# Properties

 Label 405.3.l.f Level $405$ Weight $3$ Character orbit 405.l Analytic conductor $11.035$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 405.l (of order $$12$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.0354507066$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 15) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{24}^{2} - 2 \zeta_{24}^{3} - \zeta_{24}^{4} + \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{2} + ( 4 \zeta_{24} + \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{4} + ( -1 + 2 \zeta_{24} - 3 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + \zeta_{24}^{4} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{5} + ( -\zeta_{24}^{2} + 4 \zeta_{24}^{3} - \zeta_{24}^{4} + \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{7} + ( -3 - \zeta_{24} - \zeta_{24}^{5} - 3 \zeta_{24}^{6} ) q^{8} +O(q^{10})$$ $$q + ( -\zeta_{24}^{2} - 2 \zeta_{24}^{3} - \zeta_{24}^{4} + \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{2} + ( 4 \zeta_{24} + \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{4} + ( -1 + 2 \zeta_{24} - 3 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + \zeta_{24}^{4} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{5} + ( -\zeta_{24}^{2} + 4 \zeta_{24}^{3} - \zeta_{24}^{4} + \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{7} + ( -3 - \zeta_{24} - \zeta_{24}^{5} - 3 \zeta_{24}^{6} ) q^{8} + ( 1 + 4 \zeta_{24} - 2 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 8 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{10} + ( 3 \zeta_{24} - 6 \zeta_{24}^{3} + 4 \zeta_{24}^{4} - 6 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{11} + ( 8 + 4 \zeta_{24} + 8 \zeta_{24}^{2} - 8 \zeta_{24}^{4} - 2 \zeta_{24}^{5} ) q^{13} + ( -2 \zeta_{24} - 4 \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{14} + ( 4 \zeta_{24} - 8 \zeta_{24}^{3} + 5 \zeta_{24}^{4} - 8 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{16} + ( 10 - 6 \zeta_{24}^{3} - 10 \zeta_{24}^{6} + 12 \zeta_{24}^{7} ) q^{17} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} + 6 \zeta_{24}^{6} + 12 \zeta_{24}^{7} ) q^{19} + ( 6 \zeta_{24} + 17 \zeta_{24}^{2} - 14 \zeta_{24}^{3} - 9 \zeta_{24}^{4} - 12 \zeta_{24}^{5} - 17 \zeta_{24}^{6} + 7 \zeta_{24}^{7} ) q^{20} + ( -5 + 5 \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 5 \zeta_{24}^{4} + 2 \zeta_{24}^{7} ) q^{22} + ( 14 + 4 \zeta_{24} + 14 \zeta_{24}^{2} - 14 \zeta_{24}^{4} - 2 \zeta_{24}^{5} ) q^{23} + ( -14 \zeta_{24} - 3 \zeta_{24}^{2} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{4} + 28 \zeta_{24}^{5} + 3 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{25} + ( -22 - 10 \zeta_{24} - 10 \zeta_{24}^{3} - 10 \zeta_{24}^{5} + 20 \zeta_{24}^{7} ) q^{26} + ( 11 - 2 \zeta_{24} - 2 \zeta_{24}^{5} + 11 \zeta_{24}^{6} ) q^{28} + ( -7 \zeta_{24} + 18 \zeta_{24}^{2} - 14 \zeta_{24}^{3} + 14 \zeta_{24}^{5} - 18 \zeta_{24}^{6} + 7 \zeta_{24}^{7} ) q^{29} + ( 4 + 12 \zeta_{24} - 6 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 6 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{31} + ( -19 + 19 \zeta_{24}^{2} + 7 \zeta_{24}^{3} + 19 \zeta_{24}^{4} + 7 \zeta_{24}^{7} ) q^{32} + ( -4 \zeta_{24} - 2 \zeta_{24}^{2} - 8 \zeta_{24}^{3} + 8 \zeta_{24}^{5} + 2 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{34} + ( 10 - 5 \zeta_{24} - 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} - 10 \zeta_{24}^{6} + 10 \zeta_{24}^{7} ) q^{35} + ( 16 + 18 \zeta_{24}^{3} - 16 \zeta_{24}^{6} - 36 \zeta_{24}^{7} ) q^{37} + ( 18 \zeta_{24} + 24 \zeta_{24}^{2} - 24 \zeta_{24}^{4} - 36 \zeta_{24}^{5} - 24 \zeta_{24}^{6} ) q^{38} + ( -12 - 16 \zeta_{24} + 9 \zeta_{24}^{2} + 6 \zeta_{24}^{3} + 12 \zeta_{24}^{4} + 8 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{40} + ( -14 - 12 \zeta_{24} + 6 \zeta_{24}^{3} + 14 \zeta_{24}^{4} + 6 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{41} + ( 20 \zeta_{24} - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} - 40 \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{43} + ( 5 \zeta_{24} - 5 \zeta_{24}^{3} + 5 \zeta_{24}^{5} - 32 \zeta_{24}^{6} + 10 \zeta_{24}^{7} ) q^{44} + ( -34 - 16 \zeta_{24} - 16 \zeta_{24}^{3} - 16 \zeta_{24}^{5} + 32 \zeta_{24}^{7} ) q^{46} + ( 32 \zeta_{24}^{2} - 20 \zeta_{24}^{3} + 32 \zeta_{24}^{4} - 32 \zeta_{24}^{6} + 10 \zeta_{24}^{7} ) q^{47} + ( -8 \zeta_{24} - 35 \zeta_{24}^{2} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{49} + ( 41 + 38 \zeta_{24} + 13 \zeta_{24}^{2} - 8 \zeta_{24}^{3} - 41 \zeta_{24}^{4} - 19 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{50} + ( 20 \zeta_{24}^{2} + 68 \zeta_{24}^{3} + 20 \zeta_{24}^{4} - 20 \zeta_{24}^{6} - 34 \zeta_{24}^{7} ) q^{52} + ( -14 - 12 \zeta_{24} - 12 \zeta_{24}^{5} - 14 \zeta_{24}^{6} ) q^{53} + ( -31 + 16 \zeta_{24} + 2 \zeta_{24}^{3} + 16 \zeta_{24}^{5} - 3 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{55} + ( 5 \zeta_{24} - 10 \zeta_{24}^{3} - 10 \zeta_{24}^{5} + 5 \zeta_{24}^{7} ) q^{56} + ( 3 - 8 \zeta_{24} + 3 \zeta_{24}^{2} - 3 \zeta_{24}^{4} + 4 \zeta_{24}^{5} ) q^{58} + ( -62 \zeta_{24} - 36 \zeta_{24}^{2} - 31 \zeta_{24}^{3} + 31 \zeta_{24}^{5} - 31 \zeta_{24}^{7} ) q^{59} + ( 18 \zeta_{24} - 36 \zeta_{24}^{3} - 50 \zeta_{24}^{4} - 36 \zeta_{24}^{5} + 18 \zeta_{24}^{7} ) q^{61} + ( -22 - 16 \zeta_{24}^{3} + 22 \zeta_{24}^{6} + 32 \zeta_{24}^{7} ) q^{62} + ( -10 \zeta_{24} + 10 \zeta_{24}^{3} - 10 \zeta_{24}^{5} - 79 \zeta_{24}^{6} - 20 \zeta_{24}^{7} ) q^{64} + ( 22 \zeta_{24} - 26 \zeta_{24}^{2} - 28 \zeta_{24}^{3} - 28 \zeta_{24}^{4} - 44 \zeta_{24}^{5} + 26 \zeta_{24}^{6} + 14 \zeta_{24}^{7} ) q^{65} + ( 50 - 50 \zeta_{24}^{2} - 4 \zeta_{24}^{3} - 50 \zeta_{24}^{4} - 