# Properties

 Label 729.2.e.r Level $729$ Weight $2$ Character orbit 729.e Analytic conductor $5.821$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# 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: $$3$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{10} - \beta_{9}) q^{2} + ( - \beta_{6} - \beta_{4} + \beta_{3} + 1) q^{4} + (\beta_{11} - \beta_{10} + \beta_{9} - 2 \beta_{8} + \beta_{7} + \beta_{2}) q^{5} + (\beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_1 + 1) q^{7} + (\beta_{11} - 2 \beta_{9} + 2 \beta_{7} - \beta_{2}) q^{8}+O(q^{10})$$ q + (b10 - b9) * q^2 + (-b6 - b4 + b3 + 1) * q^4 + (b11 - b10 + b9 - 2*b8 + b7 + b2) * q^5 + (b5 - b4 - 2*b3 - b1 + 1) * q^7 + (b11 - 2*b9 + 2*b7 - b2) * q^8 $$q + (\beta_{10} - \beta_{9}) q^{2} + ( - \beta_{6} - \beta_{4} + \beta_{3} + 1) q^{4} + (\beta_{11} - \beta_{10} + \beta_{9} - 2 \beta_{8} + \beta_{7} + \beta_{2}) q^{5} + (\beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_1 + 1) q^{7} + (\beta_{11} - 2 \beta_{9} + 2 \beta_{7} - \beta_{2}) q^{8} + (\beta_{6} + 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \beta_1) q^{10} + (4 \beta_{11} - 2 \beta_{10} - \beta_{9} - \beta_{8} + 2 \beta_{7}) q^{11} + ( - 3 \beta_{6} + 3 \beta_{4} + 2 \beta_{3} + 2 \beta_1 - 1) q^{13} + (\beta_{11} - \beta_{9} + \beta_{7} - 3 \beta_{2}) q^{14} + (2 \beta_{6} + \beta_{5} + \beta_{4} + \beta_1 - 2) q^{16} + ( - 2 \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + 2 \beta_{2}) q^{17} + ( - \beta_{6} - \beta_{5} - 4 \beta_{4} - \beta_{3} + 2 \beta_1 + 4) q^{19} + (\beta_{11} + \beta_{9} - 3 \beta_{8} + 3 \beta_{2}) q^{20} + (2 \beta_{6} - \beta_{5} + 3 \beta_{4} + \beta_{3} - 1) q^{22} + (3 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} + 5 \beta_{7} - 5 \beta_{2}) q^{23} + (3 \beta_{5} + 2 \beta_{4} - 5 \beta_1 - 5) q^{25} + ( - 3 \beta_{11} + 4 \beta_{10} - 2 \beta_{9} + 2 \beta_{2}) q^{26} + (\beta_{6} + \beta_{5} - 2 \beta_{3} - 2 \beta_1 - 1) q^{28} + (\beta_{11} + \beta_{10} - 2 \beta_{9} - \beta_{8} - \beta_{7} + \beta_{2}) q^{29} + (5 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 5 \beta_1) q^{31} + (\beta_{11} - \beta_{10} + 2 \beta_{9} - 4 \beta_{8} + 3 \beta_{7} + \beta_{2}) q^{32} + (5 \beta_{5} - \beta_{4} - \beta_1 + 5) q^{34} + ( - 3 \beta_{11} + 5 \beta_{10} - 4 \beta_{9} + 4 \beta_{8} - \beta_{7} - 2 \beta_{2}) q^{35} + ( - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_1) q^{37} + ( - 2 \beta_{11} + 7 \beta_{10} - 5 \beta_{9} + 6 \beta_{8} - \beta_{7} - 6 \beta_{2}) q^{38} + ( - \beta_{3} + \beta_1 - 1) q^{40} + (4 \beta_{11} - \beta_{10} - 4 \beta_{9} + \beta_{8} + 4 \beta_{7} - 3 \beta_{2}) q^{41} + (2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_1) q^{43} + (\beta_{11} + \beta_{10} - \beta_{9} + 5 \beta_{7} - 2 \beta_{2}) q^{44} + ( - 2 \beta_{6} - 2 \beta_{5} + \beta_{4} + 5 \beta_{3} - 3 \beta_1 - 1) q^{46} + (\beta_{11} + \beta_{8} + \beta_{7} - \beta_{2}) q^{47} + (3 \beta_{6} - \beta_{5} - \beta_{4} - 4 \beta_{3} + 4) q^{49} + (3 \beta_{11} - 10 \beta_{10} + 5 \beta_{9} - 5 \beta_{8} + 3 \beta_{7} + 2 \beta_{2}) q^{50} + ( - \beta_{6} - 5 \beta_{5} + \beta_{4} + \beta_{3} + 4 \beta_1 + 4) q^{52} + ( - 4 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} + 2 \beta_{2}) q^{53} + (2 \beta_{5} - 2 \beta_{3} - 2 \beta_1 - 5) q^{55} + (2 \beta_{11} + 2 \beta_{10} - 4 \beta_{9} - 2 \beta_{8} + 3 \beta_{7} - 3 \beta_{2}) q^{56} + ( - \beta_{6} - 5 \beta_{5} - \beta_{4} + \beta_{3} + 5 \beta_1 + 1) q^{58} + ( - 5 \beta_{11} + 5 \beta_{10} - 5 \beta_{9} + 7 \beta_{8} - 2 \beta_{7} - 5 \beta_{2}) q^{59} + ( - 6 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} + 4 \beta_1 - 6) q^{61} + (5 \beta_{11} - 5 \beta_{10} - 3 \beta_{8} + 5 \beta_{7}) q^{62} + (4 \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + 4 \beta_1) q^{64} + (8 \beta_{11} - 10 \beta_{10} + 5 \beta_{9} - 9 \beta_{8} + 4 \beta_{7} + 6 \beta_{2}) q^{65} + (5 \beta_{6} - 5 \beta_{4} - 8 \beta_{3} - \beta_1 - 3) q^{67} + ( - 3 \beta_{11} + 2 \beta_{10} + 3 \beta_{9} - 2 \beta_{8} - 3 \beta_{7} + 3 \beta_{2}) q^{68} + ( - 5 \beta_{6} - 7 \beta_{5} - 7 \beta_{4} + 3 \beta_1 + 4) q^{70} + (2 \beta_{11} - 2 \beta_{8} - 2 \beta_{7}) q^{71} + ( - 3 \beta_{6} - 3 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} + 6 \beta_1 - 2) q^{73} + ( - 2 \beta_{11} - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{2}) q^{74} + ( - 3 \beta_{6} - 3 \beta_{5} - 5 \beta_{4} - 2 \beta_{3} + 2) q^{76} + (2 \beta_{11} + 4 \beta_{10} - \beta_{9} + \beta_{8} - 3 \beta_{2}) q^{77} + ( - 8 \beta_{6} - 5 \beta_{5} + 3 \beta_{4} + 8 \beta_{3} + 2 \beta_1 + 2) q^{79} + (2 \beta_{11} - 6 \beta_{10} + 3 \beta_{9} - 3 \beta_{2}) q^{80} + (4 \beta_{6} - 3 \beta_{5} - \beta_{3} - \beta_1 - 2) q^{82} + ( - 