# Properties

 Label 11.3.d.a Level 11 Weight 3 Character orbit 11.d Analytic conductor 0.300 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$11$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 11.d (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.299728290796$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 \zeta_{10} + 2 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{2} + ( -2 + 3 \zeta_{10} - 3 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{3} + ( -4 + 3 \zeta_{10} - 4 \zeta_{10}^{2} ) q^{4} + 4 \zeta_{10}^{2} q^{5} + ( 6 - 7 \zeta_{10} + 3 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{6} + ( 6 - 4 \zeta_{10} + 2 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{7} + ( 3 + \zeta_{10} + \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{8} + ( -5 + 5 \zeta_{10} + 4 \zeta_{10}^{3} ) q^{9} +O(q^{10})$$ $$q + ( -2 \zeta_{10} + 2 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{2} + ( -2 + 3 \zeta_{10} - 3 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{3} + ( -4 + 3 \zeta_{10} - 4 \zeta_{10}^{2} ) q^{4} + 4 \zeta_{10}^{2} q^{5} + ( 6 - 7 \zeta_{10} + 3 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{6} + ( 6 - 4 \zeta_{10} + 2 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{7} + ( 3 + \zeta_{10} + \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{8} + ( -5 + 5 \zeta_{10} + 4 \zeta_{10}^{3} ) q^{9} + ( -4 + 8 \zeta_{10} - 8 \zeta_{10}^{2} ) q^{10} + ( -3 - 3 \zeta_{10} - 9 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{11} + ( -2 + 11 \zeta_{10}^{2} - 11 \zeta_{10}^{3} ) q^{12} + ( -2 - 6 \zeta_{10} + 4 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{13} + ( -6 + 4 \zeta_{10} - 4 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{14} + ( 4 - 12 \zeta_{10} + 4 \zeta_{10}^{2} ) q^{15} + ( 8 \zeta_{10} - 3 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{16} + ( 13 \zeta_{10} - 13 \zeta_{10}^{3} ) q^{17} + ( 5 + \zeta_{10} - 7 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{18} + ( 5 - 2 \zeta_{10} - 2 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{19} + ( 16 - 16 \zeta_{10} - 4 \zeta_{10}^{3} ) q^{20} + ( -2 + 4 \zeta_{10} + 4 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{21} + ( 6 - 16 \zeta_{10} + 29 \zeta_{10}^{2} - 14 \zeta_{10}^{3} ) q^{22} + ( -4 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{23} + ( -7 + 9 \zeta_{10} - 16 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{24} + ( 9 - 9 \zeta_{10} + 9 \zeta_{10}^{2} - 9 \zeta_{10}^{3} ) q^{25} + ( -10 + 20 \zeta_{10} - 10 \zeta_{10}^{2} ) q^{26} + ( -2 \zeta_{10} - 19 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{27} + ( -24 + 2 \zeta_{10} - 12 \zeta_{10}^{2} + 22 \zeta_{10}^{3} ) q^{28} + ( -18 + 10 \zeta_{10} - 2 \zeta_{10}^{2} + 20 \zeta_{10}^{3} ) q^{29} + ( -16 + 12 \zeta_{10} + 12 \zeta_{10}^{2} - 16 \zeta_{10}^{3} ) q^{30} + ( -18 + 18 \zeta_{10} - 4 \zeta_{10}^{3} ) q^{31} + ( 23 - 46 \zeta_{10} + 22 \zeta_{10}^{2} - 24 \zeta_{10}^{3} ) q^{32} + ( 3 + 25 \zeta_{10} - 24 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{33} + ( 13 - 39 \zeta_{10}^{2} + 39 \zeta_{10}^{3} ) q^{34} + ( 24 + 8 \zeta_{10} + 16 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{35} + ( 24 - 23 \zeta_{10} + 23 \zeta_{10}^{2} - 24 \zeta_{10}^{3} ) q^{36} + ( 36 - 18 \zeta_{10} + 36 \zeta_{10}^{2} ) q^{37} + ( -17 \zeta_{10} + 26 \zeta_{10}^{2} - 17 \zeta_{10}^{3} ) q^{38} + ( 20 - 20 \zeta_{10} + 10 \zeta_{10}^{2} ) q^{39} + ( -16 + 4 \zeta_{10} + 8 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{40} + ( -16 - 27 \zeta_{10} - 27 \zeta_{10}^{2} - 16 \zeta_{10}^{3} ) q^{41} -10 \zeta_{10}^{3} q^{42} + ( -18 + 36 \zeta_{10} - 17 \zeta_{10}^{2} + 19 \zeta_{10}^{3} ) q^{43} + ( -17 + 60 \zeta_{10} - 18 \zeta_{10}^{2} + 14 \zeta_{10}^{3} ) q^{44} + ( -16 - 20 \zeta_{10}^{2} + 20 \zeta_{10}^{3} ) q^{45} + ( -2 + 14 \zeta_{10} - 16 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{46} + ( -26 + 24 \zeta_{10} - 24 \zeta_{10}^{2} + 26 \zeta_{10}^{3} ) q^{47} + ( -19 + 17 \zeta_{10} - 19 \zeta_{10}^{2} ) q^{48} + ( -44 \zeta_{10} + 21 \zeta_{10}^{2} - 44 \zeta_{10}^{3} ) q^{49} + ( -18 + 18 \zeta_{10} - 9 \zeta_{10}^{2} ) q^{50} + ( -26 - 13 \zeta_{10} + 52 \zeta_{10}^{2} - 26 \zeta_{10}^{3} ) q^{51} + ( 22 - 4 \zeta_{10} - 4 \zeta_{10}^{2} + 22 \zeta_{10}^{3} ) q^{52} + ( 30 - 30 \zeta_{10} + 30 \zeta_{10}^{3} ) q^{53} + ( 17 - 34 \zeta_{10} + 36 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{54} + ( 8 - 36 \zeta_{10} + 24 \zeta_{10}^{2} - 48 \zeta_{10}^{3} ) q^{55} + ( 38 + 26 \zeta_{10}^{2} - 26 \zeta_{10}^{3} ) q^{56} + ( -8 + 26 \zeta_{10} - 34 \zeta_{10}^{2} + 17 \zeta_{10}^{3} ) q^{57} + ( 12 + 2 \zeta_{10} - 2 \zeta_{10}^{2} - 12 \zeta_{10}^{3} ) q^{58} + ( 15 - 22 \zeta_{10} + 15 \zeta_{10}^{2} ) q^{59} + ( 44 \zeta_{10} - 52 \zeta_{10}^{2} + 44 \zeta_{10}^{3} ) q^{60} + ( 4 + 14 \zeta_{10} + 2 \zeta_{10}^{2} - 18 \zeta_{10}^{3} ) q^{61} + ( 18 + 22 \zeta_{10} - 62 \zeta_{10}^{2} + 44 \zeta_{10}^{3} ) q^{62} + ( 18 + 26 \zeta_{10} + 26 \zeta_{10}^{2} + 18 \zeta_{10}^{3} ) q^{63} + ( -36 + 36 \zeta_{10} - 41 \zeta_{10}^{3} ) q^{64} + ( -8 + 16 \zeta_{10} - 24 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{65} + ( 49 - 83 \zeta_{10} + 37 \zeta_{10}^{2} + 14 \zeta_{10}^{3} ) q^{66} + ( -49 + 17 \zeta_{10}^{2} - 17 \zeta_{10}^{3} ) q^{67} + ( -13 - 91 \zeta_{10} + 78 \zeta_{10}^{2} - 39 \zeta_{10}^{3} ) q^{68} + ( 10 - 20 \zeta_{10} + 20 \zeta_{10}^{2} - 10 \zeta_{10}^{3} ) q^{69} + ( -8 - 16 \zeta_{10} - 8 \zeta_{10}^{2} ) q^{70} + ( 62 \zeta_{10} - 24 \zeta_{10}^{2} + 62 \zeta_{10}^{3} ) q^{71} + ( -38 + 17 \zeta_{10} - 19 \zeta_{10}^{2} + 21 \zeta_{10}^{3} ) q^{72} + ( 77 - 17 \zeta_{10} - 43 \zeta_{10}^{2} - 34 \zeta_{10}^{3} ) q^{73} + ( -54 + 18 \zeta_{10} + 18 \zeta_{10}^{2} - 54 \zeta_{10}^{3} ) q^{74} + ( 9 - 9 \zeta_{10} + 18 \zeta_{10}^{3} ) q^{75} + ( -23 + 46 \zeta_{10} - 41 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{76} + ( -82 + 28 \zeta_{10} - 26 \zeta_{10}^{2} + 74 \zeta_{10}^{3} ) q^{77} + ( -30 + 40 \zeta_{10}^{2} - 40 \zeta_{10}^{3} ) q^{78} + ( -12 + 40 \zeta_{10} - 52 \zeta_{10}^{2} + 26 \zeta_{10}^{3} ) q^{79} + ( -20 - 12 \zeta_{10} + 12 \zeta_{10}^{2} + 20 \zeta_{10}^{3} ) q^{80} + ( -60 + 64 \zeta_{10} - 60 \zeta_{10}^{2} ) q^{81} + ( 21 \zeta_{10} + 17 \zeta_{10}^{2} + 21 \zeta_{10}^{3} ) q^{82} + ( 90 - 47 \zeta_{10} + 45 \zeta_{10}^{2} - 43 \zeta_{10}^{3} ) q^{83} + ( 32 - 14 \zeta_{10} - 4 \zeta_{10}^{2} - 28 \zeta_{10}^{3} ) q^{84} + ( 52 + 52 \zeta_{10}^{3} ) q^{85} + ( 53 - 53 \zeta_{10} + 16 \zeta_{10}^{3} ) q^{86} + ( 14 - 28 \zeta_{10} + 2 \zeta_{10}^{2} - 26 \zeta_{10}^{3} ) q^{87} + ( 22 - 44 \zeta_{10} - 22 \zeta_{10}^{2} - 11 \zeta_{10}^{3} ) q^{88} + ( 72 + 83 \zeta_{10}^{2} - 83 \zeta_{10}^{3} ) q^{89} + ( 20 - 28 \zeta_{10} + 48 \zeta_{10}^{2} - 24 \zeta_{10}^{3} ) q^{90} + ( -40 + 20 \zeta_{10} - 20 \zeta_{10}^{2} + 40 \zeta_{10}^{3} ) q^{91} + ( 22 - 26 \zeta_{10} + 22 \zeta_{10}^{2} ) q^{92} + ( -58 \zeta_{10} + 84 \zeta_{10}^{2} - 58 \zeta_{10}^{3} ) q^{93} + ( 48 - 46 \zeta_{10} + 24 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{94} + ( -12 - 8 \zeta_{10} + 28 \zeta_{10}^{2} - 16 \zeta_{10}^{3} ) q^{95} + ( 68 - 21 \zeta_{10} - 21 \zeta_{10}^{2} + 68 \zeta_{10}^{3} ) q^{96} + ( -67 + 67 \zeta_{10} + 36 \zeta_{10}^{3} ) q^{97} + ( -65 + 130 \zeta_{10} - 86 \zeta_{10}^{2} + 44 \zeta_{10}^{3} ) q^{98} + ( 28 - 5 \zeta_{10} + 7 \zeta_{10}^{2} - 69 \zeta_{10}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 5q^{2} - 9q^{4} - 4q^{5} + 15q^{6} + 10q^{7} + 15q^{8} - 11q^{9} + O(q^{10})$$ $$4q - 5q^{2} - 9q^{4} - 4q^{5} + 15q^{6} + 10q^{7} + 15q^{8} - 11q^{9} + q^{11} - 30q^{12} - 20q^{13} - 10q^{14} + 19q^{16} + 30q^{18} + 25q^{19} + 44q^{20} - 35q^{22} - 20q^{23} + 5q^{24} + 9q^{25} - 10q^{26} + 15q^{27} - 60q^{28} - 40q^{29} - 80q^{30} - 58q^{31} + 65q^{33} + 130q^{34} + 80q^{35} + 26q^{36} + 90q^{37} - 60q^{38} + 50q^{39} - 60q^{40} - 80q^{41} - 10q^{42} + 24q^{44} - 24q^{45} + 30q^{46} - 30q^{47} - 40q^{48} - 109q^{49} - 45q^{50} - 195q^{51} + 110q^{52} + 120q^{53} - 76q^{55} + 100q^{56} + 45q^{57} + 40q^{58} + 23q^{59} + 140q^{60} + 10q^{61} + 200q^{62} + 90q^{63} - 149q^{64} + 90q^{66} - 230q^{67} - 260q^{68} - 10q^{69} - 40q^{70} + 148q^{71} - 95q^{72} + 