# Properties

 Label 126.9.n.d Level $126$ Weight $9$ Character orbit 126.n Analytic conductor $51.330$ Analytic rank $0$ Dimension $20$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$126 = 2 \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 126.n (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$51.3297048677$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$10$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ Defining polynomial: $$x^{20} - 8 x^{19} - 26382 x^{18} + 177344 x^{17} + 298653216 x^{16} - 1823810808 x^{15} - 1891249463672 x^{14} + 11806020599312 x^{13} + 7316478649605270 x^{12} - 52687155135219848 x^{11} - 17677778016929268548 x^{10} + 159557587043375632176 x^{9} + 25942214297922759947412 x^{8} - 305008424965479773875688 x^{7} - 20749002299789629192234200 x^{6} + 325395581603349554951740400 x^{5} + 6083917772973420120813872017 x^{4} - 144252792084909643404755532432 x^{3} + 1061914206438596097569355774114 x^{2} - 3448453709078208081753932409840 x + 4271031984253419377206782621732$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{40}\cdot 3^{18}\cdot 7^{12}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{19}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( -128 + 128 \beta_{1} ) q^{4} + ( -5 \beta_{2} + 10 \beta_{3} + \beta_{11} - \beta_{12} ) q^{5} + ( -31 + 480 \beta_{1} - \beta_{8} ) q^{7} + ( -128 \beta_{2} + 128 \beta_{3} ) q^{8} +O(q^{10})$$ $$q + \beta_{2} q^{2} + ( -128 + 128 \beta_{1} ) q^{4} + ( -5 \beta_{2} + 10 \beta_{3} + \beta_{11} - \beta_{12} ) q^{5} + ( -31 + 480 \beta_{1} - \beta_{8} ) q^{7} + ( -128 \beta_{2} + 128 \beta_{3} ) q^{8} + ( -590 - 588 \beta_{1} - 2 \beta_{10} ) q^{10} + ( -6 \beta_{2} - 157 \beta_{3} - \beta_{6} + 6 \beta_{11} - 12 \beta_{12} + \beta_{16} + 2 \beta_{17} - \beta_{18} ) q^{11} + ( 542 - 1084 \beta_{1} + 3 \beta_{4} + 3 \beta_{5} - 3 \beta_{9} + 3 \beta_{10} + 2 \beta_{13} - 2 \beta_{14} ) q^{13} + ( -33 \beta_{2} + 479 \beta_{3} - \beta_{6} - 3 \beta_{11} - 2 \beta_{12} + 2 \beta_{15} + 2 \beta_{16} - \beta_{18} + 2 \beta_{19} ) q^{14} -16384 \beta_{1} q^{16} + ( -1423 \beta_{2} + 712 \beta_{3} - \beta_{6} + \beta_{11} + 3 \beta_{15} - 5 \beta_{16} + \beta_{18} ) q^{17} + ( -16747 + 8361 \beta_{1} + 11 \beta_{4} + 18 \beta_{5} + 29 \beta_{7} - 18 \beta_{8} - 16 \beta_{9} + 9 \beta_{13} ) q^{19} + ( -512 \beta_{2} - 640 \beta_{3} + 128 \beta_{12} ) q^{20} + ( 21020 - 12 \beta_{4} + 12 \beta_{5} + 40 \beta_{7} - 40 \beta_{8} - 14 \beta_{9} - 14 \beta_{10} - 6 \beta_{13} - 6 \beta_{14} ) q^{22} + ( 3377 \beta_{2} + 23 \beta_{6} - 40 \beta_{11} - 17 \beta_{12} - \beta_{15} - \beta_{16} + \beta_{17} + \beta_{19} ) q^{23} + ( 25854 - 25814 \beta_{1} + 57 \beta_{4} + 34 \beta_{5} + 57 \beta_{7} + 23 \beta_{8} + 70 \beta_{9} - 35 \beta_{10} - 10 \beta_{13} + 5 \beta_{14} ) q^{25} + ( 493 \beta_{2} - 954 \beta_{3} + 16 \beta_{6} + 168 \beta_{11} - 152 \beta_{12} + 14 \beta_{15} - 4 \beta_{16} - 4 \beta_{17} + 32 \beta_{18} + 14 \beta_{19} ) q^{26} + ( -57472 - 3968 \beta_{1} - 128 \beta_{4} ) q^{28} + ( 1964 \beta_{2} - 2423 \beta_{3} - 877 \beta_{11} + 418 \beta_{12} - 26 \beta_{15} + 18 \beta_{16} + 9 \beta_{17} - 41 \beta_{18} - 13 \beta_{19} ) q^{29} + ( 11582 + 11432 \beta_{1} - 85 \beta_{4} + 209 \beta_{5} + 85 \beta_{7} + 124 \beta_{8} + 154 \beta_{10} + 4 \beta_{14} ) q^{31} -16384 \beta_{3} q^{32} + ( 91272 - 182544 \beta_{1} + 164 \beta_{4} + 164 \beta_{5} + 144 \beta_{7} + 144 \beta_{8} - 16 \beta_{9} + 16 \beta_{10} - 34 \beta_{13} + 34 \beta_{14} ) q^{34} + ( -10133 \beta_{2} + 15833 \beta_{3} - 103 \beta_{6} + 1270 \beta_{11} - 408 \beta_{12} - 17 \beta_{15} + 59 \beta_{16} + 27 \beta_{17} - 16 \beta_{18} - 88 \beta_{19} ) q^{35} + ( -27 - 480340 \beta_{1} + 70 \beta_{4} + 479 \beta_{5} + 409 \beta_{7} + 479 \beta_{8} - 21 \beta_{9} + 42 \beta_{10} - 48 \beta_{13} + 96 \beta_{14} ) q^{37} + ( -16469 \beta_{2} + 8683 \beta_{3} - 83 \beta_{6} + 897 \beta_{11} + 50 \beta_{15} + 22 \beta_{16} + 83 \beta_{18} ) q^{38} + ( 150784 - 75264 \beta_{1} + 256 \beta_{9} ) q^{40} + ( -5228 \beta_{2} - 1766 \beta_{3} + 88 \beta_{6} - 44 \beta_{11} - 3418 \beta_{12} - 144 \beta_{17} + 44 \beta_{18} - 126 \beta_{19} ) q^{41} + ( 347939 - 587 \beta_{4} + 587 \beta_{5} + 1311 \beta_{7} - 1311 \beta_{8} - 154 \beta_{9} - 154 \beta_{10} - 105 \beta_{13} - 105 \beta_{14} ) q^{43} + ( 21504 \beta_{2} + 128 \beta_{6} + 640 \beta_{11} + 768 \beta_{12} + 128 \beta_{16} - 128 \beta_{17} ) q^{44} + ( -436066 + 435888 \beta_{1} + 564 \beta_{4} + 440 \beta_{5} + 564 \beta_{7} + 124 \beta_{8} - 168 \beta_{9} + 84 \beta_{10} + 188 \beta_{13} - 94 \beta_{14} ) q^{46} + ( 87679 \beta_{2} - 175388 \beta_{3} - 15 \beta_{6} - 3194 \beta_{11} + 3179 \beta_{12} - 345 \beta_{15} + 165 \beta_{16} + 165 \beta_{17} - 30 \beta_{18} - 345 \beta_{19} ) q^{47} + ( -1286866 - 32025 \beta_{1} - 343 \beta_{4} + 441 \beta_{5} + 1176 \beta_{7} - 532 \beta_{8} + 343 \beta_{9} - 1029 \beta_{10} + 343 \beta_{13} - 343 \beta_{14} ) q^{49} + ( 26251 \beta_{2} - 28707 \beta_{3} - 4872 \beta_{11} + 2416 \beta_{12} - 132 \beta_{15} + 216 \beta_{16} + 108 \beta_{17} - 40 \beta_{18} - 66 \beta_{19} ) q^{50} + ( 68736 + 69376 \beta_{1} - 384 \beta_{4} + 384 \beta_{7} - 384 \beta_{8} - 384 \beta_{10} + 256 \beta_{14} ) q^{52} + ( 1511 \beta_{2} - 139287 \beta_{3} + 990 \beta_{6} - 1511 \beta_{11} + 3022 \beta_{12} - 184 \beta_{15} - 198 \beta_{16} - 396 \beta_{17} + 990 \beta_{18} - 368 \beta_{19} ) q^{53} + ( 2250147 - 4500294 \beta_{1} + 2483 \beta_{4} + 2483 \beta_{5} + 932 \beta_{7} + 932 \beta_{8} - 