4 \beta_{11} + \beta_{10} + 3 \beta_{9} + 4 \beta_{8} - 8 \beta_{7} + 8 \beta_{2}) q^{83} + (4 \beta_{6} + 3 \beta_{5} + 5 \beta_{4} - 5 \beta_{3} - 3 \beta_1 - 4) q^{85} + (2 \beta_{9} + 2 \beta_{8} - 2 \beta_{7}) q^{86} + ( - 4 \beta_{4} + 7 \beta_{3} - 4 \beta_1) q^{88} + ( - 3 \beta_{11} - 2 \beta_{10} + 10 \beta_{9} + \beta_{8} - 8 \beta_{7} + 5 \beta_{2}) q^{89} + ( - 6 \beta_{6} - \beta_{5} - \beta_{3} - 6 \beta_1) q^{91} + ( - 4 \beta_{11} + 4 \beta_{10} + 3 \beta_{9} - \beta_{8} - 2 \beta_{7} - 2 \beta_{2}) q^{92} + ( - \beta_{6} + \beta_{4} + \beta_{3} - 2 \beta_1) q^{94} + (2 \beta_{11} + 5 \beta_{10} - 2 \beta_{9} - 5 \beta_{8} + 2 \beta_{7} + 6 \beta_{2}) q^{95} + (2 \beta_{6} + 4 \beta_{5} + 4 \beta_{4} - 7 \beta_1 + 3) q^{97} + (2 \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} - 2 \beta_{2}) q^{98}+O(q^{100})$$ q + (b10 - b9) * q^2 + (-b6 - b4 + b3 + 1) * q^4 + (b11 - b10 + b9 - 2*b8 + b7 + b2) * q^5 + (b5 - b4 - 2*b3 - b1 + 1) * q^7 + (b11 - 2*b9 + 2*b7 - b2) * q^8 + (b6 + 2*b5 + 2*b4 + 2*b3 + b1) * q^10 + (4*b11 - 2*b10 - b9 - b8 + 2*b7) * q^11 + (-3*b6 + 3*b4 + 2*b3 + 2*b1 - 1) * q^13 + (b11 - b9 + b7 - 3*b2) * q^14 + (2*b6 + b5 + b4 + b1 - 2) * q^16 + (-2*b11 - b10 + b9 + b8 + b7 + 2*b2) * q^17 + (-b6 - b5 - 4*b4 - b3 + 2*b1 + 4) * q^19 + (b11 + b9 - 3*b8 + 3*b2) * q^20 + (2*b6 - b5 + 3*b4 + b3 - 1) * q^22 + (3*b11 + 2*b10 - 2*b9 + 2*b8 + 5*b7 - 5*b2) * q^23 + (3*b5 + 2*b4 - 5*b1 - 5) * q^25 + (-3*b11 + 4*b10 - 2*b9 + 2*b2) * q^26 + (b6 + b5 - 2*b3 - 2*b1 - 1) * q^28 + (b11 + b10 - 2*b9 - b8 - b7 + b2) * q^29 + (5*b5 - 2*b4 + 2*b3 - 5*b1) * q^31 + (b11 - b10 + 2*b9 - 4*b8 + 3*b7 + b2) * q^32 + (5*b5 - b4 - b1 + 5) * q^34 + (-3*b11 + 5*b10 - 4*b9 + 4*b8 - b7 - 2*b2) * q^35 + (-b6 - b5 + b4 - b3 - b1) * q^37 + (-2*b11 + 7*b10 - 5*b9 + 6*b8 - b7 - 6*b2) * q^38 + (-b3 + b1 - 1) * q^40 + (4*b11 - b10 - 4*b9 + b8 + 4*b7 - 3*b2) * q^41 + (2*b6 - 2*b5 - 2*b4 + 2*b1) * q^43 + (b11 + b10 - b9 + 5*b7 - 2*b2) * q^44 + (-2*b6 - 2*b5 + b4 + 5*b3 - 3*b1 - 1) * q^46 + (b11 + b8 + b7 - b2) * q^47 + (3*b6 - b5 - b4 - 4*b3 + 4) * q^49 + (3*b11 - 10*b10 + 5*b9 - 5*b8 + 3*b7 + 2*b2) * q^50 + (-b6 - 5*b5 + b4 + b3 + 4*b1 + 4) * q^52 + (-4*b11 + 2*b10 - 2*b9 + 2*b2) * q^53 + (2*b5 - 2*b3 - 2*b1 - 5) * q^55 + (2*b11 + 2*b10 - 4*b9 - 2*b8 + 3*b7 - 3*b2) * q^56 + (-b6 - 5*b5 - b4 + b3 + 5*b1 + 1) * q^58 + (-5*b11 + 5*b10 - 5*b9 + 7*b8 - 2*b7 - 5*b2) * q^59 + (-6*b5 + 4*b4 - 2*b3 + 4*b1 - 6) * q^61 + (5*b11 - 5*b10 - 3*b8 + 5*b7) * q^62 + (4*b6 + b5 - b4 + b3 + 4*b1) * q^64 + (8*b11 - 10*b10 + 5*b9 - 9*b8 + 4*b7 + 6*b2) * q^65 + (5*b6 - 5*b4 - 8*b3 - b1 - 3) * q^67 + (-3*b11 + 2*b10 + 3*b9 - 2*b8 - 3*b7 + 3*b2) * q^68 + (-5*b6 - 7*b5 - 7*b4 + 3*b1 + 4) * q^70 + (2*b11 - 2*b8 - 2*b7) * q^71 + (-3*b6 - 3*b5 + 2*b4 - 3*b3 + 6*b1 - 2) * q^73 + (-2*b11 - b9 + b8 - b7 - b2) * q^74 + (-3*b6 - 3*b5 - 5*b4 - 2*b3 + 2) * q^76 + (2*b11 + 4*b10 - b9 + b8 - 3*b2) * q^77 + (-8*b6 - 5*b5 + 3*b4 + 8*b3 + 2*b1 + 2) * q^79 + (2*b11 - 6*b10 + 3*b9 - 3*b2) * q^80 + (4*b6 - 3*b5 - b3 - b1 - 2) * q^82 + (-4*b11 + b10 + 3*b9 + 4*b8 - 8*b7 + 8*b2) * q^83 + (4*b6 + 3*b5 + 5*b4 - 5*b3 - 3*b1 - 4) * q^85 + (2*b9 + 2*b8 - 2*b7) * q^86 + (-4*b4 + 7*b3 - 4*b1) * q^88 + (-3*b11 - 2*b10 + 10*b9 + b8 - 8*b7 + 5*b2) * q^89 + (-6*b6 - b5 - b3 - 6*b1) * q^91 + (-4*b11 + 4*b10 + 3*b9 - b8 - 2*b7 - 2*b2) * q^92 + (-b6 + b4 + b3 - 2*b1) * q^94 + (2*b11 + 5*b10 - 2*b9 - 5*b8 + 2*b7 + 6*b2) * q^95 + (2*b6 + 4*b5 + 4*b4 - 7*b1 + 3) * q^97 + (2*b11 + b10 - b9 - b8 - b7 - 2*b2) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 6 q^{4} + 6 q^{7}+O(q^{10})$$ 12 * q + 6 * q^4 + 6 * q^7 $$12 q + 6 q^{4} + 6 q^{7} + 12 q^{10} + 6 q^{13} - 18 q^{16} + 24 q^{19} + 6 q^{22} - 48 q^{25} - 12 q^{28} - 12 q^{31} + 54 q^{34} + 6 q^{37} - 12 q^{40} - 12 q^{43} - 6 q^{46} + 42 q^{49} + 54 q^{52} - 60 q^{55} + 6 q^{58} - 48 q^{61} - 6 q^{64} - 66 q^{67} + 6 q^{70} - 12 q^{73} - 6 q^{76} + 42 q^{79} - 24 q^{82} - 18 q^{85} - 24 q^{88} + 6 q^{94} + 60 q^{97}+O(q^{100})$$ 12 * q + 6 * q^4 + 6 * q^7 + 12 * q^10 + 6 * q^13 - 18 * q^16 + 24 * q^19 + 6 * q^22 - 48 * q^25 - 12 * q^28 - 12 * q^31 + 54 * q^34 + 6 * q^37 - 12 * q^40 - 12 * q^43 - 6 * q^46 + 42 * q^49 + 54 * q^52 - 60 * q^55 + 6 * q^58 - 48 * q^61 - 6 * q^64 - 66 * q^67 + 6 * q^70 - 12 * q^73 - 6 * q^76 + 42 * q^79 - 24 * q^82 - 18 * q^85 - 24 * q^88 + 6 * q^94 + 60 * q^97

Basis of coefficient ring

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

## Character values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$\beta_{1} - \beta_{5}$$

## 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
 0.642788 + 0.766044i −0.642788 − 0.766044i −0.984808 − 0.173648i 0.984808 + 0.173648i 0.342020 − 0.939693i −0.342020 + 0.939693i 0.342020 + 0.939693i −0.342020 − 0.939693i −0.984808 + 0.173648i 0.984808 − 0.173648i 0.642788 − 0.766044i −0.642788 + 0.766044i
