# Properties

 Label 176.2.m.d Level $176$ Weight $2$ Character orbit 176.m Analytic conductor $1.405$ 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] = [176,2,Mod(49,176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(176, base_ring=CyclotomicField(10))

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

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

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

chi := DirichletCharacter("176.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$176 = 2^{4} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 176.m (of order $$5$$, degree $$4$$, not 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: $$1.40536707557$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.682515625.5 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 3x^{7} + 5x^{6} + 2x^{5} + 19x^{4} + 28x^{3} + 100x^{2} + 88x + 121$$ x^8 - 3*x^7 + 5*x^6 + 2*x^5 + 19*x^4 + 28*x^3 + 100*x^2 + 88*x + 121 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 88) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{6} - \beta_{4} - \beta_1) q^{3} + ( - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3}) q^{5} + (2 \beta_{7} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 2) q^{7} + (\beta_{5} - 2 \beta_{3} - \beta_{2} - 2) q^{9}+O(q^{10})$$ q + (b6 - b4 - b1) * q^3 + (-b6 - b5 - b4 + b3) * q^5 + (2*b7 - b5 - b4 - b3 + b2 - 2) * q^7 + (b5 - 2*b3 - b2 - 2) * q^9 $$q + (\beta_{6} - \beta_{4} - \beta_1) q^{3} + ( - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3}) q^{5} + (2 \beta_{7} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 2) q^{7} + (\beta_{5} - 2 \beta_{3} - \beta_{2} - 2) q^{9} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + 1) q^{11} + (\beta_{6} - \beta_{5} + \beta_{3} + 1) q^{13} + ( - \beta_{7} - \beta_{6} - 3 \beta_{3} + \beta_1 + 1) q^{15} + (3 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - \beta_{4} - 2 \beta_{2} - 3 \beta_1) q^{17} + ( - 4 \beta_{7} + 4 \beta_{3} - 3 \beta_{2} + 3) q^{19} + ( - 3 \beta_{7} - \beta_{5} + \beta_{4} - 2 \beta_{2} + \beta_1 + 1) q^{21} + (2 \beta_{7} + 2 \beta_{4} + 2 \beta_{2}) q^{23} + (6 \beta_{2} + \beta_1 - 6) q^{25} + (5 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{27} + ( - 2 \beta_{7} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 2) q^{29} + (\beta_{6} + \beta_{5} + \beta_{3} + 2 \beta_{2} + 1) q^{31} + (5 \beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{4} - 3 \beta_{3} + 5 \beta_{2} + \beta_1 - 1) q^{33} + (2 \beta_{6} + \beta_{5} - 5 \beta_{3} + 7 \beta_{2} - 5) q^{35} + ( - 2 \beta_{7} - 2 \beta_{6} + \beta_{5} + \beta_{4} + 5 \beta_{3} - \beta_{2} + 2 \beta_1 + 2) q^{37} + ( - 7 