# Properties

 Label 273.2.bd.b Level $273$ Weight $2$ Character orbit 273.bd Analytic conductor $2.180$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [273,2,Mod(43,273)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(273, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("273.43");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.bd (of order $$6$$, degree $$2$$, 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: $$2.17991597518$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - 4 x^{15} + 10 x^{14} - 8 x^{13} - 3 x^{12} + 32 x^{11} - 5 x^{10} - 44 x^{9} + 214 x^{8} - 132 x^{7} - 45 x^{6} + 864 x^{5} - 243 x^{4} - 1944 x^{3} + 7290 x^{2} - 8748 x + 6561$$ x^16 - 4*x^15 + 10*x^14 - 8*x^13 - 3*x^12 + 32*x^11 - 5*x^10 - 44*x^9 + 214*x^8 - 132*x^7 - 45*x^6 + 864*x^5 - 243*x^4 - 1944*x^3 + 7290*x^2 - 8748*x + 6561 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{14} q^{2} - \beta_{5} q^{3} + (\beta_{13} + \beta_{7} + 2 \beta_{5} + 2) q^{4} + (\beta_{12} - \beta_{9} - 2 \beta_{5} + \beta_{3} - \beta_{2} - \beta_1 - 1) q^{5} + ( - \beta_{14} + \beta_{4}) q^{6} + (\beta_{15} - \beta_{3}) q^{7} + (2 \beta_{13} - 2 \beta_{10} + 2 \beta_{8} - 2 \beta_{7} + 2 \beta_{5} + \beta_{2} + \beta_1 + 1) q^{8} + ( - \beta_{5} - 1) q^{9}+O(q^{10})$$ q - b14 * q^2 - b5 * q^3 + (b13 + b7 + 2*b5 + 2) * q^4 + (b12 - b9 - 2*b5 + b3 - b2 - b1 - 1) * q^5 + (-b14 + b4) * q^6 + (b15 - b3) * q^7 + (2*b13 - 2*b10 + 2*b8 - 2*b7 + 2*b5 + b2 + b1 + 1) * q^8 + (-b5 - 1) * q^9 $$q - \beta_{14} q^{2} - \beta_{5} q^{3} + (\beta_{13} + \beta_{7} + 2 \beta_{5} + 2) q^{4} + (\beta_{12} - \beta_{9} - 2 \beta_{5} + \beta_{3} - \beta_{2} - \beta_1 - 1) q^{5} + ( - \beta_{14} + \beta_{4}) q^{6} + (\beta_{15} - \beta_{3}) q^{7} + (2 \beta_{13} - 2 \beta_{10} + 2 \beta_{8} - 2 \beta_{7} + 2 \beta_{5} + \beta_{2} + \beta_1 + 1) q^{8} + ( - \beta_{5} - 1) q^{9} + ( - \beta_{15} - 2 \beta_{12} + \beta_{11} - \beta_{10} + 2 \beta_{9} - \beta_{8} + 2 \beta_{5} + \cdots + 2 \beta_1) q^{10}+ \cdots + (\beta_{13} + \beta_{11} - \beta_{10} - \beta_{6} + 2 \beta_{5} - 2 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{99}+O(q^{100})$$ q - b14 * q^2 - b5 * q^3 + (b13 + b7 + 2*b5 + 2) * q^4 + (b12 - b9 - 2*b5 + b3 - b2 - b1 - 1) * q^5 + (-b14 + b4) * q^6 + (b15 - b3) * q^7 + (2*b13 - 2*b10 + 2*b8 - 2*b7 + 2*b5 + b2 + b1 + 1) * q^8 + (-b5 - 1) * q^9 + (-b15 - 2*b12 + b11 - b10 + 2*b9 - b8 + 2*b5 + 2*b3 + 2*b1) * q^10 + (2*b15 - b13 - b12 + b11 - b9 - b8 - b7 + 2*b6 - b5 - b1) * q^11 + (b13 + b10 + b8 + b7 + 2) * q^12 + (-b14 + b13 - b12 + b11 - b9 + b8 - b7 + b5 + b4 + b3 + b2 - b1 + 1) * q^13 + (-b11 - b6) * q^14 + (b15 - b9 - b5 - b2 - 1) * q^15 + (b15 + b13 - 2*b12 - 2*b10 - b8 - b7 + 4*b5 - 2*b3 + 2*b2 + 2*b1 + 2) * q^16 + (-b13 + b12 - b10 + b9 + b8 - 2*b7 + b6 + b5 + b2 + 2*b1 - 1) * q^17 + b4 * q^18 + (b15 + b14 - b13 + 2*b12 - b10 + 2*b9 - b8 - b5 - b4 - b3 - 2*b2 - 3) * q^19 + (2*b13 + b12 + 2*b11 + 2*b10 + 2*b8 + b6 - b5 - b1) * q^20 - b3 * q^21 + (-3*b15 - 2*b13 + 3*b10 - 2*b9 - 3*b8 + b7 - 2*b6 - 5*b5 - 3*b3 - 2*b2 - 2*b1 - 3) * q^22 + (b14 - 2*b13 + b12 - b11 - 2*b8 + 2*b7 - b5 - 2*b4 - b2 - b1 - 1) * q^23 + (b12 - 2*b10 + b9 + 2*b8 + b5 + b1) * q^24 + (-2*b15 - 2*b14 + b13 - b12 + b10 - b9 - b8 - b7 + b4 + b3 - 2*b2 + 2*b1 - 2) * q^25 + (-3*b15 + b14 - b13 - 2*b12 + b11 - 2*b8 + b7 - b5 - b4 + 4*b3 - 2*b2 + 1) * q^26 - q^27 + (2*b15 - b12 + b2 - b1 + 1) * q^28 + (2*b15 - 4*b13 + 3*b12 - 2*b11 + 2*b10 - b9 - 2*b8 + 4*b7 - 3*b5 - 4*b3 - 2*b2 - 3*b1 - 2) * q^29 + (b15 + b13 - 2*b12 + 2*b9 + b7 - b6 + 2*b5 + b3 + 2*b2 + 2) * q^30 + (-b12 + b11 + b9 - b8 + b7 - b6 + 2*b5 - b4 + b3 + 2*b2 + 2*b1 + 1) * q^31 + (4*b15 - 2*b11 - 3*b9 - b6 + b5 - 4*b3 + 3*b2 - 3*b1 + 2) * q^32 + (2*b15 - b12 + 2*b11 - b10 - b9 - b8 - b7 + b6 + b5 - 2*b3 + b2 + 1) * q^33 + (-b13 + 2*b12 + b10 - 2*b9 - 2*b8 + 2*b7 + 4*b5 + 2*b4 - 7*b3 + b2 + b1 + 2) * q^34 + (-b10 + b8 - b7 + b5 + 1) * q^35 + (b10 + b8 - 2*b5) * q^36 + (-b15 - b13 + 2*b11 - 2*b9 - b8 - b7 + 4*b6 - 2*b5 - 2*b2 - 2) * q^37 + (-4*b15 + 2*b14 - b13 + 3*b12 - b11 - b10 + 3*b9 - b8 - b7 - b6 - b4 + 2*b3 - 3*b2 + 3*b1 - 6) * q^38 + (b15 - b10 - b9 + b8 - b7 - b6 + 2*b5 + b4 + b2 + 2) * q^39 + (-6*b15 - 3*b13 + 3*b12 - 2*b11 - 3*b10 + 3*b9 + b8 + b7 - 2*b6 + 3*b3 - 2*b2 + 2*b1 - 4) * q^40 + (-2*b15 + b14 - b13 + b12 + 2*b10 - 2*b9 - 3*b8 - b7 - 3*b5 - 3*b2 + b1 - 5) * q^41 - b11 * q^42 + (2*b15 + b13 - 2*b12 - 2*b10 + 2*b8 - b7 + 2*b6 + b5 + 2*b3 - 2*b1 + 3) * q^43 + (-2*b13 - 3*b12 - b11 + 2*b10 + 3*b9 + b6 - 6*b5 + b4 + 8*b3 - 3) * q^44 + (b15 - b12 + b5 - b3 + b1) * q^45 + (2*b15 + b14 + 2*b13 + b10 - b9 + b8 - b7 + 2*b5 - b4 - 2*b3 + b2 - b1 - 1) * q^46 + (-4*b13 + b12 + 4*b10 - b9 - b8 + b7 - 4*b5 - 2*b4 - 2*b3 - 2*b2 - 2*b1 - 2) * q^47 + (-b15 + 2*b13 - b10 + 2*b9 + b8 + b7 + 4*b5 - b3 + 2*b2 + 2*b1 + 2) * q^48 - b5 * q^49 + (4*b15 + b13 - b12 + b11 + 3*b9 + b8 + b7 + 2*b6 + 5*b5 + 4*b2 - b1 + 8) * q^50 + (-2*b13 + 2*b12 + b11 - 2*b10 + 2*b9 - b8 - b7 + b6 - b2 + b1 - 3) * q^51 + (-b13 + 2*b12 + 2*b11 + b10 + 3*b9 + b8 + b7 + b6 - 4*b5 - b4 + 4*b3 - 2*b1 - 2) * q^52 + (-4*b15 + 3*b13 + b12 - 2*b11 + 3*b10 + b9 + 2*b8 + 2*b7 - 2*b6 + 2*b3 + b2 - b1 + 2) * q^53 + b14 * q^54 + (-b15 - b14 + 4*b13 + b12 - b11 - 3*b9 + 4*b8 - 4*b7 + b5 + 2*b4 + 2*b3 + 2*b2 - b1 + 2) * q^55 + (-b13 + b10 - 2*b9 - b8 - 4*b5 - 2*b2 - 2*b1 - 2) * q^56 + (-b8 + b7 - 2*b5 - b4 - b3 - 2*b2 - 2*b1 - 1) * q^57 + (3*b15 + 2*b14 + 2*b13 - 4*b11 + 5*b10 + 5*b8 + 3*b7 - 2*b6 - 4*b5 - 2*b4 - 3*b3 + 4) * q^58 + (-6*b15 - b14 - b12 + 2*b11 - b10 - b9 - b8 - b7 + b6 + 2*b5 + b4 + 6*b3 + b2) * q^59 + (b12 + b11 - b9 + 2*b8 - 2*b7 - b6 - 2*b5 - b2 - b1 - 1) * q^60 + (b15 + 3*b13 - 2*b12 - 2*b10 + 3*b9 + 2*b8 + b7 - 2*b6 + 4*b5 + b3 + 3*b2 + b1 + 3) * q^61 + (-2*b15 - b14 + 3*b13 + b12 + b11 - 3*b9 + 3*b8 - 3*b7 + 4*b5 + 2*b4 + 4*b3 + 2*b2 - b1 + 2) * q^62 - b15 * q^63 + (4*b15 + 2*b14 + 2*b13 - 3*b12 - 4*b11 + 2*b10 - 3*b9 + b8 + b7 - 4*b6 - b4 - 2*b3 + 2*b2 - 2*b1 + 2) * q^64 + (-4*b15 + b14 - 2*b13 + b12 - b11 - 4*b10 + 3*b9 - 2*b7 - 3*b6 + 2*b5 - b4 + 2*b3 + 2*b2 + 2*b1 - 3) * q^65 + (-6*b15 + b13 - 2*b12 - 2*b11 + b10 - 2*b9 - 2*b8 - 2*b7 - 2*b6 + 3*b3 - 1) * q^66 + (3*b15 + b14 + 3*b12 - 3*b10 + 4*b9 + 3*b8 + 4*b5 + b2 + 3*b1 + 1) * q^67 + (b15 + b14 + 3*b13 - b12 - 5*b11 - 2*b9 + 3*b8 - 3*b7 - 3*b5 - 2*b4 - 2*b3 + 3*b2 + b1 + 3) * q^68 + (-b14 + 2*b10 - b9 - 2*b8 + 2*b7 + b6 - b5 - b4 - b2 - b1) * q^69 + (-b12 + b9 - 2*b8 + 2*b7 + 2*b5 - b4 + 1) * q^70 + (2*b15 + 2*b13 + b12 - 2*b11 - 2*b7 - b6 - 2*b3 - b1 - 1) * q^71 + (-2*b13 + b12 + b9 + 2*b7 - b5 - b2 - 1) * q^72 + (3*b13 - 3*b10 + 2*b8 - 2*b7 - 4*b5 + 3*b3 + b2 + b1 - 2) * q^73 + (-6*b15 + 2*b10 + b9 - 2*b8 + 2*b7 + b6 + b5 - 6*b3 + b2 + b1) * q^74 + (-b15 - b14 + 2*b13 - 3*b12 - b10 + 2*b9 + b8 - 2*b7 + 4*b5 + 2*b4 + 2*b3 + b2 + 3*b1 + 1) * q^75 + (-b15 + b14 + 2*b11 + b9 + 4*b6 - b5 + b2 - 3) * q^76 + (2*b14 + b12 + b9 + b8 + b7 - b4 - b2 + b1 + 1) * q^77 + (b15 + b13 - 4*b12 + b10 - b8 + b7 - b6 - b4 + 3*b3 + 2*b2 + 3) * q^78 + (-2*b15 - b13 - 2*b12 - b10 - 2*b9 + b3 + 2*b2 - 2*b1 + 9) * q^79 + (4*b15 + 2*b14 + 3*b13 + b12 + b11 - b10 + 3*b9 + 4*b8 + 3*b7 + 2*b6 + 7*b5 + 2*b2 + b1 + 10) * q^80 + b5 * q^81 + (2*b15 + b14 - 4*b13 - 2*b12 + b10 + 3*b9 - b8 - 3*b7 + 2*b6 - 5*b5 + b4 + 2*b3 + 3*b2 + b1 - 6) * q^82 + (-b13 + 2*b12 + b10 - 2*b9 + 4*b5 - b4 + 5*b3 - b2 - b1 + 2) * q^83 + (2*b15 - b9 - 2*b3 + b2 - b1 + 1) * q^84 + (-b15 - b14 - 2*b13 + 2*b12 - 4*b11 + b9 + 2*b7 - 2*b6 - 4*b5 + b4 + b3 - b2 - b1 + 1) * q^85 + (2*b11 - 4*b8 + 4*b7 - 2*b6 - 2*b5 - 3*b4 - 6*b3 - 3*b2 - 3*b1 - 1) * q^86 + (-2*b15 - 2*b13 + b12 + 4*b10 - 3*b9 - 4*b8 + 2*b7 + 2*b6 - 3*b5 - 2*b3 - 3*b2 - 2*b1 - 1) * q^87 + (-2*b15 - 3*b14 - b13 + 5*b12 + 4*b11 + 4*b10 - 2*b9 + 3*b8 + b7 - 12*b5 + 6*b4 + 4*b3 - 3*b2 - 5*b1 - 3) * q^88 + (5*b15 - b14 + 4*b13 - 2*b12 + b10 + b9 + 3*b8 + 4*b7 + 2*b5 + 3*b2 - 2*b1 + 5) * q^89 + (2*b15 + b13 - b11 + b10 + b8 + b7 - b6 - b3 + 2*b2 - 2*b1 + 2) * q^90 + (b12 - b11 + b10 - b9 + b8 + b7 - b5 - b4 + b3 - b2 - b1) * q^91 + (-b13 - 3*b12 - b10 - 3*b9 - 4*b8 - 4*b7 - 2*b2 + 2*b1 - 7) * q^92 + (b15 - b14 + b13 + b12 - b11 + b10 + 2*b9 + b7 - 2*b6 + b5 + b2 + b1 - 1) * q^93 + (-2*b15 - 7*b13 + 2*b12 - 2*b11 + 5*b10 + 3*b9 - 2*b8 + 7*b7 - 4*b5 + 4*b3 - 5*b2 - 2*b1 - 5) * q^94 + (-2*b15 - 2*b13 + b12 + b10 - 3*b9 - b8 - b7 + 3*b6 - 7*b5 - 2*b3 - 3*b2 - 2*b1 - 5) * q^95 + (3*b12 - b11 - 3*b9 + b6 + 2*b5 - 4*b3 + 1) * q^96 + (3*b15 - 3*b14 + b13 - 2*b12 - 2*b11 + b10 - b9 + b8 - b6 + b5 + 3*b4 - 3*b3 + b2 + b1 + 2) * q^97 + (-b14 + b4) * q^98 + (b13 + b11 - b10 - b6 + 2*b5 - 2*b3 + b2 + b1 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 8 q^{3} + 14 q^{4} - 8 q^{9}+O(q^{10})$$ 16 * q + 8 * q^3 + 14 * q^4 - 8 * q^9 $$16 q + 8 q^{3} + 14 q^{4} - 8 q^{9} - 4 q^{10} + 28 q^{12} - 12 q^{13} - 4 q^{14} - 12 q^{15} - 10 q^{16} - 2 q^{17} + 18 q^{20} - 18 q^{22} - 6 q^{23} - 20 q^{25} + 20 q^{26} - 16 q^{27} - 12 q^{29} + 4 q^{30} - 30 q^{32} + 6 q^{35} + 14 q^{36} - 6 q^{37} - 24 q^{38} - 28 q^{40} - 30 q^{41} - 2 q^{42} + 14 q^{43} - 12 q^{45} - 42 q^{46} + 10 q^{48} + 8 q^{49} + 84 q^{50} - 4 q^{51} + 30 q^{52} + 28 q^{53} + 2 q^{55} - 12 q^{56} + 66 q^{58} - 24 q^{59} + 2 q^{61} - 20 q^{62} - 48 q^{64} - 44 q^{65} - 36 q^{66} + 30 q^{67} + 36 q^{68} + 6 q^{69} - 6 q^{71} + 6 q^{74} - 10 q^{75} - 24 q^{76} + 32 q^{77} + 10 q^{78} + 92 q^{79} + 114 q^{80} - 8 q^{81} - 42 q^{82} + 48 q^{85} + 12 q^{87} + 62 q^{88} + 18 q^{89} + 8 q^{90} - 116 q^{92} - 6 q^{93} - 24 q^{94} - 24 q^{95} - 6 q^{97}+O(q^{100})$$ 16 * q + 8 * q^3 + 14 * q^4 - 8 * q^9 - 4 * q^10 + 28 * q^12 - 12 * q^13 - 4 * q^14 - 12 * q^15 - 10 * q^16 - 2 * q^17 + 18 * q^20 - 18 * q^22 - 6 * q^23 - 20 * q^25 + 20 * q^26 - 16 * q^27 - 12 * q^29 + 4 * q^30 - 30 * q^32 + 6 * q^35 + 14 * q^36 - 6 * q^37 - 24 * q^38 - 28 * q^40 - 30 * q^41 - 2 * q^42 + 14 * q^43 - 12 * q^45 - 42 * q^46 + 10 * q^48 + 8 * q^49 + 84 * q^50 - 4 * q^51 + 30 * q^52 + 28 * q^53 + 2 * q^55 - 12 * q^56 + 66 * q^58 - 24 * q^59 + 2 * q^61 - 20 * q^62 - 48 * q^64 - 44 * q^65 - 36 * q^66 + 30 * q^67 + 36 * q^68 + 6 * q^69 - 6 * q^71 + 6 * q^74 - 10 * q^75 - 24 * q^76 + 32 * q^77 + 10 * q^78 + 92 * q^79 + 114 * q^80 - 8 * q^81 - 42 * q^82 + 48 * q^85 + 12 * q^87 + 62 * q^88 + 18 * q^89 + 8 * q^90 - 116 * q^92 - 6 * q^93 - 24 * q^94 - 24 * q^95 - 6 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 4 x^{15} + 10 x^{14} - 8 x^{13} - 3 x^{12} + 32 x^{11} - 5 x^{10} - 44 x^{9} + 214 x^{8} - 132 x^{7} - 45 x^{6} + 864 x^{5} - 243 x^{4} - 1944 x^{3} + 7290 x^{2} - 8748 x + 6561$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{15} - 4 \nu^{14} + 10 \nu^{13} - 8 \nu^{12} - 3 \nu^{11} + 32 \nu^{10} - 5 \nu^{9} - 44 \nu^{8} + 214 \nu^{7} - 132 \nu^{6} - 45 \nu^{5} + 864 \nu^{4} - 243 \nu^{3} - 1944 \nu^{2} + 7290 \nu - 8748 ) / 2187$$ (v^15 - 4*v^14 + 10*v^13 - 8*v^12 - 3*v^11 + 32*v^10 - 5*v^9 - 44*v^8 + 214*v^7 - 132*v^6 - 45*v^5 + 864*v^4 - 243*v^3 - 1944*v^2 + 7290*v - 8748) / 2187 $$\beta_{3}$$ $$=$$ $$( 57 \nu^{15} - 5518 \nu^{14} + 7381 \nu^{13} - 18694 \nu^{12} - 28900 \nu^{11} - 3540 \nu^{10} - 97601 \nu^{9} - 227974 \nu^{8} - 68965 \nu^{7} - 678274 \nu^{6} + \cdots - 11910402 ) / 64152$$ (57*v^15 - 5518*v^14 + 7381*v^13 - 18694*v^12 - 28900*v^11 - 3540*v^10 - 97601*v^9 - 227974*v^8 - 68965*v^7 - 678274*v^6 - 899556*v^5 - 99108*v^4 - 2301399*v^3 - 4692006*v^2 + 5341869*v - 11910402) / 64152 $$\beta_{4}$$ $$=$$ $$( - 2194 \nu^{15} + 19537 \nu^{14} - 30118 \nu^{13} + 44909 \nu^{12} + 84636 \nu^{11} - 34820 \nu^{10} + 195806 \nu^{9} + 639719 \nu^{8} - 100294 \nu^{7} + \cdots + 35549685 ) / 192456$$ (-2194*v^15 + 19537*v^14 - 30118*v^13 + 44909*v^12 + 84636*v^11 - 34820*v^10 + 195806*v^9 + 639719*v^8 - 100294*v^7 + 1629003*v^6 + 2560212*v^5 - 867564*v^4 + 4904226*v^3 + 15869601*v^2 - 21366990*v + 35549685) / 192456 $$\beta_{5}$$ $$=$$ $$( 686 \nu^{15} - 4953 \nu^{14} + 7898 \nu^{13} - 10141 \nu^{12} - 20180 \nu^{11} + 11796 \nu^{10} - 40674 \nu^{9} - 152359 \nu^{8} + 47706 \nu^{7} - 369947 \nu^{6} + \cdots - 8790525 ) / 42768$$ (686*v^15 - 4953*v^14 + 7898*v^13 - 10141*v^12 - 20180*v^11 + 11796*v^10 - 40674*v^9 - 152359*v^8 + 47706*v^7 - 369947*v^6 - 611332*v^5 + 281604*v^4 - 1069686*v^3 - 4003857*v^2 + 5544450*v - 8790525) / 42768 $$\beta_{6}$$ $$=$$ $$( - 2551 \nu^{15} - 30666 \nu^{14} + 35541 \nu^{13} - 130226 \nu^{12} - 178900 \nu^{11} - 67220 \nu^{10} - 709257 \nu^{9} - 1424186 \nu^{8} - 727301 \nu^{7} + \cdots - 73435086 ) / 128304$$ (-2551*v^15 - 30666*v^14 + 35541*v^13 - 130226*v^12 - 178900*v^11 - 67220*v^10 - 709257*v^9 - 1424186*v^8 - 727301*v^7 - 4565062*v^6 - 5476404*v^5 - 1553364*v^4 - 16204887*v^3 - 25765938*v^2 + 27258525*v - 73435086) / 128304 $$\beta_{7}$$ $$=$$ $$( 515 \nu^{15} - 348 \nu^{14} + 1311 \nu^{13} + 3844 \nu^{12} + 2060 \nu^{11} + 9484 \nu^{10} + 27621 \nu^{9} + 20896 \nu^{8} + 68905 \nu^{7} + 127508 \nu^{6} + 67356 \nu^{5} + \cdots + 769824 ) / 11664$$ (515*v^15 - 348*v^14 + 1311*v^13 + 3844*v^12 + 2060*v^11 + 9484*v^10 + 27621*v^9 + 20896*v^8 + 68905*v^7 + 127508*v^6 + 67356*v^5 + 229788*v^4 + 591435*v^3 - 194724*v^2 + 827415*v + 769824) / 11664 $$\beta_{8}$$ $$=$$ $$( 533 \nu^{15} - 2036 \nu^{14} + 3545 \nu^{13} - 1876 \nu^{12} - 6828 \nu^{11} + 8356 \nu^{10} - 2509 \nu^{9} - 49216 \nu^{8} + 47039 \nu^{7} - 80580 \nu^{6} - 210588 \nu^{5} + \cdots - 2881008 ) / 11664$$ (533*v^15 - 2036*v^14 + 3545*v^13 - 1876*v^12 - 6828*v^11 + 8356*v^10 - 2509*v^9 - 49216*v^8 + 47039*v^7 - 80580*v^6 - 210588*v^5 + 196740*v^4 - 117747*v^3 - 1626156*v^2 + 2440449*v - 2881008) / 11664 $$\beta_{9}$$ $$=$$ $$( 2209 \nu^{15} - 1038 \nu^{14} + 4653 \nu^{13} + 18122 \nu^{12} + 10156 \nu^{11} + 37244 \nu^{10} + 122175 \nu^{9} + 99098 \nu^{8} + 279395 \nu^{7} + 580462 \nu^{6} + \cdots + 4543614 ) / 42768$$ (2209*v^15 - 1038*v^14 + 4653*v^13 + 18122*v^12 + 10156*v^11 + 37244*v^10 + 122175*v^9 + 99098*v^8 + 279395*v^7 + 580462*v^6 + 311100*v^5 + 902988*v^4 + 2670273*v^3 - 543510*v^2 + 2746629*v + 4543614) / 42768 $$\beta_{10}$$ $$=$$ $$( 25793 \nu^{15} - 43421 \nu^{14} + 101321 \nu^{13} + 100655 \nu^{12} - 37320 \nu^{11} + 444640 \nu^{10} + 869603 \nu^{9} - 115039 \nu^{8} + 2984339 \nu^{7} + \cdots - 21167973 ) / 384912$$ (25793*v^15 - 43421*v^14 + 101321*v^13 + 100655*v^12 - 37320*v^11 + 444640*v^10 + 869603*v^9 - 115039*v^8 + 2984339*v^7 + 2850081*v^6 - 1361736*v^5 + 10317888*v^4 + 17012997*v^3 - 33436557*v^2 + 64514313*v - 21167973) / 384912 $$\beta_{11}$$ $$=$$ $$( - 35555 \nu^{15} + 209918 \nu^{14} - 340439 \nu^{13} + 377302 \nu^{12} + 822876 \nu^{11} - 555316 \nu^{10} + 1446739 \nu^{9} + 6248686 \nu^{8} + \cdots + 358488666 ) / 384912$$ (-35555*v^15 + 209918*v^14 - 340439*v^13 + 377302*v^12 + 822876*v^11 - 555316*v^10 + 1446739*v^9 + 6248686*v^8 - 2436425*v^7 + 14366802*v^6 + 25605324*v^5 - 12546900*v^4 + 39127293*v^3 + 171173574*v^2 - 235676223*v + 358488666) / 384912 $$\beta_{12}$$ $$=$$ $$( 36175 \nu^{15} - 126178 \nu^{14} + 228019 \nu^{13} - 76154 \nu^{12} - 382332 \nu^{11} + 612740 \nu^{10} + 137617 \nu^{9} - 2689898 \nu^{8} + 3627757 \nu^{7} + \cdots - 166373838 ) / 384912$$ (36175*v^15 - 126178*v^14 + 228019*v^13 - 76154*v^12 - 382332*v^11 + 612740*v^10 + 137617*v^9 - 2689898*v^8 + 3627757*v^7 - 3487038*v^6 - 11616444*v^5 + 14749236*v^4 - 1187217*v^3 - 99205722*v^2 + 155611611*v - 166373838) / 384912 $$\beta_{13}$$ $$=$$ $$( 13197 \nu^{15} - 41905 \nu^{14} + 78133 \nu^{13} - 14485 \nu^{12} - 117448 \nu^{11} + 226080 \nu^{10} + 116311 \nu^{9} - 824875 \nu^{8} + 1357079 \nu^{7} + \cdots - 53118585 ) / 128304$$ (13197*v^15 - 41905*v^14 + 78133*v^13 - 14485*v^12 - 117448*v^11 + 226080*v^10 + 116311*v^9 - 824875*v^8 + 1357079*v^7 - 830875*v^6 - 3648648*v^5 + 5321808*v^4 + 960417*v^3 - 33076593*v^2 + 52548021*v - 53118585) / 128304 $$\beta_{14}$$ $$=$$ $$( - 47866 \nu^{15} + 127777 \nu^{14} - 247654 \nu^{13} - 30859 \nu^{12} + 311940 \nu^{11} - 813620 \nu^{10} - 843826 \nu^{9} + 2022815 \nu^{8} - 5148238 \nu^{7} + \cdots + 134472069 ) / 384912$$ (-47866*v^15 + 127777*v^14 - 247654*v^13 - 30859*v^12 + 311940*v^11 - 813620*v^10 - 843826*v^9 + 2022815*v^8 - 5148238*v^7 - 23901*v^6 + 9382500*v^5 - 19635588*v^4 - 14476806*v^3 + 98903673*v^2 - 167505246*v + 134472069) / 384912 $$\beta_{15}$$ $$=$$ $$( - 5481 \nu^{15} + 12006 \nu^{14} - 25157 \nu^{13} - 12274 \nu^{12} + 21460 \nu^{11} - 95516 \nu^{10} - 141111 \nu^{9} + 126350 \nu^{8} - 619643 \nu^{7} + \cdots + 10314378 ) / 42768$$ (-5481*v^15 + 12006*v^14 - 25157*v^13 - 12274*v^12 + 21460*v^11 - 95516*v^10 - 141111*v^9 + 126350*v^8 - 619643*v^7 - 302534*v^6 + 673252*v^5 - 2246412*v^4 - 2622393*v^3 + 9246798*v^2 - 16556157*v + 10314378) / 42768
