# Properties

 Label 252.12.k.d Level $252$ Weight $12$ Character orbit 252.k Analytic conductor $193.622$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$12$$ Character orbit: $$[\chi]$$ $$=$$ 252.k (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$193.622481501$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 581500324 x^{14} - 481772282104 x^{13} + 132272376701859942 x^{12} +$$$$18\!\cdots\!08$$$$x^{11} -$$$$14\!\cdots\!08$$$$x^{10} -$$$$25\!\cdots\!56$$$$x^{9} +$$$$80\!\cdots\!79$$$$x^{8} +$$$$11\!\cdots\!68$$$$x^{7} -$$$$19\!\cdots\!68$$$$x^{6} +$$$$59\!\cdots\!08$$$$x^{5} +$$$$21\!\cdots\!06$$$$x^{4} -$$$$37\!\cdots\!04$$$$x^{3} -$$$$31\!\cdots\!28$$$$x^{2} +$$$$25\!\cdots\!24$$$$x +$$$$79\!\cdots\!77$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{42}\cdot 3^{15}\cdot 7^{9}$$ Twist minimal: no (minimal twist has level 84) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 269 \beta_{1} - \beta_{3} ) q^{5} + ( 1064 + 4186 \beta_{1} - \beta_{4} + \beta_{5} ) q^{7} +O(q^{10})$$ $$q + ( 269 \beta_{1} - \beta_{3} ) q^{5} + ( 1064 + 4186 \beta_{1} - \beta_{4} + \beta_{5} ) q^{7} + ( 27848 - 27847 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - \beta_{7} + \beta_{9} ) q^{11} + ( 168931 + 2 \beta_{1} + 36 \beta_{2} - 4 \beta_{4} - \beta_{8} + \beta_{9} ) q^{13} + ( -639310 + 639291 \beta_{1} - 22 \beta_{2} - 19 \beta_{3} + 45 \beta_{4} - 7 \beta_{5} - 3 \beta_{6} - 6 \beta_{7} + \beta_{8} + 6 \beta_{9} + 2 \beta_{10} - \beta_{11} - \beta_{13} - \beta_{14} ) q^{17} + ( -37 + 863778 \beta_{1} + 5 \beta_{2} - 175 \beta_{3} - 34 \beta_{4} + 61 \beta_{5} + \beta_{6} + 10 \beta_{7} - 5 \beta_{10} + 2 \beta_{13} - 3 \beta_{14} - \beta_{15} ) q^{19} + ( 78 + 6423053 \beta_{1} - 8 \beta_{2} - 684 \beta_{3} - 142 \beta_{5} - 5 \beta_{6} + 7 \beta_{7} + 8 \beta_{10} + \beta_{13} - 2 \beta_{14} + 5 \beta_{15} ) q^{23} + ( -23930950 + 23930931 \beta_{1} - 1812 \beta_{2} - 1809 \beta_{3} + 12 \beta_{4} - 399 \beta_{5} - 3 \beta_{6} + 19 \beta_{7} - 2 \beta_{8} - 19 \beta_{9} - 20 \beta_{10} - 12 \beta_{11} - 5 \beta_{12} + 2 \beta_{13} - 12 \beta_{14} + 5 \beta_{15} ) q^{25} + ( -7427324 + 446 \beta_{1} - 2067 \beta_{2} - 82 \beta_{3} - 1045 \beta_{4} - 40 \beta_{5} + 31 \beta_{6} - 21 \beta_{8} - 89 \beta_{9} + 51 \beta_{10} - 11 \beta_{12} ) q^{29} + ( 20583436 - 20583280 \beta_{1} - 2049 \beta_{2} - 2041 \beta_{3} - 262 \beta_{4} - 73 \beta_{5} - 8 \beta_{6} - 136 \beta_{7} + 32 \beta_{8} + 136 \beta_{9} - 8 \beta_{10} + 42 \beta_{11} - 5 \beta_{12} - 32 \beta_{13} + 42 \beta_{14} + 5 \beta_{15} ) q^{31} + ( -1547228 - 3814468 \beta_{1} - 3443 \beta_{2} - 15150 \beta_{3} - 136 \beta_{4} + 42 \beta_{5} - 133 \beta_{6} - \beta_{7} + 89 \beta_{8} + 383 \beta_{9} + 99 \beta_{10} + 19 \beta_{11} - 15 \beta_{12} - 78 \beta_{13} - 27 \beta_{14} - 25 \beta_{15} ) q^{35} + ( -2204 + 9465150 \beta_{1} + 261 \beta_{2} + 11459 \beta_{3} - 491 \beta_{4} + 3862 \beta_{5} + 123 \beta_{6} + 116 \beta_{7} - 261 \beta_{10} - 20 \beta_{13} - 24 \beta_{14} + 10 \beta_{15} ) q^{37} + ( 113464461 + 1471 \beta_{1} + 15048 \beta_{2} - 111 \beta_{3} - 1010 \beta_{4} + 1807 \beta_{5} - 112 \beta_{6} + 398 \beta_{8} - 628 \beta_{9} + 223 \beta_{10} + 124 \beta_{11} - 45 \beta_{12} ) q^{41} + ( 649497 + 3022 \beta_{1} + 43662 \beta_{2} - 194 \beta_{3} - 4837 \beta_{4} + 1450 \beta_{5} + 82 \beta_{6} + 274 \beta_{8} + 452 \beta_{9} + 112 \beta_{10} + 33 \beta_{11} - 110 \beta_{12} ) q^{43} + ( -5536 + 129318552 \beta_{1} + 593 \beta_{2} + 29443 \beta_{3} - 7351 \beta_{4} + 10048 \beta_{5} + 30 \beta_{6} + 70 \beta_{7} - 593 \beta_{10} - 426 \beta_{13} + 162 \beta_{14} + 43 \beta_{15} ) q^{47} + ( 333639744 - 151879142 \beta_{1} + 76166 \beta_{2} + 73425 \beta_{3} - 5906 \beta_{4} + 857 \beta_{5} + 397 \beta_{6} + 803 \beta_{7} - 277 \beta_{8} - 1460 \beta_{9} + 607 \beta_{10} + 234 \beta_{11} - 51 \beta_{12} + 859 \beta_{13} + 30 \beta_{14} - 134 \beta_{15} ) q^{49} + ( 83145816 - 83143140 \beta_{1} + 10216 \beta_{2} + 10667 \beta_{3} - 5398 \beta_{4} - 11911 \beta_{5} - 451 \beta_{6} - 3088 \beta_{7} - 790 \beta_{8} + 3088 \beta_{9} - 850 \beta_{10} - 98 \beta_{11} - 325 \beta_{12} + 790 \beta_{13} - 98 \beta_{14} + 325 \beta_{15} ) q^{53} + ( -79101842 + 32483 \beta_{1} + 129383 \beta_{2} - 2354 \beta_{3} - 59708 \beta_{4} + 8261 \beta_{5} + 1045 \beta_{6} - 1034 \beta_{8} + 6182 \beta_{9} + 1309 \beta_{10} + 396 \beta_{11} - 330 \beta_{12} ) q^{55} + ( -130104384 + 130087498 \beta_{1} + 160281 \beta_{2} + 163448 \beta_{3} + 40421 \beta_{4} - 80218 \beta_{5} - 3167 \beta_{6} - 636 \beta_{7} - 122 \beta_{8} + 636 \beta_{9} - 182 \beta_{10} + 497 \beta_{11} + 181 \beta_{12} + 122 \beta_{13} + 497 \beta_{14} - 181 \beta_{15} ) q^{59} + ( 5376 - 1798876419 \beta_{1} + 437 \beta_{2} + 106323 \beta_{3} + 41573 \beta_{4} - 11350 \beta_{5} + 2967 \beta_{6} + 5353 \beta_{7} - 437 \beta_{10} + 1265 \beta_{13} + 276 \beta_{14} + 506 \beta_{15} ) q^{61} + ( -35850 + 2617237750 \beta_{1} + 4770 \beta_{2} - 266415 \beta_{3} - 144405 \beta_{4} + 60085 \beta_{5} - 3020 \beta_{6} + 18960 \beta_{7} - 4770 \beta_{10} + 3145 \beta_{13} - 2075 \beta_{14} - 285 \beta_{15} ) q^{65} + ( -4163529197 + 4163561323 \beta_{1} + 170307 \beta_{2} + 164062 \beta_{3} - 77137 \beta_{4} + 113118 \beta_{5} + 6245 \beta_{6} + 6766 \beta_{7} + 76 \beta_{8} - 6766 \beta_{9} - 572 \beta_{10} + 177 \beta_{11} - 1265 \beta_{12} - 76 \beta_{13} + 177 \beta_{14} + 1265 \beta_{15} ) q^{67} + ( -4115817150 - 25304 \beta_{1} + 472412 \beta_{2} + 9245 \beta_{3} + 115436 \beta_{4} + 51878 \beta_{5} - 6433 \beta_{6} - 2936 \beta_{8} - 6326 \beta_{9} - 2812 \beta_{10} - 2728 \beta_{11} - 479 \beta_{12} ) q^{71} + ( 2213731729 - 2213708470 \beta_{1} - 226261 \beta_{2} - 228449 \beta_{3} - 47686 \beta_{4} + 202977 \beta_{5} + 2188 \beta_{6} - 20302 \beta_{7} + 1019 \beta_{8} + 20302 \beta_{9} + 3892 \beta_{10} - 684 \beta_{11} + 1695 \beta_{12} - 1019 \beta_{13} - 684 \beta_{14} - 1695 \beta_{15} ) q^{73} + ( 3547475329 - 8168605099 \beta_{1} + 136136 \beta_{2} + 1593900 \beta_{3} - 6944 \beta_{4} + 35775 \beta_{5} - 9111 \beta_{6} - 3569 \beta_{7} - 2784 \beta_{8} + 13470 \beta_{9} + 537 \beta_{10} + 3978 \beta_{11} + 673 \beta_{12} + 407 \beta_{13} + 4052 \beta_{14} - 1263 \beta_{15} ) q^{77} + ( -69010 - 3328113107 \beta_{1} + 11510 \beta_{2} + 644839 \beta_{3} + 259309 \beta_{4} + 112690 \beta_{5} + 19878 \beta_{6} + 12572 \beta_{7} - 11510 \beta_{10} + 8548 \beta_{13} - 2310 \beta_{14} - 869 \beta_{15} ) q^{79} + ( 13144762778 + 150089 \beta_{1} - 1450452 \beta_{2} + 16864 \beta_{3} + 49910 \beta_{4} + 308789 \beta_{5} - 24710 \beta_{6} - 6391 \beta_{8} - 13937 \beta_{9} + 7846 \beta_{10} - 275 \beta_{11} + 3575 \beta_{12} ) q^{83} + ( -1615366049 - 174124 \beta_{1} - 2296594 \beta_{2} - 26783 \beta_{3} - 92861 \beta_{4} - 383938 \beta_{5} + 28840 \beta_{6} - 13133 \beta_{8} - 18601 \beta_{9} - 2057 \beta_{10} - 1548 \beta_{11} - 1860 \beta_{12} ) q^{85} + ( 37456 + 6993972508 \beta_{1} - 16320 \beta_{2} - 410250 \beta_{3} - 484670 \beta_{4} - 36232 \beta_{5} - 32012 \beta_{6} - 40478 \beta_{7} + 16320 \beta_{10} + 15202 \beta_{13} + 6040 \beta_{14} + 1590 \beta_{15} ) q^{89} + ( 8154876015 - 8053079604 \beta_{1} + 2321862 \beta_{2} - 1587888 \beta_{3} - 235318 \beta_{4} + 213447 \beta_{5} + 49928 \beta_{6} + 32818 \beta_{7} + 396 \beta_{8} + 5016 \beta_{9} + 776 \beta_{10} + 6780 \beta_{11} + 4706 \beta_{12} - 17722 \beta_{13} + 11373 \beta_{14} - 2806 \beta_{15} ) q^{91} + ( -13347829082 + 13347287704 \beta_{1} - 5050727 \beta_{2} - 4974876 \beta_{3} + 1243049 \beta_{4} - 1841478 \beta_{5} - 75851 \beta_{6} + 31608 \beta_{7} + 45186 \beta_{8} - 31608 \beta_{9} + 6665 \beta_{10} + 1926 \beta_{11} - 5555 \beta_{12} - 45186 \beta_{13} + 1926 \beta_{14} + 5555 \beta_{15} ) q^{95} + ( -9764625871 - 267288 \beta_{1} + 2715087 \beta_{2} - 52582 \beta_{3} - 38227 \beta_{4} - 469591 \beta_{5} + 40292 \beta_{6} + 25035 \beta_{8} - 148062 \beta_{9} + 12290 \beta_{10} - 10338 \beta_{11} + 6605 \beta_{12} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 2156q^{5} + 50512q^{7} + O(q^{10})$$ $$16q + 2156q^{5} + 50512q^{7} + 222796q^{11} + 2703176q^{13} - 5114600q^{17} + 6910556q^{19} + 51387712q^{23} - 191456372q^{25} - 118854616q^{29} + 164659160q^{31} - 55239344q^{35} + 75658364q^{37} + 1815568608q^{41} + 10754408q^{43} + 1034359464q^{47} + 4123496848q^{49} + 665159988q^{53} - 1264543896q^{55} - 1040514580q^{59} - 14391208024q^{61} + 20938150200q^{65} - 33307097284q^{67} - 65848902896q^{71} + 17709749204q^{73} - 8594484604q^{77} - 26626784032q^{79} + 210306955048q^{83} - 25867402032q^{85} + 55951560072q^{89} + 66078280292q^{91} - 106810047392q^{95} - 156216030712q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 581500324 x^{14} - 481772282104 x^{13} + 132272376701859942 x^{12} +$$$$18\!\cdots\!08$$$$x^{11} -$$$$14\!\cdots\!08$$$$x^{10} -$$$$25\!\cdots\!56$$$$x^{9} +$$$$80\!