# Properties

 Label 1050.3.p.b Level 1050 Weight 3 Character orbit 1050.p Analytic conductor 28.610 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1050.p (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$28.6104277578$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.3317760000.3 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 210) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{2} - \beta_{4} ) q^{2} + ( -1 - \beta_{3} ) q^{3} + ( -2 + 2 \beta_{3} ) q^{4} + ( \beta_{2} + 2 \beta_{4} ) q^{6} + ( -\beta_{1} - \beta_{2} - 3 \beta_{4} - 2 \beta_{6} ) q^{7} + 2 \beta_{2} q^{8} + 3 \beta_{3} q^{9} +O(q^{10})$$ $$q + ( -\beta_{2} - \beta_{4} ) q^{2} + ( -1 - \beta_{3} ) q^{3} + ( -2 + 2 \beta_{3} ) q^{4} + ( \beta_{2} + 2 \beta_{4} ) q^{6} + ( -\beta_{1} - \beta_{2} - 3 \beta_{4} - 2 \beta_{6} ) q^{7} + 2 \beta_{2} q^{8} + 3 \beta_{3} q^{9} + ( -1 - 2 \beta_{1} + \beta_{3} + 3 \beta_{4} + 4 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{11} + ( 4 - 2 \beta_{3} ) q^{12} + ( -2 + \beta_{1} - 4 \beta_{2} + 4 \beta_{3} - 8 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{13} + ( -6 + 2 \beta_{3} - 2 \beta_{5} - 3 \beta_{7} ) q^{14} -4 \beta_{3} q^{16} + ( -7 + 2 \beta_{2} - 7 \beta_{3} + \beta_{4} - 3 \beta_{6} - 2 \beta_{7} ) q^{17} -3 \beta_{4} q^{18} + ( 18 - 2 \beta_{1} - 7 \beta_{2} - 9 \beta_{3} + 7 \beta_{4} - \beta_{5} ) q^{19} + ( -\beta_{1} - \beta_{2} + 4 \beta_{4} + 5 \beta_{6} ) q^{21} + ( 6 - 4 \beta_{1} + \beta_{2} + \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{22} + ( -3 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} + 2 \beta_{5} + 6 \beta_{6} + 4 \beta_{7} ) q^{23} + ( -4 \beta_{2} - 2 \beta_{4} ) q^{24} + ( -16 + 2 \beta_{1} + 2 \beta_{2} + 8 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{26} + ( 3 - 6 \beta_{3} ) q^{27} + ( 6 \beta_{1} + 6 \beta_{2} + 4 \beta_{4} - 2 \beta_{6} ) q^{28} + ( 9 - \beta_{1} + 13 \beta_{2} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{29} + ( -11 - 2 \beta_{2} - 11 \beta_{3} - \beta_{4} - 2 \beta_{6} + 5 \beta_{7} ) q^{31} + 4 \beta_{4} q^{32} + ( 2 + 3 \beta_{1} + 3 \beta_{2} - \beta_{3} - 3 \beta_{4} - 6 \beta_{5} ) q^{33} + ( 2 + 4 \beta_{1} + 7 \beta_{2} - 4 \beta_{3} + 14 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} - 3 \beta_{7} ) q^{34} -6 q^{36} + ( 5 \beta_{1} - 2 \beta_{2} + 24 \beta_{3} - 2 \beta_{4} - \beta_{5} - 10 \beta_{6} - 2 \beta_{7} ) q^{37} + ( 14 - 18 \beta_{2} + 14 \beta_{3} - 9 