# Properties

 Label 1456.2.s.q Level $1456$ Weight $2$ Character orbit 1456.s Analytic conductor $11.626$ 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] = [1456,2,Mod(113,1456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1456, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 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("1456.113");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1456 = 2^{4} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1456.s (of order $$3$$, degree $$2$$, 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: $$11.6262185343$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{7} + 7x^{6} + 38x^{4} - 16x^{3} + 15x^{2} + 3x + 1$$ x^8 - x^7 + 7*x^6 + 38*x^4 - 16*x^3 + 15*x^2 + 3*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{6} - \beta_{5}) q^{3} + ( - \beta_{3} - 2) q^{5} + \beta_{4} q^{7} + ( - \beta_{7} + \beta_{5} + 2 \beta_{4}) q^{9}+O(q^{10})$$ q + (b6 - b5) * q^3 + (-b3 - 2) * q^5 + b4 * q^7 + (-b7 + b5 + 2*b4) * q^9 $$q + (\beta_{6} - \beta_{5}) q^{3} + ( - \beta_{3} - 2) q^{5} + \beta_{4} q^{7} + ( - \beta_{7} + \beta_{5} + 2 \beta_{4}) q^{9} + ( - \beta_{6} + \beta_{5}) q^{11} + ( - \beta_{7} - \beta_{5} - \beta_{4} + \beta_{2} + \beta_1) q^{13} + ( - 2 \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 + 1) q^{15} + ( - \beta_{7} - \beta_{5} - \beta_{4} + \beta_1) q^{17} + ( - \beta_{7} - 2 \beta_{5} + 3 \beta_1) q^{19} - \beta_{6} q^{21} + (\beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} + \beta_1) q^{23} + (3 \beta_{3} - \beta_{2} + 2) q^{25} + ( - 3 \beta_{6} - \beta_{3} + 7) q^{27} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{2}) q^{29} + ( - \beta_{2} + 1) q^{31} + (\beta_{7} - \beta_{5} - 5 \beta_{4}) q^{33} + ( - 2 \beta_{4} - \beta_1) q^{35} + (\beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_1 + 2) q^{37} + ( - \beta_{7} + 2 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 4) q^{39} + ( - \beta_{7} + 6 \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_1 + 6) q^{41} + ( - \beta_{4} - \beta_1) q^{43} + (2 \beta_{7} - 3 \beta_{5} - 3 \beta_{4} + 2 \beta_1) q^{45} + ( - \beta_{6} - 3 \beta_{3} - 1) q^{47} + ( - \beta_{4} - 1) q^{49} + ( - \beta_{6} - 2 \beta_{3} - 2 \beta_{2} - 2) q^{51} + ( - 4 \beta_{6} - 2 \beta_{3} - 2 \beta_{2} + 1) q^{53} + (2 \beta_{6} - 2 \beta_{5} - \beta_{4} + \beta_{3} + \beta_1 - 1) q^{55} + ( - \beta_{6} - 4 \beta_{3} - 3 \beta_{2} - 5) q^{57} + (\beta_{7} - 2 \beta_{5} + \beta_{4} - 2 \beta_1) q^{59} + (\beta_{7} + 2 \beta_{5} + 