# Properties

 Label 966.2.i.l Level $966$ Weight $2$ Character orbit 966.i Analytic conductor $7.714$ Analytic rank $0$ Dimension $8$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [966,2,Mod(277,966)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(966, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4, 0]))

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

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

chi := DirichletCharacter("966.277");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$966 = 2 \cdot 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 966.i (of order $$3$$, 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: $$7.71354883526$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.1768034304.4 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 2x^{7} - x^{6} - 6x^{5} + 14x^{4} + 18x^{3} - 31x^{2} - 14x + 49$$ x^8 - 2*x^7 - x^6 - 6*x^5 + 14*x^4 + 18*x^3 - 31*x^2 - 14*x + 49 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} - x^{6} - 6x^{5} + 14x^{4} + 18x^{3} - 31x^{2} - 14x + 49$$ :

 $$\beta_{1}$$ $$=$$ $$( 2\nu^{7} - 67\nu^{6} + 173\nu^{5} - 208\nu^{4} - 56\nu^{3} - 965\nu^{2} + 1303\nu + 3997 ) / 1659$$ (2*v^7 - 67*v^6 + 173*v^5 - 208*v^4 - 56*v^3 - 965*v^2 + 1303*v + 3997) / 1659 $$\beta_{2}$$ $$=$$ $$( 34\nu^{7} - 33\nu^{6} + 176\nu^{5} - 218\nu^{4} + 154\nu^{3} - 1474\nu^{2} - 522\nu - 623 ) / 3318$$ (34*v^7 - 33*v^6 + 176*v^5 - 218*v^4 + 154*v^3 - 1474*v^2 - 522*v - 623) / 3318 $$\beta_{3}$$ $$=$$ $$( 128\nu^{7} - 417\nu^{6} - 1094\nu^{5} - 40\nu^{4} + 5264\nu^{3} + 7918\nu^{2} + 1548\nu - 22351 ) / 9954$$ (128*v^7 - 417*v^6 - 1094*v^5 - 40*v^4 + 5264*v^3 + 7918*v^2 + 1548*v - 22351) / 9954 $$\beta_{4}$$ $$=$$ $$( 20\nu^{7} - 117\nu^{6} - 8\nu^{5} - 184\nu^{4} + 704\nu^{3} + 304\nu^{2} - 1506\nu + 1655 ) / 1422$$ (20*v^7 - 117*v^6 - 8*v^5 - 184*v^4 + 704*v^3 + 304*v^2 - 1506*v + 1655) / 1422 $$\beta_{5}$$ $$=$$ $$( -332\nu^{7} + 615\nu^{6} + 38\nu^{5} + 1348\nu^{4} - 6188\nu^{3} - 9028\nu^{2} + 21492\nu + 16135 ) / 9954$$ (-332*v^7 + 615*v^6 + 38*v^5 + 1348*v^4 - 6188*v^3 - 9028*v^2 + 21492*v + 16135) / 9954 $$\beta_{6}$$ $$=$$ $$( 338\nu^{7} + 843\nu^{6} - 1178\nu^{5} - 1972\nu^{4} - 7252\nu^{3} + 11110\nu^{2} + 666\nu - 22393 ) / 9954$$ (338*v^7 + 843*v^6 - 1178*v^5 - 1972*v^4 - 7252*v^3 + 11110*v^2 + 666*v - 22393) / 9954 $$\beta_{7}$$ $$=$$ $$( -254\nu^{7} - 339\nu^{6} + 149\nu^{5} + 3190\nu^{4} + 3241\nu^{3} - 1870\nu^{2} - 8982\nu + 4459 ) / 4977$$ (-254*v^7 - 339*v^6 + 149*v^5 + 3190*v^4 + 3241*v^3 - 1870*v^2 - 8982*v + 4459) / 4977
 $$\nu$$ $$=$$ $$( -\beta_{4} + \beta_{3} + \beta _1 + 1 ) / 2$$ (-b4 + b3 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$-\beta_{5} - \beta_{4} - 2\beta_{2} + \beta_1$$ -b5 - b4 - 2*b2 + b1 $$\nu^{3}$$ $$=$$ $$( -2\beta_{7} - 4\beta_{6} - 2\beta_{5} - 3\beta_{4} + \beta_{3} - 3\beta _1 + 9 ) / 2$$ (-2*b7 - 4*b6 - 2*b5 - 3*b4 + b3 - 3*b1 + 9) / 2 $$\nu^{4}$$ $$=$$ $$3\beta_{7} + 3\beta_{6} - 3\beta_{4} + 3\beta_{3} + 5\beta_{2} + 3\beta _1 + 8$$ 3*b7 + 3*b6 - 3*b4 + 3*b3 + 5*b2 + 3*b1 + 8 $$\nu^{5}$$ $$=$$ $$( -4\beta_{6} - 18\beta_{5} - 23\beta_{4} - 3\beta_{3} - 12\beta_{2} + 17\beta _1 - 3 ) / 2$$ (-4*b6 - 18*b5 - 23*b4 - 3*b3 - 12*b2 + 17*b1 - 3) / 2 $$\nu^{6}$$ $$=$$ $$-8\beta_{7} - 12\beta_{6} - 8\beta_{5} - 15\beta_{4} - 3\beta_{3} - 15\beta _1 + 40$$ -8*b7 - 12*b6 - 8*b5 - 15*b4 - 3*b3 - 15*b1 + 40 $$\nu^{7}$$ $$=$$ $$( 32\beta_{7} + 54\beta_{6} - 37\beta_{4} + 59\beta_{3} + 148\beta_{2} + 37\beta _1 + 207 ) / 2$$ (32*b7 + 54*b6 - 37*b4 + 59*b3 + 148*b2 + 37*b1 + 207) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/966\mathbb{Z}\right)^\times$$.

