Properties

 Label 160.6.c.d Level 160 Weight 6 Character orbit 160.c Analytic conductor 25.661 Analytic rank 0 Dimension 12 CM no Inner twists 4

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$160 = 2^{5} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 160.c (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$25.6614111701$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 6 x^{11} + 18 x^{10} + 348 x^{9} + 21226 x^{8} - 87824 x^{7} + 205428 x^{6} + 2113880 x^{5} + 27827225 x^{4} - 59798850 x^{3} + 89646050 x^{2} + 438522500 x + 1072562500$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{46}\cdot 5^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + ( -5 + \beta_{5} ) q^{5} + ( -5 \beta_{2} - 3 \beta_{3} - \beta_{6} ) q^{7} + ( -22 + \beta_{1} + 3 \beta_{4} - 3 \beta_{5} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} + ( -5 + \beta_{5} ) q^{5} + ( -5 \beta_{2} - 3 \beta_{3} - \beta_{6} ) q^{7} + ( -22 + \beta_{1} + 3 \beta_{4} - 3 \beta_{5} ) q^{9} + \beta_{8} q^{11} -\beta_{9} q^{13} + ( 15 \beta_{2} + \beta_{3} - 3 \beta_{6} + \beta_{7} ) q^{15} + ( -2 \beta_{4} - \beta_{5} - \beta_{9} - \beta_{11} ) q^{17} + ( \beta_{3} - 2 \beta_{7} - 2 \beta_{8} + \beta_{10} ) q^{19} + ( 1016 + 28 \beta_{1} - 9 \beta_{4} + 9 \beta_{5} ) q^{21} + ( -61 \beta_{2} + 8 \beta_{3} + 57 \beta_{6} ) q^{23} + ( -1370 - 5 \beta_{1} + 37 \beta_{4} - 24 \beta_{5} - \beta_{9} + \beta_{11} ) q^{25} + ( 98 \beta_{2} + 24 \beta_{3} + 30 \beta_{6} ) q^{27} + ( -676 - 62 \beta_{1} - 12 \beta_{4} + 12 \beta_{5} ) q^{29} + ( -\beta_{3} + 2 \beta_{7} - 2 \beta_{8} + 4 \beta_{10} ) q^{31} + ( 22 \beta_{4} + 21 \beta_{5} + 7 \beta_{9} + \beta_{11} ) q^{33} + ( -55 \beta_{2} - 45 \beta_{3} - 6 \beta_{6} - 4 \beta_{7} + 8 \beta_{8} + 3 \beta_{10} ) q^{35} + ( -31 \beta_{4} - 33 \beta_{5} + 6 \beta_{9} + 2 \beta_{11} ) q^{37} + ( -4 \beta_{3} + 8 \beta_{7} + 24 \beta_{8} + 6 \beta_{10} ) q^{39} + ( 5495 + 95 \beta_{1} - 113 \beta_{4} + 113 \beta_{5} ) q^{41} + ( 77 \beta_{2} - 54 \beta_{3} + 146 \beta_{6} ) q^{43} + ( -5025 + 30 \beta_{1} - 104 \beta_{4} + 15 \beta_{5} + 7 \beta_{9} - 2 \beta_{11} ) q^{45} + ( -393 \beta_{2} - 338 \beta_{3} + 135 \beta_{6} ) q^{47} + ( 1398 - 295 \beta_{1} + 159 \beta_{4} - 159 \beta_{5} ) q^{49} + ( 5 \beta_{3} - 10 \beta_{7} + 21 \beta_{8} + 15 \beta_{10} ) q^{51} + ( -154 \beta_{4} - 152 \beta_{5} - 11 \beta_{9} - 2 \beta_{11} ) q^{53} + ( 520 \beta_{2} + 325 \beta_{3} + 300 \beta_{6} - 5 \beta_{7} - 20 \beta_{8} + 10 \beta_{10} ) q^{55} + ( 228 \beta_{4} + 219 \beta_{5} - 3 \beta_{9} + 9 \beta_{11} ) q^{57} + ( -5 \beta_{3} + 10 \beta_{7} - 10 \beta_{8} + 15 \beta_{10} ) q^{59} + ( -9764 + 266 \beta_{1} + 87 \beta_{4} - 87 \beta_{5} ) q^{61} + ( 77 \beta_{2} + 222 \beta_{3} + 39 \beta_{6} ) q^{63} + ( 1145 + 595 \beta_{1} + 375 \beta_{4} - 155 \beta_{5} - 10 \beta_{9} - 10 \beta_{11} ) q^{65} + ( 617 \beta_{2} + 354 \beta_{3} + 622 \beta_{6} ) q^{67} + ( 16826 - 638 \beta_{1} - 525 \beta_{4} + 525 \beta_{5} ) q^{69} + ( \beta_{3} - 2 \beta_{7} + 34 \beta_{8} + 6 \beta_{10} ) q^{71} + ( 418 \beta_{4} + 429 \beta_{5} - 29 \beta_{9} - 11 \beta_{11} ) q^{73} + ( -2555 \beta_{2} - 273 \beta_{3} + 120 \beta_{6} + 36 \beta_{7} + 27 \beta_{8} - 3 \beta_{10} ) q^{75} + ( -491 \beta_{4} - 473 \beta_{5} - 45 \beta_{9} - 18 \beta_{11} ) q^{77} + ( 35 \beta_{3} - 70 \beta_{7} - 32 \beta_{8} + 10 \beta_{10} ) q^{79} + ( -28910 - 121 \beta_{1} + 843 \beta_{4} - 843 \beta_{5} ) q^{81} + ( 1473 \beta_{2} - 286 \beta_{3} - 594 \beta_{6} ) q^{83} + ( 8390 + 1090 \beta_{1} - 1045 \beta_{4} + 495 \beta_{5} - 35 \beta_{9} + 20 \beta_{11} ) q^{85} + ( -10 \beta_{2} - 2010 \beta_{3} - 816 \beta_{6} ) q^{87} + ( 29280 - 894 \beta_{1} + 756 \beta_{4} - 756 \beta_{5} ) q^{89} + ( -41 \beta_{3} + 82 \beta_{7} - 79 \beta_{8} - 31 \beta_{10} ) q^{91} + ( -626 \beta_{4} - 648 \beta_{5} + 100 \beta_{9} + 22 \beta_{11} ) q^{93} + ( 4440 \beta_{2} + 1630 \beta_{3} - 540 \beta_{6} - 5 \beta_{7} + 110 \beta_{8} - 30 \beta_{10} ) q^{95} + ( 576 \beta_{4} + 603 \beta_{5} + 67 \beta_{9} - 27 \beta_{11} ) q^{97} + ( -\beta_{3} + 2 \beta_{7} + 78 \beta_{8} - 51 \beta_{10} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 60q^{5} - 268q^{9} + O(q^{10})$$ $$12q - 60q^{5} - 268q^{9} + 12080q^{21} - 16420q^{25} - 7864q^{29} + 65560q^{41} - 60420q^{45} + 17956q^{49} - 118232q^{61} + 11360q^{65} + 204464q^{69} - 346436q^{81} + 96320q^{85} + 354936q^{89} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 6 x^{11} + 18 x^{10} + 348 x^{9} + 21226 x^{8} - 87824 x^{7} + 205428 x^{6} + 2113880 x^{5} + 27827225 x^{4} - 59798850 x^{3} + 89646050 x^{2} + 438522500 x + 1072562500$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$307234436488338 \nu^{11} - 8634349133755400 \nu^{10} + 192091663960648556 \nu^{9} - 3535072440302249852 \nu^{8} + 17130235630071728642 \nu^{7} - 141248900500608409604 \nu^{6} + 2720507879582880380992 \nu^{5} - 64281851585677873437336 \nu^{4} + 122142306781141996930480 \nu^{3} - 92941749509420247948800 \nu^{2} - 603348316903702480110000 \nu - 8464373013786978826329925$$$$)/$$$$91\!\cdots\!75$$ $$\beta_{2}$$ $$=$$ $$($$$$29012648376852651029539 \nu^{11} - 336349955350644341398009 \nu^{10} + 1330069159872813637571727 \nu^{9} + 8095179206972280621725247 \nu^{8} + 553816801053669539036573989 \nu^{7} - 5904338358432590439392932161 \nu^{6} + 16235640217562460443603413417 \nu^{5} + 40547627105984292925075437245 \nu^{4} + 421040030395298219604019986400 \nu^{3} - 4327556220803483078670456761150 \nu^{2} + 5147183325728664616715123982200 \nu + 5056183094605590776086552532500$$$$)/$$$$43\!