# Properties

 Label 10.11.c.c Level 10 Weight 11 Character orbit 10.c Analytic conductor 6.354 Analytic rank 0 Dimension 6 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$10 = 2 \cdot 5$$ Weight: $$k$$ = $$11$$ Character orbit: $$[\chi]$$ = 10.c (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.35357252674$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{8}\cdot 5^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -16 - 16 \beta_{1} ) q^{2} + ( 21 - 21 \beta_{1} + \beta_{2} ) q^{3} + 512 \beta_{1} q^{4} + ( 910 - 896 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{5} + ( -672 - 16 \beta_{2} - 16 \beta_{3} ) q^{6} + ( 2244 + 2244 \beta_{1} + 19 \beta_{3} + 5 \beta_{4} + 5 \beta_{5} ) q^{7} + ( 8192 - 8192 \beta_{1} ) q^{8} + ( -59175 \beta_{1} + 61 \beta_{2} - 61 \beta_{3} + 30 \beta_{5} ) q^{9} +O(q^{10})$$ $$q + ( -16 - 16 \beta_{1} ) q^{2} + ( 21 - 21 \beta_{1} + \beta_{2} ) q^{3} + 512 \beta_{1} q^{4} + ( 910 - 896 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{5} + ( -672 - 16 \beta_{2} - 16 \beta_{3} ) q^{6} + ( 2244 + 2244 \beta_{1} + 19 \beta_{3} + 5 \beta_{4} + 5 \beta_{5} ) q^{7} + ( 8192 - 8192 \beta_{1} ) q^{8} + ( -59175 \beta_{1} + 61 \beta_{2} - 61 \beta_{3} + 30 \beta_{5} ) q^{9} + ( -28896 - 224 \beta_{1} + 48 \beta_{2} - 16 \beta_{3} + 16 \beta_{4} + 16 \beta_{5} ) q^{10} + ( 108144 - 263 \beta_{2} - 263 \beta_{3} + 10 \beta_{4} ) q^{11} + ( 10752 + 10752 \beta_{1} + 512 \beta_{3} ) q^{12} + ( -123466 + 123466 \beta_{1} - 988 \beta_{2} - 65 \beta_{4} + 65 \beta_{5} ) q^{13} + ( -71808 \beta_{1} + 304 \beta_{2} - 304 \beta_{3} - 160 \beta_{5} ) q^{14} + ( 262272 + 52122 \beta_{1} + 2883 \beta_{2} - 776 \beta_{3} - 63 \beta_{4} - 27 \beta_{5} ) q^{15} -262144 q^{16} + ( -126945 - 126945 \beta_{1} + 3386 \beta_{3} - 110 \beta_{4} - 110 \beta_{5} ) q^{17} + ( -946800 + 946800 \beta_{1} - 1952 \beta_{2} + 480 \beta_{4} - 480 \beta_{5} ) q^{18} + ( -181986 \beta_{1} + 416 \beta_{2} - 416 \beta_{3} + 130 \beta_{5} ) q^{19} + ( 458752 + 465920 \beta_{1} - 1024 \beta_{2} - 512 \beta_{3} - 512 \beta_{5} ) q^{20} + ( 2050806 - 6661 \beta_{2} - 6661 \beta_{3} - 540 \beta_{4} ) q^{21} + ( -1730304 - 1730304 \beta_{1} + 8416 \beta_{3} - 160 \beta_{4} - 160 \beta_{5} ) q^{22} + ( -2511464 + 2511464 \beta_{1} + 8631 \beta_{2} - 735 \beta_{4} + 735 \beta_{5} ) q^{23} + ( -344064 \beta_{1} + 8192 \beta_{2} - 8192 \beta_{3} ) q^{24} + ( 7103915 - 1114930 \beta_{1} - 2485 \beta_{2} + 12895 \beta_{3} + 270 \beta_{4} + 1810 \beta_{5} ) q^{25} + ( 3950912 + 15808 \beta_{2} + 15808 \beta_{3} + 2080 \beta_{4} ) q^{26} + ( -7979328 - 7979328 \beta_{1} - 62996 \beta_{3} + 1920 \beta_{4} + 1920 \beta_{5} ) q^{27} + ( -1148928 + 1148928 \beta_{1} - 9728 \beta_{2} - 2560 \beta_{4} + 2560 \beta_{5} ) q^{28} + ( -3270462 \beta_{1} - 50938 \beta_{2} + 50938 \beta_{3} + 1630 \beta_{5} ) q^{29} + ( -3362400 - 5030304 \beta_{1} - 58544 \beta_{2} - 33712 \beta_{3} + 576 \beta_{4} + 1440 \beta_{5} ) q^{30} + ( 25623276 + 26719 \beta_{2} + 26719 \beta_{3} + 310 \beta_{4} ) q^{31} + ( 4194304 + 4194304 \beta_{1} ) q^{32} + ( -28862862 + 28862862 \beta_{1} + 67774 \beta_{2} + 7920 \beta_{4} - 7920 \beta_{5} ) q^{33} + ( 4062240 \beta_{1} + 54176 \beta_{2} - 54176 \beta_{3} + 3520 \beta_{5} ) q^{34} + ( -34852363 - 33003937 \beta_{1} + 38204 \beta_{2} + 49457 \beta_{3} - 1792 \beta_{4} - 13002 \beta_{5} ) q^{35} + ( 30297600 + 31232 \beta_{2} + 31232 \beta_{3} - 15360 \beta_{4} ) q^{36} + ( -5431860 - 5431860 \beta_{1} + 21066 \beta_{3} - 12735 \beta_{4} - 12735 \beta_{5} ) q^{37} + ( -2911776 + 2911776 \beta_{1} - 13312 \beta_{2} + 2080 \beta_{4} - 2080 \beta_{5} ) q^{38} + ( 117571248 \beta_{1} - 37341 \beta_{2} + 37341 \beta_{3} - 29250 \beta_{5} ) q^{39} + ( 114688 - 14794752 \beta_{1} + 8192 \beta_{2} + 24576 \beta_{3} - 8192 \beta_{4} + 8192 \beta_{5} ) q^{40} + ( 37820468 - 204319 \beta_{2} - 204319 \beta_{3} + 23870 \beta_{4} ) q^{41} + ( -32812896 - 32812896 \beta_{1} + 213152 \beta_{3} + 8640 \beta_{4} + 8640 \beta_{5} ) q^{42} + ( -10535633 + 10535633 \beta_{1} - 185267 \beta_{2} - 22210 \beta_{4} + 22210 \beta_{5} ) q^{43} + ( 55369728 \beta_{1} + 134656 \beta_{2} - 134656 \beta_{3} + 5120 \beta_{5} ) q^{44} + ( -29090754 - 289958130 \beta_{1} + 350233 \beta_{2} - 101096 \beta_{3} + 23010 \beta_{4} + 27549 \beta_{5} ) q^{45} + ( 80366848 - 138096 \beta_{2} - 138096 \beta_{3} + 23520 \beta_{4} ) q^{46} + ( -3421866 - 3421866 \beta_{1} + 383139 \beta_{3} + 32235 \beta_{4} + 32235 \beta_{5} ) q^{47} + ( -5505024 + 5505024 \beta_{1} - 262144 \beta_{2} ) q^{48} + ( 134255115 \beta_{1} + 287639 \beta_{2} - 287639 \beta_{3} + 123910 \beta_{5} ) q^{49} + ( -131501520 - 95823760 \beta_{1} + 246080 \beta_{2} - 166560 \beta_{3} + 24640 \beta_{4} - 33280 \beta_{5} ) q^{50} + ( 397993002 + 221125 \beta_{2} + 221125 \beta_{3} - 102240 \beta_{4} ) q^{51} + ( -63214592 - 63214592 \beta_{1} - 505856 \beta_{3} - 33280 \beta_{4} - 33280 \beta_{5} ) q^{52} + ( -158256646 + 158256646 \beta_{1} - 264378 \beta_{2} + 33795 \beta_{4} - 33795 \beta_{5} ) q^{53} + ( 255338496 \beta_{1} - 1007936 \beta_{2} + 1007936 \beta_{3} - 61440 \beta_{5} ) q^{54} + ( -22013600 - 190298682 \beta_{1} - 