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} ) q^{38} + ( 6 - 2 \beta_{1} - 6 \beta_{3} + 12 \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{39} + ( -3 + 19 \beta_{1} - \beta_{2} + 6 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - 19 \beta_{6} + 4 \beta_{7} ) q^{41} + ( 8 + 2 \beta_{3} + 5 \beta_{5} + 4 \beta_{7} ) q^{42} + ( 14 - 9 \beta_{1} + 5 \beta_{2} + 4 \beta_{5} - 9 \beta_{6} - 4 \beta_{7} ) q^{43} + ( 2 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} - 8 \beta_{7} ) q^{44} + ( 2 - 8 \beta_{1} - 2 \beta_{3} + 3 \beta_{4} + 6 \beta_{5} + 4 \beta_{6} + 3 \beta_{7} ) q^{46} + ( 4 + 8 \beta_{1} - 20 \beta_{2} - 2 \beta_{3} + 20 \beta_{4} - 2 \beta_{5} ) q^{47} + ( -4 + 8 \beta_{3} ) q^{48} + ( 15 + 9 \beta_{3} - 16 \beta_{5} - 10 \beta_{7} ) q^{49} + ( -3 \beta_{1} - 3 \beta_{2} + 21 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + 6 \beta_{6} + 4 \beta_{7} ) q^{51} + ( -4 + 16 \beta_{2} - 4 \beta_{3} + 8 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{52} + ( -8 - 8 \beta_{1} + 8 \beta_{3} + 26 \beta_{4} - 16 \beta_{5} + 4 \beta_{6} - 8 \beta_{7} ) q^{53} + ( -3 \beta_{2} + 3 \beta_{4} ) q^{54} + ( 8 - 12 \beta_{3} - 2 \beta_{5} + 4 \beta_{7} ) q^{56} + ( -27 + 2 \beta_{1} + 21 \beta_{2} + \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{57} + ( -4 \beta_{1} - 9 \beta_{2} - 26 \beta_{3} - 9 \beta_{4} - \beta_{5} + 8 \beta_{6} - 2 \beta_{7} ) q^{58} + ( 11 + 14 \beta_{2} + 11 \beta_{3} + 7 \beta_{4} - 21 \beta_{6} - 8 \beta_{7} ) q^{59} + ( 16 - 16 \beta_{1} - 2 \beta_{2} - 8 \beta_{3} + 2 \beta_{4} + 16 \beta_{5} ) q^{61} + ( -2 - 10 \beta_{1} + 11 \beta_{2} + 4 \beta_{3} + 22 \beta_{4} - 2 \beta_{5} + 10 \beta_{6} - 2 \beta_{7} ) q^{62} + ( 6 \beta_{1} + 6 \beta_{2} - 3 \beta_{4} - 9 \beta_{6} ) q^{63} + 8 q^{64} + ( -6 - 2 \beta_{2} - 6 \beta_{3} - \beta_{4} + 12 \beta_{6} + 3 \beta_{7} ) q^{66} + ( 30 + 6 \beta_{1} - 30 \beta_{3} + 9 \beta_{4} - 16 \beta_{5} - 3 \beta_{6} - 8 \beta_{7} ) q^{67} + ( 28 + 6 \beta_{1} - 2 \beta_{2} - 14 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} ) q^{68} + ( -3 + 9 \beta_{1} - \beta_{2} + 6 \beta_{3} - 2 \beta_{4} - 6 \beta_{5} - 9 \beta_{6} - 6 \beta_{7} ) q^{69} + ( 1 - 7 \beta_{1} + 29 \beta_{2} - 6 \beta_{5} - 7 \beta_{6} + 6 \beta_{7} ) q^{71} + ( 6 \beta_{2} + 6 \beta_{4} ) q^{72} + ( -2 + 20 \beta_{2} - 2 \beta_{3} + 10 \beta_{4} + 25 \beta_{6} - 15 \beta_{7} ) q^{73} + ( -4 + 4 \beta_{1} + 4 \beta_{3} - 24 \beta_{4} - 10 \beta_{5} - 2 \beta_{6} - 5 \beta_{7} ) q^{74} + ( -18 + 4 \beta_{1} - 14 \beta_{2} + 36 \beta_{3} - 28 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{76} + ( 31 + 7 \beta_{1} - 7 \beta_{2} - 8 \beta_{3} + 42 \beta_{4} + 8 \beta_{5} - 21 \beta_{6} - 2 \beta_{7} ) q^{77} + ( 24 - 2 \beta_{1} - 6 \beta_{2} + \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{78} + ( 2 \beta_{1} - 13 \beta_{2} + 3 \beta_{3} - 13 \beta_{4} - 19 \beta_{5} - 4 \beta_{6} - 38 \beta_{7} ) q^{79} + ( -9 + 9 \beta_{3} ) q^{81} + ( -4 - 8 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 19 \beta_{5} ) q^{82} + ( 3 - 7 \beta_{1} + 15 \beta_{2} - 6 \beta_{3} + 30 \beta_{4} - 34 \beta_{5} + 7 \beta_{6} - 34 \beta_{7} ) q^{83} + ( -8 \beta_{1} - 8 \beta_{2} - 10 \beta_{4} - 2 \beta_{6} ) q^{84} + ( 8 \beta_{1} - 14 \beta_{2} - 10 \beta_{3} - 14 \beta_{4} - 9 \beta_{5} - 16 \beta_{6} - 18 \beta_{7} ) q^{86} + ( -9 - 26 \beta_{2} - 9 \beta_{3} - 13 \beta_{4} + 3 \beta_{6} - 6 \beta_{7} ) q^{87} + ( -12 + 16 \beta_{1} + 12 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} - 8 \beta_{6} - 2 \beta_{7} ) q^{88} + ( 82 - 19 \beta_{1} + 11 \beta_{2} - 41 \beta_{3} - 11 \beta_{4} + 4 \beta_{5} ) q^{89} + ( -42 + 12 \beta_{1} + 5 \beta_{2} + 7 \beta_{3} - 13 \beta_{4} - 21 \beta_{5} - 4 \beta_{6} - 14 \beta_{7} ) q^{91} + ( 6 - 6 \beta_{1} - 2 \beta_{2} + 4 \beta_{5} - 6 \beta_{6} - 4 \beta_{7} ) q^{92} + ( -2 \beta_{1} + 3 \beta_{2} + 33 \beta_{3} + 3 \beta_{4} - 5 \beta_{5} + 4 \beta_{6} - 10 \beta_{7} ) q^{93} + ( 40 - 4 \beta_{2} + 40 \beta_{3} - 2 \beta_{4} + 4 \beta_{6} + 8 \beta_{7} ) q^{94} + ( 4 \beta_{2} - 4 \beta_{4} ) q^{96} + ( -36 - 8 \beta_{1} - 20 \beta_{2} + 72 \beta_{3} - 40 \beta_{4} - 18 \beta_{5} + 8 \beta_{6} - 18 \beta_{7} ) q^{97} + ( 20 \beta_{1} - 15 \beta_{2} - 24 \beta_{4} + 12 \beta_{6} ) q^{98} + ( -3 - 3 \beta_{1} - 9 \beta_{2} + 6 \beta_{5} - 3 \beta_{6} - 6 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 12q^{3} - 8q^{4} + 12q^{9} + O(q^{10})$$ $$8q - 12q^{3} - 8q^{4} + 12q^{9} - 4q^{11} + 24q^{12} - 40q^{14} - 16q^{16} - 84q^{17} + 108q^{19} + 48q^{22} - 12q^{23} - 96q^{26} + 72q^{29} - 132q^{31} + 12q^{33} - 48q^{36} + 96q^{37} + 168q^{38} + 24q^{39} + 72q^{42} + 112q^{43} - 8q^{44} + 8q^{46} + 24q^{47} + 156q^{49} + 84q^{51} - 48q^{52} - 32q^{53} + 16q^{56} - 216q^{57} - 104q^{58} + 132q^{59} + 96q^{61} + 64q^{64} - 72q^{66} + 120q^{67} + 168q^{68} + 8q^{71} - 24q^{73} - 16q^{74} + 216q^{77} + 192q^{78} + 12q^{79} - 36q^{81} - 24q^{82} - 