# Properties

 Label 231.2.i.e Level $231$ Weight $2$ Character orbit 231.i Analytic conductor $1.845$ 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] = [231,2,Mod(67,231)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(231, 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("231.67");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$231 = 3 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 231.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: $$1.84454428669$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.10423593216.5 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 2x^{7} + 8x^{6} + 21x^{4} - 4x^{3} + 28x^{2} + 12x + 9$$ x^8 - 2*x^7 + 8*x^6 + 21*x^4 - 4*x^3 + 28*x^2 + 12*x + 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{4} - 1) q^{3} + (\beta_{6} + \beta_{3} + \beta_{2} + \beta_1) q^{4} + (\beta_{6} - \beta_{4} - \beta_1) q^{5} - \beta_{3} q^{6} + ( - \beta_{7} - \beta_{5} - \beta_{3}) q^{7} + ( - \beta_{7} - \beta_{3} - 2 \beta_{2} + 2) q^{8} - \beta_{4} q^{9}+O(q^{10})$$ q - b1 * q^2 + (b4 - 1) * q^3 + (b6 + b3 + b2 + b1) * q^4 + (b6 - b4 - b1) * q^5 - b3 * q^6 + (-b7 - b5 - b3) * q^7 + (-b7 - b3 - 2*b2 + 2) * q^8 - b4 * q^9 $$q - \beta_1 q^{2} + (\beta_{4} - 1) q^{3} + (\beta_{6} + \beta_{3} + \beta_{2} + \beta_1) q^{4} + (\beta_{6} - \beta_{4} - \beta_1) q^{5} - \beta_{3} q^{6} + ( - \beta_{7} - \beta_{5} - \beta_{3}) q^{7} + ( - \beta_{7} - \beta_{3} - 2 \beta_{2} + 2) q^{8} - \beta_{4} q^{9} + ( - \beta_{7} - \beta_{5} + 2 \beta_{4} - 2) q^{10} + ( - \beta_{4} + 1) q^{11} + ( - \beta_{6} - \beta_1) q^{12} - \beta_{7} q^{13} + ( - \beta_{6} - 2 \beta_{4} - \beta_{3} + \cdots + 1) q^{14}+ \cdots - q^{99}+O(q^{100})$$ q - b1 * q^2 + (b4 - 1) * q^3 + (b6 + b3 + b2 + b1) * q^4 + (b6 - b4 - b1) * q^5 - b3 * q^6 + (-b7 - b5 - b3) * q^7 + (-b7 - b3 - 2*b2 + 2) * q^8 - b4 * q^9 + (-b7 - b5 + 2*b4 - 2) * q^10 + (-b4 + 1) * q^11 + (-b6 - b1) * q^12 - b7 * q^13 + (-b6 - 2*b4 - b3 + b2 - b1 + 1) * q^14 + (-b3 + b2 + 1) * q^15 + (-2*b6 - 2*b5 - b4 - 4*b1) * q^16 + (-2*b6 + 2*b4 - b3 - 2*b2 - b1 - 2) * q^17 + (b3 + b1) * q^18 + (2*b6 + b5 - b1) * q^19 + (-b3 - b2 - 1) * q^20 + (b5 + b3 + b1) * q^21 + b3 * q^22 + (2*b5 - b4 + 2*b1) * q^23 + (b7 + 2*b6 + b5 + 2*b4 + b3 + 2*b2 + b1 - 2) * q^24 + (-2*b7 - 2*b6 - 2*b5 + b4 - 2*b2 - 1) * q^25 + (-b6 + b4 + b1) * q^26 + q^27 + (b7 + 2*b6 + 