Properties

 Label 729.2.c.e 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: 12.0.1952986685049.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 6 x^{11} + 27 x^{10} - 80 x^{9} + 186 x^{8} - 330 x^{7} + 463 x^{6} - 504 x^{5} + 420 x^{4} + \cdots + 3$$ x^12 - 6*x^11 + 27*x^10 - 80*x^9 + 186*x^8 - 330*x^7 + 463*x^6 - 504*x^5 + 420*x^4 - 258*x^3 + 108*x^2 - 27*x + 3 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3^{3}$$ Twist minimal: no (minimal twist has level 27) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{10} - \beta_{8} - \beta_{6}) q^{2} + (\beta_{11} + \beta_{10} - \beta_{6} + \cdots - 1) q^{4}+ \cdots + (\beta_{5} + \beta_{4} + 2 \beta_{2} - 1) q^{8}+O(q^{10})$$ q + (b10 - b8 - b6) * q^2 + (b11 + b10 - b6 + b4 - b3 + b2 - 1) * q^4 + (b9 - b7 - b5 - b1 + 1) * q^5 + (b11 - b10 + b8 - b7) * q^7 + (b5 + b4 + 2*b2 - 1) * q^8 $$q + (\beta_{10} - \beta_{8} - \beta_{6}) q^{2} + (\beta_{11} + \beta_{10} - \beta_{6} + \cdots - 1) q^{4}+ \cdots + ( - 4 \beta_{4} + 3 \beta_{3} + \cdots + 9) q^{98}+O(q^{100})$$ q + (b10 - b8 - b6) * q^2 + (b11 + b10 - b6 + b4 - b3 + b2 - 1) * q^4 + (b9 - b7 - b5 - b1 + 1) * q^5 + (b11 - b10 + b8 - b7) * q^7 + (b5 + b4 + 2*b2 - 1) * q^8 + (b5 + b3 + b2 - b1 + 1) * q^10 + (-2*b9 - b7) * q^11 + (b9 - b7 - 2*b6 - b5 - 2*b3 - b1 - 1) * q^13 + (-b11 + 2*b10 + b9 - b8 - b5 - b4 + b2 + 1) * q^14 + (b11 - b10 - b9 + b8 + b7 + b6) * q^16 + (-b4 + b3 + b2 - 1) * q^17 + (-b4 - b3 + b2 - b1) * q^19 + (b11 + b10 - b9 - 2*b8) * q^20 + (2*b10 - b8 - b7 - 2*b6 - 2*b5 - 2*b3 + b2 - b1 - 2) * q^22 + (2*b10 + 2*b9 - 2*b8 + b7 + b6 - b5 + b3 + b1 + 3) * q^23 + (b11 - b9 + 2*b8 - b7) * q^25 + (b5 - b3 + 3*b2 - b1 - 3) * q^26 + (2*b5 + b2 - 1) * q^28 + (-2*b10 - 3*b9 - b8 + 2*b6) * q^29 + (-b11 - 2*b10 + 2*b8 - 2*b7 - b4 - 2*b1) * q^31 + (b11 + 4*b10 - 2*b8 + 2*b7 + b4 + 2*b2 + 2*b1) * q^32 + (-b11 + b10 - b9 - b8 + 2*b7 + 2*b6) * q^34 + (-b5 - 2*b4 - b2 + b1 - 2) * q^35 + (-b4 - b3 - 2*b2 + 2*b1) * q^37 + (2*b10 - 5*b8 + 3*b7 + b6) * q^38 + (b11 + b10 - 2*b9 + 2*b6 + b4 + 2*b3 + b2) * q^40 + (3*b11 + b10 + 3*b9 - b8 - b6 - b5 + 3*b4 - b3 + 2) * q^41 + (-b11 + b10 + b9 - b8 - 3*b7 - 2*b6) * q^43 + (2*b5 + 2*b4 - 3*b3 + 4*b2 - 2) * q^44 + (2*b5 + 2*b4 + 3*b3 - 2*b2 + b1 + 2) * q^46 + (-3*b11 - b10 - 4*b9 - 2*b8 + b7 + b6) * q^47 + (b11 - 4*b10 + b9 + 4*b8 + 2*b6 + 4*b5 + b4 + 2*b3 + 3) * q^49 + (3*b10 + 2*b9 - 2*b7 - 3*b6 - 2*b5 - 3*b3 + 3*b2 - 2*b1 - 1) * q^50 + (-b11 - 3*b10 + b9 - 2*b8 + 2*b7 + 2*b6) * q^52 + (3*b3 + 3*b1) * q^53 + (-4*b5 - b4 - 2*b2 - 2*b1 - 1) * q^55 + (-b11 + 3*b10 - 2*b8 + b7 + 2*b6) * q^56 + (-2*b11 + b10 + b9 - 6*b8 + 3*b7 + 2*b6 - 3*b5 - 2*b4 + 2*b3 - 5*b2 + 3*b1 + 3) * q^58 + (-3*b11 + 4*b9 - 3*b8 + 2*b7 - b5 - 3*b4 - 3*b2 + 2*b1 + 4) * q^59 + (-b11 + 2*b10 + b9 - 5*b6) * q^61 + (b5 - 2*b4 + 2*b2 - 3*b1 + 2) * q^62 + (-5*b5 + 2*b4 - 4*b3 - b1 - 4) * q^64 + (-b11 - 2*b10 + 2*b9 - b7 - 2*b6) * q^65 + (-b11 + b8 + 4*b7 + 3*b6 + 5*b5 - b4 + 3*b3 + b2 + 4*b1 + 3) * q^67 + (-b11 - b10 - 3*b9 - b8 + b7 + 3*b6 - b4 + 3*b3 - 2*b2 + b1) * q^68 + (-b11 + 2*b10 + 2*b9 + b6) * q^70 + (2*b4 + b3 - 2*b2 - 4) * q^71 + (-3*b5 + 3*b1 - 1) * q^73 + (2*b10 + 3*b9 + b8 - 3*b7 - 2*b6) * q^74 + (-b10 - 2*b9 - 4*b8 + 4*b7 + 2*b6 + 2*b3 - 5*b2 + 4*b1) * q^76 + (b11 - 2*b10 - 2*b9 + b8 - 2*b7 - b5 + b4 - b2 - 2*b1 - 2) * q^77 + (6*b10 + 2*b9 - 6*b8 + b7 - 4*b6) * q^79 + (b5 - b4 - b3 + b2 + 3) * q^80 + (b4 + 2*b3 + b2 + 3*b1) * q^82 + (3*b11 - 2*b10 + 5*b8 - 3*b7 - 4*b6) * q^83 + (-b11 + b10 - b9 + 2*b8 + 2*b7 + 2*b6 + 4*b5 - b4 + 2*b3 + 3*b2 + 2*b1 + 1) * q^85 + (b11 - b10 - 3*b9 + 6*b8 - 4*b7 - b6 + 2*b5 + b4 - b3 + 5*b2 - 4*b1 - 4) * q^86 + (-2*b11 - 2*b10 + b9 - 3*b8 + 2*b6) * q^88 + (-3*b5 + 4*b4 - b3 - b2 - 2) * q^89 + (b5 + 2*b3 + 3*b2 + b1) * q^91 + (3*b11 + 2*b10 + 4*b8 - 3*b7 - 2*b6) * q^92 + (-b11 - 3*b10 - b9 - 2*b8 + b7 - b5 - b4 - 5*b2 + b1 - 1) * q^94 + (-3*b11 - 3*b10 + 2*b9 + 3*b8 - 2*b7 + b5 - 3*b4 - 2*b1 + 2) * q^95 + (-b11 - 2*b10 + b9 - b8 + 3*b7 - 2*b6) * q^97 + (-4*b4 + 3*b3 - 2*b2 + b1 + 9) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 3 q^{2} - 3 q^{4} + 6 q^{5} - 12 q^{8}+O(q^{10})$$ 12 * q + 3 * q^2 - 3 * q^4 + 6 * q^5 - 12 * q^8 $$12 q + 3 q^{2} - 3 q^{4} + 6 q^{5} - 12 q^{8} + 6 q^{10} + 12 q^{11} + 6 q^{14} + 3 q^{16} - 18 q^{17} + 6 q^{19} + 6 q^{20} - 6 q^{22} + 15 q^{23} + 6 q^{25} - 30 q^{26} - 12 q^{28} + 12 q^{29} - 24 q^{35} + 6 q^{37} - 3 q^{38} - 6 q^{40} + 15 q^{41} - 6 q^{44} + 6 q^{46} + 21 q^{47} + 12 q^{49} + 3 q^{50} - 12 q^{52} - 18 q^{53} - 12 q^{55} - 6 q^{56} + 12 q^{58} + 24 q^{59} + 9 q^{61} + 24 q^{62} - 24 q^{64} - 6 q^{65} + 9 q^{67} - 9 q^{68} - 15 q^{70} - 54 q^{71} - 12 q^{73} - 12 q^{74} - 6 q^{76} - 12 q^{77} + 42 q^{80} - 12 q^{82} + 12 