40q^{86} - 108q^{87} - 48q^{88} + 492q^{89} - 308q^{91} + 48q^{92} + 132q^{93} + 480q^{94} - 24q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{6} + 7 x^{4} - 36 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{6} + 14 \nu^{4} + 7 \nu^{2} - 36$$$$)/63$$ $$\beta_{2}$$ $$=$$ $$($$$$-4 \nu^{7} + 7 \nu^{5} + 35 \nu^{3} + 81 \nu$$$$)/189$$ $$\beta_{3}$$ $$=$$ $$($$$$-4 \nu^{6} + 7 \nu^{4} - 28 \nu^{2} + 144$$$$)/63$$ $$\beta_{4}$$ $$=$$ $$($$$$-5 \nu^{7} - 7 \nu^{5} - 35 \nu^{3} + 180 \nu$$$$)/189$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + 13 \nu$$$$)/21$$ $$\beta_{6}$$ $$=$$ $$($$$$-8 \nu^{6} + 14 \nu^{4} + 7 \nu^{2} + 162$$$$)/63$$ $$\beta_{7}$$ $$=$$ $$($$$$19 \nu^{7} - 49 \nu^{5} + 133 \nu^{3} - 684 \nu$$$$)/189$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} - 2 \beta_{3} + 2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{7} + \beta_{5} + 7 \beta_{2}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$\beta_{3} + 4 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$($$$$-5 \beta_{7} - 19 \beta_{4}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-7 \beta_{6} + 7 \beta_{1} + 22$$ $$\nu^{7}$$ $$=$$ $$($$$$29 \beta_{5} - 13 \beta_{4} - 13 \beta_{2}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$701$$ $$\chi(n)$$ $$1$$ $$\beta_{3}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
451.1
 1.72286 − 0.178197i −1.01575 + 1.40294i −1.72286 + 0.178197i 1.01575 − 1.40294i 1.72286 + 0.178197i −1.01575 − 1.40294i −1.72286 − 0.178197i 1.01575 + 1.40294i
−0.707107 1.22474i −1.50000 0.866025i −1.00000 + 1.73205i 0 2.44949i −5.10237 4.79227i 2.82843 1.50000 + 2.59808i 0
451.2 −0.707107 1.22474i −1.50000 0.866025i −1.00000 + 1.73205i 0 2.44949i 6.51658 2.55620i 2.82843 1.50000 + 2.59808i 0
451.3 0.707107 + 1.22474i −1.50000 0.866025i −1.00000 + 1.73205i 0 2.44949i −6.51658 + 2.55620i −2.82843 1.50000 + 2.59808i 0
451.4 0.707107 + 1.22474i −1.50000 0.866025i −1.00000 + 1.73205i 0 2.44949i 5.10237 + 4.79227i −2.82843 1.50000 + 2.59808i 0
901.1 −0.707107 + 1.22474i −1.50000 + 0.866025i −1.00000 1.73205i 0 2.44949i −5.10237 + 4.79227i 2.82843 1.50000 2.59808i 0
901.2 −0.707107 + 1.22474i −1.50000 + 0.866025i −1.00000 1.73205i 0 2.44949i 6.51658 + 2.55620i 2.82843 1.50000 2.59808i 0
901.3 0.707107 1.22474i −1.50000 + 0.866025i −1.00000 1.73205i 0 2.44949i −6.51658 2.55620i −2.82843 1.50000 2.59808i 0