\cdots\!79$$$$x^{8} +$$$$11\!\cdots\!68$$$$x^{7} -$$$$19\!\cdots\!68$$$$x^{6} +$$$$59\!\cdots\!08$$$$x^{5} +$$$$21\!\cdots\!06$$$$x^{4} -$$$$37\!\cdots\!04$$$$x^{3} -$$$$31\!\cdots\!28$$$$x^{2} +$$$$25\!\cdots\!24$$$$x +$$$$79\!\cdots\!77$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-$$$$68\!\cdots\!80$$$$\nu^{15} -$$$$16\!\cdots\!78$$$$\nu^{14} +$$$$39\!\cdots\!38$$$$\nu^{13} +$$$$12\!\cdots\!31$$$$\nu^{12} -$$$$87\!\cdots\!12$$$$\nu^{11} -$$$$34\!\cdots\!47$$$$\nu^{10} +$$$$92\!\cdots\!98$$$$\nu^{9} +$$$$39\!\cdots\!72$$$$\nu^{8} -$$$$45\!\cdots\!96$$$$\nu^{7} -$$$$18\!\cdots\!03$$$$\nu^{6} +$$$$92\!\cdots\!94$$$$\nu^{5} +$$$$21\!\cdots\!32$$$$\nu^{4} -$$$$95\!\cdots\!52$$$$\nu^{3} +$$$$25\!\cdots\!91$$$$\nu^{2} +$$$$83\!\cdots\!08$$$$\nu +$$$$22\!\cdots\!73$$$$)/$$$$11\!\cdots\!32$$ $$\beta_{2}$$ $$=$$ $$($$$$-$$$$68\!\cdots\!80$$$$\nu^{15} -$$$$16\!\cdots\!78$$$$\nu^{14} +$$$$39\!\cdots\!38$$$$\nu^{13} +$$$$12\!\cdots\!31$$$$\nu^{12} -$$$$87\!\cdots\!12$$$$\nu^{11} -$$$$34\!\cdots\!47$$$$\nu^{10} +$$$$92\!\cdots\!98$$$$\nu^{9} +$$$$39\!\cdots\!72$$$$\nu^{8} -$$$$45\!\cdots\!96$$$$\nu^{7} -$$$$18\!\cdots\!03$$$$\nu^{6} +$$$$92\!\cdots\!94$$$$\nu^{5} +$$$$21\!\cdots\!32$$$$\nu^{4} -$$$$95\!\cdots\!52$$$$\nu^{3} +$$$$25\!\cdots\!91$$$$\nu^{2} +$$$$83\!\cdots\!40$$$$\nu +$$$$22\!\cdots\!73$$$$)/$$$$11\!\cdots\!32$$ $$\beta_{3}$$ $$=$$ $$($$$$54\!\cdots\!86$$$$\nu^{15} +$$$$13\!\cdots\!20$$$$\nu^{14} -$$$$31\!\cdots\!83$$$$\nu^{13} -$$$$10\!\cdots\!93$$$$\nu^{12} +$$$$70\!\cdots\!73$$$$\nu^{11} +$$$$27\!\cdots\!63$$$$\nu^{10} -$$$$74\!\cdots\!30$$$$\nu^{9} -$$$$31\!\cdots\!32$$$$\nu^{8} +$$$$36\!\cdots\!53$$$$\nu^{7} +$$$$14\!\cdots\!83$$$$\nu^{6} -$$$$73\!\cdots\!22$$$$\nu^{5} -$$$$17\!\cdots\!20$$$$\nu^{4} +$$$$76\!\cdots\!27$$$$\nu^{3} -$$$$20\!\cdots\!53$$$$\nu^{2} -$$$$66\!\cdots\!67$$$$\nu -$$$$18\!\cdots\!67$$$$)/$$$$38\!\cdots\!44$$ $$\beta_{4}$$ $$=$$ $$($$$$-$$$$40\!\cdots\!01$$$$\nu^{15} -$$$$10\!\cdots\!79$$$$\nu^{14} +$$$$23\!\cdots\!71$$$$\nu^{13} +$$$$77\!\cdots\!66$$$$\nu^{12} -$$$$51\!\cdots\!20$$$$\nu^{11} -$$$$20\!\cdots\!55$$$$\nu^{10} +$$$$54\!\cdots\!39$$$$\nu^{9} +$$$$23\!\cdots\!79$$$$\nu^{8} -$$$$26\!\cdots\!36$$$$\nu^{7} -$$$$11\!\cdots\!35$$$$\nu^{6} +$$$$53\!\cdots\!75$$$$\nu^{5} +$$$$12\!\cdots\!91$$$$\nu^{4} -$$$$55\!\cdots\!19$$$$\nu^{3} +$$$$13\!\cdots\!16$$$$\nu^{2} +$$$$47\!\cdots\!40$$$$\nu +$$$$13\!\cdots\!49$$$$)/$$$$14\!\cdots\!00$$ $$\beta_{5}$$ $$=$$ $$($$$$17\!\cdots\!43$$$$\nu^{15} +$$$$41\!\cdots\!69$$$$\nu^{14} -$$$$99\!\cdots\!71$$$$\nu^{13} -$$$$32\!\cdots\!36$$$$\nu^{12} +$$$$22\!\cdots\!62$$$$\nu^{11} +$$$$86\!\cdots\!59$$$$\nu^{10} -$$$$23\!\cdots\!29$$$$\nu^{9} -$$$$99\!\cdots\!09$$$$\nu^{8} +$$$$11\!\cdots\!86$$$$\nu^{7} +$$$$47\!\cdots\!23$$$$\nu^{6} -$$$$23\!\cdots\!69$$$$\nu^{5} -$$$$54\!\cdots\!01$$$$\nu^{4} +$$$$24\!\cdots\!39$$$$\nu^{3} -$$$$65\!\cdots\!26$$$$\nu^{2} -$$$$21\!\cdots\!98$$$$\nu -$$$$57\!\cdots\!93$$$$)/$$$$14\!\cdots\!00$$ $$\beta_{6}$$ $$=$$ $$($$$$24\!\cdots\!46$$$$\nu^{15} +$$$$55\!\cdots\!65$$$$\nu^{14} -$$$$13\!\cdots\!90$$$$\nu^{13} -$$$$43\!\cdots\!85$$$$\nu^{12} +$$$$31\!\cdots\!96$$$$\nu^{11} +$$$$11\!\cdots\!32$$$$\nu^{10} -$$$$32\!\cdots\!00$$$$\nu^{9} -$$$$13\!\cdots\!75$$$$\nu^{8} +$$$$16\!\cdots\!60$$$$\nu^{7} +$$$$64\!\cdots\!04$$$$\nu^{6} -$$$$33\!\cdots\!52$$$$\nu^{5} -$$$$75\!\cdots\!35$$$$\nu^{4} +$$$$34\!\cdots\!10$$$$\nu^{3} -$$$$10\!\cdots\!65$$$$\nu^{2} -$$$$30\!\cdots\!54$$$$\nu -$$$$84\!\cdots\!82$$$$)/$$$$10\!\cdots\!00$$ $$\beta_{7}$$ $$=$$ $$($$$$95\!\cdots\!24$$$$\nu^{15} +$$$$22\!\cdots\!47$$$$\nu^{14} -$$$$55\!\cdots\!53$$$$\nu^{13} -$$$$17\!\cdots\!33$$$$\nu^{12} +$$$$12\!\cdots\!51$$$$\nu^{11} +$$$$46\!\cdots\!82$$$$\nu^{10} -$$$$12\!\cdots\!12$$$$\nu^{9} -$$$$53\!\cdots\!67$$$$\nu^{8} +$$$$64\!\cdots\!63$$$$\nu^{7} +$$$$25\!\cdots\!14$$$$\nu^{6} -$$$$13\!\cdots\!32$$$$\nu^{5} -$$$$29\!\cdots\!83$$$$\nu^{4} +$$$$13\!\cdots\!47$$$$\nu^{3} -$$$$39\!\cdots\!13$$$$\nu^{2} -$$$$12\!\cdots\!59$$$$\nu -$$$$33\!\cdots\!74$$$$)/$$$$34\!\cdots\!00$$ $$\beta_{8}$$ $$=$$ $$($$$$-$$$$35\!\cdots\!90$$$$\nu^{15} -$$$$85\!\cdots\!31$$$$\nu^{14} +$$$$20\!\cdots\!14$$$$\nu^{13} +$$$$66\!\cdots\!14$$$$\nu^{12} -$$$$45\!\cdots\!66$$$$\nu^{11} -$$$$17\!\cdots\!97$$$$\nu^{10} +$$$$48\!\cdots\!06$$$$\nu^{9} +$$$$20\!\cdots\!11$$$$\nu^{8} -$$$$23\!\cdots\!34$$$$\nu^{7} -$$$$97\!\cdots\!29$$$$\nu^{6} +$$$$48\!\cdots\!62$$$$\nu^{5} +$$$$11\!\cdots\!39$$$$\nu^{4} -$$$$49\!\cdots\!56$$$$\nu^{3} +$$$$13\!