# Properties

 Label 81.2.e.a Level $81$ Weight $2$ Character orbit 81.e Analytic conductor $0.647$ 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] = [81,2,Mod(10,81)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(81, 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("81.10");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$81 = 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 81.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: $$0.646788256372$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{9})$$ Coefficient field: 12.0.1952986685049.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 6 x^{11} + 27 x^{10} - 80 x^{9} + 186 x^{8} - 330 x^{7} + 463 x^{6} - 504 x^{5} + 420 x^{4} - 258 x^{3} + 108 x^{2} - 27 x + 3$$ 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: $$3$$ Twist minimal: no (minimal twist has level 27) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{8} + \beta_{3} + 1) q^{2} + (\beta_{9} - \beta_{8} - \beta_{6} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{4} + ( - \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} - \beta_{5} - \beta_{3} + \beta_1 + 1) q^{5} + ( - \beta_{11} + \beta_{9} - \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 - 1) q^{7} + (2 \beta_{10} + 2 \beta_{8} + 2 \beta_{7} - \beta_{4} + \beta_{2}) q^{8}+O(q^{10})$$ q + (-b8 + b3 + 1) * q^2 + (b9 - b8 - b6 + b3 + b2 + b1 + 1) * q^4 + (-b11 - b10 + b9 - b8 - b7 - b5 - b3 + b1 + 1) * q^5 + (-b11 + b9 - b7 - b5 + b4 - b3 - b1 - 1) * q^7 + (2*b10 + 2*b8 + 2*b7 - b4 + b2) * q^8 $$q + ( - \beta_{8} + \beta_{3} + 1) q^{2} + (\beta_{9} - \beta_{8} - \beta_{6} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{4} + ( - \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} - \beta_{5} - \beta_{3} + \beta_1 + 1) q^{5} + ( - \beta_{11} + \beta_{9} - \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 - 1) q^{7} + (2 \beta_{10} + 2 \beta_{8} + 2 \beta_{7} - \beta_{4} + \beta_{2}) q^{8} + (\beta_{11} - \beta_{9} + \beta_{8} + \beta_{7} + 2 \beta_{6} - \beta_{3} - \beta_{2} - 2) q^{10} + ( - \beta_{11} - \beta_{10} - 2 \beta_{9} + 2 \beta_{8} - \beta_{7} + \beta_{5} + \beta_{4} - \beta_{2} + \cdots - 1) q^{11}+ \cdots + ( - 5 \beta_{11} + \beta_{10} - 3 \beta_{9} + 2 \beta_{8} + 3 \beta_{7} - 2 \beta_{6} + \cdots + 8) q^{98}+O(q^{100})$$ q + (-b8 + b3 + 1) * q^2 + (b9 - b8 - b6 + b3 + b2 + b1 + 1) * q^4 + (-b11 - b10 + b9 - b8 - b7 - b5 - b3 + b1 + 1) * q^5 + (-b11 + b9 - b7 - b5 + b4 - b3 - b1 - 1) * q^7 + (2*b10 + 2*b8 + 2*b7 - b4 + b2) * q^8 + (b11 - b9 + b8 + b7 + 2*b6 - b3 - b2 - 2) * q^10 + (-b11 - b10 - 2*b9 + 2*b8 - b7 + b5 + b4 - b2 - b1 - 1) * q^11 + (-b9 + b7 - b6 + b5 - b4 - b2 + b1) * q^13 + (b11 - b8 + b4 - b3 - b1 - 3) * q^14 + (-b9 + b8 + b6 + 2*b5 - b2 - 2*b1 - 2) * q^16 + (2*b11 + b9 - b8 - b7 - b6 + b5 - b4 + b3 - 1) * q^17 + (b11 + b10 - 2*b9 + b8 - b7 - b5 - 2*b4 + b3 + 2*b2 + 1) * q^19 + (3*b11 - b10 + 2*b7 + b6 + b5 - b2 - 2) * q^20 + (b11 - 2*b10 - b9 - 2*b7 - b5 + b3 - b1) * q^22 + (2*b11 + 3*b10 + 2*b8 + b6 + b4 - b3 - 1) * q^23 + (-2*b11 + 2*b7 - b6 + b5 + b3 - b2 + b1 + 2) * q^25 + (-3*b10 - 3*b9 + 2*b8 - 2*b7 - b5 - b4 + 3) * q^26 + (b10 + b9 - 2*b8 + 2*b7 - 1) * q^28 + (-3*b11 + 3*b10 - b8 + 3*b7 - 2*b3 + 1) * q^29 + (b11 + b10 + b8 + 3*b6 + 2*b4 - 3*b3 - 2*b2 - b1 - 4) * q^31 + (-3*b11 - b10 + 3*b9 - b8 - b7 - b6 + 2*b5 + b4 + 2*b3 - 2*b1 + 1) * q^32 + (-b6 + b5 + b2 + 2*b1 + 2) * q^34 + (-4*b11 - 2*b10 + b9 - 2*b8 - b7 - b6 - b5 + b4 + b3 - b2 + b1 + 2) * q^35 + (-b11 - 2*b10 + 4*b9 - 2*b8 - 4*b7 - 3*b6 - b5 + b4 + b3 + 2*b2 + 2) * q^37 + (b9 - b8 - 3*b7 - 3*b5 - 2*b4 + 3*b2 + 3*b1) * q^38 + (-b11 + 3*b9 + b8 - b7 - 2*b5 + b4 - b3 + 2*b2 - b1 + 2) * q^40 + (-3*b9 + 5*b7 - b5 - 3*b4 + 3*b3 + b2 + 3*b1 + 3) * q^41 + (b11 + b10 + 2*b9 - 2*b8 + b7 - b6 - 4*b5 - b4 + 3*b2 + 4*b1 + 4) * q^43 + (3*b10 + b9 - 4*b8 - b7 + 3*b6 - 2*b5 + 2*b4 - 2*b3 - 2) * q^44 + (-4*b11 - b10 + 3*b9 - b8 + 2*b7 - 2*b6 + b5 + 3*b4 - b3 - 3*b2 + 2*b1 + 1) * q^46 + (-b11 - 3*b10 + 6*b9 - 4*b7 - 2*b6 - 3*b5 + 2*b2 - b1 + 3) * q^47 + (-b11 + 4*b10 + b9 - b8 + 4*b7 + b6 + 2*b5 - b4 + 1) * q^49 + (-b11 - b10 - 3*b9 + 2*b8 - 2*b6 - 2*b4 + 2*b3 - b2 + 5) * q^50 + (5*b11 - b10 - 2*b8 - 5*b7 + 2*b6 - 2*b5 + b2 - b1 - 2) * q^52 + (-3*b8 + 3*b7 + 3*b5 + 3*b4) * q^53 + (-2*b10 - 2*b9 + 3*b8 - 3*b7 + 3*b5 + 2*b4 + b3 - b2 - 2*b1 - 3) * q^55 + (b11 + 2*b10 + b8 - b7 - b6 + b5 - b3 - b2 + b1 + 2) * q^56 + (-3*b11 - b10 - 5*b9 + 2*b8 - b6 - 3*b4 + b3 + b2 - 2*b1 + 4) * q^58 + (5*b11 - b10 - 5*b9 + 2*b8 - b7 + 3*b6 + 2*b5 - 3*b4 + 2*b3 - 2*b1 - 2) * q^59 + (6*b11 - 2*b10 - 5*b9 + 4*b7 + b6 + b5 - 5*b4 + 5*b3 - b2 + 1) * q^61 + (-b10 + 3*b9 - b8 + 2*b7 + 3*b6 + 3*b5 + 5*b4 - 3*b3 - 5*b2 - 3*b1 - 6) * q^62 + (-2*b11 + 3*b10 - 3*b9 + 3*b7 - 3*b6 + 2*b5 - 2*b4 + b3 - b2 + 3) * q^64 + (4*b11 + 4*b10 - b9 + b8 + 2*b7 + b6 + 