−0.223238 1.26604i 0 0.326352 0.118782i −0.342020 0.286989i 0 3.31908 + 1.20805i −1.50881 2.61334i 0 −0.286989 + 0.497079i
82.2 0.223238 + 1.26604i 0 0.326352 0.118782i 0.342020 + 0.286989i 0 3.31908 + 1.20805i 1.50881 + 2.61334i 0 −0.286989 + 0.497079i
163.1 −1.85083 + 0.673648i 0 1.43969 1.20805i −0.642788 3.64543i 0 −1.79813 1.50881i 0.118782 0.205737i 0 3.64543 + 6.31407i
163.2 1.85083 0.673648i 0 1.43969 1.20805i 0.642788 + 3.64543i 0 −1.79813 1.50881i −0.118782 + 0.205737i 0 3.64543 + 6.31407i
325.1 −0.524005 + 0.439693i 0 −0.266044 + 1.50881i 0.984808 0.358441i 0 −0.0209445 0.118782i −1.20805 2.09240i 0 −0.358441 + 0.620838i
325.2 0.524005 0.439693i 0 −0.266044 + 1.50881i −0.984808 + 0.358441i 0 −0.0209445 0.118782i 1.20805 + 2.09240i 0 −0.358441 + 0.620838i
406.1 −0.524005 0.439693i 0 −0.266044 1.50881i 0.984808 + 0.358441i 0 −0.0209445 + 0.118782i −1.20805 + 2.09240i 0 −0.358441 0.620838i
406.2 0.524005 + 0.439693i 0 −0.266044 1.50881i −0.984808 0.358441i 0 −0.0209445 + 0.118782i 1.20805 2.09240i 0 −0.358441 0.620838i
568.1 −1.85083 0.673648i 0 1.43969 + 1.20805i −0.642788 + 3.64543i 0 −1.79813 + 1.50881i 0.118782 + 0.205737i 0 3.64543 6.31407i
568.2 1.85083 + 0.673648i 0 1.43969 + 1.20805i 0.642788 3.64543i 0 −1.79813 + 1.50881i −0.118782 0.205737i 0 3.64543 6.31407i
649.1 −0.223238 + 1.26604i 0 0.326352 + 0.118782i −0.342020 + 0.286989i 0 3.31908 1.20805i −1.50881 + 2.61334i 0 −0.286989 0.497079i
649.2 0.223238 1.26604i 0 0.326352 + 0.118782i 0.342020 0.286989i 0 3.31908 1.20805i 1.50881 2.61334i 0 −0.286989 0.497079i
 $$n$$: 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
27.e even 9 1 inner
27.f odd 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 729.2.e.r 12
3.b odd 2 1 inner 729.2.e.r 12
9.c even 3 1 729.2.e.m 12
9.c even 3 1 729.2.e.q 12
9.d odd 6 1 729.2.e.m 12
9.d odd 6 1 729.2.e.q 12
27.e even 9 1 729.2.a.c 6
27.e even 9 2 729.2.c.c 12
27.e even 9 1 729.2.e.m 12
27.e even 9 1 729.2.e.q 12
27.e even 9 1 inner 729.2.e.r 12
27.f odd 18 1 729.2.a.c 6
27.f odd 18 2 729.2.c.c 12
27.f odd 18 1 729.2.e.m 12
27.f odd 18 1 729.2.e.q 12
27.f odd 18 1 inner 729.2.e.r 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
729.2.a.c 6 27.e even 9 1
729.2.a.c 6 27.f odd 18 1
729.2.c.c 12 27.e even 9 2
