# Properties

 Label 729.2.e.o Level $729$ Weight $2$ Character orbit 729.e Analytic conductor $5.821$ Analytic rank $0$ Dimension $12$ CM no Inner twists $12$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [729,2,Mod(82,729)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(729, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([8]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("729.82");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$729 = 3^{6}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 729.e (of order $$9$$, degree $$6$$, not minimal)

## 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: no Analytic conductor: $$5.82109430735$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{9})$$ Coefficient field: $$\Q(\zeta_{36})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - x^{6} + 1$$ x^12 - x^6 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3^{3}$$ Twist minimal: no (minimal twist has level 81) Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{8} q^{2} + (\beta_{7} - \beta_{2}) q^{4} + \beta_{4} q^{5} - 2 \beta_1 q^{7} - \beta_{6} q^{8}+O(q^{10})$$ q - b8 * q^2 + (b7 - b2) * q^4 + b4 * q^5 - 2*b1 * q^7 - b6 * q^8 $$q - \beta_{8} q^{2} + (\beta_{7} - \beta_{2}) q^{4} + \beta_{4} q^{5} - 2 \beta_1 q^{7} - \beta_{6} q^{8} + ( - 3 \beta_{3} + 3) q^{10} + (2 \beta_{11} - 2 \beta_{8}) q^{11} + \beta_{7} q^{13} + ( - 2 \beta_{9} + 2 \beta_{4}) q^{14} + (5 \beta_{5} - 5 \beta_1) q^{16} + (3 \beta_{10} - 3 \beta_{6}) q^{17} - 2 \beta_{3} q^{19} - \beta_{11} q^{20} - 6 \beta_{2} q^{22} + 2 \beta_{9} q^{23} - 2 \beta_{5} q^{25} + \beta_{10} q^{26} + 2 q^{28} - \beta_{8} q^{29} + (8 \beta_{7} - 8 \beta_{2}) q^{31} + 3 \beta_{4} q^{32} - 9 \beta_1 q^{34} - 2 \beta_{6} q^{35} + ( - 7 \beta_{3} + 7) q^{37} + ( - 2 \beta_{11} + 2 \beta_{8}) q^{38} - 3 \beta_{7} q^{40} + (4 \beta_{9} - 4 \beta_{4}) q^{41} + ( - 2 \beta_{5} + 2 \beta_1) q^{43} + ( - 2 \beta_{10} + 2 \beta_{6}) q^{44} - 6 \beta_{3} q^{46} + 4 \beta_{11} q^{47} - 3 \beta_{2} q^{49} - 2 \beta_{9} q^{50} - \beta_{5} q^{52} + 6 q^{55} + 2 \beta_{8} q^{56} + (3 \beta_{7} - 3 \beta_{2}) q^{58} - 8 \beta_{4} q^{59} + 7 \beta_1 q^{61} + 8 \beta_{6} q^{62} + (\beta_{3} - 1) q^{64} + ( - \beta_{11} + \beta_{8}) q^{65} + 10 \beta_{7} q^{67} + ( - 3 \beta_{9} + 3 \beta_{4}) q^{68} + (6 \beta_{5} - 6 \beta_1) q^{70} + ( - 6 \beta_{10} + 6 \beta_{6}) q^{71} + 7 \beta_{3} q^{73} - 7 \beta_{11} q^{74} + 2 \beta_{2} q^{76} - 4 \beta_{9} q^{77} + 2 \beta_{5} q^{79} - 5 \beta_{10} q^{80} - 12 q^{82} + 8 \beta_{8} q^{83} + ( - 9 \beta_{7} + 9 \beta_{2}) q^{85} - 2 \beta_{4} q^{86} - 6 \beta_1 q^{88} - 3 \beta_{6} q^{89} + ( - 