# Properties

 Label 32.18.b.a Level $32$ Weight $18$ Character orbit 32.b Analytic conductor $58.631$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$32 = 2^{5}$$ Weight: $$k$$ $$=$$ $$18$$ Character orbit: $$[\chi]$$ $$=$$ 32.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$58.6310679503$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 8 x^{15} + 83403052 x^{14} - 583821224 x^{13} + 2709232900692534 x^{12} - 16255389814479656 x^{11} + 44023334982114936772524 x^{10} - 220116525902848632203992 x^{9} + 375637035261256983364255923921 x^{8} - 1502546820346051325629336482512 x^{7} + 1580449203685347736142710540597573024 x^{6} - 4741342352143096486478474620456939520 x^{5} + 2671768262327077521053591460599154325518592 x^{4} - 5343528622418654441129876221219442847805440 x^{3} + 1662520959198696999402497079485716800591945908224 x^{2} - 1662518287435176013550209293272621607067374059520 x + 210991858777295491315579293406297919761107107446784000$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{240}\cdot 3^{14}\cdot 7$$ Twist minimal: no (minimal twist has level 8) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + ( 4 \beta_{1} - \beta_{9} ) q^{5} + ( -720600 - \beta_{2} ) q^{7} + ( -37665881 - 2 \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + ( 4 \beta_{1} - \beta_{9} ) q^{5} + ( -720600 - \beta_{2} ) q^{7} + ( -37665881 - 2 \beta_{2} + \beta_{3} ) q^{9} + ( 2151 \beta_{1} - 157 \beta_{9} - \beta_{10} ) q^{11} + ( 6871 \beta_{1} + 176 \beta_{9} - \beta_{11} ) q^{13} + ( 624580136 - 38 \beta_{2} - 8 \beta_{3} + \beta_{4} ) q^{15} + ( -468070350 + 22 \beta_{2} + 26 \beta_{3} - \beta_{5} ) q^{17} + ( -69658 \beta_{1} - 6759 \beta_{9} + \beta_{10} - \beta_{11} - \beta_{14} ) q^{19} + ( 3601303 \beta_{1} + 12495 \beta_{9} - 18 \beta_{10} + 9 \beta_{11} + \beta_{14} + \beta_{15} ) q^{21} + ( -46677834120 + 1430 \beta_{2} - 136 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{8} ) q^{23} + ( -113105151979 + 7233 \beta_{2} - 356 \beta_{3} - 17 \beta_{4} - 5 \beta_{5} + \beta_{7} + 2 \beta_{8} ) q^{25} + ( 36709029 \beta_{1} - 64286 \beta_{9} - 274 \beta_{10} + 134 \beta_{11} + \beta_{12} + 3 \beta_{14} + 8 \beta_{15} ) q^{27} + ( -9759682 \beta_{1} + 119690 \beta_{9} + 743 \beta_{10} + 58 \beta_{11} - 2 \beta_{12} - \beta_{13} - 29 \beta_{14} + 3 \beta_{15} ) q^{29} + ( 19936234912 - 17574 \beta_{2} + 4171 \beta_{3} - 43 \beta_{4} - 110 \beta_{5} + \beta_{6} - 8 \beta_{7} + \beta_{8} ) q^{31} + ( 352095386100 + 43403 \beta_{2} - 10124 \beta_{3} + 577 \beta_{4} - 12 \beta_{5} + 2 \beta_{6} + 15 \beta_{7} - 2 \beta_{8} ) q^{33} + ( 52332943 \beta_{1} - 1227551 \beta_{9} - 1831 \beta_{10} - 2802 \beta_{11} + 13 \beta_{12} - 8 \beta_{13} + 7 \beta_{14} - 24 \beta_{15} ) q^{35} + ( -276341403 \beta_{1} + 1609889 \beta_{9} - 5833 \beta_{10} + 135 \beta_{11} - 22 \beta_{12} - 27 \beta_{13} + 124 \beta_{14} - 36 \beta_{15} ) q^{37} + ( 1153606628216 - 366134 \beta_{2} + 50699 \beta_{3} - 158 \beta_{4} + 966 \beta_{5} + 25 \beta_{6} - 72 \beta_{7} - 37 \beta_{8} ) q^{39} + ( 467640721002 + 224152 \beta_{2} + 2457 \beta_{3} - 3694 \beta_{4} - 33 \beta_{5} + 46 \beta_{6} + 78 \beta_{7} - 68 \beta_{8} ) q^{41} + ( -326076851 \beta_{1} - 12703649 \beta_{9} - 4861 \beta_{10} + 16613 \beta_{11} + 53 \beta_{12} - 168 \beta_{13} - 26 \beta_{14} - 216 \beta_{15} ) q^{43} + ( -962994141 \beta_{1} + 24721556 \beta_{9} + 114658 \beta_{10} - 659 \beta_{11} - 64 \beta_{12} - 336 \beta_{13} + 399 \beta_{14} - 113 \beta_{15} ) q^{45} + ( 23543675301360 + 1149934 \beta_{2} - 477451 \beta_{3} - 459 \beta_{4} + 296 \beta_{5} + 287 \beta_{6} - 120 \beta_{7} - 36 \beta_{8} ) q^{47} + ( 7980705756345 + 3476701 \beta_{2} - 158179 \beta_{3} + 8751 \beta_{4} - 46 \beta_{5} + 484 \beta_{6} + 65 \beta_{7} + 66 \beta_{8} ) q^{49} + ( 3635840806 \beta_{1} + 30994255 \beta_{9} + 6491 \beta_{10} - 76498 \beta_{11} - 26 \beta_{12} - 1608 \beta_{13} + 284 \beta_{14} + 688 \beta_{15} ) q^{51} + ( 5900767049 \beta_{1} - 121555555 \beta_{9} - 524787 \beta_{10} - 9463 \beta_{11} + 154 \beta_{12} - 2547 \beta_{13} - 3083 \beta_{14} + 597 \beta_{15} ) q^{53} + ( -138064792982072 + 2447529 \beta_{2} + 2812752 \beta_{3} + 854 \beta_{4} - 14610 \beta_{5} + 2000 \beta_{6} + 768 \beta_{7} + 631 \beta_{8} ) q^{55} + ( -11907581143420 - 12729158 \beta_{2} - 285733 \beta_{3} + 22018 \beta_{4} - 474 \beta_{5} + 3082 \beta_{6} - 930 \beta_{7} + 1052 \beta_{8} ) q^{57} + ( 7079712398 \beta_{1} - 17449264 \beta_{9} + 168900 \beta_{10} + 72233 \beta_{11} - 802 \beta_{12} - 9296 \beta_{13} - 817 \beta_{14} + 2544 \beta_{15} ) q^{59} + ( 21820161745 \beta_{1} + 63027115 \beta_{9} + 1262809 \beta_{10} - 64035 \beta_{11} + 1194 \beta_{12} - 13083 \beta_{13} + 83 \beta_{14} + 1971 \beta_{15} ) q^{61} + ( 508193537958680 + 56226017 \beta_{2} - 15033277 \beta_{3} + 16431 \beta_{4} + 18612 \beta_{5} + 9417 \beta_{6} + 3000 \beta_{7} + 594 \beta_{8} ) q^{63} + ( 149124248459760 + 106597708 \beta_{2} + 1189001 \beta_{3} - 166164 \beta_{4} + 695 \beta_{5} + 13240 \beta_{6} - 3372 \beta_{7} - 984 \beta_{8} ) q^{65} + ( -30346744351 \beta_{1} + 480964434 \beta_{9} + 647762 \beta_{10} + 260119 \beta_{11} - 1729 \beta_{12} - 36288 \beta_{13} - 6182 \beta_{14} - 8712 \beta_{15} ) q^{67} + ( 25816105085 \beta_{1} - 284440744 \beta_{9} - 1180181 \beta_{10} - 278267 \beta_{11} + 1166 \beta_{12} - 47817 \beta_{13} + 29557 \beta_{14} - 5995 \beta_{15} ) q^{69} + ( -564120392848536 - 6512158 \beta_{2} + 27191280 \beta_{3} + 19741 \beta_{4} + 66978 \beta_{5} + 31480 \beta_{6} - 576 \beta_{7} - 6511 \beta_{8} ) q^{71} + ( 708250127882410 - 237036161 \beta_{2} - 5770718 \beta_{3} + 180459 \beta_{4} + 26458 \beta_{5} + 40388 \beta_{6} + 325 \beta_{7} - 9654 \beta_{8} ) q^{73} + ( 48719226650 \beta_{1} - 2554243704 \beta_{9} + 396444 \beta_{10} - 2355417 \beta_{11} + 2748 \beta_{12} - 100968 \beta_{13} + 19987 \beta_{14} - 16544 \beta_{15} ) q^{75} + ( -63846201013 \beta_{1} - 601162026 \beta_{9} - 2050065 \beta_{10} - 518993 \beta_{11} - 6098 \beta_{12} - 126169 \beta_{13} - 37109 \beta_{14} - 20949 \beta_{15} ) q^{77} + ( 2831229462000528 + 209668937 \beta_{2} - 45737035 \beta_{3} - 176282 \beta_{4} - 205570 \beta_{5} + 75719 \beta_{6} - 18232 \beta_{7} - 5881 \beta_{8} ) q^{79} + ( 1256368874955677 + 1143184202 \beta_{2} - 27169465 \beta_{3} + 630786 \beta_{4} + 42678 \beta_{5} + 88530 \beta_{6} + 24414 \beta_{7} + 8604 \beta_{8} ) q^{81} + ( 431705761992 \beta_{1} + 6690378451 \beta_{9} - 1650885 \beta_{10} + 1560128 \beta_{11} + 15613 \beta_{12} - 201936 \beta_{13} + 60169 \beta_{14} + 62184 \beta_{15} ) q^{83} + ( 147546288167 \beta_{1} + 3869878080 \beta_{9} + 24308737 \beta_{10} + 127949 \beta_{11} - 15906 \beta_{12} - 232929 \beta_{13} - 133954 \beta_{14} + 40158 \beta_{15} ) q^{85} + ( -1622860557552360 + 232841053 \beta_{2} + 9813197 \beta_{3} - 286327 \beta_{4} + 73110 \beta_{5} + 125335 \beta_{6} - 24120 \beta_{7} + 44627 \beta_{8} ) q^{87} + ( -4367448413047878 - 1298351507 \beta_{2} - 16908740 \beta_{3} - 1865783 \beta_{4} - 295670 \beta_{5} + 130456 \beta_{6} + 40647 \beta_{7} + 56846 \beta_{8} ) q^{89} + ( 529159774133 \beta_{1} - 29045943768 \beta_{9} + 28644 \beta_{10} + 1991229 \beta_{11} + 11428 \beta_{12} - 261096 \beta_{13} - 211599 \beta_{14} + 59040 \beta_{15} ) q^{91} + ( 545965760660 \beta_{1} - 3435884157 \beta_{9} - 39794811 \beta_{10} - 171810 \beta_{11} + 966 \beta_{12} - 246669 \beta_{13} + 332538 \beta_{14} + 149658 \beta_{15} ) q^{93} + ( -5861902772387736 - 11553771 \beta_{2} + 104575810 \beta_{3} + 1238516 \beta_{4} + 784240 \beta_{5} + 103670 \beta_{6} + 21456 \beta_{7} + 38152 \beta_{8} ) q^{95} + ( 5974587412636290 + 3079961393 \beta_{2} + 25767489 \beta_{3} + 1414493 \beta_{4} - 707967 \beta_{5} + 71960 \beta_{6} - 48525 \beta_{7} - 47258 \beta_{8} ) q^{97} + ( -1429358873670 \beta_{1} + 69527223000 \beta_{9} + 5857524 \beta_{10} - 17258025 \beta_{11} - 43446 \beta_{12} - 49128 \beta_{13} - 355047 \beta_{14} - 255024 \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 11529600q^{7} - 602654096q^{9} + O(q^{10})$$ $$16q - 11529600q^{7} - 602654096q^{9} + 9993282176q^{15} - 7489125600q^{17} - 746845345920q^{23} - 1809682431664q^{25} + 318979758592q^{31} + 5633526177600q^{33} + 18457706051456q^{39} + 7482251536032q^{41} + 376698804821760q^{47} + 127691292101520q^{49} - 2209036687713152q^{55} - 190521298294720q^{57} + 8131096607338880q^{63} + 2385987975356160q^{65} - 9025926285576576q^{71} + 11332002046118560q^{73} + 45299671392008448q^{79} + 20101901999290832q^{81} - 25965768920837760q^{87} - 69879174608766048q^{89} - 93790444358203776q^{95} + 95593398602180640q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 8 x^{15} + 83403052 x^{14} - 583821224 x^{13} + 2709232900692534 x^{12} - 16255389814479656 x^{11} + 44023334982114936772524 x^{10} - 220116525902848632203992 x^{9} + 375637035261256983364255923921 x^{8} - 1502546820346051325629336482512 x^{7} + 1580449203685347736142710540597573024 x^{6} - 4741342352143096486478474620456939520 x^{5} + 2671768262327077521053591460599154325518592 x^{4} - 5343528622418654441129876221219442847805440 x^{3} + 1662520959198696999402497079485716800591945908224 x^{2} - 1662518287435176013550209293272621607067374059520 x + 210991858777295491315579293406297919761107107446784000$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$4 \nu - 2$$ $$\beta_{2}$$ $$=$$ $$($$$$-$$$$22\!