Properties

Label 300.9.g.h
Level $300$
Weight $9$
Character orbit 300.g
Analytic conductor $122.214$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 300.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(122.213583018\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 7378 x^{14} + 23156928 x^{12} - 101588726286 x^{10} + 1484695155668670 x^{8} - 4373061557178808206 x^{6} + 42910255095788484704448 x^{4} - 588516817021165377771877458 x^{2} + 3433683820292512484657849089281\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{38}\cdot 3^{20}\cdot 5^{18} \)
Twist minimal: no (minimal twist has level 60)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} -\beta_{7} q^{7} + ( 922 + \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} -\beta_{7} q^{7} + ( 922 + \beta_{2} ) q^{9} -\beta_{3} q^{11} + ( -27 \beta_{1} + \beta_{6} + \beta_{12} ) q^{13} + ( 47 \beta_{1} + 2 \beta_{6} + \beta_{13} ) q^{17} + ( 3001 + 3 \beta_{2} + \beta_{8} ) q^{19} + ( 296 + 3 \beta_{3} + \beta_{8} - \beta_{10} ) q^{21} + ( 364 \beta_{1} + 16 \beta_{6} + \beta_{11} + 2 \beta_{13} + \beta_{15} ) q^{23} + ( -917 \beta_{1} + 2 \beta_{6} - 14 \beta_{7} - 2 \beta_{11} + 11 \beta_{12} - 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{27} + ( 5 - 23 \beta_{2} - 10 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{9} + 2 \beta_{10} ) q^{29} + ( 29381 + 8 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 5 \beta_{8} - \beta_{9} - 5 \beta_{10} ) q^{31} + ( 14 \beta_{1} - 4 \beta_{6} - 98 \beta_{7} + 5 \beta_{11} + 11 \beta_{12} - 10 \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{33} + ( -421 \beta_{1} + 6 \beta_{6} + 74 \beta_{7} + 2 \beta_{11} - 53 \beta_{12} - \beta_{13} - 6 \beta_{14} + \beta_{15} ) q^{37} + ( 178098 + 33 \beta_{2} + 39 \beta_{3} - 15 \beta_{4} + 3 \beta_{5} + 15 \beta_{8} - 6 \beta_{9} + 6 \beta_{10} ) q^{39} + ( 35 - 122 \beta_{2} - 141 \beta_{3} - 26 \beta_{4} - 7 \beta_{5} - 4 \beta_{9} - 7 \beta_{10} ) q^{41} + ( 3233 \beta_{1} - 109 \beta_{6} + 608 \beta_{7} - 4 \beta_{11} - 48 \beta_{12} + 2 \beta_{13} + 12 \beta_{14} - 2 \beta_{15} ) q^{43} + ( -962 \beta_{1} - 49 \beta_{6} + 23 \beta_{11} + 7 \beta_{13} - 14 \beta_{15} ) q^{47} + ( 1238386 + 690 \beta_{2} - 3 \beta_{3} - 24 \beta_{4} - 3 \beta_{5} + 10 \beta_{8} - 24 \beta_{9} + 15 \beta_{10} ) q^{49} + ( 314143 - 78 \beta_{2} + 380 \beta_{3} + 51 \beta_{4} - 7 \beta_{5} - 37 \beta_{8} - 15 \beta_{9} - 3 \beta_{10} ) q^{51} + ( 2260 \beta_{1} + 108 \beta_{6} - 87 \beta_{11} - 18 \beta_{13} + 4 \beta_{15} ) q^{53} + ( -3093 \beta_{1} + 699 \beta_{6} - 1188 \beta_{7} - 141 \beta_{11} + 153 \beta_{12} + 30 \beta_{13} + 6 \beta_{15} ) q^{57} + ( 412 - 1591 \beta_{2} - 176 \beta_{3} + 123 \beta_{4} + \beta_{5} - 59 \beta_{9} + \beta_{10} ) q^{59} + ( 1177939 + 654 \beta_{2} - 9 \beta_{3} - 24 \beta_{4} - 9 \beta_{5} - 10 \beta_{8} - 24 \beta_{9} + 45 \beta_{10} ) q^{61} + ( -131 \beta_{1} + 620 \beta_{6} - 5039 \beta_{7} + 274 \beta_{11} + 77 \beta_{12} + 85 \beta_{13} - 7 \beta_{14} + 2 \beta_{15} ) q^{63} + ( -33697 \beta_{1} + 1092 \beta_{6} + 2566 \beta_{7} - 10 \beta_{11} + 10 \beta_{12} + 5 \beta_{13} + 30 \beta_{14} - 5 \beta_{15} ) q^{67} + ( 2416915 - 342 \beta_{2} + 326 \beta_{3} - 609 \beta_{4} - 25 \beta_{5} - 157 \beta_{8} - 108 \beta_{9} - 36 \beta_{10} ) q^{69} + ( 252 - 1050 \beta_{2} - 452 \beta_{3} - 566 \beta_{4} + 42 \beta_{5} - 42 \beta_{9} + 42 \beta_{10} ) q^{71} + ( -57056 \beta_{1} + 1863 \beta_{6} + 9782 \beta_{7} + 34 \beta_{11} + 272 \beta_{12} - 17 \beta_{13} - 102 \beta_{14} + 17 \beta_{15} ) q^{73} + ( 106696 \beta_{1} + 4596 \beta_{6} + 1072 \beta_{11} - 132 \beta_{13} + 128 \beta_{15} ) q^{77} + ( 5672991 + 3019 \beta_{2} + 26 \beta_{3} - 122 \beta_{4} + 26 \beta_{5} - 57 \beta_{8} - 122 \beta_{9} - 130 \beta_{10} ) q^{79} + ( 1015000 + 664 \beta_{2} - 369 \beta_{3} + 1236 \beta_{4} + 63 \beta_{5} + 306 \beta_{8} - 6 \beta_{9} + 81 \beta_{10} ) q^{81} + ( 23601 \beta_{1} + 1124 \beta_{6} - 1428 \beta_{11} - 11 \beta_{13} - 77 \beta_{15} ) q^{83} + ( -1662 \beta_{1} + 5565 \beta_{6} - 6159 \beta_{7} - 1707 \beta_{11} - 1140 \beta_{12} - 177 \beta_{13} + 48 \beta_{14} - 12 \beta_{15} ) q^{87} + ( 618 - 2133 \beta_{2} + 219 \beta_{3} + 1771 \beta_{4} - 135 \beta_{5} - 69 \beta_{9} - 135 \beta_{10} ) q^{89} + ( 1398232 + 1658 \beta_{2} + 34 \beta_{3} - 22 \beta_{4} + 34 \beta_{5} + 400 \beta_{8} - 22 \beta_{9} - 170 \beta_{10} ) q^{91} + ( -28465 \beta_{1} + 2256 \beta_{6} - 22194 \beta_{7} + 2307 \beta_{11} - 1089 \beta_{12} - 141 \beta_{13} - 54 \beta_{14} - 93 \beta_{15} ) q^{93} + ( -495104 \beta_{1} + 16224 \beta_{6} + 10432 \beta_{7} - 32 \beta_{11} + 1868 \beta_{12} + 16 \beta_{13} + 96 \beta_{14} - 16 \beta_{15} ) q^{97} + ( -1060648 + 1016 \beta_{2} - 2493 \beta_{3} - 2976 \beta_{4} + 72 \beta_{5} + 630 \beta_{8} + 102 \beta_{9} - 18 \beta_{10} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 14756q^{9} + O(q^{10}) \) \( 16q + 14756q^{9} + 48024q^{19} + 4724q^{21} + 470104q^{31} + 2849664q^{39} + 19816920q^{49} + 5026040q^{51} + 18849944q^{61} + 38669180q^{69} + 90778632q^{79} + 16242056q^{81} + 22375296q^{91} - 16968560q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 7378 x^{14} + 23156928 x^{12} - 101588726286 x^{10} + 1484695155668670 x^{8} - 4373061557178808206 x^{6} + 42910255095788484704448 x^{4} - 588516817021165377771877458 x^{2} + 3433683820292512484657849089281\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 922 \)
\(\beta_{3}\)\(=\)\((\)\(-118241503 \nu^{14} + 32210391788401 \nu^{12} + 126457566128179947 \nu^{10} + 214013909643019174755 \nu^{8} + 4746195418588577798202495 \nu^{6} + 3485352495014175965316858063 \nu^{4} + 97439320345474574721245443129749 \nu^{2} + 1423431157324590957071749337819988093\)\()/ \)\(20\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(438411229 \nu^{14} + 545360564057 \nu^{12} + 5205191446634679 \nu^{10} + 134747004904914583035 \nu^{8} - 140050706881610814672285 \nu^{6} + 7126746748103003389938313191 \nu^{4} + 15988522642231144261362784701993 \nu^{2} - 2374313718998962356072521984335899\)\()/ \)\(20\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-5968589155 \nu^{14} - 101803994977442 \nu^{12} - 1605609642957550713 \nu^{10} + 284564718899432445330 \nu^{8} + 86691769095675942519262875 \nu^{6} + 257905289910296311314646868010 \nu^{4} - 1473149156225017240571482096707663 \nu^{2} - 7767748177539020226873719607705391242\)\()/ \)\(10\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\nu^{15} - 7378 \nu^{13} + 23156928 \nu^{11} - 101588726286 \nu^{9} + 1484695155668670 \nu^{7} - 4373061557178808206 \nu^{5} + 42910255095788484704448 \nu^{3} - 510983834350445298184690566 \nu\)\()/ \)\(19\!\cdots\!23\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-9990143 \nu^{15} - 13366675591 \nu^{13} - 104908177452681 \nu^{11} + 1454707853876694735 \nu^{9} + 893990708450313260895 \nu^{7} + 15889852799914567568262183 \nu^{5} - 488135239609492118517858798039 \nu^{3} + 2286102946760216024014189243114641 \nu\)\()/ \)\(14\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-56323 \nu^{14} + 84994792 \nu^{12} + 852147203931 \nu^{10} + 150878465402940 \nu^{8} - 44424226549043107605 \nu^{6} + 307048508107766448925968 \nu^{4} - 2092818668997351782154051027 \nu^{2} + 18959612788879968454203241251324\)\()/ \)\(31\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(6522599171 \nu^{14} - 51903695295257 \nu^{12} + 155990425193416521 \nu^{10} - 841907185104513957435 \nu^{8} + 10475029126320841638840285 \nu^{6} - 37567673727464881971509655591 \nu^{4} + 282710209346205493966540669557207 \nu^{2} - 3582208268407457812459773082999060101\)\()/ \)\(20\!\cdots\!00\)\( \)
\(\beta_{10}\)\(=\)\((\)\(27940379809 \nu^{14} - 7840932369085 \nu^{12} + 106687622468324331 \nu^{10} - 7006703647655295554535 \nu^{8} + 8022390700695158204032815 \nu^{6} + 96875049710885773953791899341 \nu^{4} + 1305055640077155944800803129395205 \nu^{2} - 10303770409953809798553362085042235881\)\()/ \)\(68\!\cdots\!00\)\( \)
\(\beta_{11}\)\(=\)\((\)\(20596949 \nu^{15} - 37487115353 \nu^{13} - 26193217876281 \nu^{11} - 9615576560197871115 \nu^{9} + 33111695571416626953915 \nu^{7} + 334926758600430895264046121 \nu^{5} + 1185049326849210503136732085353 \nu^{3} - 12370583909607752899704971964400389 \nu\)\()/ \)\(74\!\cdots\!00\)\( \)
\(\beta_{12}\)\(=\)\((\)\(515192467 \nu^{15} + 615081057479 \nu^{13} - 23769410150578311 \nu^{11} - 13979069008262681715 \nu^{9} + 1455452294036789810563245 \nu^{7} - 718501964677245201926306727 \nu^{5} - 39986773812708795760737886002009 \nu^{3} - 146868508805231148372599489199012429 \nu\)\()/ \)\(16\!\cdots\!00\)\( \)
\(\beta_{13}\)\(=\)\((\)\(439511153 \nu^{15} - 3923184692141 \nu^{13} - 25648690419064557 \nu^{11} - 96300953039951192655 \nu^{9} + 51322315211094545556255 \nu^{7} - 1803169386614614640336367363 \nu^{5} + 21053398589594473713522842730141 \nu^{3} - 361636306388836808583648784537192833 \nu\)\()/ \)\(37\!\cdots\!00\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-315617774114 \nu^{15} + 2906961097996085 \nu^{13} + 5341402867084775874 \nu^{11} - 91133985746362493381265 \nu^{9} + 1173175902795009552147044010 \nu^{7} - 7577842817073919295320004129661 \nu^{5} + 59132031186145636229505428318139270 \nu^{3} + 59491772755992574410359124796072221801 \nu\)\()/ \)\(22\!\cdots\!00\)\( \)