\cdots\!94$$$$\nu^{2} +$$$$43\!\cdots\!84$$$$\nu +$$$$11\!\cdots\!53$$$$)/$$$$52\!\cdots\!00$$ $$\beta_{9}$$ $$=$$ $$($$$$-$$$$26\!\cdots\!18$$$$\nu^{15} -$$$$64\!\cdots\!72$$$$\nu^{14} +$$$$15\!\cdots\!18$$$$\nu^{13} +$$$$49\!\cdots\!23$$$$\nu^{12} -$$$$34\!\cdots\!70$$$$\nu^{11} -$$$$13\!\cdots\!65$$$$\nu^{10} +$$$$35\!\cdots\!72$$$$\nu^{9} +$$$$15\!\cdots\!62$$$$\nu^{8} -$$$$17\!\cdots\!98$$$$\nu^{7} -$$$$72\!\cdots\!65$$$$\nu^{6} +$$$$35\!\cdots\!20$$$$\nu^{5} +$$$$84\!\cdots\!58$$$$\nu^{4} -$$$$37\!\cdots\!12$$$$\nu^{3} +$$$$98\!\cdots\!43$$$$\nu^{2} +$$$$32\!\cdots\!90$$$$\nu +$$$$88\!\cdots\!67$$$$)/$$$$34\!\cdots\!00$$ $$\beta_{10}$$ $$=$$ $$($$$$-$$$$14\!\cdots\!87$$$$\nu^{15} -$$$$34\!\cdots\!61$$$$\nu^{14} +$$$$82\!\cdots\!24$$$$\nu^{13} +$$$$26\!\cdots\!24$$$$\nu^{12} -$$$$18\!\cdots\!93$$$$\nu^{11} -$$$$71\!\cdots\!31$$$$\nu^{10} +$$$$19\!\cdots\!91$$$$\nu^{9} +$$$$82\!\cdots\!11$$$$\nu^{8} -$$$$94\!\cdots\!49$$$$\nu^{7} -$$$$39\!\cdots\!87$$$$\nu^{6} +$$$$19\!\cdots\!31$$$$\nu^{5} +$$$$45\!\cdots\!79$$$$\nu^{4} -$$$$19\!\cdots\!66$$$$\nu^{3} +$$$$51\!\cdots\!74$$$$\nu^{2} +$$$$17\!\cdots\!87$$$$\nu +$$$$47\!\cdots\!87$$$$)/$$$$10\!\cdots\!00$$ $$\beta_{11}$$ $$=$$ $$($$$$16\!\cdots\!19$$$$\nu^{15} +$$$$38\!\cdots\!86$$$$\nu^{14} -$$$$92\!\cdots\!54$$$$\nu^{13} -$$$$30\!\cdots\!59$$$$\nu^{12} +$$$$20\!\cdots\!15$$$$\nu^{11} +$$$$79\!\cdots\!05$$$$\nu^{10} -$$$$21\!\cdots\!01$$$$\nu^{9} -$$$$92\!\cdots\!06$$$$\nu^{8} +$$$$10\!\cdots\!79$$$$\nu^{7} +$$$$43\!\cdots\!05$$$$\nu^{6} -$$$$21\!\cdots\!25$$$$\nu^{5} -$$$$50\!\cdots\!34$$$$\nu^{4} +$$$$22\!\cdots\!16$$$$\nu^{3} -$$$$59\!\cdots\!09$$$$\nu^{2} -$$$$19\!\cdots\!35$$$$\nu -$$$$53\!\cdots\!31$$$$)/$$$$10\!\cdots\!00$$ $$\beta_{12}$$ $$=$$ $$($$$$-$$$$20\!\cdots\!17$$$$\nu^{15} -$$$$48\!\cdots\!19$$$$\nu^{14} +$$$$11\!\cdots\!16$$$$\nu^{13} +$$$$37\!\cdots\!36$$$$\nu^{12} -$$$$25\!\cdots\!51$$$$\nu^{11} -$$$$10\!\cdots\!97$$$$\nu^{10} +$$$$27\!\cdots\!69$$$$\nu^{9} +$$$$11\!\cdots\!49$$$$\nu^{8} -$$$$13\!\cdots\!11$$$$\nu^{7} -$$$$55\!\cdots\!09$$$$\nu^{6} +$$$$27\!\cdots\!37$$$$\nu^{5} +$$$$64\!\cdots\!01$$$$\nu^{4} -$$$$28\!\cdots\!94$$$$\nu^{3} +$$$$75\!\cdots\!46$$$$\nu^{2} +$$$$24\!\cdots\!09$$$$\nu +$$$$67\!\cdots\!21$$$$)/$$$$52\!\cdots\!00$$ $$\beta_{13}$$ $$=$$ $$($$$$-$$$$88\!\cdots\!88$$$$\nu^{15} -$$$$21\!\cdots\!10$$$$\nu^{14} +$$$$50\!\cdots\!55$$$$\nu^{13} +$$$$16\!\cdots\!35$$$$\nu^{12} -$$$$11\!\cdots\!63$$$$\nu^{11} -$$$$44\!\cdots\!11$$$$\nu^{10} +$$$$11\!\cdots\!60$$$$\nu^{9} +$$$$50\!\cdots\!10$$$$\nu^{8} -$$$$58\!\cdots\!35$$$$\nu^{7} -$$$$24\!\cdots\!07$$$$\nu^{6} +$$$$11\!\cdots\!76$$$$\nu^{5} +$$$$28\!\cdots\!70$$$$\nu^{4} -$$$$12\!\cdots\!95$$$$\nu^{3} +$$$$32\!\cdots\!95$$$$\nu^{2} +$$$$10\!\cdots\!57$$$$\nu +$$$$29\!\cdots\!31$$$$)/$$$$10\!\cdots\!00$$ $$\beta_{14}$$ $$=$$ $$($$$$-$$$$12\!\cdots\!84$$$$\nu^{15} -$$$$30\!\cdots\!97$$$$\nu^{14} +$$$$72\!\cdots\!18$$$$\nu^{13} +$$$$23\!\cdots\!83$$$$\nu^{12} -$$$$16\!\cdots\!66$$$$\nu^{11} -$$$$62\!\cdots\!22$$$$\nu^{10} +$$$$17\!\cdots\!22$$$$\nu^{9} +$$$$72\!\cdots\!47$$$$\nu^{8} -$$$$84\!\cdots\!78$$$$\nu^{7} -$$$$34\!\cdots\!34$$$$\nu^{6} +$$$$16\!\cdots\!02$$$$\nu^{5} +$$$$39\!\cdots\!63$$$$\nu^{4} -$$$$17\!\cdots\!42$$$$\nu^{3} +$$$$47\!\cdots\!23$$$$\nu^{2} +$$$$15\!\cdots\!64$$$$\nu +$$$$41\!\cdots\!84$$$$)/$$$$52\!\cdots\!00$$ $$\beta_{15}$$ $$=$$ $$($$$$29\!\cdots\!89$$$$\nu^{15} +$$$$70\!\cdots\!86$$$$\nu^{14} -$$$$16\!\cdots\!39$$$$\nu^{13} -$$$$54\!\cdots\!39$$$$\nu^{12} +$$$$37\!\cdots\!20$$$$\nu^{11} +$$$$14\!\cdots\!65$$$$\nu^{10} -$$$$39\!\cdots\!21$$$$\nu^{9} -$$$$16\!\cdots\!56$$$$\nu^{8} +$$$$19\!\cdots\!44$$$$\nu^{7} +$$$$80\!\cdots\!45$$$$\nu^{6} -$$$$39\!\cdots\!25$$$$\nu^{5} -$$$$93\!\cdots\!24$$$$\nu^{4} +$$$$41\!\cdots\!71$$$$\nu^{3} -$$$$11\!\cdots\!09$$$$\nu^{2} -$$$$35\!\cdots\!90$$$$\nu -$$$$98\!\cdots\!41$$$$)/$$$$10\!