# Properties

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

# 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: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} + 18x^{10} + 105x^{8} + 266x^{6} + 306x^{4} + 132x^{2} + 1$$ x^12 + 18*x^10 + 105*x^8 + 266*x^6 + 306*x^4 + 132*x^2 + 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{10} - \beta_{7} - \beta_{5}) q^{2} + (\beta_{11} - \beta_{10} - \beta_{8} - \beta_{6} + 2 \beta_{5} + \beta_{4} - 1) q^{4} + (\beta_{11} - \beta_{10} + \beta_{9} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - 1) q^{5} + ( - \beta_{11} + \beta_{8} + 2 \beta_{7} + 3 \beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} + 1) q^{7} + ( - \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} + 2 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} - 3 \beta_1 + 1) q^{8}+O(q^{10})$$ q + (b10 - b7 - b5) * q^2 + (b11 - b10 - b8 - b6 + 2*b5 + b4 - 1) * q^4 + (b11 - b10 + b9 + b6 + b5 - b4 - b3 - 1) * q^5 + (-b11 + b8 + 2*b7 + 3*b6 - b4 - b3 - b2 + 1) * q^7 + (-b11 + b10 + b9 + b8 + 2*b7 + b6 - b5 + b3 - 3*b1 + 1) * q^8 $$q + (\beta_{10} - \beta_{7} - \beta_{5}) q^{2} + (\beta_{11} - \beta_{10} - \beta_{8} - \beta_{6} + 2 \beta_{5} + \beta_{4} - 1) q^{4} + (\beta_{11} - \beta_{10} + \beta_{9} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - 1) q^{5} + ( - \beta_{11} + \beta_{8} + 2 \beta_{7} + 3 \beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} + 1) q^{7} + ( - \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} + 2 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} - 3 \beta_1 + 1) q^{8} + ( - \beta_{10} + \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + \beta_{4} + \beta_{3} - 2 \beta_1) q^{10} + (\beta_{8} - 2 \beta_{7} + \beta_{6} - 2 \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_1 + 2) q^{11} + (\beta_{11} - \beta_{10} + 2 \beta_{9} + \beta_{8} - \beta_{7} - \beta_{5} + \beta_{3} - \beta_{2} - \beta_1) q^{13} + ( - \beta_{11} + \beta_{10} + \beta_{8} - 3 \beta_{7} + 3 \beta_{6} - 5 \beta_{5} - \beta_{3} + \cdots + 3) q^{14}+ \cdots + ( - 4 \beta_{11} - 2 \beta_{10} - 5 \beta_{9} + 2 \beta_{8} + 8 \beta_{7} + 8 \beta_{6} + \cdots - 1) q^{98}+O(q^{100})$$ q + (b10 - b7 - b5) * q^2 + (b11 - b10 - b8 - b6 + 2*b5 + b4 - 1) * q^4 + (b11 - b10 + b9 + b6 + b5 - b4 - b3 - 1) * q^5 + (-b11 + b8 + 