Properties

Label 320.6.d.c
Level 320
Weight 6
Character orbit 320.d
Analytic conductor 51.323
Analytic rank 0
Dimension 12
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 320.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(51.3228223402\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{30} \)
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.

\(f(q)\) \(=\) \( q + ( 3 \beta_{1} - \beta_{4} ) q^{3} -25 \beta_{1} q^{5} + ( -22 + \beta_{2} - \beta_{3} + \beta_{11} ) q^{7} + ( -36 - 3 \beta_{2} + 12 \beta_{3} - \beta_{6} - \beta_{8} - \beta_{11} ) q^{9} +O(q^{10})\) \( q + ( 3 \beta_{1} - \beta_{4} ) q^{3} -25 \beta_{1} q^{5} + ( -22 + \beta_{2} - \beta_{3} + \beta_{11} ) q^{7} + ( -36 - 3 \beta_{2} + 12 \beta_{3} - \beta_{6} - \beta_{8} - \beta_{11} ) q^{9} + ( -63 \beta_{1} - 8 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} + 2 \beta_{9} - \beta_{10} ) q^{11} + ( 147 \beta_{1} - 4 \beta_{4} + 22 \beta_{5} + \beta_{7} - 3 \beta_{9} - \beta_{10} ) q^{13} + ( 75 - 25 \beta_{3} ) q^{15} + ( 201 + 2 \beta_{2} - 20 \beta_{3} + 5 \beta_{6} - 3 \beta_{8} - 5 \beta_{11} ) q^{17} + ( -401 \beta_{1} + 2 \beta_{4} - 22 \beta_{5} + 4 \beta_{7} - 6 \beta_{9} + 5 \beta_{10} ) q^{19} + ( 108 \beta_{1} - 16 \beta_{4} + 57 \beta_{5} + 4 \beta_{7} + 4 \beta_{9} - 2 \beta_{10} ) q^{21} + ( -682 - 62 \beta_{2} + 123 \beta_{3} - 14 \beta_{6} + \beta_{8} - 4 \beta_{11} ) q^{23} -625 q^{25} + ( -2284 \beta_{1} + 104 \beta_{4} - 109 \beta_{5} + 3 \beta_{7} - 2 \beta_{9} + 25 \beta_{10} ) q^{27} + ( -972 \beta_{1} - 80 \beta_{4} - 53 \beta_{5} - 16 \beta_{7} - 6 \beta_{9} + 22 \beta_{10} ) q^{29} + ( -654 - 56 \beta_{2} + 146 \beta_{3} + 32 \beta_{6} + 2 \beta_{8} + 2 \beta_{11} ) q^{31} + ( -1745 - 2 \beta_{2} - 124 \beta_{3} - 27 \beta_{6} - 13 \beta_{8} - 27 \beta_{11} ) q^{33} + ( 550 \beta_{1} + 25 \beta_{4} + 25 \beta_{5} + 25 \beta_{7} ) q^{35} + ( -1750 \beta_{1} - 408 \beta_{4} - 188 \beta_{5} + 10 \beta_{7} + 48 \beta_{9} - 26 \beta_{10} ) q^{37} + ( -3404 + 40 \beta_{2} + 494 \beta_{3} + 12 \beta_{6} - 40 \beta_{8} - 18 \beta_{11} ) q^{39} + ( -4663 - 85 \beta_{2} + 20 \beta_{3} + 17 \beta_{6} - 59 \beta_{8} + 37 \beta_{11} ) q^{41} + ( -3351 \beta_{1} - 385 \beta_{4} - 681 \beta_{5} + 3 \beta_{7} + 58 \beta_{9} + 3 \beta_{10} ) q^{43} + ( 900 \beta_{1} - 300 \beta_{4} - 75 \beta_{5} - 25 \beta_{7} - 25 \beta_{9} - 25 \beta_{10} ) q^{45} + ( -1784 + 401 \beta_{2} - 615 \beta_{3} - 16 \beta_{6} + 80 \beta_{8} + 37 \beta_{11} ) q^{47} + ( -482 - 1085 \beta_{2} + 780 \beta_{3} + 31 \beta_{6} - 17 \beta_{8} - 109 \beta_{11} ) q^{49} + ( 6064 \beta_{1} - 378 \beta_{4} - 822 \beta_{5} - 4 \beta_{7} - 24 \beta_{9} + 26 \beta_{10} ) q^{51} + ( -6643 \beta_{1} + 620 \beta_{4} - 200 \beta_{5} + 41 \beta_{7} - 109 \beta_{9} + 43 \beta_{10} ) q^{53} + ( -1575 + 50 \beta_{2} - 200 \beta_{3} - 50 \beta_{6} + 25 \beta_{8} - 50 \beta_{11} ) q^{55} + ( 