\cdots\!00$$ $$\beta_{3}$$ $$=$$ $$($$$$-35487438352839306343684 \nu^{11} + 342032820193123746040004 \nu^{10} - 1288624321388489938316212 \nu^{9} - 10558192013211749347811332 \nu^{8} - 708929235973517859664455884 \nu^{7} + 5936843624052729300018114516 \nu^{6} - 15996453114193488563332995852 \nu^{5} - 55267579553504090608740814220 \nu^{4} - 734502633247871175095721996400 \nu^{3} + 6869631364598617368781441411400 \nu^{2} - 7864227539297539707847320823200 \nu - 7386979791568949031077717570000$$$$)/$$$$10\!\cdots\!75$$ $$\beta_{4}$$ $$=$$ $$($$$$3819436876559129065107 \nu^{11} - 24390480814556742053692 \nu^{10} + 101907647401723816832326 \nu^{9} + 1341251957154278515508486 \nu^{8} + 79467691751665476070252957 \nu^{7} - 353513087409331781691943918 \nu^{6} + 1457441126009688154572641096 \nu^{5} + 9494693305781491970010858360 \nu^{4} + 84857668491313643941069525250 \nu^{3} - 182485468868851969384468116700 \nu^{2} + 966696128591457978513444957600 \nu + 3771591191044499990531642035000$$$$)/$$$$68\!\cdots\!50$$ $$\beta_{5}$$ $$=$$ $$($$$$3849709870770421728297 \nu^{11} - 25231493151480755942132 \nu^{10} + 120057440476131252867146 \nu^{9} + 1029602505511852694880106 \nu^{8} + 81019587808431128378848547 \nu^{7} - 367304707707933383880065978 \nu^{6} + 1729370103611386916078494216 \nu^{5} + 3160430599955282470390733760 \nu^{4} + 96884199386560477688191616050 \nu^{3} - 191592319632745966239138376700 \nu^{2} + 907437818417872766148745707600 \nu - 2036534985088695391886390890000$$$$)/$$$$68\!\cdots\!50$$ $$\beta_{6}$$ $$=$$ $$($$$$313677025240705610068319 \nu^{11} - 3239381530897085209268814 \nu^{10} + 12443769663051544989248017 \nu^{9} + 91243948800234727227622687 \nu^{8} + 6171754608673057072697438769 \nu^{7} - 56714669716932961581143325756 \nu^{6} + 154094809792834460407542013507 \nu^{5} + 470432232219088190304321996145 \nu^{4} + 5784303746828694991625105258900 \nu^{3} - 55648572657113628809418715936150 \nu^{2} + 64471944995097890103304967031200 \nu + 61438249275599206351748887432500$$$$)/$$$$32\!\cdots\!25$$ $$\beta_{7}$$ $$=$$ $$($$$$-2827084074291580965128848 \nu^{11} + 18300133453687299879499888 \nu^{10} - 104273244821536198851494064 \nu^{9} - 642612896986899411655680704 \nu^{8} - 62659217463110980019254375248 \nu^{7} + 267836139336459886618191422552 \nu^{6} - 1617031699424633555739083918144 \nu^{5} - 1519157197284422402677430336840 \nu^{4} - 140480076118818394288020475332800 \nu^{3} + 185730841052364162746840086645800 \nu^{2} + 646279583481233129360666314679600 \nu - 1885362490222329991007790081965000$$$$)/$$$$20\!