300q^{73} - 270q^{74} + 45q^{75} - 200q^{77} - 200q^{78} + 70q^{79} - 84q^{80} - 116q^{81} + 25q^{82} + 225q^{83} + 90q^{84} + 260q^{85} + 175q^{86} + 55q^{88} + 122q^{89} - 20q^{90} - 80q^{91} + 40q^{92} - 200q^{93} + 120q^{94} - 100q^{95} + 340q^{96} - 165q^{97} + 31q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/11\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$\zeta_{10}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 0.809017 + 0.587785i 0.809017 − 0.587785i −0.309017 − 0.951057i −0.309017 + 0.951057i
−0.690983 0.224514i −1.11803 + 0.812299i −2.80902 2.04087i 1.23607 + 3.80423i 0.954915 0.310271i 5.85410 8.05748i 3.19098 + 4.39201i −2.19098 + 6.74315i 2.90617i
6.1 −0.690983 + 0.224514i −1.11803 0.812299i −2.80902 + 2.04087i 1.23607 3.80423i 0.954915 + 0.310271i 5.85410 + 8.05748i 3.19098 4.39201i −2.19098 6.74315i 2.90617i
7.1 −1.80902 + 2.48990i 1.11803 3.44095i −1.69098 5.20431i −3.23607 + 2.35114i 6.54508 + 9.00854i −0.854102 + 0.277515i 4.30902 + 1.40008i −3.30902 2.40414i 12.3107i
8.1 −1.80902 2.48990i 1.11803 + 3.44095i −1.69098 + 5.20431i −3.23607 2.35114i 6.54508 9.00854i −0.854102 0.277515i 4.30902 1.40008i −3.30902 + 2.40414i 12.3107i
 $$n$$: e.g. 2-40 or 990-1000 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.d odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 11.3.d.a 4
3.b odd 2 1 99.3.k.a 4
4.b odd 2 1 176.3.n.a 4
5.b even 2 1 275.3.x.e 4
5.c odd 4 2 275.3.q.d 8
11.b odd 2 1 121.3.d.d 4
11.c even 5 1 121.3.b.b 4
11.c even 5 1 121.3.d.a 4
11.c even 5 1 121.3.d.c 4
11.c even 5 1 121.3.d.d 4
11.d odd 10 1 inner 11.3.d.a 4
11.d odd 10 1 121.3.b.b 4
11.d odd 10 1 121.3.d.a 4
11.d odd 10 1 121.3.d.c 4
33.f even 10 1 99.3.k.a 4
33.f even 10 1 1089.3.c.e 4
33.h odd 10 1 1089.3.c.e 4
44.g even 10 1 176.3.n.a 4
55.h odd 10 1 275.3.x.e 4
55.l even 20 2 275.3.q.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.3.d.a 4 1.a even 1 1 trivial
11.3.d.a 4 11.d odd 10 1 inner
99.3.k.a 4 3.b odd 2 1
99.3.k.a 4 33.f even 10 1
121.3.b.b 4 11.c even 5 1
121.3.b.b 4 11.d odd 10 1
121.3.d.a 4 11.c even 5 1
121.3.d.a 4 11.d odd 10 1
121.3.d.c 4 11.c even 5 1
121.3.d.c 4 11.d odd 10 1
121.3.d.d 4 11.b odd 2 1
121.3.d.d 4 11.c even 5 1
176.3.n.a 4 4.b odd 2 1
176.3.n.a 4 44.g even 10 1
275.3.q.d 8 5.c odd 4 2
275.3.q.d 8 55.l even 20 2
275.3.x.e 4 5.b even 2 1
275.3.x.e 4 55.h odd 10 1
1089.3.c.e 4 33.f even 10 1
1089.3.c.e 4 33.h odd 10 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(11, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 5 T + 19 T^{2} + 55 T^{3} + 121 T^{4} + 220 T^{5} + 304 T^{6} + 320 T^{7} + 256 T^{8}$$
$3$ $$1 + T^{2} - 20 T^{3} + 61 T^{4} - 180 T^{5} + 81 T^{6} + 6561 T^{8}$$
$5$ $$1 + 4 T - 9 T^{2} - 136 T^{3} - 319 T^{4} - 3400 T^{5} - 5625 T^{6} + 62500 T^{7} + 390625 T^{8}$$