3321 - 998 \beta_{2} - 492 \beta_{3} + 3 \beta_{6} + 91 \beta_{8} + 53 \beta_{11} ) q^{57} + ( 1535 \beta_{1} + 578 \beta_{4} - 608 \beta_{5} + 26 \beta_{7} - 154 \beta_{9} - 65 \beta_{10} ) q^{59} + ( -5244 \beta_{1} - 64 \beta_{4} + 390 \beta_{5} + 28 \beta_{7} + 58 \beta_{9} - 32 \beta_{10} ) q^{61} + ( -14906 + 422 \beta_{2} - 257 \beta_{3} - 54 \beta_{6} - 87 \beta_{8} + 128 \beta_{11} ) q^{63} + ( 3675 - 550 \beta_{2} - 100 \beta_{3} + 75 \beta_{6} + 25 \beta_{8} - 25 \beta_{11} ) q^{65} + ( 4629 \beta_{1} + 1175 \beta_{4} - 1491 \beta_{5} - 27 \beta_{7} + 118 \beta_{9} - 27 \beta_{10} ) q^{67} + ( -29846 \beta_{1} + 2976 \beta_{4} + 709 \beta_{5} - 28 \beta_{7} + 22 \beta_{9} + 70 \beta_{10} ) q^{69} + ( -30590 + 406 \beta_{2} - 626 \beta_{3} + 20 \beta_{6} - 146 \beta_{8} - 10 \beta_{11} ) q^{71} + ( 4893 - 1948 \beta_{2} - 636 \beta_{3} - 187 \beta_{6} + 71 \beta_{8} + 253 \beta_{11} ) q^{73} + ( -1875 \beta_{1} + 625 \beta_{4} ) q^{75} + ( -26819 \beta_{1} - 1516 \beta_{4} - 180 \beta_{5} + 63 \beta_{7} - 59 \beta_{9} + 21 \beta_{10} ) q^{77} + ( -2254 - 2240 \beta_{2} + 332 \beta_{3} + 92 \beta_{6} + 88 \beta_{8} - 218 \beta_{11} ) q^{79} + ( 34150 - 3775 \beta_{2} - 2500 \beta_{3} - 217 \beta_{6} + 299 \beta_{8} + 43 \beta_{11} ) q^{81} + ( 21101 \beta_{1} - 1743 \beta_{4} - 4237 \beta_{5} - 317 \beta_{7} - 2 \beta_{9} - 123 \beta_{10} ) q^{83} + ( -5025 \beta_{1} + 500 \beta_{4} + 50 \beta_{5} - 125 \beta_{7} + 125 \beta_{9} - 75 \beta_{10} ) q^{85} + ( -15306 - 1981 \beta_{2} - 1292 \beta_{3} - 30 \beta_{6} + 221 \beta_{8} + 133 \beta_{11} ) q^{87} + ( 7428 - 5528 \beta_{2} + 2792 \beta_{3} + 322 \beta_{6} + 18 \beta_{8} + 62 \beta_{11} ) q^{89} + ( -24698 \beta_{1} - 2938 \beta_{4} - 5022 \beta_{5} - 420 \beta_{7} + 220 \beta_{9} - 192 \beta_{10} ) q^{91} + ( -36566 \beta_{1} - 928 \beta_{4} + 284 \beta_{5} + 252 \beta_{7} + 354 \beta_{9} + 28 \beta_{10} ) q^{93} + ( -10025 + 550 \beta_{2} + 50 \beta_{3} + 150 \beta_{6} - 125 \beta_{8} - 100 \beta_{11} ) q^{95} + ( -14139 - 4518 \beta_{2} + 3796 \beta_{3} + 121 \beta_{6} - 225 \beta_{8} + 321 \beta_{11} ) q^{97} + ( 14133 \beta_{1} + 2892 \beta_{4} - 3442 \beta_{5} + 156 \beta_{7} + 26 \beta_{9} - 119 \beta_{10} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 268q^{7} - 428q^{9} + O(q^{10}) \) \( 12q - 268q^{7} - 428q^{9} + 900q^{15} + 2400q^{17} - 8108q^{23} - 7500q^{25} - 7976q^{31} - 20776q^{33} - 40984q^{39} - 56408q^{41} - 21172q^{47} - 5540q^{49} - 18400q^{55} + 39992q^{57} - 179516q^{63} + 44000q^{65} - 367704q^{71} + 58736q^{73} - 26192q^{79} + 411692q^{81} - 183200q^{87} + 87672q^{89} - 121000q^{95} - 172336q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 343 x^{10} - 696 x^{9} + 44406 x^{8} + 179640 x^{7} - 2401691 x^{6} - 15554592 x^{5} + 26901210 x^{4} + 434775816 x^{3} + 1271335685 x^{2} + 1475231592 x + 653157349\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-1114169565385359 \nu^{11} + 5148378830255625 \nu^{10} + 344320662557978084 \nu^{9} - 777687620581938609 \nu^{8} - 42209525222022135489 \nu^{7} - 7591386010812523011 \nu^{6} + 2363407957163362867206 \nu^{5} + 5974186120174593456795 \nu^{4} - 45036396282092927536469 \nu^{3} - 239134506183485171752707 \nu^{2} - 385502990089768818965820 \nu - 217987432168313171400201\)\()/ \)\(19\!\cdots\!56\)\( \)