\cdots\!17$$$$\nu^{14} +$$$$16\!\cdots\!19$$$$\nu^{13} -$$$$15\!\cdots\!02$$$$\nu^{12} +$$$$92\!\cdots\!65$$$$\nu^{11} -$$$$35\!\cdots\!40$$$$\nu^{10} +$$$$17\!\cdots\!07$$$$\nu^{9} -$$$$33\!\cdots\!52$$$$\nu^{8} +$$$$13\!\cdots\!19$$$$\nu^{7} -$$$$79\!\cdots\!35$$$$\nu^{6} +$$$$23\!\cdots\!78$$$$\nu^{5} +$$$$35\!\cdots\!86$$$$\nu^{4} -$$$$71\!\cdots\!48$$$$\nu^{3} +$$$$67\!\cdots\!36$$$$\nu^{2} -$$$$67\!\cdots\!16$$$$\nu +$$$$17\!\cdots\!40$$$$)/$$$$40\!\cdots\!70$$ $$\beta_{3}$$ $$=$$ $$($$$$-$$$$22\!\cdots\!17$$$$\nu^{14} +$$$$16\!\cdots\!19$$$$\nu^{13} -$$$$15\!\cdots\!02$$$$\nu^{12} +$$$$92\!\cdots\!65$$$$\nu^{11} -$$$$35\!\cdots\!40$$$$\nu^{10} +$$$$17\!\cdots\!07$$$$\nu^{9} -$$$$33\!\cdots\!52$$$$\nu^{8} +$$$$13\!\cdots\!19$$$$\nu^{7} -$$$$79\!\cdots\!35$$$$\nu^{6} +$$$$23\!\cdots\!78$$$$\nu^{5} +$$$$35\!\cdots\!86$$$$\nu^{4} -$$$$71\!\cdots\!48$$$$\nu^{3} +$$$$70\!\cdots\!96$$$$\nu^{2} -$$$$70\!\cdots\!76$$$$\nu +$$$$51\!\cdots\!20$$$$)/$$$$20\!\cdots\!85$$ $$\beta_{4}$$ $$=$$ $$($$$$81\!\cdots\!73$$$$\nu^{14} -$$$$56\!\cdots\!11$$$$\nu^{13} +$$$$58\!\cdots\!02$$$$\nu^{12} -$$$$34\!\cdots\!69$$$$\nu^{11} +$$$$15\!\cdots\!76$$$$\nu^{10} -$$$$75\!\cdots\!43$$$$\nu^{9} +$$$$18\!\cdots\!92$$$$\nu^{8} -$$$$72\!\cdots\!91$$$$\nu^{7} +$$$$10\!\cdots\!27$$$$\nu^{6} -$$$$30\!\cdots\!66$$$$\nu^{5} +$$$$22\!\cdots\!46$$$$\nu^{4} -$$$$44\!\cdots\!60$$$$\nu^{3} +$$$$17\!\cdots\!12$$$$\nu^{2} -$$$$17\!\cdots\!88$$$$\nu +$$$$28\!\cdots\!40$$$$)/$$$$13\!\cdots\!90$$ $$\beta_{5}$$ $$=$$ $$($$$$39\!\cdots\!81$$$$\nu^{14} -$$$$27\!\cdots\!67$$$$\nu^{13} +$$$$27\!\cdots\!54$$$$\nu^{12} -$$$$16\!\cdots\!53$$$$\nu^{11} +$$$$68\!\cdots\!12$$$$\nu^{10} -$$$$34\!\cdots\!71$$$$\nu^{9} +$$$$77\!\cdots\!92$$$$\nu^{8} -$$$$31\!\cdots\!59$$$$\nu^{7} +$$$$37\!\cdots\!03$$$$\nu^{6} -$$$$11\!\cdots\!62$$$$\nu^{5} +$$$$50\!\cdots\!22$$$$\nu^{4} -$$$$10\!\cdots\!24$$$$\nu^{3} -$$$$17\!\cdots\!56$$$$\nu^{2} +$$$$17\!\cdots\!28$$$$\nu -$$$$29\!\cdots\!80$$$$)/$$$$61\!\cdots\!55$$ $$\beta_{6}$$ $$=$$ $$($$$$32\!\cdots\!83$$$$\nu^{14} -$$$$22\!\cdots\!81$$$$\nu^{13} +$$$$23\!\cdots\!10$$$$\nu^{12} -$$$$13\!\cdots\!07$$$$\nu^{11} +$$$$58\!\cdots\!76$$$$\nu^{10} -$$$$29\!\cdots\!13$$$$\nu^{9} +$$$$66\!\cdots\!44$$$$\nu^{8} -$$$$26\!\cdots\!37$$$$\nu^{7} +$$$$32\!\cdots\!85$$$$\nu^{6} -$$$$96\!\cdots\!58$$$$\nu^{5} +$$$$44\!\cdots\!70$$$$\nu^{4} -$$$$89\!\cdots\!44$$$$\nu^{3} +$$$$18\!\cdots\!60$$$$\nu^{2} -$$$$18\!\cdots\!88$$$$\nu +$$$$31\!\cdots\!80$$$$)/$$$$12\!\cdots\!10$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$84\!\cdots\!25$$$$\nu^{14} +$$$$59\!\cdots\!75$$$$\nu^{13} -$$$$60\!\cdots\!74$$$$\nu^{12} +$$$$36\!\cdots\!69$$$$\nu^{11} -$$$$15\!\cdots\!20$$$$\nu^{10} +$$$$79\!\cdots\!55$$$$\nu^{9} -$$$$18\!\cdots\!80$$$$\nu^{8} +$$$$75\!\cdots\!99$$$$\nu^{7} -$$$$10\!\cdots\!31$$$$\nu^{6} +$$$$30\!\cdots\!26$$$$\nu^{5} -$$$$19\!\cdots\!22$$$$\nu^{4} +$$$$39\!\cdots\!44$$$$\nu^{3} -$$$$14\!\cdots\!64$$$$\nu^{2} +$$$$14\!\cdots\!48$$$$\nu -$$$$20\!\cdots\!80$$$$)/$$$$61\!\cdots\!55$$ $$\beta_{8}$$ $$=$$ $$($$$$23\!\cdots\!64$$$$\nu^{14} -$$$$16\!\cdots\!48$$$$\nu^{13} +$$$$16\!\cdots\!92$$$$\nu^{12} -$$$$10\!\cdots\!28$$$$\nu^{11} +$$$$43\!\cdots\!44$$$$\nu^{10} -$$$$21\!\cdots\!24$$$$\nu^{9} +$$$$51\!\cdots\!52$$$$\nu^{8} -$$$$20\!\cdots\!68$$$$\nu^{7} +$$$$27\!\cdots\!72$$$$\nu^{6} -$$$$81\!\cdots\!80$$$$\nu^{5} +$$$$51\!\cdots\!84$$$$\nu^{4} -$$$$10\!\cdots\!80$$$$\nu^{3} +$$$$33\!\cdots\!64$$$$\nu^{2} -$$$$33\!\cdots\!44$$$$\nu +$$$$38\!\cdots\!60$$$$)/$$$$16\!\cdots\!15$$ $$\beta_{9}$$ $$=$$ $$($$$$-$$$$50\!\cdots\!82$$$$\nu^{15} +$$$$38\!\cdots\!65$$$$\nu^{14} -$$$$38\!\cdots\!37$$$$\nu^{13} +$$$$25\!\cdots\!88$$$$\nu^{12} -$$$$11\!\cdots\!15$$$$\nu^{11} +$$$$60\!\cdots\!10$$$$\nu^{10} -$$$$15\!\cdots\!45$$$$\nu^{9} +$$$$69\!\cdots\!82$$$$\nu^{8} -$$$$10\!\cdots\!07$$$$\nu^{7} +$$$$37\!\cdots\!29$$$$\nu^{6} -$$$$35\!\cdots\!10$$$$\nu^{5} +$$$$87\!\cdots\!50$$$$\nu^{4} -$$$$40\!\cdots\!60$$$$\nu^{3} +$$$$61\!\cdots\!32$$$$\nu^{2} -$$$$13\!\cdots\!20$$$$\nu +$$$$65\!\cdots\!60$$$$)/$$$$24\!\cdots\!80$$ $$\beta_{10}$$ $$=$$ $$($$$$32\!\cdots\!14$$$$\nu^{15} -$$$$24\!\cdots\!05$$$$\nu^{14} +$$$$23\!\cdots\!57$$$$\nu^{13} -$$$$15\!\cdots\!28$$$$\nu^{12} +$$$$64\!\cdots\!87$$$$\nu^{11} -$$$$35\!\cdots\!74$$$$\nu^{10} +$$$$81\!\cdots\!73$$$$\nu^{9} -$$$$36\!\cdots\!78$$$$\nu^{8} +$$$$49\!\cdots\!87$$$$\nu^{7} -$$$$17\!\cdots\!53$$$$\nu^{6} +$$$$12\!\cdots\!66$$$$\nu^{5} -$$$$31\!\cdots\!94$$$$\nu^{4} +$$$$12\!\cdots\!04$$$$\nu^{3} -$$$$18\!