# Properties

 Label 450.2.l.a Level $450$ Weight $2$ Character orbit 450.l Analytic conductor $3.593$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 450.l (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.59326809096$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\Q(\zeta_{20})$$ Defining polynomial: $$x^{8} - x^{6} + x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 150) Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{20}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{20} q^{2} + \zeta_{20}^{2} q^{4} + ( \zeta_{20} + \zeta_{20}^{2} + \zeta_{20}^{4} - \zeta_{20}^{5} ) q^{5} + ( -1 + 2 \zeta_{20}^{2} + \zeta_{20}^{5} + 2 \zeta_{20}^{6} ) q^{7} + \zeta_{20}^{3} q^{8} +O(q^{10})$$ $$q + \zeta_{20} q^{2} + \zeta_{20}^{2} q^{4} + ( \zeta_{20} + \zeta_{20}^{2} + \zeta_{20}^{4} - \zeta_{20}^{5} ) q^{5} + ( -1 + 2 \zeta_{20}^{2} + \zeta_{20}^{5} + 2 \zeta_{20}^{6} ) q^{7} + \zeta_{20}^{3} q^{8} + ( \zeta_{20}^{2} + \zeta_{20}^{3} + \zeta_{20}^{5} - \zeta_{20}^{6} ) q^{10} + ( -1 + \zeta_{20}^{2} - \zeta_{20}^{5} - 2 \zeta_{20}^{6} - \zeta_{20}^{7} ) q^{11} + ( -4 - \zeta_{20} + 2 \zeta_{20}^{2} - 2 \zeta_{20}^{4} + 2 \zeta_{20}^{6} + \zeta_{20}^{7} ) q^{13} + ( -\zeta_{20} + 2 \zeta_{20}^{3} + \zeta_{20}^{6} + 2 \zeta_{20}^{7} ) q^{14} + \zeta_{20}^{4} q^{16} + ( -1 + \zeta_{20} - \zeta_{20}^{2} - 2 \zeta_{20}^{3} - \zeta_{20}^{4} + \zeta_{20}^{5} - \zeta_{20}^{6} ) q^{17} + ( -2 + 3 \zeta_{20} + \zeta_{20}^{2} - \zeta_{20}^{3} - \zeta_{20}^{4} - \zeta_{20}^{5} + 2 \zeta_{20}^{6} - 2 \zeta_{20}^{7} ) q^{19} + ( \zeta_{20}^{3} + \zeta_{20}^{4} + \zeta_{20}^{6} - \zeta_{20}^{7} ) q^{20} + ( 1 - \zeta_{20} - \zeta_{20}^{2} + \zeta_{20}^{3} + \zeta_{20}^{4} - 2 \zeta_{20}^{6} - 2 \zeta_{20}^{7} ) q^{22} + ( -2 + \zeta_{20} + 4 \zeta_{20}^{2} - 6 \zeta_{20}^{4} + 2 \zeta_{20}^{5} + 3 \zeta_{20}^{6} - 2 \zeta_{20}^{7} ) q^{23} + ( -2 + 2 \zeta_{20} + 2 \zeta_{20}^{2} + 4 \zeta_{20}^{5} + \zeta_{20}^{6} - 4 \zeta_{20}^{7} ) q^{25} + ( -1 - 4 \zeta_{20} + 2 \zeta_{20}^{3} - \zeta_{20}^{4} - 2 \zeta_{20}^{5} + \zeta_{20}^{6} + 2 \zeta_{20}^{7} ) q^{26} + ( -2 + \zeta_{20}^{2} + 