1012 \beta_{9} + 1012 \beta_{10} - 302 \beta_{13} + 302 \beta_{14} ) q^{55} + ( -57600 \beta_{2} - 3584 \beta_{3} + 128 \beta_{6} + 512 \beta_{11} - 384 \beta_{12} - 256 \beta_{16} - 256 \beta_{17} - 256 \beta_{19} ) q^{56} + ( 1116 + 244188 \beta_{1} + 360 \beta_{4} - 4 \beta_{5} - 364 \beta_{7} - 4 \beta_{8} - 882 \beta_{9} + 1764 \beta_{10} + 234 \beta_{13} - 468 \beta_{14} ) q^{58} + ( 165329 \beta_{2} - 80397 \beta_{3} - 14 \beta_{6} + 4535 \beta_{11} + 645 \beta_{15} + 1420 \beta_{16} + 14 \beta_{18} ) q^{59} + ( -1026666 + 512450 \beta_{1} - 1020 \beta_{4} - 582 \beta_{5} - 1602 \beta_{7} + 582 \beta_{8} - 1256 \beta_{9} + 510 \beta_{13} ) q^{61} + ( 5999 \beta_{2} + 15281 \beta_{3} + 354 \beta_{6} - 177 \beta_{11} - 9105 \beta_{12} + 110 \beta_{17} + 177 \beta_{18} - 604 \beta_{19} ) q^{62} + 2097152 q^{64} + ( -556703 \beta_{2} - 577 \beta_{6} + 4908 \beta_{11} + 4331 \beta_{12} + 1059 \beta_{15} + 409 \beta_{16} - 409 \beta_{17} - 1059 \beta_{19} ) q^{65} + ( -7002834 + 7005371 \beta_{1} - 2968 \beta_{4} - 2577 \beta_{5} - 2968 \beta_{7} - 391 \beta_{8} + 4592 \beta_{9} - 2296 \beta_{10} - 482 \beta_{13} + 241 \beta_{14} ) q^{67} + ( 91008 \beta_{2} - 182272 \beta_{3} - 128 \beta_{6} - 128 \beta_{12} - 384 \beta_{15} + 640 \beta_{16} + 640 \beta_{17} - 256 \beta_{18} - 384 \beta_{19} ) q^{68} + ( -640220 - 1301124 \beta_{1} - 1228 \beta_{4} - 5096 \beta_{5} - 2156 \beta_{7} - 356 \beta_{8} + 2058 \beta_{9} - 2744 \beta_{10} - 980 \beta_{13} + 98 \beta_{14} ) q^{70} + ( 8546 \beta_{2} - 18442 \beta_{3} - 20883 \beta_{11} + 10987 \beta_{12} - 34 \beta_{15} + 2826 \beta_{16} + 1413 \beta_{17} + 1091 \beta_{18} - 17 \beta_{19} ) q^{71} + ( 7344178 + 7341924 \beta_{1} + 3421 \beta_{4} + 12 \beta_{5} - 3421 \beta_{7} + 3433 \beta_{8} + 1947 \beta_{10} - 307 \beta_{14} ) q^{73} + ( -560 \beta_{2} - 479173 \beta_{3} - 264 \beta_{6} + 560 \beta_{11} - 1120 \beta_{12} - 1010 \beta_{15} + 524 \beta_{16} + 1048 \beta_{17} - 264 \beta_{18} - 2020 \beta_{19} ) q^{74} + ( 1070208 - 2140416 \beta_{1} - 3712 \beta_{4} - 3712 \beta_{5} - 1408 \beta_{7} - 1408 \beta_{8} + 2048 \beta_{9} - 2048 \beta_{10} - 1152 \beta_{13} + 1152 \beta_{14} ) q^{76} + ( -363230 \beta_{2} - 905988 \beta_{3} + 1533 \beta_{6} + 13151 \beta_{11} - 29796 \beta_{12} - 613 \beta_{15} + 3549 \beta_{16} + 977 \beta_{17} + 674 \beta_{18} - 1547 \beta_{19} ) q^{77} + ( 3180 + 1278880 \beta_{1} - 1273 \beta_{4} - 5603 \beta_{5} - 4330 \beta_{7} - 5603 \beta_{8} - 3794 \beta_{9} + 7588 \beta_{10} - 614 \beta_{13} + 1228 \beta_{14} ) q^{79} + ( 147456 \beta_{2} - 81920 \beta_{3} - 16384 \beta_{11} ) q^{80} + ( 804028 - 405432 \beta_{1} + 3136 \beta_{4} - 5552 \beta_{5} - 2416 \beta_{7} + 5552 \beta_{8} - 6524 \beta_{9} + 312 \beta_{13} ) q^{82} + ( 396238 \beta_{2} + 389375 \beta_{3} + 2880 \beta_{6} - 1440 \beta_{11} + 8303 \beta_{12} - 4854 \beta_{17} + 1440 \beta_{18} - 5607 \beta_{19} ) q^{83} + ( 1466964 + 3120 \beta_{4} - 3120 \beta_{5} - 11348 \beta_{7} + 11348 \beta_{8} + 4718 \beta_{9} + 4718 \beta_{10} - 214 \beta_{13} - 214 \beta_{14} ) q^{85} + ( 349610 \beta_{2} + 2383 \beta_{6} + 2932 \beta_{11} + 5315 \beta_{12} + 754 \beta_{15} + 3462 \beta_{16} - 3462 \beta_{17} - 754 \beta_{19} ) q^{86} + ( -2691584 + 2692608 \beta_{1} - 3584 \beta_{4} - 5120 \beta_{5} - 3584 \beta_{7} + 1536 \beta_{8} + 3584 \beta_{9} - 1792 \beta_{10} + 1536 \beta_{13} - 768 \beta_{14} ) q^{88} + ( 1390952 \beta_{2} - 2773204 \beta_{3} + 4350 \beta_{6} - 7218 \beta_{11} + 11568 \beta_{12} - 3618 \beta_{15} - 490 \beta_{16} - 490 \beta_{17} + 8700 \beta_{18} - 3618 \beta_{19} ) q^{89} + ( -10325205 - 10460399 \beta_{1} + 3859 \beta_{4} + 6566 \beta_{5} - 6615 \beta_{7} + 848 \beta_{8} - 6174 \beta_{9} - 9604 \beta_{10} + 833 \beta_{13} + 1666 \beta_{14} ) q^{91} + ( -434432 \beta_{2} + 439552 \beta_{3} + 7296 \beta_{11} - 2176 \beta_{12} + 256 \beta_{15} + 256 \beta_{16} + 128 \beta_{17} + 2944 \beta_{18} + 128 \beta_{19} ) q^{92} + ( 11074702 + 11067024 \beta_{1} + 5100 \beta_{4} - 13200 \beta_{5} - 5100 \beta_{7} - 8100 \beta_{8} + 5788 \beta_{10} - 1890 \beta_{14} ) q^{94} + ( 22525 \beta_{2} - 2905744 \beta_{3} - 2401 \beta_{6} - 22525 \beta_{11} + 45050 \beta_{12} - 5431 \beta_{15} - 935 \beta_{16} - 1870 \beta_{17} - 2401 \beta_{18} - 10862 \beta_{19} ) q^{95} + ( -2327111 + 4654222 \beta_{1} - 7147 \beta_{4} - 7147 \beta_{5} - 11064 \beta_{7} - 11064 \beta_{8} + 13645 \beta_{9} - 13645 \beta_{10} + 6637 \beta_{13} - 6637 \beta_{14} ) q^{97} + ( -1250641 \beta_{2} - 68719 \beta_{3} + 2996 \beta_{6} - 25116 \beta_{11} + 65184 \beta_{12} + 966 \beta_{15} - 1484 \beta_{16} - 4900 \beta_{17} + 6132 \beta_{18} - 602 \beta_{19} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20 q - 1280 q^{4} + 4186 q^{7} + O(q^{10})$$ $$20 q - 1280 q^{4} + 4186 q^{7} - 17664 q^{10} - 163840 q^{16} - 250890 q^{19} + 420864 q^{22} + 258962 q^{25} - 1189888 q^{28} + 342762 q^{31} - 4806598 q^{37} + 2260992 q^{40} + 6968252 q^{43} - 4357632 q^{46} - 26046538 q^{49} + 2075904 q^{52} + 2455296 q^{58} - 15410424 q^{61} + 41943040 q^{64} - 70041074 q^{67} - 25804800 q^{70} + 220264098 q^{73} + 12860578 q^{79} + 12085248 q^{82} + 29161632 q^{85} - 26935296 q^{88} - 311022894 q^{91} + 332230656 q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 8 x^{19} - 26382 x^{18} + 177344 x^{17} + 298653216 x^{16} - 1823810808 x^{15} - 1891249463672 x^{14} + 11806020599312 x^{13} + 7316478649605270 x^{12} - 52687155135219848 x^{11} - 17677778016929268548 x^{10} + 159557587043375632176 x^{9} + 25942214297922759947412 x^{8} - 305008424965479773875688 x^{7} - 20749002299789629192234200 x^{6} + 325395581603349554951740400 x^{5} + 6083917772973420120813872017 x^{4} - 144252792084909643404755532432 x^{3} + 1061914206438596097569355774114 x^{2} - 3448453709078208081753932409840 x + 4271031984253419377206782621732$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-$$$$13\!