\cdots\!76$$$$\nu^{2} +$$$$31\!\cdots\!80$$$$\nu -$$$$15\!\cdots\!80$$$$)/$$$$62\!\cdots\!80$$ $$\beta_{11}$$ $$=$$ $$($$$$-$$$$34\!\cdots\!42$$$$\nu^{15} +$$$$25\!\cdots\!65$$$$\nu^{14} -$$$$24\!\cdots\!57$$$$\nu^{13} +$$$$15\!\cdots\!68$$$$\nu^{12} -$$$$63\!\cdots\!55$$$$\nu^{11} +$$$$34\!\cdots\!90$$$$\nu^{10} -$$$$74\!\cdots\!85$$$$\nu^{9} +$$$$33\!\cdots\!82$$$$\nu^{8} -$$$$39\!\cdots\!67$$$$\nu^{7} +$$$$13\!\cdots\!09$$$$\nu^{6} -$$$$72\!\cdots\!30$$$$\nu^{5} +$$$$18\!\cdots\!70$$$$\nu^{4} -$$$$26\!\cdots\!20$$$$\nu^{3} +$$$$39\!\cdots\!52$$$$\nu^{2} +$$$$16\!\cdots\!00$$$$\nu -$$$$82\!\cdots\!40$$$$)/$$$$26\!\cdots\!80$$ $$\beta_{12}$$ $$=$$ $$($$$$-$$$$25\!\cdots\!86$$$$\nu^{15} +$$$$19\!\cdots\!45$$$$\nu^{14} -$$$$19\!\cdots\!73$$$$\nu^{13} +$$$$12\!\cdots\!92$$$$\nu^{12} -$$$$57\!\cdots\!55$$$$\nu^{11} +$$$$31\!\cdots\!62$$$$\nu^{10} -$$$$81\!\cdots\!93$$$$\nu^{9} +$$$$36\!\cdots\!54$$$$\nu^{8} -$$$$58\!\cdots\!71$$$$\nu^{7} +$$$$20\!\cdots\!21$$$$\nu^{6} -$$$$19\!\cdots\!58$$$$\nu^{5} +$$$$48\!\cdots\!14$$$$\nu^{4} -$$$$24\!\cdots\!80$$$$\nu^{3} +$$$$37\!\cdots\!20$$$$\nu^{2} -$$$$14\!\cdots\!92$$$$\nu +$$$$74\!\cdots\!00$$$$)/$$$$47\!\cdots\!40$$ $$\beta_{13}$$ $$=$$ $$($$$$51\!\cdots\!58$$$$\nu^{15} -$$$$38\!\cdots\!35$$$$\nu^{14} +$$$$37\!\cdots\!19$$$$\nu^{13} -$$$$24\!\cdots\!76$$$$\nu^{12} +$$$$98\!\cdots\!89$$$$\nu^{11} -$$$$54\!\cdots\!18$$$$\nu^{10} +$$$$11\!\cdots\!51$$$$\nu^{9} -$$$$53\!\cdots\!46$$$$\nu^{8} +$$$$65\!\cdots\!49$$$$\nu^{7} -$$$$23\!\cdots\!71$$$$\nu^{6} +$$$$13\!\cdots\!82$$$$\nu^{5} -$$$$34\!\cdots\!78$$$$\nu^{4} +$$$$84\!\cdots\!48$$$$\nu^{3} -$$$$12\!\cdots\!32$$$$\nu^{2} -$$$$59\!\cdots\!00$$$$\nu +$$$$29\!\cdots\!80$$$$)/$$$$84\!\cdots\!30$$ $$\beta_{14}$$ $$=$$ $$($$$$-$$$$12\!\cdots\!10$$$$\nu^{15} +$$$$96\!\cdots\!25$$$$\nu^{14} -$$$$89\!\cdots\!17$$$$\nu^{13} +$$$$58\!\cdots\!48$$$$\nu^{12} -$$$$22\!\cdots\!63$$$$\nu^{11} +$$$$12\!\cdots\!46$$$$\nu^{10} -$$$$24\!\cdots\!69$$$$\nu^{9} +$$$$11\!\cdots\!70$$$$\nu^{8} -$$$$11\!\cdots\!07$$$$\nu^{7} +$$$$39\!\cdots\!13$$$$\nu^{6} -$$$$11\!\cdots\!74$$$$\nu^{5} +$$$$29\!\cdots\!82$$$$\nu^{4} -$$$$11\!\cdots\!64$$$$\nu^{3} +$$$$17\!\cdots\!72$$$$\nu^{2} -$$$$34\!\cdots\!12$$$$\nu +$$$$17\!\cdots\!80$$$$)/$$$$18\!\cdots\!60$$ $$\beta_{15}$$ $$=$$ $$($$$$13\!\cdots\!46$$$$\nu^{15} -$$$$97\!\cdots\!45$$$$\nu^{14} +$$$$93\!\cdots\!37$$$$\nu^{13} -$$$$60\!\cdots\!08$$$$\nu^{12} +$$$$24\!\cdots\!35$$$$\nu^{11} -$$$$13\!\cdots\!66$$$$\nu^{10} +$$$$29\!\cdots\!21$$$$\nu^{9} -$$$$13\!\cdots\!14$$$$\nu^{8} +$$$$16\!\cdots\!71$$$$\nu^{7} -$$$$56\!\cdots\!37$$$$\nu^{6} +$$$$33\!\cdots\!94$$$$\nu^{5} -$$$$82\!\cdots\!98$$$$\nu^{4} +$$$$25\!\cdots\!28$$$$\nu^{3} -$$$$38\!\cdots\!92$$$$\nu^{2} +$$$$51\!\cdots\!68$$$$\nu -$$$$25\!\cdots\!20$$$$)/$$$$18\!\cdots\!60$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1} + 2$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 2 \beta_{2} + 4 \beta_{1} - 166806040$$$$)/16$$ $$\nu^{3}$$ $$=$$ $$($$$$-8 \beta_{15} - 3 \beta_{14} - \beta_{12} - 134 \beta_{11} + 274 \beta_{10} + 64286 \beta_{9} + 6 \beta_{3} - 12 \beta_{2} - 294989343 \beta_{1} - 1000836256$$$$)/64$$ $$\nu^{4}$$ $$=$$ $$($$$$-32 \beta_{15} - 12 \beta_{14} - 4 \beta_{12} - 536 \beta_{11} + 1096 \beta_{10} + 257144 \beta_{9} + 4302 \beta_{8} + 12207 \beta_{7} + 44265 \beta_{6} + 21339 \beta_{5} + 315393 \beta_{4} - 207294965 \beta_{3} + 959012566 \beta_{2} - 1179957404 \beta_{1} + 24601631153289792$$$$)/128$$ $$\nu^{5}$$ $$=$$ $$($$$$2566225536 \beta_{15} + 836634771 \beta_{14} + 246342348 \beta_{13} + 217858872 \beta_{12} + 44543917995 \beta_{11} - 155189497134 \beta_{10} - 56178676364514 \beta_{9} + 43020 \beta_{8} + 122070 \beta_{7} + 442650 \beta_{6} + 213390 \beta_{5} + 3153930 \beta_{4} - 2072949730 \beta_{3} + 9590125820 \beta_{2} + 52735675177445143 \beta_{1} + 246016324877381376$$$$)/512$$ $$\nu^{6}$$ $$=$$ $$($$$$30794707712 \beta_{15} + 10039617732 \beta_{14} + 2956108176 \beta_{13} + 2614306624 \beta_{12} + 534527037380 \beta_{11} - 1862274009448 \beta_{10} - 674144126659928 \beta_{9} - 3663483745302 \beta_{8} - 9028745691915 \beta_{7} - 28976118776688 \beta_{6} - 12809893276638 \beta_{5} - 250300088045637 \beta_{4} + 89808814001298485 \beta_{3} - 539076789550118011 \beta_{2} + 632828149327638132 \beta_{1} - 8794232718841355952278528$$$$)/2048$$ $$\nu^{7}$$ $$=$$ $$($$$$-1328484331885823496 \beta_{15} - 402274961741910456 \beta_{14} - 217126205926476336 \beta_{13} - 98138858673699297 \beta_{12} - 24554973311041428891 \beta_{11} + 97535811787115815755 \beta_{10} + 40520465449040699055135 \beta_{9} - 51288774843348 \beta_{8} - 