2 \zeta_{20}^{6} + \zeta_{20}^{7} ) q^{28} + ( 3 + 2 \zeta_{20}^{2} - 6 \zeta_{20}^{3} + 3 \zeta_{20}^{4} + 6 \zeta_{20}^{5} - 3 \zeta_{20}^{7} ) q^{29} + ( 6 + \zeta_{20} - 3 \zeta_{20}^{2} + \zeta_{20}^{3} + 3 \zeta_{20}^{4} - 3 \zeta_{20}^{5} - 6 \zeta_{20}^{6} + 2 \zeta_{20}^{7} ) q^{31} + \zeta_{20}^{5} q^{32} + ( -\zeta_{20} + \zeta_{20}^{2} - \zeta_{20}^{3} - 2 \zeta_{20}^{4} - \zeta_{20}^{5} + \zeta_{20}^{6} - \zeta_{20}^{7} ) q^{34} + ( -3 + \zeta_{20}^{2} + 3 \zeta_{20}^{3} - \zeta_{20}^{4} + 5 \zeta_{20}^{6} + 2 \zeta_{20}^{7} ) q^{35} + ( -4 + 2 \zeta_{20} + 4 \zeta_{20}^{2} + \zeta_{20}^{3} - 2 \zeta_{20}^{4} - \zeta_{20}^{5} - 2 \zeta_{20}^{7} ) q^{37} + ( 2 - 2 \zeta_{20} + \zeta_{20}^{2} + \zeta_{20}^{3} + \zeta_{20}^{4} - \zeta_{20}^{5} - 3 \zeta_{20}^{6} + 2 \zeta_{20}^{7} ) q^{38} + ( 1 - \zeta_{20}^{2} + 2 \zeta_{20}^{4} + \zeta_{20}^{5} - \zeta_{20}^{6} + \zeta_{20}^{7} ) q^{40} + ( -\zeta_{20} - 3 \zeta_{20}^{2} - \zeta_{20}^{3} + 5 \zeta_{20}^{4} - \zeta_{20}^{5} - 3 \zeta_{20}^{6} - \zeta_{20}^{7} ) q^{41} + ( -2 + 4 \zeta_{20}^{2} - 5 \zeta_{20}^{3} - 3 \zeta_{20}^{4} - \zeta_{20}^{5} + \zeta_{20}^{6} - 5 \zeta_{20}^{7} ) q^{43} + ( 2 + \zeta_{20} - 3 \zeta_{20}^{2} - \zeta_{20}^{3} + 3 \zeta_{20}^{4} + \zeta_{20}^{5} - 2 \zeta_{20}^{6} - 2 \zeta_{20}^{7} ) q^{44} + ( 2 - 2 \zeta_{20} - \zeta_{20}^{2} + 4 \zeta_{20}^{3} + 2 \zeta_{20}^{4} - 6 \zeta_{20}^{5} + 3 \zeta_{20}^{7} ) q^{46} + ( -1 + 5 \zeta_{20} - 5 \zeta_{20}^{3} + \zeta_{20}^{4} - 5 \zeta_{20}^{7} ) q^{47} + ( -1 - 4 \zeta_{20} - 4 \zeta_{20}^{4} - 2 \zeta_{20}^{5} + 4 \zeta_{20}^{6} + 4 \zeta_{20}^{7} ) q^{49} + ( 4 - 2 \zeta_{20} - 2 \zeta_{20}^{2} + 2 \zeta_{20}^{3} + 4 \zeta_{20}^{4} + \zeta_{20}^{7} ) q^{50} + ( -2 - \zeta_{20} - 2 \zeta_{20}^{2} - \zeta_{20}^{5} + \zeta_{20}^{7} ) q^{52} + ( -\zeta_{20} + 3 \zeta_{20}^{2} + \zeta_{20}^{3} - 6 \zeta_{20}^{4} + 6 \zeta_{20}^{6} + 4 \zeta_{20}^{7} ) q^{53} + ( 4 - 5 \zeta_{20}^{2} - \zeta_{20}^{3} + 3 \zeta_{20}^{4} + 3 \zeta_{20}^{5} - 3 \zeta_{20}^{6} - 6 \zeta_{20}^{7} ) q^{55} + ( -1 - 2 \zeta_{20} + \zeta_{20}^{2} + \zeta_{20}^{3} - \zeta_{20}^{4} + \zeta_{20}^{6} + 