\cdots\!18$$$$\nu^{19} +$$$$91\!\cdots\!47$$$$\nu^{18} +$$$$31\!\cdots\!43$$$$\nu^{17} -$$$$21\!\cdots\!78$$$$\nu^{16} -$$$$29\!\cdots\!68$$$$\nu^{15} +$$$$20\!\cdots\!53$$$$\nu^{14} +$$$$14\!\cdots\!39$$$$\nu^{13} -$$$$11\!\cdots\!30$$$$\nu^{12} -$$$$39\!\cdots\!00$$$$\nu^{11} +$$$$36\!\cdots\!85$$$$\nu^{10} +$$$$46\!\cdots\!85$$$$\nu^{9} -$$$$68\!\cdots\!46$$$$\nu^{8} +$$$$17\!\cdots\!68$$$$\nu^{7} +$$$$73\!\cdots\!11$$$$\nu^{6} -$$$$99\!\cdots\!75$$$$\nu^{5} -$$$$35\!\cdots\!82$$$$\nu^{4} +$$$$74\!\cdots\!26$$$$\nu^{3} +$$$$28\!\cdots\!88$$$$\nu^{2} -$$$$61\!\cdots\!16$$$$\nu +$$$$18\!\cdots\!92$$$$)/$$$$16\!\cdots\!00$$ $$\beta_{2}$$ $$=$$ $$($$$$-$$$$68\!\cdots\!09$$$$\nu^{19} +$$$$37\!\cdots\!01$$$$\nu^{18} +$$$$15\!\cdots\!57$$$$\nu^{17} -$$$$84\!\cdots\!93$$$$\nu^{16} -$$$$15\!\cdots\!79$$$$\nu^{15} +$$$$79\!\cdots\!15$$$$\nu^{14} +$$$$79\!\cdots\!41$$$$\nu^{13} -$$$$41\!\cdots\!25$$$$\nu^{12} -$$$$23\!\cdots\!29$$$$\nu^{11} +$$$$12\!\cdots\!69$$$$\nu^{10} +$$$$39\!\cdots\!23$$$$\nu^{9} -$$$$23\!\cdots\!51$$$$\nu^{8} -$$$$28\!\cdots\!09$$$$\nu^{7} +$$$$23\!\cdots\!65$$$$\nu^{6} -$$$$50\!\cdots\!25$$$$\nu^{5} -$$$$89\!\cdots\!51$$$$\nu^{4} +$$$$14\!\cdots\!54$$$$\nu^{3} -$$$$90\!\cdots\!54$$$$\nu^{2} -$$$$49\!\cdots\!80$$$$\nu +$$$$33\!\cdots\!64$$$$)/$$$$95\!\cdots\!00$$ $$\beta_{3}$$ $$=$$ $$($$$$-$$$$96\!\cdots\!17$$$$\nu^{19} +$$$$23\!\cdots\!88$$$$\nu^{18} +$$$$23\!\cdots\!66$$$$\nu^{17} -$$$$51\!\cdots\!09$$$$\nu^{16} -$$$$24\!\cdots\!77$$$$\nu^{15} +$$$$48\!\cdots\!70$$$$\nu^{14} +$$$$14\!\cdots\!58$$$$\nu^{13} -$$$$26\!\cdots\!25$$$$\nu^{12} -$$$$48\!\cdots\!27$$$$\nu^{11} +$$$$86\!\cdots\!22$$$$\nu^{10} +$$$$98\!\cdots\!74$$$$\nu^{9} -$$$$17\!\cdots\!63$$$$\nu^{8} -$$$$11\!\cdots\!67$$$$\nu^{7} +$$$$21\!\cdots\!70$$$$\nu^{6} +$$$$68\!\cdots\!50$$$$\nu^{5} -$$$$13\!\cdots\!63$$$$\nu^{4} -$$$$11\!\cdots\!48$$$$\nu^{3} +$$$$26\!\cdots\!98$$$$\nu^{2} -$$$$13\!\cdots\!40$$$$\nu +$$$$23\!\cdots\!32$$$$)/$$$$47\!\cdots\!00$$ $$\beta_{4}$$ $$=$$ $$($$$$-$$$$56\!\cdots\!67$$$$\nu^{19} +$$$$74\!\cdots\!84$$$$\nu^{18} +$$$$10\!\cdots\!97$$$$\nu^{17} -$$$$17\!\cdots\!40$$$$\nu^{16} -$$$$78\!\cdots\!65$$$$\nu^{15} +$$$$16\!\cdots\!68$$$$\nu^{14} +$$$$22\!\cdots\!09$$$$\nu^{13} -$$$$89\!\cdots\!84$$$$\nu^{12} +$$$$18\!\cdots\!49$$$$\nu^{11} +$$$$28\!\cdots\!96$$$$\nu^{10} -$$$$30\!\cdots\!65$$$$\nu^{9} -$$$$53\!\cdots\!16$$$$\nu^{8} +$$$$88\!\cdots\!21$$$$\nu^{7} +$$$$55\!\cdots\!80$$$$\nu^{6} -$$$$11\!\cdots\!09$$$$\nu^{5} -$$$$24\!\cdots\!88$$$$\nu^{4} +$$$$60\!\cdots\!62$$$$\nu^{3} +$$$$84\!\cdots\!16$$$$\nu^{2} -$$$$29\!\cdots\!76$$$$\nu +$$$$95\!\cdots\!08$$$$)/$$$$39\!\cdots\!00$$ $$\beta_{5}$$ $$=$$ $$($$$$-$$$$63\!\cdots\!87$$$$\nu^{19} +$$$$79\!\cdots\!14$$$$\nu^{18} +$$$$12\!\cdots\!47$$$$\nu^{17} -$$$$18\!\cdots\!90$$$$\nu^{16} -$$$$93\!\cdots\!65$$$$\nu^{15} +$$$$17\!\cdots\!38$$$$\nu^{14} +$$$$30\!\cdots\!79$$$$\nu^{13} -$$$$95\!\cdots\!54$$$$\nu^{12} -$$$$28\!\cdots\!51$$$$\nu^{11} +$$$$30\!\cdots\!86$$$$\nu^{10} -$$$$28\!\cdots\!55$$$$\nu^{9} -$$$$56\!\cdots\!46$$$$\nu^{8} +$$$$87\!\cdots\!81$$$$\nu^{7} +$$$$58\!\cdots\!90$$$$\nu^{6} -$$$$11\!\cdots\!39$$$$\nu^{5} -$$$$25\!\cdots\!38$$$$\nu^{4} +$$$$62\!\cdots\!22$$$$\nu^{3} -$$$$63\!\cdots\!84$$$$\nu^{2} -$$$$28\!\cdots\!16$$$$\nu +$$$$88\!\cdots\!28$$$$)/$$$$39\!\cdots\!00$$ $$\beta_{6}$$ $$=$$ $$($$$$11\!\cdots\!90$$$$\nu^{19} +$$$$65\!\cdots\!33$$$$\nu^{18} -$$$$32\!\cdots\!48$$$$\nu^{17} -$$$$15\!\cdots\!31$$$$\nu^{16} +$$$$38\!\cdots\!74$$$$\nu^{15} +$$$$15\!\cdots\!39$$$$\nu^{14} -$$$$24\!\cdots\!68$$$$\nu^{13} -$$$$83\!\cdots\!39$$$$\nu^{12} +$$$$94\!\cdots\!82$$$$\nu^{11} +$$$$25\!\cdots\!81$$$$\nu^{10} -$$$$22\!\cdots\!48$$$$\nu^{9} -$$$$41\!\cdots\!73$$$$\nu^{8} +$$$$33\!\cdots\!46$$$$\nu^{7} +$$$$30\!\cdots\!77$$$$\nu^{6} -$$$$27\!\cdots\!88$$$$\nu^{5} +$$$$19\!\cdots\!83$$$$\nu^{4} +$$$$91\!\cdots\!40$$$$\nu^{3} -$$$$10\!\cdots\!18$$$$\nu^{2} +$$$$75\!\cdots\!88$$$$\nu -$$$$20\!\cdots\!28$$$$)/$$$$55\!\cdots\!00$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$47\!\cdots\!46$$$$\nu^{19} +$$$$23\!\cdots\!38$$$$\nu^{18} +$$$$11\!\cdots\!45$$$$\nu^{17} -$$$$53\!\cdots\!82$$$$\nu^{16} -$$$$11\!\cdots\!32$$$$\nu^{15} +$$$$52\!\cdots\!42$$$$\nu^{14} +$$$$60\!\cdots\!21$$$$\nu^{13} -$$$$27\!\cdots\!50$$$$\nu^{12} -$$$$18\!\cdots\!84$$$$\nu^{11} +$$$$84\!\cdots\!70$$$$\nu^{10} +$$$$33\!\cdots\!99$$$$\nu^{9} -$$$$14\!\cdots\!34$$$$\nu^{8} -$$$$29\!\cdots\!00$$$$\nu^{7} +$$$$13\!\cdots\!74$$$$\nu^{6} +$$$$74\!\cdots\!47$$$$\nu^{5} -$$$$30\!\cdots\!98$$$$\nu^{4} +$$$$19\!\cdots\!66$$$$\nu^{3} -$$$$19\!\cdots\!88$$$$\nu^{2} +$$$$18\!\cdots\!28$$$$\nu -$$$$48\!\cdots\!92$$$$)/$$$$19\!\cdots\!00$$ $$\beta_{8}$$ $$=$$ $$($$$$-$$$$54\!\cdots\!61$$$$\nu^{19} +$$$$28\!\cdots\!41$$$$\nu^{18} +$$$$12\!\cdots\!72$$$$\nu^{17} -$$$$64\!\cdots\!13$$$$\nu^{16} -$$$$12\!\cdots\!13$$$$\nu^{15} +$$$$62\!\cdots\!11$$$$\nu^{14} +$$$$67\!\cdots\!08$$$$\nu^{13} -$$$$32\!\cdots\!49$$$$\nu^{12} -$$$$20\!\cdots\!87$$$$\nu^{11} +$$$$10\!\cdots\!61$$$$\nu^{10} +$$$$35\!\cdots\!56$$$$\nu^{9} -$$$$17\!\cdots\!87$$$$\nu^{8} -$$$$29\!\cdots\!59$$$$\nu^{7} +$$$$16\!\cdots\!81$$$$\nu^{6} +$$$$47\!\cdots\!04$$$$\nu^{5} -$$$$41\!\cdots\!35$$$$\nu^{4} +$$$$41\!