2 \beta_{3} + 2) q^{91} + ( - 2 \beta_{11} + 2 \beta_{8}) q^{92} - 12 \beta_{7} q^{94} + (2 \beta_{9} - 2 \beta_{4}) q^{95} + ( - 2 \beta_{5} + 2 \beta_1) q^{97} + ( - 3 \beta_{10} + 3 \beta_{6}) q^{98}+O(q^{100})$$ q - b8 * q^2 + (b7 - b2) * q^4 + b4 * q^5 - 2*b1 * q^7 - b6 * q^8 + (-3*b3 + 3) * q^10 + (2*b11 - 2*b8) * q^11 + b7 * q^13 + (-2*b9 + 2*b4) * q^14 + (5*b5 - 5*b1) * q^16 + (3*b10 - 3*b6) * q^17 - 2*b3 * q^19 - b11 * q^20 - 6*b2 * q^22 + 2*b9 * q^23 - 2*b5 * q^25 + b10 * q^26 + 2 * q^28 - b8 * q^29 + (8*b7 - 8*b2) * q^31 + 3*b4 * q^32 - 9*b1 * q^34 - 2*b6 * q^35 + (-7*b3 + 7) * q^37 + (-2*b11 + 2*b8) * q^38 - 3*b7 * q^40 + (4*b9 - 4*b4) * q^41 + (-2*b5 + 2*b1) * q^43 + (-2*b10 + 2*b6) * q^44 - 6*b3 * q^46 + 4*b11 * q^47 - 3*b2 * q^49 - 2*b9 * q^50 - b5 * q^52 + 6 * q^55 + 2*b8 * q^56 + (3*b7 - 3*b2) * q^58 - 8*b4 * q^59 + 7*b1 * q^61 + 8*b6 * q^62 + (b3 - 1) * q^64 + (-b11 + b8) * q^65 + 10*b7 * q^67 + (-3*b9 + 3*b4) * q^68 + (6*b5 - 6*b1) * q^70 + (-6*b10 + 6*b6) * q^71 + 7*b3 * q^73 - 7*b11 * q^74 + 2*b2 * q^76 - 4*b9 * q^77 + 2*b5 * q^79 - 5*b10 * q^80 - 12 * q^82 + 8*b8 * q^83 + (-9*b7 + 9*b2) * q^85 - 2*b4 * q^86 - 6*b1 * q^88 - 3*b6 * q^89 + (-2*b3 + 2) * q^91 + (-2*b11 + 2*b8) * q^92 - 12*b7 * q^94 + (2*b9 - 2*b4) * q^95 + (-2*b5 + 2*b1) * q^97 + (-3*b10 + 3*b6) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q+O(q^{10})$$ 12 * q $$12 q + 18 q^{10} - 12 q^{19} + 24 q^{28} + 42 q^{37} - 36 q^{46} + 72 q^{55} - 6 q^{64} + 42 q^{73} - 144 q^{82} + 12 q^{91}+O(q^{100})$$ 12 * q + 18 * q^10 - 12 * q^19 + 24 * q^28 + 42 * q^37 - 36 * q^46 + 72 * q^55 - 6 * q^64 + 42 * q^73 - 144 * q^82 + 12 * q^91

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{36}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{36}^{4}$$ v^4 $$\beta_{3}$$ $$=$$ $$\zeta_{36}^{6}$$ v^6 $$\beta_{4}$$ $$=$$ $$\zeta_{36}^{7} + \zeta_{36}$$ v^7 + v $$\beta_{5}$$ $$=$$ $$\zeta_{36}^{8}$$ v^8 $$\beta_{6}$$ $$=$$ $$\zeta_{36}^{9} + \zeta_{36}^{3}$$ v^9 + v^3 $$\beta_{7}$$ $$=$$ $$\zeta_{36}^{10}$$ v^10 $$\beta_{8}$$ $$=$$ $$\zeta_{36}^{11} + \zeta_{36}^{5}$$ v^11 + v^5 $$\beta_{9}$$ $$=$$ $$-\zeta_{36}^{7} + 2\zeta_{36}$$ -v^7 + 2*v $$\beta_{10}$$ $$=$$ $$-\zeta_{36}^{9} + 2\zeta_{36}^{3}$$ -v^9 + 2*v^3 $$\beta_{11}$$ $$=$$ $$-\zeta_{36}^{11} + 2\zeta_{36}^{5}$$ -v^11 + 2*v^5