2 \zeta_{20}^{7} ) q^{56} + ( 3 + 3 \zeta_{20} - 3 \zeta_{20}^{2} + 2 \zeta_{20}^{3} - 3 \zeta_{20}^{4} + 3 \zeta_{20}^{5} + 3 \zeta_{20}^{6} ) q^{58} + ( 2 \zeta_{20}^{2} - 6 \zeta_{20}^{3} - 6 \zeta_{20}^{4} - 6 \zeta_{20}^{5} + 2 \zeta_{20}^{6} ) q^{59} + ( -\zeta_{20} + 2 \zeta_{20}^{3} + 5 \zeta_{20}^{5} + 7 \zeta_{20}^{7} ) q^{61} + ( -2 + 6 \zeta_{20} + 3 \zeta_{20}^{2} - 3 \zeta_{20}^{3} - \zeta_{20}^{4} + 3 \zeta_{20}^{5} - \zeta_{20}^{6} - 6 \zeta_{20}^{7} ) q^{62} + \zeta_{20}^{6} q^{64} + ( -3 - 6 \zeta_{20} - 3 \zeta_{20}^{2} + 4 \zeta_{20}^{3} - 3 \zeta_{20}^{4} - 2 \zeta_{20}^{5} + 2 \zeta_{20}^{6} + 3 \zeta_{20}^{7} ) q^{65} + ( 1 + 5 \zeta_{20} + 4 \zeta_{20}^{2} - 6 \zeta_{20}^{3} + 4 \zeta_{20}^{4} + 5 \zeta_{20}^{5} + \zeta_{20}^{6} ) q^{67} + ( 1 - 2 \zeta_{20}^{2} + \zeta_{20}^{3} - 2 \zeta_{20}^{5} - 2 \zeta_{20}^{6} + \zeta_{20}^{7} ) q^{68} + ( -2 - 3 \zeta_{20} + 2 \zeta_{20}^{2} + \zeta_{20}^{3} + \zeta_{20}^{4} - \zeta_{20}^{5} + 2 \zeta_{20}^{6} + 5 \zeta_{20}^{7} ) q^{70} + ( -1 - \zeta_{20} - 7 \zeta_{20}^{2} + 3 \zeta_{20}^{3} - \zeta_{20}^{4} - 4 \zeta_{20}^{5} + 2 \zeta_{20}^{7} ) q^{71} + ( -1 - 4 \zeta_{20} - 3 \zeta_{20}^{2} + 2 \zeta_{20}^{4} - 4 \zeta_{20}^{5} - \zeta_{20}^{6} + 4 \zeta_{20}^{7} ) q^{73} + ( 2 - 4 \zeta_{20} + 4 \zeta_{20}^{3} + 3 \zeta_{20}^{4} - 2 \zeta_{20}^{5} - 3 \zeta_{20}^{6} ) q^{74} + ( -2 + 2 \zeta_{20} + \zeta_{20}^{3} - \zeta_{20}^{4} + \zeta_{20}^{5} + \zeta_{20}^{6} - 3 \zeta_{20}^{7} ) q^{76} + ( 4 + 6 \zeta_{20} + 4 \zeta_{20}^{4} + 2 \zeta_{20}^{5} - 2 \zeta_{20}^{6} - 2 \zeta_{20}^{7} ) q^{77} + ( 6 - 2 \zeta_{20} - 10 \zeta_{20}^{2} - 2 \zeta_{20}^{3} + 6 \zeta_{20}^{4} ) q^{79} + ( -1 + \zeta_{20} + \zeta_{20}^{2} - \zeta_{20}^{3} - \zeta_{20}^{4} + 2 \zeta_{20}^{5} + 2 \zeta_{20}^{6} - \zeta_{20}^{7} ) q^{80} + ( 1 - 2 \zeta_{20}^{2} - 3 \zeta_{20}^{3} + 5 \zeta_{20}^{5} - 2 \zeta_{20}^{6} - 3 \zeta_{20}^{7} ) q^{82} + ( -7 - \zeta_{20}^{2} + 3 \zeta_{20}^{3} - \zeta_{20}^{4} - 7 \zeta_{20}^{6} ) q^{83} + ( 2 - 4 \zeta_{20} + 2 \zeta_{20}^{3} - 4 \zeta_{20}^{4} - 3 \zeta_{20}^{5} - 2 \zeta_{20}^{6} + \zeta_{20}^{7} ) q^{85} + ( 5 - 2 \zeta_{20} - 5 \zeta_{20}^{2} + 4 \zeta_{20}^{3} - 3 \zeta_{20}^{5} - 6 \zeta_{20}^{6} + \zeta_{20}^{7} ) q^{86} + ( 2 + 2 \zeta_{20} - \zeta_{20}^{2} - 3 \zeta_{20}^{3} + \zeta_{20}^{4} + 3 \zeta_{20}^{5} - \zeta_{20}^{6} - 2 \zeta_{20}^{7} ) q^{88} + ( 4 + 3 \zeta_{20} - 4 \zeta_{20}^{2} - 6 \zeta_{20}^{3} + \zeta_{20}^{5} + 5 \zeta_{20}^{6} - 5 \zeta_{20}^{7} ) q^{89} + ( -\zeta_{20} - 7 \zeta_{20}^{2} - 4 \zeta_{20}^{3} - 2 \zeta_{20}^{4} - 4 \zeta_{20}^{5} - 7 \zeta_{20}^{6} - \zeta_{20}^{7} ) q^{91} + ( -3 + 2 \zeta_{20} + \zeta_{20}^{2} - \zeta_{20}^{3} + \zeta_{20}^{4} + 2 \zeta_{20}^{5} - 3 \zeta_{20}^{6} ) q^{92} + ( 5 - \zeta_{20} + \zeta_{20}^{5} - 5 \zeta_{20}^{6} ) q^{94} + ( -3 + 4 \zeta_{20} - \zeta_{20}^{2} + 2 \zeta_{20}^{3} - 2 \zeta_{20}^{4} + 5 \zeta_{20}^{5} - 4 \zeta_{20}^{6} - 3 \zeta_{20}^{7} ) q^{95} + ( 9 - 3 \zeta_{20}^{2} - 3 \zeta_{20}^{4} - 6 \zeta_{20}^{6} - 2 \zeta_{20}^{7} ) q^{97} + ( -4 - \zeta_{20} - 4 \zeta_{20}^{4} - 4 \zeta_{20}^{5} + 2 \zeta_{20}^{6} + 4 \zeta_{20}^{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 2q^{4} + O(q^{10})$$ $$8q + 2q^{4} - 10q^{11} - 20q^{13} + 2q^{14} - 2q^{16} - 10q^{17} - 8q^{19} + 10q^{23} - 10q^{25} - 4q^{26} - 10q^{28} + 22q^{29} + 24q^{31} + 8q^{34} - 10q^{35} - 20q^{37} + 10q^{38} - 22q^{41} + 10q^{46} - 10q^{47} + 8q^{49} + 20q^{50} - 20q^{52} + 30q^{53} + 10q^{55} - 2q^{56} + 30q^{58} + 20q^{59} - 10q^{62} + 2q^{64} - 20q^{65} + 10q^{67} - 10q^{70} - 20q^{71} - 20q^{73} + 4q^{74} - 12q^{76} + 20q^{77} + 16q^{79} - 70q^{83} + 20q^{85} + 18q^{86} + 10q^{88} + 34q^{89} - 24q^{91} - 30q^{92} + 30q^{94} - 30q^{95} + 60q^{97} - 20q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$\zeta_{20}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 −0.951057 + 0.309017i 0.951057 − 0.309017i −0.587785 + 0.809017i 0.587785 − 0.809017i −0.587785 − 0.809017i 0.587785 + 0.809017i −0.951057 − 0.309017i 0.951057 + 0.309017i
−0.951057 + 0.309017i 0 0.809017 0.587785i 0.166977 2.22982i 0 2.07768i −0.587785 + 0.809017i 0 0.530249 + 2.17229i
19.2 0.951057 0.309017i 0 0.809017 0.587785i 2.06909 0.847859i 0 4.07768i 0.587785 0.809017i 0 1.70582 1.44575i