2*b5 - b4 - b2 - 2*b1 - 4) * q^65 + (-b11 - 3*b9 + b8 - 2*b7 + 4*b6 + b5 + 3*b4 + b3 - b2 - 3*b1 - 4) * q^67 + (-2*b11 - 2*b9 + 2*b8 - b7 - b6 + 2*b5 - b3 - 2*b2 - 1) * q^68 + (2*b11 + 2*b10 - 2*b9 + 2*b8 - b6 - b5 - b4 + b1 - 1) * q^70 + (2*b11 - 3*b10 + b9 + 2*b8 - b7 - b6 - 2*b5 + 2*b4 - 2*b3 - 4) * q^71 + (b11 - 3*b10 + 3*b9 - 3*b8 + 3*b6 + 3*b5 + 3*b4 - 3*b3 - 3*b2 - 3*b1 - 6) * q^73 + (3*b10 - b9 + 3*b7 - 3*b6 - b4 + b3 + 3*b2 + 3*b1 + 1) * q^74 + (-3*b11 + 3*b9 + 2*b8 - 2*b5 - 4*b3 + 4*b1 - 2) * q^76 + (2*b11 - b10 + b9 + b8 - b6 - 2*b4 + b3 + 2*b2 - b1 - 1) * q^77 + (-b10 + 2*b8 + b6 - b5 - 5*b3 - 6) * q^79 + (-b10 - b9 + 3*b8 - 3*b7 - b5 - b3 + b2 - b1 - 4) * q^80 + (b10 + b9 + 3*b8 - 3*b7 - 4*b5 - 3*b4 - b3 + b2 + b1 + 1) * q^82 + (3*b11 - 3*b10 - b8 - 3*b7 + 3*b6 - 3*b5 + b3 + 3*b2 - 3*b1 - 5) * q^83 + (3*b11 + 3*b9 - b6 - 2*b4 + b3 - b1 - 3) * q^85 + (-3*b11 + 3*b10 + 3*b9 + 2*b8 + 3*b7 - b6 - 3*b5 + b4 - 4*b3 + 4*b1 - 2) * q^86 + (-3*b11 + 5*b10 + b9 + 2*b7 + 2*b6 + 2*b5 + 2*b4 - 2*b3 - 2*b2 - 4) * q^88 + (2*b11 + 5*b10 - 6*b9 + 5*b8 - b7 - b6 - 4*b4 + 4*b2 + b1 + 1) * q^89 + (b10 - 4*b9 + 3*b8 + 4*b7 + b6 + b3 + b2 + 1) * q^91 + (-6*b11 - 6*b10 + b9 - b8 - 3*b6 + b4 - 3*b2 + 6) * q^92 + (5*b11 - 5*b8 + 2*b7 + b6 + b5 + b3 - b2 - 4) * q^94 + (4*b9 - b7 + 2*b6 + 2*b5 + 5*b4 - 3*b3 - 2*b2 - 5*b1 - 4) * q^95 + (-8*b11 - 8*b10 + 2*b9 - 2*b8 - 5*b7 - b6 + 2*b5 + 5*b4 - 3*b2 - 2*b1 + 1) * q^97 + (-5*b11 + b10 - 3*b9 + 2*b8 + 3*b7 - 2*b6 + 4*b5 - 4*b4 + 3*b3 - b2 + 8) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 6 q^{2} - 6 q^{4} + 3 q^{5} - 6 q^{7} - 6 q^{8}+O(q^{10})$$ 12 * q + 6 * q^2 - 6 * q^4 + 3 * q^5 - 6 * q^7 - 6 * q^8 $$12 q + 6 q^{2} - 6 q^{4} + 3 q^{5} - 6 q^{7} - 6 q^{8} - 3 q^{10} - 3 q^{11} - 6 q^{13} - 15 q^{14} - 9 q^{17} - 3 q^{19} + 3 q^{20} + 3 q^{22} + 12 q^{23} + 3 q^{25} + 30 q^{26} - 12 q^{28} + 6 q^{29} + 3 q^{31} + 9 q^{34} - 12 q^{35} - 3 q^{37} - 42 q^{38} + 21 q^{40} - 15 q^{41} + 3 q^{43} - 3 q^{44} - 3 q^{46} + 15 q^{47} + 12 q^{49} + 33 q^{50} + 9 q^{52} + 18 q^{53} - 12 q^{55} + 33 q^{56} + 21 q^{58} + 12 q^{59} + 12 q^{61} + 12 q^{62} + 12 q^{64} - 3 q^{65} - 15 q^{67} - 9 q^{68} - 15 q^{70} - 27 q^{71} + 6 q^{73} - 33 q^{74} - 48 q^{76} - 15 q^{77} - 42 q^{79} - 42 q^{80} - 12 q^{82} - 39 q^{83} - 27 q^{85} - 51 q^{86} - 30 q^{88} - 9 q^{89} + 6 q^{91} + 39 q^{92} - 15 q^{94} + 33 q^{95} + 3 q^{97} + 45 q^{98}+O(q^{100})$$ 12 * q + 6 * q^2 - 6 * q^4 + 3 * q^5 - 6 * q^7 - 6 * q^8 - 3 * q^10 - 3 * q^11 - 6 * q^13 - 15 * q^14 - 9 * q^17 - 3 * q^19 + 3 * q^20 + 3 * q^22 + 12 * q^23 + 3 * q^25 + 30 * q^26 - 12 * q^28 + 6 * q^29 + 3 * q^31 + 9 * q^34 - 12 * q^35 - 3 * q^37 - 42 * q^38 + 21 * q^40 - 15 * q^41 + 3 * q^43 - 3 * q^44 - 3 * q^46 + 15 * q^47 + 12 * q^49 + 33 * q^50 + 9 * q^52 + 18 * q^53 - 12 * q^55 + 33 * q^56 + 21 * q^58 + 12 * q^59 + 12 * q^61 + 12 * q^62 + 12 * q^64 - 3 * q^65 - 15 * q^67 - 9 * q^68 - 15 * q^70 - 27 * q^71 + 6 * q^73 - 33 * q^74 - 48 * q^76 - 15 * q^77 - 42 * q^79 - 42 * q^80 - 12 * q^82 - 39 * q^83 - 27 * q^85 - 51 * q^86 - 30 * q^88 - 9 * q^89 + 6 * q^91 + 39 * q^92 - 15 * q^94 + 33 * q^95 + 3 * q^97 + 45 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 6 x^{11} + 27 x^{10} - 80 x^{9} + 186 x^{8} - 330 x^{7} + 463 x^{6} - 504 x^{5} + 420 x^{4} - 258 x^{3} + 108 x^{2} - 27 x + 3$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{10} - 5 \nu^{9} + 22 \nu^{8} - 58 \nu^{7} + 127 \nu^{6} - 199 \nu^{5} + 249 \nu^{4} - 224 \nu^{3} + 145 \nu^{2} - 58 \nu + 9$$ 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 $$\beta_{2}$$ $$=$$ $$3 \nu^{11} - 16 \nu^{10} + 71 \nu^{9} - 197 \nu^{8} + 445 \nu^{7} - 747 \nu^{6} + 1006 \nu^{5} - 1030 \nu^{4} + 803 \nu^{3} - 445 \nu^{2} + 155 \nu - 25$$ 3*v^11 - 16*v^10 + 71*v^9 - 197*v^8 + 445*v^7 - 747*v^6 + 1006*v^5 - 1030*v^4 + 803*v^3 - 445*v^2 + 155*v - 25 $$\beta_{3}$$ $$=$$ $$- 6 \nu^{11} + 32 \nu^{10} - 140 \nu^{9} + 384 \nu^{8} - 849 \nu^{7} + 1390 \nu^{6} - 1805 \nu^{5} + 1762 \nu^{4} - 1285 \nu^{3} + 649 \nu^{2} - 195 \nu + 25$$ -6*v^11 + 32*v^10 - 140*v^9 + 384*v^8 - 849*v^7 + 1390*v^6 - 1805*v^5 + 1762*v^4 - 1285*v^3 + 649*v^2 - 195*v + 25 $$\beta_{4}$$ $$=$$ $$- 9 \nu^{11} + 49 \nu^{10} - 216 \nu^{9} + 601 \nu^{8} - 1344 \nu^{7} + 2232 \nu^{6} - 2942 \nu^{5} + 2918 \nu^{4} - 2170 \nu^{3} + 1118 \nu^{2} - 348 \nu + 49$$ -9*v^11 + 49*v^10 - 216*v^9 + 601*v^8 - 1344*v^7 + 2232*v^6 - 2942*v^5 + 2918*v^4 - 2170*v^3 + 1118*v^2 - 348*v + 49 $$\beta_{5}$$ $$=$$ $$9 \nu^{11} - 50 \nu^{10} + 221 \nu^{9} - 623 \nu^{8} + 1402 \nu^{7} - 2360 \nu^{6} + 3144 \nu^{5} - 3178 \nu^{4} + 2411 \nu^{3} - 1286 \nu^{2} + 421 \nu - 62$$ 9*v^11 - 50*v^10 + 221*v^9 - 623*v^8 + 1402*v^7 - 2360*v^6 + 3144*v^5 - 3178*v^4 + 2411*v^3 - 1286*v^2 + 421*v - 62 $$\beta_{6}$$ $$=$$ $$11 \nu^{11} - 60 \nu^{10} + 265 \nu^{9} - 739 \nu^{8} + 1657 \nu^{7} - 2761 \nu^{6} + 3653 \nu^{5} - 3643 \nu^{4} + 2724 \nu^{3} - 1417 \nu^{2} + 442 \nu - 61$$ 11*v^11 - 60*v^10 + 265*v^9 - 739*v^8 + 1657*v^7 - 2761*v^6 + 3653*v^5 - 3643*v^4 + 2724*v^3 - 1417*v^2 + 442*v - 61 $$\beta_{7}$$ $$=$$ $$- 16 \nu^{11} + 87 \nu^{10} - 383 \nu^{9} + 1064 \nu^{8} - 2375 \nu^{7} + 3936 \nu^{6} - 5176 \nu^{5} + 5122 \nu^{4} - 3802 \nu^{3} + 1958 \nu^{2} - 610 \nu + 85$$ -16*v^11 + 87*v^10 - 383*v^9 + 1064*v^8 - 2375*v^7 + 3936*v^6 - 5176*v^5 + 5122*v^4 - 3802*v^3 + 1958*v^2 - 610*v + 85 $$\beta_{8}$$ $$=$$ $$- 16 \nu^{11} + 89 \nu^{10} - 393 \nu^{9} + 1108 \nu^{8} - 2491 \nu^{7} + 4191 \nu^{6} - 5577 \nu^{5} + 5631 \nu^{4} - 4267 \nu^{3} + 2272 \nu^{2} - 742 \nu + 110$$ -16*v^11 + 89*v^10 - 393*v^9 + 1108*v^8 - 2491*v^7 + 4191*v^6 - 5577*v^5 + 5631*v^4 - 4267*v^3 + 2272*v^2 - 742*v + 110 $$\beta_{9}$$ $$=$$ $$36 \nu^{11} - 198 \nu^{10} + 873 \nu^{9} - 2443 \nu^{8} + 5472 \nu^{7} - 9134 \nu^{6} + 12076 \nu^{5} - 12058 \nu^{4} + 9024 \nu^{3} - 4708 \nu^{2} + 1486 \nu - 209$$ 36*v^11 - 198*v^10 + 873*v^9 - 2443*v^8 + 5472*v^7 - 9134*v^6 + 12076*v^5 - 12058*v^4 + 9024*v^3 - 4708*v^2 + 1486*v - 209 $$\beta_{10}$$ $$=$$ $$- 36 \nu^{11} + 198 \nu^{10} - 873 \nu^{9} + 2444 \nu^{8} - 5476 \nu^{7} + 9150 \nu^{6} - 12110 \nu^{5} + 12120 \nu^{4} - 9096 \nu^{3} + 4772 \nu^{2} - 1519 \nu + 217$$ -36*v^11 + 198*v^10 - 873*v^9 + 2444*v^8 - 5476*v^7 + 9150*v^6 - 12110*v^5 + 12120*v^4 - 9096*v^3 + 4772*v^2 - 1519*v + 217 $$\beta_{11}$$ $$=$$ $$- 42 \nu^{11} + 231 \nu^{10} - 1019 \nu^{9} + 2853 \nu^{8} - 6396 \nu^{7} + 10689 \nu^{6} - 14157 \nu^{5} + 14172 \nu^{4} - 10648 \nu^{3} + 5589 \nu^{2} - 1785 \nu + 257$$ -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
 $$\nu$$ $$=$$ $$( \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} + \beta _1 + 3 ) / 3$$ (b11 - b10 + b9 + b8 + b7 - 2*b6 + b5 - 2*b4 + b3 + b2 + b1 + 3) / 3 $$\nu^{2}$$ $$=$$ $$( \beta_{11} - \beta_{10} + \beta_{9} + 4 \beta_{8} - 2 \beta_{7} - 2 \beta_{6} + 4 \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta _1 - 6 ) / 3$$ (b11 - b10 + b9 + 4*b8 - 2*b7 - 2*b6 + 4*b5 + b4 + b3 + b2 - 2*b1 - 6) / 3 $$\nu^{3}$$ $$=$$ $$( - 5 \beta_{11} + 5 \beta_{10} - 5 \beta_{9} + \beta_{8} - 8 \beta_{7} + 7 \beta_{6} + 4 \beta_{5} + 10 \beta_{4} - 5 \beta_{3} - 5 \beta_{2} - 8 \beta _1 - 18 ) / 3$$ (-5*b11 + 5*b10 - 5*b9 + b8 - 8*b7 + 7*b6 + 4*b5 + 10*b4 - 5*b3 - 5*b2 - 8*b1 - 18) / 3 $$\nu^{4}$$ $$=$$ $$( - 11 \beta_{11} + 17 \beta_{10} - 5 \beta_{9} - 20 \beta_{8} + 4 \beta_{7} + 16 \beta_{6} - 14 \beta_{5} + 4 \beta_{4} - 14 \beta_{3} - 8 \beta_{2} + \beta _1 + 6 ) / 3$$ (-11*b11 + 17*b10 - 5*b9 - 20*b8 + 4*b7 + 16*b6 - 14*b5 + 4*b4 - 14*b3 - 8*b2 + b1 + 6) / 3 $$\nu^{5}$$ $$=$$ $$( 19 \beta_{11} - \beta_{10} + 31 \beta_{9} - 32 \beta_{8} + 43 \beta_{7} - 20 \beta_{6} - 41 \beta_{5} - 44 \beta_{4} + 10 \beta_{3} + 25 \beta_{2} + 40 \beta _1 + 87 ) / 3$$ (19*b11 - b10 + 31*b9 - 32*b8 + 43*b7 - 20*b6 - 41*b5 - 44*b4 + 10*b3 + 25*b2 + 40*b1 + 87) / 3 $$\nu^{6}$$ $$=$$ $$( 85 \beta_{11} - 97 \beta_{10} + 55 \beta_{9} + 64 \beta_{8} + 10 \beta_{7} - 101 \beta_{6} + 19 \beta_{5} - 62 \beta_{4} + 91 \beta_{3} + 70 \beta_{2} + 31 \beta _1 + 60 ) / 3$$ (85*b11 - 97*b10 + 55*b9 + 64*b8 + 10*b7 - 101*b6 + 19*b5 - 62*b4 + 91*b3 + 70*b2 + 31*b1 + 60) / 3 $$\nu^{7}$$ $$=$$ $$( - 20 \beta_{11} - 118 \beta_{10} - 134 \beta_{9} + 244 \beta_{8} - 218 \beta_{7} + \beta_{6} + 232 \beta_{5} + 157 \beta_{4} + 52 \beta_{3} - 74 \beta_{2} - 179 \beta _1 - 357 ) / 3$$ (-20*b11 - 118*b10 - 134*b9 + 244*b8 - 218*b7 + b6 + 232*b5 + 157*b4 + 52*b3 - 74*b2 - 179*b1 - 357) / 3 $$\nu^{8}$$ $$=$$ $$( - 503 \beta_{11} + 386 \beta_{10} - 440 \beta_{9} - 47 \beta_{8} - 233 \beta_{7} + 514 \beta_{6} + 163 \beta_{5} + 466 \beta_{4} - 431 \beta_{3} - 461 \beta_{2} - 329 \beta _1 - 639 ) / 3$$ (-503*b11 + 386*b10 - 440*b9 - 47*b8 - 233*b7 + 514*b6 + 163*b5 + 466*b4 - 431*b3 - 461*b2 - 329*b1 - 639) / 3 $$\nu^{9}$$ $$=$$ $$( - 425 \beta_{11} + 1076 \beta_{10} + 319 \beta_{9} - 1313 \beta_{8} + 955 \beta_{7} + 502 \beta_{6} - 1013 \beta_{5} - 332 \beta_{4} - 743 \beta_{3} - 59 \beta_{2} + 631 \beta _1 + 1164 ) / 3$$ (-425*b11 + 1076*b10 + 319*b9 - 1313*b8 + 955*b7 + 502*b6 - 1013*b5 - 332*b4 - 743*b3 - 59*b2 + 631*b1 + 1164) / 3 $$\nu^{10}$$ $$=$$ $$( 2299 \beta_{11} - 862 \beta_{10} + 2725 \beta_{9} - 1193 \beta_{8} + 2104 \beta_{7} - 2135 \beta_{6} - 1907 \beta_{5} - 2705 \beta_{4} + 1495 \beta_{3} + 2425 \beta_{2} + 2344 \beta _1 + 4356 ) / 3$$ (2299*b11 - 862*b10 + 2725*b9 - 1193*b8 + 2104*b7 - 2135*b6 - 1907*b5 - 2705*b4 + 1495*b3 + 2425*b2 + 2344*b1 + 4356) / 3 $$\nu^{11}$$ $$=$$ $$( 4708 \beta_{11} - 6628 \beta_{10} + 985 \beta_{9} + 5506 \beta_{8} - 3107 \beta_{7} - 4679 \beta_{6} + 3238 \beta_{5} - 992 \beta_{4} + 5476 \beta_{3} + 2770 \beta_{2} - 1043 \beta _1 - 1698 ) / 3$$ (4708*b11 - 6628*b10 + 985*b9 + 5506*b8 - 3107*b7 - 4679*b6 + 3238*b5 - 992*b4 + 5476*b3 + 2770*b2 - 1043*b1 - 1698) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/81\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$\beta_{10}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