126402446522730 \beta_{7} - 405665687662032 \beta_{6} - 179338517822772 \beta_{5} - 3504201409258998 \beta_{4} + 1257323512103364566 \beta_{3} - 7547075590748699866 \beta_{2} - 21337036245341218178702884 \beta_{1} - 123119271840693325923471360$$$$)/8192$$ $$\nu^{8}$$ $$=$$ $$($$$$-2656968951188921360 \beta_{15} - 804550017186920640 \beta_{14} - 434252439443295648 \beta_{13} - 196277741747594050 \beta_{12} - 49109951611001913350 \beta_{11} + 195071640955455801526 \beta_{10} + 81040937190093266136350 \beta_{9} + 271927878074222494176 \beta_{8} + 629871610470737670600 \beta_{7} + 1888584626773213762299 \beta_{6} + 583596742641264309744 \beta_{5} + 18356234880609458224536 \beta_{4} - 5065777647178138716491631 \beta_{3} + 33599249481887410112439945 \beta_{2} - 42674078397078584852181848 \beta_{1} + 444677255474762441871038762616832$$$$)/4096$$ $$\nu^{9}$$ $$=$$ $$($$$$40371757038516530915981104 \beta_{15} + 11659586541318538458251424 \beta_{14} + 8387680453473674176012854 \beta_{13} + 2846494302656463989485178 \beta_{12} + 785889801770961236410746886 \beta_{11} - 3258102082081554512747262200 \beta_{10} - 1455119948005817585821451494132 \beta_{9} + 2447351210400654398616 \beta_{8} + 5668845252651326374884 \beta_{7} + 16997264074953079578963 \beta_{6} + 5252371759802500064136 \beta_{5} + 165206134950693791518908 \beta_{4} - 45592006368544460370835467 \beta_{3} + 302393290619440879960616877 \beta_{2} + 580640039789804835621146150906376 \beta_{1} + 4002096037988509553296592378490880$$$$)/8192$$ $$\nu^{10}$$ $$=$$ $$($$$$201858825047117784665552096 \beta_{15} + 58297944774843231204367056 \beta_{14} + 41938408781155045300527918 \beta_{13} + 14232474457448519361923392 \beta_{12} + 3929449745504095313839652840 \beta_{11} - 16290513336482439039245949154 \beta_{10} - 7275600955643164656541878279974 \beta_{9} - 17880126318198461381728394232 \beta_{8} - 40037185647943119961630334724 \beta_{7} - 115170035923668247562184130947 \beta_{6} - 21999758191494546976375880520 \beta_{5} - 1203268748028257287379306693724 \beta_{4} + 291795199917823933607362758050459 \beta_{3} - 2014684767959008157473468542809629 \beta_{2} + 2903200839060217853473062297499648 \beta_{1} - 24198048773507166734910892049678204579840$$$$)/8192$$ $$\nu^{11}$$ $$=$$ $$($$$$-$$$$12\!\cdots\!00$$$$\beta_{15} -$$$$33\!\cdots\!20$$$$\beta_{14} -$$$$28\!\cdots\!68$$$$\beta_{13} -$$$$83\!\cdots\!95$$$$\beta_{12} -$$$$24\!\cdots\!61$$$$\beta_{11} +$$$$10\!\cdots\!41$$$$\beta_{10} +$$$$47\!\cdots\!41$$$$\beta_{9} -$$$$98\!\cdots\!04$$$$\beta_{8} -$$$$22\!\cdots\!02$$$$\beta_{7} -$$$$63\!\cdots\!88$$$$\beta_{6} -$$$$12\!\cdots\!72$$$$\beta_{5} -$$$$66\!\cdots\!30$$$$\beta_{4} +$$$$16\!\cdots\!02$$$$\beta_{3} -$$$$11\!\cdots\!26$$$$\beta_{2} -$$$$16\!\cdots\!64$$$$\beta_{1} -$$$$13\!\cdots\!60$$$$)/8192$$ $$\nu^{12}$$ $$=$$ $$($$$$-$$$$72\!\cdots\!84$$$$\beta_{15} -$$$$20\!\cdots\!44$$$$\beta_{14} -$$$$16\!\cdots\!78$$$$\beta_{13} -$$$$50\!\cdots\!68$$$$\beta_{12} -$$$$14\!\cdots\!12$$$$\beta_{11} +$$$$61\!\cdots\!98$$$$\beta_{10} +$$$$28\!\cdots\!38$$$$\beta_{9} +$$$$55\!\cdots\!16$$$$\beta_{8} +$$$$12\!\cdots\!44$$$$\beta_{7} +$$$$34\!\cdots\!83$$$$\beta_{6} +$$$$36\!\cdots\!16$$$$\beta_{5} +$$$$37\!\cdots\!88$$$$\beta_{4} -$$$$84\!\cdots\!91$$$$\beta_{3} +$$$$59\!\cdots\!05$$$$\beta_{2} -$$$$98\!\cdots\!76$$$$\beta_{1} +$$$$68\!\cdots\!72$$$$)/8192$$ $$\nu^{13}$$ $$=$$ $$($$$$35\!\cdots\!60$$$$\beta_{15} +$$$$97\!\cdots\!40$$$$\beta_{14} +$$$$89\!\cdots\!62$$$$\beta_{13} +$$$$24\!\cdots\!40$$$$\beta_{12} +$$$$73\!\cdots\!80$$$$\beta_{11} -$$$$30\!\cdots\!14$$$$\beta_{10} -$$$$14\!\cdots\!30$$$$\beta_{9} +$$$$36\!\cdots\!36$$$$\beta_{8} +$$$$79\!\cdots\!52$$$$\beta_{7} +$$$$22\!\cdots\!31$$$$\beta_{6} +$$$$23\!\cdots\!00$$$$\beta_{5} +$$$$24\!\cdots\!92$$$$\beta_{4} -$$$$55\!\cdots\!11$$$$\beta_{3} +$$$$38\!\cdots\!85$$$$\beta_{2} +$$$$47\!\cdots\!76$$$$\beta_{1} +$$$$44\!\cdots\!96$$$$)/8192$$ $$\nu^{14}$$ $$=$$ $$($$$$61\!\cdots\!88$$$$\beta_{15} +$$$$17\!\cdots\!88$$$$\beta_{14} +$$$$15\!\cdots\!58$$$$\beta_{13} +$$$$42\!\cdots\!26$$$$\beta_{12} +$$$$12\!\cdots\!98$$$$\beta_{11} -$$$$54\!\cdots\!60$$$$\beta_{10} -$$$$25\!\cdots\!80$$$$\beta_{9} -$$$$42\!\cdots\!38$$$$\beta_{8} -$$$$91\!\cdots\!23$$$$\beta_{7} -$$$$25\!\cdots\!50$$$$\beta_{6} -$$$$12\!\cdots\!06$$$$\beta_{5} -$$$$28\!\cdots\!57$$$$\beta_{4} +$$$$62\!\cdots\!15$$$$\beta_{3} -$$$$43\!\cdots\!49$$$$\beta_{2} +$$$$82\!\cdots\!48$$$$\beta_{1} -$$$$49\!\cdots\!48$$$$)/2048$$ $$\nu^{15}$$ $$=$$ $$($$$$-$$$$10\!\cdots\!64$$$$\beta_{15} -$$$$28\!\cdots\!24$$$$\beta_{14} -$$$$27\!\cdots\!56$$$$\beta_{13} -$$$$72\!\cdots\!53$$$$\beta_{12} -$$$$21\!