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - \beta_{4} + 5 \beta_{3} - 7 \beta_{2} - 3 \beta_1) q^{39} + ( - 2 \beta_{7} - \beta_{6} + \beta_{4} + 2 \beta_{3} - 4 \beta_{2} - \beta_1 + 4) q^{41} + ( - \beta_{5} + \beta_{4} + \beta_{2} + \beta_1 - 1) q^{43} + ( - \beta_{7} - \beta_{5} + 2 \beta_{4} + \beta_1 + 6) q^{45} + ( - 3 \beta_{7} - 2 \beta_{6} + 2 \beta_{4} + 3 \beta_{3} - 3 \beta_{2} + \beta_1 + 3) q^{47} + ( - \beta_{7} + \beta_{6} + \beta_{5} + 3 \beta_{4} - \beta_{3} + 2 \beta_1) q^{49} + (8 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - 9 \beta_{3} - 2 \beta_{2} + \cdots - 8) q^{51}+ \cdots + ( - 3 \beta_{7} + 4 \beta_{6} - \beta_{5} - \beta_{4} + 2 \beta_{3} + 4 \beta_{2} - 4 \beta_1 + 4) q^{99}+O(q^{100})$$ q + (b6 - b4 - b1) * q^3 + (-b6 - b5 - b4 + b3) * q^5 + (2*b7 - b5 - b4 - b3 + b2 - 2) * q^7 + (b5 - 2*b3 - b2 - 2) * q^9 + (-b7 - b6 + b5 + b4 + 2*b3 - 3*b2 + 1) * q^11 + (b6 - b5 + b3 + 1) * q^13 + (-b7 - b6 - 3*b3 + b1 + 1) * q^15 + (3*b7 + 2*b6 + 2*b5 - b4 - 2*b2 - 3*b1) * q^17 + (-4*b7 + 4*b3 - 3*b2 + 3) * q^19 + (-3*b7 - b5 + b4 - 2*b2 + b1 + 1) * q^21 + (2*b7 + 2*b4 + 2*b2) * q^23 + (6*b2 + b1 - 6) * q^25 + (5*b7 - 2*b6 - 2*b5 - 2*b3 + 4*b2 + 2*b1) * q^27 + (-2*b7 - b5 - b4 - b3 + b2 + 2) * q^29 + (b6 + b5 + b3 + 2*b2 + 1) * q^31 + (5*b7 - b6 + b5 + 2*b4 - 3*b3 + 5*b2 + b1 - 1) * q^33 + (2*b6 + b5 - 5*b3 + 7*b2 - 5) * q^35 + (-2*b7 - 2*b6 + b5 + b4 + 5*b3 - b2 + 2*b1 + 2) * q^37 + (-7*b7 + 2*b6 + 2*b5 - b4 + 5*b3 - 7*b2 - 3*b1) * q^39 + (-2*b7 - b6 + b4 + 2*b3 - 4*b2 - b1 + 4) * q^41 + (-b5 + b4 + b2 + b1 - 1) * q^43 + (-b7 - b5 + 2*b4 + b1 + 6) * q^45 + (-3*b7 - 2*b6 + 2*b4 + 3*b3 - 3*b2 + b1 + 3) * q^47 + (-b7 + b6 + b5 + 3*b4 - b3 + 2*b1) * q^49 + (8*b7 + 2*b6 + 2*b5 + 2*b4 - 9*b3 - 2*b2 - 2*b1 - 8) * q^51 + (-b6 - b5 - 5*b3 - 5) * q^53 + (6*b7 + b6 - b5 - b4 + 3*b3 + 2*b2 + 2*b1 - 4) * q^55 + (-b6 + 4*b5 - 4*b2) * q^57 + (-5*b7 + 5) * q^59 + (2*b7 - 3*b6 - 3*b5 + b4 + 5*b3 - 2*b2 + 4*b1) * q^61 + (-4*b7 - b6 + b4 + 4*b3 - 5*b2 + 5) * q^63 + (-2*b7 - 3*b4 - 2*b2 - 1) * q^65 + (-b7 - 2*b5 - 3*b4 + b2 + 2*b1) * q^67 + (-4*b7 + 2*b6 - 2*b4 + 4*b3 + 2*b2 - 6*b1 - 2) * q^69 + (2*b7 + 3*b6 + 3*b5 - b4 - 7*b3 + 4*b2 - 4*b1) * q^71 + (2*b7 + 3*b6 - 4*b3 - 3*b1 - 2) * q^73 + (-5*b6 + 3*b3 - b2 + 3) * q^75 + (8*b7 + 2*b6 + b5 - 3*b4 - 3*b3 + 7*b2 - 2*b1 - 6) * q^77 + (-3*b6 - 3*b5 + b3 + 2*b2 + 1) * q^79 + (2*b6 - 4*b5 - 4*b4 - 2*b3 + 4*b2 - 2*b1) * q^81 + (5*b7 - 2*b6 - 2*b5 - 2*b4 + 3*b3 - b2) * q^83 + (b7 + 2*b6 - 2*b4 - b3 - 3*b2 + 3*b1 + 3) * q^85 + (-3*b7 - b5 - 3*b4 - 2*b2 + b1 + 1) * q^87 + (b5 - b4 - b2 - b1 + 3) * q^89 + (-3*b7 - b6 + b4 + 3*b3 + 3*b1) * q^91 + (3*b7 + 2*b6 + 2*b5 - 3*b4 + b3 - 3*b2 - 5*b1) * q^93 + (3*b7 - 3*b6 - 2*b5 - 2*b4 - b3 + 2*b2 + 3*b1 - 3) * q^95 + (2*b6 - 2*b5 + b3 + 1) * q^97 + (-3*b7 + 4*b6 - b5 - b4 + 2*b3 + 4*b2 - 4*b1 + 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + q^{3} - 3 q^{5} - 7 q^{7} - 13 q^{9}+O(q^{10})$$ 8 * q + q^3 - 3 * q^5 - 7 * q^7 - 13 * q^9 $$8 q + q^{3} - 3 q^{5} - 7 q^{7} - 13 q^{9} - 7 q^{11} + 7 q^{13} + 13 q^{15} + q^{17} + 2 q^{19} - 2 q^{21} + 4 q^{23} - 33 q^{25} + 22 q^{27} + 17 q^{29} + 13 q^{31} + 16 q^{33} - 11 q^{35} + q^{37} - 39 q^{39} + 9 q^{41} - 6 q^{43} + 44 q^{45} + q^{47} + 3 q^{49} - 38 q^{51} - 33 q^{53} - 13 q^{55} - 6 q^{57} + 30 q^{59} - 9 q^{61} + 10 q^{63} - 10 q^{65} + 10 q^{67} - 38 q^{69} + 25 q^{71} - 7 q^{73} + 6 q^{75} - 7 q^{77} + q^{79} + 14 q^{81} + 39 q^{85} + 6 q^{87} + 22 q^{89} - 7 q^{91} - 5 q^{93} - 7 q^{95} + 8 q^{97} + 27 q^{99}+O(q^{100})$$ 8 * q + q^3 - 3 * q^5 - 7 * q^7 - 13 * q^9 - 7 * q^11 + 7 * q^13 + 13 * q^15 + q^17 + 2 * q^19 - 2 * q^21 + 4 * q^23 - 33 * q^25 + 22 * q^27 + 17 * q^29 + 13 * q^31 + 16 * q^33 - 11 * q^35 + q^37 - 39 * q^39 + 9 * q^41 - 6 * q^43 + 44 * q^45 + q^47 + 3 * q^49 - 38 * q^51 - 33 * q^53 - 13 * q^55 - 6 * q^57 + 30 * q^59 - 9 * q^61 + 10 * q^63 - 10 * q^65 + 10 * q^67 - 38 * q^69 + 25 * q^71 - 7 * q^73 + 6 * q^75 - 7 * q^77 + q^79 + 14 * q^81 + 39 * q^85 + 6 * q^87 + 22 * q^89 - 7 * q^91 - 5 * q^93 - 7 * q^95 + 8 * q^97 + 27 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3x^{7} + 5x^{6} + 2x^{5} + 19x^{4} + 28x^{3} + 100x^{2} + 88x + 121$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 528 \nu^{7} + 2098 \nu^{6} - 15725 \nu^{5} + 33439 \nu^{4} + 71401 \nu^{3} - 332708 \nu^{2} + 319181 \nu + 440220 ) / 1168519$$ (528*v^7 + 2098*v^6 - 15725*v^5 + 33439*v^4 + 71401*v^3 - 332708*v^2 + 319181*v + 440220) / 1168519 $$\beta_{3}$$ $$=$$ $$( 5794 \nu^{7} - 9973 \nu^{6} - 30517 \nu^{5} + 195125 \nu^{4} - 61888 \nu^{3} + 104068 \nu^{2} + 501961 \nu + 528473 ) / 1168519$$ (5794*v^7 - 9973*v^6 - 30517*v^5 + 195125*v^4 - 61888*v^3 + 104068*v^2 + 501961*v + 528473) / 1168519 $$\beta_{4}$$ $$=$$ $$( 7409 \nu^{7} - 59487 \nu^{6} + 183537 \nu^{5} - 171974 \nu^{4} - 58164 \nu^{3} - 77439 \nu^{2} + 18601 \nu - 701074 ) / 1168519$$ (7409*v^7 - 59487*v^6 + 183537*v^5 - 171974*v^4 - 58164*v^3 - 77439*v^2 + 18601*v - 701074) / 1168519 $$\beta_{5}$$ $$=$$ $$( 8817 \nu^{7} + 16927 \nu^{6} - 106264 \nu^{5} + 200474 \nu^{4} + 521745 \nu^{3} + 380907 \nu^{2} + 2179908 \nu + 2809884 ) / 1168519$$ (8817*v^7 + 16927*v^6 - 106264*v^5 + 200474*v^4 + 521745*v^3 + 380907*v^2 + 2179908*v + 2809884) / 1168519 $$\beta_{6}$$ $$=$$ $$( - 11971 \nu^{7} + 3536 \nu^{6} + 58156 \nu^{5} - 228404 \nu^{4} - 102852 \nu^{3} - 979996 \nu^{2} - 1085964 \nu - 2305776 ) / 1168519$$ (-11971*v^7 + 3536*v^6 + 58156*v^5 - 228404*v^4 - 102852*v^3 - 979996*v^2 - 1085964*v - 2305776) / 1168519 $$\beta_{7}$$ $$=$$ $$( - 13790 \nu^{7} + 57068 \nu^{6} - 113608 \nu^{5} + 65418 \nu^{4} - 266949 \nu^{3} + 6060 \nu^{2} - 742824 \nu + 665808 ) / 1168519$$ (-13790*v^7 + 57068*v^6 - 113608*v^5 + 65418*v^4 - 266949*v^3 + 6060*v^2 - 742824*v + 665808) / 1168519