\cdots\!00$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{2} - \beta_{1}$$ $$\nu^{2}$$ $$=$$ $$5 \beta_{12} + 12 \beta_{11} + 23 \beta_{10} + 19 \beta_{9} + 2 \beta_{8} - 20 \beta_{6} + 367 \beta_{5} - 376 \beta_{4} - \beta_{3} + 1271 \beta_{2} + 387 \beta_{1} + 72686726$$ $$\nu^{3}$$ $$=$$ $$-15 \beta_{15} + 36 \beta_{14} - 6 \beta_{13} - 6215 \beta_{12} + 69038 \beta_{11} + 20558 \beta_{10} + 202806 \beta_{9} - 109157 \beta_{8} - 57 \beta_{7} + 10334 \beta_{6} - 404016 \beta_{5} - 877977 \beta_{4} - 27013 \beta_{3} + 126629123 \beta_{2} - 217809794 \beta_{1} + 90378252009$$ $$\nu^{4}$$ $$=$$ $$24890 \beta_{15} + 276080 \beta_{14} + 436640 \beta_{13} + 643637515 \beta_{12} + 1754420396 \beta_{11} + 4025476974 \beta_{10} + 6031651252 \beta_{9} + 273003141 \beta_{8} - 811110 \beta_{7} - 3790045230 \beta_{6} + 76374085911 \beta_{5} - 77261681693 \beta_{4} + 271159228 \beta_{3} + 394588846773 \beta_{2} - 281603733669 \beta_{1} + 9199680358352342$$ $$\nu^{5}$$ $$=$$ $$-3218249875 \beta_{15} + 8771411960 \beta_{14} - 1366107335 \beta_{13} - 1460566381880 \beta_{12} + 14242035566564 \beta_{11} + 15636881447401 \beta_{10} + 54584348465293 \beta_{9} - 20940546595026 \beta_{8} - 30156228770 \beta_{7} + 1294989857905 \beta_{6} + 68709487126579 \beta_{5} - 488322419943257 \beta_{4} - 14940066144606 \beta_{3} + 17628689608115725 \beta_{2} - 45721151844183187 \beta_{1} + 28259692437719391999$$ $$\nu^{6}$$ $$=$$ $$8773053227580 \beta_{15} + 85425901234104 \beta_{14} + 125647381166961 \beta_{13} + 72815001702102170 \beta_{12} + 278551431443853452 \beta_{11} + 667577344298859007 \beta_{10} + 1286838026296585024 \beta_{9} + 27114591278759112 \beta_{8} - 327415628188773 \beta_{7} - 604888633842308449 \beta_{6} + 13103784385572315526 \beta_{5} - 14246103601469100073 \beta_{4} + 43171276864778397 \beta_{3} + 84555601172402729953 \beta_{2} - 155334999644262246407 \beta_{1} + 1279266236680765484604515$$ $$\nu^{7}$$ $$=$$ $$-509735751392635565 \beta_{15} + 1949561121552480758 \beta_{14} - 190241919129525228 \beta_{13} - 245105831210677023290 \beta_{12} + 2596818447840323503846 \beta_{11} + 4364532885897707384822 \beta_{10} + 12199084410116948439077 \beta_{9} - 3570963345181939003449 \beta_{8} - 9006720546017837146 \beta_{7} - 370821133820043928573 \beta_{6} + 39325998974079021254304 \beta_{5} - 128538326524604236915807 \beta_{4} - 3397519991502422756792 \beta_{3} + 2586169072836155249014800 \beta_{2} - 8870894206794174860552757 \beta_{1} + 6070626374179100007085274947$$ $$\nu^{8}$$ $$=$$ $$1962885715513732437390 \beta_{15} + 20766750534199001182872 \beta_{14} + 28568469488195375635398 \beta_{13} + 7727968103595048859809185 \beta_{12} + 45911113498479044290410996 \beta_{11} + 111009646866411186310465166 \beta_{10} + 246345925449675004416294502 \beta_{9} - 119308358206630480604899 \beta_{8} - 97556652982570327840164 \beta_{7} - 93816986524936276249369832 \beta_{6} + 2169149903321084794167821353 \beta_{5} - 2552988401677547782215050719 \beta_{4} + 3529225219299045764319632 \beta_{3} + 16349355204929160473913929357 \beta_{2} - 46106992006851514336762409121 \beta_{1} + 187424379926477240886005279083259$$ $$\nu^{9}$$ $$=$$ $$-69560555093318898251537175 \beta_{15} + 413106606201008022612403716 \beta_{14} + 945213686808199164773109 \beta_{13} - 39769348123402665226563070130 \beta_{12} + 462376505121435333100859564444 \beta_{11} + 954781589123528763684626286917 \beta_{10} + 2481501497998719648715113126603 \beta_{9} - 585829519692711070740774021446 \beta_{8} - 2216674486229624441497888992 \beta_{7} - 176724857511084299918187288361 \beta_{6} + 10716134432867102452372887421395 \beta_{5} - 27429243162339137943008122636785 \beta_{4} - 630006685699856035981065941044 \beta_{3} + 394487405039435137424277543508214 \beta_{2} - 1667606956017170900982378070494368 \beta_{1} + 1175266726244393105607754882141554399$$ $$\nu^{10}$$ $$=$$ $$39\!\cdots\!40$$$$\beta_{15} +$$$$46\!\cdots\!32$$$$\beta_{14} +$$$$58\!\cdots\!13$$$$\beta_{13} +$$$$76\!\cdots\!45$$$$\beta_{12} +$$$$76\!\cdots\!56$$$$\beta_{11} +$$$$18\!\cdots\!76$$$$\beta_{10} +$$$$44\!\cdots\!97$$$$\beta_{9} -$$$$85\!\cdots\!34$$$$\beta_{8} -$$$$24\!\cdots\!09$$$$\beta_{7} -$$$$14\!\cdots\!37$$$$\beta_{6} +$$$$35\!\cdots\!43$$$$\beta_{5} -$$$$45\!\cdots\!09$$$$\beta_{4} -$$$$17\!\cdots\!26$$$$\beta_{3} +$$$$30\!\cdots\!26$$$$\beta_{2} -$$$$11\!\cdots\!22$$$$\beta_{1} +$$$$28\!\cdots\!39$$ $$\nu^{11}$$ $$=$$ $$-$$$$83\!\cdots\!70$$$$\beta_{15} +$$$$84\!\cdots\!30$$$$\beta_{14} +$$$$93\!\cdots\!10$$$$\beta_{13} -$$$$65\!\cdots\!75$$$$\beta_{12} +$$$$81\!\cdots\!12$$$$\beta_{11} +$$$$18\!\cdots\!68$$$$\beta_{10} +$$$$47\!\cdots\!19$$$$\beta_{9} -$$$$94\!\cdots\!88$$$$\beta_{8} -$$$$49\!\cdots\!85$$$$\beta_{7} -$$$$47\!\cdots\!95$$$$\beta_{6} +$$$$23\!\cdots\!52$$$$\beta_{5} -$$$$53\!\cdots\!66$$$$\beta_{4} -$$$$10\!\cdots\!55$$$$\beta_{3} +$$$$62\!\cdots\!07$$$$\beta_{2} -$$$$31\!\cdots\!65$$$$\beta_{1} +$$$$21\!\cdots\!50$$ $$\nu^{12}$$ $$=$$ $$78\!\cdots\!60$$$$\beta_{15} +$$$$98\!\cdots\!08$$$$\beta_{14} +$$$$11\!