{92} + 6 q^{94} - 12 q^{95} + 45 q^{98}+O(q^{100})$$ 6 * q - 3 * q^2 + 3 * q^4 - 6 * q^5 - 6 * q^8 + 3 * q^10 - 12 * q^11 - 6 * q^14 - 3 * q^16 - 9 * q^17 + 3 * q^19 - 6 * q^20 + 6 * q^22 - 15 * q^23 - 6 * q^25 - 15 * q^26 - 6 * q^28 - 12 * q^29 - 12 * q^35 + 3 * q^37 + 3 * q^38 + 6 * q^40 - 15 * q^41 - 3 * q^44 + 3 * q^46 - 21 * q^47 - 12 * q^49 - 3 * q^50 + 12 * q^52 - 9 * q^53 - 6 * q^55 + 6 * q^56 - 12 * q^58 - 24 * q^59 - 9 * q^61 + 12 * q^62 - 12 * q^64 + 6 * q^65 - 9 * q^67 + 9 * q^68 + 15 * q^70 - 27 * q^71 - 6 * q^73 + 12 * q^74 + 6 * q^76 + 12 * q^77 + 21 * q^80 - 6 * q^82 - 12 * q^83 + 21 * q^86 + 12 * q^88 - 9 * q^89 - 6 * q^91 - 6 * q^92 + 6 * q^94 - 12 * q^95 + 45 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3x^{5} - 3x^{4} + 10x^{3} + 3x^{2} - 6x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ v^2 - v - 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 2\nu^{2} - 2\nu + 2$$ v^3 - 2*v^2 - 2*v + 2 $$\beta_{4}$$ $$=$$ $$\nu^{4} - 3\nu^{3} - \nu^{2} + 6\nu - 1$$ v^4 - 3*v^3 - v^2 + 6*v - 1 $$\beta_{5}$$ $$=$$ $$\nu^{5} - 3\nu^{4} - 3\nu^{3} + 9\nu^{2} + 4\nu - 3$$ v^5 - 3*v^4 - 3*v^3 + 9*v^2 + 4*v - 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 2$$ b2 + b1 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 2\beta_{2} + 4\beta _1 + 2$$ b3 + 2*b2 + 4*b1 + 2 $$\nu^{4}$$ $$=$$ $$\beta_{4} + 3\beta_{3} + 7\beta_{2} + 7\beta _1 + 9$$ b4 + 3*b3 + 7*b2 + 7*b1 + 9 $$\nu^{5}$$ $$=$$ $$\beta_{5} + 3\beta_{4} + 12\beta_{3} + 18\beta_{2} + 20\beta _1 + 18$$ b5 + 3*b4 + 12*b3 + 18*b2 + 20*b1 + 18

## 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
 2.05432 0.584534 2.68091 −1.11662 −1.40162 0.198473
−2.40162 0 3.76778 0.0930834 0 −0.579861 −4.24555 0 −0.223551
1.2 −2.11662 0 2.48009 −2.68310 0 0.972333 −1.01617 0 5.67911
1.3 −0.801527 0 −1.35755 −2.74984 0 2.37683 2.69117 0 2.20407
1.4 −0.415466 0 −1.82739 2.21519 0 −1.31963 1.59015 0 −0.920335
1.5 1.05432 0 −0.888399 −1.74579 0 2.45925 −3.04531 0 −1.84063
1.6 1.68091 0 0.825466 −1.12954 0 −3.90892 −1.97429 0 −1.89866
 $$n$$: 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.a 6
3.b odd 2 1 729.2.a.d 6
9.c even 3 2 729.2.c.e 12
9.d odd 6 2 729.2.c.b 12
27.e even 9 2 27.2.e.a 12
27.e even 9 2 243.2.e.c 12
27.e even 9 2 243.2.e.d 12
27.f odd 18 2 81.2.e.a 12
27.f odd 18 2 243.2.e.a 12
27.f odd 18 2 243.2.e.b 12
108.j odd 18 2 432.2.u.c 12
135.p even 18 2 675.2.l.c 12
135.r odd 36 4 675.2.u.b 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.2.e.a 12 27.e even 9 2
81.2.e.a 12 27.f odd 18 2
243.2.e.a 12 27.f odd 18 2
243.2.e.b 12 27.f odd 18 2
243.2.e.c 12 27.e even 9 2
243.2.e.d 12 27.e even 9 2
432.2.u.c 12 108.j odd 18 2
675.2.l.c 12 135.p even 18 2
675.2.u.b 24 135.r odd 36 4
729.2.a.a 6 1.a even 1 1 trivial
729.2.a.d 6 3.b odd 2 1
729.2.c.b 12 9.d odd 6 2
729.2.c.e 12 9.c even 3 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 3T_{2}^{5} - 3T_{2}^{4} - 12T_{2}^{3} + 9T_{2} + 3$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(729))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 3 T^{5} + \cdots + 3$$
$3$ $$T^{6}$$
$5$ $$T^{6} + 6 T^{5} + \cdots + 3$$
$7$ $$T^{6} - 15 T^{4} + \cdots - 17$$
$11$ $$T^{6} + 12 T^{5} + \cdots + 3$$
$13$ $$T^{6} - 24 T^{4} + \cdots + 1$$
$17$ $$T^{6} + 9 T^{5} + \cdots + 27$$
$19$ $$T^{6} - 3 T^{5} + \cdots + 19$$
$23$ $$T^{6} + 15 T^{5} + \cdots + 327$$
$29$ $$T^{6} + 12 T^{5} + \cdots - 213$$
$31$ $$T^{6} - 51 T^{4} + \cdots + 163$$
$37$ $$T^{6} - 3 T^{5} + \cdots + 4933$$
$41$ $$T^{6} + 15 T^{5} + \cdots + 3351$$
$43$ $$T^{6} - 96 T^{4} + \cdots + 1819$$
$47$ $$T^{6} + 21 T^{5} + \cdots + 6537$$
$53$ $$T^{6} + 9 T^{5} + \cdots - 12393$$
$59$ $$T^{6} + 24 T^{5} + \cdots - 13281$$
$61$ $$T^{6} + 9 T^{5} + \cdots + 16543$$
$67$ $$T^{6} + 9 T^{5} + \cdots - 2879$$
$71$ $$T^{6} + 27 T^{5} + \cdots + 27$$
$73$ $$T^{6} + 6 T^{5} + \cdots - 431$$
$79$ $$T^{6} - 177 T^{4} + \cdots + 1873$$
$83$ $$T^{6} + 12 T^{5} + \cdots - 83373$$
$89$ $$T^{6} + 9 T^{5} + \cdots + 32589$$
$97$ $$T^{6} - 204 T^{4} + \cdots - 8171$$