# Properties

 Label 76.4.e.a Level $76$ Weight $4$ Character orbit 76.e Analytic conductor $4.484$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 76.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.48414516044$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - 2 x^{9} + 90 x^{8} - 212 x^{7} + 7012 x^{6} - 14448 x^{5} + 100896 x^{4} - 25920 x^{3} + 783360 x^{2} - 774144 x + 1016064$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{3} + ( \beta_{3} + \beta_{7} + \beta_{8} ) q^{5} + ( -2 - \beta_{2} + \beta_{5} ) q^{7} + ( -10 - 2 \beta_{1} - 10 \beta_{3} - \beta_{4} - \beta_{6} + \beta_{9} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{3} + ( \beta_{3} + \beta_{7} + \beta_{8} ) q^{5} + ( -2 - \beta_{2} + \beta_{5} ) q^{7} + ( -10 - 2 \beta_{1} - 10 \beta_{3} - \beta_{4} - \beta_{6} + \beta_{9} ) q^{9} + ( -6 - \beta_{4} - \beta_{5} + \beta_{7} ) q^{11} + ( -11 - \beta_{1} - 11 \beta_{3} - \beta_{4} + \beta_{8} + \beta_{9} ) q^{13} + ( 3 + 3 \beta_{3} + \beta_{6} + 3 \beta_{8} ) q^{15} + ( -\beta_{1} - \beta_{2} - 5 \beta_{3} - \beta_{5} + \beta_{6} + 3 \beta_{7} + 3 \beta_{8} ) q^{17} + ( 13 + 2 \beta_{1} - 5 \beta_{2} + 16 \beta_{3} + \beta_{4} + \beta_{5} + 3 \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{19} + ( -9 \beta_{1} - 9 \beta_{2} - 26 \beta_{3} + 6 \beta_{5} - 6 \beta_{6} - 4 \beta_{7} - 4 \beta_{8} + 3 \beta_{9} ) q^{21} + ( 39 - 4 \beta_{1} + 39 \beta_{3} + 2 \beta_{4} - \beta_{6} + \beta_{8} - 2 \beta_{9} ) q^{23} + ( -20 - 5 \beta_{1} - 20 \beta_{3} + 2 \beta_{4} + 7 \beta_{6} - 7 \beta_{8} - 2 \beta_{9} ) q^{25} + ( -19 + 15 \beta_{2} - \beta_{4} - 7 \beta_{5} ) q^{27} + ( 63 + 22 \beta_{1} + 63 \beta_{3} + 2 \beta_{4} + 3 \beta_{8} - 2 \beta_{9} ) q^{29} + ( 26 - 9 \beta_{2} + 2 \beta_{4} - \beta_{5} - 14 \beta_{7} ) q^{31} + ( 35 \beta_{1} + 35 \beta_{2} - 28 \beta_{3} - 6 \beta_{5} + 6 \beta_{6} + 3 \beta_{7} + 3 \beta_{8} + \beta_{9} ) q^{33} + ( 18 \beta_{1} + 18 \beta_{2} + 16 \beta_{3} + \beta_{5} - \beta_{6} - 20 \beta_{7} - 20 \beta_{8} - 3 \beta_{9} ) q^{35} + ( -72 - 5 \beta_{2} - \beta_{4} - 6 \beta_{5} + 16 \beta_{7} ) q^{37} + ( -17 + 33 \beta_{2} + 2 \beta_{4} - 7 \beta_{7} ) q^{39} + ( -30 \beta_{1} - 30 \beta_{2} + 19 \beta_{3} + \beta_{5} - \beta_{6} - 18 \beta_{7} - 18 \beta_{8} + \beta_{9} ) q^{41} + ( 35 \beta_{1} + 35 \beta_{2} + 89 \beta_{3} + 6 \beta_{5} - 6 \beta_{6} - \beta_{7} - \beta_{8} ) q^{43} + ( -26 - 10 \beta_{2} + 2 \beta_{4} + 8 \beta_{5} + 14 \beta_{7} ) q^{45} + ( -73 - 30 \beta_{1} - 73 \beta_{3} + 4 \beta_{4} - \beta_{6} + 3 \beta_{8} - 4 \beta_{9} ) q^{47} + ( 189 - 56 \beta_{2} + \beta_{4} - \beta_{5} + 12 \beta_{7} ) q^{49} + ( -43 + \beta_{1} - 43 \beta_{3} + \beta_{4} + 7 \beta_{6} + 13 \beta_{8} - \beta_{9} ) q^{51} + ( 165 - 5 \beta_{1} + 165 \beta_{3} + \beta_{4} + 2 \beta_{6} + 29 \beta_{8} - \beta_{9} ) q^{53} + ( -15 \beta_{1} - 15 \beta_{2} - 110 \beta_{3} - \beta_{5} + \beta_{6} + 16 \beta_{7} + 16 \beta_{8} ) q^{55} + ( 30 + 5 \beta_{1} - 54 \beta_{2} - 191 \beta_{3} - 4 \beta_{4} + 8 \beta_{5} - 7 \beta_{6} + 12 \beta_{7} + \beta_{8} + 10 \beta_{9} ) q^{57} + ( -13 \beta_{1} - 13 \beta_{2} + 87 \beta_{3} - 17 \beta_{5} + 17 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 15 \beta_{9} ) q^{59} + ( 123 - 60 \beta_{1} + 123 \beta_{3} - 12 \beta_{4} - 8 \beta_{6} - \beta_{8} + 12 \beta_{9} ) q^{61} + ( -280 - 124 \beta_{1} - 280 \beta_{3} - 12 \beta_{4} - 16 \beta_{6} - 24 \beta_{8} + 12 \beta_{9} ) q^{63} + ( -187 + 13 \beta_{2} + \beta_{4} + 16 \beta_{5} + \beta_{7} ) q^{65} + ( 127 + 73 \beta_{1} + 127 \beta_{3} - 9 \beta_{4} - \beta_{6} + 22 \beta_{8} + 9 \beta_{9} ) q^{67} + ( -151 - 64 \beta_{2} - 12 \beta_{4} - 8 \beta_{5} + 9 \beta_{7} ) q^{69} + ( 101 \beta_{1} + 101 \beta_{2} - 139 \beta_{3} + 16 \beta_{5} - 16 \beta_{6} + 9 \beta_{7} + 9 \beta_{8} - 2 \beta_{9} ) q^{71} + ( 8 \beta_{1} + 8 \beta_{2} - 301 \beta_{3} + 7 \beta_{5} - 7 \beta_{6} + 4 \beta_{7} + 4 \beta_{8} + 13 \beta_{9} ) q^{73} + ( -326 - 92 \beta_{2} + 3 \beta_{4} + 23 \beta_{5} + \beta_{7} ) q^{75} + ( -250 + 121 \beta_{2} - 9 \beta_{4} - 4 \beta_{7} ) q^{77} + ( -67 \beta_{1} - 67 \beta_{2} + 425 \beta_{3} + 19 \beta_{7} + 19 \beta_{8} - 2 \beta_{9} ) q^{79} + ( 70 \beta_{1} + 70 \beta_{2} + 205 \beta_{3} - 23 \beta_{5} + 23 \beta_{6} + 24 \beta_{7} + 24 \beta_{8} + \beta_{9} ) q^{81} + ( -70 + 38 \beta_{2} - 12 \beta_{4} - 40 \beta_{5} - 27 \beta_{7} ) q^{83} + ( -441 - 33 \beta_{1} - 441 \beta_{3} + 3 \beta_{4} + 24 \beta_{6} + 9 \beta_{8} - 3 \beta_{9} ) q^{85} + ( 805 - 122 \beta_{2} + 16 \beta_{4} + 25 \beta_{5} - \beta_{7} ) q^{87} + ( 85 - 23 \beta_{1} + 85 \beta_{3} + 18 \beta_{4} - 15 \beta_{6} - 7 \beta_{8} - 18 \beta_{9} ) q^{89} + ( 268 - 50 \beta_{1} + 268 \beta_{3} - 8 \beta_{4} - 18 \beta_{6} - 56 \beta_{8} + 8 \beta_{9} ) q^{91} + ( -51 \beta_{1} - 51 \beta_{2} - 330 \beta_{3} + 18 \beta_{5} - 18 \beta_{6} - 30 \beta_{7} - 30 \beta_{8} + \beta_{9} ) q^{93} + ( 236 + 11 \beta_{1} - 10 \beta_{2} - 353 \beta_{3} - 41 \beta_{5} + 16 \beta_{6} - 25 \beta_{7} - 23 \beta_{8} - 10 \beta_{9} ) q^{95} + ( 28 \beta_{1} + 28 \beta_{2} - 197 \beta_{3} - 4 \beta_{5} + 4 \beta_{6} - 22 \beta_{7} - 22 \beta_{8} + 16 \beta_{9} ) q^{97} + ( 1056 + 44 \beta_{1} + 