# Properties

 Label 729.2.c.a Level $729$ Weight $2$ Character orbit 729.c Analytic conductor $5.821$ Analytic rank $0$ Dimension $12$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [729,2,Mod(244,729)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(729, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2]))

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

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

chi := DirichletCharacter("729.244");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$729 = 3^{6}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 729.c (of order $$3$$, degree $$2$$, 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: $$6$$ over $$\Q(\zeta_{3})$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

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

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

 $$\beta_{1}$$ $$=$$ $$( \nu^{8} + 15\nu^{6} + 58\nu^{4} + 63\nu^{2} + 8 ) / 4$$ (v^8 + 15*v^6 + 58*v^4 + 63*v^2 + 8) / 4 $$\beta_{2}$$ $$=$$ $$( \nu^{11} + 16\nu^{9} + \nu^{8} + 70\nu^{7} + 15\nu^{6} + 76\nu^{5} + 58\nu^{4} - 95\nu^{3} + 63\nu^{2} - 149\nu + 8 ) / 8$$ (v^11 + 16*v^9 + v^8 + 70*v^7 + 15*v^6 + 76*v^5 + 58*v^4 - 95*v^3 + 63*v^2 - 149*v + 8) / 8 $$\beta_{3}$$ $$=$$ $$( \nu^{11} - 2 \nu^{10} + 19 \nu^{9} - 34 \nu^{8} + 119 \nu^{7} - 175 \nu^{6} + 310 \nu^{5} - 343 \nu^{4} + \cdots + 3 ) / 8$$ (v^11 - 2*v^10 + 19*v^9 - 34*v^8 + 119*v^7 - 175*v^6 + 310*v^5 - 343*v^4 + 322*v^3 - 222*v^2 + 95*v + 3) / 8 $$\beta_{4}$$ $$=$$ $$( 2\nu^{10} + 33\nu^{8} + 160\nu^{6} + 285\nu^{4} + 163\nu^{2} + 1 ) / 4$$ (2*v^10 + 33*v^8 + 160*v^6 + 285*v^4 + 163*v^2 + 1) / 4 $$\beta_{5}$$ $$=$$ $$( -2\nu^{10} - 34\nu^{8} - 175\nu^{6} - 343\nu^{4} - 222\nu^{2} + 3 ) / 4$$ (-2*v^10 - 34*v^8 - 175*v^6 - 343*v^4 - 222*v^2 + 3) / 4 $$\beta_{6}$$ $$=$$ $$( - \nu^{11} - 5 \nu^{10} - 16 \nu^{9} - 83 \nu^{8} - 73 \nu^{7} - 409 \nu^{6} - 121 \nu^{5} - 760 \nu^{4} + \cdots - 5 ) / 8$$ (-v^11 - 5*v^10 - 16*v^9 - 83*v^8 - 73*v^7 - 409*v^6 - 121*v^5 - 760*v^4 - 79*v^3 - 474*v^2 - 28*v - 5) / 8 $$\beta_{7}$$ $$=$$ $$( - 6 \nu^{10} + 3 \nu^{9} - 99 \nu^{8} + 49 \nu^{7} - 482 \nu^{6} + 234 \nu^{5} - 881 \nu^{4} + \cdots - 17 ) / 8$$ (-6*v^10 + 3*v^9 - 99*v^8 + 49*v^7 - 482*v^6 + 234*v^5 - 881*v^4 + 417*v^3 - 549*v^2 + 256*v - 17) / 8 $$\beta_{8}$$ $$=$$ $$( 3\nu^{11} + 50\nu^{9} + 249\nu^{7} + 477\nu^{5} + 335\nu^{3} + 42\nu - 2 ) / 4$$ (3*v^11 + 50*v^9 + 249*v^7 + 477*v^5 + 335*v^3 + 42*v - 2) / 4 $$\beta_{9}$$ $$=$$ $$( -5\nu^{10} - 83\nu^{8} - 409\nu^{6} - 760\nu^{4} - 474\nu^{2} - 5 ) / 4$$ (-5*v^10 - 83*v^8 - 409*v^6 - 760*v^4 - 474*v^2 - 5) / 4 $$\beta_{10}$$ $$=$$ $$( -6\nu^{10} - 99\nu^{8} - 482\nu^{6} - 881\nu^{4} - 549\nu^{2} - 13 ) / 4$$ (-6*v^10 - 99*v^8 - 482*v^6 - 881*v^4 - 549*v^2 - 13) / 4 $$\beta_{11}$$ $$=$$ $$( 16 \nu^{11} + 2 \nu^{10} + 265 \nu^{9} + 33 \nu^{8} + 1301 \nu^{7} + 160 \nu^{6} + 2416 \nu^{5} + \cdots + 5 ) / 8$$ (16*v^11 + 2*v^10 + 265*v^9 + 33*v^8 + 1301*v^7 + 160*v^6 + 2416*v^5 + 285*v^4 + 1555*v^3 + 163*v^2 + 86*v + 5) / 8
 $$\nu$$ $$=$$ $$( -\beta_{10} + 2\beta_{7} + \beta_{5} - 2\beta_{3} + 2\beta_{2} - \beta _1 + 1 ) / 3$$ (-b10 + 2*b7 + b5 - 2*b3 + 2*b2 - b1 + 1) / 3 $$\nu^{2}$$ $$=$$ $$\beta_{5} + \beta_{4} + \beta _1 - 3$$ b5 + b4 + b1 - 3 $$\nu^{3}$$ $$=$$ $$( 6 \beta_{11} + 4 \beta_{10} + 4 \beta_{9} - 18 \beta_{8} - 8 \beta_{7} - 8 \beta_{6} - 7 \beta_{5} + \cdots - 16 ) / 3$$ (6*b11 + 4*b10 + 4*b9 - 18*b8 - 8*b7 - 8*b6 - 7*b5 - 3*b4 + 14*b3 - 10*b2 + 5*b1 - 16) / 3 $$\nu^{4}$$ $$=$$ $$-3\beta_{10} + 4\beta_{9} - 10\beta_{5} - 9\beta_{4} - 8\beta _1 + 21$$ -3*b10 + 4*b9 - 10*b5 - 9*b4 - 8*b1 + 21 $$\nu^{5}$$ $$=$$ $$( - 76 \beta_{11} - 27 \beta_{10} - 38 \beta_{9} + 222 \beta_{8} + 54 \beta_{7} + 76 \beta_{6} + \cdots + 176 ) / 3$$ (-76*b11 - 27*b10 - 38*b9 + 222*b8 + 54*b7 + 76*b6 + 61*b5 + 38*b4 - 122*b3 + 82*b2 - 41*b1 + 176) / 3 $$\nu^{6}$$ $$=$$ $$37\beta_{10} - 52\beta_{9} + 100\beta_{5} + 81\beta_{4} + 74\beta _1 - 188$$ 37*b10 - 52*b9 + 100*b5 + 81*b4 + 74*b1 - 188 $$\nu^{7}$$ $$=$$ $$( 792 \beta_{11} + 232 \beta_{10} + 342 \beta_{9} - 2286 \beta_{8} - 464 \beta_{7} - 684 \beta_{6} + \cdots - 1771 ) / 3$$ (792*b11 + 232*b10 + 342*b9 - 2286*b8 - 464*b7 - 684*b6 - 568*b5 - 396*b4 + 1136*b3 - 776*b2 + 388*b1 - 1771) / 3 $$\nu^{8}$$ $$=$$ $$-381\beta_{10} + 548\beta_{9} - 983\beta_{5} - 756\beta_{4} - 705\beta _1 + 1783$$ -381*b10 + 548*b9 - 983*b5 - 756*b4 - 705*b1 + 1783 $$\nu^{9}$$ $$=$$ $$( - 7842 \beta_{11} - 2158 \beta_{10} - 3178 \beta_{9} + 22524 \beta_{8} + 4316 \beta_{7} + 6356 \beta_{6} + \cdots + 17341 ) / 3$$ (-7842*b11 - 2158*b10 - 3178*b9 + 22524*b8 + 4316*b7 + 6356*b6 + 5407*b5 + 3921*b4 - 10814*b3 + 7498*b2 - 3749*b1 + 17341) / 3 $$\nu^{10}$$ $$=$$ $$3754\beta_{10} - 5452\beta_{9} + 9563\beta_{5} + 7197\beta_{4} + 6771\beta _1 - 17128$$ 3754*b10 - 5452*b9 + 9563*b5 + 7197*b4 + 6771*b1 - 17128 $$\nu^{11}$$ $$=$$ $$( 76378 \beta_{11} + 20571 \beta_{10} + 30176 \beta_{9} - 218946 \beta_{8} - 41142 \beta_{7} + \cdots - 168233 ) / 3$$ (76378*b11 + 20571*b10 + 30176*b9 - 218946*b8 - 41142*b7 - 60352*b6 - 51904*b5 - 38189*b4 + 103808*b3 - 72508*b2 + 36254*b1 - 168233) / 3