q^{83} - 21 q^{86} - 12 q^{88} - 18 q^{89} - 12 q^{91} + 6 q^{92} - 6 q^{94} + 12 q^{95} + 90 q^{98}+O(q^{100})$$ 12 * q + 3 * q^2 - 3 * q^4 + 6 * q^5 - 12 * q^8 + 6 * q^10 + 12 * q^11 + 6 * q^14 + 3 * q^16 - 18 * q^17 + 6 * q^19 + 6 * q^20 - 6 * q^22 + 15 * q^23 + 6 * q^25 - 30 * q^26 - 12 * q^28 + 12 * q^29 - 24 * q^35 + 6 * q^37 - 3 * q^38 - 6 * q^40 + 15 * q^41 - 6 * q^44 + 6 * q^46 + 21 * q^47 + 12 * q^49 + 3 * q^50 - 12 * q^52 - 18 * q^53 - 12 * q^55 - 6 * q^56 + 12 * q^58 + 24 * q^59 + 9 * q^61 + 24 * q^62 - 24 * q^64 - 6 * q^65 + 9 * q^67 - 9 * q^68 - 15 * q^70 - 54 * q^71 - 12 * q^73 - 12 * q^74 - 6 * q^76 - 12 * q^77 + 42 * q^80 - 12 * q^82 + 12 * q^83 - 21 * q^86 - 12 * q^88 - 18 * q^89 - 12 * q^91 + 6 * q^92 - 6 * q^94 + 12 * q^95 + 90 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 6 x^{11} + 27 x^{10} - 80 x^{9} + 186 x^{8} - 330 x^{7} + 463 x^{6} - 504 x^{5} + 420 x^{4} + \cdots + 3$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} - \nu + 2$$ v^2 - v + 2 $$\beta_{2}$$ $$=$$ $$\nu^{8} - 4\nu^{7} + 16\nu^{6} - 34\nu^{5} + 62\nu^{4} - 72\nu^{3} + 64\nu^{2} - 33\nu + 8$$ v^8 - 4*v^7 + 16*v^6 - 34*v^5 + 62*v^4 - 72*v^3 + 64*v^2 - 33*v + 8 $$\beta_{3}$$ $$=$$ $$\nu^{10} - 5 \nu^{9} + 22 \nu^{8} - 58 \nu^{7} + 127 \nu^{6} - 199 \nu^{5} + 249 \nu^{4} - 224 \nu^{3} + \cdots + 9$$ v^10 - 5*v^9 + 22*v^8 - 58*v^7 + 127*v^6 - 199*v^5 + 249*v^4 - 224*v^3 + 145*v^2 - 58*v + 9 $$\beta_{4}$$ $$=$$ $$- 2 \nu^{10} + 10 \nu^{9} - 43 \nu^{8} + 112 \nu^{7} - 239 \nu^{6} + 367 \nu^{5} - 448 \nu^{4} + \cdots - 20$$ -2*v^10 + 10*v^9 - 43*v^8 + 112*v^7 - 239*v^6 + 367*v^5 - 448*v^4 + 395*v^3 - 256*v^2 + 104*v - 20 $$\beta_{5}$$ $$=$$ $$- 2 \nu^{10} + 10 \nu^{9} - 44 \nu^{8} + 116 \nu^{7} - 255 \nu^{6} + 401 \nu^{5} - 509 \nu^{4} + \cdots - 25$$ -2*v^10 + 10*v^9 - 44*v^8 + 116*v^7 - 255*v^6 + 401*v^5 - 509*v^4 + 465*v^3 - 314*v^2 + 132*v - 25 $$\beta_{6}$$ $$=$$ $$11 \nu^{11} - 61 \nu^{10} + 270 \nu^{9} - 761 \nu^{8} + 1715 \nu^{7} - 2888 \nu^{6} + 3852 \nu^{5} + \cdots - 71$$ 11*v^11 - 61*v^10 + 270*v^9 - 761*v^8 + 1715*v^7 - 2888*v^6 + 3852*v^5 - 3892*v^4 + 2948*v^3 - 1562*v^2 + 500*v - 71 $$\beta_{7}$$ $$=$$ $$- 14 \nu^{11} + 78 \nu^{10} - 344 \nu^{9} + 970 \nu^{8} - 2177 \nu^{7} + 3659 \nu^{6} - 4853 \nu^{5} + \cdots + 85$$ -14*v^11 + 78*v^10 - 344*v^9 + 970*v^8 - 2177*v^7 + 3659*v^6 - 4853*v^5 + 4884*v^4 - 3671*v^3 + 