# Properties

 Label 210.3.l.a Level 210 Weight 3 Character orbit 210.l Analytic conductor 5.722 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$210 = 2 \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 210.l (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.72208555157$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.12745506816.1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{2} ) q^{2} + \beta_{7} q^{3} + 2 \beta_{2} q^{4} + ( -2 \beta_{3} - \beta_{4} + 2 \beta_{7} ) q^{5} + ( \beta_{1} + \beta_{7} ) q^{6} + ( -\beta_{3} - \beta_{4} ) q^{7} + ( -2 + 2 \beta_{2} ) q^{8} -3 \beta_{2} q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{2} ) q^{2} + \beta_{7} q^{3} + 2 \beta_{2} q^{4} + ( -2 \beta_{3} - \beta_{4} + 2 \beta_{7} ) q^{5} + ( \beta_{1} + \beta_{7} ) q^{6} + ( -\beta_{3} - \beta_{4} ) q^{7} + ( -2 + 2 \beta_{2} ) q^{8} -3 \beta_{2} q^{9} + ( -3 \beta_{3} + \beta_{4} + 3 \beta_{7} ) q^{10} + ( 4 \beta_{1} + 2 \beta_{4} - \beta_{5} + \beta_{6} + 6 \beta_{7} ) q^{11} + 2 \beta_{1} q^{12} + ( 2 + 2 \beta_{6} + 2 \beta_{7} ) q^{13} + ( -\beta_{1} - 2 \beta_{3} + \beta_{7} ) q^{14} + ( -4 \beta_{2} + 2 \beta_{5} - \beta_{6} ) q^{15} -4 q^{16} + ( -4 + 3 \beta_{1} - 4 \beta_{2} - \beta_{3} - \beta_{4} ) q^{17} + ( 3 - 3 \beta_{2} ) q^{18} + ( -5 \beta_{1} - 2 \beta_{2} - 8 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} + 5 \beta_{7} ) q^{19} + ( -2 \beta_{3} + 4 \beta_{4} + 2 \beta_{7} ) q^{20} + ( -1 + \beta_{5} - \beta_{6} ) q^{21} + ( 10 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{22} + ( -7 + 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{6} - 2 \beta_{7} ) q^{23} + ( 2 \beta_{1} - 2 \beta_{7} ) q^{24} + ( -10 + 11 \beta_{2} + 7 \beta_{5} - \beta_{6} ) q^{25} + ( 4 + 2 \beta_{1} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{26} -3 \beta_{1} q^{27} + ( -2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{7} ) q^{28} + ( 6 \beta_{2} + 4 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} ) q^{29} + ( 5 - \beta_{2} + 3 \beta_{5} + \beta_{6} ) q^{30} + ( 22 + 5 \beta_{1} - 6 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} - \beta_{7} ) q^{31} + ( -4 - 4 \beta_{2} ) q^{32} + ( 16 - 3 \beta_{1} - 14 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{33} + ( 2 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} - 2 \beta_{7} ) q^{34} + ( -5 + 11 \beta_{2} + 2 \beta_{5} - \beta_{6} ) q^{35} + 6 q^{36} + ( 21 - 8 \beta_{1} + 19 \beta_{2} + 12 \beta_{3} + 12 \beta_{4} - 2 \beta_{5} ) q^{37} + ( -2 - 8 \beta_{1} - 2 \beta_{2} - 8 \beta_{3} + 8 \beta_{4} - 4 \beta_{6} + 10 \beta_{7} ) q^{38} + ( -4 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} + 4 \beta_{7} ) q^{39} + ( 2 \beta_{3} + 6 \beta_{4} - 2 \beta_{7} ) q^{40} + ( -36 - 9 \beta_{1} - 12 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} - 21 \beta_{7} ) q^{41} + ( -1 + \beta_{2} + 2 \beta_{5} ) q^{42} + ( -2 + 14 \beta_{1} + 6 \beta_{2} + 14 \beta_{3} - 14 \beta_{4} + 4 \beta_{6} - 8 \beta_{7} ) q^{43} + ( 12 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} - 12 \beta_{7} ) q^{44} + ( 3 \beta_{3} - 6 \beta_{4} - 3 \beta_{7} ) q^{45} + ( -14 + 2 \beta_{1} - 4 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} ) q^{46} + ( -14 - 10 \beta_{1} - 22 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 8 \beta_{5} ) q^{47} -4 \beta_{7} q^{48} + 7 \beta_{2} q^{49} + ( -15 + 9 \beta_{2} + 8 \beta_{5} + 6 \beta_{6} ) q^{50} + ( 8 - 4 \beta_{1} + \beta_{5} - \beta_{6} - 4 \beta_{7} ) q^{51} + ( 4 + 4 \beta_{1} - 4 \beta_{5} ) q^{52} + ( -24 + 2 \beta_{1} + 26 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - 14 \beta_{7} ) q^{53} + ( -3 \beta_{1} + 3 \beta_{7} ) q^{54} + ( 20 - 15 \beta_{1} - 18 \beta_{2} - \beta_{3} + 7 \beta_{4} + 14 \beta_{5} + 8 \beta_{6} + 6 \beta_{7} ) q^{55} + ( -2 \beta_{1} + 4 \beta_{4} + 2 \beta_{7} ) q^{56} + ( -7 + \beta_{2} + 6 \beta_{3} + 6 \beta_{4} + 8 \beta_{5} ) q^{57} + ( -10 + 4 \beta_{1} + 6 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} - 4 \beta_{6} ) q^{58} + ( -6 \beta_{1} - 8 \beta_{2} + 12 \beta_{3} - 16 \beta_{5} - 16 \beta_{6} + 6 \beta_{7} ) q^{59} + ( 10 + 6 \beta_{2} + 2 \beta_{5} + 4 \beta_{6} ) q^{60} + ( 50 - 6 \beta_{1} + 4 \beta_{4} - 14 \beta_{5} + 14 \beta_{6} - 2 \beta_{7} ) q^{61} + ( 22 + 4 \beta_{1} + 14 \beta_{2} - 6 \beta_{3} - 6 \beta_{4} - 8 \beta_{5} ) q^{62} + ( 3 \beta_{1} + 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{7} ) q^{63} -8 \beta_{2} q^{64} + ( -20 \beta_{1} - 8 \beta_{2} - 10 \beta_{3} + 4 \beta_{5} - 2 \beta_{6} ) q^{65} + ( 32 - 4 \beta_{1} + 6 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{66} + ( -2 - 12 \beta_{1} + 22 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + 24 \beta_{5} ) q^{67} + ( 8 - 2 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 4 \beta_{7} ) q^{68} + ( 11 \beta_{1} - 2 \beta_{2} + 12 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} - 11 \beta_{7} ) q^{69} + ( -15 + 9 \beta_{2} + 3 \beta_{5} + \beta_{6} ) q^{70} + ( -30 + 2 \beta_{1} - 2 \beta_{4} - 5 \beta_{5} + 5 \beta_{6} ) q^{71} + ( 6 + 6 \beta_{2} ) q^{72} + ( 34 + 20 \beta_{1} - 44 \beta_{2} + 20 \beta_{3} - 20 \beta_{4} - 10 \beta_{6} - 42 \beta_{7} ) q^{73} + ( 4 \beta_{1} + 38 \beta_{2} + 24 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} ) q^{74} + ( 20 \beta_{1} + 3 \beta_{3} - 21 \beta_{4} - 18 \beta_{7} ) q^{75} + ( -4 - 6 \beta_{1} + 16 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} + 10 \beta_{7} ) q^{76} + ( -12 - 7 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + 10 \beta_{5} ) q^{77} + ( 6 - 6 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} + 6 \beta_{4} + 8 \beta_{7} ) q^{78} + ( -26 \beta_{1} - 40 \beta_{2} - 40 \beta_{3} - 16 \beta_{5} - 16 \beta_{6} + 26 \beta_{7} ) q^{79} + ( 8 \beta_{3} + 4 \beta_{4} - 8 \beta_{7} ) q^{80} -9 q^{81} + ( -36 - 30 \beta_{1} - 44 \beta_{2} - 12 \beta_{3} - 12 \beta_{4} - 8 \beta_{5} ) q^{82} + ( -38 - 10 \beta_{1} + 42 \beta_{2} - 10 \beta_{3} + 10 \beta_{4} + 4 \beta_{6} + 48 \beta_{7} ) q^{83} + ( 2 \beta_{2} + 2 \beta_{5} + 2 \beta_{6} ) q^{84} + ( 10 + 20 \beta_{2} + 12 \beta_{3} - 4 \beta_{4} + 5 \beta_{5} + 5 \beta_{6} - 12 \beta_{7} ) q^{85} + ( -4 + 20 \beta_{1} - 28 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} - 8 \beta_{7} ) q^{86} + ( -4 + 8 \beta_{1} - 8 \beta_{2} + 6 \beta_{3} + 6 \beta_{4} - 4 \beta_{5} ) q^{87} + ( 4 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} - 4 \beta_{6} - 24 \beta_{7} ) q^{88} + ( 25 \beta_{1} - 12 \beta_{2} + 30 \beta_{3} - 16 \beta_{5} - 16 \beta_{6} - 25 \beta_{7} ) q^{89} + ( -3 \beta_{3} - 9 \beta_{4} + 3 \beta_{7} ) q^{90} + ( -2 - 6 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 8 \beta_{7} ) q^{91} + ( -14 - 6 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + 8 \beta_{5} ) q^{92} + ( 3 - 12 \beta_{1} - 9 \beta_{2} - 12 \beta_{3} + 12 \beta_{4} - 6 \beta_{6} + 30 \beta_{7} ) q^{93} + ( -8 \beta_{1} - 44 \beta_{2} + 4 \beta_{3} - 8 \beta_{5} - 8 \beta_{6} + 8 \beta_{7} ) q^{94} + ( -65 + 10 \beta_{1} + 29 \beta_{2} + 12 \beta_{3} + 6 \beta_{4} + 13 \beta_{5} + \beta_{6} + 18 \beta_{7} ) q^{95} + ( -4 \beta_{1} - 4 \beta_{7} ) q^{96} + ( -20 - 6 \beta_{1} + 14 \beta_{2} + 8 \beta_{3} + 8 \beta_{4} + 34 \beta_{5} ) q^{97} + ( -7 + 7 \beta_{2} ) q^{98} + ( -18 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} + 3 \beta_{5} + 3 \beta_{6} + 18 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 8q^{2} - 16q^{8} + O(q^{10})$$ $$8q + 8q^{2} - 16q^{8} - 8q^{11} + 8q^{13} + 12q^{15} - 32q^{16} - 32q^{17} + 24q^{18} - 8q^{22} - 40q^{23} - 48q^{25} + 16q^{26} + 48q^{30} + 144q^{31} - 32q^{32} + 120q^{33} - 28q^{35} + 48q^{36} + 160q^{37} - 320q^{41} - 32q^{43} - 80q^{46} - 144q^{47} - 112q^{50} + 72q^{51} + 16q^{52} - 200q^{53} + 184q^{55} - 24q^{57} - 64q^{58} + 72q^{60} + 288q^{61} + 144q^{62} + 24q^{65} + 240q^{66} + 80q^{67} + 64q^{68} - 112q^{70} - 280q^{71} + 48q^{72} + 312q^{73} - 56q^{77} + 48q^{78} - 72q^{81} - 320q^{82} - 320q^{83} + 80q^{85} - 64q^{86} - 48q^{87} + 16q^{88} - 80q^{92} + 48q^{93} - 472q^{95} - 24q^{97} - 56q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 23 x^{4} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{5} + 19 \nu$$$$)/5$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + 24 \nu^{2}$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + 24 \nu^{3} + 5 \nu$$$$)/5$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{5} - 24 \nu^{3} + 24 \nu$$$$)/5$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{6} - \nu^{4} - 67 \nu^{2} - 9$$$$)/5$$ $$\beta_{6}$$ $$=$$ $$($$$$-3 \nu^{6} + \nu^{4} - 67 \nu^{2} + 9$$$$)/5$$ $$\beta_{7}$$ $$=$$ $$($$$$-4 \nu^{7} - 91 \nu^{3}$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} + \beta_{3} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} + \beta_{5} + 6 \beta_{2}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$($$$$5 \beta_{6} - 5 \beta_{5} - 18$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-19 \beta_{4} - 19 \beta_{3} + 29 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-12 \beta_{6} - 12 \beta_{5} - 67 \beta_{2}$$ $$\nu^{7}$$ $$=$$ $$($$$$-48 \beta_{7} + 91 \beta_{4} - 91 \beta_{3} - 91 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/210\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$71$$ $$127$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1
 1.54779 − 1.54779i −0.323042 + 0.323042i 0.323042 − 0.323042i −1.54779 + 1.54779i 1.54779 + 1.54779i −0.323042 − 0.323042i 0.323042 + 0.323042i −1.54779 − 1.54779i
1.00000 1.00000i −1.22474 1.22474i 2.00000i −4.32032 + 2.51691i −2.44949 −1.87083 + 1.87083i −2.00000 2.00000i 3.00000i −1.80341 + 6.83723i
43.2 1.00000 1.00000i −1.22474 1.22474i 2.00000i −0.578661 4.96640i −2.44949 1.87083 1.87083i −2.00000 2.00000i 3.00000i −5.54506 4.38774i
43.3 1.00000 1.00000i 1.22474 + 1.22474i 2.00000i 0.578661 + 4.96640i 2.44949 −1.87083 + 1.87083i −2.00000 2.00000i 3.00000i 5.54506 + 4.38774i
43.4 1.00000 1.00000i 1.22474 + 1.22474i 2.00000i 4.32032 2.51691i 2.44949 1.87083 1.87083i −2.00000 2.00000i 3.00000i 1.80341 6.83723i
127.1 1.00000 + 1.00000i −1.22474 + 1.22474i 2.00000i −4.32032 2.51691i −2.44949 −1.87083 1.87083i −2.00000 + 2.00000i 3.00000i −1.80341 6.83723i
127.2 1.00000 + 1.00000i −1.22474 + 1.22474i 2.00000i −0.578661 + 4.96640i −2.44949 1.87083 + 1.87083i −2.00000 + 2.00000i 3.00000i −5.54506 + 4.38774i
127.3 1.00000 + 1.00000i 1.22474 1.22474i 2.00000i 0.578661 4.96640i 2.44949 −1.87083 1.87083i −2.00000 + 2.00000i 3.00000i 5.54506 4.38774i
127.4 1.00000 + 1.00000i 1.22474 1.22474i 2.00000i 4.32032 + 2.51691i 2.44949 1.87083 + 1.87083i −2.00000 + 2.00000i 3.00000i 1.80341 + 6.83723i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 127.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.3.l.a 8
3.b odd 2 1 630.3.o.b 8
5.b even 2 1 1050.3.l.b 8
5.c odd 4 1 inner 210.3.l.a 8
5.c odd 4 1 1050.3.l.b 8
15.e even 4 1 630.3.o.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.l.a 8 1.a even 1 1 trivial
210.3.l.a 8 5.c odd 4 1 inner
630.3.o.b 8 3.b odd 2 1
630.3.o.b 8 15.e even 4 1
1050.3.l.b 8 5.b even 2 1
1050.3.l.b 8 5.c odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{4} + 4 T_{11}^{3} - 364 T_{11}^{2} - 2416 T_{11} + 10000$$ acting on $$S_{3}^{\mathrm{new}}(210, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T + 2 T^{2} )^{4}$$
$3$ $$( 1 + 9 T^{4} )^{2}$$
$5$ $$1 + 24 T^{2} + 50 T^{4} + 15000 T^{6} + 390625 T^{8}$$
$7$ $$( 1 + 49 T^{4} )^{2}$$
$11$ $$( 1 + 4 T + 120 T^{2} - 964 T^{3} + 9758 T^{4} - 116644 T^{5} + 1756920 T^{6} + 7086244 T^{7} + 214358881 T^{8} )^{2}$$