\(\beta_{2}\)\(=\)\((\)\(36815915 \nu^{11} - 1614964016 \nu^{10} - 8316126004 \nu^{9} + 393167792559 \nu^{8} + 1368899772199 \nu^{7} - 32730922343066 \nu^{6} - 139410432367826 \nu^{5} + 857644800821275 \nu^{4} + 5250130473321761 \nu^{3} + 6141622707594396 \nu^{2} + 506366224717298 \nu + 4388000374446049\)\()/ 484488895993706 \)
\(\beta_{3}\)\(=\)\((\)\(3513742031841606386 \nu^{11} - 70661177934391715597 \nu^{10} - 970562338568173639224 \nu^{9} + 16294561626189934266384 \nu^{8} + 132142088648168857632184 \nu^{7} - 1218425713243063389228323 \nu^{6} - 9763097509765769583985496 \nu^{5} + 20901286382332442400657520 \nu^{4} + 283737884722866991445184958 \nu^{3} + 687852158936708631963167595 \nu^{2} + 681999693930617945570246238 \nu + 581426451064245854644667842\)\()/ \)\(36\!\cdots\!04\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-10231513249461913599 \nu^{11} + 95045593935047337780 \nu^{10} + 3043275071165687287774 \nu^{9} - 18879843136352678124641 \nu^{8} - 388795639100860195034671 \nu^{7} + 923710347683365989128438 \nu^{6} + 24321066494443697292639176 \nu^{5} + 30706837610558187280253691 \nu^{4} - 552996627940027869804602545 \nu^{3} - 2594800642456820605873067720 \nu^{2} - 4343313783885789675179200592 \nu - 2560494130722577331427101863\)\()/ \)\(36\!\cdots\!04\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-74273403 \nu^{11} + 389497022 \nu^{10} + 22748159420 \nu^{9} - 69843390937 \nu^{8} - 2729723240483 \nu^{7} + 2122604298688 \nu^{6} + 150021763825354 \nu^{5} + 211252396213243 \nu^{4} - 3022005704518913 \nu^{3} - 10571516347535470 \nu^{2} - 8667704717034982 \nu - 1430082952317763\)\()/ 143817507231506 \)
\(\beta_{6}\)\(=\)\((\)\(13639266356667327 \nu^{11} - 91124133229166981 \nu^{10} - 4139261263854231198 \nu^{9} + 16865639858706048357 \nu^{8} + 511744645096093141965 \nu^{7} - 602191962960693180693 \nu^{6} - 29802757647914065725084 \nu^{5} - 48050332016629088674543 \nu^{4} + 625668140384878066459881 \nu^{3} + 2817604227319846900601423 \nu^{2} + 3893131488180761585810778 \nu - 98920471395888825472533\)\()/ \)\(95\!\cdots\!78\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-54754580421881446865 \nu^{11} + 782892543987925631975 \nu^{10} + 12418662926401850962132 \nu^{9} - 177728667037626808988547 \nu^{8} - 1136470451040249814781311 \nu^{7} + 13647693775930350342127611 \nu^{6} + 55345622743396298138630626 \nu^{5} - 339095214197795010763360823 \nu^{4} - 1230442815340743268840968667 \nu^{3} - 1400843290371205317628670133 \nu^{2} - 5580760055468273501014891708 \nu - 5166027282445433905581974531\)\()/ \)\(36\!\cdots\!04\)\( \)