\cdots\!16$$$$\nu^{3} -$$$$20\!\cdots\!66$$$$\nu^{2} +$$$$18\!\cdots\!56$$$$\nu -$$$$43\!\cdots\!20$$$$)/$$$$19\!\cdots\!00$$ $$\beta_{9}$$ $$=$$ $$($$$$-$$$$23\!\cdots\!29$$$$\nu^{19} +$$$$14\!\cdots\!61$$$$\nu^{18} +$$$$52\!\cdots\!76$$$$\nu^{17} -$$$$33\!\cdots\!66$$$$\nu^{16} -$$$$49\!\cdots\!81$$$$\nu^{15} +$$$$31\!\cdots\!45$$$$\nu^{14} +$$$$25\!\cdots\!00$$$$\nu^{13} -$$$$16\!\cdots\!42$$$$\nu^{12} -$$$$72\!\cdots\!35$$$$\nu^{11} +$$$$49\!\cdots\!03$$$$\nu^{10} +$$$$10\!\cdots\!32$$$$\nu^{9} -$$$$89\!\cdots\!70$$$$\nu^{8} -$$$$46\!\cdots\!87$$$$\nu^{7} +$$$$89\!\cdots\!47$$$$\nu^{6} -$$$$69\!\cdots\!96$$$$\nu^{5} -$$$$37\!\cdots\!78$$$$\nu^{4} +$$$$70\!\cdots\!04$$$$\nu^{3} -$$$$55\!\cdots\!76$$$$\nu^{2} -$$$$25\!\cdots\!04$$$$\nu +$$$$71\!\cdots\!68$$$$)/$$$$68\!\cdots\!00$$ $$\beta_{10}$$ $$=$$ $$($$$$54\!\cdots\!83$$$$\nu^{19} -$$$$35\!\cdots\!75$$$$\nu^{18} -$$$$12\!\cdots\!49$$$$\nu^{17} +$$$$79\!\cdots\!53$$$$\nu^{16} +$$$$11\!\cdots\!73$$$$\nu^{15} -$$$$74\!\cdots\!89$$$$\nu^{14} -$$$$59\!\cdots\!17$$$$\nu^{13} +$$$$38\!\cdots\!13$$$$\nu^{12} +$$$$17\!\cdots\!43$$$$\nu^{11} -$$$$11\!\cdots\!27$$$$\nu^{10} -$$$$25\!\cdots\!31$$$$\nu^{9} +$$$$21\!\cdots\!43$$$$\nu^{8} +$$$$10\!\cdots\!23$$$$\nu^{7} -$$$$21\!\cdots\!91$$$$\nu^{6} +$$$$16\!\cdots\!45$$$$\nu^{5} +$$$$87\!\cdots\!23$$$$\nu^{4} -$$$$16\!\cdots\!18$$$$\nu^{3} +$$$$26\!\cdots\!50$$$$\nu^{2} +$$$$46\!\cdots\!20$$$$\nu -$$$$18\!\cdots\!08$$$$)/$$$$13\!\cdots\!00$$ $$\beta_{11}$$ $$=$$ $$($$$$22\!\cdots\!86$$$$\nu^{19} -$$$$16\!\cdots\!93$$$$\nu^{18} -$$$$55\!\cdots\!40$$$$\nu^{17} +$$$$33\!\cdots\!97$$$$\nu^{16} +$$$$59\!\cdots\!62$$$$\nu^{15} -$$$$29\!\cdots\!87$$$$\nu^{14} -$$$$34\!\cdots\!16$$$$\nu^{13} +$$$$16\!\cdots\!85$$$$\nu^{12} +$$$$12\!\cdots\!54$$$$\nu^{11} -$$$$64\!\cdots\!05$$$$\nu^{10} -$$$$25\!\cdots\!44$$$$\nu^{9} +$$$$17\!\cdots\!19$$$$\nu^{8} +$$$$31\!\cdots\!30$$$$\nu^{7} -$$$$28\!\cdots\!09$$$$\nu^{6} -$$$$20\!\cdots\!32$$$$\nu^{5} +$$$$24\!\cdots\!03$$$$\nu^{4} +$$$$44\!\cdots\!64$$$$\nu^{3} -$$$$74\!\cdots\!42$$$$\nu^{2} +$$$$34\!\cdots\!12$$$$\nu -$$$$52\!\cdots\!68$$$$)/$$$$55\!\cdots\!00$$ $$\beta_{12}$$ $$=$$ $$($$$$-$$$$22\!\cdots\!10$$$$\nu^{19} +$$$$14\!\cdots\!03$$$$\nu^{18} +$$$$56\!\cdots\!52$$$$\nu^{17} -$$$$27\!\cdots\!01$$$$\nu^{16} -$$$$59\!\cdots\!86$$$$\nu^{15} +$$$$24\!\cdots\!09$$$$\nu^{14} +$$$$34\!\cdots\!32$$$$\nu^{13} -$$$$13\!\cdots\!89$$$$\nu^{12} -$$$$12\!\cdots\!38$$$$\nu^{11} +$$$$55\!\cdots\!31$$$$\nu^{10} +$$$$25\!\cdots\!32$$$$\nu^{9} -$$$$15\!\cdots\!03$$$$\nu^{8} -$$$$32\!\cdots\!94$$$$\nu^{7} +$$$$26\!\cdots\!27$$$$\nu^{6} +$$$$20\!\cdots\!92$$$$\nu^{5} -$$$$23\!\cdots\!27$$$$\nu^{4} -$$$$45\!\cdots\!20$$$$\nu^{3} +$$$$75\!\cdots\!62$$$$\nu^{2} -$$$$37\!\cdots\!12$$$$\nu +$$$$65\!\cdots\!72$$$$)/$$$$55\!\cdots\!00$$ $$\beta_{13}$$ $$=$$ $$($$$$-$$$$28\!\cdots\!45$$$$\nu^{19} +$$$$12\!\cdots\!16$$$$\nu^{18} +$$$$67\!\cdots\!09$$$$\nu^{17} -$$$$27\!\cdots\!32$$$$\nu^{16} -$$$$67\!\cdots\!67$$$$\nu^{15} +$$$$25\!\cdots\!48$$$$\nu^{14} +$$$$36\!\cdots\!89$$$$\nu^{13} -$$$$12\!\cdots\!68$$$$\nu^{12} -$$$$11\!\cdots\!81$$$$\nu^{11} +$$$$36\!\cdots\!12$$$$\nu^{10} +$$$$21\!\cdots\!99$$$$\nu^{9} -$$$$61\!\cdots\!36$$$$\nu^{8} -$$$$21\!\cdots\!53$$$$\nu^{7} +$$$$52\!\cdots\!44$$$$\nu^{6} +$$$$91\!\cdots\!59$$$$\nu^{5} -$$$$11\!\cdots\!64$$$$\nu^{4} -$$$$62\!\cdots\!30$$$$\nu^{3} -$$$$79\!\cdots\!16$$$$\nu^{2} +$$$$82\!\cdots\!16$$$$\nu -$$$$18\!\cdots\!56$$$$)/$$$$19\!\cdots\!00$$ $$\beta_{14}$$ $$=$$ $$($$$$33\!\cdots\!45$$$$\nu^{19} -$$$$15\!\cdots\!94$$$$\nu^{18} -$$$$77\!\cdots\!11$$$$\nu^{17} +$$$$33\!\cdots\!98$$$$\nu^{16} +$$$$76\!\cdots\!63$$$$\nu^{15} -$$$$30\!\cdots\!62$$$$\nu^{14} -$$$$41\!\cdots\!11$$$$\nu^{13} +$$$$15\!\cdots\!62$$$$\nu^{12} +$$$$12\!\cdots\!89$$$$\nu^{11} -$$$$45\!\cdots\!98$$$$\nu^{10} -$$$$23\!\cdots\!81$$$$\nu^{9} +$$$$76\!\cdots\!14$$$$\nu^{8} +$$$$22\!\cdots\!17$$$$\nu^{7} -$$$$67\!\cdots\!86$$$$\nu^{6} -$$$$82\!\cdots\!21$$$$\nu^{5} +$$$$16\!\cdots\!46$$$$\nu^{4} -$$$$37\!\cdots\!10$$$$\nu^{3} +$$$$86\!\cdots\!24$$$$\nu^{2} -$$$$84\!\cdots\!24$$$$\nu +$$$$24\!\cdots\!84$$$$)/$$$$19\!\cdots\!00$$ $$\beta_{15}$$ $$=$$ $$($$$$89\!\cdots\!93$$$$\nu^{19} -$$$$10\!\cdots\!37$$$$\nu^{18} -$$$$19\!\cdots\!16$$$$\nu^{17} +$$$$25\!\cdots\!10$$$$\nu^{16} +$$$$17\!\cdots\!09$$$$\nu^{15} -$$$$25\!\cdots\!17$$$$\nu^{14} -$$$$84\!\cdots\!52$$$$\nu^{13} +$$$$13\!\cdots\!74$$$$\nu^{12} +$$$$19\!\cdots\!11$$$$\nu^{11} -$$$$44\!\cdots\!19$$$$\nu^{10} -$$$$11\!\cdots\!44$$$$\nu^{9} +$$$$82\!\cdots\!66$$$$\nu^{8} -$$$$49\!\cdots\!49$$$$\nu^{7} -$$$$82\!\cdots\!43$$$$\nu^{6} +$$$$11\!\cdots\!16$$$$\nu^{5} +$$$$29\!\cdots\!26$$$$\nu^{4} -$$$$71\!\cdots\!64$$$$\nu^{3} +$$$$55\!\cdots\!88$$$$\nu^{2} -$$$$19\!\cdots\!76$$$$\nu +$$$$24\!\cdots\!08$$$$)/$$$$31\!\cdots\!00$$ $$\beta_{16}$$ $$=$$ $$($$$$-$$$$29\!\cdots\!52$$$$\nu^{19} +$$$$21\!\cdots\!51$$$$\nu^{18} +$$$$73\!\cdots\!10$$$$\nu^{17} -$$$$52\!\cdots\!59$$$$\nu^{16} -$$$$80\!\cdots\!24$$$$\nu^{15} +$$$$53\!\cdots\!09$$$$\nu^{14} +$$$$49\!\cdots\!22$$$$\nu^{13} -$$$$29\!\cdots\!75$$$$\nu^{12} -$$$$18\!\cdots\!48$$$$\nu^{11} +$$$$99\!\cdots\!75$$$$\nu^{10} +$$$$40\!\cdots\!98$$$$\nu^{9} -$$$$19\!\cdots\!93$$$$\nu^{8} -$$$$52\!\cdots\!60$$$$\nu^{7} +$$$$22\!\cdots\!63$$$$\nu^{6} +$$$$34\!\cdots\!74$$$$\nu^{5} -$$$$12\!\cdots\!81$$$$\nu^{4} -$$$$61\!\cdots\!88$$$$\nu^{3} +$$$$21\!\cdots\!54$$$$\nu^{2} -$$$$11\!\cdots\!84$$$$\nu +$$$$17\!\cdots\!56$$$$)/$$$$91\!\cdots\!00$$ $$\beta_{17}$$ $$=$$ $$($$$$-$$$$30\!\cdots\!90$$$$\nu^{19} +$$$$23\!