2*b7 + 3*b6 - b4 - b3 - b2 + 1) * q^7 + (-b11 + b10 + b9 + b8 + 2*b7 + b6 - b5 + b3 - 3*b1 + 1) * q^8 + (-b10 + b8 + 2*b7 + 2*b6 + b4 + b3 - 2*b1) * q^10 + (b8 - 2*b7 + b6 - 2*b5 + b4 + b3 - 2*b1 + 2) * q^11 + (b11 - b10 + 2*b9 + b8 - b7 - b5 + b3 - b2 - b1) * q^13 + (-b11 + b10 + b8 - 3*b7 + 3*b6 - 5*b5 - b3 - b2 + b1 + 3) * q^14 + (-4*b9 - b8 - b7 - 2*b6 + 2*b5 - b4 - 2*b3 - b2 + 4*b1 - 2) * q^16 + (b11 - b10 + 2*b9 + b8 + b7 + b6 + 2*b5 - b4 - b3 - b2 - 2*b1 + 1) * q^17 + (-2*b11 + b10 + 2*b9 + 2*b8 + 2*b7 + 2*b6 - b5 - b2 - b1 + 2) * q^19 + (-b11 - 4*b9 - b7 - 2*b6 - 4*b5 - b4 - b2 - 2) * q^20 + (b10 - 2*b9 + b7 - 2*b6 - b5 + b4 - 2*b2 - b1 + 2) * q^22 + (b11 + 2*b9 - b8 - b7 - b6 - 2*b5 + b4 + b3 - 2*b1 - 1) * q^23 + (-2*b10 + b9 - b8 + b7 - b6 + b4 - 2*b1 - 1) * q^25 + (-b10 - b8 + 3*b7 - 5*b6 + b4 + b2 + 3*b1 + 4) * q^26 + (-4*b7 + 2*b6 - 2*b5 + 3*b4 - 4*b1 + 2) * q^28 + (-2*b10 - 2*b9 - b8 + 2*b7 - b6 + 3*b5 - b4 - 2*b3 + 2*b2 - 3*b1 - 1) * q^29 + (b10 + 2*b8 - b7 + 2*b6 + b5 + b3 - 2*b1) * q^31 + (-3*b11 + 2*b10 + 4*b9 - 2*b7 + 4*b6 - 2*b5 + 2*b4 + 3*b3 + 2*b2 - 5*b1 + 7) * q^32 + (b9 - 2*b8 + b7 - 2*b6 + b5 + b4 + 2*b3 + b2 + 2) * q^34 + (2*b11 - b10 + 4*b9 - 2*b8 + 2*b7 - 4*b6 + 3*b5 - 2*b3 + b2 + b1 - 2) * q^35 + (-2*b11 + b10 - 2*b9 - b8 - b7 - b6 - 4*b5 + 2*b2 + 5*b1) * q^37 + (2*b11 - 2*b10 - 2*b9 - b8 - 3*b7 - 3*b6 - b4 - 2*b3 - b2 + 3*b1) * q^38 + (-3*b11 + b10 + 3*b9 - 2*b8 - 2*b6 - b5 + 2*b4 - b3 + 2*b2 + 7*b1 + 6) * q^40 + (b11 - 2*b10 + 2*b9 - b8 + b7 + b6 + 3*b5 + b4 + 2*b3 + b2 - b1 + 3) * q^41 + (-2*b11 + 2*b10 + 2*b9 + 2*b8 - 2*b7 - 6*b5 + 2*b4 + 2*b3 + 6) * q^43 + (-b11 + 3*b10 + 4*b9 - 3*b8 - b7 - b6 - b5 + b4 + b3 + b2 + 3*b1 + 5) * q^44 + (2*b11 - 2*b10 - 2*b8 + 2*b7 - 6*b6 + 6*b5 - b3 + 3*b1 - 2) * q^46 + (b11 - 2*b8 + b7 - 6*b6 + 2*b4 + 2*b3 + 2*b2 + 3) * q^47 + (2*b11 - 3*b10 - 2*b9 + 3*b7 - 2*b6 + 3*b5 - 3*b4 - 2*b3 + b2 + 3*b1 - 6) * q^49 + (-b11 + b10 + b9 - b6 - 4*b5 - b4 + b1 + 2) * q^50 + (3*b10 + b8 - 4*b7 + b6 - 6*b5 - 2*b4 - b3 + b2 + b1 - 3) * q^52 + (-2*b11 + b10 - b8 - 2*b7 + b6 - 3*b5 - b2 - 2*b1 - 1) * q^53 + (-b10 - b8 + 2*b7 + b6 + 5*b5 - 4*b4 + b2 + 2*b1) * q^55 + (2*b9 + b8 + b7 + b6 - 2*b5 + 2*b4 + 3*b3 - 3*b2 - 4*b1) * q^56 + (-2*b11 + 2*b10 - 3*b9 - b8 + 5*b6 - 2*b5 - 2*b4 - 4) * q^58 + (b11 - b10 + 4*b9 + 3*b7 + 4*b6 + b5 - b4 - b3 - 3*b2 - 1) * q^59 + (2*b11 + 3*b9 + 2*b6 + 3*b5 + b4 + b2 + 1) * q^61 + (b11 - 3*b10 - 2*b9 - b8 + 4*b7 - 3*b6 + 5*b5 + b3 - 2*b2 - b1 - 1) * q^62 + (4*b11 + b10 - b9 - b8 - 3*b7 - 3*b6 + b5 - 3*b4 - 3*b3 - 4*b2 + 3*b1 - 5) * q^64 + (-2*b11 + 2*b10 - b9 - 2*b8 + 3*b6 - 2*b4 - b3 + b2) * q^65 + (2*b11 - 4*b9 + b8 - 3*b7 + 3*b6 - 2*b5 - 2*b4 - b2 - 5*b1 - 4) * q^67 + (2*b11 - b9 - 2*b8 - 8*b7 - 5*b6 - 4*b5 - 2*b4 + 2*b2 + b1 - 6) * q^68 + (4*b11 - 4*b10 - b8 + 3*b7 - b6 + 10*b5 - b4 + 2*b3 + 3*b2 - 5*b1 - 10) * q^70 + (-2*b10 + 2*b9 + 2*b8 + 2*b5 + 2*b4 + 2*b3 - 6*b1 + 2) * q^71 + (3*b11 + 2*b9 - 3*b8 - 3*b7 - 3*b6 + 3*b3 + 3*b2 - 3*b1 - 3) * q^73 + (b11 - 5*b9 + 5*b8 - 4*b7 + 7*b6 - 5*b5 - b4 - 5*b3 - b2 - 5) * q^74 + (-3*b11 + 2*b10 + 6*b9 - 2*b7 + 6*b6 - 2*b5 + 2*b4 + 3*b3 + 5*b2 - 7*b1 + 9) * q^76 + (-3*b11 - 6*b9 + 4*b8 + 3*b7 + 10*b6 + 2*b5 - 3*b4 - 3*b3 - b1 - 3) * q^77 + (b10 - 2*b9 + 3*b8 + 3*b7 + 3*b6 + 3*b5 - b4 + 2*b3 - 2*b2 - 6*b1 + 2) * q^79 + (4*b11 + 2*b10 + 6*b8 - 2*b7 + 3*b6 - 3*b5 - 3*b4 - 2*b2 - 2*b1 - 4) * q^80 + (b11 + 3*b10 + 4*b8 - b7 - 2*b6 - 9*b5 - 4*b4 - 3*b2 - b1 - 4) * q^82 + (b10 + 2*b9 + b8 - b7 + b6 + b5 + b4 + 2*b3 - 2*b2 + 4) * q^83 + (-2*b11 - 2*b10 + 4*b9 + 4*b7 + b6 - b5 - 2*b4 - 4*b3 - 3*b1 + 1) * q^85 + (4*b11 - 6*b9 - 4*b7 - 6*b6 - 4*b3 - 2*b2 + 2*b1 - 2) * q^86 + (4*b11 + 3*b8 - 3*b7 - 3*b6 - b4 - 3*b3 - b2 - 4) * q^88 + (-4*b11 + 4*b10 - 4*b9 + 4*b8 - b7 + 7*b6 - 7*b5 - b3 - b1 + 4) * q^89 + (b10 - 4*b9 - b8 + 3*b7 + 3*b6 + 7*b5 - b4 - b3 - 5*b1 - 4) * q^91 + (-4*b11 + 4*b10 + 2*b9 + b8 - b6 - 2*b5 + b4 + 5*b3 + 4*b2 + 2) * q^92 + (2*b11 + 3*b10 - 4*b9 - 2*b8 - b7 - 6*b6 + 3*b5 - 5*b4 - 3*b3 + 2*b2 + 4*b1 - 10) * q^94 + (-3*b11 + 2*b10 + 2*b9 + 7*b7 - 2*b6 + 4*b5 + b4 - 2*b3 + 6*b1 + 3) * q^95 + (b11 - b10 + 2*b9 + 2*b8 - 2*b7 + 3*b6 - 3*b5 + 2*b4 + 2*b3 - 6*b1 + 3) * q^97 + (-4*b11 - 2*b10 - 5*b9 + 2*b8 + 8*b7 + 8*b6 - 2*b5 - 2*b4 - 2*b3 + 4*b2 + 2*b1 - 1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 3 q^{2} - 3 q^{4} - 6 q^{5} + 6 q^{7} - 6 q^{8}+O(q^{10})$$ 12 * q - 3 * q^2 - 3 * q^4 - 6 * q^5 + 6 * q^7 - 6 * q^8 $$12 q - 3 q^{2} - 3 q^{4} - 6 q^{5} + 6 q^{7} - 6 q^{8} - 6 q^{10} + 15 q^{11} - 3 q^{13} + 21 q^{14} + 9 q^{16} + 9 q^{17} - 12 q^{19} + 3 q^{20} + 33 q^{22} - 15 q^{23} - 12 q^{25} + 48 q^{26} + 6 q^{28} + 6 