1059677 \beta_{2} - 351921 \beta_{3} - 120142 \beta_{4} + 14170 \beta_{5} ) q^{55} + ( 36765696 + 155648 \beta_{2} + 155648 \beta_{3} + 81920 \beta_{4} ) q^{56} + ( -56184198 - 56184198 \beta_{1} - 466556 \beta_{3} + 12870 \beta_{4} + 12870 \beta_{5} ) q^{57} + ( -52327392 + 52327392 \beta_{1} + 1630016 \beta_{2} + 26080 \beta_{4} - 26080 \beta_{5} ) q^{58} + ( -1396934 \beta_{1} + 605484 \beta_{2} - 605484 \beta_{3} - 230490 \beta_{5} ) q^{59} + ( -26686464 + 134283264 \beta_{1} + 397312 \beta_{2} + 1476096 \beta_{3} + 13824 \beta_{4} - 32256 \beta_{5} ) q^{60} + ( 266705180 + 892993 \beta_{2} + 892993 \beta_{3} + 90670 \beta_{4} ) q^{61} + ( -409972416 - 409972416 \beta_{1} - 855008 \beta_{3} - 4960 \beta_{4} - 4960 \beta_{5} ) q^{62} + ( -856315332 + 856315332 \beta_{1} + 1439815 \beta_{2} - 97035 \beta_{4} + 97035 \beta_{5} ) q^{63} -134217728 \beta_{1} q^{64} + ( 233605367 - 349511323 \beta_{1} - 2857977 \beta_{2} + 835469 \beta_{3} + 294442 \beta_{4} - 82022 \beta_{5} ) q^{65} + ( 923611584 - 1084384 \beta_{2} - 1084384 \beta_{3} - 253440 \beta_{4} ) q^{66} + ( 417379583 + 417379583 \beta_{1} - 530403 \beta_{3} - 284490 \beta_{4} - 284490 \beta_{5} ) q^{67} + ( 64995840 - 64995840 \beta_{1} - 1733632 \beta_{2} + 56320 \beta_{4} - 56320 \beta_{5} ) q^{68} + ( -947419494 \beta_{1} - 745469 \beta_{2} + 745469 \beta_{3} + 263340 \beta_{5} ) q^{69} + ( 29574816 + 1085700800 \beta_{1} + 180048 \beta_{2} - 1402576 \beta_{3} - 179360 \beta_{4} + 236704 \beta_{5} ) q^{70} + ( -1190882180 + 1849663 \beta_{2} + 1849663 \beta_{3} + 180310 \beta_{4} ) q^{71} + ( -484761600 - 484761600 \beta_{1} - 999424 \beta_{3} + 245760 \beta_{4} + 245760 \beta_{5} ) q^{72} + ( 384852279 - 384852279 \beta_{1} - 2445988 \beta_{2} + 120370 \beta_{4} - 120370 \beta_{5} ) q^{73} + ( 173819520 \beta_{1} + 337056 \beta_{2} - 337056 \beta_{3} + 407520 \beta_{5} ) q^{74} + ( 1582122255 + 76966365 \beta_{1} + 6998405 \beta_{2} - 3998460 \beta_{3} - 380610 \beta_{4} - 69930 \beta_{5} ) q^{75} + ( 93176832 + 212992 \beta_{2} + 212992 \beta_{3} - 66560 \beta_{4} ) q^{76} + ( 97585072 + 97585072 \beta_{1} + 5376026 \beta_{3} + 853690 \beta_{4} + 853690 \beta_{5} ) q^{77} + ( 1881139968 - 1881139968 \beta_{1} + 1194912 \beta_{2} - 468000 \beta_{4} + 468000 \beta_{5} ) q^{78} + ( -948328216 \beta_{1} + 195656 \beta_{2} - 195656 \beta_{3} - 193640 \beta_{5} ) q^{79} + ( -238551040 + 234881024 \beta_{1} + 262144 \beta_{2} - 524288 \beta_{3} + 262144 \beta_{4} ) q^{80} + ( -4337792793 - 10608539 \beta_{2} - 10608539 \beta_{3} + 129930 \beta_{4} ) q^{81} + ( -605127488 - 605127488 \beta_{1} + 