# Properties

 Label 322.2.i.b Level $322$ Weight $2$ Character orbit 322.i Analytic conductor $2.571$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [322,2,Mod(29,322)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(322, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([0, 18]))

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

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

chi := DirichletCharacter("322.29");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$322 = 2 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 322.i (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: $$2.57118294509$$ 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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/322\mathbb{Z}\right)^\times$$.

 $$n$$ $$185$$ $$281$$ $$\chi(n)$$ $$1$$ $$\zeta_{22}^{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1
 0.959493 − 0.281733i 0.142315 − 0.989821i −0.841254 − 0.540641i 0.142315 + 0.989821i −0.415415 − 0.909632i −0.415415 + 0.909632i −0.841254 + 0.540641i 0.959493 + 0.281733i 0.654861 + 0.755750i 0.654861 − 0.755750i
−0.415415 + 0.909632i 2.30075 + 0.675560i −0.654861 0.755750i 1.74412 1.12088i −1.57028 + 1.81219i 0.142315 0.989821i 0.959493 0.281733i 2.31329 + 1.48666i 0.295053 + 2.05214i
71.1 −0.841254 0.540641i −0.317178 2.20602i 0.415415 + 0.909632i 1.95204 + 0.573170i −0.925839 + 2.02730i 0.654861 0.755750i 0.142315 0.989821i −1.88745 + 0.554206i −1.33228 1.53753i
85.1 0.654861 0.755750i 0.0741615 0.0476607i −0.142315 0.989821i 1.57773 + 3.45475i 0.0125459 0.0872586i 0.959493 0.281733i −0.841254 0.540641i −1.24302 + 2.72183i 3.64412 + 1.07001i
127.1 −0.841254 + 0.540641i −0.317178 + 2.20602i 0.415415 0.909632i 1.95204 0.573170i −0.925839 2.02730i 0.654861 + 0.755750i 0.142315 + 0.989821i −1.88745 0.554206i −1.33228 + 1.53753i
141.1 0.142315 + 0.989821i −0.570276 + 1.24873i −0.959493 + 0.281733i −1.59792 + 1.84410i −1.31718 0.386758i −0.841254 + 0.540641i −0.415415 0.909632i 0.730471 + 0.843008i −2.05274 1.31921i
169.1 0.142315 0.989821i −0.570276 1.24873i −0.959493 0.281733i −1.59792 1.84410i −1.31718 + 0.386758i −0.841254 0.540641i −0.415415 + 0.909632i 0.730471 0.843008i −2.05274 + 1.31921i
197.1 0.654861 + 0.755750i 0.0741615 + 0.0476607i −0.142315 + 0.989821i 1.57773 3.45475i 0.0125459 + 0.0872586i 0.959493 + 0.281733i −0.841254 + 0.540641i −1.24302 2.72183i 3.64412 1.07001i
211.1 −0.415415 0.909632i 2.30075 0.675560i −0.654861 + 0.755750i 1.74412 + 1.12088i −1.57028 1.81219i 0.142315 + 0.989821i 0.959493 + 0.281733i 2.31329 1.48666i 0.295053 2.05214i
225.1 0.959493 + 0.281733i 1.01255 1.16854i 0.841254 + 0.540641i 0.324031 2.25368i 1.30075 0.835939i −0.415415 0.909632i 0.654861 + 0.755750i 0.0867074 + 0.603063i 0.945841 2.07110i
239.1 0.959493 0.281733i 1.01255 + 1.16854i 0.841254 0.540641i 0.324031 + 2.25368i 1.30075 + 0.835939i −0.415415 + 0.909632i 0.654861 0.755750i 0.0867074 0.603063i 0.945841 + 2.07110i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 239.1 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 322.2.i.b 10
23.c even 11 1 inner 322.2.i.b 10
23.c even 11 1 7406.2.a.bf 5
23.d odd 22 1 7406.2.a.be 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.i.b 10 1.a even 1 1 trivial
322.2.i.b 10 23.c even 11 1 inner
7406.2.a.be 5 23.d odd 22 1
7406.2.a.bf 5 23.c even 11 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{10} - 5T_{3}^{9} + 14T_{3}^{8} - 37T_{3}^{7} + 75T_{3}^{6} - 100T_{3}^{5} + 126T_{3}^{4} - 135T_{3}^{3} + 147T_{3}^{2} - 20T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(322, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} - T^{9} + T^{8} - T^{7} + T^{6} - T^{5} + \cdots + 1$$
$3$ $$T^{10} - 5 T^{9} + 14 T^{8} - 37 T^{7} + \cdots + 1$$
$5$ $$T^{10} - 8 T^{9} + 42 T^{8} + \cdots + 7921$$
$7$ $$T^{10} - T^{9} + T^{8} - T^{7} + T^{6} - T^{5} + \cdots + 1$$
$11$ $$T^{10} - 11 T^{9} + 55 T^{8} + \cdots + 121$$
$13$ $$T^{10} + 6 T^{9} + 14 T^{8} + 18 T^{7} + \cdots + 1$$
$17$ $$T^{10} + 2 T^{9} + 4 T^{8} + 85 T^{7} + \cdots + 4489$$
$19$ $$T^{10} + 3 T^{9} + 31 T^{8} + 38 T^{7} + \cdots + 1$$
$23$ $$T^{10} + 21 T^{9} + 177 T^{8} + \cdots + 6436343$$
$29$ $$T^{10} - 2 T^{9} + 15 T^{8} + \cdots + 124609$$
$31$ $$T^{10} - 3 T^{9} + 20 T^{8} + \cdots + 11881$$
$37$ $$T^{10} + 6 T^{9} - 30 T^{8} + \cdots + 7921$$
$41$ $$T^{10} + 23 T^{9} + 221 T^{8} + \cdots + 73599241$$
$43$ $$T^{10} + T^{9} - 32 T^{8} + 254 T^{7} + \cdots + 4489$$
$47$ $$(T^{5} - 19 T^{4} + 85 T^{3} + 293 T^{2} + \cdots + 4643)^{2}$$
$53$ $$T^{10} + 49 T^{9} + 1125 T^{8} + \cdots + 49266361$$
$59$ $$T^{10} - 9 T^{9} + 4 T^{8} + \cdots + 17161$$
$61$ $$T^{10} - 16 T^{9} + 14 T^{8} + \cdots + 89510521$$
$67$ $$T^{10} - 7 T^{9} + 214 T^{8} + \cdots + 529$$
$71$ $$T^{10} + 16 T^{9} + \cdots + 310499641$$
$73$ $$T^{10} + 9 T^{9} + 268 T^{8} + \cdots + 802985569$$
$79$ $$T^{10} - 6 T^{9} + 102 T^{8} + \cdots + 189090001$$
$83$ $$T^{10} + 19 T^{9} + 229 T^{8} + \cdots + 79798489$$
$89$ $$T^{10} - 25 T^{9} + 504 T^{8} + \cdots + 30724849$$
$97$ $$T^{10} - 16 T^{9} + 25 T^{8} + \cdots + 436921$$