 $$n$$ $$323$$ $$829$$ $$925$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{2}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
277.1
 −0.885685 + 1.90520i 2.09279 + 0.185573i 0.972165 − 0.800426i −1.17927 + 0.441707i −0.885685 − 1.90520i 2.09279 − 0.185573i 0.972165 + 0.800426i −1.17927 − 0.441707i
−0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.914214 1.58346i 1.00000 −1.75255 1.98206i 1.00000 −0.500000 0.866025i −0.914214 + 1.58346i
277.2 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.914214 1.58346i 1.00000 2.45965 0.974732i 1.00000 −0.500000 0.866025i −0.914214 + 1.58346i
277.3 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.91421 + 3.31552i 1.00000 −1.87485 + 1.86680i 1.00000 −0.500000 0.866025i 1.91421 3.31552i
277.4 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.91421 + 3.31552i 1.00000 1.16774 2.37411i 1.00000 −0.500000 0.866025i 1.91421 3.31552i
415.1 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i −0.914214 + 1.58346i 1.00000 −1.75255 + 1.98206i 1.00000 −0.500000 + 0.866025i −0.914214 1.58346i
415.2 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i −0.914214 + 1.58346i 1.00000 2.45965 + 0.974732i 1.00000 −0.500000 + 0.866025i −0.914214 1.58346i
415.3 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.91421 3.31552i 1.00000 −1.87485 1.86680i 1.00000 −0.500000 + 0.866025i 1.91421 + 3.31552i
415.4 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.91421 3.31552i 1.00000 1.16774 + 2.37411i 1.00000 −0.500000 + 0.866025i 1.91421 + 3.31552i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 277.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.2.i.l 8
7.c even 3 1 inner 966.2.i.l 8
7.c even 3 1 6762.2.a.cp 4
7.d odd 6 1 6762.2.a.cm 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.i.l 8 1.a even 1 1 trivial
966.2.i.l 8 7.c even 3 1 inner
6762.2.a.cm 4 7.d odd 6 1
6762.2.a.cp 4 7.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(966, [\chi])$$:

 $$T_{5}^{4} - 2T_{5}^{3} + 11T_{5}^{2} + 14T_{5} + 49$$ T5^4 - 2*T5^3 + 11*T5^2 + 14*T5 + 49 $$T_{11}^{8} + 6T_{11}^{7} + 55T_{11}^{6} + 198T_{11}^{5} + 1485T_{11}^{4} + 5220T_{11}^{3} + 20764T_{11}^{2} + 29328T_{11} + 35344$$ T11^8 + 6*T11^7 + 55*T11^6 + 198*T11^5 + 1485*T11^4 + 5220*T11^3 + 20764*T11^2 + 29328*T11 + 35344

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{4}$$
$3$ $$(T^{2} + T + 1)^{4}$$
$5$ $$(T^{4} - 2 T^{3} + 11 T^{2} + \cdots + 49)^{2}$$
$7$ $$T^{8} - 12 T^{5} + \cdots + 2401$$
$11$ $$T^{8} + 6 T^{7} + \cdots + 35344$$
$13$ $$(T^{4} - 6 T^{3} - T^{2} + \cdots - 2)^{2}$$
$17$ $$T^{8} - 10 T^{7} + \cdots + 21316$$
$19$ $$T^{8} - 4 T^{7} + \cdots + 4096$$
$23$ $$(T^{2} + T + 1)^{4}$$
$29$ $$(T^{4} - 6 T^{3} - T^{2} + \cdots + 16)^{2}$$
$31$ $$T^{8} + 2 T^{7} + \cdots + 35344$$
$37$ $$T^{8} + 58 T^{6} + \cdots + 18496$$
$41$ $$(T^{4} - 4 T^{3} - 30 T^{2} + \cdots + 64)^{2}$$
$43$ $$(T^{4} - 20 T^{3} + \cdots - 5696)^{2}$$
$47$ $$T^{8} + 10 T^{7} + \cdots + 1817104$$
$53$ $$T^{8} + 58 T^{6} + \cdots + 305809$$
$59$ $$T^{8} + 6 T^{7} + \cdots + 12544$$
$61$ $$T^{8} + 232 T^{6} + \cdots + 78287104$$
$67$ $$T^{8} + 18 T^{7} + \cdots + 125350416$$
$71$ $$(T^{4} - 42 T^{3} + \cdots + 10654)^{2}$$
$73$ $$T^{8} + 6 T^{7} + \cdots + 1024$$
$79$ $$T^{8} - 6 T^{7} + \cdots + 256$$
$83$ $$(T^{4} + 10 T^{3} + \cdots + 1252)^{2}$$
$89$ $$T^{8} + 106 T^{6} + \cdots + 399424$$
$97$ $$(T^{4} + 10 T^{3} + \cdots + 64)^{2}$$