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{14} + \beta_{9} + \beta_{8} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1$$ b14 + b9 + b8 + b5 - b4 - b3 + b2 + b1 $$\nu^{3}$$ $$=$$ $$-2\beta_{15} - \beta_{13} - \beta_{12} - 2\beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} - 2\beta_{4} + \beta_{3} - \beta _1 - 2$$ -2*b15 - b13 - b12 - 2*b8 + b7 + b6 - b5 - 2*b4 + b3 - b1 - 2 $$\nu^{4}$$ $$=$$ $$\beta_{15} - 3 \beta_{14} + 5 \beta_{13} - 4 \beta_{12} + 2 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} - 3 \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} + 3 \beta_{2} - \beta_1$$ b15 - 3*b14 + 5*b13 - 4*b12 + 2*b11 - b10 - b9 + b8 - 3*b7 + b5 + b4 + b3 + 3*b2 - b1 $$\nu^{5}$$ $$=$$ $$7 \beta_{15} - 2 \beta_{14} + \beta_{12} - 5 \beta_{11} - 5 \beta_{10} + 5 \beta_{9} + \beta_{8} - 6 \beta_{6} + 3 \beta_{5} + 4 \beta_{4} + 4 \beta_{2} - 2 \beta_1$$ 7*b15 - 2*b14 + b12 - 5*b11 - 5*b10 + 5*b9 + b8 - 6*b6 + 3*b5 + 4*b4 + 4*b2 - 2*b1 $$\nu^{6}$$ $$=$$ $$10 \beta_{15} - 5 \beta_{14} - 9 \beta_{13} + 13 \beta_{12} - 6 \beta_{11} + 12 \beta_{10} - 10 \beta_{9} - 5 \beta_{8} + 3 \beta_{7} - 20 \beta_{5} + 4 \beta_{4} - 12 \beta_{3} - 11 \beta_{2} - 11 \beta _1 - 16$$ 10*b15 - 5*b14 - 9*b13 + 13*b12 - 6*b11 + 12*b10 - 10*b9 - 5*b8 + 3*b7 - 20*b5 + 4*b4 - 12*b3 - 11*b2 - 11*b1 - 16 $$\nu^{7}$$ $$=$$ $$- 15 \beta_{15} - 2 \beta_{14} - 8 \beta_{13} - 2 \beta_{12} + 9 \beta_{11} + 9 \beta_{10} - 6 \beta_{9} - 5 \beta_{8} - 8 \beta_{7} + 10 \beta_{6} + 21 \beta_{5} + 16 \beta_{4} + 4 \beta_{3} + \beta_{2} + 2 \beta _1 + 14$$ -15*b15 - 2*b14 - 8*b13 - 2*b12 + 9*b11 + 9*b10 - 6*b9 - 5*b8 - 8*b7 + 10*b6 + 21*b5 + 16*b4 + 4*b3 + b2 + 2*b1 + 14 $$\nu^{8}$$ $$=$$ $$- 37 \beta_{15} + 22 \beta_{14} - 10 \beta_{13} + 26 \beta_{12} + 18 \beta_{11} - 12 \beta_{10} - 18 \beta_{9} + 6 \beta_{8} + \beta_{7} + 4 \beta_{6} - 18 \beta_{5} + 39 \beta_{3} - 20 \beta_{2} - 10$$ -37*b15 + 22*b14 - 10*b13 + 26*b12 + 18*b11 - 12*b10 - 18*b9 + 6*b8 + b7 + 4*b6 - 18*b5 + 39*b3 - 20*b2 - 10 $$\nu^{9}$$ $$=$$ $$22 \beta_{15} + 40 \beta_{14} + 46 \beta_{13} - 18 \beta_{12} + 14 \beta_{11} + 2 \beta_{10} + 26 \beta_{9} + 36 \beta_{8} + 44 \beta_{7} + 2 \beta_{6} + 48 \beta_{5} - 22 \beta_{4} + 15 \beta_{3} - 26 \beta_{2} + 22 \beta _1 + 36$$ 22*b15 + 40*b14 + 46*b13 - 18*b12 + 14*b11 + 2*b10 + 26*b9 + 36*b8 + 44*b7 + 2*b6 + 48*b5 - 22*b4 + 15*b3 - 26*b2 + 22*b1 + 36 $$\nu^{10}$$ $$=$$ $$- 11 \beta_{15} + 14 \beta_{14} - 24 \beta_{13} - 18 \beta_{12} - 44 \beta_{11} - 50 \beta_{10} + 34 \beta_{9} + 15 \beta_{8} + 14 \beta_{7} - 22 \beta_{6} - 22 \beta_{5} - 62 \beta_{4} - 62 \beta_{3} - 40 \beta_{2} + 4 \beta _1 + 112$$ -11*b15 + 14*b14 - 24*b13 - 18*b12 - 44*b11 - 50*b10 + 34*b9 + 15*b8 + 14*b7 - 22*b6 - 22*b5 - 62*b4 - 62*b3 - 40*b2 + 4*b1 + 112 $$\nu^{11}$$ $$=$$ $$41 \beta_{15} - 78 \beta_{14} + 89 \beta_{13} - 90 \beta_{12} + 2 \beta_{11} + 68 \beta_{10} - 42 \beta_{9} - 92 \beta_{8} - 5 \beta_{7} + 61 \beta_{6} - 117 \beta_{5} - 18 \beta_{4} - 73 \beta_{3} + 66 \beta_{2} + 59 \beta _1 + 153$$ 41*b15 - 78*b14 + 89*b13 - 90*b12 + 2*b11 + 68*b10 - 42*b9 - 92*b8 - 5*b7 + 61*b6 - 117*b5 - 18*b4 - 73*b3 + 66*b2 + 59*b1 + 153 $$\nu^{12}$$ $$=$$ $$- 30 \beta_{15} + 5 \beta_{14} + 24 \beta_{13} - 78 \beta_{12} - 28 \beta_{11} - 211 \beta_{10} + 267 \beta_{9} + 101 \beta_{8} - 299 \beta_{7} - 122 \beta_{6} + 356 \beta_{5} + 19 \beta_{4} - 74 \beta_{3} + 