\cdots\!25$$ $$\beta_{8}$$ $$=$$ $$($$$$12786843029867194402942 \nu^{11} - 60559280090711923940446 \nu^{10} + 94358023410900278385750 \nu^{9} + 4831533237809968297831574 \nu^{8} + 277867258573519992440636490 \nu^{7} - 796267271369866131175206782 \nu^{6} + 264731673056464903910310842 \nu^{5} + 31174837622563674885812742458 \nu^{4} + 405832246012641969136425210680 \nu^{3} - 417822243601734527667600236100 \nu^{2} - 2360453584231989109516797320000 \nu + 6633573388185658237146481110800$$$$)/$$$$19\!\cdots\!25$$ $$\beta_{9}$$ $$=$$ $$($$$$1359694548862667227062 \nu^{11} - 8732287291322962879072 \nu^{10} + 37623336599970765419816 \nu^{9} + 423985868687745515382976 \nu^{8} + 28586049715341475715859912 \nu^{7} - 127457878720636111217731788 \nu^{6} + 539937291429219360503765336 \nu^{5} + 2276022108763635022823599460 \nu^{4} + 33537718366826554483626329450 \nu^{3} - 69247673344582608488167548700 \nu^{2} + 341531046229990296123113512600 \nu + 314645192407482428284256072500$$$$)/$$$$13\!\cdots\!75$$ $$\beta_{10}$$ $$=$$ $$($$$$-39481333059848761968766 \nu^{11} + 176470660700834971420510 \nu^{10} - 43514089925929985152662 \nu^{9} - 16287254467035875384325206 \nu^{8} - 849725811911438899185204234 \nu^{7} + 2277415417449224518013982398 \nu^{6} + 3597301278365766222912816646 \nu^{5} - 111961871411699812034142689978 \nu^{4} - 1080067789772402685377392336760 \nu^{3} + 1165078204978013309001819908100 \nu^{2} + 6455560390324419119268423320000 \nu - 18805046405068574273020085934800$$$$)/$$$$19\!\cdots\!25$$ $$\beta_{11}$$ $$=$$ $$($$$$-1418494773144224753357511 \nu^{11} + 8880905548636125402851916 \nu^{10} - 33968730326415030686260798 \nu^{9} - 455572105548067210950465278 \nu^{8} - 30083810747371655673807140561 \nu^{7} + 130371814028530750368743915414 \nu^{6} - 461098401547767991077060769608 \nu^{5} - 2504491796754790920407588779880 \nu^{4} - 39878523044853549731715328046850 \nu^{3} + 83183849708845419343480937771100 \nu^{2} - 390477593838046001533139635172800 \nu - 368330390114519124189420498405000$$$$)/$$$$34\!\cdots\!50$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-2 \beta_{11} + \beta_{10} - 8 \beta_{9} + \beta_{8} + 8 \beta_{7} + 28 \beta_{5} + 26 \beta_{4} + 16 \beta_{3} + 640$$$$)/1280$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{11} - 4 \beta_{9} + 240 \beta_{6} + 14 \beta_{5} + 13 \beta_{4} + 590 \beta_{3} - 800 \beta_{2}$$$$)/320$$ $$\nu^{3}$$ $$=$$ $$($$$$-94 \beta_{11} - 207 \beta_{10} - 976 \beta_{9} - 807 \beta_{8} - 976 \beta_{7} + 1440 \beta_{6} + 4236 \beta_{5} + 7022 \beta_{4} + 4008 \beta_{3} - 4800 \beta_{2} + 1200 \beta_{1} - 122480$$$$)/1280$$ $$\nu^{4}$$ $$=$$ $$($$$$-103 \beta_{10} - 403 \beta_{8} - 484 \beta_{7} - 16680 \beta_{5} + 16680 \beta_{4} + 242 \beta_{3} + 19300 \beta_{1} - 1245620$$$$)/160$$ $$\nu^{5}$$ $$=$$ $$($$$$8758 \beta_{11} - 36389 \beta_{10} + 135832 \beta_{9} - 137189 \beta_{8} - 135832 \beta_{7} - 328800 \beta_{6} - 1265972 \beta_{5} - 599614 \beta_{4} - 583764 \beta_{3} + 1528000 \beta_{2} + 382000 \beta_{1} - 24504560$$$$)/1280$$ $$\nu^{6}$$ $$=$$ $$($$$$12679 \beta_{11} + 198916 \beta_{9} - 4843680 \beta_{6} - 1377546 \beta_{5} - 1364867 \beta_{4} - 9187400 \beta_{3} + 24742400 \beta_{2}$$$$)/320$$ $$\nu^{7}$$ $$=$$ $$($$$$1123186 \beta_{11} + 5765693 \beta_{10} + 20122144 \beta_{9} + 21395093 \beta_{8} + 20122144 \beta_{7} - 65516640 \beta_{6} - 80361764 \beta_{5} - 210271858 \beta_{4} - 134139312 \beta_{3} + 335720000 \beta_{2} - 83930000 \beta_{1} + 4799527120$$$$)/1280$$ $$\nu^{8}$$ $$=$$ $$($$$$1379183 \beta_{10} + 5114333 \beta_{8} + 4799594 \beta_{7} + 91172610 \beta_{5} - 91172610 \beta_{4} - 2399797 \beta_{3} - 119831925 \beta_{1} + 6655524745$$$$)/40$$ $$\nu^{9}$$ $$=$$ $$($$$$-163277722 \beta_{11} + 894814071 \beta_{10} - 3062582488 \beta_{9} + 3304285671 \beta_{8} + 3062582488 \beta_{7} + 12348156960 \beta_{6} + 34873263228 \beta_{5} + 10013671586 \beta_{4} + 21571462596 \beta_{3} - 65020142400 \beta_{2} - 16255035600 \beta_{1} + 901210957840$$$$)/1280$$ $$\nu^{10}$$ $$=$$ $$($$$$-376612341 \beta_{11} - 7086035964 \beta_{9} + 111951031200 \beta_{6} + 52166826094 \beta_{5} + 51790213753 \beta_{4} + 208809453560 \beta_{3} - 593572832000 \beta_{2}$$$$)/320$$ $$\nu^{11}$$ $$=$$ $$($$$$-24877018814 \beta_{11} - 139055063647 \beta_{10} - 473195849456 \beta_{9} - 512742837847 \beta_{8} - 473195849456 \beta_{7} + 2239393733280 \beta_{6} + 1255812720556 \beta_{5} + 5709723168302 \beta_{4} + 4412257819928 \beta_{3} - 11881203998400 \beta_{2} + 2970300999600 \beta_{1} - 163277454708080$$$$)/1280$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/160\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$97$$ $$101$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
129.1
 −8.04538 − 8.04538i 9.04538 + 9.04538i −3.92310 + 3.92310i 4.92310 − 4.92310i −1.27837 − 1.27837i 2.27837 + 2.27837i 2.27837 − 2.27837i −1.27837 + 1.27837i 4.92310 + 4.92310i −3.92310 − 3.92310i 9.04538 − 9.04538i −8.04538 + 8.04538i
0 21.9637i 0 18.7585 52.6604i 0 29.6983i 0 −239.406 0
129.2 0 21.9637i 0 18.7585 + 52.6604i 0 29.6983i 0 −239.406 0
129.3 0 16.9825i 0 12.3893 54.5115i 0 211.691i 0 −45.4059 0
129.4 0 16.9825i 0 12.3893 + 54.5115i 0 211.691i 0 −45.4059 0
129.5 0 5.01877i 0 −46.1478 31.5496i 0 15.3895i 0 217.812 0
129.6 0 5.01877i 0 −46.1478 + 31.5496i 0 15.3895i 0 217.812 0