3*b3 + 2*b2 + 3*b1 - 2) * q^28 + (b7 + b3 + 2) * q^29 + (b5 - 2*b4) * q^30 + (2*b7 - b6 + 2*b5 - b4 - b3 - b2 - b1 + 1) * q^31 + (4*b6 + 2*b4 + 5*b3 + 4*b2 + 5*b1 - 2) * q^32 + b4 * q^33 + (2*b7 + 3*b3 + 3*b2 - 2) * q^34 + (2*b7 + b5 - 2*b4 - 2*b3 + 2*b2) * q^35 + (-b3 - b2) * q^36 + (b6 + b1) * q^37 + (-2*b7 - 2*b6 - 2*b5 + 3*b4 - 2*b3 - 2*b2 - 2*b1 - 3) * q^38 + (b7 + b5) * q^39 + (-2*b6 - 3*b5 + 2*b4 - 2*b1) * q^40 + (b7 + 2*b3 - 2*b2) * q^41 + (-b6 + b4 - 2*b2 + b1 + 1) * q^42 + (-3*b3 + 2*b2 + 4) * q^43 + (b6 + b1) * q^44 + (-b6 + b4 + b3 - b2 + b1 - 1) * q^45 + (-4*b6 - 2*b4 + b3 - 4*b2 + b1 + 2) * q^46 + (-b6 - 2*b4 - b1) * q^47 + (-2*b7 - 4*b3 - 2*b2 + 1) * q^48 + (-4*b4 + b3 - 3*b2 + 2*b1) * q^49 + (2*b7 + b3 + 4*b2 + 2) * q^50 + (2*b6 - 2*b4 + b1) * q^51 + (-b7 - b5 - 2*b4 + 2) * q^52 + (4*b7 + b6 + 4*b5 - 3*b4 + 3*b3 + b2 + 3*b1 + 3) * q^53 - b1 * q^54 + (b3 - b2 - 1) * q^55 + (-2*b7 + 3*b6 - 3*b4 - 7*b3 - b2 - b1 + 4) * q^56 + (b7 - b3 + 2*b2) * q^57 + (2*b6 + b4 - 2*b1) * q^58 + (2*b7 + 2*b5 + 5*b4 - 2*b3 - 2*b1 - 5) * q^59 + (b6 - b4 + b3 + b2 + b1 + 1) * q^60 + (-b5 - 2*b1) * q^61 + (b7 + 6*b3 - 4) * q^62 + (b7 - b1) * q^63 + (-7*b3 - 5*b2 + 8) * q^64 + (-2*b6 - 2*b5 + 2*b1) * q^65 + (-b3 - b1) * q^66 + (2*b7 + b6 + 2*b5 + 9*b4 - 3*b3 + b2 - 3*b1 - 9) * q^67 + (4*b6 + 3*b5 + 8*b4 + 7*b1) * q^68 + (2*b7 + 2*b3 + 1) * q^69 + (b6 + 2*b5 - 5*b4 + 3*b3 - b2 + 3*b1 - 1) * q^70 + (-2*b7 + b3 - b2 + 4) * q^71 + (-2*b6 - b5 - 2*b4 - b1) * q^72 + (-b7 + 6*b6 - b5 - 2*b4 + 6*b2 + 2) * q^73 + (-b7 - 2*b6 - b5 - 2*b4 - 3*b3 - 2*b2 - 3*b1 + 2) * q^74 + (2*b6 + 2*b5 - b4) * q^75 + (3*b3 + 2*b2 - 2) * q^76 + (-b5 - b3 - b1) * q^77 + (b3 - b2 - 1) * q^78 + (-2*b6 + 3*b5 + 4*b1) * q^79 + (2*b7 + 5*b6 + 2*b5 + 3*b4 - b3 + 5*b2 - b1 - 3) * q^80 + (b4 - 1) * q^81 + (b6 - 2*b5 + 3*b4 - 3*b1) * q^82 + (-4*b7 - 2*b3 - 2) * q^83 + (-b7 - 2*b6 - b5 - 2*b4 - 3*b1 + 2) * q^84 + (b7 + 4*b2 + 6) * q^85 + (-b6 + 2*b5 - 6*b4 - 3*b1) * q^86 + (-b7 - b5 + 2*b4 - b3 - b1 - 2) * q^87 + (-b7 - 2*b6 - b5 - 2*b4 - b3 - 2*b2 - b1 + 2) * q^88 + (-6*b6 + 2*b1) * q^89 + (b7 + 2) * q^90 + (2*b6 - 6*b4 + b3 + b2 + 5) * q^91 + (5*b3 + 3*b2) * q^92 + (b6 - 2*b5 + b4 + b1) * q^93 + (b7 + 2*b6 + b5 + 2*b4 + 5*b3 + 2*b2 + 5*b1 - 2) * q^94 + (-5*b7 - 4*b6 - 5*b5 + 8*b4 - 4*b2 - 8) * q^95 + (-4*b6 - 2*b4 - 5*b1) * q^96 + (-2*b7 - b3 + b2 - 10) * q^97 + (-4*b6 - 