6538208 \beta_{3} - 381920 \beta_{4} - 381920 \beta_{5} ) q^{82} + ( -12859987 + 12859987 \beta_{1} - 5994475 \beta_{2} + 484120 \beta_{4} - 484120 \beta_{5} ) q^{83} + ( 1050012672 \beta_{1} + 3410432 \beta_{2} - 3410432 \beta_{3} - 276480 \beta_{5} ) q^{84} + ( 284105332 + 1721564848 \beta_{1} + 2583799 \beta_{2} + 9721417 \beta_{3} + 299593 \beta_{4} + 203853 \beta_{5} ) q^{85} + ( 337140256 + 2964272 \beta_{2} + 2964272 \beta_{3} + 710720 \beta_{4} ) q^{86} + ( 5863997874 + 5863997874 \beta_{1} - 2346212 \beta_{3} - 1523250 \beta_{4} - 1523250 \beta_{5} ) q^{87} + ( 885915648 - 885915648 \beta_{1} - 4308992 \beta_{2} + 81920 \beta_{4} - 81920 \beta_{5} ) q^{88} + ( -4350280900 \beta_{1} + 1011780 \beta_{2} - 1011780 \beta_{3} - 882300 \beta_{5} ) q^{89} + ( -4173878016 + 5104782144 \beta_{1} - 7221264 \beta_{2} - 3986192 \beta_{3} + 72624 \beta_{4} - 808944 \beta_{5} ) q^{90} + ( -7290495682 + 7179181 \beta_{2} + 7179181 \beta_{3} - 1998020 \beta_{4} ) q^{91} + ( -1285869568 - 1285869568 \beta_{1} + 4419072 \beta_{3} - 376320 \beta_{4} - 376320 \beta_{5} ) q^{92} + ( 3664888554 - 3664888554 \beta_{1} + 27161566 \beta_{2} - 800640 \beta_{4} + 800640 \beta_{5} ) q^{93} + ( 109499712 \beta_{1} + 6130224 \beta_{2} - 6130224 \beta_{3} - 1031520 \beta_{5} ) q^{94} + ( -5971980 - 1214821790 \beta_{1} + 1675620 \beta_{2} - 1073840 \beta_{3} + 86060 \beta_{4} + 44030 \beta_{5} ) q^{95} + ( 176160768 + 4194304 \beta_{2} + 4194304 \beta_{3} ) q^{96} + ( 5553402337 + 5553402337 \beta_{1} - 21140480 \beta_{3} + 1477760 \beta_{4} + 1477760 \beta_{5} ) q^{97} + ( 2148081840 - 2148081840 \beta_{1} - 9204448 \beta_{2} + 1982560 \beta_{4} - 1982560 \beta_{5} ) q^{98} + ( 77635512 \beta_{1} - 25931375 \beta_{2} + 25931375 \beta_{3} + 2576190 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 96q^{2} + 128q^{3} + 5460q^{5} - 4096q^{6} + 13512q^{7} + 49152q^{8} + O(q^{10})$$ $$6q - 96q^{2} + 128q^{3} + 5460q^{5} - 4096q^{6} + 13512q^{7} + 49152q^{8} - 173280q^{10} + 647832q^{11} + 65536q^{12} - 742902q^{13} + 1577720q^{15} - 1572864q^{16} - 755118q^{17} - 5683744q^{18} + 2749440q^{20} + 12277112q^{21} - 10365312q^{22} - 15052992q^{23} + 42644850q^{25} + 23772864q^{26} - 47998120q^{27} - 6918144q^{28} - 20357760q^{30} + 153847152q^{31} + 25165824q^{32} - 173025784q^{33} - 208942440q^{35} + 181879808q^{36} - 32574498q^{37} - 17493120q^{38} + 737280q^{40} + 226153272q^{41} - 196433792q^{42} - 63628752q^{43} - 174000230q^{45} + 481695744q^{46} - 19700448q^{47} - 33554432q^{48} - 788800800q^{50} + 2388638032q^{51} - 380365824q^{52} - 950001042q^{53} - 