1934*v^2 - 613*v + 85 $$\beta_{8}$$ $$=$$ $$- 20 \nu^{11} + 111 \nu^{10} - 490 \nu^{9} + 1379 \nu^{8} - 3097 \nu^{7} + 5198 \nu^{6} - 6900 \nu^{5} + \cdots + 124$$ -20*v^11 + 111*v^10 - 490*v^9 + 1379*v^8 - 3097*v^7 + 5198*v^6 - 6900*v^5 + 6936*v^4 - 5222*v^3 + 2750*v^2 - 876*v + 124 $$\beta_{9}$$ $$=$$ $$42 \nu^{11} - 231 \nu^{10} + 1019 \nu^{9} - 2853 \nu^{8} + 6396 \nu^{7} - 10689 \nu^{6} + 14157 \nu^{5} + \cdots - 257$$ 42*v^11 - 231*v^10 + 1019*v^9 - 2853*v^8 + 6396*v^7 - 10689*v^6 + 14157*v^5 - 14172*v^4 + 10648*v^3 - 5589*v^2 + 1785*v - 257 $$\beta_{10}$$ $$=$$ $$68 \nu^{11} - 374 \nu^{10} + 1649 \nu^{9} - 4616 \nu^{8} + 10342 \nu^{7} - 17277 \nu^{6} + 22863 \nu^{5} + \cdots - 412$$ 68*v^11 - 374*v^10 + 1649*v^9 - 4616*v^8 + 10342*v^7 - 17277*v^6 + 22863*v^5 - 22873*v^4 + 17165*v^3 - 9002*v^2 + 2871*v - 412 $$\beta_{11}$$ $$=$$ $$- 98 \nu^{11} + 540 \nu^{10} - 2381 \nu^{9} + 6671 \nu^{8} - 14949 \nu^{7} + 24987 \nu^{6} - 33072 \nu^{5} + \cdots + 595$$ -98*v^11 + 540*v^10 - 2381*v^9 + 6671*v^8 - 14949*v^7 + 24987*v^6 - 33072*v^5 + 33097*v^4 - 24840*v^3 + 13028*v^2 - 4153*v + 595
 $$\nu$$ $$=$$ $$( 2\beta_{11} + 2\beta_{10} + \beta_{9} - 2\beta_{8} - 2\beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 1 ) / 3$$ (2*b11 + 2*b10 + b9 - 2*b8 - 2*b6 - b5 + b4 - b3 + 1) / 3 $$\nu^{2}$$ $$=$$ $$( 2\beta_{11} + 2\beta_{10} + \beta_{9} - 2\beta_{8} - 2\beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 3\beta _1 - 5 ) / 3$$ (2*b11 + 2*b10 + b9 - 2*b8 - 2*b6 - b5 + b4 - b3 + 3*b1 - 5) / 3 $$\nu^{3}$$ $$=$$ $$( - 4 \beta_{11} - 5 \beta_{10} + \beta_{9} + 9 \beta_{8} - 3 \beta_{7} + 4 \beta_{6} + 3 \beta_{5} + \cdots - 5 ) / 3$$ (-4*b11 - 5*b10 + b9 + 9*b8 - 3*b7 + 4*b6 + 3*b5 - 2*b4 + 2*b3 + 2*b2 + 3*b1 - 5) / 3 $$\nu^{4}$$ $$=$$ $$( - 10 \beta_{11} - 12 \beta_{10} + \beta_{9} + 20 \beta_{8} - 6 \beta_{7} + 10 \beta_{6} + 10 \beta_{5} + \cdots + 16 ) / 3$$ (-10*b11 - 12*b10 + b9 + 20*b8 - 6*b7 + 10*b6 + 10*b5 - 8*b4 + 5*b3 + 7*b2 - 12*b1 + 16) / 3 $$\nu^{5}$$ $$=$$ $$( 7 \beta_{11} + 14 \beta_{10} - 12 \beta_{9} - 25 \beta_{8} + 14 \beta_{7} - 6 \beta_{6} + 2 \beta_{5} + \cdots + 30 ) / 3$$ (7*b11 + 14*b10 - 12*b9 - 25*b8 + 14*b7 - 6*b6 + 2*b5 - 4*b4 - 3*b3 + 2*b2 - 23*b1 + 30) / 3 $$\nu^{6}$$ $$=$$ $$( 47 \beta_{11} + 73 \beta_{10} - 38 \beta_{9} - 126 \beta_{8} + 57 \beta_{7} - 44 \beta_{6} - 48 \beta_{5} + \cdots - 