3*b5 - 2*b4 + 2*b3 - 2*b2 - 3*b1 + 4) * q^98 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 2 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} + 4 q^{6} + 2 q^{7} + 24 q^{8} - 4 q^{9}+O(q^{10})$$ 8 * q - 2 * q^2 - 4 * q^3 - 4 * q^4 - 4 * q^5 + 4 * q^6 + 2 * q^7 + 24 * q^8 - 4 * q^9 $$8 q - 2 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} + 4 q^{6} + 2 q^{7} + 24 q^{8} - 4 q^{9} - 10 q^{10} + 4 q^{11} - 4 q^{12} - 4 q^{13} - 4 q^{14} + 8 q^{15} - 12 q^{16} - 2 q^{17} - 2 q^{18} - 4 q^{21} - 4 q^{22} - 4 q^{23} - 12 q^{24} - 4 q^{25} + 4 q^{26} + 8 q^{27} - 22 q^{28} + 16 q^{29} - 10 q^{30} + 12 q^{31} - 26 q^{32} + 4 q^{33} - 32 q^{34} - 2 q^{35} + 8 q^{36} + 4 q^{37} - 8 q^{38} + 2 q^{39} + 6 q^{40} + 4 q^{41} + 20 q^{42} + 36 q^{43} + 4 q^{44} - 4 q^{45} + 14 q^{46} - 12 q^{47} + 24 q^{48} - 4 q^{49} + 4 q^{50} - 2 q^{51} + 6 q^{52} + 12 q^{53} - 2 q^{54} - 8 q^{55} + 48 q^{56} + 4 q^{58} - 12 q^{59} - 2 q^{61} - 52 q^{62} + 2 q^{63} + 112 q^{64} + 4 q^{65} + 2 q^{66} - 28 q^{67} + 48 q^{68} + 8 q^{69} - 32 q^{70} + 24 q^{71} - 12 q^{72} - 6 q^{73} + 16 q^{74} - 4 q^{75} - 36 q^{76} + 4 q^{77} - 8 q^{78} - 2 q^{79} - 16 q^{80} - 4 q^{81} + 12 q^{82} - 24 q^{83} - 4 q^{84} + 36 q^{85} - 36 q^{86} - 8 q^{87} + 12 q^{88} - 8 q^{89} + 20 q^{90} + 12 q^{91} - 32 q^{92} + 12 q^{93} - 20 q^{94} - 34 q^{95} - 26 q^{96} - 88 q^{97} + 16 q^{98} - 8 q^{99}+O(q^{100})$$ 8 * q - 2 * q^2 - 4 * q^3 - 4 * q^4 - 4 * q^5 + 4 * q^6 + 2 * q^7 + 24 * q^8 - 4 * q^9 - 10 * q^10 + 4 * q^11 - 4 * q^12 - 4 * q^13 - 4 * q^14 + 8 * q^15 - 12 * q^16 - 2 * q^17 - 2 * q^18 - 4 * q^21 - 4 * q^22 - 4 * q^23 - 12 * q^24 - 4 * q^25 + 4 * q^26 + 8 * q^27 - 22 * q^28 + 16 * q^29 - 10 * q^30 + 12 * q^31 - 26 * q^32 + 4 * q^33 - 32 * q^34 - 2 * q^35 + 8 * q^36 + 4 * q^37 - 8 * q^38 + 2 * q^39 + 6 * q^40 + 4 * q^41 + 20 * q^42 + 36 * q^43 + 4 * q^44 - 4 * q^45 + 14 * q^46 - 12 * q^47 + 24 * q^48 - 4 * q^49 + 4 * q^50 - 2 * q^51 + 6 * q^52 + 12 * q^53 - 2 * q^54 - 8 * q^55 + 48 * q^56 + 4 * q^58 - 12 * q^59 - 2 * q^61 - 52 * q^62 + 2 * q^63 + 112 * q^64 + 4 * q^65 + 2 * q^66 - 28 * q^67 + 48 * q^68 + 8 * q^69 - 32 * q^70 + 24 * q^71 - 12 * q^72 - 6 * q^73 + 16 * q^74 - 4 * q^75 - 36 * q^76 + 4 * q^77 - 8 * q^78 - 2 * q^79 - 16 * q^80 - 4 * q^81 + 12 * q^82 - 24 * q^83 - 4 * q^84 + 36 * q^85 - 36 * q^86 - 8 * q^87 + 12 * q^88 - 8 * q^89 + 20 * q^90 + 12 * q^91 - 32 * q^92 + 12 * q^93 - 20 * q^94 - 34 * q^95 - 26 * q^96 - 88 * q^97 + 16 * q^98 - 8 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} + 8x^{6} + 21x^{4} - 4x^{3} + 28x^{2} + 12x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 68\nu^{7} - 215\nu^{6} + 357\nu^{5} + 646\nu^{4} - 1444\nu^{3} + 1156\nu^{2} + 561\nu + 5468 ) / 4243$$ (68*v^7 - 215*v^6 + 357*v^5 + 646*v^4 - 1444*v^3 + 1156*v^2 + 561*v + 5468) / 4243 $$\beta_{3}$$ $$=$$ $$( 84\nu^{7} - 16\nu^{6} + 441\nu^{5} + 798\nu^{4} + 3208\nu^{3} + 1428\nu^{2} + 693\nu + 2262 ) / 4243$$ (84*v^7 - 16*v^6 + 441*v^5 + 798*v^4 + 3208*v^3 + 1428*v^2 + 693*v + 2262) / 4243 $$\beta_{4}$$ $$=$$ $$( -754\nu^{7} + 1760\nu^{6} - 6080\nu^{5} + 1323\nu^{4} - 13440\nu^{3} + 12640\nu^{2} - 16828\nu + 5760 ) / 12729$$ (-754*v^7 + 1760*v^6 - 6080*v^5 + 1323*v^4 - 13440*v^3 + 12640*v^2 - 16828*v + 5760) / 12729 $$\beta_{5}$$ $$=$$ $$( -815\nu^{7} + 3388\nu^{6} - 11704\nu^{5} + 15594\nu^{4} - 25872\nu^{3} + 24332\nu^{2} - 56579\nu + 11088 ) / 12729$$ (-815*v^7 + 3388*v^6 - 11704*v^5 + 15594*v^4 - 25872*v^3 + 24332*v^2 - 56579*v + 11088) / 12729 $$\beta_{6}$$ $$=$$ $$( 1052\nu^{7} - 2827\nu^{6} + 9766\nu^{5} - 6978\nu^{4} + 21588\nu^{3} - 20303\nu^{2} + 17165\nu - 9252 ) / 12729$$ (1052*v^7 - 2827*v^6 + 9766*v^5 - 6978*v^4 + 21588*v^3 - 20303*v^2 + 17165*v - 9252) / 12729 $$\beta_{7}$$ $$=$$ $$( -556\nu^{7} + 510\nu^{6} - 2919\nu^{5} - 5282\nu^{4} - 8909\nu^{3} - 9452\nu^{2} - 4587\nu - 13760 ) / 4243$$ (-556*v^7 + 510*v^6 - 2919*v^5 - 5282*v^4 - 8909*v^3 - 9452*v^2 - 4587*v - 13760) / 4243
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{6} + 2\beta_{4} + \beta_{3} + \beta_{2} + \beta _1 - 2$$ b6 + 2*b4 + b3 + b2 + b1 - 2 $$\nu^{3}$$ $$=$$ $$\beta_{7} + 5\beta_{3} + 2\beta_{2} - 2$$ b7 + 5*b3 + 2*b2 - 2 $$\nu^{4}$$ $$=$$ $$-8\beta_{6} - 2\beta_{5} - 9\beta_{4} - 10\beta_1$$ -8*b6 - 2*b5 - 9*b4 - 10*b1 $$\nu^{5}$$ $$=$$ $$-8\beta_{7} - 20\beta_{6} - 8\beta_{5} - 18\beta_{4} - 33\beta_{3} - 20\beta_{2} - 33\beta _1 + 18$$ -8*b7 - 20*b6 - 8*b5 - 18*b4 - 33*b3 - 20*b2 - 33*b1 + 18 $$\nu^{6}$$ $$=$$ $$-20\beta_{7} - 83\beta_{3} - 61\beta_{2} + 58$$ -20*b7 - 83*b3 - 61*b2 + 58 $$\nu^{7}$$ $$=$$ $$164\beta_{6} + 61\beta_{5} + 146\beta_{4} + 243\beta_1$$ 164*b6 + 61*b5 + 146*b4 + 243*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/231\mathbb{Z}\right)^\times$$.