\(\beta_{8}\)\(=\)\((\)\(34003364205222503302 \nu^{11} + 143280450754515900289 \nu^{10} - 6753967617680930083028 \nu^{9} - 52414456244141235315222 \nu^{8} + 169635384594321812887328 \nu^{7} + 4739366617868354835284603 \nu^{6} + 37468431433912499527525892 \nu^{5} - 5360134902433571636730090 \nu^{4} - 1889413878050639052509821774 \nu^{3} - 9335236695460745980003599107 \nu^{2} - 13789237970643232634050789618 \nu - 5357881289418969973280396130\)\()/ \)\(18\!\cdots\!02\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-44234630164150217 \nu^{11} - 115775673391474869 \nu^{10} + 17502036736459030668 \nu^{9} + 51793124112605240505 \nu^{8} - 2490352575452209485519 \nu^{7} - 9537599746470372114465 \nu^{6} + 149603854094149565872106 \nu^{5} + 772142929035544358218749 \nu^{4} - 2630413554028126583041875 \nu^{3} - 22389716409000305804366985 \nu^{2} - 44397370695649112120155436 \nu - 28286273382402104580263775\)\()/ \)\(19\!\cdots\!56\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-128705849992829120893 \nu^{11} + 644642326512253546581 \nu^{10} + 44639957700360428213656 \nu^{9} - 128556339007628903695591 \nu^{8} - 6171859732685479040894591 \nu^{7} + 3329158345812296085750717 \nu^{6} + 398242724666323927919815670 \nu^{5} + 702991763010290290224807677 \nu^{4} - 9502399212145153462872180943 \nu^{3} - 38939536065376814251435508591 \nu^{2} - 43460313045814929858639489068 \nu - 16459470084154113895305812859\)\()/ \)\(36\!\cdots\!04\)\( \)
\(\beta_{11}\)\(=\)\((\)\(78237484165190888991 \nu^{11} - 418685991849346335709 \nu^{10} - 21909708977731768086998 \nu^{9} + 63587200441678776729387 \nu^{8} + 2448055765108875426834705 \nu^{7} - 628606524459807419585633 \nu^{6} - 126185264131728728881735944 \nu^{5} - 288746807766053051174364153 \nu^{4} + 2249132229357742391094901349 \nu^{3} + 11481762909904075317986816135 \nu^{2} + 16450443817427961744959054578 \nu + 7133995622365770276605081365\)\()/ \)\(18\!\cdots\!02\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-2 \beta_{3} + \beta_{2} - 2 \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} + \beta_{5} + 2 \beta_{4} - 6 \beta_{3} + 3 \beta_{2} + 229\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{9} - 2 \beta_{8} + \beta_{6} + 9 \beta_{5} + 18 \beta_{4} - 338 \beta_{3} + 149 \beta_{2} - 689 \beta_{1} + 1393\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(2 \beta_{11} - 2 \beta_{10} + \beta_{9} + 86 \beta_{6} + 150 \beta_{5} + 340 \beta_{4} - 856 \beta_{3} + 500 \beta_{2} - 1393 \beta_{1} + 19254\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-92 \beta_{11} + 435 \beta_{9} - 236 \beta_{8} + 10 \beta_{7} + 423 \beta_{6} + 2515 \beta_{5} + 4310 \beta_{4} - 30494 \beta_{3} + 13393 \beta_{2} - 97419 \beta_{1} + 197149\)\()/8\)