$13$ $$1 - 8 T + 32 T^{2} - 1000 T^{3} + 57188 T^{4} - 379192 T^{5} + 1703520 T^{6} - 57464472 T^{7} + 1905455878 T^{8} - 9711495768 T^{9} + 48654234720 T^{10} - 1830287358328 T^{11} + 46650008472548 T^{12} - 137858491849000 T^{13} + 745538723919392 T^{14} - 31499011085594312 T^{15} + 665416609183179841 T^{16}$$
$17$ $$1 + 32 T + 512 T^{2} + 13280 T^{3} + 494308 T^{4} + 9125408 T^{5} + 127106560 T^{6} + 2946229728 T^{7} + 67506528198 T^{8} + 851460391392 T^{9} + 10616066997760 T^{10} + 220265165253152 T^{11} + 3448172709145828 T^{12} + 26772398997962720 T^{13} + 298302585461637632 T^{14} + 5388090449900829728 T^{15} + 48661191875666868481 T^{16}$$
$19$ $$1 - 1632 T^{2} + 1354756 T^{4} - 767600928 T^{6} + 320946269766 T^{8} - 100034520537888 T^{10} + 23008583931172996 T^{12} - 3612129947915974752 T^{14} +$$$$28\!\cdots\!81$$$$T^{16}$$
$23$ $$1 + 40 T + 800 T^{2} + 23320 T^{3} + 928156 T^{4} + 22803400 T^{5} + 441522400 T^{6} + 12864455160 T^{7} + 373563445830 T^{8} + 6805296779640 T^{9} + 123556069938400 T^{10} + 3375721591222600 T^{11} + 72684810854471836 T^{12} + 966066241502294680 T^{13} + 17531699545616256800 T^{14} +$$$$46\!\cdots\!60$$$$T^{15} +$$$$61\!\cdots\!61$$$$T^{16}$$
$29$ $$1 - 5816 T^{2} + 15494364 T^{4} - 24578727304 T^{6} + 25344087778310 T^{8} - 17384066826300424 T^{10} + 7751000012112051804 T^{12} -$$$$20\!\cdots\!56$$$$T^{14} +$$$$25\!\cdots\!21$$$$T^{16}$$
$31$ $$( 1 - 72 T + 4816 T^{2} - 187848 T^{3} + 7076706 T^{4} - 180521928 T^{5} + 4447677136 T^{6} - 63900265032 T^{7} + 852891037441 T^{8} )^{2}$$
$37$ $$1 - 160 T + 12800 T^{2} - 692576 T^{3} + 28393916 T^{4} - 973559072 T^{5} + 32158084608 T^{6} - 1118111280864 T^{7} + 40611983255110 T^{8} - 1530694343502816 T^{9} + 60269428007013888 T^{10} - 2497886221751932448 T^{11} + 99733046566358744636 T^{12} -$$$$33\!\cdots\!24$$$$T^{13} +$$$$84\!\cdots\!00$$$$T^{14} -$$$$14\!\cdots\!40$$$$T^{15} +$$$$12\!\cdots\!41$$$$T^{16}$$
$41$ $$( 1 + 160 T + 11944 T^{2} + 591520 T^{3} + 24800530 T^{4} + 994345120 T^{5} + 33750889384 T^{6} + 760016678560 T^{7} + 7984925229121 T^{8} )^{2}$$
$43$ $$1 + 32 T + 512 T^{2} - 44896 T^{3} - 2147132 T^{4} + 116117984 T^{5} + 5822932480 T^{6} + 160684182624 T^{7} + 3183857489478 T^{8} + 297105053671776 T^{9} + 19907447385556480 T^{10} + 734023933381973216 T^{11} - 25096108838445990332 T^{12} -$$$$97\!\cdots\!04$$$$T^{13} +$$$$20\!\cdots\!12$$$$T^{14} +$$$$23\!\cdots\!68$$$$T^{15} +$$$$13\!\cdots\!01$$$$T^{16}$$
$47$ $$1 + 144 T + 10368 T^{2} + 613776 T^{3} + 40380676 T^{4} + 2550145200 T^{5} + 136914549120 T^{6} + 6692132733360 T^{7} + 317421876290310 T^{8} + 14782921207992240 T^{9} + 668099323964430720 T^{10} + 27488564231015770800 T^{11} +$$$$96\!\cdots\!36$$$$T^{12} +$$$$32\!\cdots\!24$$$$T^{13} +$$$$12\!\cdots\!88$$$$T^{14} +$$$$36\!\cdots\!36$$$$T^{15} +$$$$56\!\cdots\!21$$$$T^{16}$$