109.1 −0.587785 + 0.809017i 0 −0.309017 0.951057i −1.70582 + 1.44575i 0 1.72654i 0.951057 + 0.309017i 0 −0.166977 2.22982i
109.2 0.587785 0.809017i 0 −0.309017 0.951057i −0.530249 2.17229i 0 0.273457i −0.951057 0.309017i 0 −2.06909 0.847859i
289.1 −0.587785 0.809017i 0 −0.309017 + 0.951057i −1.70582 1.44575i 0 1.72654i 0.951057 0.309017i 0 −0.166977 + 2.22982i
289.2 0.587785 + 0.809017i 0 −0.309017 + 0.951057i −0.530249 + 2.17229i 0 0.273457i −0.951057 + 0.309017i 0 −2.06909 + 0.847859i
379.1 −0.951057 0.309017i 0 0.809017 + 0.587785i 0.166977 + 2.22982i 0 2.07768i −0.587785 0.809017i 0 0.530249 2.17229i
379.2 0.951057 + 0.309017i 0 0.809017 + 0.587785i 2.06909 + 0.847859i 0 4.07768i 0.587785 + 0.809017i 0 1.70582 + 1.44575i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 379.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.2.l.a 8
3.b odd 2 1 150.2.h.a 8
15.d odd 2 1 750.2.h.c 8
15.e even 4 1 750.2.g.c 8
15.e even 4 1 750.2.g.e 8
25.e even 10 1 inner 450.2.l.a 8
75.h odd 10 1 150.2.h.a 8
75.h odd 10 1 3750.2.c.e 8
75.j odd 10 1 750.2.h.c 8
75.j odd 10 1 3750.2.c.e 8
75.l even 20 1 750.2.g.c 8
75.l even 20 1 750.2.g.e 8
75.l even 20 1 3750.2.a.m 4
75.l even 20 1 3750.2.a.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.2.h.a 8 3.b odd 2 1
150.2.h.a 8 75.h odd 10 1
450.2.l.a 8 1.a even 1 1 trivial
450.2.l.a 8 25.e even 10 1 inner
750.2.g.c 8 15.e even 4 1
750.2.g.c 8 75.l even 20 1
750.2.g.e 8 15.e even 4 1
750.2.g.e 8 75.l even 20 1
750.2.h.c 8 15.d odd 2 1
750.2.h.c 8 75.j odd 10 1
3750.2.a.m 4 75.l even 20 1
3750.2.a.o 4 75.l even 20 1
3750.2.c.e 8 75.h odd 10 1
3750.2.c.e 8 75.j odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} + 24 T_{7}^{6} + 136 T_{7}^{4} + 224 T_{7}^{2} + 16$$ acting on $$S_{2}^{\mathrm{new}}(450, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$625 + 125 T^{2} - 50 T^{3} + 5 T^{4} - 10 T^{5} + 5 T^{6} + T^{8}$$
$7$ $$16 + 224 T^{2} + 136 T^{4} + 24 T^{6} + T^{8}$$
$11$ $$400 + 800 T + 2000 T^{2} + 1400 T^{3} + 560 T^{4} + 160 T^{5} + 50 T^{6} + 10 T^{7} + T^{8}$$
