Properties

Label 14.16.c.b
Level $14$
Weight $16$
Character orbit 14.c
Analytic conductor $19.977$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 14.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.9770907140\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - x^{9} + 7979102 x^{8} - 3342530557 x^{7} + 48610066963550 x^{6} - 20531463882664153 x^{5} + 122945882607722292337 x^{4} - 89586690298994071428648 x^{3} + 238785864099387585455090736 x^{2} - 108333121326673799746890760320 x + 51761574501460447109413728819456\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{3}\cdot 5^{4}\cdot 7^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 128 + 128 \beta_{1} ) q^{2} + ( -479 \beta_{1} + \beta_{3} ) q^{3} + 16384 \beta_{1} q^{4} + ( 2194 + 2194 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{5} ) q^{5} + ( 61312 - 128 \beta_{2} ) q^{6} + ( 102354 - 270343 \beta_{1} - 38 \beta_{2} - 146 \beta_{3} - 9 \beta_{4} - 3 \beta_{5} + \beta_{7} - \beta_{9} ) q^{7} -2097152 q^{8} + ( -9101064 - 9101067 \beta_{1} - 174 \beta_{2} - 179 \beta_{3} - 5 \beta_{4} - 71 \beta_{5} + 7 \beta_{6} - 20 \beta_{7} + 16 \beta_{8} + \beta_{9} ) q^{9} +O(q^{10})\) \( q + ( 128 + 128 \beta_{1} ) q^{2} + ( -479 \beta_{1} + \beta_{3} ) q^{3} + 16384 \beta_{1} q^{4} + ( 2194 + 2194 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{5} ) q^{5} + ( 61312 - 128 \beta_{2} ) q^{6} + ( 102354 - 270343 \beta_{1} - 38 \beta_{2} - 146 \beta_{3} - 9 \beta_{4} - 3 \beta_{5} + \beta_{7} - \beta_{9} ) q^{7} -2097152 q^{8} + ( -9101064 - 9101067 \beta_{1} - 174 \beta_{2} - 179 \beta_{3} - 5 \beta_{4} - 71 \beta_{5} + 7 \beta_{6} - 20 \beta_{7} + 16 \beta_{8} + \beta_{9} ) q^{9} + ( 280832 \beta_{1} - 256 \beta_{3} - 128 \beta_{4} - 128 \beta_{5} ) q^{10} + ( -32 + 681087 \beta_{1} - 16 \beta_{2} + 4124 \beta_{3} - 176 \beta_{4} - 192 \beta_{5} - 15 \beta_{6} + 17 \beta_{7} - 64 \beta_{8} - 21 \beta_{9} ) q^{11} + ( 7847936 + 7847936 \beta_{1} - 16384 \beta_{2} - 16384 \beta_{3} ) q^{12} + ( 36941034 + 111 \beta_{1} + 9623 \beta_{2} + 23 \beta_{3} + 896 \beta_{4} + 23 \beta_{5} + 65 \beta_{6} + 178 \beta_{7} - 86 \beta_{8} + 132 \beta_{9} ) q^{13} + ( 47705216 + 13101440 \beta_{1} + 13824 \beta_{2} - 4864 \beta_{3} - 384 \beta_{4} + 768 \beta_{5} + 128 \beta_{6} + 128 \beta_{7} - 128 \beta_{8} ) q^{14} + ( 45367751 + 285 \beta_{1} + 64597 \beta_{2} + 31 \beta_{3} + 3609 \beta_{4} + 31 \beta_{5} + 223 \beta_{6} + 443 \beta_{7} - 319 \beta_{8} + 381 \beta_{9} ) q^{15} + ( -268435456 - 268435456 \beta_{1} ) q^{16} + ( -1502 + 640340751 \beta_{1} - 751 \beta_{2} - 47198 \beta_{3} + 1829 \beta_{4} + 1078 \beta_{5} - 846 \beta_{6} + 656 \beta_{7} - 3004 \beta_{8} - 1033 \beta_{9} ) q^{17} + ( 1280 - 1164935424 \beta_{1} + 640 \beta_{2} - 22272 \beta_{3} - 9088 \beta_{4} - 8448 \beta_{5} + 768 \beta_{6} - 512 \beta_{7} + 2560 \beta_{8} + 896 \beta_{9} ) q^{18} + ( -212685622 - 212686707 \beta_{1} - 196642 \beta_{2} - 197696 \beta_{3} - 1054 \beta_{4} - 13040 \beta_{5} + 1023 \beta_{6} - 4216 \beta_{7} + 2015 \beta_{8} + 1116 \beta_{9} ) q^{19} + ( -35946496 + 32768 \beta_{2} - 16384 \beta_{4} ) q^{20} + ( 3261988216 + 2312235495 \beta_{1} + 513934 \beta_{2} + 833627 \beta_{3} - 56104 \beta_{4} - 29374 \beta_{5} - 2773 \beta_{6} + 4004 \beta_{7} - 4234 \beta_{8} + 1553 \beta_{9} ) q^{21} + ( -87185152 - 3328 \beta_{1} - 529920 \beta_{2} - 2048 \beta_{3} - 24576 \beta_{4} - 2048 \beta_{5} + 768 \beta_{6} - 6016 \beta_{7} - 2176 \beta_{8} - 1920 \beta_{9} ) q^{22} + ( -6336070562 - 6336075695 \beta_{1} + 1073599 \beta_{2} + 1069046 \beta_{3} - 