\cdots\!22$$$$\beta_{13} +$$$$64\!\cdots\!40$$$$\beta_{12} +$$$$12\!\cdots\!40$$$$\beta_{11} +$$$$31\!\cdots\!18$$$$\beta_{10} +$$$$80\!\cdots\!80$$$$\beta_{9} -$$$$27\!\cdots\!40$$$$\beta_{8} -$$$$57\!\cdots\!46$$$$\beta_{7} -$$$$22\!\cdots\!78$$$$\beta_{6} +$$$$59\!\cdots\!48$$$$\beta_{5} -$$$$80\!\cdots\!54$$$$\beta_{4} -$$$$15\!\cdots\!90$$$$\beta_{3} +$$$$54\!\cdots\!10$$$$\beta_{2} -$$$$25\!\cdots\!26$$$$\beta_{1} +$$$$44\!\cdots\!81$$ $$\nu^{13}$$ $$=$$ $$-$$$$83\!\cdots\!30$$$$\beta_{15} +$$$$16\!\cdots\!32$$$$\beta_{14} +$$$$36\!\cdots\!28$$$$\beta_{13} -$$$$11\!\cdots\!60$$$$\beta_{12} +$$$$14\!\cdots\!48$$$$\beta_{11} +$$$$35\!\cdots\!04$$$$\beta_{10} +$$$$89\!\cdots\!26$$$$\beta_{9} -$$$$15\!\cdots\!82$$$$\beta_{8} -$$$$10\!\cdots\!84$$$$\beta_{7} -$$$$10\!\cdots\!42$$$$\beta_{6} +$$$$47\!\cdots\!60$$$$\beta_{5} -$$$$99\!\cdots\!10$$$$\beta_{4} -$$$$17\!\cdots\!60$$$$\beta_{3} +$$$$10\!\cdots\!31$$$$\beta_{2} -$$$$57\!\cdots\!89$$$$\beta_{1} +$$$$39\!\cdots\!78$$ $$\nu^{14}$$ $$=$$ $$15\!\cdots\!40$$$$\beta_{15} +$$$$20\!\cdots\!00$$$$\beta_{14} +$$$$21\!\cdots\!20$$$$\beta_{13} +$$$$33\!\cdots\!25$$$$\beta_{12} +$$$$21\!\cdots\!88$$$$\beta_{11} +$$$$54\!\cdots\!17$$$$\beta_{10} +$$$$14\!\cdots\!81$$$$\beta_{9} -$$$$67\!\cdots\!12$$$$\beta_{8} -$$$$12\!\cdots\!00$$$$\beta_{7} -$$$$36\!\cdots\!80$$$$\beta_{6} +$$$$99\!\cdots\!63$$$$\beta_{5} -$$$$14\!\cdots\!34$$$$\beta_{4} -$$$$41\!\cdots\!91$$$$\beta_{3} +$$$$98\!\cdots\!31$$$$\beta_{2} -$$$$54\!\cdots\!41$$$$\beta_{1} +$$$$72\!\cdots\!32$$ $$\nu^{15}$$ $$=$$ $$-$$$$50\!\cdots\!55$$$$\beta_{15} +$$$$32\!\cdots\!20$$$$\beta_{14} +$$$$10\!\cdots\!40$$$$\beta_{13} -$$$$18\!\cdots\!05$$$$\beta_{12} +$$$$25\!\cdots\!06$$$$\beta_{11} +$$$$64\!\cdots\!04$$$$\beta_{10} +$$$$16\!\cdots\!72$$$$\beta_{9} -$$$$24\!\cdots\!59$$$$\beta_{8} -$$$$21\!\cdots\!15$$$$\beta_{7} -$$$$22\!\cdots\!60$$$$\beta_{6} +$$$$90\!\cdots\!86$$$$\beta_{5} -$$$$17\!\cdots\!73$$$$\beta_{4} -$$$$27\!\cdots\!23$$$$\beta_{3} +$$$$16\!\cdots\!93$$$$\beta_{2} -$$$$10\!\cdots\!66$$$$\beta_{1} +$$$$70\!\cdots\!57$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$73$$ $$127$$ $$\chi(n)$$ $$1$$ $$-\beta_{1}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 −11427.1 + 0.866025i −10168.9 + 0.866025i −6748.51 + 0.866025i −438.744 + 0.866025i 2396.04 + 0.866025i 2410.13 + 0.866025i 10810.2 + 0.866025i 13166.9 + 0.866025i −11427.1 − 0.866025i −10168.9 − 0.866025i −6748.51 − 0.866025i −438.744 − 0.866025i 2396.04 − 0.866025i 2410.13 − 0.866025i 10810.2 − 0.866025i 13166.9 − 0.866025i
0 0 0 −5578.82 + 9662.79i 0 24922.9 36826.3i 0 0 0
37.2 0 0 0 −4949.72 + 8573.16i 0 29163.1 + 33568.4i 0 0 0
37.3 0 0 0 −3239.51 + 5610.99i 0 −44011.6 + 6348.37i 0 0 0
37.4 0 0 0 −84.6221 + 146.570i 0 −9851.70 43362.1i 0 0 0
37.5 0 0 0 1332.77 2308.42i 0 43648.5 8493.30i 0 0 0
37.6 0 0 0 1339.81 2320.63i 0 −28039.7 + 34512.4i 0 0 0
37.7 0 0 0 5539.86 9595.32i 0 −32020.8 30854.4i 0 0 0
37.8 0 0 0 6718.22 11636.3i 0 41445.4 + 16112.4i 0 0 0
109.1 0 0 0 −5578.82 9662.79i 0 24922.9 + 36826.3i 0 0 0
109.2 0 0 0 −4949.72 8573.16i 0 29163.1 33568.4i 0 0 0
109.3 0 0 0 −3239.51 5610.99i 0 −44011.6 6348.37i 0 0 0
109.4 0 0 0 −84.6221 146.570i 0 −9851.70 + 43362.1i 0 0 0
109.5 0 0 0 1332.77 + 2308.42i 0 43648.5 + 8493.30i 0 0 0
109.6 0 0 0 1339.81 + 2320.63i 0 −28039.7 34512.4i 0 0 0
109.7 0 0 0 5539.86 + 9595.32i 0 −32020.8 + 30854.4i 0 0 0
109.8 0 0 0 6718.22 + 11636.3i 0 41445.4 16112.4i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 109.8 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.12.k.d 16
3.b odd 2 1 84.12.i.b 16
7.c even 3 1 inner 252.12.k.d 16
21.h odd 6 1 84.12.i.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.12.i.b 16 3.b odd 2 1
84.12.i.b 16 21.h odd 6 1
252.12.k.d 16 1.a even 1 1 trivial
252.12.k.d 16 7.c even 3 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$59\!\cdots\!19$$$$T_{5}^{12} +$$$$36\!\cdots\!40$$$$T_{5}^{11} +$$$$60\!\cdots\!50$$$$T_{5}^{10} +$$$$65\!\cdots\!00$$$$T_{5}^{9} +$$$$44\!\cdots\!25$$$$T_{5}^{8} -$$$$23\!\cdots\!00$$$$T_{5}^{7} +$$$$10\!\cdots\!00$$$$T_{5}^{6} -$$$$32\!\cdots\!00$$$$T_{5}^{5} +$$$$18\!\cdots\!00$$$$T_{5}^{4} -$$$$29\!\cdots\!00$$$$T_{5}^{3} +$$$$52\!\cdots\!00$$$$T_{5}^{2} +$$$$88\!\cdots\!00$$$$T_{5} +$$$$16\!\cdots\!00$$">$$T_{5}^{16} - \cdots$$ acting on $$S_{12}^{\mathrm{new}}(252, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16}$$
