# Properties

 Label 475.2.e.e Level $475$ Weight $2$ Character orbit 475.e Analytic conductor $3.793$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 475.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.79289409601$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.4601315889.1 Defining polynomial: $$x^{8} - x^{7} + 6 x^{6} - 3 x^{5} + 26 x^{4} - 14 x^{3} + 31 x^{2} + 12 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 6 x^{6} - 3 x^{5} + 26 x^{4} - 14 x^{3} + 31 x^{2} + 12 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-26 \nu^{7} - 189 \nu^{6} + 729 \nu^{5} - 911 \nu^{4} + 3051 \nu^{3} - 3618 \nu^{2} + 14317 \nu - 1215$$$$)/4243$$ $$\beta_{3}$$ $$=$$ $$($$$$115 \nu^{7} + 20 \nu^{6} + 529 \nu^{5} + 276 \nu^{4} + 3314 \nu^{3} + 989 \nu^{2} + 483 \nu + 1947$$$$)/4243$$ $$\beta_{4}$$ $$=$$ $$($$$$-135 \nu^{7} + 161 \nu^{6} - 621 \nu^{5} - 324 \nu^{4} - 2599 \nu^{3} - 1161 \nu^{2} - 567 \nu - 11694$$$$)/4243$$ $$\beta_{5}$$ $$=$$ $$($$$$-649 \nu^{7} + 994 \nu^{6} - 3834 \nu^{5} + 3534 \nu^{4} - 16046 \nu^{3} + 19028 \nu^{2} - 17152 \nu + 6390$$$$)/12729$$ $$\beta_{6}$$ $$=$$ $$($$$$434 \nu^{7} - 109 \nu^{6} + 2845 \nu^{5} + 193 \nu^{4} + 12064 \nu^{3} + 338 \nu^{2} + 16249 \nu + 6573$$$$)/4243$$ $$\beta_{7}$$ $$=$$ $$($$$$514 \nu^{7} - 833 \nu^{6} + 3213 \nu^{5} - 3858 \nu^{4} + 13447 \nu^{3} - 15946 \nu^{2} + 16585 \nu - 5355$$$$)/4243$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} + 3 \beta_{5} - \beta_{4} - 3$$ $$\nu^{3}$$ $$=$$ $$-\beta_{6} + 4 \beta_{3} + \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$-5 \beta_{7} - 12 \beta_{5} + \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$-\beta_{7} + 6 \beta_{6} + \beta_{4} - 17 \beta_{3} - 17 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$7 \beta_{6} + 23 \beta_{4} - \beta_{3} - 7 \beta_{2} + 51$$ $$\nu^{7}$$ $$=$$ $$8 \beta_{7} + 3 \beta_{5} - 30 \beta_{2} + 74 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 1.07988 + 1.87040i −1.02359 − 1.77290i 0.689667 + 1.19454i −0.245959 − 0.426014i 1.07988 − 1.87040i −1.02359 + 1.77290i 0.689667 − 1.19454i −0.245959 + 0.426014i
−0.832272 1.44154i −0.579878 1.00438i −0.385355 + 0.667454i 0 −0.965233 + 1.67183i 2.43525 −2.04621 0.827483 1.43324i 0
26.2 −0.595455 1.03136i 1.52359 + 2.63893i 0.290867 0.503797i 0 1.81445 3.14272i 0.609175 −3.07461 −3.14263 + 5.44319i 0
26.3 0.548719 + 0.950409i −0.189667 0.328513i 0.397815 0.689035i 0 0.208148 0.360522i −1.89307 3.06803 1.42805 2.47346i 0
26.4 1.37901 + 2.38851i 0.745959 + 1.29204i −2.80333 + 4.85550i 0 −2.05737 + 3.56347i 2.84864 −9.94721 0.387090 0.670459i 0
201.1 −0.832272 + 1.44154i −0.579878 + 1.00438i −0.385355 0.667454i 0 −0.965233 1.67183i 2.43525 −2.04621 0.827483 + 1.43324i 0
201.2 −0.595455 + 1.03136i 1.52359 2.63893i 0.290867 + 0.503797i 0 1.81445 + 3.14272i 0.609175 −3.07461 −3.14263 5.44319i 0
201.3 0.548719 0.950409i −0.189667 + 0.328513i 0.397815 + 0.689035i 0 0.208148 + 0.360522i −1.89307 3.06803 1.42805 + 2.47346i 0
201.4 1.37901 2.38851i 0.745959 1.29204i −2.80333 4.85550i 0 −2.05737 3.56347i 2.84864 −9.94721 0.387090 + 0.670459i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 201.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.e.e 8