\(\beta_{15}\)\(=\)\((\)\(360231947 \nu^{15} + 1695521995441 \nu^{13} - 3519056907251343 \nu^{11} + 116929110967281142155 \nu^{9} + 396517034323738594366245 \nu^{7} + 3446460383777940257068507263 \nu^{5} + 47389009058756312349490136702559 \nu^{3} - 16852169989705515576706347509262267 \nu\)\()/ \)\(12\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 922\)
\(\nu^{3}\)\(=\)\(\beta_{15} + \beta_{14} + 2 \beta_{13} - 11 \beta_{12} + 2 \beta_{11} + 14 \beta_{7} - 2 \beta_{6} + 917 \beta_{1}\)
\(\nu^{4}\)\(=\)\(81 \beta_{10} - 6 \beta_{9} + 306 \beta_{8} + 63 \beta_{5} + 1236 \beta_{4} - 369 \beta_{3} + 664 \beta_{2} + 1015000\)
\(\nu^{5}\)\(=\)\(4624 \beta_{15} - 6959 \beta_{14} - 7519 \beta_{13} - 22595 \beta_{12} + 113252 \beta_{11} + 465362 \beta_{7} - 143816 \beta_{6} + 1051022 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-164916 \beta_{10} + 446349 \beta_{9} - 37224 \beta_{8} + 609156 \beta_{5} - 1832559 \beta_{4} + 11296188 \beta_{3} + 1559200 \beta_{2} + 24228485347\)
\(\nu^{7}\)\(=\)\(14029492 \beta_{15} + 41799856 \beta_{14} - 69784480 \beta_{13} + 438941944 \beta_{12} + 253173212 \beta_{11} + 3868482764 \beta_{7} + 3025466029 \beta_{6} + 23670718700 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-6583202100 \beta_{10} + 674791752 \beta_{9} - 7360658640 \beta_{8} - 455274180 \beta_{5} + 53815623996 \beta_{4} + 6595207740 \beta_{3} + 16287378400 \beta_{2} - 493406497298627\)
\(\nu^{9}\)\(=\)\(372515274100 \beta_{15} - 191909050640 \beta_{14} - 181942319680 \beta_{13} + 341871542440 \beta_{12} - 3560043027652 \beta_{11} + 14206187118740 \beta_{7} + 13550675701912 \beta_{6} - 497682836598727 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-4284421754940 \beta_{10} + 37589920269960 \beta_{9} + 76428779934600 \beta_{8} - 17864334358740 \beta_{5} + 116051575465164 \beta_{4} + 340033287412260 \beta_{3} - 441150705624935 \beta_{2} - 2734248649531353554\)
\(\nu^{11}\)\(=\)\(-718952827444355 \beta_{15} - 526320068789135 \beta_{14} - 8400999370274950 \beta_{13} - 10145890256279315 \beta_{12} + 1525742933722718 \beta_{11} - 60538389884256790 \beta_{7} + 191269996924912870 \beta_{6} - 2769509565289291003 \beta_{1}\)
\(\nu^{12}\)\(=\)\(78326388969301125 \beta_{10} - 217705289330581350 \beta_{9} - 290780653654009710 \beta_{8} - 25054327119139245 \beta_{5} - 643445293264229976 \beta_{4} + 3378812378443150275 \beta_{3} - 1865253820476150464 \beta_{2} - 35943471852211625378108\)
\(\nu^{13}\)\(=\)\(1212546650043993916 \beta_{15} + 7354618417090964641 \beta_{14} - 24686342839711451863 \beta_{13} + 80995809963092854789 \beta_{12} - 7561995861217751200 \beta_{11} - 1182534817833708789706 \beta_{7} - 1283996327823107379512 \beta_{6} - 35593092956130168296362 \beta_{1}\)
