# Properties

 Label 234.3.bb.f Level $234$ Weight $3$ Character orbit 234.bb Analytic conductor $6.376$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [234,3,Mod(19,234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(234, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 5]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("234.19");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$234 = 2 \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 234.bb (of order $$12$$, degree $$4$$, 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: $$6.37603818603$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ Coefficient field: 8.0.612074651904.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 74x^{6} + 2067x^{4} - 25778x^{2} + 121801$$ x^8 - 74*x^6 + 2067*x^4 - 25778*x^2 + 121801 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{5} - \beta_{3} + 1) q^{2} + 2 \beta_{4} q^{4} + (\beta_{7} - 2 \beta_{5} - 2 \beta_{3} - \beta_1) q^{5} + (\beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1) q^{7} + ( - 2 \beta_{5} + 2 \beta_{4} + 2) q^{8}+O(q^{10})$$ q + (b5 - b3 + 1) * q^2 + 2*b4 * q^4 + (b7 - 2*b5 - 2*b3 - b1) * q^5 + (b7 - b6 + b5 - 2*b4 - 2*b3 + b2 + b1) * q^7 + (-2*b5 + 2*b4 + 2) * q^8 $$q + (\beta_{5} - \beta_{3} + 1) q^{2} + 2 \beta_{4} q^{4} + (\beta_{7} - 2 \beta_{5} - 2 \beta_{3} - \beta_1) q^{5} + (\beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1) q^{7} + ( - 2 \beta_{5} + 2 \beta_{4} + 2) q^{8} + ( - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{10} + ( - \beta_{7} - 5 \beta_{5} - 5 \beta_{3} + \beta_{2} + \beta_1 + 5) q^{11} + (2 \beta_{7} - \beta_{6} + 7 \beta_{5} - 9 \beta_{4} - 3 \beta_{3} + \beta_1 + 5) q^{13} + (2 \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + \beta_1 - 3) q^{14} + ( - 4 \beta_{3} + 4) q^{16} + ( - \beta_{7} + \beta_{5} - 8 \beta_{4} - 3 \beta_{3} + \beta_{2} + 2 \beta_1 + 7) q^{17} + ( - 3 \beta_{7} + \beta_{6} - 9 \beta_{5} + 6 \beta_{4} - 6 \beta_{3} + 3 \beta_{2} + \cdots + 10) q^{19}+ \cdots + ( - 3 \beta_{7} + 4 \beta_{6} + 41 \beta_{5} - 22 \beta_{4} - 22 \beta_{3} + \cdots + 45) q^{98}+O(q^{100})$$ q + (b5 - b3 + 1) * q^2 + 2*b4 * q^4 + (b7 - 2*b5 - 2*b3 - b1) * q^5 + (b7 - b6 + b5 - 2*b4 - 