2 \beta_{4} - 2 \beta_1) q^{61} + (\beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_{2} - 2) q^{63} + (\beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{65} + ( - \beta_{7} + 4 \beta_{6} - 4 \beta_{5} - \beta_{4} + 6 \beta_{3} + \beta_{2} + 6 \beta_1 - 1) q^{67} + (2 \beta_{7} + 2 \beta_{5} - 4 \beta_{4}) q^{69} + (2 \beta_{7} + 2 \beta_{5} + 4 \beta_{4}) q^{71} + ( - 2 \beta_{6} - 2 \beta_{3} - 3 \beta_{2} - 2) q^{73} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{75} + \beta_{6} q^{77} + (3 \beta_{6} + \beta_{3} - \beta_{2} + 6) q^{79} + (7 \beta_{6} - 7 \beta_{5} - 8 \beta_{4} - \beta_{3} - \beta_1 - 8) q^{81} + (4 \beta_{3} + \beta_{2} + 1) q^{83} + (\beta_{7} + \beta_{5} + 2 \beta_{4} + 2 \beta_1) q^{85} + (2 \beta_{7} + 2 \beta_{5} - 3 \beta_{4} - \beta_1) q^{87} + (\beta_{7} + 2 \beta_{6} - 2 \beta_{5} + \beta_{3} - \beta_{2} + \beta_1) q^{89} + ( - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{91} + (\beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{93} + ( - \beta_{7} + 3 \beta_{5} - 5 \beta_{4} - 2 \beta_1) q^{95} + ( - \beta_{7} - 5 \beta_{5} - \beta_{4} - 2 \beta_1) q^{97} + (6 \beta_{6} + \beta_{3} - 7) q^{99}+O(q^{100})$$ q + (b6 - b5) * q^3 + (-b3 - 2) * q^5 + b4 * q^7 + (-b7 + b5 + 2*b4) * q^9 + (-b6 + b5) * q^11 + (-b7 - b5 - b4 + b2 + b1) * q^13 + (-2*b6 + 2*b5 + b4 - b3 - b1 + 1) * q^15 + (-b7 - b5 - b4 + b1) * q^17 + (-b7 - 2*b5 + 3*b1) * q^19 - b6 * q^21 + (b7 - b6 + b5 + b3 - b2 + b1) * q^23 + (3*b3 - b2 + 2) * q^25 + (-3*b6 - b3 + 7) * q^27 + (b7 - b6 + b5 - b2) * q^29 + (-b2 + 1) * q^31 + (b7 - b5 - 5*b4) * q^33 + (-2*b4 - b1) * q^35 + (b6 - b5 + 2*b4 - b3 - b1 + 2) * q^37 + (-b7 + 2*b6 - 3*b5 - 2*b4 - b3 - b2 + b1 - 4) * q^39 + (-b7 + 6*b4 + 2*b3 + b2 + 2*b1 + 6) * q^41 + (-b4 - b1) * q^43 + (2*b7 - 3*b5 - 3*b4 + 2*b1) * q^45 + (-b6 - 3*b3 - 1) * q^47 + (-b4 - 1) * q^49 + (-b6 - 2*b3 - 2*b2 - 2) * q^51 + (-4*b6 - 2*b3 - 2*b2 + 1) * q^53 + (2*b6 - 2*b5 - b4 + b3 + b1 - 1) * q^55 + (-b6 - 4*b3 - 3*b2 - 5) * q^57 + (b7 - 2*b5 + b4 - 2*b1) * q^59 + (b7 + 2*b5 + 2*b4 - 2*b1) * q^61 + (b7 + b6 - b5 - 2*b4 - b2 - 2) * q^63 + (b7 + b6 + b5 + 2*b4 + 3*b3 - 2*b2 + 2*b1 + 2) * q^65 + (-b7 + 4*b6 - 4*b5 - b4 + 6*b3 + b2 + 6*b1 - 1) * q^67 + (2*b7 + 2*b5 - 4*b4) * q^69 + (2*b7 + 2*b5 + 4*b4) * q^71 + (-2*b6 - 2*b3 - 3*b2 - 2) * q^73 + (b7 - b6 + b5 - b4 + 2*b3 - b2 + 2*b1 - 1) * q^75 + b6 * q^77 + (3*b6 + b3 - b2 + 6) * q^79 + (7*b6 - 7*b5 - 8*b4 - b3 - b1 - 8) * q^81 + (4*b3 + b2 + 1) * q^83 + (b7 + b5 + 2*b4 + 2*b1) * q^85 + (2*b7 + 2*b5 - 3*b4 - b1) * q^87 + (b7 + 2*b6 - 2*b5 + b3 - b2 + b1) * q^89 + (-b6 + b5 + b4 - b3 - b2 - b1 + 1) * q^91 + (b7 - 2*b6 + 2*b5 + 2*b4 - b3 - b2 - b1 + 2) * q^93 + (-b7 + 3*b5 - 5*b4 - 2*b1) * q^95 + (-b7 - 5*b5 - b4 - 2*b1) * q^97 + (6*b6 + b3 - 7) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + q^{3} - 14 q^{5} - 4 q^{7} - 7 q^{9}+O(q^{10})$$ 8 * q + q^3 - 14 * q^5 - 4 * q^7 - 7 * q^9 $$8 q + q^{3} - 14 q^{5} - 4 q^{7} - 7 q^{9} - q^{11} + 4 q^{13} + 3 q^{15} + 4 q^{17} + q^{19} - 2 q^{21} - 2 q^{23} + 10 q^{25} + 52 q^{27} - q^{29} + 8 q^{31} + 19 q^{33} + 7 q^{35} + 10 q^{37} - 20 q^{39} + 22 q^{41} + 3 q^{43} + 11 q^{45} - 4 q^{47} - 4 q^{49} - 14 q^{51} + 4 q^{53} - 3 q^{55} - 34 q^{57} - 8 q^{59} - 8 q^{61} - 7 q^{63} + 7 q^{65} - 6 q^{67} + 18 q^{69} - 14 q^{71} - 16 q^{73} - 7 q^{75} + 2 q^{77} + 52 q^{79} - 24 q^{81} - 5 q^{85} + 13 q^{87} + q^{89} + 4 q^{91} + 7 q^{93} + 21 q^{95} - 3 q^{97} - 46 q^{99}+O(q^{100})$$ 8 * q + q^3 - 14 * q^5 - 4 * q^7 - 7 * q^9 - q^11 + 4 * q^13 + 3 * q^15 + 4 * q^17 + q^19 - 2 * q^21 - 2 * q^23 + 10 * q^25 + 52 * q^27 - q^29 + 8 * q^31 + 19 * q^33 + 7 * q^35 + 10 * q^37 - 20 * q^39 + 22 * q^41 + 3 * q^43 + 11 * q^45 - 4 * q^47 - 4 * q^49 - 14 * q^51 + 4 * q^53 - 3 * q^55 - 34 * q^57 - 8 * q^59 - 8 * q^61 - 7 * q^63 + 7 * q^65 - 6 * q^67 + 18 * q^69 - 14 * q^71 - 16 * q^73 - 7 * q^75 + 2 * q^77 + 52 * q^79 - 24 * q^81 - 5 * q^85 + 13 * q^87 + q^89 + 4 * q^91 + 7 * q^93 + 21 * q^95 - 3 * q^97 - 46 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 7x^{6} + 38x^{4} - 16x^{3} + 15x^{2} + 3x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -12\nu^{7} - 7\nu^{6} - 76\nu^{5} - 44\nu^{4} - 602\nu^{3} - 36\nu^{2} - 8\nu + 1249 ) / 458$$ (-12*v^7 - 7*v^6 - 76*v^5 - 44*v^4 - 602*v^3 - 36*v^2 - 8*v + 1249) / 458 $$\beta_{3}$$ $$=$$ $$( 57\nu^{7} - 24\nu^{6} + 361\nu^{5} + 209\nu^{4} + 2287\nu^{3} + 171\nu^{2} + 38\nu + 193 ) / 916$$ (57*v^7 - 24*v^6 + 361*v^5 + 209*v^4 + 2287*v^3 + 171*v^2 + 38*v + 193) / 916 $$\beta_{4}$$ $$=$$ $$( 193\nu^{7} - 250\nu^{6} + 1375\nu^{5} - 361\nu^{4} + 7125\nu^{3} - 5375\nu^{2} + 2724\nu - 375 ) / 916$$ (193*v^7 - 250*v^6 + 1375*v^5 - 361*v^4 + 7125*v^3 - 5375*v^2 + 2724*v - 375) / 916 $$\beta_{5}$$ $$=$$ $$( 174\nu^{7} - 242\nu^{6} + 1331\nu^{5} - 507\nu^{4} + 6897\nu^{3} - 5203\nu^{2} + 5383\nu - 363 ) / 458$$ (174*v^7 - 242*v^6 + 1331*v^5 - 507*v^4 + 6897*v^3 - 5203*v^2 + 5383*v - 363) / 458 $$\beta_{6}$$ $$=$$ $$( -375\nu^{7} + 182\nu^{6} - 2375\nu^{5} - 1375\nu^{4} - 13889\nu^{3} - 1125\nu^{2} - 250\nu - 2933 ) / 916$$ (-375*v^7 + 182*v^6 - 2375*v^5 - 1375*v^4 - 13889*v^3 - 1125*v^2 - 250*v - 2933) / 916 $$\beta_{7}$$ $$=$$ $$( -273\nu^{7} + 356\nu^{6} - 1958\nu^{5} + 602\nu^{4} - 10146\nu^{3} + 7654\nu^{2} - 3617\nu + 534 ) / 458$$ (-273*v^7 + 356*v^6 - 1958*v^5 + 602*v^4 - 10146*v^3 + 7654*v^2 - 3617*v + 534) / 458
