Properties

 Label 138.2.e.b Level $138$ Weight $2$ Character orbit 138.e Analytic conductor $1.102$ Analytic rank $0$ Dimension $10$ Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [138,2,Mod(13,138)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(138, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([0, 14]))

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

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

chi := DirichletCharacter("138.13");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$138 = 2 \cdot 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 138.e (of order $$11$$, degree $$10$$, 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: $$1.10193554789$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\Q(\zeta_{22})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$$ x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-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 primitive root of unity $$\zeta_{22}$$. We also show the integral $$q$$-expansion of the trace form.

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

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/138\mathbb{Z}\right)^\times$$.

 $$n$$ $$47$$ $$97$$ $$\chi(n)$$ $$1$$ $$-\zeta_{22}$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 0.654861 + 0.755750i −0.841254 − 0.540641i 0.142315 − 0.989821i 0.142315 + 0.989821i 0.959493 − 0.281733i −0.415415 − 0.909632i 0.654861 − 0.755750i −0.415415 + 0.909632i −0.841254 + 0.540641i 0.959493 + 0.281733i
−0.654861 0.755750i −0.841254 0.540641i −0.142315 + 0.989821i −1.61435 + 3.53494i 0.142315 + 0.989821i −3.99283 1.17240i 0.841254 0.540641i 0.415415 + 0.909632i 3.72871 1.09485i
25.1 0.841254 + 0.540641i 0.142315 + 0.989821i 0.415415 + 0.909632i 0.186393 + 0.0547299i −0.415415 + 0.909632i −1.27819 + 1.47511i −0.142315 + 0.989821i −0.959493 + 0.281733i 0.127214 + 0.146813i
31.1 −0.142315 + 0.989821i −0.415415 0.909632i −0.959493 0.281733i 0.698939 + 0.806618i 0.959493 0.281733i 3.72270 + 2.39243i 0.415415 0.909632i −0.654861 + 0.755750i −0.897877 + 0.577031i
49.1 −0.142315 0.989821i −0.415415 + 0.909632i −0.959493 + 0.281733i 0.698939 0.806618i 0.959493 + 0.281733i 3.72270 2.39243i 0.415415 + 0.909632i −0.654861 0.755750i −0.897877 0.577031i
55.1 −0.959493 + 0.281733i 0.654861 + 0.755750i 0.841254 0.540641i −0.544078 3.78415i −0.841254 0.540641i 1.20217 2.63238i −0.654861 + 0.755750i −0.142315 + 0.989821i 1.58816 + 3.47758i
73.1 0.415415 + 0.909632i 0.959493 0.281733i −0.654861 + 0.755750i 0.273100 + 0.175511i 0.654861 + 0.755750i 0.346156 + 2.40757i −0.959493 0.281733i 0.841254 0.540641i −0.0462003 + 0.321330i
85.1 −0.654861 + 0.755750i −0.841254 + 0.540641i −0.142315 0.989821i −1.61435 3.53494i 0.142315 0.989821i −3.99283 + 1.17240i 0.841254 + 0.540641i 0.415415 0.909632i 3.72871 + 1.09485i
121.1 0.415415 0.909632i 0.959493 + 0.281733i −0.654861 0.755750i 0.273100 0.175511i 0.654861 0.755750i 0.346156 2.40757i −0.959493 + 0.281733i 0.841254 + 0.540641i −0.0462003 0.321330i
127.1 0.841254 0.540641i 0.142315 0.989821i 0.415415 0.909632i 0.186393 0.0547299i −0.415415 0.909632i −1.27819 1.47511i −0.142315 0.989821i −0.959493 0.281733i 0.127214 0.146813i
133.1 −0.959493 0.281733i 0.654861 0.755750i 0.841254 + 0.540641i −0.544078 + 3.78415i −0.841254 + 0.540641i 1.20217 + 2.63238i −0.654861 0.755750i −0.142315 0.989821i 1.58816 3.47758i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 13.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 138.2.e.b 10
3.b odd 2 1 414.2.i.e 10
23.c even 11 1 inner 138.2.e.b 10
23.c even 11 1 3174.2.a.ba 5
23.d odd 22 1 3174.2.a.bb 5
69.g even 22 1 9522.2.a.br 5
69.h odd 22 1 414.2.i.e 10
69.h odd 22 1 9522.2.a.bs 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.2.e.b 10 1.a even 1 1 trivial
138.2.e.b 10 23.c even 11 1 inner
414.2.i.e 10 3.b odd 2 1
414.2.i.e 10 69.h odd 22 1
3174.2.a.ba 5 23.c even 11 1
3174.2.a.bb 5 23.d odd 22 1
9522.2.a.br 5 69.g even 22 1
9522.2.a.bs 5 69.h odd 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{10} + 2T_{5}^{9} + 26T_{5}^{8} - 3T_{5}^{7} + 159T_{5}^{6} - 386T_{5}^{5} + 526T_{5}^{4} - 323T_{5}^{3} + 102T_{5}^{2} - 16T_{5} + 1$$ acting on $$S_{2}^{\mathrm{new}}(138, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} + T^{9} + \cdots + 1$$
$3$ $$T^{10} - T^{9} + \cdots + 1$$
$5$ $$T^{10} + 2 T^{9} + \cdots + 1$$
$7$ $$T^{10} - 11 T^{8} + \cdots + 64009$$
$11$ $$T^{10} - 11 T^{9} + \cdots + 64009$$
$13$ $$T^{10} - 13 T^{9} + \cdots + 11881$$
$17$ $$T^{10} + 24 T^{9} + \cdots + 279841$$
$19$ $$T^{10} + 14 T^{9} + \cdots + 4489$$
$23$ $$T^{10} + 10 T^{9} + \cdots + 6436343$$
$29$ $$T^{10} - 13 T^{9} + \cdots + 529$$
$31$ $$T^{10} - 8 T^{9} + \cdots + 529$$
$37$ $$T^{10} + 13 T^{9} + \cdots + 139129$$
$41$ $$T^{10} + 10 T^{9} + \cdots + 434281$$
$43$ $$T^{10} + 8 T^{9} + \cdots + 50794129$$
$47$ $$(T^{5} + 4 T^{4} + \cdots - 4817)^{2}$$
$53$ $$T^{10} + 35 T^{9} + \cdots + 279841$$
$59$ $$T^{10} + \cdots + 127893481$$
$61$ $$T^{10} + 2 T^{9} + \cdots + 11485321$$
$67$ $$T^{10} + \cdots + 1051640041$$
$71$ $$T^{10} - 44 T^{9} + \cdots + 25938649$$
$73$ $$T^{10} + \cdots + 310499641$$
$79$ $$T^{10} + 8 T^{9} + \cdots + 23203489$$
$83$ $$T^{10} + 17 T^{9} + \cdots + 1$$
$89$ $$T^{10} + \cdots + 3044501329$$
$97$ $$T^{10} + 21 T^{9} + \cdots + 9078169$$