2*b3 + b2 + b1) * q^7 + (-2*b5 + 2*b4 + 2) * q^8 + (-b7 + b6 - b5 - b4 - b3 + 2*b2 + b1 - 1) * q^10 + (-b7 - 5*b5 - 5*b3 + b2 + b1 + 5) * q^11 + (2*b7 - b6 + 7*b5 - 9*b4 - 3*b3 + b1 + 5) * q^13 + (2*b7 - b6 + b5 - b3 + b2 + b1 - 3) * q^14 + (-4*b3 + 4) * q^16 + (-b7 + b5 - 8*b4 - 3*b3 + b2 + 2*b1 + 7) * q^17 + (-3*b7 + b6 - 9*b5 + 6*b4 - 6*b3 + 3*b2 + b1 + 10) * q^19 + (-2*b6 - 2*b5 + 2*b1 - 4) * q^20 + (b7 - 2*b6 - b5 + b4 - 11*b3 - b2 - 1) * q^22 + (-2*b7 + 2*b6 + 14*b5 - b4 + 4*b3 + 2*b2 + b1 + 4) * q^23 + (3*b6 + 11*b5 - 11*b4 - 25*b3 + 14) * q^25 + (2*b7 + b6 + 13*b5 - b4 + b2 + b1 - 5) * q^26 + (2*b7 - 2*b5 + 2*b3 + 2*b2 + 2*b1 - 2) * q^28 + (b7 + b6 - b5 - 3*b4 - 2*b3 - 5*b1 + 2) * q^29 + (3*b6 + 3*b5 - 2*b4 + b3 - 3*b2 - 3*b1 + 5) * q^31 + (4*b4 - 4*b3) * q^32 + (2*b7 - 2*b6 + 11*b5 - 3*b4 - 8*b3 - 2*b2 - 5) * q^34 + (-3*b7 + 6*b6 + 7*b5 - 3*b4 + 21*b3 - b2 + 3) * q^35 + (-b7 - 5*b6 - 3*b5 + 12*b4 - 3*b3 + 6*b2 + b1 - 14) * q^37 + (-5*b6 - 5*b5 + 6*b4 - 19*b3 - b2 + b1 + 7) * q^38 + (4*b7 - 2*b6 - 4*b5 - 2*b4 - 2*b3 - 2*b2 - 2*b1 - 4) * q^40 + (-2*b7 - 6*b6 + 8*b5 - 17*b4 - 8*b3 - 8*b2 - 2*b1 - 15) * q^41 + (-6*b7 + b5 - 17*b4 + 12*b3 + b2 + 2*b1 - 18) * q^43 + (2*b7 - 12*b5 + 10*b4 - 2*b3 - 2*b1 - 10) * q^44 + (-b7 - 2*b6 + 7*b5 + 11*b4 + 11*b3 - b2 + 2*b1 + 5) * q^46 + (-2*b7 + b6 + 11*b5 - 4*b4 - 7*b3 + b2 - b1 - 3) * q^47 + (-b7 + b6 - 3*b5 + 2*b4 + 22*b3 - 4*b2 - 2*b1 + 22) * q^49 + (-3*b7 + 3*b6 + 25*b4 + 3*b1 - 22) * q^50 + (2*b6 - 2*b5 + 10*b4 + 18*b3 + 2*b2 + 4*b1 - 4) * q^52 + (10*b7 - 5*b6 - 24*b5 - 26*b4 - 5*b3 + 2*b2 + 2*b1 + 14) * q^53 + (5*b7 + 5*b6 - 8*b5 + 13*b4 + 38*b3 - 3*b1 - 38) * q^55 + (2*b7 + 2*b5 - 6*b4 - 2*b3 + 2*b2 + 4*b1 + 2) * q^56 + (-6*b7 + 2*b6 + 4*b5 - 3*b4 + 3*b3 + 6*b2 + 2*b1 - 2) * q^58 + (-7*b7 + b6 - 23*b5 + 44*b4 + 44*b3 - 7*b2 - b1 - 22) * q^59 + (-4*b7 + 8*b6 + 6*b5 - 2*b4 + 11*b3 - 2*b2 + 4) * q^61 + (-6*b7 + 6*b6 + 8*b5 + 4*b3 + 4) * q^62 + (-8*b5 + 8*b4) * q^64 + (-6*b7 + 14*b6 - 2*b5 - 10*b4 - 10*b3 + 4*b2 - 9*b1 + 51) * q^65 + (b7 - 12*b6 + 9*b5 - 8*b4 - 9*b3 - 11*b2 + b1 - 11) * q^67 + (2*b7 + 2*b6 - 8*b5 + 14*b4 + 14*b3 - 2*b1 - 14) * q^68 + (-6*b7 + 4*b6 + 24*b5 - 14*b4 + 10*b3 - 4*b2 + 2*b1 + 18) * q^70 + (5*b7 - 11*b6 - 21*b5 - 10*b4 + 10*b3 - 5*b2 - 11*b1 + 10) * q^71 + (-b7 - 2*b6 - 36*b5 + 8*b4 + 28*b3 - 2*b2 - 3*b1 + 6) * q^73 + (6*b7 - 12*b6 - 30*b5 + 13*b4 + 5*b3 + 4*b2 - 6) * q^74 + (6*b7 - 4*b6 - 18*b5 + 20*b4 - 18*b3 - 2*b2 - 6*b1 - 6) * q^76 + (4*b6 + 35*b5 - 28*b4 - 7*b2 + 7*b1 + 2) * q^77 + (-40*b5 - 24*b4 - 16*b2 - 16*b1 - 12) * q^79 + (4*b6 - 8*b4 + 4*b2 - 4) * q^80 + (4*b7 - 8*b5 + b4 + 19*b3 - 8*b2 - 16*b1 - 42) * q^82 + (8*b7 - 3*b6 + 47*b5 - 12*b4 + 35*b3 + 3*b2 - 5*b1 + 9) * q^83 + (2*b7 + 11*b6 - 29*b5 + 40*b4 + 40*b3 + 2*b2 - 11*b1 - 18) * q^85 + (2*b7 - 7*b6 + 5*b5 - 22*b4 + 17*b3 - 7*b2 - 5*b1 - 29) * q^86 + (-2*b7 + 2*b6 - 20*b5 - 2*b4 + 4*b2 + 2*b1) * q^88 + (-7*b7 + 6*b6 - 56*b5 + 66*b4 - 56*b3 + b2 + 7*b1 - 4) * q^89 + (3*b7 - 9*b6 + 31*b5 + 40*b4 - 20*b3 - b2 - 7*b1 + 42) * q^91 + (4*b7 - 2*b6 + 4*b5 + 8*b4 - 2*b3 - 4*b2 - 4*b1 + 26) * q^92 + (-2*b7 - 2*b6 - 8*b5 + 16*b4 + 16*b3 - 16) * q^94 + (13*b7 + b5 + 31*b4 + 53*b3 + b2 + 2*b1 - 119) * q^95 + (3*b7 + 7*b6 + b5 + 21*b4 - 21*b3 - 3*b2 + 7*b1 + 6) * q^97 + (-3*b7 + 4*b6 + 41*b5 - 22*b4 - 22*b3 - 3*b2 - 4*b1 + 45) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{2} - 6 q^{5} - 2 q^{7} + 16 q^{8}+O(q^{10})$$ 8 * q + 4 * q^2 - 6 * q^5 - 2 * q^7 + 16 * q^8 $$8 q + 4 q^{2} - 6 q^{5} - 2 q^{7} + 16 q^{8} - 18 q^{10} + 18 q^{11} + 36 q^{13} - 20 q^{14} + 16 q^{16} + 42 q^{17} + 46 q^{19} - 24 q^{20} - 42 q^{22} + 36 q^{23} - 40 q^{26} - 4 q^{28} + 6 q^{29} + 32 q^{31} - 16 q^{32} - 60 q^{34} + 78 q^{35} - 106 q^{37} - 24 q^{40} - 132 q^{41} - 108 q^{43} - 84 q^{44} + 90 q^{46} - 60 q^{47} + 258 q^{49} - 194 q^{50} + 32 q^{52} + 132 q^{53} - 162 q^{55} + 12 q^{56} - 24 q^{58} - 18 q^{59} + 36 q^{61} + 12 q^{62} + 300 q^{65} - 74 q^{67} - 60 q^{68} + 156 q^{70} + 174 q^{71} + 166 q^{73} + 32 q^{74} - 92 q^{76} - 96 q^{79} - 48 q^{80} - 252 q^{82} + 240 q^{83} - 24 q^{85} - 132 q^{86} - 12 q^{88} - 294 q^{89} + 298 q^{91} + 216 q^{92} - 60 q^{94} - 714 q^{95} - 58 q^{97} + 250 q^{98}+O(q^{100})$$ 8 * q + 4 * q^2 - 6 * q^5 - 2 * q^7 + 16 * q^8 - 18 * q^10 + 18 * q^11 + 36 * q^13 - 20 * q^14 + 16 * q^16 + 42 * q^17 + 46 * q^19 - 24 * q^20 - 42 * q^22 + 36 * q^23 - 40 * q^26 - 4 * q^28 + 6 * q^29 + 32 * q^31 - 16 * q^32 - 60 * q^34 + 78 * q^35 - 106 * q^37 - 24 * q^40 - 132 * q^41 - 108 * q^43 - 84 * q^44 + 90 * q^46 - 60 * q^47 + 258 * q^49 - 194 * q^50 + 32 * q^52 + 132 * q^53 - 162 * q^55 + 12 * q^56 - 24 * q^58 - 18 * q^59 + 36 * q^61 + 12 * q^62 + 300 * q^65 - 74 * q^67 - 60 * q^68 + 156 * q^70 + 174 * q^71 + 166 * q^73 + 32 * q^74 - 92 * q^76 - 96 * q^79 - 48 * q^80 - 252 * q^82 + 240 * q^83 - 24 * q^85 - 132 * q^86 - 12 * q^88 - 294 * q^89 + 298 * q^91 + 216 * q^92 - 60 * q^94 - 714 * q^95 - 58 * q^97 + 250 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 74x^{6} + 2067x^{4} - 25778x^{2} + 121801$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{4} - 37\nu^{2} + 4\nu + 349 ) / 8$$ (v^4 - 37*v^2 + 4*v + 349) / 8 $$\beta_{2}$$ $$=$$ $$( -\nu^{4} + 37\nu^{2} + 4\nu - 349 ) / 8$$ (-v^4 + 37*v^2 + 4*v - 349) / 8 $$\beta_{3}$$ $$=$$ $$( \nu^{7} - 74\nu^{5} + 1718\nu^{3} - 12865\nu + 1396 ) / 2792$$ (v^7 - 74*v^5 + 1718*v^3 - 12865*v + 1396) / 2792 $$\beta_{4}$$ $$=$$ $$( 40\nu^{7} - 349\nu^{6} - 2262\nu^{5} + 19195\nu^{4} + 44290\nu^{3} - 345859\nu^{2} - 296126\nu + 2024898 ) / 86552$$ (40*v^7 - 349*v^6 - 2262*v^5 + 19195*v^4 + 44290*v^3 - 345859*v^2 - 296126*v + 2024898) / 86552 $$\beta_{5}$$ $$=$$ $$( -40\nu^{7} - 349\nu^{6} + 2262\nu^{5} + 19195\nu^{4} - 44290\nu^{3} - 345859\nu^{2} + 296126\nu + 2024898 ) / 86552$$ (-40*v^7 - 349*v^6 + 2262*v^5 + 19195*v^4 - 44290*v^3 - 345859*v^2 + 296126*v + 2024898) / 86552 $$\beta_{6}$$ $$=$$ $$( \nu^{6} - 55\nu^{4} + 1053\nu^{2} - 6980 ) / 62$$ (v^6 - 55*v^4 + 1053*v^2 - 6980) / 62 $$\beta_{7}$$ $$=$$ $$( - 318 \nu^{7} + 698 \nu^{6} + 16901 \nu^{5} - 38390 \nu^{4} - 292601 \nu^{3} + 734994 \nu^{2} + 1626083 \nu - 4828764 ) / 86552$$ (-318*v^7 + 698*v^6 + 16901*v^5 - 38390*v^4 - 292601*v^3 + 734994*v^2 + 1626083*v - 4828764) / 86552
 $$\nu$$ $$=$$ $$\beta_{2} + \beta_1$$ b2 + b1 $$\nu^{2}$$ $$=$$ $$\beta_{6} + 2\beta_{5} + 2\beta_{4} + 19$$ b6 + 2*b5 + 2*b4 + 19 $$\nu^{3}$$ $$=$$ $$4\beta_{7} - 2\beta_{6} - 19\beta_{5} + 19\beta_{4} - 8\beta_{3} + 18\beta_{2} + 18\beta _1 + 2$$ 4*b7 - 2*b6 - 19*b5 + 19*b4 - 8*b3 + 18*b2 + 18*b1 + 2 $$\nu^{4}$$ $$=$$ $$37\beta_{6} + 74\beta_{5} + 74\beta_{4} - 4\beta_{2} + 4\beta _1 + 354$$ 37*b6 + 74*b5 + 74*b4 - 4*b2 + 4*b1 + 354 $$\nu^{5}$$ $$=$$ $$140\beta_{7} - 70\beta_{6} - 727\beta_{5} + 727\beta_{4} - 440\beta_{3} + 317\beta_{2} + 317\beta _1 + 150$$ 140*b7 - 70*b6 - 727*b5 + 727*b4 - 440*b3 + 317*b2 + 317*b1 + 150 $$\nu^{6}$$ $$=$$ $$1044\beta_{6} + 1964\beta_{5} + 1964\beta_{4} - 220\beta_{2} + 220\beta _1 + 6443$$ 1044*b6 + 1964*b5 + 1964*b4 - 220*b2 + 220*b1 + 6443 $$\nu^{7}$$ $$=$$ $$3488 \beta_{7} - 1744 \beta_{6} - 21156 \beta_{5} + 21156 \beta_{4} - 16024 \beta_{3} + 5399 \beta_{2} + 5399 \beta _1 + 6268$$ 3488*b7 - 1744*b6 - 21156*b5 + 21156*b4 - 16024*b3 + 5399*b2 + 5399*b1 + 6268

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/234\mathbb{Z}\right)^\times$$.

 $$n$$ $$145$$ $$209$$ $$\chi(n)$$ $$\beta_{4}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 3.90972 + 0.500000i −3.90972 + 0.500000i 3.90972 − 0.500000i −3.90972 − 0.500000i −4.71318 + 0.500000i 4.71318 + 0.500000i −4.71318 − 0.500000i 4.71318 − 0.500000i
−0.366025 1.36603i 0 −1.73205 + 1.00000i −4.79174 + 4.79174i 0 1.13983 4.25390i 2.00000 + 2.00000i 0 8.29953 + 4.79174i
19.2 −0.366025 1.36603i 0 −1.73205 + 1.00000i 5.88981 5.88981i 0 0.0922225 0.344179i 2.00000 + 2.00000i 0 −10.2015 5.88981i
37.1 −0.366025 + 1.36603i 0 −1.73205 1.00000i −4.79174 4.79174i 0 1.13983 + 4.25390i 2.00000 2.00000i 0 8.29953 4.79174i
37.2 −0.366025 + 1.36603i 0 −1.73205 1.00000i 5.88981 + 5.88981i 0 0.0922225 + 0.344179i 2.00000 2.00000i 0 −10.2015 + 5.88981i
145.1 1.36603 + 0.366025i 0 1.73205 + 1.00000i −3.77418 + 3.77418i 0 −9.91095 + 2.65563i 2.00000 + 2.00000i 0 −6.53708 + 3.77418i
145.2 1.36603 + 0.366025i 0 1.73205 + 1.00000i −0.323893 + 0.323893i 0 7.67890 2.05755i 2.00000 + 2.00000i 0 −0.560999 + 0.323893i
163.1 1.36603 0.366025i 0 1.73205 1.00000i −3.77418 3.77418i 0 −9.91095 2.65563i 2.00000 2.00000i 0 −6.53708 3.77418i
163.2 1.36603 0.366025i 0 1.73205 1.00000i −0.323893 0.323893i 0 7.67890 + 2.05755i 2.00000 2.00000i 0 −0.560999 0.323893i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 19.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.f odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 234.3.bb.f 8
3.b odd 2 1 26.3.f.b 8
12.b even 2 1 208.3.bd.f 8
13.f odd 12 1 inner 234.3.bb.f 8
39.d odd 2 1 338.3.f.i 8