\(\nu^{6}\)\(=\)\((\)\(337 \beta_{11} - 359 \beta_{10} + 637 \beta_{9} - 348 \beta_{8} - 138 \beta_{7} + 7422 \beta_{6} + 20465 \beta_{5} + 46592 \beta_{4} - 100358 \beta_{3} + 58831 \beta_{2} - 299206 \beta_{1} + 1730389\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(-6986 \beta_{11} - 1218 \beta_{10} + 26740 \beta_{9} - 12629 \beta_{8} + 1197 \beta_{7} + 32355 \beta_{6} + 210315 \beta_{5} + 358806 \beta_{4} - 1416094 \beta_{3} + 649064 \beta_{2} - 6227247 \beta_{1} + 11481466\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(25936 \beta_{11} - 52196 \beta_{10} + 132395 \beta_{9} - 69564 \beta_{8} - 28588 \beta_{7} + 661484 \beta_{6} + 2692110 \beta_{5} + 5882600 \beta_{4} - 10862912 \beta_{3} + 6333844 \beta_{2} - 47325409 \beta_{1} + 160121062\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(-1550544 \beta_{11} - 640692 \beta_{10} + 6276069 \beta_{9} - 2721428 \beta_{8} + 247830 \beta_{7} + 7884109 \beta_{6} + 59538957 \beta_{5} + 102090018 \beta_{4} - 267071198 \beta_{3} + 129270893 \beta_{2} - 1516227061 \beta_{1} + 2470443355\)\()/8\)
\(\nu^{10}\)\(=\)\((\)\(798437 \beta_{11} - 7197563 \beta_{10} + 20707699 \beta_{9} - 10001826 \beta_{8} - 4091736 \beta_{7} + 60493781 \beta_{6} + 343512064 \beta_{5} + 712125342 \beta_{4} - 1123315444 \beta_{3} + 654736640 \beta_{2} - 6533150362 \beta_{1} + 15038933230\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(-156982508 \beta_{11} - 115919892 \beta_{10} + 723555305 \beta_{9} - 294490040 \beta_{8} + 11081026 \beta_{7} + 869353185 \beta_{6} + 7752531457 \beta_{5} + 13300236290 \beta_{4} - 25331289514 \beta_{3} + 12947121299 \beta_{2} - 179465088425 \beta_{1} + 253907779519\)\()/8\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
10.0457 + 0.500000i
−8.35048 0.500000i
10.2435 + 0.500000i
−1.02698 0.500000i
−7.49748 0.500000i
−3.41426 + 0.500000i
−3.41426 0.500000i
−7.49748 + 0.500000i
−1.02698 + 0.500000i
10.2435 0.500000i
−8.35048 + 0.500000i
10.0457 0.500000i
0 30.0196i 0 25.0000i 0 45.4594 0 −658.179 0
161.2 0 20.6292i 0 25.0000i 0 93.5254 0 −182.562 0
161.3 0 16.5588i 0 25.0000i 0 −55.4690 0 −31.1927 0
161.4 0 5.98217i 0 25.0000i 0 −86.0039 0 207.214 0
161.5 0 5.06676i 0 25.0000i 0 −250.333 0 217.328 0
161.6 0 3.09968i 0 25.0000i 0 118.821 0 233.392 0
161.7 0 3.09968i 0 25.0000i 0 118.821 0 233.392 0
161.8 0 5.06676i 0 25.0000i 0 −250.333 0 217.328 0
161.9 0 5.98217i 0 25.0000i 0 −86.0039 0 207.214 0
161.10 0 16.5588i 0 25.0000i 0 −55.4690 0 −31.1927 0
161.11 0 20.6292i 0 25.0000i 0 93.5254 0 −182.562 0
161.12 0 30.0196i 0 25.0000i 0 45.4594 0 −658.179 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.12
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 320.6.d.c 12
4.b odd 2 1 320.6.d.d yes 12
8.b even 2 1 inner 320.6.d.c 12
8.d odd 2 1 320.6.d.d yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.6.d.c 12 1.a even 1 1 trivial
320.6.d.c 12 8.b even 2 1 inner
320.6.d.d yes 12 4.b odd 2 1