135145080q^{55} + 221380608q^{56} - 338012560q^{57} - 310652160q^{58} - 156344320q^{60} + 1603984392q^{61} - 2461554432q^{62} - 5135206432q^{63} + 1398176070q^{65} + 5536825088q^{66} + 2502647712q^{67} + 386620416q^{68} + 174645120q^{70} - 7137533808q^{71} - 2910076928q^{72} + 2304462438q^{73} + 9497972200q^{75} + 559779840q^{76} + 597969864q^{77} + 11288293632q^{78} - 1431306240q^{80} - 26068931054q^{81} - 3618452352q^{82} - 88180632q^{83} + 1729841610q^{85} + 2036120064q^{86} + 35176248320q^{87} + 5307039744q^{88} - 25065537760q^{90} - 43718253408q^{91} - 7707131904q^{92} + 22042053176q^{93} - 34456200q^{95} + 1073741824q^{96} + 33281088582q^{97} + 12874047264q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 1148 x^{3} + 68121 x^{2} - 299628 x + 658952$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$68121 \nu^{5} + 149814 \nu^{4} + 329476 \nu^{3} - 39101454 \nu^{2} + 4554477405 \nu - 10394598718$$$$)/ 10016360270$$ $$\beta_{2}$$ $$=$$ $$($$$$60546049 \nu^{5} - 58729384 \nu^{4} + 638377544 \nu^{3} - 84835233476 \nu^{2} + 4158168070345 \nu - 18141291569772$$$$)/ 30049080810$$ $$\beta_{3}$$ $$=$$ $$($$$$-60844529 \nu^{5} - 325695636 \nu^{4} - 716280824 \nu^{3} + 85006560996 \nu^{2} - 4158168070345 \nu + 822290385952$$$$)/ 30049080810$$ $$\beta_{4}$$ $$=$$ $$($$$$-31840 \nu^{5} - 738610 \nu^{4} - 8310240 \nu^{3} + 18276160 \nu^{2} - 20819537159$$$$)/10470063$$ $$\beta_{5}$$ $$=$$ $$($$$$-517677579 \nu^{5} - 1138493986 \nu^{4} + 16684615276 \nu^{3} + 797964943846 \nu^{2} - 29603038676095 \nu + 67978379651882$$$$)/ 30049080810$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} - 10 \beta_{3} + 3 \beta_{1} + 3$$$$)/400$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + 25 \beta_{3} - 25 \beta_{2} + 17383 \beta_{1}$$$$)/100$$ $$\nu^{3}$$ $$=$$ $$($$$$261 \beta_{5} - 261 \beta_{4} + 2610 \beta_{2} - 228817 \beta_{1} + 228817$$$$)/400$$ $$\nu^{4}$$ $$=$$ $$($$$$13 \beta_{4} - 3980 \beta_{3} - 3980 \beta_{2} - 2268051$$$$)/50$$ $$\nu^{5}$$ $$=$$ $$($$$$-13165 \beta_{5} - 13165 \beta_{4} + 159202 \beta_{3} + 19926521 \beta_{1} + 19926521$$$$)/80$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/10\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$\chi(n)$$ $$\beta_{1}$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 2.29143 − 2.29143i 10.1043 − 10.1043i −12.3957 + 12.3957i 2.29143 + 2.29143i 10.1043 + 10.1043i −12.3957 − 12.3957i
−16.0000 + 16.0000i −266.441 266.441i 512.000i 2583.39 + 1758.32i 8526.10 −13021.6 + 13021.6i 8192.00 + 8192.00i 82932.3i −69467.5 + 13201.2i