$5$ $$16\!\cdots\!00$$$$+$$$$88\!\cdots\!00$$$$T +$$$$52\!\cdots\!00$$$$T^{2} -$$$$29\!\cdots\!00$$$$T^{3} +$$$$18\!\cdots\!00$$$$T^{4} -$$$$32\!\cdots\!00$$$$T^{5} +$$$$10\!\cdots\!00$$$$T^{6} -$$$$23\!\cdots\!00$$$$T^{7} +$$$$44\!\cdots\!25$$$$T^{8} +$$$$65\!\cdots\!00$$$$T^{9} +$$$$60\!\cdots\!50$$$$T^{10} + 36135501613992262840 T^{11} + 59972577625827719 T^{12} + 166151332768 T^{13} + 293364854 T^{14} - 2156 T^{15} + T^{16}$$
$7$ $$23\!\cdots\!01$$$$-$$$$59\!\cdots\!84$$$$T -$$$$46\!\cdots\!48$$$$T^{2} +$$$$12\!\cdots\!48$$$$T^{3} +$$$$69\!\cdots\!42$$$$T^{4} -$$$$94\!\cdots\!24$$$$T^{5} -$$$$43\!\cdots\!64$$$$T^{6} +$$$$64\!\cdots\!24$$$$T^{7} +$$$$88\!\cdots\!39$$$$T^{8} +$$$$32\!\cdots\!68$$$$T^{9} -$$$$11\!\cdots\!36$$$$T^{10} -$$$$12\!\cdots\!32$$$$T^{11} + 4524021951195991342 T^{12} + 39745685096936 T^{13} - 786017352 T^{14} - 50512 T^{15} + T^{16}$$
$11$ $$11\!\cdots\!84$$$$+$$$$19\!\cdots\!00$$$$T +$$$$41\!\cdots\!04$$$$T^{2} -$$$$87\!\cdots\!64$$$$T^{3} +$$$$10\!\cdots\!08$$$$T^{4} -$$$$14\!\cdots\!00$$$$T^{5} +$$$$13\!\cdots\!84$$$$T^{6} -$$$$20\!\cdots\!40$$$$T^{7} +$$$$10\!\cdots\!77$$$$T^{8} -$$$$93\!\cdots\!16$$$$T^{9} +$$$$40\!\cdots\!34$$$$T^{10} -$$$$28\!\cdots\!80$$$$T^{11} +$$$$10\!\cdots\!67$$$$T^{12} - 335100284799754240 T^{13} + 1215430595858 T^{14} - 222796 T^{15} + T^{16}$$
$13$ $$($$$$10\!\cdots\!00$$$$+$$$$25\!\cdots\!00$$$$T -$$$$36\!\cdots\!44$$$$T^{2} -$$$$15\!\cdots\!08$$$$T^{3} +$$$$30\!\cdots\!37$$$$T^{4} + 9797327173900188700 T^{5} - 9885557561546 T^{6} - 1351588 T^{7} + T^{8} )^{2}$$
$17$ $$39\!\cdots\!36$$$$-$$$$75\!\cdots\!40$$$$T +$$$$17\!\cdots\!80$$$$T^{2} +$$$$48\!\cdots\!40$$$$T^{3} +$$$$42\!\cdots\!36$$$$T^{4} +$$$$32\!\cdots\!40$$$$T^{5} +$$$$34\!\cdots\!36$$$$T^{6} +$$$$22\!\cdots\!00$$$$T^{7} +$$$$18\!\cdots\!32$$$$T^{8} +$$$$67\!\cdots\!40$$$$T^{9} +$$$$57\!\cdots\!68$$$$T^{10} +$$$$23\!\cdots\!80$$$$T^{11} +$$$$12\!\cdots\!68$$$$T^{12} +$$$$32\!\cdots\!40$$$$T^{13} + 145405856269568 T^{14} + 5114600 T^{15} + T^{16}$$
$19$ $$11\!\cdots\!36$$$$+$$$$90\!\cdots\!76$$$$T +$$$$51\!\cdots\!04$$$$T^{2} +$$$$16\!\cdots\!28$$$$T^{3} +$$$$42\!\cdots\!08$$$$T^{4} +$$$$63\!\cdots\!20$$$$T^{5} +$$$$83\!\cdots\!28$$$$T^{6} +$$$$51\!\cdots\!36$$$$T^{7} +$$$$82\!\cdots\!45$$$$T^{8} +$$$$26\!\cdots\!80$$$$T^{9} +$$$$66\!\cdots\!38$$$$T^{10} -$$$$79\!\cdots\!88$$$$T^{11} +$$$$31\!\cdots\!39$$$$T^{12} -$$$$25\!\cdots\!64$$$$T^{13} + 678373754866794 T^{14} - 6910556 T^{15} + T^{16}$$
$23$ $$55\!\cdots\!44$$$$-$$$$82\!\cdots\!80$$$$T +$$$$11\!\cdots\!80$$$$T^{2} -$$$$49\!\cdots\!00$$$$T^{3} +$$$$34\!\cdots\!36$$$$T^{4} -$$$$94\!\cdots\!12$$$$T^{5} +$$$$62\!\cdots\!52$$$$T^{6} -$$$$13\!\cdots\!44$$$$T^{7} +$$$$66\!\cdots\!52$$$$T^{8} -$$$$11\!\cdots\!92$$$$T^{9} +$$$$49\!\cdots\!20$$$$T^{10} -$$$$70\!\cdots\!00$$$$T^{11} +$$$$22\!\cdots\!00$$$$T^{12} -$$$$22\!\cdots\!44$$$$T^{13} + 6666026666325056 T^{14} - 51387712 T^{15} + T^{16}$$
$29$ $$( -$$$$52\!\cdots\!48$$$$-$$$$92\!\cdots\!20$$$$T -$$$$28\!\cdots\!08$$$$T^{2} +$$$$19\!\cdots\!72$$$$T^{3} +$$$$76\!\cdots\!33$$$$T^{4} -$$$$22\!\cdots\!72$$$$T^{5} - 51378044248844398 T^{6} + 59427308 T^{7} + T^{8} )^{2}$$
$31$ $$15\!\cdots\!69$$$$+$$$$10\!\cdots\!04$$$$T +$$$$51\!\cdots\!56$$$$T^{2} +$$$$11\!\cdots\!84$$$$T^{3} +$$$$20\!\cdots\!90$$$$T^{4} +$$$$19\!\cdots\!00$$$$T^{5} +$$$$18\!\cdots\!28$$$$T^{6} +$$$$89\!\cdots\!48$$$$T^{7} +$$$$85\!\cdots\!31$$$$T^{8} +$$$$26\!\cdots\!44$$$$T^{9} +$$$$29\!\cdots\!40$$$$T^{10} +$$$$24\!\cdots\!68$$$$T^{11} +$$$$68\!\cdots\!34$$$$T^{12} -$$$$21\!\cdots\!84$$$$T^{13} + 120948747053994744 T^{14} - 164659160 T^{15} + T^{16}$$
$37$ $$20\!\cdots\!00$$$$+$$$$28\!\cdots\!00$$$$T +$$$$17\!\cdots\!00$$$$T^{2} +$$$$56\!\cdots\!00$$$$T^{3} +$$$$98\!\cdots\!76$$$$T^{4} +$$$$18\!\cdots\!44$$$$T^{5} +$$$$29\!\cdots\!96$$$$T^{6} -$$$$34\!\cdots\!20$$$$T^{7} +$$$$62\!\cdots\!45$$$$T^{8} -$$$$49\!\cdots\!96$$$$T^{9} +$$$$76\!\cdots\!82$$$$T^{10} -$$$$96\!\cdots\!64$$$$T^{11} +$$$$66\!\cdots\!55$$$$T^{12} -$$$$66\!\cdots\!40$$$$T^{13} + 317539801427310666 T^{14} - 75658364 T^{15} + T^{16}$$
$41$ $$( -$$$$46\!\cdots\!68$$$$-$$$$43\!\cdots\!16$$$$T +$$$$52\!\cdots\!96$$$$T^{2} -$$$$13\!\cdots\!32$$$$T^{3} +$$$$27\!\cdots\!92$$$$T^{4} +$$$$26\!\cdots\!60$$$$T^{5} - 2126262369348540072 T^{6} - 907784304 T^{7} + T^{8} )^{2}$$
$43$ $$($$$$45\!\cdots\!92$$$$+$$$$31\!\cdots\!72$$$$T -$$$$10\!\cdots\!48$$$$T^{2} -$$$$62\!\cdots\!92$$$$T^{3} +$$$$35\!\cdots\!41$$$$T^{4} +$$$$55\!\cdots\!20$$$$T^{5} - 3524898654205841186 T^{6} - 5377204 T^{7} + T^{8} )^{2}$$