 $$\zeta_{36}$$ $$=$$ $$( \beta_{9} + \beta_{4} ) / 3$$ (b9 + b4) / 3 $$\zeta_{36}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{36}^{3}$$ $$=$$ $$( \beta_{10} + \beta_{6} ) / 3$$ (b10 + b6) / 3 $$\zeta_{36}^{4}$$ $$=$$ $$\beta_{2}$$ b2 $$\zeta_{36}^{5}$$ $$=$$ $$( \beta_{11} + \beta_{8} ) / 3$$ (b11 + b8) / 3 $$\zeta_{36}^{6}$$ $$=$$ $$\beta_{3}$$ b3 $$\zeta_{36}^{7}$$ $$=$$ $$( -\beta_{9} + 2\beta_{4} ) / 3$$ (-b9 + 2*b4) / 3 $$\zeta_{36}^{8}$$ $$=$$ $$\beta_{5}$$ b5 $$\zeta_{36}^{9}$$ $$=$$ $$( -\beta_{10} + 2\beta_{6} ) / 3$$ (-b10 + 2*b6) / 3 $$\zeta_{36}^{10}$$ $$=$$ $$\beta_{7}$$ b7 $$\zeta_{36}^{11}$$ $$=$$ $$( -\beta_{11} + 2\beta_{8} ) / 3$$ (-b11 + 2*b8) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/729\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-\beta_{2} + \beta_{7}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
82.1
 0.984808 + 0.173648i −0.984808 − 0.173648i 0.342020 − 0.939693i −0.342020 + 0.939693i −0.642788 + 0.766044i 0.642788 − 0.766044i −0.642788 − 0.766044i 0.642788 + 0.766044i 0.342020 + 0.939693i −0.342020 − 0.939693i 0.984808 − 0.173648i −0.984808 + 0.173648i
−0.300767 1.70574i 0 −0.939693 + 0.342020i 1.32683 + 1.11334i 0 −1.87939 0.684040i −0.866025 1.50000i 0 1.50000 2.59808i
82.2 0.300767 + 1.70574i 0 −0.939693 + 0.342020i −1.32683 1.11334i 0 −1.87939 0.684040i 0.866025 + 1.50000i 0 1.50000 2.59808i
163.1 −1.62760 + 0.592396i 0 0.766044 0.642788i −0.300767 1.70574i 0 1.53209 + 1.28558i 0.866025 1.50000i 0 1.50000 + 2.59808i
163.2 1.62760 0.592396i 0 0.766044 0.642788i 0.300767 + 1.70574i 0 1.53209 + 1.28558i −0.866025 + 1.50000i 0 1.50000 + 2.59808i
325.1 −1.32683 + 1.11334i 0 0.173648 0.984808i −1.62760 + 0.592396i 0 0.347296 + 1.96962i −0.866025 1.50000i 0 1.50000 2.59808i
325.2 1.32683 1.11334i 0 0.173648 0.984808i 1.62760 0.592396i 0 0.347296 + 1.96962i 0.866025 + 1.50000i 0 1.50000 2.59808i
406.1 −1.32683 1.11334i 0 0.173648 + 0.984808i −1.62760 0.592396i 0 0.347296 1.96962i −0.866025 + 1.50000i 0 1.50000 + 2.59808i
406.2 1.32683 + 1.11334i 0 0.173648 + 0.984808i 1.62760 + 0.592396i 0 0.347296 1.96962i 0.866025 1.50000i 0 1.50000 + 2.59808i
568.1 −1.62760 0.592396i 0 0.766044 + 0.642788i −0.300767 + 1.70574i 0 1.53209 1.28558i 0.866025 + 1.50000i 0 1.50000 2.59808i
568.2 1.62760 + 0.592396i 0 0.766044 + 0.642788i 0.300767 1.70574i 0 1.53209 1.28558i −0.866025 1.50000i 0 1.50000 2.59808i
649.1 −0.300767 + 1.70574i 0 −0.939693 0.342020i 1.32683 1.11334i 0 −1.87939 + 0.684040i −0.866025 + 1.50000i 0 1.50000 + 2.59808i
649.2 0.300767 1.70574i 0 −0.939693 0.342020i −1.32683 + 1.11334i 0 −1.87939 + 0.684040i 0.866025 1.50000i 0 1.50000 + 2.59808i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 82.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 2 inner
9.d odd 6 2 inner
27.e even 9 3 inner
27.f odd 18 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 729.2.e.o 12
3.b odd 2 1 inner 729.2.e.o 12
9.c even 3 2 inner 729.2.e.o 12
9.d odd 6 2 inner 729.2.e.o 12
27.e even 9 1 81.2.a.a 2