4 \zeta_{24}^{7} ) q^{67} + ( -26 + 68 \zeta_{24} - 26 \zeta_{24}^{2} + 26 \zeta_{24}^{4} - 34 \zeta_{24}^{5} ) q^{68} + ( -10 \zeta_{24} - 5 \zeta_{24}^{2} + 15 \zeta_{24}^{4} + 20 \zeta_{24}^{5} + 5 \zeta_{24}^{6} ) q^{70} + 68 q^{71} + ( 19 - 48 \zeta_{24} - 48 \zeta_{24}^{5} + 19 \zeta_{24}^{6} ) q^{73} + ( -34 \zeta_{24} - 86 \zeta_{24}^{2} - 68 \zeta_{24}^{3} + 68 \zeta_{24}^{5} + 86 \zeta_{24}^{6} + 34 \zeta_{24}^{7} ) q^{74} + ( -78 - 36 \zeta_{24} + 18 \zeta_{24}^{3} + 78 \zeta_{24}^{4} + 18 \zeta_{24}^{5} + 18 \zeta_{24}^{7} ) q^{76} + ( 22 - 22 \zeta_{24}^{2} + 14 \zeta_{24}^{3} - 22 \zeta_{24}^{4} + 14 \zeta_{24}^{7} ) q^{77} + ( -10 \zeta_{24} - 20 \zeta_{24}^{3} + 20 \zeta_{24}^{5} + 10 \zeta_{24}^{7} ) q^{79} + ( -41 + 21 \zeta_{24} + 2 \zeta_{24}^{3} + 21 \zeta_{24}^{5} - 3 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{80} + ( 32 + 26 \zeta_{24}^{3} - 32 \zeta_{24}^{6} - 52 \zeta_{24}^{7} ) q^{82} + ( 14 \zeta_{24} + 4 \zeta_{24}^{2} - 4 \zeta_{24}^{4} - 28 \zeta_{24}^{5} - 4 \zeta_{24}^{6} ) q^{83} + ( -58 + 16 \zeta_{24} + 16 \zeta_{24}^{2} - 36 \zeta_{24}^{3} + 58 \zeta_{24}^{4} - 8 \zeta_{24}^{5} - 36 \zeta_{24}^{7} ) q^{85} + ( -56 - 36 \zeta_{24} + 18 \zeta_{24}^{3} + 56 \zeta_{24}^{4} + 18 \zeta_{24}^{5} + 18 \zeta_{24}^{7} ) q^{86} + ( -14 \zeta_{24} + 3 \zeta_{24}^{2} - 3 \zeta_{24}^{4} + 28 \zeta_{24}^{5} - 3 \zeta_{24}^{6} ) q^{88} + ( -36 \zeta_{24} + 36 \zeta_{24}^{3} - 36 \zeta_{24}^{5} - 6 \zeta_{24}^{6} - 72 \zeta_{24}^{7} ) q^{89} + ( -4 + 14 \zeta_{24} + 14 \zeta_{24}^{3} + 14 \zeta_{24}^{5} - 28 \zeta_{24}^{7} ) q^{91} + ( 26 \zeta_{24}^{2} + 116 \zeta_{24}^{3} + 26 \zeta_{24}^{4} - 26 \zeta_{24}^{6} - 58 \zeta_{24}^{7} ) q^{92} + ( -44 \zeta_{24} - 34 \zeta_{24}^{2} - 22 \zeta_{24}^{3} + 22 \zeta_{24}^{5} - 22 \zeta_{24}^{7} ) q^{94} + ( 36 + 48 \zeta_{24} + 48 \zeta_{24}^{2} - 18 \zeta_{24}^{3} - 36 \zeta_{24}^{4} - 24 \zeta_{24}^{5} - 18 \zeta_{24}^{7} ) q^{95} + ( 5 \zeta_{24}^{2} - 32 \zeta_{24}^{3} + 5 \zeta_{24}^{4} - 5 \zeta_{24}^{6} + 16 \zeta_{24}^{7} ) q^{97} + ( 47 + 43 \zeta_{24} + 43 \zeta_{24}^{5} + 47 \zeta_{24}^{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{2} - 4q^{5} - 4q^{7} - 24q^{8} + O(q^{10})$$ $$8q - 4q^{2} - 4q^{5} - 4q^{7} - 24q^{8} + 8q^{10} + 16q^{11} + 32q^{13} + 20q^{16} + 80q^{17} - 36q^{20} - 20q^{22} + 56q^{23} - 16q^{25} - 176q^{26} + 88q^{28} + 16q^{31} - 76q^{32} + 80q^{35} + 128q^{37} - 96q^{38} - 48q^{40} - 56q^{41} + 8q^{43} - 272q^{46} + 128q^{47} + 164q^{50} + 80q^{52} - 112q^{53} - 248q^{55} + 12q^{58} - 200q^{61} - 176q^{62} - 112q^{65} + 200q^{67} - 104q^{68} + 60q^{70} + 544q^{71} + 152q^{73} - 312q^{76} + 88q^{77} - 328q^{80} + 256q^{82} - 16q^{83} - 232q^{85} - 224q^{86} - 12q^{88} - 32q^{91} + 104q^{92} + 144q^{95} + 20q^{97} + 376q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$$.