901.4 0.707107 1.22474i −1.50000 + 0.866025i −1.00000 1.73205i 0 2.44949i 5.10237 4.79227i −2.82843 1.50000 2.59808i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 901.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.3.p.b 8
5.b even 2 1 210.3.o.a 8
5.c odd 4 2 1050.3.q.c 16
7.d odd 6 1 inner 1050.3.p.b 8
15.d odd 2 1 630.3.v.b 8
35.i odd 6 1 210.3.o.a 8
35.i odd 6 1 1470.3.f.a 8
35.j even 6 1 1470.3.f.a 8
35.k even 12 2 1050.3.q.c 16
105.p even 6 1 630.3.v.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.o.a 8 5.b even 2 1
210.3.o.a 8 35.i odd 6 1
630.3.v.b 8 15.d odd 2 1
630.3.v.b 8 105.p even 6 1
1050.3.p.b 8 1.a even 1 1 trivial
1050.3.p.b 8 7.d odd 6 1 inner
1050.3.q.c 16 5.c odd 4 2
1050.3.q.c 16 35.k even 12 2
1470.3.f.a 8 35.i odd 6 1
1470.3.f.a 8 35.j even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1050, [\chi])$$:

 $$T_{11}^{8} + \cdots$$ $$T_{17}^{8} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T^{2} + 4 T^{4} )^{2}$$
$3$ $$( 1 + 3 T + 3 T^{2} )^{4}$$
$5$ 1
$7$ $$1 - 78 T^{2} + 5243 T^{4} - 187278 T^{6} + 5764801 T^{8}$$
$11$ $$1 + 4 T - 168 T^{2} + 1928 T^{3} + 16250 T^{4} - 341700 T^{5} + 2353408 T^{6} + 32486668 T^{7} - 296735661 T^{8} + 3930886828 T^{9} + 34456246528 T^{10} - 605342393700 T^{11} + 3483331816250 T^{12} + 50007354630728 T^{13} - 527255967289128 T^{14} + 1518999334332964 T^{15} + 45949729863572161 T^{16}$$
$13$ $$1 - 860 T^{2} + 357498 T^{4} - 95438800 T^{6} + 18541152803 T^{8} - 2725827566800 T^{10} + 291622101296058 T^{12} - 20036353205333660 T^{14} + 665416609183179841 T^{16}$$
$17$ $$1 + 84 T + 4208 T^{2} + 155904 T^{3} + 4746250 T^{4} + 123749124 T^{5} + 2818194464 T^{6} + 56792172564 T^{7} + 1020121281523 T^{8} + 16412937870996 T^{9} + 235378419827744 T^{10} + 2987003019239556 T^{11} + 33108688754346250 T^{12} + 314301513055600896 T^{13} + 2451674374262834288 T^{14} + 14143737430989678036 T^{15} + 48661191875666868481 T^{16}$$
$19$ $$1 - 108 T + 6142 T^{2} - 243432 T^{3} + 7548345 T^{4} - 198492408 T^{5} + 4660679966 T^{6} - 100240172052 T^{7} + 1986343374068 T^{8} - 36186702110772 T^{9} + 607384473849086 T^{10} - 9338250206171448 T^{11} + 128197793162717145 T^{12} - 1492497721269013032 T^{13} + 13594180232904360862 T^{14} - 86292722064551485068 T^{15} +$$$$28\!\cdots\!81$$$$T^{16}$$
$23$ $$1 + 12 T - 1512 T^{2} - 15144 T^{3} + 1340890 T^{4} + 9662580 T^{5} - 850652928 T^{6} - 2479269564 T^{7} + 454065720819 T^{8} - 1311533599356 T^{9} - 238047566024448 T^{10} + 1430408620333620 T^{11} + 105006417053440090 T^{12} - 627363085819500456 T^{13} - 33134912141214725352 T^{14} +$$$$13\!\cdots\!08$$$$T^{15} +$$$$61\!\cdots\!61$$$$T^{16}$$