$7$ $$1 - 10 T + 129 T^{2} - 1310 T^{3} + 8361 T^{4} - 64190 T^{5} + 309729 T^{6} - 1176490 T^{7} + 5764801 T^{8}$$
$11$ $$1 - T - 209 T^{2} - 121 T^{3} + 14641 T^{4}$$
$13$ $$1 + 20 T + 369 T^{2} + 6070 T^{3} + 81261 T^{4} + 1025830 T^{5} + 10539009 T^{6} + 96536180 T^{7} + 815730721 T^{8}$$
$17$ $$1 + 289 T^{2} + 7800 T^{3} - 17879 T^{4} + 2254200 T^{5} + 24137569 T^{6} + 6975757441 T^{8}$$
$19$ $$1 - 25 T + 561 T^{2} - 5855 T^{3} + 141756 T^{4} - 2113655 T^{5} + 73110081 T^{6} - 1176147025 T^{7} + 16983563041 T^{8}$$
$23$ $$( 1 + 10 T + 1078 T^{2} + 5290 T^{3} + 279841 T^{4} )^{2}$$
$29$ $$1 + 40 T + 1881 T^{2} + 81410 T^{3} + 2180301 T^{4} + 68465810 T^{5} + 1330395561 T^{6} + 23792932840 T^{7} + 500246412961 T^{8}$$
$31$ $$1 + 58 T + 423 T^{2} - 17434 T^{3} - 243175 T^{4} - 16754074 T^{5} + 390649383 T^{6} + 51475213498 T^{7} + 852891037441 T^{8}$$
$37$ $$1 - 90 T + 3491 T^{2} - 145800 T^{3} + 6716341 T^{4} - 199600200 T^{5} + 6542696051 T^{6} - 230915376810 T^{7} + 3512479453921 T^{8}$$
$41$ $$1 + 80 T + 6401 T^{2} + 349320 T^{3} + 15534121 T^{4} + 587206920 T^{5} + 18087696161 T^{6} + 380008339280 T^{7} + 7984925229121 T^{8}$$
$43$ $$1 - 5771 T^{2} + 15084961 T^{4} - 19729900571 T^{6} + 11688200277601 T^{8}$$
$47$ $$1 + 30 T - 1569 T^{2} - 98050 T^{3} + 581001 T^{4} - 216592450 T^{5} - 7656219489 T^{6} + 323376459870 T^{7} + 23811286661761 T^{8}$$
$53$ $$1 - 120 T + 2591 T^{2} + 279810 T^{3} - 24164819 T^{4} + 785986290 T^{5} + 20444236271 T^{6} - 2659723335480 T^{7} + 62259690411361 T^{8}$$
$59$ $$1 - 23 T - 2427 T^{2} - 104641 T^{3} + 14691380 T^{4} - 364255321 T^{5} - 29408835147 T^{6} - 970152273743 T^{7} + 146830437604321 T^{8}$$
$61$ $$1 - 10 T + 3601 T^{2} + 243430 T^{3} + 7253641 T^{4} + 905803030 T^{5} + 49858873441 T^{6} - 515203743610 T^{7} + 191707312997281 T^{8}$$
$67$ $$( 1 + 115 T + 11923 T^{2} + 516235 T^{3} + 20151121 T^{4} )^{2}$$
$71$ $$1 - 148 T + 9423 T^{2} - 841046 T^{3} + 82451165 T^{4} - 4239712886 T^{5} + 239454270063 T^{6} - 18958842020308 T^{7} + 645753531245761 T^{8}$$
$73$ $$1 - 300 T + 46429 T^{2} - 5018400 T^{3} + 415041241 T^{4} - 26743053600 T^{5} + 1318501931389 T^{6} - 45400267886700 T^{7} + 806460091894081 T^{8}$$
$79$ $$1 - 70 T + 10021 T^{2} - 1216880 T^{3} + 75589621 T^{4} - 7594548080 T^{5} + 390318761701 T^{6} - 17016121886470 T^{7} + 1517108809906561 T^{8}$$
$83$ $$1 - 225 T + 27589 T^{2} - 2589975 T^{3} + 221504596 T^{4} - 17842337775 T^{5} + 1309327618069 T^{6} - 73561584008025 T^{7} + 2252292232139041 T^{8}$$
$89$ $$( 1 - 61 T + 8161 T^{2} - 483181 T^{3} + 62742241 T^{4} )^{2}$$
$97$ $$1 + 165 T + 7431 T^{2} + 999895 T^{3} + 180445176 T^{4} + 9408012055 T^{5} + 657861087111 T^{6} + 137440380813285 T^{7} + 7837433594376961 T^{8}$$