\cdots\!91$$$$\nu^{18} +$$$$77\!\cdots\!24$$$$\nu^{17} -$$$$55\!\cdots\!37$$$$\nu^{16} -$$$$84\!\cdots\!82$$$$\nu^{15} +$$$$56\!\cdots\!53$$$$\nu^{14} +$$$$50\!\cdots\!04$$$$\nu^{13} -$$$$31\!\cdots\!13$$$$\nu^{12} -$$$$18\!\cdots\!06$$$$\nu^{11} +$$$$10\!\cdots\!87$$$$\nu^{10} +$$$$41\!\cdots\!44$$$$\nu^{9} -$$$$20\!\cdots\!91$$$$\nu^{8} -$$$$53\!\cdots\!18$$$$\nu^{7} +$$$$23\!\cdots\!59$$$$\nu^{6} +$$$$33\!\cdots\!24$$$$\nu^{5} -$$$$13\!\cdots\!99$$$$\nu^{4} -$$$$55\!\cdots\!80$$$$\nu^{3} +$$$$20\!\cdots\!94$$$$\nu^{2} -$$$$12\!\cdots\!84$$$$\nu +$$$$23\!\cdots\!84$$$$)/$$$$91\!\cdots\!00$$ $$\beta_{18}$$ $$=$$ $$($$$$-$$$$99\!\cdots\!72$$$$\nu^{19} +$$$$14\!\cdots\!42$$$$\nu^{18} +$$$$20\!\cdots\!34$$$$\nu^{17} -$$$$35\!\cdots\!31$$$$\nu^{16} -$$$$15\!\cdots\!56$$$$\nu^{15} +$$$$35\!\cdots\!72$$$$\nu^{14} +$$$$43\!\cdots\!26$$$$\nu^{13} -$$$$19\!\cdots\!63$$$$\nu^{12} +$$$$67\!\cdots\!96$$$$\nu^{11} +$$$$62\!\cdots\!12$$$$\nu^{10} -$$$$85\!\cdots\!78$$$$\nu^{9} -$$$$11\!\cdots\!89$$$$\nu^{8} +$$$$25\!\cdots\!52$$$$\nu^{7} +$$$$10\!\cdots\!32$$$$\nu^{6} -$$$$34\!\cdots\!42$$$$\nu^{5} -$$$$26\!\cdots\!65$$$$\nu^{4} +$$$$17\!\cdots\!12$$$$\nu^{3} -$$$$18\!\cdots\!02$$$$\nu^{2} +$$$$78\!\cdots\!92$$$$\nu -$$$$11\!\cdots\!40$$$$)/$$$$27\!\cdots\!00$$ $$\beta_{19}$$ $$=$$ $$($$$$24\!\cdots\!35$$$$\nu^{19} -$$$$26\!\cdots\!55$$$$\nu^{18} -$$$$54\!\cdots\!93$$$$\nu^{17} +$$$$61\!\cdots\!57$$$$\nu^{16} +$$$$49\!\cdots\!21$$$$\nu^{15} -$$$$61\!\cdots\!29$$$$\nu^{14} -$$$$23\!\cdots\!53$$$$\nu^{13} +$$$$33\!\cdots\!49$$$$\nu^{12} +$$$$58\!\cdots\!31$$$$\nu^{11} -$$$$10\!\cdots\!39$$$$\nu^{10} -$$$$48\!\cdots\!39$$$$\nu^{9} +$$$$19\!\cdots\!15$$$$\nu^{8} -$$$$94\!\cdots\!61$$$$\nu^{7} -$$$$19\!\cdots\!67$$$$\nu^{6} +$$$$25\!\cdots\!61$$$$\nu^{5} +$$$$70\!\cdots\!39$$$$\nu^{4} -$$$$16\!\cdots\!18$$$$\nu^{3} +$$$$12\!\cdots\!86$$$$\nu^{2} -$$$$44\!\cdots\!12$$$$\nu +$$$$59\!\cdots\!60$$$$)/$$$$63\!\cdots\!00$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{14} - \beta_{13} + 4 \beta_{8} - 4 \beta_{7} - 14 \beta_{5} + 14 \beta_{4} - 126 \beta_{2} + 398$$$$)/1008$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{19} + \beta_{15} - 5 \beta_{14} - 5 \beta_{13} + 28 \beta_{10} + 28 \beta_{9} - 21 \beta_{8} + 21 \beta_{7} + 18 \beta_{5} - 18 \beta_{4} - 12 \beta_{2} + 252 \beta_{1} + 332715$$$$)/126$$ $$\nu^{3}$$ $$=$$ $$($$$$16 \beta_{19} + 82 \beta_{17} - 82 \beta_{16} - 16 \beta_{15} - 1675 \beta_{14} - 1669 \beta_{13} + 1821 \beta_{12} + 1956 \beta_{11} + 1960 \beta_{10} + 1960 \beta_{9} + 9704 \beta_{8} - 9752 \beta_{7} - 135 \beta_{6} - 16694 \beta_{5} + 16682 \beta_{4} - 84 \beta_{3} - 331281 \beta_{2} + 792 \beta_{1} + 1706020$$$$)/336$$ $$\nu^{4}$$ $$=$$ $$($$$$-7522 \beta_{19} - 1140 \beta_{17} + 1140 \beta_{16} + 7510 \beta_{15} - 18450 \beta_{14} - 18270 \beta_{13} + 18150 \beta_{12} + 21240 \beta_{11} + 166103 \beta_{10} + 165095 \beta_{9} - 34843 \beta_{8} + 35023 \beta_{7} - 3090 \beta_{6} + 12920 \beta_{5} - 13208 \beta_{4} - 48 \beta_{3} - 300018 \beta_{2} + 3994308 \beta_{1} + 1248623035$$$$)/126$$ $$\nu^{5}$$ $$=$$ $$($$$$-235760 \beta_{19} - 2700 \beta_{18} + 1087830 \beta_{17} - 1082910 \beta_{16} + 236720 \beta_{15} - 23952621 \beta_{14} - 23651541 \beta_{13} + 53091015 \beta_{12} + 55142280 \beta_{11} + 40540080 \beta_{10} + 40187280 \beta_{9} + 154451844 \beta_{8} - 155871684 \beta_{7} - 2163225 \beta_{6} - 199216614 \beta_{5} + 199382454 \beta_{4} - 6702336 \beta_{3} - 6211221555 \beta_{2} + 101964960 \beta_{1} + 82160942238$$$$)/1008$$ $$\nu^{6}$$ $$=$$ $$($$$$-46482399 \beta_{19} - 30900 \beta_{18} - 11269770 \beta_{17} + 11235570 \beta_{16} + 46256895 \beta_{15} - 83551115 \beta_{14} - 81896915 \beta_{13} + 189420855 \beta_{12} + 219624240 \beta_{11} + 843176488 \beta_{10} + 828262498 \beta_{9} + 9999879 \beta_{8} - 10262229 \beta_{7} - 30778785 \beta_{6} - 181221552 \beta_{5} + 179514612 \beta_{4} - 3394416 \beta_{3} - 7515712401 \beta_{2} + 37547523810 \beta_{1} + 5125264972995$$$$)/126$$ $$\nu^{7}$$ $$=$$ $$($$$$-4849687696 \beta_{19} - 30273810 \beta_{18} + 4528730178 \beta_{17} - 4483130862 \beta_{16} + 4839768304 \beta_{15} - 113760132557 \beta_{14} - 110756853131 \beta_{13} + 373797341589 \beta_{12} + 383028077166 \beta_{11} + 230462713320 \beta_{10} + 225371950440 \beta_{9} + 737867811272 \beta_{8} - 748112072600 \beta_{7} - 11488843485 \beta_{6} - 839046839722 \beta_{5} + 843712165030 \beta_{4} - 88538055732 \beta_{3} - 35736677418267 \beta_{2} + 3448938700872 \beta_{1} + 645119109051556$$$$)/1008$$ $$\nu^{8}$$ $$=$$ $$($$$$-268640003668 \beta_{19} - 574364280 \beta_{18} - 74014415160 \beta_{17} + 73384137960 \beta_{16} + 266042461084 \beta_{15} - 421868358140 \beta_{14} - 407939803220 \beta_{13} + 1465888002180 \beta_{12} + 1674885694440 \beta_{11} + 4090394581637 \beta_{10} + 3949715077133 \beta_{9} + 639403619365 \beta_{8} - 659042360725 \beta_{7} - 220166411220 \beta_{6} - 1684312080272 \beta_{5} + 1675314215648 \beta_{4} - 147973162176 \beta_{3} - 79482477095124 \beta_{2} + 288665122660992 \beta_{1} + 22176163503637631$$$$)/126$$ $$\nu^{9}$$ $$=$$ $$($$$$-45463978287120 \beta_{19} - 275514462720 \beta_{18} + 18457946828070 \beta_{17} - 18133300606758 \beta_{16} + 45115110325776 \beta_{15} - 538661795807113 \beta_{14} - 514341756179449 \beta_{13} + 2309086053824175 \beta_{12} + 2347606208775564 \beta_{11} + 1239139479795840 \beta_{10} + 1189787009624640 \beta_{9} + 3404048959677988 \beta_{8} - 3471958790913700 \beta_{7} - 65639577294045 \beta_{6} - 3668188062725918 \beta_{5} + 3714711327626750 \beta_{4} - 877331980355040 \beta_{3} - 199374098456527047 \beta_{2} + 46606569682074048 \beta_{1} + 4072190148704183054$$$$)/1008$$ $$\nu^{10}$$ $$=$$ $$($$$$-1502728620405037 \beta_{19} - 6599826691920 \beta_{18} - 421785264094830 \beta_{17} + 415148543519310 \beta_{16} + 1478652297714829 \beta_{15} - 2211050376490625 \beta_{14} - 2098607259781025 \beta_{13} + 10063835416774845 \beta_{12} + 11293625769529140 \beta_{11} + 19583298705415858 \beta_{10} + 18493653074658748 \beta_{9} + 5334708153373539 \beta_{8} - 5579233796449689 \beta_{7} - 1367906983843095 \beta_{6} - 10913847784493292 \beta_{5} + 10879253608046832 \beta_{4} - 2481805359347712 \beta_{3} - 631462758082871895 \beta_{2} + 2016732597700089258 \beta_{1} + 98921283650228492427$$$$)/126$$ $$\nu^{11}$$ $$=$$ $$($$$$-112898031986832400 \beta_{19} - 801253145693550 \beta_{18} + 23751411880662070 \beta_{17} - 23079775955307786 \beta_{16} + 111236561962971632 \beta_{15} - 850225519254332067 \beta_{14} - 792011977348658429 \beta_{13} + 4454649452907642615 \beta_{12} + 4500538795369438338 \beta_{11} + 2170270293150143240 \beta_{10} + 2036074272274919240 \beta_{9} + 5175430193838617480 \beta_{8} - 5320910129959790584 \beta_{7} - 130930701623282055 \beta_{6} - 5475819580446654470 \beta_{5} + 5593157037675086394 \beta_{4} - 2501221181440482444 \beta_{3} - 363895879662535760001 \beta_{2} + 150611322398253504216 \beta_{1} + 7750251013364403361236$$$$)/336$$ $$\nu^{12}$$ $$=$$ $$($$$$-8238287463891497206 \beta_{19} - 60100948808596740 \beta_{18} - 2255324847374818500 \beta_{17} + 2200028638198397100 \beta_{16} + 8041303731485317810 \beta_{15} - 11653788673828433430 \beta_{14} - 10795505968904613210 \beta_{13} + 64587480035193394350 \beta_{12} + 71107784892062520780 \beta_{11} + 93594367379283240827 \beta_{10} + 86006954274481355087 \beta_{9} + 34755357758524736201 \beta_{8} - 36922518508032802601 \beta_{7} - 7901312256553780710 \beta_{6} - 62593726661296812388 \beta_{5} + 62594485302393507628 \beta_{4} - 28800570273055723104 \beta_{3} - 4355247183468131725734 \beta_{2} + 13274331603349513312440 \beta_{1} + 449265044690976929634967$$$$)/126$$ $$\nu^{13}$$ $$=$$ $$($$$$-2254955736344511911984 \beta_{19} - 20387801154870338280 \beta_{18} + 244732017708090399942 \beta_{17} - 233752434675747385398 \beta_{16} + 2202456263798022320816 \beta_{15} - 12108543531985654335765 \beta_{14} - 10946553487901442489261 \beta_{13} + 74571730587868322927391 \beta_{12} + 74854883594453031491604 \beta_{11} + 33775324135224478163280 \beta_{10} + 30801417969407566062960 \beta_{9} + 70656226676119431284676 \beta_{8} - 73403220976963149085188 \beta_{7} - 2371243608919196511045 \beta_{6} - 74735528185870857996726 \beta_{5} + 77076090524232050537958 \beta_{4} - 58438251326937813681168 \beta_{3} - 5892912322880345161394415 \beta_{2} + 3683229885570756800549088 \beta_{1} + 125690710444442465864418750$$$$)/1008$$ $$\nu^{14}$$ $$=$$ $$($$$$-44535520560429805820859 \beta_{19} - 478256140319239835280 \beta_{18} - 11679227262222379421730 \beta_{17} + 11273115446219793608610 \beta_{16} + 43051674191205008882043 \beta_{15} - 61101561614513387825975 \beta_{14} - 54920459466200447282735 \beta_{13} + 396023324354351161208475 \beta_{12} + 427682254714085043228780 \beta_{11} + 448441761898797985005604 \beta_{10} + 398946658455499700234878 \beta_{9} + 204131598498945467717703 \beta_{8} - 220525457183481627534693 \beta_{7} - 43788006120884707981425 \beta_{6} - 338449036813282503633192 \beta_{5} + 340102213357752218381676 \beta_{4} - 273674964235442354099424 \beta_{3} - 27666843522204014034347985 \beta_{2} + 83749475482693703394325362 \beta_{1} + 2063273071994518523660125659$$$$)/126$$ $$\nu^{15}$$ $$=$$ $$($$$$-14050671171741826581820624 \beta_{19} - 165560747106601944194370 \beta_{18} + 633692801565696730027122 \beta_{17} - 583296678678742428869694 \beta_{16} + 13581904107923706735337648 \beta_{15} - 57712292314140335392775477 \beta_{14} - 50367801743582607644116115 \beta_{13} + 406975588499177099759503821 \beta_{12} + 405478180643840408582702142 \beta_{11} + 173704132506463173349769160 \beta_{10} + 153152451322078176472480200 \beta_{9} + 321979809628056886492009256 \beta_{8} - 338928286004270353436876792 \beta_{7} - 14140248624745820279922645 \beta_{6} - 343634799825820077917208346 \beta_{5} + 358112506975513878210573142 \beta_{4} - 425832938488900893621814164 \beta_{3} - 31446576381976658235264433779 \beta_{2} + 27023269671833478947163326664 \beta_{1} + 657235313702583111702165107956$$$$)/1008$$ $$\nu^{16}$$ $$=$$ $$($$$$-$$$$23\!\cdots\!72$$$$\beta_{19} -$$$$34\!\cdots\!80$$$$\beta_{18} -$$$$59\!\cdots\!80$$$$\beta_{17} +$$$$56\!\cdots\!80$$$$\beta_{16} +$$$$22\!\cdots\!56$$$$\beta_{15} -$$$$31\!\cdots\!80$$$$\beta_{14} -$$$$27\!\cdots\!00$$$$\beta_{13} +$$$$23\!\cdots\!40$$$$\beta_{12} +$$$$24\!\cdots\!00$$$$\beta_{11} +$$$$21\!\cdots\!33$$$$\beta_{10} +$$$$18\!\cdots\!77$$$$\beta_{9} +$$$$11\!\cdots\!65$$$$\beta_{8} -$$$$12\!\cdots\!05$$$$\beta_{7} -$$$$23\!\cdots\!20$$$$\beta_{6} -$$$$17\!\cdots\!48$$$$\beta_{5} +$$$$17\!\cdots\!12$$$$\beta_{4} -$$$$22\!\cdots\!04$$$$\beta_{3} -$$$$16\!\cdots\!56$$$$\beta_{2} +$$$$51\!\cdots\!08$$$$\beta_{1} +$$$$95\!\cdots\!23$$$$)/126$$ $$\nu^{17}$$ $$=$$ $$($$$$-$$$$83\!\cdots\!80$$$$\beta_{19} -$$$$12\!\cdots\!00$$$$\beta_{18} +$$$$12\!\cdots\!10$$$$\beta_{17} +$$$$46\!\cdots\!18$$$$\beta_{16} +$$$$80\!\cdots\!24$$$$\beta_{15} -$$$$27\!\cdots\!01$$$$\beta_{14} -$$$$23\!\cdots\!05$$$$\beta_{13} +$$$$21\!\cdots\!35$$$$\beta_{12} +$$$$21\!\cdots\!96$$$$\beta_{11} +$$$$88\!\cdots\!40$$$$\beta_{10} +$$$$75\!