293 \beta_{2} + 481 \beta _1 - 232$$ -30*b15 + 5*b14 + 24*b13 - 78*b12 - 28*b11 - 211*b10 + 267*b9 + 101*b8 - 299*b7 - 122*b6 + 356*b5 + 19*b4 - 74*b3 + 293*b2 + 481*b1 - 232 $$\nu^{13}$$ $$=$$ $$27 \beta_{15} + 238 \beta_{14} - 349 \beta_{13} + 305 \beta_{12} - 146 \beta_{11} + 220 \beta_{10} + 121 \beta_{9} - 164 \beta_{8} + 265 \beta_{7} - 17 \beta_{6} - 271 \beta_{5} - 486 \beta_{4} - 3 \beta_{3} + 166 \beta_{2} - 384 \beta _1 - 643$$ 27*b15 + 238*b14 - 349*b13 + 305*b12 - 146*b11 + 220*b10 + 121*b9 - 164*b8 + 265*b7 - 17*b6 - 271*b5 - 486*b4 - 3*b3 + 166*b2 - 384*b1 - 643 $$\nu^{14}$$ $$=$$ $$- 289 \beta_{15} - 753 \beta_{14} + 145 \beta_{13} - 851 \beta_{12} + 332 \beta_{11} + 383 \beta_{10} - 608 \beta_{9} - 481 \beta_{8} - 73 \beta_{7} + 314 \beta_{6} + 209 \beta_{5} + 146 \beta_{4} + 626 \beta_{3} - 9 \beta_{2} + \cdots - 687$$ -289*b15 - 753*b14 + 145*b13 - 851*b12 + 332*b11 + 383*b10 - 608*b9 - 481*b8 - 73*b7 + 314*b6 + 209*b5 + 146*b4 + 626*b3 - 9*b2 - 653*b1 - 687 $$\nu^{15}$$ $$=$$ $$786 \beta_{15} - 652 \beta_{14} + 1048 \beta_{13} - 316 \beta_{12} + 169 \beta_{11} - 773 \beta_{10} - 385 \beta_{9} + 1261 \beta_{8} - 590 \beta_{7} - 644 \beta_{6} + 33 \beta_{5} + 1406 \beta_{4} + 493 \beta_{3} + 1169 \beta_{2} + \cdots + 1595$$ 786*b15 - 652*b14 + 1048*b13 - 316*b12 + 169*b11 - 773*b10 - 385*b9 + 1261*b8 - 590*b7 - 644*b6 + 33*b5 + 1406*b4 + 493*b3 + 1169*b2 - 1460*b1 + 1595

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/273\mathbb{Z}\right)^\times$$.

 $$n$$ $$92$$ $$106$$ $$157$$ $$\chi(n)$$ $$1$$ $$1 + \beta_{5}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1
 −1.67549 + 0.438998i 1.35201 − 1.08262i 1.33452 − 1.10411i −1.36010 − 1.07244i 0.750089 + 1.56121i −0.306536 − 1.70471i 0.590887 − 1.62814i 1.31463 + 1.12772i −1.67549 − 0.438998i 1.35201 + 1.08262i 1.33452 + 1.10411i −1.36010 + 1.07244i 0.750089 − 1.56121i −0.306536 + 1.70471i 0.590887 + 1.62814i 1.31463 − 1.12772i
−2.20165 1.27113i 0.500000 0.866025i 2.23152 + 3.86511i 4.07309i −2.20165 + 1.27113i 0.866025 0.500000i 6.26168i −0.500000 0.866025i −5.17741 + 8.96754i
43.2 −1.98604 1.14664i 0.500000 0.866025i 1.62956 + 2.82249i 0.692320i −1.98604 + 1.14664i −0.866025 + 0.500000i 2.88753i −0.500000 0.866025i 0.793841 1.37497i
43.3 −1.44724 0.835563i 0.500000 0.866025i 0.396329 + 0.686463i 2.68351i −1.44724 + 0.835563i 0.866025 0.500000i 2.01762i −0.500000 0.866025i 2.24224 3.88368i
43.4 −0.654865 0.378087i 0.500000 0.866025i −0.714101 1.23686i 4.01537i −0.654865 + 0.378087i −0.866025 + 0.500000i 2.59231i −0.500000 0.866025i −1.51816 + 2.62953i
43.5 0.924500 + 0.533760i 0.500000 0.866025i −0.430200 0.745128i 0.994065i 0.924500 0.533760i 0.866025 0.500000i 3.05354i −0.500000 0.866025i 0.530592 0.919013i
43.6 1.15033 + 0.664145i 0.500000 0.866025i −0.117823 0.204075i 1.55828i 1.15033 0.664145i −0.866025 + 0.500000i 2.96959i −0.500000 0.866025i 1.03492 1.79254i
43.7 1.85837 + 1.07293i 0.500000 0.866025i 1.30235 + 2.25573i 1.91954i 1.85837 1.07293i 0.866025 0.500000i 1.29759i −0.500000 0.866025i −2.05953 + 3.56721i
43.8 2.35660 + 1.36058i 0.500000 0.866025i 2.70236 + 4.68063i 1.58278i 2.35660 1.36058i −0.866025 + 0.500000i 9.26480i −0.500000 0.866025i 2.15350 3.72996i
127.1 −2.20165 + 1.27113i 0.500000 + 0.866025i 2.23152 3.86511i 4.07309i −2.20165 1.27113i 0.866025 + 0.500000i 6.26168i −0.500000 + 0.866025i −5.17741 8.96754i