56 ) / 3$$ (47*b11 + 73*b10 - 38*b9 - 126*b8 + 57*b7 - 44*b6 - 48*b5 + 34*b4 - 28*b3 - 37*b2 + 42*b1 - 56) / 3 $$\nu^{7}$$ $$=$$ $$( 8 \beta_{11} - 6 \beta_{10} + 28 \beta_{9} + 2 \beta_{8} - 15 \beta_{7} - 14 \beta_{6} - 80 \beta_{5} + \cdots - 191 ) / 3$$ (8*b11 - 6*b10 + 28*b9 + 2*b8 - 15*b7 - 14*b6 - 80*b5 + 67*b4 - 28*b3 - 65*b2 + 150*b1 - 191) / 3 $$\nu^{8}$$ $$=$$ $$( - 212 \beta_{11} - 394 \beta_{10} + 291 \beta_{9} + 644 \beta_{8} - 340 \beta_{7} + 174 \beta_{6} + \cdots + 129 ) / 3$$ (-212*b11 - 394*b10 + 291*b9 + 644*b8 - 340*b7 + 174*b6 + 143*b5 - 91*b4 + 99*b3 + 113*b2 - 86*b1 + 129) / 3 $$\nu^{9}$$ $$=$$ $$( - 226 \beta_{11} - 359 \beta_{10} + 199 \beta_{9} + 624 \beta_{8} - 288 \beta_{7} + 214 \beta_{6} + \cdots + 1108 ) / 3$$ (-226*b11 - 359*b10 + 199*b9 + 624*b8 - 288*b7 + 214*b6 + 600*b5 - 455*b4 + 287*b3 + 488*b2 - 846*b1 + 1108) / 3 $$\nu^{10}$$ $$=$$ $$( 842 \beta_{11} + 1734 \beta_{10} - 1457 \beta_{9} - 2695 \beta_{8} + 1539 \beta_{7} - 596 \beta_{6} + \cdots + 358 ) / 3$$ (842*b11 + 1734*b10 - 1457*b9 - 2695*b8 + 1539*b7 - 596*b6 - 23*b5 - 44*b4 - 115*b3 - 14*b2 - 324*b1 + 358) / 3 $$\nu^{11}$$ $$=$$ $$( 1918 \beta_{11} + 3656 \beta_{10} - 2790 \beta_{9} - 5875 \beta_{8} + 3164 \beta_{7} - 1515 \beta_{6} + \cdots - 5451 ) / 3$$ (1918*b11 + 3656*b10 - 2790*b9 - 5875*b8 + 3164*b7 - 1515*b6 - 3154*b5 + 2312*b4 - 1632*b3 - 2578*b2 + 4090*b1 - 5451) / 3

Character values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$-1 - \beta_{9}$$

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
 0.5 − 0.0126039i 0.5 − 1.68614i 0.5 − 0.258654i 0.5 − 1.27297i 0.5 + 2.22827i 0.5 + 1.00210i 0.5 + 0.0126039i 0.5 + 1.68614i 0.5 + 0.258654i 0.5 + 1.27297i 0.5 − 2.22827i 0.5 − 1.00210i
−0.840456 1.45571i 0 −0.412733 + 0.714874i 0.564772 0.978214i 0 1.95446 + 3.38523i −1.97429 0 −1.89866
244.2 −0.527162 0.913072i 0 0.444200 0.769376i 0.872894 1.51190i 0 −1.22962 2.12977i −3.04531 0 −1.84063
244.3 0.207733 + 0.359804i 0 0.913694 1.58256i −1.10759 + 1.91841i 0 0.659815 + 1.14283i 1.59015 0 −0.920335
244.4 0.400763 + 0.694143i 0 0.678777 1.17568i 1.37492 2.38143i 0 −1.18842 2.05840i 2.69117 0 2.20407
244.5 1.05831 + 1.83305i 0 −1.24005 + 2.14782i 1.34155 2.32363i 0 −0.486166 0.842065i −1.01617 0 5.67911
244.6 1.20081 + 2.07986i 0 −1.88389 + 3.26300i −0.0465417 + 0.0806126i 0 0.289931 + 0.502175i −4.24555 0 −0.223551