129.7 0 5.01877i 0 −46.1478 31.5496i 0 15.3895i 0 217.812 0
129.8 0 5.01877i 0 −46.1478 + 31.5496i 0 15.3895i 0 217.812 0
129.9 0 16.9825i 0 12.3893 54.5115i 0 211.691i 0 −45.4059 0
129.10 0 16.9825i 0 12.3893 + 54.5115i 0 211.691i 0 −45.4059 0
129.11 0 21.9637i 0 18.7585 52.6604i 0 29.6983i 0 −239.406 0
129.12 0 21.9637i 0 18.7585 + 52.6604i 0 29.6983i 0 −239.406 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 129.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.6.c.d 12
4.b odd 2 1 inner 160.6.c.d 12
5.b even 2 1 inner 160.6.c.d 12
5.c odd 4 1 800.6.a.w 6
5.c odd 4 1 800.6.a.bb 6
8.b even 2 1 320.6.c.l 12
8.d odd 2 1 320.6.c.l 12
20.d odd 2 1 inner 160.6.c.d 12
20.e even 4 1 800.6.a.w 6
20.e even 4 1 800.6.a.bb 6
40.e odd 2 1 320.6.c.l 12
40.f even 2 1 320.6.c.l 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.c.d 12 1.a even 1 1 trivial
160.6.c.d 12 4.b odd 2 1 inner
160.6.c.d 12 5.b even 2 1 inner
160.6.c.d 12 20.d odd 2 1 inner
320.6.c.l 12 8.b even 2 1
320.6.c.l 12 8.d odd 2 1
320.6.c.l 12 40.e odd 2 1
320.6.c.l 12 40.f even 2 1
800.6.a.w 6 5.c odd 4 1
800.6.a.w 6 20.e even 4 1
800.6.a.bb 6 5.c odd 4 1
800.6.a.bb 6 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} + 796 T_{3}^{4} + 158544 T_{3}^{2} + 3504384$$ acting on $$S_{6}^{\mathrm{new}}(160, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - 662 T^{2} + 270567 T^{4} - 78508116 T^{6} + 15976710783 T^{8} - 2308251273462 T^{10} + 205891132094649 T^{12} )^{2}$$
$5$ $$( 1 + 30 T + 4555 T^{2} + 273300 T^{3} + 14234375 T^{4} + 292968750 T^{5} + 30517578125 T^{6} )^{2}$$
$7$ $$( 1 - 54910 T^{2} + 1199559327 T^{4} - 18786317442980 T^{6} + 338845819584597423 T^{8} -$$$$43\!\cdots\!10$$$$T^{10} +$$$$22\!\cdots\!49$$$$T^{12} )^{2}$$
$11$ $$( 1 + 311650 T^{2} + 88422679991 T^{4} + 16030422126529084 T^{6} +$$$$22\!\cdots\!91$$$$T^{8} +$$$$20\!\cdots\!50$$$$T^{10} +$$$$17\!\cdots\!01$$$$T^{12} )^{2}$$
$13$ $$( 1 - 183262 T^{2} + 260173900423 T^{4} - 19831011157232516 T^{6} +$$$$35\!\cdots\!27$$$$T^{8} -$$$$34\!\cdots\!62$$$$T^{10} +$$$$26\!\cdots\!49$$$$T^{12} )^{2}$$
$17$ $$( 1 + 1156442 T^{2} + 502148271983 T^{4} - 1980542974409409364 T^{6} +$$$$10\!\cdots\!67$$$$T^{8} +$$$$47\!\cdots\!42$$$$T^{10} +$$$$81\!\cdots\!49$$$$T^{12} )^{2}$$
$19$ $$( 1 + 3143250 T^{2} + 17341223849991 T^{4} + 36134402435803900316 T^{6} +$$$$10\!\cdots\!91$$$$T^{8} +$$$$11\!\cdots\!50$$$$T^{10} +$$$$23\!\cdots\!01$$$$T^{12} )^{2}$$
$23$ $$( 1 - 2071710 T^{2} + 29683977204927 T^{4} -$$$$44\!\cdots\!80$$$$T^{6} +$$$$12\!\cdots\!23$$$$T^{8} -$$$$35\!\cdots\!10$$$$T^{10} +$$$$71\!\cdots\!49$$$$T^{12} )^{2}$$
$29$ $$( 1 + 1966 T + 38511987 T^{2} + 51339089236 T^{3} + 789925103643063 T^{4} + 827110420668195166 T^{5} +$$$$86\!\cdots\!49$$$$T^{6} )^{4}$$