1056 \beta_{3} + 23 \beta_{4} + 41 \beta_{6} + 10 \beta_{8} - 23 \beta_{9} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q + 7q^{3} - 4q^{5} - 20q^{7} - 50q^{9} + O(q^{10})$$ $$10q + 7q^{3} - 4q^{5} - 20q^{7} - 50q^{9} - 50q^{11} - 56q^{13} + 10q^{15} + 32q^{17} + 77q^{19} + 126q^{21} + 184q^{23} - 121q^{25} - 218q^{27} + 352q^{29} + 264q^{31} + 83q^{33} - 132q^{35} - 640q^{37} - 324q^{39} - 57q^{41} - 528q^{43} - 232q^{45} - 434q^{47} + 2138q^{49} - 242q^{51} + 780q^{53} + 598q^{55} + 1482q^{57} - 343q^{59} + 536q^{61} - 1568q^{63} - 1988q^{65} + 779q^{67} - 1156q^{69} + 474q^{71} + 1453q^{73} - 2994q^{75} - 2956q^{77} - 1968q^{79} - 1097q^{81} - 698q^{83} - 2334q^{85} + 8372q^{87} + 380q^{89} + 1348q^{91} + 1684q^{93} + 4312q^{95} + 883q^{97} + 5230q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 2 x^{9} + 90 x^{8} - 212 x^{7} + 7012 x^{6} - 14448 x^{5} + 100896 x^{4} - 25920 x^{3} + 783360 x^{2} - 774144 x + 1016064$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$1165047 \nu^{9} - 9250992 \nu^{8} + 143004918 \nu^{7} - 836736016 \nu^{6} + 11390283900 \nu^{5} - 63805633656 \nu^{4} + 380680235696 \nu^{3} - 545271970464 \nu^{2} + 550952079552 \nu - 3174016519296$$$$)/ 2610376990464$$ $$\beta_{3}$$ $$=$$ $$($$$$393603239 \nu^{9} - 640410556 \nu^{8} + 34258666518 \nu^{7} - 65425267000 \nu^{6} + 2654517173852 \nu^{5} - 4251603825672 \nu^{4} + 31673482561488 \nu^{3} + 37763513742816 \nu^{2} + 239628765024576 \nu - 235285623828864$$$$)/ 328907500798464$$ $$\beta_{4}$$ $$=$$ $$($$$$-24604755 \nu^{9} + 228981360 \nu^{8} - 3539670190 \nu^{7} + 21170339856 \nu^{6} - 281933299500 \nu^{5} + 1579322603480 \nu^{4} - 10227111378288 \nu^{3} + 13496619321120 \nu^{2} - 13637213876160 \nu - 127823859374976$$$$)/ 7831130971392$$ $$\beta_{5}$$ $$=$$ $$($$$$45367449 \nu^{9} - 343433424 \nu^{8} + 5308908346 \nu^{7} - 30833262864 \nu^{6} + 422852403300 \nu^{5} - 2368717564232 \nu^{4} + 11772333234960 \nu^{3} - 20242652877408 \nu^{2} + 20453520999744 \nu - 150545568650112$$$$)/ 7831130971392$$ $$\beta_{6}$$ $$=$$ $$($$$$-430238075 \nu^{9} + 257384052 \nu^{8} - 36268548014 \nu^{7} + 37431389032 \nu^{6} - 2896690749132 \nu^{5} + 1768422847240 \nu^{4} - 35310473511696 \nu^{3} - 46254702763872 \nu^{2} - 803825764282944 \nu - 109947804333696$$$$)/ 54817916799744$$ $$\beta_{7}$$ $$=$$ $$($$$$65962847 \nu^{9} - 378416976 \nu^{8} + 5849695754 \nu^{7} - 35865870088 \nu^{6} + 465925901700 \nu^{5} - 2610004952968 \nu^{4} + 6520126725504 \nu^{3} - 22304653399392 \nu^{2} + 22537001422656 \nu - 179178099194880$$$$)/ 7831130971392$$ $$\beta_{8}$$ $$=$$ $$($$$$1518356977 \nu^{9} + 423607099 \nu^{8} + 