 $$n$$ $$155$$ $$199$$ $$211$$ $$\chi(n)$$ $$1$$ $$-\beta_{4}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 1.39083 + 2.40898i 0.643668 + 1.11487i −0.276205 − 0.478401i −0.758290 − 1.31340i 1.39083 − 2.40898i 0.643668 − 1.11487i −0.276205 + 0.478401i −0.758290 + 1.31340i
−1.39083 2.40898i −0.500000 + 0.866025i −2.86880 + 4.96890i −0.412855 0.715087i 2.78165 2.63323 + 0.257073i 10.3967 −0.500000 0.866025i −1.14842 + 1.98912i
67.2 −0.643668 1.11487i −0.500000 + 0.866025i 0.171383 0.296844i −1.95872 3.39260i 1.28734 −0.234193 + 2.63537i −3.01593 −0.500000 0.866025i −2.52153 + 4.36742i
67.3 0.276205 + 0.478401i −0.500000 + 0.866025i 0.847422 1.46778i −0.795012 1.37700i −0.552409 0.886763 2.49272i 2.04107 −0.500000 0.866025i 0.439172 0.760669i
67.4 0.758290 + 1.31340i −0.500000 + 0.866025i −0.150007 + 0.259820i 1.16659 + 2.02059i −1.51658 −2.28580 + 1.33233i 2.57816 −0.500000 0.866025i −1.76922 + 3.06438i
100.1 −1.39083 + 2.40898i −0.500000 0.866025i −2.86880 4.96890i −0.412855 + 0.715087i 2.78165 2.63323 0.257073i 10.3967 −0.500000 + 0.866025i −1.14842 1.98912i
100.2 −0.643668 + 1.11487i −0.500000 0.866025i 0.171383 + 0.296844i −1.95872 + 3.39260i 1.28734 −0.234193 2.63537i −3.01593 −0.500000 + 0.866025i −2.52153 4.36742i
100.3 0.276205 0.478401i −0.500000 0.866025i 0.847422 + 1.46778i −0.795012 + 1.37700i −0.552409 0.886763 + 2.49272i 2.04107 −0.500000 + 0.866025i 0.439172 + 0.760669i
100.4 0.758290 1.31340i −0.500000 0.866025i −0.150007 0.259820i 1.16659 2.02059i −1.51658 −2.28580 1.33233i 2.57816 −0.500000 + 0.866025i −1.76922 3.06438i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 67.4 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 231.2.i.e 8
3.b odd 2 1 693.2.i.i 8
7.c even 3 1 inner 231.2.i.e 8
7.c even 3 1 1617.2.a.z 4
7.d odd 6 1 1617.2.a.x 4
21.g even 6 1 4851.2.a.bu 4
21.h odd 6 1 693.2.i.i 8
21.h odd 6 1 4851.2.a.bt 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.i.e 8 1.a even 1 1 trivial
231.2.i.e 8 7.c even 3 1 inner
693.2.i.i 8 3.b odd 2 1
693.2.i.i 8 21.h odd 6 1
1617.2.a.x 4 7.d odd 6 1
1617.2.a.z 4 7.c even 3 1
4851.2.a.bt 4 21.h odd 6 1
4851.2.a.bu 4 21.g even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 2T_{2}^{7} + 8T_{2}^{6} + 21T_{2}^{4} + 4T_{2}^{3} + 28T_{2}^{2} - 12T_{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(231, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 2 T^{7} + \cdots + 9$$
$3$ $$(T^{2} + T + 1)^{4}$$
$5$ $$T^{8} + 4 T^{7} + \cdots + 144$$
$7$ $$T^{8} - 2 T^{7} + \cdots + 2401$$
$11$ $$(T^{2} - T + 1)^{4}$$
$13$ $$(T^{4} + 2 T^{3} - 8 T^{2} + \cdots - 4)^{2}$$
$17$ $$T^{8} + 2 T^{7} + \cdots + 225$$
$19$ $$T^{8} + 40 T^{6} + \cdots + 7921$$
$23$ $$T^{8} + 4 T^{7} + \cdots + 216225$$
$29$ $$(T^{4} - 8 T^{3} + 12 T^{2} + \cdots + 3)^{2}$$
$31$ $$T^{8} - 12 T^{7} + \cdots + 1948816$$
$37$ $$T^{8} - 4 T^{7} + \cdots + 1$$
$41$ $$(T^{4} - 2 T^{3} - 44 T^{2} + \cdots + 60)^{2}$$
$43$ $$(T^{4} - 18 T^{3} + \cdots - 1385)^{2}$$
$47$ $$T^{8} + 12 T^{7} + \cdots + 81$$
$53$ $$T^{8} - 12 T^{7} + \cdots + 39087504$$
$59$ $$T^{8} + 12 T^{7} + \cdots + 62001$$
$61$ $$T^{8} + 2 T^{7} + \cdots + 400$$
$67$ $$T^{8} + 28 T^{7} + \cdots + 2131600$$
$71$ $$(T^{4} - 12 T^{3} + \cdots - 699)^{2}$$
$73$ $$T^{8} + 6 T^{7} + \cdots + 294328336$$
$79$ $$T^{8} + 2 T^{7} + \cdots + 150544$$
$83$ $$(T^{4} + 12 T^{3} + \cdots + 2592)^{2}$$
$89$ $$T^{8} + 8 T^{7} + \cdots + 145154304$$
$97$ $$(T^{4} + 44 T^{3} + \cdots + 8501)^{2}$$