10.1
 0.5 + 1.00210i 0.5 − 1.68614i 0.5 + 0.0126039i 0.5 + 1.27297i 0.5 + 2.22827i 0.5 − 0.258654i 0.5 − 2.22827i 0.5 + 0.258654i 0.5 − 0.0126039i 0.5 − 1.27297i 0.5 − 1.00210i 0.5 + 1.68614i
−0.183082 1.03831i 0 0.834822 0.303850i 1.33735 + 1.12217i 0 −2.31094 0.841112i −1.52266 2.63732i 0 0.920313 1.59403i
10.2 0.417037 + 2.36514i 0 −3.54056 + 1.28866i −0.0713060 0.0598329i 0 0.544891 + 0.198324i −2.12277 3.67675i 0 0.111776 0.193601i
19.1 −0.753189 + 0.274138i 0 −1.03995 + 0.872619i 0.477505 + 2.70806i 0 1.82076 + 1.52780i 1.34559 2.33062i 0 −1.10204 1.90878i
19.2 1.57954 0.574906i 0 0.632343 0.530599i 0.196143 + 1.11238i 0 −2.99441 2.51261i −0.987144 + 1.70978i 0 0.949332 + 1.64429i
37.1 0.318266 0.267057i 0 −0.317323 + 1.79963i 2.08159 0.757639i 0 −0.229151 1.29958i 0.795075 + 1.37711i 0 0.460168 0.797034i
37.2 1.62143 1.36054i 0 0.430663 2.44241i −2.52129 + 0.917674i 0 0.168844 + 0.957561i −0.508086 0.880031i 0 −2.83955 + 4.91825i
46.1 0.318266 + 0.267057i 0 −0.317323 1.79963i 2.08159 + 0.757639i 0 −0.229151 + 1.29958i 0.795075 1.37711i 0 0.460168 + 0.797034i
46.2 1.62143 + 1.36054i 0 0.430663 + 2.44241i −2.52129 0.917674i 0 0.168844 0.957561i −0.508086 + 0.880031i 0 −2.83955 4.91825i
64.1 −0.753189 0.274138i 0 −1.03995 0.872619i 0.477505 2.70806i 0 1.82076 1.52780i 1.34559 + 2.33062i 0 −1.10204 + 1.90878i
64.2 1.57954 + 0.574906i 0 0.632343 + 0.530599i 0.196143 1.11238i 0 −2.99441 + 2.51261i −0.987144 1.70978i 0 0.949332 1.64429i
73.1 −0.183082 + 1.03831i 0 0.834822 + 0.303850i 1.33735 1.12217i 0 −2.31094 + 0.841112i −1.52266 + 2.63732i 0 0.920313 + 1.59403i
73.2 0.417037 2.36514i 0 −3.54056 1.28866i −0.0713060 + 0.0598329i 0 0.544891 0.198324i −2.12277 + 3.67675i 0 0.111776 + 0.193601i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 73.2 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 81.2.e.a 12
3.b odd 2 1 27.2.e.a 12
9.c even 3 1 243.2.e.a 12
9.c even 3 1 243.2.e.b 12
9.d odd 6 1 243.2.e.c 12
9.d odd 6 1 243.2.e.d 12
12.b even 2 1 432.2.u.c 12
15.d odd 2 1 675.2.l.c 12
15.e even 4 2 675.2.u.b 24
27.e even 9 1 inner 81.2.e.a 12
27.e even 9 1 243.2.e.a 12
27.e even 9 1 243.2.e.b 12
27.e even 9 1 729.2.a.d 6