q^{29} - 12 q^{31} + 27 q^{32} + 27 q^{34} - 30 q^{35} - 3 q^{37} + 39 q^{38} + 24 q^{40} + 39 q^{41} + 24 q^{43} + 33 q^{44} + 3 q^{46} + 42 q^{47} - 30 q^{49} + 15 q^{50} - 45 q^{52} - 18 q^{53} + 30 q^{55} - 12 q^{56} - 30 q^{58} - 15 q^{59} - 3 q^{61} + 30 q^{62} - 6 q^{64} + 6 q^{65} - 3 q^{67} - 36 q^{68} - 75 q^{70} - 12 q^{73} - 60 q^{74} + 30 q^{76} - 33 q^{77} + 33 q^{79} - 42 q^{80} - 42 q^{82} + 33 q^{83} - 18 q^{85} + 30 q^{86} - 42 q^{88} + 9 q^{89} - 18 q^{91} - 33 q^{92} - 66 q^{94} - 12 q^{95} + 15 q^{97} - 18 q^{98}+O(q^{100})$$ 12 * q - 3 * q^2 - 3 * q^4 - 6 * q^5 + 6 * q^7 - 6 * q^8 - 6 * q^10 + 15 * q^11 - 3 * q^13 + 21 * q^14 + 9 * q^16 + 9 * q^17 - 12 * q^19 + 3 * q^20 + 33 * q^22 - 15 * q^23 - 12 * q^25 + 48 * q^26 + 6 * q^28 + 6 * q^29 - 12 * q^31 + 27 * q^32 + 27 * q^34 - 30 * q^35 - 3 * q^37 + 39 * q^38 + 24 * q^40 + 39 * q^41 + 24 * q^43 + 33 * q^44 + 3 * q^46 + 42 * q^47 - 30 * q^49 + 15 * q^50 - 45 * q^52 - 18 * q^53 + 30 * q^55 - 12 * q^56 - 30 * q^58 - 15 * q^59 - 3 * q^61 + 30 * q^62 - 6 * q^64 + 6 * q^65 - 3 * q^67 - 36 * q^68 - 75 * q^70 - 12 * q^73 - 60 * q^74 + 30 * q^76 - 33 * q^77 + 33 * q^79 - 42 * q^80 - 42 * q^82 + 33 * q^83 - 18 * q^85 + 30 * q^86 - 42 * q^88 + 9 * q^89 - 18 * q^91 - 33 * q^92 - 66 * q^94 - 12 * q^95 + 15 * q^97 - 18 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 18x^{10} + 105x^{8} + 266x^{6} + 306x^{4} + 132x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{11} - 16\nu^{9} + \nu^{8} - 72\nu^{7} + 15\nu^{6} - 106\nu^{5} + 58\nu^{4} - 21\nu^{3} + 63\nu^{2} + 31\nu + 8 ) / 8$$ (-v^11 - 16*v^9 + v^8 - 72*v^7 + 15*v^6 - 106*v^5 + 58*v^4 - 21*v^3 + 63*v^2 + 31*v + 8) / 8 $$\beta_{2}$$ $$=$$ $$( 2 \nu^{10} + \nu^{9} + 33 \nu^{8} + 15 \nu^{7} + 160 \nu^{6} + 58 \nu^{5} + 285 \nu^{4} + 63 \nu^{3} + 163 \nu^{2} + 16 \nu + 1 ) / 8$$ (2*v^10 + v^9 + 33*v^8 + 15*v^7 + 160*v^6 + 58*v^5 + 285*v^4 + 63*v^3 + 163*v^2 + 16*v + 1) / 8 $$\beta_{3}$$ $$=$$ $$( - 2 \nu^{10} + 3 \nu^{9} - 33 \nu^{8} + 47 \nu^{7} - 160 \nu^{6} + 204 \nu^{5} - 285 \nu^{4} + 301 \nu^{3} - 159 \nu^{2} + 126 \nu + 11 ) / 8$$ (-2*v^10 + 3*v^9 - 33*v^8 + 47*v^7 - 160*v^6 + 204*v^5 - 285*v^4 + 301*v^3 - 159*v^2 + 126*v + 11) / 8 $$\beta_{4}$$ $$=$$ $$( 2\nu^{10} + 34\nu^{8} + 175\nu^{6} + 343\nu^{4} + 222\nu^{2} - 3 ) / 4$$ (2*v^10 + 34*v^8 + 175*v^6 + 