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{7} - 3\beta_{4} + \beta_{3} + \beta_{2} + \beta _1 - 3$$ -b7 - 3*b4 + b3 + b2 + b1 - 3 $$\nu^{3}$$ $$=$$ $$\beta_{6} + 7\beta_{3} + \beta_{2} - 1$$ b6 + 7*b3 + b2 - 1 $$\nu^{4}$$ $$=$$ $$7\beta_{7} + \beta_{5} + 18\beta_{4} - 10\beta_1$$ 7*b7 + b5 + 18*b4 - 10*b1 $$\nu^{5}$$ $$=$$ $$10\beta_{7} - 7\beta_{6} + 7\beta_{5} + 15\beta_{4} - 48\beta_{3} - 10\beta_{2} - 48\beta _1 + 15$$ 10*b7 - 7*b6 + 7*b5 + 15*b4 - 48*b3 - 10*b2 - 48*b1 + 15 $$\nu^{6}$$ $$=$$ $$-10\beta_{6} - 86\beta_{3} - 48\beta_{2} + 117$$ -10*b6 - 86*b3 - 48*b2 + 117 $$\nu^{7}$$ $$=$$ $$-86\beta_{7} - 48\beta_{5} - 152\beta_{4} + 337\beta_1$$ -86*b7 - 48*b5 - 152*b4 + 337*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times$$.

 $$n$$ $$561$$ $$911$$ $$1093$$ $$1249$$ $$\chi(n)$$ $$\beta_{4}$$ $$1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
113.1
 −0.115680 − 0.200364i −1.11000 − 1.92258i 1.37054 + 2.37385i 0.355143 + 0.615126i −0.115680 + 0.200364i −1.11000 + 1.92258i 1.37054 − 2.37385i 0.355143 − 0.615126i
0 −1.66113 + 2.87716i 0 −2.23136 0 −0.500000 0.866025i 0 −4.01868 6.96056i 0
113.2 0 0.274776 0.475925i 0 −4.22001 0 −0.500000 0.866025i 0 1.34900 + 2.33653i 0
113.3 0 0.682410 1.18197i 0 0.741082 0 −0.500000 0.866025i 0 0.568634 + 0.984903i 0
113.4 0 1.20394 2.08529i 0 −1.28971 0 −0.500000 0.866025i 0 −1.39895 2.42305i 0
1121.1 0 −1.66113 2.87716i 0 −2.23136 0 −0.500000 + 0.866025i 0 −4.01868 + 6.96056i 0
1121.2 0 0.274776 + 0.475925i 0 −4.22001 0 −0.500000 + 0.866025i 0 1.34900 2.33653i 0
1121.3 0 0.682410 + 1.18197i 0 0.741082 0 −0.500000 + 0.866025i 0 0.568634 0.984903i 0
1121.4 0 1.20394 + 2.08529i 0 −1.28971 0 −0.500000 + 0.866025i 0 −1.39895 + 2.42305i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 113.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1456.2.s.q 8
4.b odd 2 1 91.2.f.c 8
12.b even 2 1 819.2.o.h 8
13.c even 3 1 inner 1456.2.s.q 8
28.d even 2 1 637.2.f.i 8
28.f even 6 1 637.2.g.j 8
28.f even 6 1 637.2.h.i 8
28.g odd 6 1 637.2.g.k 8
28.g odd 6 1 637.2.h.h 8
52.i odd 6 1 1183.2.a.l 4
52.j odd 6 1 91.2.f.c 8
52.j odd 6 1 1183.2.a.k 4
52.l even 12 2 1183.2.c.g 8