3.2 −16.0000 + 16.0000i 4.29207 + 4.29207i 512.000i −2978.33 + 946.124i −137.346 21284.7 21284.7i 8192.00 + 8192.00i 59012.2i 32515.4 62791.3i
3.3 −16.0000 + 16.0000i 326.149 + 326.149i 512.000i 3124.94 19.4459i −10436.8 −1507.13 + 1507.13i 8192.00 + 8192.00i 153697.i −49687.9 + 50310.2i
7.1 −16.0000 16.0000i −266.441 + 266.441i 512.000i 2583.39 1758.32i 8526.10 −13021.6 13021.6i 8192.00 8192.00i 82932.3i −69467.5 13201.2i
7.2 −16.0000 16.0000i 4.29207 4.29207i 512.000i −2978.33 946.124i −137.346 21284.7 + 21284.7i 8192.00 8192.00i 59012.2i 32515.4 + 62791.3i
7.3 −16.0000 16.0000i 326.149 326.149i 512.000i 3124.94 + 19.4459i −10436.8 −1507.13 1507.13i 8192.00 8192.00i 153697.i −49687.9 50310.2i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 7.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 10.11.c.c 6
3.b odd 2 1 90.11.g.c 6
4.b odd 2 1 80.11.p.c 6
5.b even 2 1 50.11.c.e 6
5.c odd 4 1 inner 10.11.c.c 6
5.c odd 4 1 50.11.c.e 6
15.e even 4 1 90.11.g.c 6
20.e even 4 1 80.11.p.c 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.11.c.c 6 1.a even 1 1 trivial
10.11.c.c 6 5.c odd 4 1 inner
50.11.c.e 6 5.b even 2 1
50.11.c.e 6 5.c odd 4 1
80.11.p.c 6 4.b odd 2 1
80.11.p.c 6 20.e even 4 1
90.11.g.c 6 3.b odd 2 1
90.11.g.c 6 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} - 128 T_{3}^{5} + 8192 T_{3}^{4} + 20688696 T_{3}^{3} + 30028037796 T_{3}^{2} - 258527462832 T_{3} +$$$$11\!\cdots\!72$$ acting on $$S_{11}^{\mathrm{new}}(10, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 32 T + 512 T^{2} )^{3}$$
$3$ $$1 - 128 T + 8192 T^{2} + 13130424 T^{3} - 441069057 T^{4} + 422886187368 T^{5} + 35687822941728 T^{6} + 24971006477893032 T^{7} - 1537912707711379857 T^{8} +$$$$27\!\cdots\!76$$$$T^{9} +$$$$99\!\cdots\!92$$$$T^{10} -$$$$91\!\cdots\!72$$$$T^{11} +$$$$42\!\cdots\!01$$$$T^{12}$$
$5$ $$1 - 5460 T - 6416625 T^{2} + 85710975000 T^{3} - 62662353515625 T^{4} - 520706176757812500 T^{5} +$$$$93\!\cdots\!25$$$$T^{6}$$
$7$ $$1 - 13512 T + 91287072 T^{2} + 5680562811056 T^{3} - 79588467032526417 T^{4} -$$$$39\!\cdots\!08$$$$T^{5} +$$$$28\!\cdots\!88$$$$T^{6} -$$$$11\!\cdots\!92$$$$T^{7} -$$$$63\!\cdots\!17$$$$T^{8} +$$$$12\!\cdots\!44$$$$T^{9} +$$$$58\!\cdots\!72$$$$T^{10} -$$$$24\!\cdots\!88$$$$T^{11} +$$$$50\!\cdots\!01$$$$T^{12}$$
$11$ $$( 1 - 323916 T + 86871291855 T^{2} - 14136588581335480 T^{3} +$$$$22\!\cdots\!55$$$$T^{4} -$$$$21\!\cdots\!16$$$$T^{5} +$$$$17\!\cdots\!01$$$$T^{6} )^{2}$$
$13$ $$1 + 742902 T + 275951690802 T^{2} + 104807015418799014 T^{3} -$$$$65\!\cdots\!77$$$$T^{4} -$$$$19\!\cdots\!52$$$$T^{5} -$$$$72\!\cdots\!52$$$$T^{6} -$$$$26\!\cdots\!48$$$$T^{7} -$$$$12\!\cdots\!77$$$$T^{8} +$$$$27\!\cdots\!86$$$$T^{9} +$$$$99\!\cdots\!02$$$$T^{10} +$$$$36\!\cdots\!98$$$$T^{11} +$$$$68\!\cdots\!01$$$$T^{12}$$