343*v^4 + 222*v^2 - 3) / 4 $$\beta_{5}$$ $$=$$ $$( -5\nu^{10} - 82\nu^{8} + \nu^{7} - 394\nu^{6} + 15\nu^{5} - 702\nu^{4} + 58\nu^{3} - 411\nu^{2} + 59\nu + 3 ) / 8$$ (-5*v^10 - 82*v^8 + v^7 - 394*v^6 + 15*v^5 - 702*v^4 + 58*v^3 - 411*v^2 + 59*v + 3) / 8 $$\beta_{6}$$ $$=$$ $$( -5\nu^{10} - 82\nu^{8} - \nu^{7} - 394\nu^{6} - 15\nu^{5} - 702\nu^{4} - 58\nu^{3} - 411\nu^{2} - 59\nu + 3 ) / 8$$ (-5*v^10 - 82*v^8 - v^7 - 394*v^6 - 15*v^5 - 702*v^4 - 58*v^3 - 411*v^2 - 59*v + 3) / 8 $$\beta_{7}$$ $$=$$ $$( \nu^{11} + 5 \nu^{10} + 16 \nu^{9} + 83 \nu^{8} + 71 \nu^{7} + 409 \nu^{6} + 91 \nu^{5} + 760 \nu^{4} - 37 \nu^{3} + 474 \nu^{2} - 90 \nu + 5 ) / 8$$ (v^11 + 5*v^10 + 16*v^9 + 83*v^8 + 71*v^7 + 409*v^6 + 91*v^5 + 760*v^4 - 37*v^3 + 474*v^2 - 90*v + 5) / 8 $$\beta_{8}$$ $$=$$ $$( - 3 \nu^{11} + 4 \nu^{10} - 49 \nu^{9} + 66 \nu^{8} - 232 \nu^{7} + 321 \nu^{6} - 389 \nu^{5} + 583 \nu^{4} - 158 \nu^{3} + 356 \nu^{2} + 78 \nu + 7 ) / 4$$ (-3*v^11 + 4*v^10 - 49*v^9 + 66*v^8 - 232*v^7 + 321*v^6 - 389*v^5 + 583*v^4 - 158*v^3 + 356*v^2 + 78*v + 7) / 4 $$\beta_{9}$$ $$=$$ $$( -3\nu^{11} - 50\nu^{9} - 249\nu^{7} - 477\nu^{5} - 335\nu^{3} - 42\nu - 2 ) / 4$$ (-3*v^11 - 50*v^9 - 249*v^7 - 477*v^5 - 335*v^3 - 42*v - 2) / 4 $$\beta_{10}$$ $$=$$ $$( 6 \nu^{11} + 6 \nu^{10} + 99 \nu^{9} + 99 \nu^{8} + 481 \nu^{7} + 482 \nu^{6} + 866 \nu^{5} + 881 \nu^{4} + 495 \nu^{3} + 549 \nu^{2} - 22 \nu + 13 ) / 8$$ (6*v^11 + 6*v^10 + 99*v^9 + 99*v^8 + 481*v^7 + 482*v^6 + 866*v^5 + 881*v^4 + 495*v^3 + 549*v^2 - 22*v + 13) / 8 $$\beta_{11}$$ $$=$$ $$( 3 \nu^{11} + 4 \nu^{10} + 49 \nu^{9} + 66 \nu^{8} + 232 \nu^{7} + 321 \nu^{6} + 389 \nu^{5} + 583 \nu^{4} + 158 \nu^{3} + 356 \nu^{2} - 78 \nu + 11 ) / 4$$ (3*v^11 + 4*v^10 + 49*v^9 + 66*v^8 + 232*v^7 + 321*v^6 + 389*v^5 + 583*v^4 + 158*v^3 + 356*v^2 - 78*v + 11) / 4
 $$\nu$$ $$=$$ $$( - 2 \beta_{11} + \beta_{10} - 2 \beta_{9} + \beta_{8} + \beta_{7} + 3 \beta_{6} - 2 \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta _1 + 1 ) / 3$$ (-2*b11 + b10 - 2*b9 + b8 + b7 + 3*b6 - 2*b5 - b4 - 2*b3 + b2 + b1 + 1) / 3 $$\nu^{2}$$ $$=$$ $$\beta_{11} - \beta_{10} + \beta_{7} - \beta_{6} + 2\beta_{5} - \beta_{4} + \beta_{2} + \beta _1 - 4$$ b11 - b10 + b7 - b6 + 2*b5 - b4 + b2 + b1 - 4 $$\nu^{3}$$ $$=$$ $$( 11 \beta_{11} - \beta_{10} + 20 \beta_{9} - 10 \beta_{8} - 7 \beta_{7} - 27 \beta_{6} + 20 \beta_{5} + 7 \beta_{4} + 14 \beta_{3} - \beta_{2} - 7 \beta _1 - 1 ) / 3$$ (11*b11 - b10 + 20*b9 - 10*b8 - 7*b7 - 27*b6 + 20*b5 + 7*b4 + 14*b3 - b2 - 7*b1 - 1) / 3 $$\nu^{4}$$ $$=$$ $$- 9 \beta_{11} + 12 \beta_{10} + 3 \beta_{8} - 12 \beta_{7} + 16 \beta_{6} - 20 \beta_{5} + 10 \beta_{4} - 12 \beta_{2} - 12 \beta _1 + 30$$ -9*b11 + 12*b10 + 3*b8 - 12*b7 + 16*b6 - 20*b5 + 10*b4 - 12*b2 - 12*b1 + 30 $$\nu^{5}$$ $$=$$ $$( - 92 \beta_{11} - 11 \beta_{10} - 200 \beta_{9} + 103 \beta_{8} + 79 \beta_{7} + 261 \beta_{6} - 182 \beta_{5} - 61 \beta_{4} - 122 \beta_{3} - 11 \beta_{2} + 43 \beta _1 - 8 ) / 3$$ (-92*b11 - 11*b10 - 200*b9 + 103*b8 + 79*b7 + 261*b6 - 182*b5 - 61*b4 - 122*b3 - 11*b2 + 43*b1 - 8) / 3 $$\nu^{6}$$ $$=$$ $$81 \beta_{11} - 118 \beta_{10} - 37 \beta_{8} + 126 \beta_{7} - 170 \beta_{6} + 192 \beta_{5} - 100 \beta_{4} + 118 \beta_{2} + 126 \beta _1 - 269$$ 81*b11 - 118*b10 - 37*b8 + 126*b7 - 170*b6 + 192*b5 - 100*b4 + 118*b2 + 126*b1 - 269 $$\nu^{7}$$ $$=$$ $$( 860 \beta_{11} + 164 \beta_{10} + 1958 \beta_{9} - 1024 \beta_{8} - 838 \beta_{7} - 2538 \beta_{6} + 1700 \beta_{5} + 568 \beta_{4} + 1136 \beta_{3} + 164 \beta_{2} - 298 \beta _1 + 119 ) / 3$$ (860*b11 + 164*b10 + 1958*b9 - 1024*b8 - 838*b7 - 2538*b6 + 1700*b5 + 568*b4 + 1136*b3 + 164*b2 - 298*b1 + 119) / 3 $$\nu^{8}$$ $$=$$ $$- 756 \beta_{11} + 1137 \beta_{10} + 381 \beta_{8} - 1253 \beta_{7} + 1685 \beta_{6} - 1842 \beta_{5} + 983 \beta_{4} - 1137 \beta_{2} - 1253 \beta _1 + 2539$$ -756*b11 + 1137*b10 + 381*b8 - 1253*b7 + 1685*b6 - 1842*b5 + 983*b4 - 1137*b2 - 1253*b1 + 2539 $$\nu^{9}$$ $$=$$ $$( - 8237 \beta_{11} - 1763 \beta_{10} - 18998 \beta_{9} + 10000 \beta_{8} + 8413 \beta_{7} + 24597 \beta_{6} - 16184 \beta_{5} - 5407 \beta_{4} - 10814 \beta_{3} - 1763 \beta_{2} + 2401 \beta _1 - 1262 ) / 3$$ (-8237*b11 - 1763*b10 - 18998*b9 + 10000*b8 + 8413*b7 + 24597*b6 - 16184*b5 - 5407*b4 - 10814*b3 - 1763*b2 + 2401*b1 - 1262) / 3 $$\nu^{10}$$ $$=$$ $$7197 \beta_{11} - 10951 \beta_{10} - 3754 \beta_{8} + 12223 \beta_{7} - 16403 \beta_{6} + 17722 \beta_{5} - 9563 \beta_{4} + 10951 \beta_{2} + 12223 \beta _1 - 24325$$ 7197*b11 - 10951*b10 - 3754*b8 + 12223*b7 - 16403*b6 + 17722*b5 - 9563*b4 + 10951*b2 + 12223*b1 - 24325 $$\nu^{11}$$ $$=$$ $$( 79331 \beta_{11} + 17618 \beta_{10} + 183710 \beta_{9} - 96949 \beta_{8} - 82456 \beta_{7} - 237822 \beta_{6} + 155366 \beta_{5} + 51904 \beta_{4} + 103808 \beta_{3} + 17618 \beta_{2} + \cdots + 12524 ) / 3$$ (79331*b11 + 17618*b10 + 183710*b9 - 96949*b8 - 82456*b7 - 237822*b6 + 155366*b5 + 51904*b4 + 103808*b3 + 17618*b2 - 21352*b1 + 12524) / 3