$53$ $$1 + 200 T + 20000 T^{2} + 1719464 T^{3} + 146284516 T^{4} + 10449435128 T^{5} + 642474929248 T^{6} + 39041749119576 T^{7} + 2230315686063750 T^{8} + 109668273276888984 T^{9} + 5069436222207688288 T^{10} +$$$$23\!\cdots\!12$$$$T^{11} +$$$$91\!\cdots\!76$$$$T^{12} +$$$$30\!\cdots\!36$$$$T^{13} +$$$$98\!\cdots\!00$$$$T^{14} +$$$$27\!\cdots\!00$$$$T^{15} +$$$$38\!\cdots\!21$$$$T^{16}$$
$59$ $$1 - 616 T^{2} + 1419676 T^{4} + 5207600552 T^{6} - 204890323221242 T^{8} + 63102375832383272 T^{10} +$$$$20\!\cdots\!96$$$$T^{12} -$$$$10\!\cdots\!96$$$$T^{14} +$$$$21\!\cdots\!41$$$$T^{16}$$
$61$ $$( 1 - 144 T + 14124 T^{2} - 1141872 T^{3} + 82373126 T^{4} - 4248905712 T^{5} + 195558658284 T^{6} - 7418933907984 T^{7} + 191707312997281 T^{8} )^{2}$$
$67$ $$1 - 80 T + 3200 T^{2} - 105296 T^{3} - 42100604 T^{4} + 2558251408 T^{5} - 64394556032 T^{6} - 2106545111664 T^{7} + 886628492165190 T^{8} - 9456281006259696 T^{9} - 1297622490342111872 T^{10} +$$$$23\!\cdots\!52$$$$T^{11} -$$$$17\!\cdots\!64$$$$T^{12} -$$$$19\!\cdots\!04$$$$T^{13} +$$$$26\!\cdots\!00$$$$T^{14} -$$$$29\!\cdots\!20$$$$T^{15} +$$$$16\!\cdots\!81$$$$T^{16}$$
$71$ $$( 1 + 140 T + 26424 T^{2} + 2214100 T^{3} + 216062270 T^{4} + 11161278100 T^{5} + 671478258744 T^{6} + 17934039748940 T^{7} + 645753531245761 T^{8} )^{2}$$
$73$ $$1 - 312 T + 48672 T^{2} - 5493144 T^{3} + 450793636 T^{4} - 23206886280 T^{5} + 386836170336 T^{6} + 68692339480344 T^{7} - 8375186112407610 T^{8} + 366061477090753176 T^{9} + 10985466792718778976 T^{10} -$$$$35\!\cdots\!20$$$$T^{11} +$$$$36\!\cdots\!16$$$$T^{12} -$$$$23\!\cdots\!56$$$$T^{13} +$$$$11\!\cdots\!12$$$$T^{14} -$$$$38\!\cdots\!08$$$$T^{15} +$$$$65\!\cdots\!61$$$$T^{16}$$
$79$ $$1 - 2856 T^{2} - 59627684 T^{4} + 132589015272 T^{6} + 2412904555054278 T^{8} + 5164352884554637032 T^{10} -$$$$90\!\cdots\!24$$$$T^{12} -$$$$16\!\cdots\!96$$$$T^{14} +$$$$23\!\cdots\!21$$$$T^{16}$$
$83$ $$1 + 320 T + 51200 T^{2} + 5915840 T^{3} + 553217092 T^{4} + 45528000320 T^{5} + 3742826444800 T^{6} + 317655699460800 T^{7} + 26711748349549254 T^{8} + 2188330113585451200 T^{9} +$$$$17\!\cdots\!00$$$$T^{10} +$$$$14\!\cdots\!80$$$$T^{11} +$$$$12\!\cdots\!72$$$$T^{12} +$$$$91\!\cdots\!60$$$$T^{13} +$$$$54\!\cdots\!00$$$$T^{14} +$$$$23\!\cdots\!80$$$$T^{15} +$$$$50\!\cdots\!81$$$$T^{16}$$
$89$ $$1 - 26800 T^{2} + 323387236 T^{4} - 2700009379600 T^{6} + 20962417894146886 T^{8} -$$$$16\!\cdots\!00$$$$T^{10} +$$$$12\!\cdots\!16$$$$T^{12} -$$$$66\!\cdots\!00$$$$T^{14} +$$$$15\!\cdots\!61$$$$T^{16}$$
$97$ $$1 + 24 T + 288 T^{2} + 197496 T^{3} - 223582556 T^{4} - 4223145048 T^{5} - 17461370016 T^{6} + 856985885448 T^{7} + 24034841148321606 T^{8} + 8063380196180232 T^{9} - 1545842532791438496 T^{10} -$$$$35\!\cdots\!92$$$$T^{11} -$$$$17\!\cdots\!16$$$$T^{12} +$$$$14\!\cdots\!04$$$$T^{13} +$$$$19\!\cdots\!08$$$$T^{14} +$$$$15\!\cdots\!56$$$$T^{15} +$$$$61\!\cdots\!21$$$$T^{16}$$