133515913632 \nu^{7} - 29537922338 \nu^{6} + 10208000900824 \nu^{5} + 446702419380 \nu^{4} + 126687690620448 \nu^{3} + 178212334254336 \nu^{2} + 1256670693632160 \nu + 410923373422464$$$$)/ 82226875199616$$ $$\beta_{9}$$ $$=$$ $$($$$$403452304 \nu^{9} - 295261329 \nu^{8} + 33030273168 \nu^{7} - 34821843230 \nu^{6} + 2517611367720 \nu^{5} - 1790044289292 \nu^{4} + 18081022303128 \nu^{3} + 66801758258784 \nu^{2} + 122693044716576 \nu - 116960641689024$$$$)/ 13704479199936$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{9} + \beta_{6} - \beta_{5} + 36 \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$-4 \beta_{5} + 2 \beta_{4} + 66 \beta_{2} + 36$$ $$\nu^{4}$$ $$=$$ $$82 \beta_{9} - 24 \beta_{8} - 82 \beta_{6} - 82 \beta_{4} - 2400 \beta_{3} + 36 \beta_{1} - 2400$$ $$\nu^{5}$$ $$=$$ $$-200 \beta_{9} - 48 \beta_{8} - 48 \beta_{7} - 316 \beta_{6} + 316 \beta_{5} + 4200 \beta_{3} - 4836 \beta_{2} - 4836 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-2160 \beta_{7} + 6148 \beta_{5} + 6268 \beta_{4} + 5088 \beta_{2} + 177264$$ $$\nu^{7}$$ $$=$$ $$17864 \beta_{9} + 3840 \beta_{8} + 21664 \beta_{6} - 17864 \beta_{4} - 403536 \beta_{3} + 361944 \beta_{1} - 403536$$ $$\nu^{8}$$ $$=$$ $$-476728 \beta_{9} + 165792 \beta_{8} + 165792 \beta_{7} + 452440 \beta_{6} - 452440 \beta_{5} + 13364544 \beta_{3} - 548016 \beta_{2} - 548016 \beta_{1}$$ $$\nu^{9}$$ $$=$$ $$234432 \beta_{7} - 1427536 \beta_{5} + 1550048 \beta_{4} + 27257712 \beta_{2} + 36364896$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$39$$ $$\chi(n)$$ $$-1 - \beta_{3}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
45.1
 −4.43560 + 7.68268i −1.41615 + 2.45284i 0.576829 − 0.999096i 2.06508 − 3.57683i 4.20983 − 7.29165i −4.43560 − 7.68268i −1.41615 − 2.45284i 0.576829 + 0.999096i 2.06508 + 3.57683i 4.20983 + 7.29165i
0 −3.93560 6.81665i 0 0.0670670 + 0.116164i 0 −9.41300 0 −17.4778 + 30.2725i 0
45.2 0 −0.916150 1.58682i 0 2.48769 + 4.30880i 0 19.2102 0 11.8213 20.4752i 0
45.3 0 1.07683 + 1.86512i 0 −10.6285 18.4091i 0 −18.8370 0 11.1809 19.3659i 0
45.4 0 2.56508 + 4.44285i 0 7.98141 + 13.8242i 0 −31.8733 0 0.340695 0.590101i 0
45.5 0 4.70983 + 8.15767i 0 −1.90769 3.30421i 0 30.9131 0 −30.8651 + 53.4599i 0
49.1 0 −3.93560 + 6.81665i 0 0.0670670 0.116164i 0 −9.41300 0 −17.4778 30.2725i 0
49.2 0 −0.916150 + 1.58682i 0 2.48769 4.30880i 0 19.2102 0 11.8213 + 20.4752i 0
49.3 0 1.07683 1.86512i 0 −10.6285 + 18.4091i 0 −18.8370 0 11.1809 + 19.3659i 0
49.4 0 2.56508 4.44285i 0 7.98141 13.8242i 0 −31.8733 0 0.340695 + 0.590101i 0