## Character values

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

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

## 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
 − 1.91182i 1.22778i 1.37340i − 0.0878222i − 1.13697i 3.10658i 1.13697i − 3.10658i − 1.37340i 0.0878222i 1.91182i − 1.22778i
−0.426791 2.42045i 0 −3.79704 + 1.38201i −2.35962 1.97995i 0 2.49833 + 0.909318i 2.50784 + 4.34371i 0 −3.78532 + 6.55636i
82.2 0.274087 + 1.55442i 0 −0.461727 + 0.168055i −1.28581 1.07892i 0 −2.61167 0.950570i 1.19062 + 2.06222i 0 1.32468 2.29442i
163.1 −2.54193 + 0.925187i 0 4.07335 3.41794i −0.290407 1.64698i 0 0.383475 + 0.321774i −4.48686 + 7.77147i 0 2.26195 + 3.91782i
163.2 0.162544 0.0591613i 0 −1.50917 + 1.26634i 0.648847 + 3.67980i 0 2.32226 + 1.94861i −0.343364 + 0.594724i 0 0.323168 + 0.559743i
325.1 −0.595778 + 0.499917i 0 −0.242262 + 1.37394i 2.23304 0.812759i 0 −0.434359 2.46337i −1.32025 2.28674i 0 −0.924081 + 1.60056i
325.2 1.62787 1.36594i 0 0.436855 2.47753i −1.94605 + 0.708303i 0 0.841963 + 4.77501i −0.547989 0.949144i 0 −2.20040 + 3.81121i
406.1 −0.595778 0.499917i 0 −0.242262 1.37394i 2.23304 + 0.812759i 0 −0.434359 + 2.46337i −1.32025 + 2.28674i 0 −0.924081 1.60056i
406.2 1.62787 + 1.36594i 0 0.436855 + 2.47753i −1.94605 0.708303i 0 0.841963 4.77501i −0.547989 + 0.949144i 0 −2.20040 3.81121i
568.1 −2.54193 0.925187i 0 4.07335 + 3.41794i −0.290407 + 1.64698i 0 0.383475 0.321774i −4.48686 7.77147i 0 2.26195 3.91782i
568.2 0.162544 + 0.0591613i 0 −1.50917 1.26634i 0.648847 3.67980i 0 2.32226 1.94861i −0.343364 0.594724i 0 0.323168 0.559743i
649.1 −0.426791 + 2.42045i 0 −3.79704 1.38201i −2.35962 + 1.97995i 0 2.49833 0.909318i 2.50784 4.34371i 0 −3.78532 6.55636i