\cdots\!00$$$$\beta_{9} +$$$$14\!\cdots\!08$$$$\beta_{8} -$$$$15\!\cdots\!36$$$$\beta_{7} -$$$$82\!\cdots\!05$$$$\beta_{6} -$$$$15\!\cdots\!38$$$$\beta_{5} +$$$$16\!\cdots\!46$$$$\beta_{4} -$$$$29\!\cdots\!40$$$$\beta_{3} -$$$$16\!\cdots\!47$$$$\beta_{2} +$$$$18\!\cdots\!92$$$$\beta_{1} +$$$$33\!\cdots\!18$$$$)/1008$$ $$\nu^{18}$$ $$=$$ $$($$$$-$$$$12\!\cdots\!97$$$$\beta_{19} -$$$$24\!\cdots\!20$$$$\beta_{18} -$$$$29\!\cdots\!50$$$$\beta_{17} +$$$$28\!\cdots\!70$$$$\beta_{16} +$$$$11\!\cdots\!05$$$$\beta_{15} -$$$$16\!\cdots\!45$$$$\beta_{14} -$$$$13\!\cdots\!25$$$$\beta_{13} +$$$$13\!\cdots\!25$$$$\beta_{12} +$$$$14\!\cdots\!00$$$$\beta_{11} +$$$$10\!\cdots\!74$$$$\beta_{10} +$$$$85\!\cdots\!44$$$$\beta_{9} +$$$$60\!\cdots\!47$$$$\beta_{8} -$$$$67\!\cdots\!97$$$$\beta_{7} -$$$$12\!\cdots\!55$$$$\beta_{6} -$$$$90\!\cdots\!76$$$$\beta_{5} +$$$$92\!\cdots\!76$$$$\beta_{4} -$$$$17\!\cdots\!08$$$$\beta_{3} -$$$$96\!\cdots\!23$$$$\beta_{2} +$$$$30\!\cdots\!50$$$$\beta_{1} +$$$$44\!\cdots\!43$$$$)/126$$ $$\nu^{19}$$ $$=$$ $$($$$$-$$$$16\!\cdots\!76$$$$\beta_{19} -$$$$31\!\cdots\!30$$$$\beta_{18} -$$$$49\!\cdots\!42$$$$\beta_{17} +$$$$49\!\cdots\!58$$$$\beta_{16} +$$$$15\!\cdots\!64$$$$\beta_{15} -$$$$44\!\cdots\!63$$$$\beta_{14} -$$$$35\!\cdots\!97$$$$\beta_{13} +$$$$38\!\cdots\!39$$$$\beta_{12} +$$$$37\!\cdots\!26$$$$\beta_{11} +$$$$15\!\cdots\!40$$$$\beta_{10} +$$$$12\!\cdots\!60$$$$\beta_{9} +$$$$22\!\cdots\!36$$$$\beta_{8} -$$$$24\!\cdots\!84$$$$\beta_{7} -$$$$15\!\cdots\!75$$$$\beta_{6} -$$$$24\!\cdots\!26$$$$\beta_{5} +$$$$26\!\cdots\!54$$$$\beta_{4} -$$$$65\!\cdots\!72$$$$\beta_{3} -$$$$29\!\cdots\!73$$$$\beta_{2} +$$$$40\!\cdots\!52$$$$\beta_{1} +$$$$56\!\cdots\!64$$$$)/336$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/126\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$73$$ $$\chi(n)$$ $$1$$ $$\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 42.0992 − 1.22474i −49.5297 − 1.22474i 5.49836 − 1.22474i −63.1006 − 1.22474i 70.5683 − 1.22474i 69.1541 + 1.22474i −64.5148 + 1.22474i 4.08415 + 1.22474i −50.9439 + 1.22474i 40.6850 + 1.22474i 42.0992 + 1.22474i −49.5297 + 1.22474i 5.49836 + 1.22474i −63.1006 + 1.22474i 70.5683 + 1.22474i 69.1541 − 1.22474i −64.5148 − 1.22474i 4.08415 − 1.22474i −50.9439 − 1.22474i 40.6850 − 1.22474i
−5.65685 + 9.79796i 0 −64.0000 110.851i −565.865 326.703i 0 2120.69 1125.82i 1448.15 0 6402.04 3696.22i
19.2 −5.65685 + 9.79796i 0 −64.0000 110.851i −268.774 155.177i 0 −1576.44 + 1810.98i 1448.15 0 3040.83 1755.63i
19.3 −5.65685 + 9.79796i 0 −64.0000 110.851i −254.021 146.659i 0 −1180.12 2090.96i 1448.15 0 2873.92 1659.26i
19.4 −5.65685 + 9.79796i 0 −64.0000 110.851i 664.367 + 383.572i 0 1667.57 + 1727.43i 1448.15 0 −7516.45 + 4339.62i
19.5 −5.65685 + 9.79796i 0 −64.0000 110.851i 814.617 + 470.320i 0 14.7985 2400.95i 1448.15 0 −9216.34 + 5321.06i
19.6 5.65685 9.79796i 0 −64.0000 110.851i −814.617 470.320i 0 14.7985 2400.95i −1448.15 0 −9216.34 + 5321.06i
19.7 5.65685 9.79796i 0 −64.0000 110.851i −664.367 383.572i 0 1667.57 + 1727.43i −1448.15 0 −7516.45 + 4339.62i
19.8 5.65685 9.79796i 0 −64.0000 110.851i 254.021 + 146.659i 0 −1180.12 2090.96i −1448.15 0 2873.92 1659.26i
19.9 5.65685 9.79796i 0 −64.0000 110.851i 268.774 + 155.177i 0 −1576.44 + 1810.98i −1448.15 0 3040.83 1755.63i
19.10 5.65685 9.79796i 0 −64.0000 110.851i 565.865 + 326.703i 0 2120.69 1125.82i −1448.15 0 6402.04 3696.22i
73.1 −5.65685 9.79796i 0 −64.0000 + 110.851i −565.865 + 326.703i 0 2120.69 + 1125.82i 1448.15 0 6402.04 + 3696.22i
73.2 −5.65685 9.79796i 0 −64.0000 + 110.851i −268.774 + 155.177i 0 −1576.44 1810.98i 1448.15 0 3040.83 + 1755.63i
73.3 −5.65685 9.79796i 0 −64.0000 + 110.851i −254.021 + 146.659i 0 −1180.12 + 2090.96i 1448.15 0 2873.92 + 1659.26i
73.4 −5.65685 9.79796i 0 −64.0000 + 110.851i 664.367 383.572i 0 1667.57 1727.43i 1448.15 0 −7516.45 4339.62i
73.5 −5.65685 9.79796i 0 −64.0000 + 110.851i 814.617 470.320i 0 14.7985 + 2400.95i 1448.15 0 −9216.34 5321.06i
73.6 5.65685 + 9.79796i 0 −64.0000 + 110.851i −814.617 + 470.320i 0 14.7985 + 2400.95i −1448.15 0 −9216.34 5321.06i
73.7 5.65685 + 9.79796i 0 −64.0000 + 110.851i −664.367 + 383.572i 0 1667.57 1727.43i −1448.15 0 −7516.45 4339.62i
73.8 5.65685 + 9.79796i 0 −64.0000 + 110.851i 254.021 146.659i 0 −1180.12 + 2090.96i −1448.15 0 2873.92 + 1659.26i
73.9 5.65685 + 9.79796i 0 −64.0000 + 110.851i 268.774 155.177i 0 −1576.44 1810.98i −1448.15 0 3040.83 + 1755.63i
73.10 5.65685 + 9.79796i 0 −64.0000 + 110.851i 565.865 326.703i 0 2120.69 + 1125.82i −1448.15 0 6402.04 + 3696.22i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 73.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.9.n.d 20
3.b odd 2 1 inner 126.9.n.d 20
7.d odd 6 1 inner 126.9.n.d 20
21.g even 6 1 inner 126.9.n.d 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.9.n.d 20 1.a even 1 1 trivial
126.9.n.d 20 3.b odd 2 1 inner
126.9.n.d 20 7.d odd 6 1 inner
126.9.n.d 20 21.g even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$28\!\cdots\!27$$$$T_{5}^{16} -$$$$22\!\cdots\!66$$$$T_{5}^{14} +$$$$12\!\cdots\!17$$$$T_{5}^{12} -$$$$46\!\cdots\!96$$$$T_{5}^{10} +$$$$12\!\cdots\!36$$$$T_{5}^{8} -$$$$16\!\cdots\!00$$$$T_{5}^{6} +$$$$16\!\cdots\!00$$$$T_{5}^{4} -$$$$92\!\cdots\!00$$$$T_{5}^{2} +$$$$33\!\cdots\!00$$">$$T_{5}^{20} - \cdots$$ acting on $$S_{9}^{\mathrm{new}}(126, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 16384 + 128 T^{2} + T^{4} )^{5}$$
$3$ $$T^{20}$$
$5$ $$33\!\cdots\!00$$$$-$$$$92\!\cdots\!00$$$$T^{2} +$$$$16\!\cdots\!00$$$$T^{4} -$$$$16\!\cdots\!00$$$$T^{6} +$$$$12\!\cdots\!36$$$$T^{8} -$$$$46\!\cdots\!96$$$$T^{10} +$$$$12\!\cdots\!17$$$$T^{12} - 2237917280863850766 T^{14} + 2832711417027 T^{16} - 2082606 T^{18} + T^{20}$$