49.5 0 4.70983 8.15767i 0 −1.90769 + 3.30421i 0 30.9131 0 −30.8651 53.4599i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 49.5 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 76.4.e.a 10
3.b odd 2 1 684.4.k.c 10
4.b odd 2 1 304.4.i.f 10
19.c even 3 1 inner 76.4.e.a 10
19.c even 3 1 1444.4.a.f 5
19.d odd 6 1 1444.4.a.g 5
57.h odd 6 1 684.4.k.c 10
76.g odd 6 1 304.4.i.f 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.4.e.a 10 1.a even 1 1 trivial
76.4.e.a 10 19.c even 3 1 inner
304.4.i.f 10 4.b odd 2 1
304.4.i.f 10 76.g odd 6 1
684.4.k.c 10 3.b odd 2 1
684.4.k.c 10 57.h odd 6 1
1444.4.a.f 5 19.c even 3 1
1444.4.a.g 5 19.d odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(76, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10}$$
$3$ $$2253001 - 208639 T + 661749 T^{2} - 144644 T^{3} + 182129 T^{4} - 28659 T^{5} + 7481 T^{6} - 380 T^{7} + 117 T^{8} - 7 T^{9} + T^{10}$$
$5$ $$746496 - 5531328 T + 41281092 T^{2} + 1558764 T^{3} + 2457150 T^{4} - 176910 T^{5} + 125455 T^{6} - 2144 T^{7} + 381 T^{8} + 4 T^{9} + T^{10}$$
$7$ $$( 3356160 + 356736 T - 12664 T^{2} - 1342 T^{3} + 10 T^{4} + T^{5} )^{2}$$
$11$ $$( -14618352 + 2107752 T - 24079 T^{2} - 2647 T^{3} + 25 T^{4} + T^{5} )^{2}$$
$13$ $$2828994831383616 + 39489331235424 T + 6855206311648 T^{2} + 116566238648 T^{3} + 12496639720 T^{4} + 197952374 T^{5} + 9592717 T^{6} + 129356 T^{7} + 5059 T^{8} + 56 T^{9} + T^{10}$$
$17$ $$261619974144 + 219784347648 T + 136392035328 T^{2} + 35642702592 T^{3} + 6827509476 T^{4} + 422771922 T^{5} + 20250105 T^{6} + 341580 T^{7} + 5803 T^{8} - 32 T^{9} + T^{10}$$
$19$ $$15181127029874798299 - 170425248768094397 T + 1177164721497792 T^{2} + 16973730451871 T^{3} + 117097197809 T^{4} - 1249353132 T^{5} + 17072051 T^{6} + 360791 T^{7} + 3648 T^{8} - 77 T^{9} + T^{10}$$
$23$ $$24620264819571600 - 6393426944838120 T + 1467932535485124 T^{2} - 49409306373432 T^{3} + 1461974009878 T^{4} - 16920158740 T^{5} + 269160887 T^{6} - 2138776 T^{7} + 35555 T^{8} - 184 T^{9} + T^{10}$$
$29$ $$75\!\cdots\!36$$$$+ 5556038395520543616 T + 235215650132949636 T^{2} - 1473264345256668 T^{3} + 40031220384654 T^{4} - 175731472410 T^{5} + 2689032727 T^{6} - 13821892 T^{7} + 124773 T^{8} - 352 T^{9} + T^{10}$$
$31$ $$( -11017558272 + 938899744 T + 4041756 T^{2} - 75578 T^{3} - 132 T^{4} + T^{5} )^{2}$$
$37$ $$( -309775727920 - 7573884632 T - 56949700 T^{2} - 109142 T^{3} + 320 T^{4} + T^{5} )^{2}$$
$41$ $$23\!\cdots\!25$$$$+$$$$15\!\cdots\!85$$$$T +$$$$13\!\cdots\!51$$$$T^{2} + 382863699049589886 T^{3} + 2330479330251349 T^{4} + 4650533764791 T^{5} + 29853515757 T^{6} + 32192878 T^{7} + 199551 T^{8} + 57 T^{9} + T^{10}$$