## Character values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$\beta_{8}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
244.1
 0.0878222i − 1.13697i − 1.91182i − 1.37340i 3.10658i 1.22778i − 0.0878222i 1.13697i 1.91182i 1.37340i − 3.10658i − 1.22778i
−1.35253 2.34265i 0 −2.65869 + 4.60498i 0.836192 1.44833i 0 −0.250296 0.433525i 8.97372 0 −4.52391
244.2 −1.06251 1.84033i 0 −1.25787 + 2.17870i −1.03547 + 1.79349i 0 −2.42434 4.19907i 1.09598 0 4.40081
244.3 −0.789202 1.36694i 0 −0.245680 + 0.425530i 0.839254 1.45363i 0 −1.38964 2.40693i −2.38124 0 −2.64936
244.4 0.0864880 + 0.149802i 0 0.985040 1.70614i −1.86828 + 3.23596i 0 −1.51575 2.62535i 0.686728 0 −0.646335
244.5 0.388866 + 0.673536i 0 0.697566 1.20822i 1.18817 2.05798i 0 1.25069 + 2.16626i 2.64050 0 1.84816
244.6 1.22889 + 2.12851i 0 −2.02036 + 3.49937i 1.54013 2.66759i 0 1.32933 + 2.30247i −5.01568 0 7.57064
487.1 −1.35253 + 2.34265i 0 −2.65869 4.60498i 0.836192 + 1.44833i 0 −0.250296 + 0.433525i 8.97372 0 −4.52391
487.2 −1.06251 + 1.84033i 0 −1.25787 2.17870i −1.03547 1.79349i 0 −2.42434 + 4.19907i 1.09598 0 4.40081
487.3 −0.789202 + 1.36694i 0 −0.245680 0.425530i 0.839254 + 1.45363i 0 −1.38964 + 2.40693i −2.38124 0 −2.64936
487.4 0.0864880 0.149802i 0 0.985040 + 1.70614i −1.86828 3.23596i 0 −1.51575 + 2.62535i 0.686728 0 −0.646335
487.5 0.388866 0.673536i 0 0.697566 + 1.20822i 1.18817 + 2.05798i 0 1.25069 2.16626i 2.64050 0 1.84816
487.6 1.22889 2.12851i 0 −2.02036 3.49937i 1.54013 + 2.66759i 0 1.32933 2.30247i −5.01568 0 7.57064
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 244.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 729.2.c.a 12
3.b odd 2 1 729.2.c.d 12
9.c even 3 1 729.2.a.e yes 6
9.c even 3 1 inner 729.2.c.a 12
9.d odd 6 1 729.2.a.b 6
9.d odd 6 1 729.2.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 9.d odd 6 1
729.2.a.e yes 6 9.c even 3 1
729.2.c.a 12 1.a even 1 1 trivial
729.2.c.a 12 9.c even 3 1 inner
729.2.c.d 12 3.b odd 2 1
729.2.c.d 12 9.d odd 6 1
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 $$T_{2}^{12} + 3 T_{2}^{11} + 15 T_{2}^{10} + 24 T_{2}^{9} + 99 T_{2}^{8} + 144 T_{2}^{7} + 381 T_{2}^{6} + \cdots + 9$$ acting on $$S_{2}^{\mathrm{new}}(729, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} + 3 T^{11} + \cdots + 9$$
$3$ $$T^{12}$$
$5$ $$T^{12} - 3 T^{11} + \cdots + 25281$$
$7$ $$T^{12} + 6 T^{11} + \cdots + 18496$$
$11$ $$T^{12} - 6 T^{11} + \cdots + 788544$$
$13$ $$T^{12} + 6 T^{11} + \cdots + 7921$$
$17$ $$(T^{6} + 9 T^{5} + \cdots + 459)^{2}$$
$19$ $$(T^{6} - 12 T^{5} + \cdots - 296)^{2}$$
$23$ $$T^{12} - 12 T^{11} + \cdots + 207936$$
$29$ $$T^{12} + 21 T^{11} + \cdots + 97594641$$
$31$ $$T^{12} + 15 T^{11} + \cdots + 1032256$$
$37$ $$(T^{6} - 3 T^{5} + \cdots - 16109)^{2}$$
$41$ $$T^{12} + \cdots + 534025881$$
$43$ $$T^{12} + \cdots + 300814336$$
$47$ $$T^{12} + \cdots + 314565696$$
$53$ $$(T^{6} + 9 T^{5} + \cdots + 1944)^{2}$$
$59$ $$T^{12} + 6 T^{11} + \cdots + 166464$$
$61$ $$T^{12} + 24 T^{11} + \cdots + 2483776$$
$67$ $$T^{12} + \cdots + 298874944$$
$71$ $$(T^{6} - 180 T^{4} + \cdots + 29376)^{2}$$
$73$ $$(T^{6} - 12 T^{5} + \cdots + 57601)^{2}$$
$79$ $$T^{12} + \cdots + 8681766976$$
$83$ $$T^{12} - 6 T^{11} + \cdots + 18870336$$
$89$ $$(T^{6} + 9 T^{5} + \cdots - 123957)^{2}$$
$97$ $$T^{12} + \cdots + 333099001$$