\(\nu^{14}\)\(=\)\(607395000117700370376 \beta_{10} - 142763328134049426531 \beta_{9} - 3269572223624975677104 \beta_{8} - 446326529173083519912 \beta_{5} + 9112531606003942407525 \beta_{4} - 660305187878822012664 \beta_{3} - 44213012442513461020976 \beta_{2} + 191856289957555519437810535\)
\(\nu^{15}\)\(=\)\(19919522998163068870744 \beta_{15} - 88447899581114690345984 \beta_{14} - 21170669082847405668304 \beta_{13} + 588773532910800051195280 \beta_{12} - 419231745548751852435400 \beta_{11} - 10188874813885804821015688 \beta_{7} + 1822300630910515632596029 \beta_{6} + 192056041250389533913791308 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
79.3447 + 16.2916i
79.3447 16.2916i
74.2942 + 32.2702i
74.2942 32.2702i
51.0255 + 62.9079i
51.0255 62.9079i
23.4026 + 77.5456i
23.4026 77.5456i
−23.4026 + 77.5456i
−23.4026 77.5456i
−51.0255 + 62.9079i
−51.0255 62.9079i
−74.2942 + 32.2702i
−74.2942 32.2702i
−79.3447 + 16.2916i
−79.3447 16.2916i
0 −79.3447 16.2916i 0 0 0 −3405.57 0 6030.17 + 2585.30i 0
101.2 0 −79.3447 + 16.2916i 0 0 0 −3405.57 0 6030.17 2585.30i 0
101.3 0 −74.2942 32.2702i 0 0 0 4017.33 0 4478.26 + 4794.99i 0
101.4 0 −74.2942 + 32.2702i 0 0 0 4017.33 0 4478.26 4794.99i 0
101.5 0 −51.0255 62.9079i 0 0 0 −459.019 0 −1353.80 + 6419.81i 0
101.6 0 −51.0255 + 62.9079i 0 0 0 −459.019 0 −1353.80 6419.81i 0
101.7 0 −23.4026 77.5456i 0 0 0 −256.799 0 −5465.63 + 3629.54i 0
101.8 0 −23.4026 + 77.5456i 0 0 0 −256.799 0 −5465.63 3629.54i 0
101.9 0 23.4026 77.5456i 0 0 0 256.799 0 −5465.63 3629.54i 0
101.10 0 23.4026 + 77.5456i 0 0 0 256.799 0 −5465.63 + 3629.54i 0
101.11 0 51.0255 62.9079i 0 0 0 459.019 0 −1353.80 6419.81i 0
101.12 0 51.0255 + 62.9079i 0 0 0 459.019 0 −1353.80 + 6419.81i 0
101.13 0 74.2942 32.2702i 0 0 0 −4017.33 0 4478.26 4794.99i 0
101.14 0 74.2942 + 32.2702i 0 0 0 −4017.33 0 4478.26 + 4794.99i 0
101.15 0 79.3447 16.2916i 0 0 0 3405.57 0 6030.17 2585.30i 0
101.16 0 79.3447 + 16.2916i 0 0 0 3405.57 0 6030.17 + 2585.30i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.9.g.h 16
3.b odd 2 1 inner 300.9.g.h 16
5.b even 2 1 inner 300.9.g.h 16
5.c odd 4 2 60.9.b.a 16
15.d odd 2 1 inner 300.9.g.h 16
15.e even 4 2 60.9.b.a 16
20.e even 4 2 240.9.c.d 16
60.l odd 4 2 240.9.c.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.9.b.a 16 5.c odd 4 2
60.9.b.a 16 15.e even 4 2
240.9.c.d 16 20.e even 4 2
240.9.c.d 16 60.l odd 4 2
300.9.g.h 16 1.a even 1 1 trivial
300.9.g.h 16 3.b odd 2 1 inner
300.9.g.h 16 5.b even 2 1 inner
300.9.g.h 16 15.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} - 28013434 T_{7}^{6} + \)\(19\!\cdots\!96\)\( T_{7}^{4} - \)\(52\!\cdots\!44\)\( T_{7}^{2} + \)\(26\!\cdots\!56\)\( \) acting on \(S_{9}^{\mathrm{new}}(300, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( \)\(34\!\cdots\!81\)\( - \)\(58\!\cdots\!58\)\( T^{2} + \)\(42\!\cdots\!48\)\( T^{4} - 4373061557178808206 T^{6} + 1484695155668670 T^{8} - 101588726286 T^{10} + 23156928 T^{12} - 7378 T^{14} + T^{16} \)