\cdots\!47$$$$\beta_{11} +$$$$92\!\cdots\!99$$$$\beta_{10} +$$$$44\!\cdots\!03$$$$\beta_{9} -$$$$12\!\cdots\!88$$$$\beta_{8} -$$$$27\!\cdots\!86$$$$\beta_{7} -$$$$75\!\cdots\!18$$$$\beta_{6} -$$$$38\!\cdots\!40$$$$\beta_{5} -$$$$85\!\cdots\!26$$$$\beta_{4} +$$$$18\!\cdots\!56$$$$\beta_{3} -$$$$13\!\cdots\!36$$$$\beta_{2} -$$$$13\!\cdots\!48$$$$\beta_{1} -$$$$14\!\cdots\!88$$$$)/8192$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/32\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$31$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 0.5 + 5420.67i 0.5 + 4000.58i 0.5 + 3466.80i 0.5 + 3446.66i 0.5 + 3370.46i 0.5 + 1208.53i 0.5 + 1062.13i 0.5 + 409.736i 0.5 − 409.736i 0.5 − 1062.13i 0.5 − 1208.53i 0.5 − 3370.46i 0.5 − 3446.66i 0.5 − 3466.80i 0.5 − 4000.58i 0.5 − 5420.67i
0 21682.7i 0 465248.i 0 −2.24795e7 0 −3.40998e8 0
17.2 0 16002.3i 0 1.27253e6i 0 −5.50569e6 0 −1.26934e8 0
17.3 0 13867.2i 0 665059.i 0 7.57536e6 0 −6.31593e7 0
17.4 0 13786.7i 0 96356.3i 0 1.47728e7 0 −6.09318e7 0
17.5 0 13481.8i 0 1.59197e6i 0 1.66055e7 0 −5.26193e7 0
17.6 0 4834.14i 0 524871.i 0 −1.57495e7 0 1.05771e8 0
17.7 0 4248.51i 0 663971.i 0 1.66742e7 0 1.11090e8 0
17.8 0 1638.94i 0 1.21254e6i 0 −1.76580e7 0 1.26454e8 0
17.9 0 1638.94i 0 1.21254e6i 0 −1.76580e7 0 1.26454e8 0
17.10 0 4248.51i 0 663971.i 0 1.66742e7 0 1.11090e8 0
17.11 0 4834.14i 0 524871.i 0 −1.57495e7 0 1.05771e8 0
17.12 0 13481.8i 0 1.59197e6i 0 1.66055e7 0 −5.26193e7 0
17.13 0 13786.7i 0 96356.3i 0 1.47728e7 0 −6.09318e7 0
17.14 0 13867.2i 0 665059.i 0 7.57536e6 0 −6.31593e7 0
17.15 0 16002.3i 0 1.27253e6i 0 −5.50569e6 0 −1.26934e8 0
17.16 0 21682.7i 0 465248.i 0 −2.24795e7 0 −3.40998e8 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 17.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 32.18.b.a 16
4.b odd 2 1 8.18.b.a 16
8.b even 2 1 inner 32.18.b.a 16
8.d odd 2 1 8.18.b.a 16
12.b even 2 1 72.18.d.b 16
24.f even 2 1 72.18.d.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.18.b.a 16 4.b odd 2 1
8.18.b.a 16 8.d odd 2 1
32.18.b.a 16 1.a even 1 1 trivial
32.18.b.a 16 8.b even 2 1 inner
72.18.d.b 16 12.b even 2 1
72.18.d.b 16 24.f even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{18}^{\mathrm{new}}(32, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 731794256 T^{2} + 282192653104988472 T^{4} -$$$$75\!\cdots\!72$$$$T^{6} +$$$$15\!\cdots\!00$$$$T^{8} -$$$$24\!\cdots\!24$$$$T^{10} +$$$$32\!\cdots\!00$$$$T^{12} -$$$$39\!\cdots\!56$$$$T^{14} +$$$$49\!\cdots\!54$$$$T^{16} -$$$$66\!\cdots\!64$$$$T^{18} +$$$$91\!\cdots\!00$$$$T^{20} -$$$$11\!\cdots\!16$$$$T^{22} +$$$$11\!\cdots\!00$$$$T^{24} -$$$$97\!\cdots\!28$$$$T^{26} +$$$$60\!\cdots\!32$$$$T^{28} -$$$$26\!\cdots\!84$$$$T^{30} +$$$$59\!\cdots\!41$$$$T^{32}$$
$5$ $$1 - 5198674409168 T^{2} +$$$$13\!\cdots\!56$$$$T^{4} -$$$$24\!\cdots\!00$$$$T^{6} +$$$$33\!\cdots\!00$$$$T^{8} -$$$$37\!\cdots\!00$$$$T^{10} +$$$$36\!\cdots\!00$$$$T^{12} -$$$$31\!\cdots\!00$$$$T^{14} +$$$$25\!\cdots\!50$$$$T^{16} -$$$$18\!\cdots\!00$$$$T^{18} +$$$$12\!\cdots\!00$$$$T^{20} -$$$$73\!\cdots\!00$$$$T^{22} +$$$$38\!\cdots\!00$$$$T^{24} -$$$$16\!\cdots\!00$$$$T^{26} +$$$$53\!\cdots\!00$$$$T^{28} -$$$$11\!\cdots\!00$$$$T^{30} +$$$$13\!\cdots\!25$$$$T^{32}$$
$7$ $$( 1 + 5764800 T + 915215692443448 T^{2} +$$$$65\!\cdots\!20$$$$T^{3} +$$$$50\!\cdots\!24$$$$T^{4} +$$$$34\!\cdots\!80$$$$T^{5} +$$$$18\!\cdots\!60$$$$T^{6} +$$$$12\!\cdots\!60$$$$T^{7} +$$$$50\!\cdots\!90$$$$T^{8} +$$$$27\!\cdots\!20$$$$T^{9} +$$$$10\!\cdots\!40$$$$T^{10} +$$$$44\!\cdots\!40$$$$T^{11} +$$$$14\!\cdots\!24$$$$T^{12} +$$$$44\!\cdots\!40$$$$T^{13} +$$$$14\!\cdots\!52$$$$T^{14} +$$$$21\!\cdots\!00$$$$T^{15} +$$$$85\!\cdots\!01$$$$T^{16} )^{2}$$
$11$ $$1 - 3877608573076448976 T^{2} +$$$$79\!\cdots\!84$$$$T^{4} -$$$$11\!\cdots\!04$$$$T^{6} +$$$$12\!\cdots\!88$$$$T^{8} -$$$$11\!\cdots\!72$$$$T^{10} +$$$$81\!\cdots\!96$$$$T^{12} -$$$$51\!\cdots\!28$$$$T^{14} +$$$$28\!\cdots\!22$$$$T^{16} -$$$$13\!\cdots\!48$$$$T^{18} +$$$$53\!\cdots\!76$$$$T^{20} -$$$$18\!\cdots\!12$$$$T^{22} +$$$$52\!\cdots\!68$$$$T^{24} -$$$$12\!\cdots\!04$$$$T^{26} +$$$$22\!\cdots\!44$$$$T^{28} -$$$$27\!\cdots\!56$$$$T^{30} +$$$$18\!\cdots\!21$$$$T^{32}$$
$13$ $$1 - 63371249746529137488 T^{2} +$$$$20\!\cdots\!16$$$$T^{4} -$$$$48\!\cdots\!88$$$$T^{6} +$$$$85\!\cdots\!68$$$$T^{8} -$$$$12\!\cdots\!16$$$$T^{10} +$$$$15\!\cdots\!64$$$$T^{12} -$$$$16\!\cdots\!56$$$$T^{14} +$$$$15\!\cdots\!62$$$$T^{16} -$$$$12\!\cdots\!84$$$$T^{18} +$$$$85\!\cdots\!44$$$$T^{20} -$$$$52\!\cdots\!04$$$$T^{22} +$$$$26\!\cdots\!88$$$$T^{24} -$$$$11\!\cdots\!12$$$$T^{26} +$$$$36\!\cdots\!76$$$$T^{28} -$$$$83\!\cdots\!52$$$$T^{30} +$$$$98\!\cdots\!81$$$$T^{32}$$