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{5} + \beta_{3} - 5\beta_{2} + 1$$ b6 + b5 + b3 - 5*b2 + 1 $$\nu^{3}$$ $$=$$ $$\beta_{7} + 6\beta_{6} + 6\beta_{5} + 2\beta_{4} + 4\beta_{3} - 10\beta_{2} - 4\beta_1$$ b7 + 6*b6 + 6*b5 + 2*b4 + 4*b3 - 10*b2 - 4*b1 $$\nu^{4}$$ $$=$$ $$12\beta_{7} + 10\beta_{6} + 13\beta_{5} + 13\beta_{4} + 14\beta_{3} - 13\beta_{2} - 10\beta _1 - 12$$ 12*b7 + 10*b6 + 13*b5 + 13*b4 + 14*b3 - 13*b2 - 10*b1 - 12 $$\nu^{5}$$ $$=$$ $$43\beta_{7} + 25\beta_{5} + 49\beta_{4} + 18\beta_{2} - 25\beta _1 - 62$$ 43*b7 + 25*b5 + 49*b4 + 18*b2 - 25*b1 - 62 $$\nu^{6}$$ $$=$$ $$97\beta_{7} - 92\beta_{6} + 92\beta_{4} - 97\beta_{3} + 221\beta_{2} - 44\beta _1 - 221$$ 97*b7 - 92*b6 + 92*b4 - 97*b3 + 221*b2 - 44*b1 - 221 $$\nu^{7}$$ $$=$$ $$-449\beta_{6} - 260\beta_{5} - 412\beta_{3} + 896\beta_{2} - 412$$ -449*b6 - 260*b5 - 412*b3 + 896*b2 - 412

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/176\mathbb{Z}\right)^\times$$.

 $$n$$ $$111$$ $$133$$ $$145$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{3}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −1.20316 − 0.874145i 2.51217 + 1.82520i 0.581882 − 1.79085i −0.390899 + 1.20306i −1.20316 + 0.874145i 2.51217 − 1.82520i 0.581882 + 1.79085i −0.390899 − 1.20306i
0 −0.743592 0.540251i 0 −1.24359 + 3.82738i 0 −3.01217 + 2.18847i 0 −0.665993 2.04972i 0
49.2 0 1.55261 + 1.12804i 0 1.05261 3.23960i 0 0.703158 0.510874i 0 0.211078 + 0.649631i 0
81.1 0 −0.941506 + 2.89766i 0 −1.44151 + 1.04732i 0 −0.109101 0.335777i 0 −5.08293 3.69296i 0
81.2 0 0.632489 1.94660i 0 0.132489 0.0962586i 0 −1.08188 3.32969i 0 −0.962157 0.699048i 0
97.1 0 −0.743592 + 0.540251i 0 −1.24359 3.82738i 0 −3.01217 2.18847i 0 −0.665993 + 2.04972i 0
97.2 0 1.55261 1.12804i 0 1.05261 + 3.23960i 0 0.703158 + 0.510874i 0 0.211078 0.649631i 0
113.1 0 −0.941506 2.89766i 0 −1.44151 1.04732i 0 −0.109101 + 0.335777i 0 −5.08293 + 3.69296i 0
113.2 0 0.632489 + 1.94660i 0 0.132489 + 0.0962586i 0 −1.08188 + 3.32969i 0 −0.962157 + 0.699048i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 49.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 176.2.m.d 8
4.b odd 2 1 88.2.i.b 8
8.b even 2 1 704.2.m.i 8
8.d odd 2 1 704.2.m.l 8
11.c even 5 1 inner 176.2.m.d 8
11.c even 5 1 1936.2.a.bb 4
11.d odd 10 1 1936.2.a.bc 4
12.b even 2 1 792.2.r.g 8
44.c even 2 1 968.2.i.p 8