487.1 −0.840456 + 1.45571i 0 −0.412733 0.714874i 0.564772 + 0.978214i 0 1.95446 3.38523i −1.97429 0 −1.89866
487.2 −0.527162 + 0.913072i 0 0.444200 + 0.769376i 0.872894 + 1.51190i 0 −1.22962 + 2.12977i −3.04531 0 −1.84063
487.3 0.207733 0.359804i 0 0.913694 + 1.58256i −1.10759 1.91841i 0 0.659815 1.14283i 1.59015 0 −0.920335
487.4 0.400763 0.694143i 0 0.678777 + 1.17568i 1.37492 + 2.38143i 0 −1.18842 + 2.05840i 2.69117 0 2.20407
487.5 1.05831 1.83305i 0 −1.24005 2.14782i 1.34155 + 2.32363i 0 −0.486166 + 0.842065i −1.01617 0 5.67911
487.6 1.20081 2.07986i 0 −1.88389 3.26300i −0.0465417 0.0806126i 0 0.289931 0.502175i −4.24555 0 −0.223551
 $$n$$: 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.e 12
3.b odd 2 1 729.2.c.b 12
9.c even 3 1 729.2.a.a 6
9.c even 3 1 inner 729.2.c.e 12
9.d odd 6 1 729.2.a.d 6
9.d odd 6 1 729.2.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 9.c even 3 1
729.2.a.d 6 9.d odd 6 1
729.2.c.b 12 3.b odd 2 1
729.2.c.b 12 9.d odd 6 1
729.2.c.e 12 1.a even 1 1 trivial
729.2.c.e 12 9.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} - 3 T_{2}^{11} + 12 T_{2}^{10} - 15 T_{2}^{9} + 45 T_{2}^{8} - 45 T_{2}^{7} + 123 T_{2}^{6} + \cdots + 9$$ acting on $$S_{2}^{\mathrm{new}}(729, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 3 T^{11} + \cdots + 9$$
$3$ $$T^{12}$$
$5$ $$T^{12} - 6 T^{11} + \cdots + 9$$
$7$ $$T^{12} + 15 T^{10} + \cdots + 289$$
$11$ $$T^{12} - 12 T^{11} + \cdots + 9$$
$13$ $$T^{12} + 24 T^{10} + \cdots + 1$$
$17$ $$(T^{6} + 9 T^{5} + 9 T^{4} + \cdots + 27)^{2}$$
$19$ $$(T^{6} - 3 T^{5} - 30 T^{4} + \cdots + 19)^{2}$$
$23$ $$T^{12} - 15 T^{11} + \cdots + 106929$$
$29$ $$T^{12} - 12 T^{11} + \cdots + 45369$$
$31$ $$T^{12} + 51 T^{10} + \cdots + 26569$$
$37$ $$(T^{6} - 3 T^{5} + \cdots + 4933)^{2}$$
$41$ $$T^{12} - 15 T^{11} + \cdots + 11229201$$
$43$ $$T^{12} + 96 T^{10} + \cdots + 3308761$$
$47$ $$T^{12} - 21 T^{11} + \cdots + 42732369$$
$53$ $$(T^{6} + 9 T^{5} + \cdots - 12393)^{2}$$
$59$ $$T^{12} + \cdots + 176384961$$
$61$ $$T^{12} + \cdots + 273670849$$
$67$ $$T^{12} - 9 T^{11} + \cdots + 8288641$$
$71$ $$(T^{6} + 27 T^{5} + \cdots + 27)^{2}$$
$73$ $$(T^{6} + 6 T^{5} + \cdots - 431)^{2}$$
$79$ $$T^{12} + 177 T^{10} + \cdots + 3508129$$
$83$ $$T^{12} + \cdots + 6951057129$$
$89$ $$(T^{6} + 9 T^{5} + \cdots + 32589)^{2}$$
$97$ $$T^{12} + 204 T^{10} + \cdots + 66765241$$