$5$ \( T^{16} \)
$7$ \( ( \)\(26\!\cdots\!56\)\( - 52166784840953810944 T^{2} + 194864255690496 T^{4} - 28013434 T^{6} + T^{8} )^{2} \)
$11$ \( ( \)\(10\!\cdots\!00\)\( + \)\(28\!\cdots\!00\)\( T^{2} + 265474129795728000 T^{4} + 971194160 T^{6} + T^{8} )^{2} \)
$13$ \( ( \)\(36\!\cdots\!36\)\( - \)\(33\!\cdots\!84\)\( T^{2} + 2136237732098603136 T^{4} - 3173472864 T^{6} + T^{8} )^{2} \)
$17$ \( ( \)\(17\!\cdots\!00\)\( + \)\(15\!\cdots\!00\)\( T^{2} + \)\(41\!\cdots\!00\)\( T^{4} + 36442812140 T^{6} + T^{8} )^{2} \)
$19$ \( ( 14325224454022538016 + 306871114764024 T - 14875364124 T^{2} - 12006 T^{3} + T^{4} )^{4} \)
$23$ \( ( \)\(58\!\cdots\!00\)\( + \)\(31\!\cdots\!00\)\( T^{2} + \)\(57\!\cdots\!00\)\( T^{4} + 416302278710 T^{6} + T^{8} )^{2} \)
$29$ \( ( \)\(31\!\cdots\!00\)\( + \)\(12\!\cdots\!00\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{4} + 2076533641440 T^{6} + T^{8} )^{2} \)
$31$ \( ( \)\(11\!\cdots\!96\)\( + 42008109391780664 T - 2411512956084 T^{2} - 117526 T^{3} + T^{4} )^{4} \)
$37$ \( ( \)\(49\!\cdots\!56\)\( - \)\(49\!\cdots\!56\)\( T^{2} + \)\(79\!\cdots\!96\)\( T^{4} - 17166889014016 T^{6} + T^{8} )^{2} \)
$41$ \( ( \)\(90\!\cdots\!00\)\( + \)\(43\!\cdots\!00\)\( T^{2} + \)\(72\!\cdots\!00\)\( T^{4} + 47937327156740 T^{6} + T^{8} )^{2} \)
$43$ \( ( \)\(11\!\cdots\!76\)\( - \)\(18\!\cdots\!04\)\( T^{2} + \)\(68\!\cdots\!56\)\( T^{4} - 49663114389754 T^{6} + T^{8} )^{2} \)
$47$ \( ( \)\(22\!\cdots\!00\)\( + \)\(46\!\cdots\!00\)\( T^{2} + \)\(10\!\cdots\!00\)\( T^{4} + 67977556756310 T^{6} + T^{8} )^{2} \)
$53$ \( ( \)\(49\!\cdots\!00\)\( + \)\(17\!\cdots\!00\)\( T^{2} + \)\(44\!\cdots\!00\)\( T^{4} + 37034825790060 T^{6} + T^{8} )^{2} \)
$59$ \( ( \)\(76\!\cdots\!00\)\( + \)\(26\!\cdots\!00\)\( T^{2} + \)\(24\!\cdots\!00\)\( T^{4} + 867038659162160 T^{6} + T^{8} )^{2} \)
$61$ \( ( -\)\(21\!\cdots\!64\)\( - \)\(14\!\cdots\!16\)\( T - 246246167429364 T^{2} - 4712486 T^{3} + T^{4} )^{4} \)
$67$ \( ( \)\(32\!\cdots\!76\)\( - \)\(15\!\cdots\!04\)\( T^{2} + \)\(54\!\cdots\!56\)\( T^{4} - 433265383552954 T^{6} + T^{8} )^{2} \)
$71$ \( ( \)\(21\!\cdots\!00\)\( + \)\(19\!\cdots\!00\)\( T^{2} + \)\(13\!\cdots\!00\)\( T^{4} + 2138475782597760 T^{6} + T^{8} )^{2} \)
$73$ \( ( \)\(89\!\cdots\!16\)\( - \)\(15\!\cdots\!76\)\( T^{2} + \)\(75\!\cdots\!76\)\( T^{4} - 5313866871866656 T^{6} + T^{8} )^{2} \)
$79$ \( ( \)\(62\!\cdots\!16\)\( + \)\(29\!\cdots\!68\)\( T - 2904371819036676 T^{2} - 22694658 T^{3} + T^{4} )^{4} \)
$83$ \( ( \)\(74\!\cdots\!00\)\( + \)\(74\!\cdots\!00\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{4} + 6035256522659990 T^{6} + T^{8} )^{2} \)
$89$ \( ( \)\(12\!\cdots\!00\)\( + \)\(17\!\cdots\!00\)\( T^{2} + \)\(83\!\cdots\!00\)\( T^{4} + 15617313982451840 T^{6} + T^{8} )^{2} \)
$97$ \( ( \)\(93\!\cdots\!76\)\( - \)\(77\!\cdots\!04\)\( T^{2} + \)\(22\!\cdots\!56\)\( T^{4} - 27061263847686304 T^{6} + T^{8} )^{2} \)
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