$17$ $$1 + 755118 T + 285101596962 T^{2} + 2224663782493800046 T^{3} -$$$$91\!\cdots\!97$$$$T^{4} -$$$$64\!\cdots\!28$$$$T^{5} -$$$$21\!\cdots\!32$$$$T^{6} -$$$$12\!\cdots\!72$$$$T^{7} -$$$$37\!\cdots\!97$$$$T^{8} +$$$$18\!\cdots\!54$$$$T^{9} +$$$$47\!\cdots\!62$$$$T^{10} +$$$$25\!\cdots\!82$$$$T^{11} +$$$$67\!\cdots\!01$$$$T^{12}$$
$19$ $$1 - 36166867207206 T^{2} +$$$$54\!\cdots\!15$$$$T^{4} -$$$$44\!\cdots\!20$$$$T^{6} +$$$$20\!\cdots\!15$$$$T^{8} -$$$$51\!\cdots\!06$$$$T^{10} +$$$$53\!\cdots\!01$$$$T^{12}$$
$23$ $$1 + 15052992 T + 113296284076032 T^{2} +$$$$91\!\cdots\!84$$$$T^{3} +$$$$71\!\cdots\!63$$$$T^{4} +$$$$42\!\cdots\!88$$$$T^{5} +$$$$24\!\cdots\!08$$$$T^{6} +$$$$17\!\cdots\!12$$$$T^{7} +$$$$12\!\cdots\!63$$$$T^{8} +$$$$65\!\cdots\!16$$$$T^{9} +$$$$33\!\cdots\!32$$$$T^{10} +$$$$18\!\cdots\!08$$$$T^{11} +$$$$50\!\cdots\!01$$$$T^{12}$$
$29$ $$1 - 634112330650806 T^{2} +$$$$25\!\cdots\!15$$$$T^{4} -$$$$13\!\cdots\!20$$$$T^{6} +$$$$45\!\cdots\!15$$$$T^{8} -$$$$19\!\cdots\!06$$$$T^{10} +$$$$55\!\cdots\!01$$$$T^{12}$$
$31$ $$( 1 - 76923576 T + 4180287219854295 T^{2} -$$$$13\!\cdots\!40$$$$T^{3} +$$$$34\!\cdots\!95$$$$T^{4} -$$$$51\!\cdots\!76$$$$T^{5} +$$$$55\!\cdots\!01$$$$T^{6} )^{2}$$
$37$ $$1 + 32574498 T + 530548959976002 T^{2} -$$$$35\!\cdots\!14$$$$T^{3} +$$$$13\!\cdots\!23$$$$T^{4} +$$$$17\!\cdots\!52$$$$T^{5} +$$$$50\!\cdots\!48$$$$T^{6} +$$$$84\!\cdots\!48$$$$T^{7} +$$$$30\!\cdots\!23$$$$T^{8} -$$$$39\!\cdots\!86$$$$T^{9} +$$$$28\!\cdots\!02$$$$T^{10} +$$$$83\!\cdots\!02$$$$T^{11} +$$$$12\!\cdots\!01$$$$T^{12}$$
$41$ $$( 1 - 113076636 T + 22611520248205335 T^{2} -$$$$10\!\cdots\!00$$$$T^{3} +$$$$30\!\cdots\!35$$$$T^{4} -$$$$20\!\cdots\!36$$$$T^{5} +$$$$24\!\cdots\!01$$$$T^{6} )^{2}$$
$43$ $$1 + 63628752 T + 2024309040538752 T^{2} +$$$$11\!\cdots\!64$$$$T^{3} +$$$$28\!\cdots\!23$$$$T^{4} -$$$$12\!\cdots\!52$$$$T^{5} +$$$$20\!\cdots\!48$$$$T^{6} -$$$$26\!\cdots\!48$$$$T^{7} +$$$$13\!\cdots\!23$$$$T^{8} +$$$$11\!\cdots\!36$$$$T^{9} +$$$$44\!\cdots\!52$$$$T^{10} +$$$$29\!\cdots\!48$$$$T^{11} +$$$$10\!\cdots\!01$$$$T^{12}$$
$47$ $$1 + 19700448 T + 194053825700352 T^{2} -$$$$41\!\cdots\!64$$$$T^{3} +$$$$61\!\cdots\!23$$$$T^{4} +$$$$52\!\cdots\!52$$$$T^{5} +$$$$18\!\cdots\!48$$$$T^{6} +$$$$27\!\cdots\!48$$$$T^{7} +$$$$16\!\cdots\!23$$$$T^{8} -$$$$59\!\cdots\!36$$$$T^{9} +$$$$14\!\cdots\!52$$$$T^{10} +$$$$79\!\cdots\!52$$$$T^{11} +$$$$21\!\cdots\!01$$$$T^{12}$$