649.2 0.274087 1.55442i 0 −0.461727 0.168055i −1.28581 + 1.07892i 0 −2.61167 + 0.950570i 1.19062 2.06222i 0 1.32468 + 2.29442i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 649.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{2}^{12} + 3 T_{2}^{11} + 6 T_{2}^{10} + 15 T_{2}^{9} + 18 T_{2}^{8} + 45 T_{2}^{7} + 228 T_{2}^{6} + 126 T_{2}^{5} + 495 T_{2}^{4} + 387 T_{2}^{3} + 135 T_{2}^{2} - 81 T_{2} + 9$$ T2^12 + 3*T2^11 + 6*T2^10 + 15*T2^9 + 18*T2^8 + 45*T2^7 + 228*T2^6 + 126*T2^5 + 495*T2^4 + 387*T2^3 + 135*T2^2 - 81*T2 + 9 $$T_{5}^{12} + 6 T_{5}^{11} + 24 T_{5}^{10} + 75 T_{5}^{9} + 162 T_{5}^{8} - 81 T_{5}^{7} - 1203 T_{5}^{6} - 1854 T_{5}^{5} + 2547 T_{5}^{4} + 15624 T_{5}^{3} + 34290 T_{5}^{2} + 41499 T_{5} + 25281$$ T5^12 + 6*T5^11 + 24*T5^10 + 75*T5^9 + 162*T5^8 - 81*T5^7 - 1203*T5^6 - 1854*T5^5 + 2547*T5^4 + 15624*T5^3 + 34290*T5^2 + 41499*T5 + 25281 $$T_{7}^{12} - 6 T_{7}^{11} + 33 T_{7}^{10} - 83 T_{7}^{9} - 18 T_{7}^{8} + 621 T_{7}^{7} - 861 T_{7}^{6} - 198 T_{7}^{5} + 5832 T_{7}^{4} - 38936 T_{7}^{3} + 101712 T_{7}^{2} - 65280 T_{7} + 18496$$ T7^12 - 6*T7^11 + 33*T7^10 - 83*T7^9 - 18*T7^8 + 621*T7^7 - 861*T7^6 - 198*T7^5 + 5832*T7^4 - 38936*T7^3 + 101712*T7^2 - 65280*T7 + 18496

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} + 3 T^{11} + 6 T^{10} + 15 T^{9} + \cdots + 9$$
$3$ $$T^{12}$$
$5$ $$T^{12} + 6 T^{11} + 24 T^{10} + \cdots + 25281$$
$7$ $$T^{12} - 6 T^{11} + 33 T^{10} + \cdots + 18496$$
$11$ $$T^{12} - 15 T^{11} + 96 T^{10} + \cdots + 788544$$
$13$ $$T^{12} + 3 T^{11} + 51 T^{10} + \cdots + 7921$$
$17$ $$T^{12} - 9 T^{11} + 81 T^{10} + \cdots + 210681$$
$19$ $$T^{12} + 12 T^{11} + 129 T^{10} + \cdots + 87616$$
$23$ $$T^{12} + 15 T^{11} + 132 T^{10} + \cdots + 207936$$
$29$ $$T^{12} - 6 T^{11} + 33 T^{10} + \cdots + 97594641$$
$31$ $$T^{12} + 12 T^{11} + 24 T^{10} + \cdots + 1032256$$
$37$ $$T^{12} + 3 T^{11} + 138 T^{10} + \cdots + 259499881$$
$41$ $$T^{12} - 39 T^{11} + \cdots + 534025881$$
$43$ $$T^{12} - 24 T^{11} + \cdots + 300814336$$
$47$ $$T^{12} - 42 T^{11} + \cdots + 314565696$$
$53$ $$(T^{6} + 9 T^{5} - 81 T^{4} - 729 T^{3} + \cdots + 1944)^{2}$$
$59$ $$T^{12} + 15 T^{11} + 222 T^{10} + \cdots + 166464$$
$61$ $$T^{12} + 3 T^{11} - 156 T^{10} + \cdots + 2483776$$
$67$ $$T^{12} + 3 T^{11} - 84 T^{10} + \cdots + 298874944$$
$71$ $$T^{12} + 180 T^{10} + \cdots + 862949376$$
$73$ $$T^{12} + 12 T^{11} + \cdots + 3317875201$$
$79$ $$T^{12} - 33 T^{11} + \cdots + 8681766976$$
$83$ $$T^{12} - 33 T^{11} + 510 T^{10} + \cdots + 18870336$$
$89$ $$T^{12} - 9 T^{11} + \cdots + 15365337849$$
$97$ $$T^{12} - 15 T^{11} + \cdots + 333099001$$