27.e even 9 2 81.2.c.b 4
27.e even 9 3 inner 729.2.e.o 12
27.f odd 18 1 81.2.a.a 2
27.f odd 18 2 81.2.c.b 4
27.f odd 18 3 inner 729.2.e.o 12
108.j odd 18 1 1296.2.a.o 2
108.j odd 18 2 1296.2.i.s 4
108.l even 18 1 1296.2.a.o 2
108.l even 18 2 1296.2.i.s 4
135.n odd 18 1 2025.2.a.j 2
135.p even 18 1 2025.2.a.j 2
135.q even 36 2 2025.2.b.k 4
135.r odd 36 2 2025.2.b.k 4
189.y odd 18 1 3969.2.a.i 2
189.be even 18 1 3969.2.a.i 2
216.r odd 18 1 5184.2.a.bq 2
216.t even 18 1 5184.2.a.br 2
216.v even 18 1 5184.2.a.bq 2
216.x odd 18 1 5184.2.a.br 2
297.o even 18 1 9801.2.a.v 2
297.q odd 18 1 9801.2.a.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
81.2.a.a 2 27.e even 9 1
81.2.a.a 2 27.f odd 18 1
81.2.c.b 4 27.e even 9 2
81.2.c.b 4 27.f odd 18 2
729.2.e.o 12 1.a even 1 1 trivial
729.2.e.o 12 3.b odd 2 1 inner
729.2.e.o 12 9.c even 3 2 inner
729.2.e.o 12 9.d odd 6 2 inner
729.2.e.o 12 27.e even 9 3 inner
729.2.e.o 12 27.f odd 18 3 inner
1296.2.a.o 2 108.j odd 18 1
1296.2.a.o 2 108.l even 18 1
1296.2.i.s 4 108.j odd 18 2
1296.2.i.s 4 108.l even 18 2
2025.2.a.j 2 135.n odd 18 1
2025.2.a.j 2 135.p even 18 1
2025.2.b.k 4 135.q even 36 2
2025.2.b.k 4 135.r odd 36 2
3969.2.a.i 2 189.y odd 18 1
3969.2.a.i 2 189.be even 18 1
5184.2.a.bq 2 216.r odd 18 1
5184.2.a.bq 2 216.v even 18 1
5184.2.a.br 2 216.t even 18 1
5184.2.a.br 2 216.x odd 18 1
9801.2.a.v 2 297.o even 18 1
9801.2.a.v 2 297.q odd 18 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(729, [\chi])$$:

 $$T_{2}^{12} + 27T_{2}^{6} + 729$$ T2^12 + 27*T2^6 + 729 $$T_{5}^{12} + 27T_{5}^{6} + 729$$ T5^12 + 27*T5^6 + 729 $$T_{7}^{6} + 8T_{7}^{3} + 64$$ T7^6 + 8*T7^3 + 64

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} + 27T^{6} + 729$$
$3$ $$T^{12}$$
$5$ $$T^{12} + 27T^{6} + 729$$
$7$ $$(T^{6} + 8 T^{3} + 64)^{2}$$
$11$ $$T^{12} + 1728 T^{6} + \cdots + 2985984$$
$13$ $$(T^{6} - T^{3} + 1)^{2}$$
$17$ $$(T^{4} + 27 T^{2} + 729)^{3}$$
$19$ $$(T^{2} + 2 T + 4)^{6}$$
$23$ $$T^{12} + 1728 T^{6} + \cdots + 2985984$$
$29$ $$T^{12} + 27T^{6} + 729$$
$31$ $$(T^{6} + 512 T^{3} + 262144)^{2}$$
$37$ $$(T^{2} - 7 T + 49)^{6}$$
$41$ $$T^{12} + 110592 T^{6} + \cdots + 12230590464$$
$43$ $$(T^{6} + 8 T^{3} + 64)^{2}$$
$47$ $$T^{12} + 110592 T^{6} + \cdots + 12230590464$$
$53$ $$T^{12}$$
$59$ $$T^{12} + 7077888 T^{6} + \cdots + 50096498540544$$
$61$ $$(T^{6} - 343 T^{3} + 117649)^{2}$$
$67$ $$(T^{6} - 1000 T^{3} + 1000000)^{2}$$
$71$ $$(T^{4} + 108 T^{2} + 11664)^{3}$$
$73$ $$(T^{2} - 7 T + 49)^{6}$$
$79$ $$(T^{6} + 8 T^{3} + 64)^{2}$$
$83$ $$T^{12} + 7077888 T^{6} + \cdots + 50096498540544$$
$89$ $$(T^{4} + 27 T^{2} + 729)^{3}$$
$97$ $$(T^{6} + 8 T^{3} + 64)^{2}$$