$17$ $$( 1 + 3744562800 T +$$$$34\!\cdots\!44$$$$T^{2} -$$$$10\!\cdots\!00$$$$T^{3} +$$$$63\!\cdots\!60$$$$T^{4} -$$$$36\!\cdots\!00$$$$T^{5} +$$$$81\!\cdots\!48$$$$T^{6} -$$$$51\!\cdots\!00$$$$T^{7} +$$$$77\!\cdots\!98$$$$T^{8} -$$$$42\!\cdots\!00$$$$T^{9} +$$$$55\!\cdots\!92$$$$T^{10} -$$$$20\!\cdots\!00$$$$T^{11} +$$$$29\!\cdots\!60$$$$T^{12} -$$$$39\!\cdots\!00$$$$T^{13} +$$$$11\!\cdots\!16$$$$T^{14} +$$$$99\!\cdots\!00$$$$T^{15} +$$$$21\!\cdots\!81$$$$T^{16} )^{2}$$
$19$ $$1 -$$$$50\!\cdots\!16$$$$T^{2} +$$$$12\!\cdots\!60$$$$T^{4} -$$$$21\!\cdots\!48$$$$T^{6} +$$$$27\!\cdots\!00$$$$T^{8} -$$$$27\!\cdots\!68$$$$T^{10} +$$$$22\!\cdots\!64$$$$T^{12} -$$$$15\!\cdots\!44$$$$T^{14} +$$$$94\!\cdots\!82$$$$T^{16} -$$$$47\!\cdots\!24$$$$T^{18} +$$$$20\!\cdots\!24$$$$T^{20} -$$$$74\!\cdots\!48$$$$T^{22} +$$$$22\!\cdots\!00$$$$T^{24} -$$$$52\!\cdots\!48$$$$T^{26} +$$$$94\!\cdots\!60$$$$T^{28} -$$$$11\!\cdots\!56$$$$T^{30} +$$$$66\!\cdots\!61$$$$T^{32}$$
$23$ $$( 1 + 373422672960 T +$$$$64\!\cdots\!76$$$$T^{2} +$$$$17\!\cdots\!80$$$$T^{3} +$$$$19\!\cdots\!52$$$$T^{4} +$$$$41\!\cdots\!20$$$$T^{5} +$$$$40\!\cdots\!92$$$$T^{6} +$$$$69\!\cdots\!60$$$$T^{7} +$$$$63\!\cdots\!74$$$$T^{8} +$$$$98\!\cdots\!80$$$$T^{9} +$$$$79\!\cdots\!28$$$$T^{10} +$$$$11\!\cdots\!40$$$$T^{11} +$$$$77\!\cdots\!12$$$$T^{12} +$$$$96\!\cdots\!40$$$$T^{13} +$$$$50\!\cdots\!04$$$$T^{14} +$$$$41\!\cdots\!20$$$$T^{15} +$$$$15\!\cdots\!61$$$$T^{16} )^{2}$$
$29$ $$1 -$$$$62\!\cdots\!20$$$$T^{2} +$$$$20\!\cdots\!80$$$$T^{4} -$$$$44\!\cdots\!20$$$$T^{6} +$$$$75\!\cdots\!16$$$$T^{8} -$$$$10\!\cdots\!80$$$$T^{10} +$$$$11\!\cdots\!20$$$$T^{12} -$$$$10\!\cdots\!80$$$$T^{14} +$$$$81\!\cdots\!06$$$$T^{16} -$$$$54\!\cdots\!80$$$$T^{18} +$$$$30\!\cdots\!20$$$$T^{20} -$$$$14\!\cdots\!80$$$$T^{22} +$$$$57\!\cdots\!36$$$$T^{24} -$$$$18\!\cdots\!20$$$$T^{26} +$$$$43\!\cdots\!80$$$$T^{28} -$$$$70\!\cdots\!20$$$$T^{30} +$$$$59\!\cdots\!41$$$$T^{32}$$
$31$ $$( 1 - 159489879296 T +$$$$10\!\cdots\!04$$$$T^{2} -$$$$16\!\cdots\!28$$$$T^{3} +$$$$51\!\cdots\!12$$$$T^{4} -$$$$11\!\cdots\!72$$$$T^{5} +$$$$18\!\cdots\!80$$$$T^{6} -$$$$39\!\cdots\!28$$$$T^{7} +$$$$49\!\cdots\!98$$$$T^{8} -$$$$88\!\cdots\!08$$$$T^{9} +$$$$93\!\cdots\!80$$$$T^{10} -$$$$13\!\cdots\!32$$$$T^{11} +$$$$13\!\cdots\!92$$$$T^{12} -$$$$94\!\cdots\!28$$$$T^{13} +$$$$13\!\cdots\!44$$$$T^{14} -$$$$47\!\cdots\!16$$$$T^{15} +$$$$66\!\cdots\!81$$$$T^{16} )^{2}$$
$37$ $$1 -$$$$39\!\cdots\!44$$$$T^{2} +$$$$80\!\cdots\!76$$$$T^{4} -$$$$10\!\cdots\!72$$$$T^{6} +$$$$11\!\cdots\!64$$$$T^{8} -$$$$90\!\cdots\!04$$$$T^{10} +$$$$60\!\cdots\!24$$$$T^{12} -$$$$34\!\cdots\!60$$$$T^{14} +$$$$17\!\cdots\!90$$$$T^{16} -$$$$72\!\cdots\!40$$$$T^{18} +$$$$26\!\cdots\!04$$$$T^{20} -$$$$81\!\cdots\!76$$$$T^{22} +$$$$20\!\cdots\!24$$$$T^{24} -$$$$42\!\cdots\!28$$$$T^{26} +$$$$65\!\cdots\!36$$$$T^{28} -$$$$67\!\cdots\!76$$$$T^{30} +$$$$35\!\cdots\!81$$$$T^{32}$$
$41$ $$( 1 - 3741125768016 T +$$$$70\!\cdots\!80$$$$T^{2} -$$$$14\!\cdots\!84$$$$T^{3} +$$$$29\!\cdots\!00$$$$T^{4} +$$$$14\!\cdots\!12$$$$T^{5} +$$$$99\!\cdots\!52$$$$T^{6} +$$$$61\!\cdots\!84$$$$T^{7} +$$$$26\!\cdots\!42$$$$T^{8} +$$$$15\!\cdots\!04$$$$T^{9} +$$$$67\!\cdots\!72$$$$T^{10} +$$$$25\!\cdots\!92$$$$T^{11} +$$$$13\!\cdots\!00$$$$T^{12} -$$$$17\!\cdots\!84$$$$T^{13} +$$$$22\!\cdots\!80$$$$T^{14} -$$$$31\!\cdots\!76$$$$T^{15} +$$$$21\!\cdots\!41$$$$T^{16} )^{2}$$
$43$ $$1 -$$$$41\!\cdots\!32$$$$T^{2} +$$$$84\!\cdots\!12$$$$T^{4} -$$$$11\!\cdots\!28$$$$T^{6} +$$$$11\!\cdots\!40$$$$T^{8} -$$$$97\!\cdots\!68$$$$T^{10} +$$$$74\!\cdots\!32$$$$T^{12} -$$$$50\!\cdots\!72$$$$T^{14} +$$$$31\!\cdots\!30$$$$T^{16} -$$$$17\!\cdots\!28$$$$T^{18} +$$$$88\!\cdots\!32$$$$T^{20} -$$$$40\!\cdots\!32$$$$T^{22} +$$$$16\!\cdots\!40$$$$T^{24} -$$$$54\!\cdots\!72$$$$T^{26} +$$$$14\!\cdots\!12$$$$T^{28} -$$$$24\!\cdots\!68$$$$T^{30} +$$$$20\!\cdots\!01$$$$T^{32}$$
$47$ $$( 1 - 188349402410880 T +$$$$19\!\cdots\!72$$$$T^{2} -$$$$31\!\cdots\!00$$$$T^{3} +$$$$17\!\cdots\!52$$$$T^{4} -$$$$23\!\cdots\!00$$$$T^{5} +$$$$87\!\cdots\!84$$$$T^{6} -$$$$99\!\cdots\!60$$$$T^{7} +$$$$28\!\cdots\!54$$$$T^{8} -$$$$26\!\cdots\!20$$$$T^{9} +$$$$62\!\cdots\!96$$$$T^{10} -$$$$43\!\cdots\!00$$$$T^{11} +$$$$85\!\cdots\!72$$$$T^{12} -$$$$42\!\cdots\!00$$$$T^{13} +$$$$69\!\cdots\!48$$$$T^{14} -$$$$17\!\cdots\!40$$$$T^{15} +$$$$25\!\cdots\!21$$$$T^{16} )^{2}$$