$53$ $$1 + 950001042 T + 451250989900542882 T^{2} +$$$$26\!\cdots\!34$$$$T^{3} +$$$$18\!\cdots\!63$$$$T^{4} +$$$$79\!\cdots\!88$$$$T^{5} +$$$$28\!\cdots\!08$$$$T^{6} +$$$$13\!\cdots\!12$$$$T^{7} +$$$$55\!\cdots\!63$$$$T^{8} +$$$$14\!\cdots\!66$$$$T^{9} +$$$$42\!\cdots\!82$$$$T^{10} +$$$$15\!\cdots\!58$$$$T^{11} +$$$$28\!\cdots\!01$$$$T^{12}$$
$59$ $$1 - 1661016969037924806 T^{2} +$$$$15\!\cdots\!15$$$$T^{4} -$$$$96\!\cdots\!20$$$$T^{6} +$$$$40\!\cdots\!15$$$$T^{8} -$$$$11\!\cdots\!06$$$$T^{10} +$$$$17\!\cdots\!01$$$$T^{12}$$
$61$ $$( 1 - 801992196 T + 1994865806990658975 T^{2} -$$$$98\!\cdots\!60$$$$T^{3} +$$$$14\!\cdots\!75$$$$T^{4} -$$$$40\!\cdots\!96$$$$T^{5} +$$$$36\!\cdots\!01$$$$T^{6} )^{2}$$
$67$ $$1 - 2502647712 T + 3131622785189417472 T^{2} -$$$$49\!\cdots\!44$$$$T^{3} +$$$$47\!\cdots\!83$$$$T^{4} -$$$$16\!\cdots\!08$$$$T^{5} +$$$$15\!\cdots\!88$$$$T^{6} -$$$$29\!\cdots\!92$$$$T^{7} +$$$$15\!\cdots\!83$$$$T^{8} -$$$$30\!\cdots\!56$$$$T^{9} +$$$$34\!\cdots\!72$$$$T^{10} -$$$$50\!\cdots\!88$$$$T^{11} +$$$$36\!\cdots\!01$$$$T^{12}$$
$71$ $$( 1 + 3568766904 T + 12497161629854598375 T^{2} +$$$$23\!\cdots\!40$$$$T^{3} +$$$$40\!\cdots\!75$$$$T^{4} +$$$$37\!\cdots\!04$$$$T^{5} +$$$$34\!\cdots\!01$$$$T^{6} )^{2}$$
$73$ $$1 - 2304462438 T + 2655273564076451922 T^{2} -$$$$10\!\cdots\!06$$$$T^{3} +$$$$46\!\cdots\!83$$$$T^{4} -$$$$57\!\cdots\!92$$$$T^{5} +$$$$64\!\cdots\!88$$$$T^{6} -$$$$24\!\cdots\!08$$$$T^{7} +$$$$85\!\cdots\!83$$$$T^{8} -$$$$82\!\cdots\!94$$$$T^{9} +$$$$90\!\cdots\!22$$$$T^{10} -$$$$33\!\cdots\!62$$$$T^{11} +$$$$63\!\cdots\!01$$$$T^{12}$$
$79$ $$1 - 53253337181723476806 T^{2} +$$$$12\!\cdots\!15$$$$T^{4} -$$$$15\!\cdots\!20$$$$T^{6} +$$$$10\!\cdots\!15$$$$T^{8} -$$$$42\!\cdots\!06$$$$T^{10} +$$$$72\!\cdots\!01$$$$T^{12}$$
$83$ $$1 + 88180632 T + 3887911929959712 T^{2} -$$$$16\!\cdots\!96$$$$T^{3} +$$$$11\!\cdots\!03$$$$T^{4} +$$$$59\!\cdots\!28$$$$T^{5} +$$$$18\!\cdots\!68$$$$T^{6} +$$$$92\!\cdots\!72$$$$T^{7} +$$$$28\!\cdots\!03$$$$T^{8} -$$$$60\!\cdots\!04$$$$T^{9} +$$$$22\!\cdots\!12$$$$T^{10} +$$$$79\!\cdots\!68$$$$T^{11} +$$$$13\!\cdots\!01$$$$T^{12}$$
$89$ $$1 -$$$$11\!\cdots\!06$$$$T^{2} +$$$$62\!\cdots\!15$$$$T^{4} -$$$$22\!\cdots\!20$$$$T^{6} +$$$$60\!\cdots\!15$$$$T^{8} -$$$$10\!\cdots\!06$$$$T^{10} +$$$$91\!\cdots\!01$$$$T^{12}$$
$97$ $$1 - 33281088582 T +$$$$55\!\cdots\!62$$$$T^{2} -$$$$56\!\cdots\!54$$$$T^{3} +$$$$34\!\cdots\!03$$$$T^{4} -$$$$76\!\cdots\!28$$$$T^{5} -$$$$25\!\cdots\!32$$$$T^{6} -$$$$56\!\cdots\!72$$$$T^{7} +$$$$18\!\cdots\!03$$$$T^{8} -$$$$22\!\cdots\!46$$$$T^{9} +$$$$16\!\cdots\!62$$$$T^{10} -$$$$72\!\cdots\!18$$$$T^{11} +$$$$16\!\cdots\!01$$$$T^{12}$$