$13$ $$3721 + 8540 T + 9581 T^{2} + 6600 T^{3} + 2986 T^{4} + 900 T^{5} + 176 T^{6} + 20 T^{7} + T^{8}$$
$17$ $$3481 + 1180 T - 601 T^{2} - 580 T^{3} - 74 T^{4} + 80 T^{5} + 44 T^{6} + 10 T^{7} + T^{8}$$
$19$ $$5776 - 3952 T + 2088 T^{2} - 704 T^{3} + 480 T^{4} - 4 T^{5} + 28 T^{6} + 8 T^{7} + T^{8}$$
$23$ $$10000 - 30000 T + 23000 T^{2} + 4000 T^{3} - 1100 T^{4} + 200 T^{5} + 70 T^{6} - 10 T^{7} + T^{8}$$
$29$ $$12952801 - 6830902 T + 1815003 T^{2} - 293324 T^{3} + 38750 T^{4} - 3974 T^{5} + 348 T^{6} - 22 T^{7} + T^{8}$$
$31$ $$2085136 - 987696 T + 372552 T^{2} - 85728 T^{3} + 15280 T^{4} - 2292 T^{5} + 292 T^{6} - 24 T^{7} + T^{8}$$
$37$ $$143641 + 7580 T - 19464 T^{2} - 7380 T^{3} + 786 T^{4} + 920 T^{5} + 201 T^{6} + 20 T^{7} + T^{8}$$
$41$ $$192721 + 95702 T + 54283 T^{2} + 17804 T^{3} + 5430 T^{4} + 1334 T^{5} + 228 T^{6} + 22 T^{7} + T^{8}$$
$43$ $$2835856 + 465504 T^{2} + 16876 T^{4} + 224 T^{6} + T^{8}$$
$47$ $$144400 - 182400 T + 29200 T^{2} + 27000 T^{3} + 2960 T^{4} + 120 T^{5} + 70 T^{6} + 10 T^{7} + T^{8}$$
$53$ $$1739761 - 1305810 T + 551684 T^{2} - 98850 T^{3} + 9006 T^{4} - 1500 T^{5} + 319 T^{6} - 30 T^{7} + T^{8}$$
$59$ $$102400 - 921600 T + 3200000 T^{2} - 454400 T^{3} + 72960 T^{4} - 5120 T^{5} + 360 T^{6} - 20 T^{7} + T^{8}$$
$61$ $$17682025 - 231275 T^{2} + 21610 T^{4} + 240 T^{6} + T^{8}$$
$67$ $$29637136 - 12630080 T + 2197424 T^{2} - 134360 T^{3} - 12304 T^{4} + 2360 T^{5} - 46 T^{6} - 10 T^{7} + T^{8}$$
$71$ $$1392400 - 165200 T + 579800 T^{2} + 204000 T^{3} + 32960 T^{4} + 3060 T^{5} + 260 T^{6} + 20 T^{7} + T^{8}$$
$73$ $$175561 + 96370 T + 33421 T^{2} - 12520 T^{3} - 2814 T^{4} + 130 T^{5} + 136 T^{6} + 20 T^{7} + T^{8}$$
$79$ $$6885376 - 1091584 T - 147968 T^{2} + 48128 T^{3} + 26880 T^{4} - 2528 T^{5} + 432 T^{6} - 16 T^{7} + T^{8}$$
$83$ $$167857936 + 98724720 T + 28134696 T^{2} + 4928480 T^{3} + 569956 T^{4} + 44280 T^{5} + 2266 T^{6} + 70 T^{7} + T^{8}$$
$89$ $$32478601 - 12161666 T + 3488312 T^{2} - 583418 T^{3} + 69650 T^{4} - 6532 T^{5} + 587 T^{6} - 34 T^{7} + T^{8}$$
$97$ $$85396081 - 60158910 T + 18838061 T^{2} - 3457440 T^{3} + 413566 T^{4} - 33390 T^{5} + 1796 T^{6} - 60 T^{7} + T^{8}$$