$31$ $$( 1 + 47574970 T^{2} + 2163796989249871 T^{4} +$$$$78\!\cdots\!04$$$$T^{6} +$$$$17\!\cdots\!71$$$$T^{8} +$$$$31\!\cdots\!70$$$$T^{10} +$$$$55\!\cdots\!01$$$$T^{12} )^{2}$$
$37$ $$( 1 - 301201198 T^{2} + 42883739211630103 T^{4} -$$$$37\!\cdots\!24$$$$T^{6} +$$$$20\!\cdots\!47$$$$T^{8} -$$$$69\!\cdots\!98$$$$T^{10} +$$$$11\!\cdots\!49$$$$T^{12} )^{2}$$
$41$ $$( 1 - 16390 T + 318388423 T^{2} - 3510238606580 T^{3} + 36887273131161023 T^{4} -$$$$21\!\cdots\!90$$$$T^{5} +$$$$15\!\cdots\!01$$$$T^{6} )^{4}$$
$43$ $$( 1 - 640358182 T^{2} + 187647756282389367 T^{4} -$$$$33\!\cdots\!76$$$$T^{6} +$$$$40\!\cdots\!83$$$$T^{8} -$$$$29\!\cdots\!82$$$$T^{10} +$$$$10\!\cdots\!49$$$$T^{12} )^{2}$$
$47$ $$( 1 - 764642190 T^{2} + 310900354010297967 T^{4} -$$$$85\!\cdots\!20$$$$T^{6} +$$$$16\!\cdots\!83$$$$T^{8} -$$$$21\!\cdots\!90$$$$T^{10} +$$$$14\!\cdots\!49$$$$T^{12} )^{2}$$
$53$ $$( 1 - 1611835534 T^{2} + 1232324270520828407 T^{4} -$$$$61\!\cdots\!84$$$$T^{6} +$$$$21\!\cdots\!43$$$$T^{8} -$$$$49\!\cdots\!34$$$$T^{10} +$$$$53\!\cdots\!49$$$$T^{12} )^{2}$$
$59$ $$( 1 + 2333761794 T^{2} + 3044701823751957015 T^{4} +$$$$26\!\cdots\!80$$$$T^{6} +$$$$15\!\cdots\!15$$$$T^{8} +$$$$60\!\cdots\!94$$$$T^{10} +$$$$13\!\cdots\!01$$$$T^{12} )^{2}$$
$61$ $$( 1 + 29558 T + 2350852083 T^{2} + 48603849831556 T^{3} + 1985520973499944983 T^{4} +$$$$21\!\cdots\!58$$$$T^{5} +$$$$60\!\cdots\!01$$$$T^{6} )^{4}$$
$67$ $$( 1 - 3741380758 T^{2} + 9572756841213379047 T^{4} -$$$$14\!\cdots\!84$$$$T^{6} +$$$$17\!\cdots\!03$$$$T^{8} -$$$$12\!\cdots\!58$$$$T^{10} +$$$$60\!\cdots\!49$$$$T^{12} )^{2}$$
$71$ $$( 1 + 9793576042 T^{2} + 41626938509482688159 T^{4} +$$$$98\!\cdots\!36$$$$T^{6} +$$$$13\!\cdots\!59$$$$T^{8} +$$$$10\!\cdots\!42$$$$T^{10} +$$$$34\!\cdots\!01$$$$T^{12} )^{2}$$
$73$ $$( 1 - 5477783158 T^{2} + 15527894807038343935 T^{4} -$$$$34\!\cdots\!40$$$$T^{6} +$$$$66\!\cdots\!15$$$$T^{8} -$$$$10\!\cdots\!58$$$$T^{10} +$$$$79\!\cdots\!49$$$$T^{12} )^{2}$$
$79$ $$( 1 + 8854539610 T^{2} + 48426464203018517551 T^{4} +$$$$17\!\cdots\!76$$$$T^{6} +$$$$45\!\cdots\!51$$$$T^{8} +$$$$79\!\cdots\!10$$$$T^{10} +$$$$84\!\cdots\!01$$$$T^{12} )^{2}$$
$83$ $$( 1 - 17886738486 T^{2} +$$$$14\!\cdots\!27$$$$T^{4} -$$$$73\!\cdots\!08$$$$T^{6} +$$$$22\!\cdots\!23$$$$T^{8} -$$$$43\!\cdots\!86$$$$T^{10} +$$$$37\!\cdots\!49$$$$T^{12} )^{2}$$
$89$ $$( 1 - 88734 T + 11682221271 T^{2} - 581994460142148 T^{3} + 65234218073636339679 T^{4} -$$$$27\!\cdots\!34$$$$T^{5} +$$$$17\!\cdots\!49$$$$T^{6} )^{4}$$
$97$ $$( 1 - 22620660742 T^{2} +$$$$25\!\cdots\!35$$$$T^{4} -$$$$22\!\cdots\!60$$$$T^{6} +$$$$18\!\cdots\!15$$$$T^{8} -$$$$12\!\cdots\!42$$$$T^{10} +$$$$40\!\cdots\!49$$$$T^{12} )^{2}$$