 $$n$$ $$82$$ $$326$$ $$\chi(n)$$ $$-\zeta_{24}^{4}$$ $$-1 + \zeta_{24}^{2}$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
28.1
 0.965926 + 0.258819i −0.965926 − 0.258819i 0.965926 − 0.258819i −0.965926 + 0.258819i −0.258819 − 0.965926i 0.258819 + 0.965926i −0.258819 + 0.965926i 0.258819 − 0.965926i
−3.03906 0.814313i 0 5.10867 + 2.94949i −2.32162 4.42833i 0 1.98004 + 0.530550i −4.22474 4.22474i 0 3.44949 + 15.3485i
28.2 0.307007 + 0.0822623i 0 −3.37662 1.94949i −3.87453 + 3.16038i 0 −4.71209 1.26260i −1.77526 1.77526i 0 −1.44949 + 0.651531i
217.1 −3.03906 + 0.814313i 0 5.10867 2.94949i −2.32162 + 4.42833i 0 1.98004 0.530550i −4.22474 + 4.22474i 0 3.44949 15.3485i
217.2 0.307007 0.0822623i 0 −3.37662 + 1.94949i −3.87453 3.16038i 0 −4.71209 + 1.26260i −1.77526 + 1.77526i 0 −1.44949 0.651531i
298.1 −0.0822623 0.307007i 0 3.37662 1.94949i −0.799701 4.93563i 0 1.26260 + 4.71209i −1.77526 1.77526i 0 −1.44949 + 0.651531i
298.2 0.814313 + 3.03906i 0 −5.10867 + 2.94949i 4.99585 + 0.203583i 0 −0.530550 1.98004i −4.22474 4.22474i 0 3.44949 + 15.3485i
352.1 −0.0822623 + 0.307007i 0 3.37662 + 1.94949i −0.799701 + 4.93563i 0 1.26260 4.71209i −1.77526 + 1.77526i 0 −1.44949 0.651531i
352.2 0.814313 3.03906i 0 −5.10867 2.94949i 4.99585 0.203583i 0 −0.530550 + 1.98004i −4.22474 + 4.22474i 0 3.44949 15.3485i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 352.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
9.c even 3 1 inner
45.k odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.3.l.f 8
3.b odd 2 1 405.3.l.h 8
5.c odd 4 1 inner 405.3.l.f 8
9.c even 3 1 45.3.g.b 4
9.c even 3 1 inner 405.3.l.f 8
9.d odd 6 1 15.3.f.a 4
9.d odd 6 1 405.3.l.h 8
15.e even 4 1 405.3.l.h 8
36.f odd 6 1 720.3.bh.k 4
36.h even 6 1 240.3.bg.a 4
45.h odd 6 1 75.3.f.c 4
45.j even 6 1 225.3.g.a 4
45.k odd 12 1 45.3.g.b 4
45.k odd 12 1 225.3.g.a 4
45.k odd 12 1 inner 405.3.l.f 8
45.l even 12 1 15.3.f.a 4
45.l even 12 1 75.3.f.c 4
45.l even 12 1 405.3.l.h 8
72.j odd 6 1 960.3.bg.i 4
72.l even 6 1 960.3.bg.h 4
180.n even 6 1 1200.3.bg.k 4
180.v odd 12 1 240.3.bg.a 4
180.v odd 12 1 1200.3.bg.k 4
180.x even 12 1 720.3.bh.k 4
360.br even 12 1 960.3.bg.i 4
360.bt odd 12 1 960.3.bg.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.3.f.a 4 9.d odd 6 1
15.3.f.a 4 45.l even 12 1
45.3.g.b 4 9.c even 3 1
45.3.g.b 4 45.k odd 12 1
75.3.f.c 4 45.h odd 6 1
75.3.f.c 4 45.l even 12 1
225.3.g.a 4 45.j even 6 1
225.3.g.a 4 45.k odd 12 1
240.3.bg.a 4 36.h even 6 1
240.3.bg.a 4 180.v odd 12 1
405.3.l.f 8 1.a even 1 1 trivial
405.3.l.f 8 5.c odd 4 1 inner
405.3.l.f 8 9.c even 3 1 inner
405.3.l.f 8 45.k odd 12 1 inner