$7$ $$($$$$63\!\cdots\!01$$$$-$$$$23\!\cdots\!93$$$$T +$$$$16\!\cdots\!59$$$$T^{2} -$$$$86\!\cdots\!10$$$$T^{3} +$$$$51\!\cdots\!77$$$$T^{4} - 164466667856067843 T^{5} + 88956264577677 T^{6} - 26060237910 T^{7} + 8701959 T^{8} - 2093 T^{9} + T^{10} )^{2}$$
$11$ $$19\!\cdots\!64$$$$+$$$$19\!\cdots\!52$$$$T^{2} +$$$$17\!\cdots\!28$$$$T^{4} +$$$$10\!\cdots\!56$$$$T^{6} +$$$$48\!\cdots\!40$$$$T^{8} +$$$$90\!\cdots\!76$$$$T^{10} +$$$$12\!\cdots\!25$$$$T^{12} +$$$$43\!\cdots\!26$$$$T^{14} + 1140972317302793067 T^{16} + 1238615082 T^{18} + T^{20}$$
$13$ $$($$$$68\!\cdots\!52$$$$+$$$$23\!\cdots\!00$$$$T^{2} +$$$$19\!\cdots\!55$$$$T^{4} + 4635262920836029635 T^{6} + 3923939265 T^{8} + T^{10} )^{2}$$
$17$ $$30\!\cdots\!16$$$$-$$$$18\!\cdots\!24$$$$T^{2} +$$$$86\!\cdots\!48$$$$T^{4} -$$$$14\!\cdots\!88$$$$T^{6} +$$$$17\!\cdots\!80$$$$T^{8} -$$$$10\!\cdots\!12$$$$T^{10} +$$$$46\!\cdots\!40$$$$T^{12} -$$$$11\!\cdots\!88$$$$T^{14} +$$$$20\!\cdots\!08$$$$T^{16} - 16964573424 T^{18} + T^{20}$$
$19$ $$($$$$55\!\cdots\!00$$$$-$$$$19\!\cdots\!00$$$$T +$$$$22\!\cdots\!16$$$$T^{2} +$$$$36\!\cdots\!20$$$$T^{3} -$$$$20\!\cdots\!61$$$$T^{4} +$$$$53\!\cdots\!05$$$$T^{5} +$$$$17\!\cdots\!52$$$$T^{6} - 5874234523674255 T^{7} - 41581688784 T^{8} + 125445 T^{9} + T^{10} )^{2}$$
$23$ $$13\!\cdots\!64$$$$+$$$$30\!\cdots\!48$$$$T^{2} +$$$$46\!\cdots\!68$$$$T^{4} +$$$$38\!\cdots\!84$$$$T^{6} +$$$$22\!\cdots\!60$$$$T^{8} +$$$$79\!\cdots\!84$$$$T^{10} +$$$$20\!\cdots\!80$$$$T^{12} +$$$$32\!\cdots\!84$$$$T^{14} +$$$$36\!\cdots\!92$$$$T^{16} + 236225063448 T^{18} + T^{20}$$
$29$ $$( -$$$$18\!\cdots\!48$$$$+$$$$76\!\cdots\!76$$$$T^{2} -$$$$85\!\cdots\!96$$$$T^{4} +$$$$10\!\cdots\!77$$$$T^{6} - 2207351967498 T^{8} + T^{10} )^{2}$$
$31$ $$($$$$38\!\cdots\!27$$$$-$$$$36\!\cdots\!33$$$$T +$$$$87\!\cdots\!43$$$$T^{2} +$$$$27\!\cdots\!54$$$$T^{3} -$$$$34\!\cdots\!85$$$$T^{4} -$$$$12\!\cdots\!71$$$$T^{5} +$$$$22\!\cdots\!29$$$$T^{6} + 278996045311023390 T^{7} - 1618138198803 T^{8} - 171381 T^{9} + T^{10} )^{2}$$
$37$ $$($$$$25\!\cdots\!00$$$$-$$$$90\!\cdots\!40$$$$T +$$$$28\!\cdots\!76$$$$T^{2} -$$$$21\!\cdots\!84$$$$T^{3} +$$$$20\!\cdots\!73$$$$T^{4} -$$$$35\!\cdots\!05$$$$T^{5} +$$$$46\!\cdots\!34$$$$T^{6} + 889448420133079095 T^{7} + 12413043871638 T^{8} + 2403299 T^{9} + T^{10} )^{2}$$
$41$ $$($$$$20\!\cdots\!96$$$$+$$$$20\!\cdots\!16$$$$T^{2} +$$$$16\!\cdots\!08$$$$T^{4} +$$$$40\!\cdots\!08$$$$T^{6} + 35286842698296 T^{8} + T^{10} )^{2}$$
$43$ $$( -$$$$81\!\cdots\!20$$$$+$$$$30\!\cdots\!76$$$$T + 91455532616021966927 T^{2} - 40989097528793 T^{3} - 1742063 T^{4} + T^{5} )^{4}$$
$47$ $$34\!\cdots\!00$$$$-$$$$58\!\cdots\!00$$$$T^{2} +$$$$98\!\cdots\!56$$$$T^{4} -$$$$91\!\cdots\!00$$$$T^{6} +$$$$74\!\cdots\!12$$$$T^{8} -$$$$91\!\cdots\!20$$$$T^{10} +$$$$80\!\cdots\!08$$$$T^{12} -$$$$36\!\cdots\!20$$$$T^{14} +$$$$11\!\cdots\!32$$$$T^{16} - 122678746205640 T^{18} + T^{20}$$
$53$ $$18\!\cdots\!00$$$$+$$$$25\!\cdots\!00$$$$T^{2} +$$$$30\!\cdots\!00$$$$T^{4} +$$$$69\!\cdots\!00$$$$T^{6} +$$$$12\!\cdots\!64$$$$T^{8} +$$$$45\!\cdots\!04$$$$T^{10} +$$$$11\!\cdots\!21$$$$T^{12} +$$$$12\!\cdots\!94$$$$T^{14} +$$$$95\!\cdots\!99$$$$T^{16} + 369606619643994 T^{18} + T^{20}$$
$59$ $$12\!\cdots\!16$$$$-$$$$19\!\cdots\!84$$$$T^{2} +$$$$29\!\cdots\!28$$$$T^{4} -$$$$61\!\cdots\!88$$$$T^{6} +$$$$80\!\cdots\!00$$$$T^{8} -$$$$65\!\cdots\!12$$$$T^{10} +$$$$38\!\cdots\!85$$$$T^{12} -$$$$15\!\cdots\!78$$$$T^{14} +$$$$47\!\cdots\!23$$$$T^{16} - 860197517576574 T^{18} + T^{20}$$
$61$ $$($$$$76\!\cdots\!32$$$$+$$$$45\!\cdots\!12$$$$T +$$$$26\!\cdots\!96$$$$T^{2} -$$$$37\!\cdots\!88$$$$T^{3} -$$$$11\!\cdots\!32$$$$T^{4} +$$$$23\!\cdots\!72$$$$T^{5} +$$$$33\!\cdots\!24$$$$T^{6} -$$$$15\!\cdots\!12$$$$T^{7} - 178305275604228 T^{8} + 7705212 T^{9} + T^{10} )^{2}$$
$67$ $$($$$$16\!\cdots\!84$$$$+$$$$18\!\cdots\!12$$$$T +$$$$15\!\cdots\!68$$$$T^{2} +$$$$43\!\cdots\!96$$$$T^{3} +$$$$97\!\cdots\!05$$$$T^{4} +$$$$48\!\cdots\!41$$$$T^{5} +$$$$43\!\cdots\!30$$$$T^{6} +$$$$14\!\cdots\!21$$$$T^{7} + 1401124490666322 T^{8} + 35020537 T^{9} + T^{10} )^{2}$$
$71$ $$( -$$$$16\!\cdots\!32$$$$+$$$$66\!\cdots\!72$$$$T^{2} -$$$$30\!\cdots\!16$$$$T^{4} +$$$$53\!\cdots\!68$$$$T^{6} - 3899058060734424 T^{8} + T^{10} )^{2}$$
$73$ $$($$$$47\!\cdots\!00$$$$-$$$$18\!\cdots\!80$$$$T +$$$$21\!\cdots\!16$$$$T^{2} +$$$$15\!\cdots\!72$$$$T^{3} +$$$$15\!\cdots\!63$$$$T^{4} -$$$$12\!\cdots\!85$$$$T^{5} +$$$$18\!\cdots\!48$$$$T^{6} -$$$$13\!\cdots\!45$$$$T^{7} + 5232684730192572 T^{8} - 110132049 T^{9} + T^{10} )^{2}$$
$79$ $$($$$$38\!\cdots\!29$$$$-$$$$44\!\cdots\!85$$$$T +$$$$75\!\cdots\!23$$$$T^{2} -$$$$37\!\cdots\!86$$$$T^{3} +$$$$31\!\cdots\!93$$$$T^{4} -$$$$91\!\cdots\!23$$$$T^{5} +$$$$58\!\cdots\!77$$$$T^{6} -$$$$59\!\cdots\!02$$$$T^{7} + 2554158425902035 T^{8} - 6430289 T^{9} + T^{10} )^{2}$$
$83$ $$($$$$18\!\cdots\!96$$$$+$$$$66\!\cdots\!60$$$$T^{2} +$$$$48\!\cdots\!80$$$$T^{4} +$$$$14\!\cdots\!65$$$$T^{6} + 19686425940527550 T^{8} + T^{10} )^{2}$$
$89$ $$18\!\cdots\!00$$$$-$$$$34\!\cdots\!00$$$$T^{2} +$$$$45\!\cdots\!56$$$$T^{4} -$$$$27\!\cdots\!20$$$$T^{6} +$$$$11\!\cdots\!12$$$$T^{8} -$$$$28\!\cdots\!80$$$$T^{10} +$$$$52\!\cdots\!08$$$$T^{12} -$$$$64\!\cdots\!80$$$$T^{14} +$$$$56\!\cdots\!32$$$$T^{16} - 29621585487765240 T^{18} + T^{20}$$
$97$ $$($$$$54\!\cdots\!48$$$$+$$$$90\!\cdots\!12$$$$T^{2} +$$$$43\!\cdots\!84$$$$T^{4} +$$$$79\!\cdots\!53$$$$T^{6} + 49891734078363606 T^{8} + T^{10} )^{2}$$