39.f even 4 1 338.3.f.h 8
39.f even 4 1 338.3.f.j 8
39.h odd 6 1 338.3.d.f 8
39.h odd 6 1 338.3.f.j 8
39.i odd 6 1 338.3.d.g 8
39.i odd 6 1 338.3.f.h 8
39.k even 12 1 26.3.f.b 8
39.k even 12 1 338.3.d.f 8
39.k even 12 1 338.3.d.g 8
39.k even 12 1 338.3.f.i 8
156.v odd 12 1 208.3.bd.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.3.f.b 8 3.b odd 2 1
26.3.f.b 8 39.k even 12 1
208.3.bd.f 8 12.b even 2 1
208.3.bd.f 8 156.v odd 12 1
234.3.bb.f 8 1.a even 1 1 trivial
234.3.bb.f 8 13.f odd 12 1 inner
338.3.d.f 8 39.h odd 6 1
338.3.d.f 8 39.k even 12 1
338.3.d.g 8 39.i odd 6 1
338.3.d.g 8 39.k even 12 1
338.3.f.h 8 39.f even 4 1
338.3.f.h 8 39.i odd 6 1
338.3.f.i 8 39.d odd 2 1
338.3.f.i 8 39.k even 12 1
338.3.f.j 8 39.f even 4 1
338.3.f.j 8 39.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + 6T_{5}^{7} + 18T_{5}^{6} + 90T_{5}^{5} + 4245T_{5}^{4} + 30312T_{5}^{3} + 109512T_{5}^{2} + 64584T_{5} + 19044$$ acting on $$S_{3}^{\mathrm{new}}(234, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4)^{2}$$
$3$ $$T^{8}$$
$5$ $$T^{8} + 6 T^{7} + 18 T^{6} + \cdots + 19044$$
$7$ $$T^{8} + 2 T^{7} - 127 T^{6} + \cdots + 16384$$
$11$ $$T^{8} - 18 T^{7} + 105 T^{6} + \cdots + 389376$$
$13$ $$T^{8} - 36 T^{7} + \cdots + 815730721$$
$17$ $$T^{8} - 42 T^{7} + 570 T^{6} + \cdots + 471969$$
$19$ $$T^{8} - 46 T^{7} + \cdots + 1228362304$$
$23$ $$T^{8} - 36 T^{7} + \cdots + 2508807744$$
$29$ $$T^{8} - 6 T^{7} + \cdots + 25455883401$$
$31$ $$T^{8} - 32 T^{7} + \cdots + 8111524096$$
$37$ $$T^{8} + 106 T^{7} + \cdots + 321419829721$$
$41$ $$T^{8} + 132 T^{7} + \cdots + 326485389321$$
$43$ $$T^{8} + 108 T^{7} + \cdots + 325666531584$$
$47$ $$T^{8} + 60 T^{7} + \cdots + 1853819136$$
$53$ $$(T^{4} - 66 T^{3} - 5319 T^{2} + \cdots - 5234376)^{2}$$
$59$ $$T^{8} + 18 T^{7} + \cdots + 70410089309184$$
$61$ $$T^{8} - 36 T^{7} + \cdots + 313453297161$$
$67$ $$T^{8} + 74 T^{7} + \cdots + 2456391674944$$
$71$ $$T^{8} - 174 T^{7} + \cdots + 950999436864$$
$73$ $$T^{8} - 166 T^{7} + \cdots + 3554348548804$$
$79$ $$(T^{4} + 48 T^{3} - 14880 T^{2} + \cdots + 2312448)^{2}$$
$83$ $$T^{8} + \cdots + 154848357540864$$
$89$ $$T^{8} + 294 T^{7} + \cdots + 14950765690884$$
$97$ $$T^{8} + 58 T^{7} + \cdots + 9988090235236$$