405.3.l.h 8 3.b odd 2 1
405.3.l.h 8 9.d odd 6 1
405.3.l.h 8 15.e even 4 1
405.3.l.h 8 45.l even 12 1
720.3.bh.k 4 36.f odd 6 1
720.3.bh.k 4 180.x even 12 1
960.3.bg.h 4 72.l even 6 1
960.3.bg.h 4 360.bt odd 12 1
960.3.bg.i 4 72.j odd 6 1
960.3.bg.i 4 360.br even 12 1
1200.3.bg.k 4 180.n even 6 1
1200.3.bg.k 4 180.v odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + \cdots$$ acting on $$S_{3}^{\mathrm{new}}(405, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 4 T + 8 T^{2} - 40 T^{3} + 79 T^{4} + 40 T^{5} + 8 T^{6} + 4 T^{7} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$390625 + 62500 T + 10000 T^{2} - 5000 T^{3} - 1025 T^{4} - 200 T^{5} + 16 T^{6} + 4 T^{7} + T^{8}$$
$7$ $$10000 - 4000 T + 800 T^{2} - 1120 T^{3} + 124 T^{4} + 112 T^{5} + 8 T^{6} + 4 T^{7} + T^{8}$$
$11$ $$( 1444 + 304 T + 102 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$13$ $$181063936 - 49948672 T + 6889472 T^{2} - 1039360 T^{3} + 129904 T^{4} - 8960 T^{5} + 512 T^{6} - 32 T^{7} + T^{8}$$
$17$ $$( 8464 - 3680 T + 800 T^{2} - 40 T^{3} + T^{4} )^{2}$$
$19$ $$( 32400 + 504 T^{2} + T^{4} )^{2}$$
$23$ $$20851360000 - 3072832000 T + 226419200 T^{2} - 17194240 T^{3} + 1122544 T^{4} - 45248 T^{5} + 1568 T^{6} - 56 T^{7} + T^{8}$$
$29$ $$810000 - 1112400 T^{2} + 1526796 T^{4} - 1236 T^{6} + T^{8}$$
$31$ $$( 40000 + 1600 T + 264 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$37$ $$( 211600 + 29440 T + 2048 T^{2} - 64 T^{3} + T^{4} )^{2}$$
$41$ $$( 400 - 560 T + 804 T^{2} + 28 T^{3} + T^{4} )^{2}$$
$43$ $$2018854506496 + 13549359104 T + 45467648 T^{2} + 23038976 T^{3} - 1343552 T^{4} - 19328 T^{5} + 32 T^{6} - 8 T^{7} + T^{8}$$
$47$ $$9336104694016 - 683650686976 T + 25030688768 T^{2} - 1050701824 T^{3} + 35414128 T^{4} - 601088 T^{5} + 8192 T^{6} - 128 T^{7} + T^{8}$$
$53$ $$( 1600 - 2240 T + 1568 T^{2} + 56 T^{3} + T^{4} )^{2}$$
$59$ $$399236364810000 - 282210231600 T^{2} + 179506476 T^{4} - 14124 T^{6} + T^{8}$$
$61$ $$( 309136 + 55600 T + 9444 T^{2} + 100 T^{3} + T^{4} )^{2}$$
$67$ $$601343393468416 - 24286889881600 T + 490446080000 T^{2} - 9999078400 T^{3} + 177397696 T^{4} - 2019200 T^{5} + 20000 T^{6} - 200 T^{7} + T^{8}$$
$71$ $$( -68 + T )^{8}$$
$73$ $$( 38316100 + 470440 T + 2888 T^{2} - 76 T^{3} + T^{4} )^{2}$$
$79$ $$( 360000 - 600 T^{2} + T^{4} )^{2}$$
$83$ $$95565066496 - 2750073856 T + 39569408 T^{2} - 11031040 T^{3} - 150416 T^{4} + 19840 T^{5} + 128 T^{6} + 16 T^{7} + T^{8}$$
$89$ $$( 59907600 + 15624 T^{2} + T^{4} )^{2}$$
$97$ $$265764994576 + 7402924640 T + 103104800 T^{2} + 23492960 T^{3} - 188324 T^{4} - 32720 T^{5} + 200 T^{6} - 20 T^{7} + T^{8}$$