320.6.d.d yes 12 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(320, [\chi])\):

\( T_{3}^{12} + 1672 T_{3}^{10} + 862572 T_{3}^{8} + 160687168 T_{3}^{6} + 8614933552 T_{3}^{4} + 165296582784 T_{3}^{2} + 928201218624 \)
\( T_{7}^{6} + 134 T_{7}^{5} - 40058 T_{7}^{4} - 1534896 T_{7}^{3} + 328685460 T_{7}^{2} + 4498079256 T_{7} - 603298353096 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 1244 T^{2} + 696846 T^{4} - 230065580 T^{6} + 47479013407 T^{8} - 5765072774040 T^{10} + 634724537410692 T^{12} - 340421782234287960 T^{14} + \)\(16\!\cdots\!07\)\( T^{16} - \)\(47\!\cdots\!20\)\( T^{18} + \)\(84\!\cdots\!46\)\( T^{20} - \)\(89\!\cdots\!56\)\( T^{22} + \)\(42\!\cdots\!01\)\( T^{24} \)
$5$ \( ( 1 + 625 T^{2} )^{6} \)
$7$ \( ( 1 + 134 T + 60784 T^{2} + 9725794 T^{3} + 1872794971 T^{4} + 305623921700 T^{5} + 37504003751552 T^{6} + 5136621252011900 T^{7} + 529018225759172779 T^{8} + 46173805248034569742 T^{9} + \)\(48\!\cdots\!84\)\( T^{10} + \)\(17\!\cdots\!38\)\( T^{11} + \)\(22\!\cdots\!49\)\( T^{12} )^{2} \)
$11$ \( 1 - 1002172 T^{2} + 493219533810 T^{4} - 163075311940514028 T^{6} + \)\(41\!\cdots\!27\)\( T^{8} - \)\(86\!\cdots\!44\)\( T^{10} + \)\(15\!\cdots\!36\)\( T^{12} - \)\(22\!\cdots\!44\)\( T^{14} + \)\(27\!\cdots\!27\)\( T^{16} - \)\(28\!\cdots\!28\)\( T^{18} + \)\(22\!\cdots\!10\)\( T^{20} - \)\(11\!\cdots\!72\)\( T^{22} + \)\(30\!\cdots\!01\)\( T^{24} \)
$13$ \( 1 - 2253844 T^{2} + 2609423635634 T^{4} - 2090949123359826660 T^{6} + \)\(12\!\cdots\!95\)\( T^{8} - \)\(64\!\cdots\!44\)\( T^{10} + \)\(26\!\cdots\!16\)\( T^{12} - \)\(88\!\cdots\!56\)\( T^{14} + \)\(24\!\cdots\!95\)\( T^{16} - \)\(54\!\cdots\!40\)\( T^{18} + \)\(94\!\cdots\!34\)\( T^{20} - \)\(11\!\cdots\!56\)\( T^{22} + \)\(68\!\cdots\!01\)\( T^{24} \)
$17$ \( ( 1 - 1200 T + 5829762 T^{2} - 7194221040 T^{3} + 17474250252687 T^{4} - 17877633531638880 T^{5} + 31833151505919656476 T^{6} - \)\(25\!\cdots\!60\)\( T^{7} + \)\(35\!\cdots\!63\)\( T^{8} - \)\(20\!\cdots\!20\)\( T^{9} + \)\(23\!\cdots\!62\)\( T^{10} - \)\(69\!\cdots\!00\)\( T^{11} + \)\(81\!\cdots\!49\)\( T^{12} )^{2} \)
$19$ \( 1 - 19893836 T^{2} + 197127587113970 T^{4} - \)\(12\!\cdots\!52\)\( T^{6} + \)\(60\!\cdots\!07\)\( T^{8} - \)\(21\!\cdots\!16\)\( T^{10} + \)\(60\!\cdots\!52\)\( T^{12} - \)\(13\!\cdots\!16\)\( T^{14} + \)\(22\!\cdots\!07\)\( T^{16} - \)\(29\!\cdots\!52\)\( T^{18} + \)\(27\!\cdots\!70\)\( T^{20} - \)\(17\!\cdots\!36\)\( T^{22} + \)\(53\!\cdots\!01\)\( T^{24} \)
$23$ \( ( 1 + 4054 T + 24767560 T^{2} + 85869926290 T^{3} + 327319542066779 T^{4} + 869764424865766180 T^{5} + \)\(26\!\cdots\!28\)\( T^{6} + \)\(55\!\cdots\!40\)\( T^{7} + \)\(13\!\cdots\!71\)\( T^{8} + \)\(22\!\cdots\!30\)\( T^{9} + \)\(42\!\cdots\!60\)\( T^{10} + \)\(44\!\cdots\!22\)\( T^{11} + \)\(71\!\cdots\!49\)\( T^{12} )^{2} \)