127.2 −1.98604 + 1.14664i 0.500000 + 0.866025i 1.62956 2.82249i 0.692320i −1.98604 1.14664i −0.866025 0.500000i 2.88753i −0.500000 + 0.866025i 0.793841 + 1.37497i
127.3 −1.44724 + 0.835563i 0.500000 + 0.866025i 0.396329 0.686463i 2.68351i −1.44724 0.835563i 0.866025 + 0.500000i 2.01762i −0.500000 + 0.866025i 2.24224 + 3.88368i
127.4 −0.654865 + 0.378087i 0.500000 + 0.866025i −0.714101 + 1.23686i 4.01537i −0.654865 0.378087i −0.866025 0.500000i 2.59231i −0.500000 + 0.866025i −1.51816 2.62953i
127.5 0.924500 0.533760i 0.500000 + 0.866025i −0.430200 + 0.745128i 0.994065i 0.924500 + 0.533760i 0.866025 + 0.500000i 3.05354i −0.500000 + 0.866025i 0.530592 + 0.919013i
127.6 1.15033 0.664145i 0.500000 + 0.866025i −0.117823 + 0.204075i 1.55828i 1.15033 + 0.664145i −0.866025 0.500000i 2.96959i −0.500000 + 0.866025i 1.03492 + 1.79254i
127.7 1.85837 1.07293i 0.500000 + 0.866025i 1.30235 2.25573i 1.91954i 1.85837 + 1.07293i 0.866025 + 0.500000i 1.29759i −0.500000 + 0.866025i −2.05953 3.56721i
127.8 2.35660 1.36058i 0.500000 + 0.866025i 2.70236 4.68063i 1.58278i 2.35660 + 1.36058i −0.866025 0.500000i 9.26480i −0.500000 + 0.866025i 2.15350 + 3.72996i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 127.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.bd.b 16
3.b odd 2 1 819.2.ct.c 16
13.e even 6 1 inner 273.2.bd.b 16
13.f odd 12 1 3549.2.a.ba 8
13.f odd 12 1 3549.2.a.bc 8
39.h odd 6 1 819.2.ct.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bd.b 16 1.a even 1 1 trivial
273.2.bd.b 16 13.e even 6 1 inner
819.2.ct.c 16 3.b odd 2 1
819.2.ct.c 16 39.h odd 6 1
3549.2.a.ba 8 13.f odd 12 1
3549.2.a.bc 8 13.f odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} - 15 T_{2}^{14} + 152 T_{2}^{12} + 6 T_{2}^{11} - 839 T_{2}^{10} - 24 T_{2}^{9} + 3348 T_{2}^{8} - 78 T_{2}^{7} - 7502 T_{2}^{6} + 768 T_{2}^{5} + 11883 T_{2}^{4} - 3072 T_{2}^{3} - 7616 T_{2}^{2} + 1464 T_{2} + 3721$$ acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} - 15 T^{14} + 152 T^{12} + \cdots + 3721$$
$3$ $$(T^{2} - T + 1)^{8}$$
$5$ $$T^{16} + 50 T^{14} + 943 T^{12} + \cdots + 20449$$
$7$ $$(T^{4} - T^{2} + 1)^{4}$$
$11$ $$T^{16} - 68 T^{14} + 3295 T^{12} + \cdots + 583696$$
$13$ $$T^{16} + 12 T^{15} + \cdots + 815730721$$
$17$ $$T^{16} + 2 T^{15} + 95 T^{14} + \cdots + 238177489$$
$19$ $$T^{16} - 79 T^{14} + 4962 T^{12} + \cdots + 9412624$$
$23$ $$T^{16} + 6 T^{15} + 93 T^{14} + \cdots + 22886656$$
$29$ $$T^{16} + 12 T^{15} + \cdots + 81511963009$$
$31$ $$T^{16} + 238 T^{14} + \cdots + 1539149824$$
$37$ $$T^{16} + 6 T^{15} + \cdots + 223550241721$$
$41$ $$T^{16} + 30 T^{15} + \cdots + 7256313856$$
$43$ $$T^{16} - 14 T^{15} + \cdots + 599528101264$$
$47$ $$T^{16} + 566 T^{14} + \cdots + 1683953296$$
$53$ $$(T^{8} - 14 T^{7} - 124 T^{6} + 2434 T^{5} + \cdots - 8807)^{2}$$
$59$ $$T^{16} + 24 T^{15} + \cdots + 2315546542864$$
$61$ $$T^{16} - 2 T^{15} + \cdots + 114864732889$$
$67$ $$T^{16} - 30 T^{15} + \cdots + 119295633664$$
$71$ $$T^{16} + 6 T^{15} - 224 T^{14} + \cdots + 672053776$$
$73$ $$T^{16} + 550 T^{14} + \cdots + 9572078569$$
$79$ $$(T^{8} - 46 T^{7} + 741 T^{6} - 4238 T^{5} + \cdots - 7916)^{2}$$
$83$ $$T^{16} + 572 T^{14} + 107158 T^{12} + \cdots + 1430416$$
$89$ $$T^{16} - 18 T^{15} + \cdots + 29787842814976$$
$97$ $$T^{16} + 6 T^{15} - 229 T^{14} + \cdots + 43264$$