156.p even 6 1 819.2.o.h 8
364.q odd 6 1 637.2.h.h 8
364.v even 6 1 637.2.f.i 8
364.v even 6 1 8281.2.a.bp 4
364.ba even 6 1 637.2.g.j 8
364.bc even 6 1 8281.2.a.bt 4
364.bi odd 6 1 637.2.g.k 8
364.br even 6 1 637.2.h.i 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.c 8 4.b odd 2 1
91.2.f.c 8 52.j odd 6 1
637.2.f.i 8 28.d even 2 1
637.2.f.i 8 364.v even 6 1
637.2.g.j 8 28.f even 6 1
637.2.g.j 8 364.ba even 6 1
637.2.g.k 8 28.g odd 6 1
637.2.g.k 8 364.bi odd 6 1
637.2.h.h 8 28.g odd 6 1
637.2.h.h 8 364.q odd 6 1
637.2.h.i 8 28.f even 6 1
637.2.h.i 8 364.br even 6 1
819.2.o.h 8 12.b even 2 1
819.2.o.h 8 156.p even 6 1
1183.2.a.k 4 52.j odd 6 1
1183.2.a.l 4 52.i odd 6 1
1183.2.c.g 8 52.l even 12 2
1456.2.s.q 8 1.a even 1 1 trivial
1456.2.s.q 8 13.c even 3 1 inner
8281.2.a.bp 4 364.v even 6 1
8281.2.a.bt 4 364.bc even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1456, [\chi])$$:

 $$T_{3}^{8} - T_{3}^{7} + 10T_{3}^{6} - 23T_{3}^{5} + 103T_{3}^{4} - 156T_{3}^{3} + 202T_{3}^{2} - 96T_{3} + 36$$ T3^8 - T3^7 + 10*T3^6 - 23*T3^5 + 103*T3^4 - 156*T3^3 + 202*T3^2 - 96*T3 + 36 $$T_{5}^{4} + 7T_{5}^{3} + 12T_{5}^{2} - T_{5} - 9$$ T5^4 + 7*T5^3 + 12*T5^2 - T5 - 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} - T^{7} + 10 T^{6} - 23 T^{5} + \cdots + 36$$
$5$ $$(T^{4} + 7 T^{3} + 12 T^{2} - T - 9)^{2}$$
$7$ $$(T^{2} + T + 1)^{4}$$
$11$ $$T^{8} + T^{7} + 10 T^{6} + 23 T^{5} + \cdots + 36$$
$13$ $$T^{8} - 4 T^{7} + 16 T^{6} + \cdots + 28561$$
$17$ $$T^{8} - 4 T^{7} + 28 T^{6} + \cdots + 2809$$
$19$ $$T^{8} - T^{7} + 56 T^{6} + \cdots + 250000$$
$23$ $$T^{8} + 2 T^{7} + 42 T^{6} + \cdots + 5184$$
$29$ $$T^{8} + T^{7} + 23 T^{6} - 64 T^{5} + \cdots + 25$$
$31$ $$(T^{4} - 4 T^{3} - 13 T^{2} + 26 T + 54)^{2}$$
$37$ $$T^{8} - 10 T^{7} + 83 T^{6} + \cdots + 256$$
$41$ $$T^{8} - 22 T^{7} + 335 T^{6} + \cdots + 318096$$
$43$ $$T^{8} - 3 T^{7} + 12 T^{6} - 7 T^{5} + \cdots + 4$$
$47$ $$(T^{4} + 2 T^{3} - 51 T^{2} + 12 T + 100)^{2}$$
$53$ $$(T^{4} - 2 T^{3} - 140 T^{2} + 890 T - 1389)^{2}$$
$59$ $$T^{8} + 8 T^{7} + 141 T^{6} + \cdots + 498436$$
$61$ $$T^{8} + 8 T^{7} + 79 T^{6} + \cdots + 10000$$
$67$ $$T^{8} + 6 T^{7} + 247 T^{6} + \cdots + 121220100$$
$71$ $$T^{8} + 14 T^{7} + 212 T^{6} + \cdots + 4129024$$
$73$ $$(T^{4} + 8 T^{3} - 119 T^{2} - 88 T + 1772)^{2}$$
$79$ $$(T^{4} - 26 T^{3} + 134 T^{2} + 1024 T - 7680)^{2}$$
$83$ $$(T^{4} - 97 T^{2} + 442 T - 426)^{2}$$
$89$ $$T^{8} - T^{7} + 72 T^{6} - 297 T^{5} + \cdots + 11664$$
$97$ $$T^{8} + 3 T^{7} + 314 T^{6} + \cdots + 246238864$$