$29$ \( 1 - 118766236 T^{2} + 8086517092300050 T^{4} - \)\(38\!\cdots\!40\)\( T^{6} + \)\(13\!\cdots\!91\)\( T^{8} - \)\(38\!\cdots\!40\)\( T^{10} + \)\(88\!\cdots\!84\)\( T^{12} - \)\(16\!\cdots\!40\)\( T^{14} + \)\(24\!\cdots\!91\)\( T^{16} - \)\(28\!\cdots\!40\)\( T^{18} + \)\(25\!\cdots\!50\)\( T^{20} - \)\(15\!\cdots\!36\)\( T^{22} + \)\(55\!\cdots\!01\)\( T^{24} \)
$31$ \( ( 1 + 3988 T + 109571794 T^{2} + 302364450332 T^{3} + 5668865865566767 T^{4} + 12320621141215714888 T^{5} + \)\(19\!\cdots\!04\)\( T^{6} + \)\(35\!\cdots\!88\)\( T^{7} + \)\(46\!\cdots\!67\)\( T^{8} + \)\(70\!\cdots\!32\)\( T^{9} + \)\(73\!\cdots\!94\)\( T^{10} + \)\(76\!\cdots\!88\)\( T^{11} + \)\(55\!\cdots\!01\)\( T^{12} )^{2} \)
$37$ \( 1 - 298070820 T^{2} + 53469062376708978 T^{4} - \)\(69\!\cdots\!56\)\( T^{6} + \)\(71\!\cdots\!03\)\( T^{8} - \)\(61\!\cdots\!04\)\( T^{10} + \)\(45\!\cdots\!00\)\( T^{12} - \)\(29\!\cdots\!96\)\( T^{14} + \)\(16\!\cdots\!03\)\( T^{16} - \)\(77\!\cdots\!44\)\( T^{18} + \)\(28\!\cdots\!78\)\( T^{20} - \)\(76\!\cdots\!80\)\( T^{22} + \)\(12\!\cdots\!01\)\( T^{24} \)
$41$ \( ( 1 + 28204 T + 645547054 T^{2} + 9891921342268 T^{3} + 144344267066611295 T^{4} + \)\(17\!\cdots\!96\)\( T^{5} + \)\(20\!\cdots\!08\)\( T^{6} + \)\(19\!\cdots\!96\)\( T^{7} + \)\(19\!\cdots\!95\)\( T^{8} + \)\(15\!\cdots\!68\)\( T^{9} + \)\(11\!\cdots\!54\)\( T^{10} + \)\(58\!\cdots\!04\)\( T^{11} + \)\(24\!\cdots\!01\)\( T^{12} )^{2} \)
$43$ \( 1 - 854218268 T^{2} + 379569874740918830 T^{4} - \)\(11\!\cdots\!80\)\( T^{6} + \)\(26\!\cdots\!71\)\( T^{8} - \)\(50\!\cdots\!20\)\( T^{10} + \)\(79\!\cdots\!96\)\( T^{12} - \)\(10\!\cdots\!80\)\( T^{14} + \)\(12\!\cdots\!71\)\( T^{16} - \)\(11\!\cdots\!20\)\( T^{18} + \)\(82\!\cdots\!30\)\( T^{20} - \)\(40\!\cdots\!32\)\( T^{22} + \)\(10\!\cdots\!01\)\( T^{24} \)
$47$ \( ( 1 + 10586 T + 559351288 T^{2} + 141995130734 T^{3} + 85696824820019915 T^{4} - \)\(20\!\cdots\!52\)\( T^{5} + \)\(41\!\cdots\!80\)\( T^{6} - \)\(45\!\cdots\!64\)\( T^{7} + \)\(45\!\cdots\!35\)\( T^{8} + \)\(17\!\cdots\!62\)\( T^{9} + \)\(15\!\cdots\!88\)\( T^{10} + \)\(67\!\cdots\!02\)\( T^{11} + \)\(14\!\cdots\!49\)\( T^{12} )^{2} \)
$53$ \( 1 - 2472111540 T^{2} + 3272109664551065298 T^{4} - \)\(29\!\cdots\!96\)\( T^{6} + \)\(20\!\cdots\!23\)\( T^{8} - \)\(11\!\cdots\!44\)\( T^{10} + \)\(51\!\cdots\!20\)\( T^{12} - \)\(19\!\cdots\!56\)\( T^{14} + \)\(62\!\cdots\!23\)\( T^{16} - \)\(15\!\cdots\!04\)\( T^{18} + \)\(30\!\cdots\!98\)\( T^{20} - \)\(40\!\cdots\!60\)\( T^{22} + \)\(28\!\cdots\!01\)\( T^{24} \)
$59$ \( 1 - 3285269100 T^{2} + 3759201793217414610 T^{4} - \)\(13\!\cdots\!44\)\( T^{6} + \)\(12\!\cdots\!35\)\( T^{8} - \)\(13\!\cdots\!20\)\( T^{10} + \)\(18\!\cdots\!36\)\( T^{12} - \)\(70\!\cdots\!20\)\( T^{14} + \)\(33\!\cdots\!35\)\( T^{16} - \)\(18\!\cdots\!44\)\( T^{18} + \)\(25\!\cdots\!10\)\( T^{20} - \)\(11\!\cdots\!00\)\( T^{22} + \)\(17\!\cdots\!01\)\( T^{24} \)
