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 $2$

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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
0.500000 + 5420.67i
0.500000 + 4000.58i
0.500000 + 3466.80i
0.500000 + 3446.66i
0.500000 + 3370.46i
0.500000 + 1208.53i
0.500000 + 1062.13i
0.500000 + 409.736i
0.500000 409.736i
0.500000 1062.13i
0.500000 1208.53i
0.500000 3370.46i
0.500000 3446.66i
0.500000 3466.80i
0.500000 4000.58i
0.500000 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:

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