$53$ $$1 -$$$$12\!\cdots\!88$$$$T^{2} +$$$$92\!\cdots\!64$$$$T^{4} -$$$$47\!\cdots\!52$$$$T^{6} +$$$$18\!\cdots\!96$$$$T^{8} -$$$$62\!\cdots\!40$$$$T^{10} +$$$$17\!\cdots\!08$$$$T^{12} -$$$$43\!\cdots\!64$$$$T^{14} +$$$$94\!\cdots\!02$$$$T^{16} -$$$$18\!\cdots\!16$$$$T^{18} +$$$$31\!\cdots\!88$$$$T^{20} -$$$$46\!\cdots\!60$$$$T^{22} +$$$$59\!\cdots\!16$$$$T^{24} -$$$$63\!\cdots\!48$$$$T^{26} +$$$$52\!\cdots\!84$$$$T^{28} -$$$$30\!\cdots\!32$$$$T^{30} +$$$$10\!\cdots\!41$$$$T^{32}$$
$59$ $$1 -$$$$11\!\cdots\!84$$$$T^{2} +$$$$69\!\cdots\!76$$$$T^{4} -$$$$25\!\cdots\!84$$$$T^{6} +$$$$69\!\cdots\!16$$$$T^{8} -$$$$14\!\cdots\!00$$$$T^{10} +$$$$24\!\cdots\!32$$$$T^{12} -$$$$35\!\cdots\!68$$$$T^{14} +$$$$46\!\cdots\!62$$$$T^{16} -$$$$56\!\cdots\!48$$$$T^{18} +$$$$63\!\cdots\!72$$$$T^{20} -$$$$60\!\cdots\!00$$$$T^{22} +$$$$47\!\cdots\!56$$$$T^{24} -$$$$28\!\cdots\!84$$$$T^{26} +$$$$12\!\cdots\!36$$$$T^{28} -$$$$34\!\cdots\!64$$$$T^{30} +$$$$46\!\cdots\!81$$$$T^{32}$$
$61$ $$1 -$$$$16\!\cdots\!44$$$$T^{2} +$$$$12\!\cdots\!56$$$$T^{4} -$$$$64\!\cdots\!84$$$$T^{6} +$$$$24\!\cdots\!36$$$$T^{8} -$$$$82\!\cdots\!80$$$$T^{10} +$$$$24\!\cdots\!72$$$$T^{12} -$$$$64\!\cdots\!28$$$$T^{14} +$$$$15\!\cdots\!02$$$$T^{16} -$$$$32\!\cdots\!48$$$$T^{18} +$$$$61\!\cdots\!32$$$$T^{20} -$$$$10\!\cdots\!80$$$$T^{22} +$$$$15\!\cdots\!96$$$$T^{24} -$$$$20\!\cdots\!84$$$$T^{26} +$$$$20\!\cdots\!96$$$$T^{28} -$$$$13\!\cdots\!64$$$$T^{30} +$$$$40\!\cdots\!21$$$$T^{32}$$
$67$ $$1 -$$$$82\!\cdots\!12$$$$T^{2} +$$$$35\!\cdots\!44$$$$T^{4} -$$$$10\!\cdots\!88$$$$T^{6} +$$$$24\!\cdots\!96$$$$T^{8} -$$$$47\!\cdots\!60$$$$T^{10} +$$$$75\!\cdots\!48$$$$T^{12} -$$$$10\!\cdots\!76$$$$T^{14} +$$$$12\!\cdots\!82$$$$T^{16} -$$$$12\!\cdots\!04$$$$T^{18} +$$$$11\!\cdots\!68$$$$T^{20} -$$$$85\!\cdots\!40$$$$T^{22} +$$$$55\!\cdots\!76$$$$T^{24} -$$$$29\!\cdots\!12$$$$T^{26} +$$$$11\!\cdots\!24$$$$T^{28} -$$$$33\!\cdots\!08$$$$T^{30} +$$$$49\!\cdots\!61$$$$T^{32}$$
$71$ $$( 1 + 4512963142788288 T +$$$$90\!\cdots\!40$$$$T^{2} +$$$$19\!\cdots\!68$$$$T^{3} +$$$$31\!\cdots\!80$$$$T^{4} -$$$$14\!\cdots\!12$$$$T^{5} +$$$$78\!\cdots\!64$$$$T^{6} -$$$$17\!\cdots\!64$$$$T^{7} +$$$$22\!\cdots\!90$$$$T^{8} -$$$$51\!\cdots\!24$$$$T^{9} +$$$$68\!\cdots\!84$$$$T^{10} -$$$$36\!\cdots\!52$$$$T^{11} +$$$$24\!\cdots\!80$$$$T^{12} +$$$$43\!\cdots\!68$$$$T^{13} +$$$$60\!\cdots\!40$$$$T^{14} +$$$$89\!\cdots\!28$$$$T^{15} +$$$$59\!\cdots\!21$$$$T^{16} )^{2}$$
$73$ $$( 1 - 5666001023059280 T +$$$$17\!\cdots\!28$$$$T^{2} -$$$$10\!\cdots\!40$$$$T^{3} +$$$$12\!\cdots\!64$$$$T^{4} -$$$$66\!\cdots\!60$$$$T^{5} +$$$$61\!\cdots\!08$$$$T^{6} -$$$$24\!\cdots\!40$$$$T^{7} +$$$$27\!\cdots\!70$$$$T^{8} -$$$$11\!\cdots\!20$$$$T^{9} +$$$$13\!\cdots\!72$$$$T^{10} -$$$$71\!\cdots\!20$$$$T^{11} +$$$$65\!\cdots\!84$$$$T^{12} -$$$$24\!\cdots\!20$$$$T^{13} +$$$$19\!\cdots\!12$$$$T^{14} -$$$$30\!\cdots\!60$$$$T^{15} +$$$$25\!\cdots\!61$$$$T^{16} )^{2}$$
$79$ $$( 1 - 22649835696004224 T +$$$$78\!\cdots\!40$$$$T^{2} -$$$$10\!\cdots\!12$$$$T^{3} +$$$$22\!\cdots\!48$$$$T^{4} -$$$$15\!\cdots\!08$$$$T^{5} +$$$$30\!\cdots\!08$$$$T^{6} +$$$$78\!\cdots\!52$$$$T^{7} +$$$$34\!\cdots\!66$$$$T^{8} +$$$$14\!\cdots\!68$$$$T^{9} +$$$$10\!\cdots\!48$$$$T^{10} -$$$$94\!\cdots\!32$$$$T^{11} +$$$$24\!\cdots\!28$$$$T^{12} -$$$$20\!\cdots\!88$$$$T^{13} +$$$$28\!\cdots\!40$$$$T^{14} -$$$$14\!\cdots\!56$$$$T^{15} +$$$$11\!\cdots\!21$$$$T^{16} )^{2}$$
$83$ $$1 -$$$$26\!\cdots\!92$$$$T^{2} +$$$$34\!\cdots\!32$$$$T^{4} -$$$$29\!\cdots\!68$$$$T^{6} +$$$$21\!\cdots\!00$$$$T^{8} -$$$$13\!\cdots\!28$$$$T^{10} +$$$$71\!\cdots\!12$$$$T^{12} -$$$$34\!\cdots\!12$$$$T^{14} +$$$$15\!\cdots\!30$$$$T^{16} -$$$$61\!\cdots\!48$$$$T^{18} +$$$$22\!\cdots\!92$$$$T^{20} -$$$$73\!\cdots\!92$$$$T^{22} +$$$$21\!\cdots\!00$$$$T^{24} -$$$$52\!\cdots\!32$$$$T^{26} +$$$$10\!\cdots\!72$$$$T^{28} -$$$$14\!\cdots\!28$$$$T^{30} +$$$$97\!\cdots\!61$$$$T^{32}$$
$89$ $$( 1 + 34939587304383024 T +$$$$39\!\cdots\!00$$$$T^{2} +$$$$75\!\cdots\!32$$$$T^{3} +$$$$62\!\cdots\!48$$$$T^{4} +$$$$60\!\cdots\!48$$$$T^{5} +$$$$64\!\cdots\!08$$$$T^{6} +$$$$76\!\cdots\!28$$$$T^{7} +$$$$64\!\cdots\!66$$$$T^{8} +$$$$10\!\cdots\!12$$$$T^{9} +$$$$12\!\cdots\!28$$$$T^{10} +$$$$15\!\cdots\!72$$$$T^{11} +$$$$22\!\cdots\!88$$$$T^{12} +$$$$37\!\cdots\!68$$$$T^{13} +$$$$27\!\cdots\!00$$$$T^{14} +$$$$33\!\cdots\!16$$$$T^{15} +$$$$13\!\cdots\!61$$$$T^{16} )^{2}$$
$97$ $$( 1 - 47796699301090320 T +$$$$34\!\cdots\!40$$$$T^{2} -$$$$19\!\cdots\!20$$$$T^{3} +$$$$54\!\cdots\!44$$$$T^{4} -$$$$32\!\cdots\!40$$$$T^{5} +$$$$55\!\cdots\!60$$$$T^{6} -$$$$31\!\cdots\!00$$$$T^{7} +$$$$39\!\cdots\!50$$$$T^{8} -$$$$18\!\cdots\!00$$$$T^{9} +$$$$19\!\cdots\!40$$$$T^{10} -$$$$68\!\cdots\!20$$$$T^{11} +$$$$68\!\cdots\!84$$$$T^{12} -$$$$14\!\cdots\!40$$$$T^{13} +$$$$15\!\cdots\!60$$$$T^{14} -$$$$12\!\cdots\!60$$$$T^{15} +$$$$15\!\cdots\!21$$$$T^{16} )^{2}$$