$29$ $$( 1 - 36 T + 2904 T^{2} - 82956 T^{3} + 3490070 T^{4} - 69765996 T^{5} + 2053944024 T^{6} - 21413639556 T^{7} + 500246412961 T^{8} )^{2}$$
$31$ $$1 + 132 T + 11278 T^{2} + 722040 T^{3} + 38602473 T^{4} + 1778529960 T^{5} + 72524765390 T^{6} + 2637318955548 T^{7} + 86278260959732 T^{8} + 2534463516281628 T^{9} + 66978143857738190 T^{10} + 1578451886268782760 T^{11} + 32923703244758191593 T^{12} +$$$$59\!\cdots\!40$$$$T^{13} +$$$$88\!\cdots\!58$$$$T^{14} +$$$$99\!\cdots\!72$$$$T^{15} +$$$$72\!\cdots\!81$$$$T^{16}$$
$37$ $$1 - 96 T + 1110 T^{2} + 46464 T^{3} + 7652401 T^{4} - 360085920 T^{5} - 3532027338 T^{6} - 170035213440 T^{7} + 29111429055396 T^{8} - 232778207199360 T^{9} - 6619587887813418 T^{10} - 923881954453061280 T^{11} + 26878901285664514321 T^{12} +$$$$22\!\cdots\!36$$$$T^{13} +$$$$73\!\cdots\!10$$$$T^{14} -$$$$86\!\cdots\!44$$$$T^{15} +$$$$12\!\cdots\!41$$$$T^{16}$$
$41$ $$1 - 5456 T^{2} + 19963884 T^{4} - 49821420976 T^{6} + 96897341176550 T^{8} - 140783428358562736 T^{10} +$$$$15\!\cdots\!64$$$$T^{12} -$$$$12\!\cdots\!36$$$$T^{14} +$$$$63\!\cdots\!41$$$$T^{16}$$
$43$ $$( 1 - 56 T + 5082 T^{2} - 180688 T^{3} + 11078435 T^{4} - 334092112 T^{5} + 17374346682 T^{6} - 353996330744 T^{7} + 11688200277601 T^{8} )^{2}$$
$47$ $$1 - 24 T + 3580 T^{2} - 81312 T^{3} + 3430986 T^{4} - 200307384 T^{5} + 5424417392 T^{6} - 942687331128 T^{7} + 25778360177363 T^{8} - 2082396314461752 T^{9} + 26469426483811952 T^{10} - 2159156424124689336 T^{11} + 81696191178488726346 T^{12} -$$$$42\!\cdots\!88$$$$T^{13} +$$$$41\!\cdots\!80$$$$T^{14} -$$$$61\!\cdots\!56$$$$T^{15} +$$$$56\!\cdots\!21$$$$T^{16}$$
$53$ $$1 + 32 T - 3572 T^{2} - 570944 T^{3} - 366070 T^{4} + 2018388320 T^{5} + 123180800432 T^{6} - 3809526158624 T^{7} - 445561676340941 T^{8} - 10700958979574816 T^{9} + 971955765373487792 T^{10} + 44736287623035613280 T^{11} - 22791404868886921270 T^{12} -$$$$99\!\cdots\!56$$$$T^{13} -$$$$17\!\cdots\!52$$$$T^{14} +$$$$44\!\cdots\!08$$$$T^{15} +$$$$38\!\cdots\!21$$$$T^{16}$$
$59$ $$1 - 132 T + 15632 T^{2} - 1296768 T^{3} + 88851370 T^{4} - 4629146772 T^{5} + 218386477856 T^{6} - 7743785317668 T^{7} + 393482793133843 T^{8} - 26956116690802308 T^{9} + 2646267789699658016 T^{10} -$$$$19\!\cdots\!52$$$$T^{11} +$$$$13\!\cdots\!70$$$$T^{12} -$$$$66\!\cdots\!68$$$$T^{13} +$$$$27\!\cdots\!92$$$$T^{14} -$$$$81\!\cdots\!52$$$$T^{15} +$$$$21\!\cdots\!41$$$$T^{16}$$