$47$ $$94\!\cdots\!76$$$$-$$$$20\!\cdots\!12$$$$T +$$$$32\!\cdots\!32$$$$T^{2} -$$$$25\!\cdots\!28$$$$T^{3} +$$$$16\!\cdots\!56$$$$T^{4} -$$$$47\!\cdots\!48$$$$T^{5} +$$$$12\!\cdots\!04$$$$T^{6} +$$$$62\!\cdots\!56$$$$T^{7} +$$$$61\!\cdots\!60$$$$T^{8} +$$$$30\!\cdots\!12$$$$T^{9} +$$$$10\!\cdots\!68$$$$T^{10} +$$$$49\!\cdots\!48$$$$T^{11} +$$$$12\!\cdots\!92$$$$T^{12} +$$$$34\!\cdots\!72$$$$T^{13} + 4910740742278397664 T^{14} - 1034359464 T^{15} + T^{16}$$
$53$ $$42\!\cdots\!44$$$$+$$$$70\!\cdots\!12$$$$T +$$$$21\!\cdots\!48$$$$T^{2} -$$$$13\!\cdots\!28$$$$T^{3} +$$$$24\!\cdots\!56$$$$T^{4} -$$$$57\!\cdots\!08$$$$T^{5} +$$$$71\!\cdots\!40$$$$T^{6} -$$$$84\!\cdots\!80$$$$T^{7} +$$$$15\!\cdots\!85$$$$T^{8} -$$$$71\!\cdots\!32$$$$T^{9} +$$$$15\!\cdots\!10$$$$T^{10} -$$$$59\!\cdots\!20$$$$T^{11} +$$$$11\!\cdots\!99$$$$T^{12} -$$$$21\!\cdots\!96$$$$T^{13} + 40070196965083418118 T^{14} - 665159988 T^{15} + T^{16}$$
$59$ $$16\!\cdots\!96$$$$+$$$$14\!\cdots\!80$$$$T +$$$$10\!\cdots\!36$$$$T^{2} +$$$$13\!\cdots\!60$$$$T^{3} +$$$$18\!\cdots\!88$$$$T^{4} +$$$$48\!\cdots\!40$$$$T^{5} +$$$$80\!\cdots\!32$$$$T^{6} +$$$$19\!\cdots\!20$$$$T^{7} +$$$$24\!\cdots\!17$$$$T^{8} +$$$$55\!\cdots\!80$$$$T^{9} +$$$$16\!\cdots\!66$$$$T^{10} +$$$$14\!\cdots\!00$$$$T^{11} +$$$$71\!\cdots\!47$$$$T^{12} +$$$$24\!\cdots\!40$$$$T^{13} + 93740801099549624642 T^{14} + 1040514580 T^{15} + T^{16}$$
$61$ $$20\!\cdots\!76$$$$+$$$$12\!\cdots\!32$$$$T +$$$$59\!\cdots\!88$$$$T^{2} +$$$$70\!\cdots\!04$$$$T^{3} +$$$$64\!\cdots\!20$$$$T^{4} +$$$$28\!\cdots\!68$$$$T^{5} +$$$$12\!\cdots\!92$$$$T^{6} +$$$$36\!\cdots\!72$$$$T^{7} +$$$$12\!\cdots\!32$$$$T^{8} +$$$$30\!\cdots\!48$$$$T^{9} +$$$$73\!\cdots\!96$$$$T^{10} +$$$$10\!\cdots\!88$$$$T^{11} +$$$$16\!\cdots\!56$$$$T^{12} +$$$$14\!\cdots\!40$$$$T^{13} +$$$$23\!\cdots\!84$$$$T^{14} + 14391208024 T^{15} + T^{16}$$
$67$ $$39\!\cdots\!64$$$$-$$$$19\!\cdots\!68$$$$T +$$$$16\!\cdots\!28$$$$T^{2} -$$$$13\!\cdots\!40$$$$T^{3} +$$$$13\!\cdots\!56$$$$T^{4} +$$$$41\!\cdots\!52$$$$T^{5} +$$$$21\!\cdots\!36$$$$T^{6} +$$$$48\!\cdots\!72$$$$T^{7} +$$$$90\!\cdots\!13$$$$T^{8} +$$$$10\!\cdots\!28$$$$T^{9} +$$$$98\!\cdots\!70$$$$T^{10} +$$$$63\!\cdots\!56$$$$T^{11} +$$$$41\!\cdots\!67$$$$T^{12} +$$$$19\!\cdots\!68$$$$T^{13} +$$$$10\!\cdots\!42$$$$T^{14} + 33307097284 T^{15} + T^{16}$$
$71$ $$( -$$$$14\!\cdots\!24$$$$-$$$$48\!\cdots\!76$$$$T +$$$$47\!\cdots\!32$$$$T^{2} +$$$$42\!\cdots\!08$$$$T^{3} -$$$$16\!\cdots\!40$$$$T^{4} -$$$$28\!\cdots\!56$$$$T^{5} -$$$$51\!\cdots\!12$$$$T^{6} + 32924451448 T^{7} + T^{8} )^{2}$$
$73$ $$51\!\cdots\!00$$$$+$$$$10\!\cdots\!00$$$$T +$$$$20\!\cdots\!00$$$$T^{2} -$$$$10\!\cdots\!60$$$$T^{3} +$$$$37\!\cdots\!04$$$$T^{4} -$$$$10\!\cdots\!08$$$$T^{5} +$$$$41\!\cdots\!44$$$$T^{6} -$$$$13\!\cdots\!64$$$$T^{7} +$$$$21\!\cdots\!77$$$$T^{8} -$$$$68\!\cdots\!72$$$$T^{9} +$$$$80\!\cdots\!26$$$$T^{10} -$$$$23\!\cdots\!76$$$$T^{11} +$$$$15\!\cdots\!43$$$$T^{12} -$$$$25\!\cdots\!08$$$$T^{13} +$$$$15\!\cdots\!74$$$$T^{14} - 17709749204 T^{15} + T^{16}$$
$79$ $$34\!\cdots\!09$$$$-$$$$53\!\cdots\!76$$$$T +$$$$17\!\cdots\!24$$$$T^{2} +$$$$45\!\cdots\!56$$$$T^{3} +$$$$26\!\cdots\!38$$$$T^{4} -$$$$23\!\cdots\!88$$$$T^{5} +$$$$12\!\cdots\!72$$$$T^{6} +$$$$39\!\cdots\!40$$$$T^{7} +$$$$31\!\cdots\!23$$$$T^{8} +$$$$13\!\cdots\!24$$$$T^{9} +$$$$51\!\cdots\!08$$$$T^{10} +$$$$49\!\cdots\!36$$$$T^{11} +$$$$53\!\cdots\!06$$$$T^{12} +$$$$35\!\cdots\!08$$$$T^{13} +$$$$31\!\cdots\!76$$$$T^{14} + 26626784032 T^{15} + T^{16}$$
$83$ $$($$$$15\!\cdots\!36$$$$+$$$$70\!\cdots\!32$$$$T +$$$$29\!\cdots\!84$$$$T^{2} -$$$$23\!\cdots\!52$$$$T^{3} -$$$$53\!\cdots\!63$$$$T^{4} +$$$$32\!\cdots\!36$$$$T^{5} -$$$$52\!\cdots\!14$$$$T^{6} - 105153477524 T^{7} + T^{8} )^{2}$$
$89$ $$10\!\cdots\!76$$$$-$$$$51\!\cdots\!96$$$$T +$$$$11\!\cdots\!08$$$$T^{2} -$$$$86\!\cdots\!96$$$$T^{3} +$$$$10\!\cdots\!76$$$$T^{4} -$$$$60\!\cdots\!08$$$$T^{5} +$$$$34\!\cdots\!64$$$$T^{6} -$$$$11\!\cdots\!20$$$$T^{7} +$$$$39\!\cdots\!00$$$$T^{8} -$$$$89\!\cdots\!28$$$$T^{9} +$$$$27\!\cdots\!84$$$$T^{10} -$$$$45\!\cdots\!48$$$$T^{11} +$$$$96\!\cdots\!96$$$$T^{12} -$$$$55\!\cdots\!80$$$$T^{13} +$$$$11\!\cdots\!68$$$$T^{14} - 55951560072 T^{15} + T^{16}$$
$97$ $$($$$$60\!\cdots\!52$$$$-$$$$79\!\cdots\!92$$$$T -$$$$29\!\cdots\!04$$$$T^{2} +$$$$28\!\cdots\!60$$$$T^{3} +$$$$54\!\cdots\!01$$$$T^{4} -$$$$29\!\cdots\!20$$$$T^{5} -$$$$42\!\cdots\!54$$$$T^{6} + 78108015356 T^{7} + T^{8} )^{2}$$