$43$ $$61\!\cdots\!00$$$$- 12814266095586520320 T + 33497102894679593536 T^{2} + 82441705504023360 T^{3} + 1406923675848680 T^{4} + 3025276339056 T^{5} + 25267469361 T^{6} + 57871440 T^{7} + 330869 T^{8} + 528 T^{9} + T^{10}$$
$47$ $$13\!\cdots\!04$$$$+$$$$29\!\cdots\!72$$$$T + 45147375024025018020 T^{2} + 375120905469682092 T^{3} + 2407804242437278 T^{4} + 9613227727418 T^{5} + 33982804439 T^{6} + 79017230 T^{7} + 246275 T^{8} + 434 T^{9} + T^{10}$$
$53$ $$98\!\cdots\!56$$$$-$$$$51\!\cdots\!72$$$$T +$$$$26\!\cdots\!20$$$$T^{2} - 1927998176522876280 T^{3} + 12642108045170488 T^{4} - 34524919587822 T^{5} + 114037423509 T^{6} - 176213152 T^{7} + 684411 T^{8} - 780 T^{9} + T^{10}$$
$59$ $$36\!\cdots\!61$$$$-$$$$21\!\cdots\!65$$$$T +$$$$12\!\cdots\!89$$$$T^{2} - 27664855232794074756 T^{3} + 80440966944929889 T^{4} - 79218489565413 T^{5} + 312614679417 T^{6} - 166914012 T^{7} + 763117 T^{8} + 343 T^{9} + T^{10}$$
$61$ $$23\!\cdots\!04$$$$+$$$$11\!\cdots\!32$$$$T +$$$$73\!\cdots\!72$$$$T^{2} - 6942321968062301052 T^{3} + 57787826625923790 T^{4} - 146751114868794 T^{5} + 372880212807 T^{6} - 282536700 T^{7} + 742365 T^{8} - 536 T^{9} + T^{10}$$
$67$ $$31\!\cdots\!21$$$$+$$$$49\!\cdots\!17$$$$T +$$$$11\!\cdots\!45$$$$T^{2} - 67881078462815856468 T^{3} + 393064551442535649 T^{4} - 632731511514231 T^{5} + 1137610219873 T^{6} - 795841220 T^{7} + 1328373 T^{8} - 779 T^{9} + T^{10}$$
$71$ $$10\!\cdots\!56$$$$+$$$$46\!\cdots\!84$$$$T +$$$$19\!\cdots\!96$$$$T^{2} +$$$$32\!\cdots\!76$$$$T^{3} + 639879596418496576 T^{4} + 342561177873978 T^{5} + 1151600768985 T^{6} + 276247382 T^{7} + 1464009 T^{8} - 474 T^{9} + T^{10}$$
$73$ $$92\!\cdots\!25$$$$+$$$$68\!\cdots\!75$$$$T +$$$$77\!\cdots\!31$$$$T^{2} + 616844121036204198 T^{3} + 12765540669545677 T^{4} - 31517653778215 T^{5} + 329913303461 T^{6} - 792035338 T^{7} + 1686011 T^{8} - 1453 T^{9} + T^{10}$$
$79$ $$78\!\cdots\!36$$$$-$$$$13\!\cdots\!48$$$$T +$$$$23\!\cdots\!96$$$$T^{2} - 6435705802760487616 T^{3} + 44545883831867104 T^{4} + 210998172001892 T^{5} + 1053858319585 T^{6} + 1971545432 T^{7} + 2855833 T^{8} + 1968 T^{9} + T^{10}$$
$83$ $$( 138587802934320 + 453068281152 T - 631751235 T^{2} - 2138943 T^{3} + 349 T^{4} + T^{5} )^{2}$$
$89$ $$12\!\cdots\!16$$$$+$$$$13\!\cdots\!52$$$$T +$$$$12\!\cdots\!72$$$$T^{2} +$$$$28\!\cdots\!16$$$$T^{3} + 802020242543319396 T^{4} - 375618620841138 T^{5} + 2064213670593 T^{6} - 357053016 T^{7} + 1650379 T^{8} - 380 T^{9} + T^{10}$$
$97$ $$74\!\cdots\!25$$$$-$$$$18\!\cdots\!55$$$$T +$$$$32\!\cdots\!51$$$$T^{2} - 28611103832130803678 T^{3} + 185278527681553853 T^{4} - 394058276597457 T^{5} + 830424626021 T^{6} - 423852062 T^{7} + 1368243 T^{8} - 883 T^{9} + T^{10}$$