27.e even 9 2 729.2.c.b 12
27.f odd 18 1 27.2.e.a 12
27.f odd 18 1 243.2.e.c 12
27.f odd 18 1 243.2.e.d 12
27.f odd 18 1 729.2.a.a 6
27.f odd 18 2 729.2.c.e 12
108.l even 18 1 432.2.u.c 12
135.n odd 18 1 675.2.l.c 12
135.q even 36 2 675.2.u.b 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.2.e.a 12 3.b odd 2 1
27.2.e.a 12 27.f odd 18 1
81.2.e.a 12 1.a even 1 1 trivial
81.2.e.a 12 27.e even 9 1 inner
243.2.e.a 12 9.c even 3 1
243.2.e.a 12 27.e even 9 1
243.2.e.b 12 9.c even 3 1
243.2.e.b 12 27.e even 9 1
243.2.e.c 12 9.d odd 6 1
243.2.e.c 12 27.f odd 18 1
243.2.e.d 12 9.d odd 6 1
243.2.e.d 12 27.f odd 18 1
432.2.u.c 12 12.b even 2 1
432.2.u.c 12 108.l even 18 1
675.2.l.c 12 15.d odd 2 1
675.2.l.c 12 135.n odd 18 1
675.2.u.b 24 15.e even 4 2
675.2.u.b 24 135.q even 36 2
729.2.a.a 6 27.f odd 18 1
729.2.a.d 6 27.e even 9 1
729.2.c.b 12 27.e even 9 2
729.2.c.e 12 27.f odd 18 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(81, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 6 T^{11} + 21 T^{10} - 48 T^{9} + \cdots + 9$$
$3$ $$T^{12}$$
$5$ $$T^{12} - 3 T^{11} + 3 T^{10} + 12 T^{9} + \cdots + 9$$
$7$ $$T^{12} + 6 T^{11} + 12 T^{10} - 11 T^{9} + \cdots + 289$$
$11$ $$T^{12} + 3 T^{11} - 15 T^{10} + 6 T^{9} + \cdots + 9$$
$13$ $$T^{12} + 6 T^{11} + 48 T^{10} + 214 T^{9} + \cdots + 1$$
$17$ $$T^{12} + 9 T^{11} + 72 T^{10} + 189 T^{9} + \cdots + 729$$
$19$ $$T^{12} + 3 T^{11} + 39 T^{10} - 14 T^{9} + \cdots + 361$$
$23$ $$T^{12} - 12 T^{11} + 48 T^{10} + \cdots + 106929$$
$29$ $$T^{12} - 6 T^{11} + 21 T^{10} + \cdots + 45369$$
$31$ $$T^{12} - 3 T^{11} + 84 T^{10} + \cdots + 26569$$
$37$ $$T^{12} + 3 T^{11} + 66 T^{10} + \cdots + 24334489$$
$41$ $$T^{12} + 15 T^{11} + 93 T^{10} + \cdots + 11229201$$
$43$ $$T^{12} - 3 T^{11} - 60 T^{10} + \cdots + 3308761$$
$47$ $$T^{12} - 15 T^{11} + 111 T^{10} + \cdots + 42732369$$
$53$ $$(T^{6} - 9 T^{5} - 108 T^{4} + 513 T^{3} + \cdots - 12393)^{2}$$
$59$ $$T^{12} - 12 T^{11} + \cdots + 176384961$$
$61$ $$T^{12} - 12 T^{11} + \cdots + 273670849$$
$67$ $$T^{12} + 15 T^{11} + 255 T^{10} + \cdots + 8288641$$
$71$ $$T^{12} + 27 T^{11} + 504 T^{10} + \cdots + 729$$
$73$ $$T^{12} - 6 T^{11} + 210 T^{10} + \cdots + 185761$$
$79$ $$T^{12} + 42 T^{11} + 813 T^{10} + \cdots + 3508129$$
$83$ $$T^{12} + 39 T^{11} + \cdots + 6951057129$$
$89$ $$T^{12} + 9 T^{11} + \cdots + 1062042921$$
$97$ $$T^{12} - 3 T^{11} + 102 T^{10} + \cdots + 66765241$$