729.2.c.c 12 27.f odd 18 2
729.2.e.m 12 9.c even 3 1
729.2.e.m 12 9.d odd 6 1
729.2.e.m 12 27.e even 9 1
729.2.e.m 12 27.f odd 18 1
729.2.e.q 12 9.c even 3 1
729.2.e.q 12 9.d odd 6 1
729.2.e.q 12 27.e even 9 1
729.2.e.q 12 27.f odd 18 1
729.2.e.r 12 1.a even 1 1 trivial
729.2.e.r 12 3.b odd 2 1 inner
729.2.e.r 12 27.e even 9 1 inner
729.2.e.r 12 27.f odd 18 1 inner

## Hecke kernels

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

 $$T_{2}^{12} - 3T_{2}^{10} + 30T_{2}^{6} + 36T_{2}^{4} + 9$$ T2^12 - 3*T2^10 + 30*T2^6 + 36*T2^4 + 9 $$T_{5}^{12} + 24T_{5}^{10} + 144T_{5}^{8} - 294T_{5}^{6} + 252T_{5}^{4} - 27T_{5}^{2} + 9$$ T5^12 + 24*T5^10 + 144*T5^8 - 294*T5^6 + 252*T5^4 - 27*T5^2 + 9 $$T_{7}^{6} - 3T_{7}^{5} - 6T_{7}^{4} + 8T_{7}^{3} + 69T_{7}^{2} + 3T_{7} + 1$$ T7^6 - 3*T7^5 - 6*T7^4 + 8*T7^3 + 69*T7^2 + 3*T7 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 3 T^{10} + 30 T^{6} + 36 T^{4} + \cdots + 9$$
$3$ $$T^{12}$$
$5$ $$T^{12} + 24 T^{10} + 144 T^{8} - 294 T^{6} + \cdots + 9$$
$7$ $$(T^{6} - 3 T^{5} - 6 T^{4} + 8 T^{3} + 69 T^{2} + \cdots + 1)^{2}$$
$11$ $$T^{12} - 39 T^{10} + 648 T^{8} + \cdots + 1172889$$
$13$ $$(T^{6} - 3 T^{5} - 6 T^{4} - 28 T^{3} + \cdots + 5041)^{2}$$
$17$ $$T^{12} + 81 T^{10} + 4806 T^{8} + \cdots + 95004009$$
$19$ $$(T^{6} - 12 T^{5} + 105 T^{4} - 430 T^{3} + \cdots + 361)^{2}$$
$23$ $$T^{12} + 33 T^{10} - 2772 T^{8} + \cdots + 751689$$
$29$ $$T^{12} - 3 T^{10} + 468 T^{8} + \cdots + 16867449$$
$31$ $$(T^{6} + 6 T^{5} + 84 T^{4} + 179 T^{3} + \cdots + 5329)^{2}$$
$37$ $$(T^{6} - 3 T^{5} + 15 T^{4} - 16 T^{3} + \cdots + 289)^{2}$$
$41$ $$T^{12} + 15 T^{10} + 2745 T^{8} + \cdots + 71014329$$
$43$ $$(T^{6} + 6 T^{5} + 12 T^{4} - 64 T^{3} + \cdots + 64)^{2}$$
$47$ $$T^{12} - 3 T^{10} - 36 T^{8} + 435 T^{6} + \cdots + 9$$
$53$ $$(T^{6} - 108 T^{4} + 2592 T^{2} + \cdots - 15552)^{2}$$
$59$ $$T^{12} + 330 T^{10} + 30465 T^{8} + \cdots + 9$$
$61$ $$(T^{6} + 24 T^{5} + 336 T^{4} + \cdots + 87616)^{2}$$
$67$ $$(T^{6} + 33 T^{5} + 471 T^{4} + \cdots + 63001)^{2}$$
$71$ $$T^{12} + 72 T^{10} + 3888 T^{8} + \cdots + 2985984$$
$73$ $$(T^{6} + 6 T^{5} + 105 T^{4} - 592 T^{3} + \cdots + 7921)^{2}$$
$79$ $$(T^{6} - 21 T^{5} + 228 T^{4} + \cdots + 11449)^{2}$$
$83$ $$T^{12} + 285 T^{10} + \cdots + 2405238079689$$
$89$ $$T^{12} + 387 T^{10} + \cdots + 60886809$$
$97$ $$(T^{6} - 30 T^{5} + 417 T^{4} + \cdots + 418609)^{2}$$