5.b even 2 1 95.2.e.c 8
5.c odd 4 2 475.2.j.c 16
15.d odd 2 1 855.2.k.h 8
19.c even 3 1 inner 475.2.e.e 8
19.c even 3 1 9025.2.a.bg 4
19.d odd 6 1 9025.2.a.bp 4
20.d odd 2 1 1520.2.q.o 8
95.h odd 6 1 1805.2.a.i 4
95.i even 6 1 95.2.e.c 8
95.i even 6 1 1805.2.a.o 4
95.m odd 12 2 475.2.j.c 16
285.n odd 6 1 855.2.k.h 8
380.p odd 6 1 1520.2.q.o 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.e.c 8 5.b even 2 1
95.2.e.c 8 95.i even 6 1
475.2.e.e 8 1.a even 1 1 trivial
475.2.e.e 8 19.c even 3 1 inner
475.2.j.c 16 5.c odd 4 2
475.2.j.c 16 95.m odd 12 2
855.2.k.h 8 15.d odd 2 1
855.2.k.h 8 285.n odd 6 1
1520.2.q.o 8 20.d odd 2 1
1520.2.q.o 8 380.p odd 6 1
1805.2.a.i 4 95.h odd 6 1
1805.2.a.o 4 95.i even 6 1
9025.2.a.bg 4 19.c even 3 1
9025.2.a.bp 4 19.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(475, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$36 + 6 T + 37 T^{2} + 6 T^{3} + 31 T^{4} + 4 T^{5} + 7 T^{6} - T^{7} + T^{8}$$
$3$ $$4 + 10 T + 29 T^{2} + 2 T^{3} + 17 T^{4} - 4 T^{5} + 11 T^{6} - 3 T^{7} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$( -8 + 15 T - T^{2} - 4 T^{3} + T^{4} )^{2}$$
$11$ $$( 3 - 19 T - 25 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$13$ $$256 - 368 T + 641 T^{2} - 63 T^{3} + 226 T^{4} - 95 T^{5} + 42 T^{6} - 7 T^{7} + T^{8}$$
$17$ $$11664 + 1836 T + 3529 T^{2} - 726 T^{3} + 775 T^{4} - 64 T^{5} + 31 T^{6} + T^{7} + T^{8}$$
$19$ $$130321 - 34295 T + 11191 T^{2} - 1273 T^{3} + 395 T^{4} - 67 T^{5} + 31 T^{6} - 5 T^{7} + T^{8}$$
$23$ $$36 - 42 T + 151 T^{2} + 143 T^{3} + 269 T^{4} + 48 T^{5} + 21 T^{6} - 2 T^{7} + T^{8}$$
$29$ $$19881 + 27354 T + 28753 T^{2} + 11940 T^{3} + 3916 T^{4} + 451 T^{5} + 64 T^{6} - T^{7} + T^{8}$$
$31$ $$( 1063 - 5 T - 67 T^{2} + T^{4} )^{2}$$
$37$ $$( -118 + 123 T - 31 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$41$ $$5008644 - 2173098 T + 748135 T^{2} - 120285 T^{3} + 17575 T^{4} - 1246 T^{5} + 151 T^{6} - 8 T^{7} + T^{8}$$
$43$ $$630436 + 156418 T + 116621 T^{2} - 17718 T^{3} + 9007 T^{4} - 296 T^{5} + 99 T^{6} - T^{7} + T^{8}$$
$47$ $$5363856 + 2151564 T + 735661 T^{2} + 106679 T^{3} + 16489 T^{4} + 1198 T^{5} + 199 T^{6} + 12 T^{7} + T^{8}$$
$53$ $$2916 + 12258 T + 46885 T^{2} + 20062 T^{3} + 8585 T^{4} + 24 T^{5} + 111 T^{6} + 5 T^{7} + T^{8}$$
$59$ $$3515625 + 937500 T + 484375 T^{2} - 43750 T^{3} + 16250 T^{4} - 375 T^{5} + 150 T^{6} - 5 T^{7} + T^{8}$$
$61$ $$9296401 - 268312 T + 404114 T^{2} + 11440 T^{3} + 13851 T^{4} + 176 T^{5} + 130 T^{6} + T^{8}$$
$67$ $$4096 - 1536 T + 2880 T^{2} + 1376 T^{3} + 1136 T^{4} + 192 T^{5} + 52 T^{6} - 4 T^{7} + T^{8}$$
$71$ $$59049 + 23085 T + 31138 T^{2} + 1075 T^{3} + 10424 T^{4} + 2010 T^{5} + 309 T^{6} + 20 T^{7} + T^{8}$$
$73$ $$2979076 + 377994 T + 211931 T^{2} + 48235 T^{3} + 15131 T^{4} + 2338 T^{5} + 305 T^{6} + 20 T^{7} + T^{8}$$
$79$ $$33856 + 3680 T + 13648 T^{2} + 4816 T^{3} + 5708 T^{4} + 1264 T^{5} + 217 T^{6} + 17 T^{7} + T^{8}$$
$83$ $$( 366 - 55 T - 62 T^{2} + T^{3} + T^{4} )^{2}$$
$89$ $$14561856 + 5296608 T + 1583104 T^{2} + 208872 T^{3} + 27184 T^{4} + 1786 T^{5} + 211 T^{6} + 11 T^{7} + T^{8}$$
$97$ $$55383364 - 5990810 T + 2627597 T^{2} + 229014 T^{3} + 62509 T^{4} + 1876 T^{5} + 267 T^{6} - T^{7} + T^{8}$$