$61$ $$1 - 96 T + 11380 T^{2} - 797568 T^{3} + 60840138 T^{4} - 3862327392 T^{5} + 197844384848 T^{6} - 13588740778464 T^{7} + 649379857320947 T^{8} - 50563704436664544 T^{9} + 2739321895348217168 T^{10} -$$$$19\!\cdots\!12$$$$T^{11} +$$$$11\!\cdots\!78$$$$T^{12} -$$$$56\!\cdots\!68$$$$T^{13} +$$$$30\!\cdots\!80$$$$T^{14} -$$$$94\!\cdots\!36$$$$T^{15} +$$$$36\!\cdots\!61$$$$T^{16}$$
$67$ $$1 - 120 T - 4522 T^{2} + 749040 T^{3} + 46360561 T^{4} - 4026690720 T^{5} - 240439443082 T^{6} + 2848051223400 T^{7} + 1781423144538724 T^{8} + 12784901941842600 T^{9} - 4845124310717994922 T^{10} -$$$$36\!\cdots\!80$$$$T^{11} +$$$$18\!\cdots\!01$$$$T^{12} +$$$$13\!\cdots\!60$$$$T^{13} -$$$$37\!\cdots\!42$$$$T^{14} -$$$$44\!\cdots\!80$$$$T^{15} +$$$$16\!\cdots\!81$$$$T^{16}$$
$71$ $$( 1 - 4 T + 13176 T^{2} - 338828 T^{3} + 79144886 T^{4} - 1708031948 T^{5} + 334824308856 T^{6} - 512401135684 T^{7} + 645753531245761 T^{8} )^{2}$$
$73$ $$1 + 24 T + 9630 T^{2} + 226512 T^{3} + 47274361 T^{4} - 2709573216 T^{5} - 18252715698 T^{6} - 34150744827912 T^{7} - 1287593588654412 T^{8} - 181989319187943048 T^{9} - 518345019296287218 T^{10} -$$$$41\!\cdots\!24$$$$T^{11} +$$$$38\!\cdots\!41$$$$T^{12} +$$$$97\!\cdots\!88$$$$T^{13} +$$$$22\!\cdots\!30$$$$T^{14} +$$$$29\!\cdots\!16$$$$T^{15} +$$$$65\!\cdots\!61$$$$T^{16}$$
$79$ $$1 - 12 T - 2418 T^{2} + 386472 T^{3} - 58917335 T^{4} - 324907032 T^{5} + 66125106126 T^{6} - 10581169939908 T^{7} + 1991860414252788 T^{8} - 66037081594965828 T^{9} + 2575578239741296206 T^{10} - 78980823689760123672 T^{11} -$$$$89\!\cdots\!35$$$$T^{12} +$$$$36\!\cdots\!72$$$$T^{13} -$$$$14\!\cdots\!38$$$$T^{14} -$$$$44\!\cdots\!72$$$$T^{15} +$$$$23\!\cdots\!21$$$$T^{16}$$
$83$ $$1 - 2384 T^{2} + 54393900 T^{4} - 170213772976 T^{6} + 3204332319622694 T^{8} - 8078059876516133296 T^{10} +$$$$12\!\cdots\!00$$$$T^{12} -$$$$25\!\cdots\!24$$$$T^{14} +$$$$50\!\cdots\!81$$$$T^{16}$$
$89$ $$1 - 492 T + 137248 T^{2} - 27827520 T^{3} + 4512312618 T^{4} - 615183153660 T^{5} + 72837035721440 T^{6} - 7634393359602348 T^{7} + 716618506839255347 T^{8} - 60472029801410198508 T^{9} +$$$$45\!\cdots\!40$$$$T^{10} -$$$$30\!\cdots\!60$$$$T^{11} +$$$$17\!\cdots\!58$$$$T^{12} -$$$$86\!\cdots\!20$$$$T^{13} +$$$$33\!\cdots\!08$$$$T^{14} -$$$$96\!\cdots\!72$$$$T^{15} +$$$$15\!\cdots\!61$$$$T^{16}$$
$97$ $$1 - 35880 T^{2} + 724155484 T^{4} - 9811041724440 T^{6} + 103976225413471686 T^{8} -$$$$86\!\cdots\!40$$$$T^{10} +$$$$56\!\cdots\!24$$$$T^{12} -$$$$24\!\cdots\!80$$$$T^{14} +$$$$61\!\cdots\!21$$$$T^{16}$$