$61$ \( 1 - 9275968036 T^{2} + 40037810273961512978 T^{4} - \)\(10\!\cdots\!60\)\( T^{6} + \)\(19\!\cdots\!03\)\( T^{8} - \)\(25\!\cdots\!00\)\( T^{10} + \)\(24\!\cdots\!52\)\( T^{12} - \)\(18\!\cdots\!00\)\( T^{14} + \)\(98\!\cdots\!03\)\( T^{16} - \)\(38\!\cdots\!60\)\( T^{18} + \)\(10\!\cdots\!78\)\( T^{20} - \)\(17\!\cdots\!36\)\( T^{22} + \)\(13\!\cdots\!01\)\( T^{24} \)
$67$ \( 1 - 7932747836 T^{2} + 31498024738605168590 T^{4} - \)\(85\!\cdots\!12\)\( T^{6} + \)\(17\!\cdots\!55\)\( T^{8} - \)\(30\!\cdots\!40\)\( T^{10} + \)\(44\!\cdots\!12\)\( T^{12} - \)\(55\!\cdots\!60\)\( T^{14} + \)\(59\!\cdots\!55\)\( T^{16} - \)\(51\!\cdots\!88\)\( T^{18} + \)\(34\!\cdots\!90\)\( T^{20} - \)\(15\!\cdots\!64\)\( T^{22} + \)\(36\!\cdots\!01\)\( T^{24} \)
$71$ \( ( 1 + 183852 T + 22719949218 T^{2} + 1935102444145668 T^{3} + \)\(13\!\cdots\!79\)\( T^{4} + \)\(73\!\cdots\!52\)\( T^{5} + \)\(34\!\cdots\!16\)\( T^{6} + \)\(13\!\cdots\!52\)\( T^{7} + \)\(43\!\cdots\!79\)\( T^{8} + \)\(11\!\cdots\!68\)\( T^{9} + \)\(24\!\cdots\!18\)\( T^{10} + \)\(35\!\cdots\!52\)\( T^{11} + \)\(34\!\cdots\!01\)\( T^{12} )^{2} \)
$73$ \( ( 1 - 29368 T + 4461233378 T^{2} - 258540669925656 T^{3} + 16168785780639245407 T^{4} - \)\(68\!\cdots\!32\)\( T^{5} + \)\(45\!\cdots\!60\)\( T^{6} - \)\(14\!\cdots\!76\)\( T^{7} + \)\(69\!\cdots\!43\)\( T^{8} - \)\(23\!\cdots\!92\)\( T^{9} + \)\(82\!\cdots\!78\)\( T^{10} - \)\(11\!\cdots\!24\)\( T^{11} + \)\(79\!\cdots\!49\)\( T^{12} )^{2} \)
$79$ \( ( 1 + 13096 T + 12178064842 T^{2} - 46215677031496 T^{3} + 67636133741719889263 T^{4} - \)\(78\!\cdots\!08\)\( T^{5} + \)\(24\!\cdots\!84\)\( T^{6} - \)\(24\!\cdots\!92\)\( T^{7} + \)\(64\!\cdots\!63\)\( T^{8} - \)\(13\!\cdots\!04\)\( T^{9} + \)\(10\!\cdots\!42\)\( T^{10} + \)\(36\!\cdots\!04\)\( T^{11} + \)\(84\!\cdots\!01\)\( T^{12} )^{2} \)
$83$ \( 1 - 15441152812 T^{2} + \)\(13\!\cdots\!30\)\( T^{4} - \)\(79\!\cdots\!24\)\( T^{6} + \)\(37\!\cdots\!87\)\( T^{8} - \)\(15\!\cdots\!88\)\( T^{10} + \)\(63\!\cdots\!52\)\( T^{12} - \)\(24\!\cdots\!12\)\( T^{14} + \)\(91\!\cdots\!87\)\( T^{16} - \)\(29\!\cdots\!76\)\( T^{18} + \)\(77\!\cdots\!30\)\( T^{20} - \)\(13\!\cdots\!88\)\( T^{22} + \)\(13\!\cdots\!01\)\( T^{24} \)
$89$ \( ( 1 - 43836 T + 13171715490 T^{2} - 211972966345068 T^{3} + 95738414377062864255 T^{4} - \)\(28\!\cdots\!20\)\( T^{5} + \)\(69\!\cdots\!56\)\( T^{6} - \)\(16\!\cdots\!80\)\( T^{7} + \)\(29\!\cdots\!55\)\( T^{8} - \)\(36\!\cdots\!32\)\( T^{9} + \)\(12\!\cdots\!90\)\( T^{10} - \)\(23\!\cdots\!64\)\( T^{11} + \)\(30\!\cdots\!01\)\( T^{12} )^{2} \)
$97$ \( ( 1 + 86168 T + 30789268338 T^{2} + 1787720936745144 T^{3} + \)\(45\!\cdots\!83\)\( T^{4} + \)\(19\!\cdots\!32\)\( T^{5} + \)\(44\!\cdots\!92\)\( T^{6} + \)\(16\!\cdots\!24\)\( T^{7} + \)\(33\!\cdots\!67\)\( T^{8} + \)\(11\!\cdots\!92\)\( T^{9} + \)\(16\!\cdots\!38\)\( T^{10} + \)\(40\!\cdots\!76\)\( T^{11} + \)\(40\!\cdots\!49\)\( T^{12} )^{2} \)
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