44.g even 10 1 968.2.a.m 4
44.g even 10 1 968.2.i.p 8
44.g even 10 2 968.2.i.t 8
44.h odd 10 1 88.2.i.b 8
44.h odd 10 1 968.2.a.n 4
44.h odd 10 2 968.2.i.s 8
88.k even 10 1 7744.2.a.dh 4
88.l odd 10 1 704.2.m.l 8
88.l odd 10 1 7744.2.a.di 4
88.o even 10 1 704.2.m.i 8
88.o even 10 1 7744.2.a.dr 4
88.p odd 10 1 7744.2.a.ds 4
132.n odd 10 1 8712.2.a.cd 4
132.o even 10 1 792.2.r.g 8
132.o even 10 1 8712.2.a.ce 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.2.i.b 8 4.b odd 2 1
88.2.i.b 8 44.h odd 10 1
176.2.m.d 8 1.a even 1 1 trivial
176.2.m.d 8 11.c even 5 1 inner
704.2.m.i 8 8.b even 2 1
704.2.m.i 8 88.o even 10 1
704.2.m.l 8 8.d odd 2 1
704.2.m.l 8 88.l odd 10 1
792.2.r.g 8 12.b even 2 1
792.2.r.g 8 132.o even 10 1
968.2.a.m 4 44.g even 10 1
968.2.a.n 4 44.h odd 10 1
968.2.i.p 8 44.c even 2 1
968.2.i.p 8 44.g even 10 1
968.2.i.s 8 44.h odd 10 2
968.2.i.t 8 44.g even 10 2
1936.2.a.bb 4 11.c even 5 1
1936.2.a.bc 4 11.d odd 10 1
7744.2.a.dh 4 88.k even 10 1
7744.2.a.di 4 88.l odd 10 1
7744.2.a.dr 4 88.o even 10 1
7744.2.a.ds 4 88.p odd 10 1
8712.2.a.cd 4 132.n odd 10 1
8712.2.a.ce 4 132.o even 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} - T_{3}^{7} + 10T_{3}^{6} - 19T_{3}^{5} + 49T_{3}^{4} - 29T_{3}^{3} + 20T_{3}^{2} + 99T_{3} + 121$$ acting on $$S_{2}^{\mathrm{new}}(176, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} - T^{7} + 10 T^{6} - 19 T^{5} + \cdots + 121$$
$5$ $$T^{8} + 3 T^{7} + 26 T^{6} + 54 T^{5} + \cdots + 16$$
$7$ $$T^{8} + 7 T^{7} + 30 T^{6} + 62 T^{5} + \cdots + 16$$
$11$ $$T^{8} + 7 T^{7} + 23 T^{6} + \cdots + 14641$$
$13$ $$T^{8} - 7 T^{7} + 20 T^{6} + \cdots + 1936$$
$17$ $$T^{8} - T^{7} + 22 T^{6} - 153 T^{5} + \cdots + 39601$$
$19$ $$(T^{4} - T^{3} + 16 T^{2} - 66 T + 121)^{2}$$
$23$ $$(T^{4} - 2 T^{3} - 52 T^{2} + 128 T - 64)^{2}$$
$29$ $$T^{8} - 17 T^{7} + 130 T^{6} + \cdots + 1936$$
$31$ $$T^{8} - 13 T^{7} + 96 T^{6} + \cdots + 55696$$
$37$ $$T^{8} - T^{7} + 110 T^{6} + 726 T^{5} + \cdots + 1936$$
$41$ $$T^{8} - 9 T^{7} + 14 T^{6} + 267 T^{5} + \cdots + 121$$
$43$ $$(T^{4} + 3 T^{3} - 15 T^{2} - 44 T - 16)^{2}$$
$47$ $$T^{8} - T^{7} + 12 T^{6} - 78 T^{5} + \cdots + 30976$$
$53$ $$T^{8} + 33 T^{7} + 536 T^{6} + \cdots + 30976$$
$59$ $$(T^{4} - 15 T^{3} + 100 T^{2} - 250 T + 625)^{2}$$
$61$ $$T^{8} + 9 T^{7} + 164 T^{6} + \cdots + 3748096$$
$67$ $$(T^{4} - 5 T^{3} - 101 T^{2} - 260 T - 176)^{2}$$
$71$ $$T^{8} - 25 T^{7} + 402 T^{6} + \cdots + 512656$$
$73$ $$T^{8} + 7 T^{7} + 130 T^{6} + \cdots + 609961$$
$79$ $$T^{8} - T^{7} + 204 T^{6} + \cdots + 712336$$
$83$ $$T^{8} + 167 T^{6} - 1680 T^{5} + \cdots + 121$$
$89$ $$(T^{4} - 11 T^{3} + 27 T^{2} + 36 T - 124)^{2}$$
$97$ $$T^{8} - 8 T^{7} + 11 T^{6} + \cdots + 241081$$