4553 \beta_{4} - 53836 \beta_{5} + 3973 \beta_{6} - 18212 \beta_{7} + 7366 \beta_{8} + 5713 \beta_{9} ) q^{23} + ( 1004535808 \beta_{1} - 2097152 \beta_{3} ) q^{24} + ( -9188 - 7802505532 \beta_{1} - 4594 \beta_{2} - 2718484 \beta_{3} - 15026 \beta_{4} - 19620 \beta_{5} - 3744 \beta_{6} + 5444 \beta_{7} - 18376 \beta_{8} - 5842 \beta_{9} ) q^{25} + ( 4728446464 + 4728443776 \beta_{1} + 1228800 \beta_{2} + 1231744 \beta_{3} + 2944 \beta_{4} - 111744 \beta_{5} - 8576 \beta_{6} + 11776 \beta_{7} - 22784 \beta_{8} + 8320 \beta_{9} ) q^{26} + ( 6642770438 + 25206 \beta_{1} + 8763820 \beta_{2} + 352 \beta_{3} + 55980 \beta_{4} + 352 \beta_{5} + 24502 \beta_{6} + 37985 \beta_{7} - 36577 \beta_{8} + 37281 \beta_{9} ) q^{27} + ( 4429299712 + 6106284032 \beta_{1} + 2392064 \beta_{2} + 1769472 \beta_{3} + 98304 \beta_{4} + 147456 \beta_{5} + 16384 \beta_{6} - 16384 \beta_{8} + 16384 \beta_{9} ) q^{28} + ( 29493453040 + 4627 \beta_{1} - 10859985 \beta_{2} - 17677 \beta_{3} - 296382 \beta_{4} - 17677 \beta_{5} + 39981 \beta_{6} - 1898 \beta_{7} - 68810 \beta_{8} + 33456 \beta_{9} ) q^{29} + ( 5807064192 + 5807051904 \beta_{1} + 8264448 \beta_{2} + 8268416 \beta_{3} + 3968 \beta_{4} - 457984 \beta_{5} - 20224 \beta_{6} + 15872 \beta_{7} - 56704 \beta_{8} + 28544 \beta_{9} ) q^{30} + ( -13710 - 6040164650 \beta_{1} - 6855 \beta_{2} - 25726862 \beta_{3} - 939178 \beta_{4} - 946033 \beta_{5} + 16269 \beta_{6} + 29979 \beta_{7} - 27420 \beta_{8} - 1432 \beta_{9} ) q^{31} -34359738368 \beta_{1} q^{32} + ( -96512157699 - 96512172273 \beta_{1} - 19468536 \beta_{2} - 19477718 \beta_{3} - 9182 \beta_{4} - 1436252 \beta_{5} + 3790 \beta_{6} - 36728 \beta_{7} + 2188 \beta_{8} + 19966 \beta_{9} ) q^{33} + ( -81963916672 - 168320 \beta_{1} + 5945216 \beta_{2} - 96128 \beta_{3} + 137984 \beta_{4} - 96128 \beta_{5} + 23936 \beta_{6} - 300544 \beta_{7} - 83968 \beta_{8} - 108288 \beta_{9} ) q^{34} + ( -214920263019 - 121778116551 \beta_{1} + 22147419 \beta_{2} + 51925566 \beta_{3} - 555254 \beta_{4} - 297556 \beta_{5} + 70385 \beta_{6} - 160027 \beta_{7} + 131642 \beta_{8} - 15519 \beta_{9} ) q^{35} + ( 149111996416 + 147456 \beta_{1} + 2932736 \beta_{2} + 81920 \beta_{3} - 1081344 \beta_{4} + 81920 \beta_{5} - 16384 \beta_{6} + 262144 \beta_{7} + 65536 \beta_{8} + 98304 \beta_{9} ) q^{36} + ( 273937613074 + 273937726183 \beta_{1} - 34975194 \beta_{2} - 34886025 \beta_{3} + 89169 \beta_{4} + 1302306 \beta_{5} - 65229 \beta_{6} + 356676 \beta_{7} - 106518 \beta_{8} - 137049 \beta_{9} ) q^{37} + ( 269824 - 27223771520 \beta_{1} + 134912 \beta_{2} - 25170176 \beta_{3} - 1669120 \beta_{4} - 1534208 \beta_{5} - 11904 \beta_{6} - 281728 \beta_{7} + 539648 \beta_{8} + 130944 \beta_{9} ) q^{38} + ( -106136 + 209753434286 \beta_{1} - 53068 \beta_{2} + 78013919 \beta_{3} + 8284390 \beta_{4} + 8231322 \beta_{5} + 17079 \beta_{6} + 123215 \beta_{7} - 212272 \beta_{8} - 47375 \beta_{9} ) q^{39} + ( -4601151488 - 4601151488 \beta_{1} + 4194304 \beta_{2} + 4194304 \beta_{3} + 2097152 \beta_{5} ) q^{40} + ( 363485019110 + 335817 \beta_{1} + 54942255 \beta_{2} + 380837 \beta_{3} + 4457634 \beta_{4} + 380837 \beta_{5} - 425857 \beta_{6} + 694144 \beta_{7} + 829204 \beta_{8} - 67530 \beta_{9} ) q^{41} + ( 121567993344 + 417533937920 \beta_{1} - 40920704 \beta_{2} + 65783552 \beta_{3} - 3759872 \beta_{4} + 3421440 \beta_{5} - 553728 \beta_{6} - 29440 \beta_{7} - 512512 \beta_{8} - 354944 \beta_{9} ) q^{42} + ( 41560149198 - 479002 \beta_{1} - 48274366 \beta_{2} + 103390 \beta_{3} - 708250 \beta_{4} + 103390 \beta_{5} - 685782 \beta_{6} - 666808 \beta_{7} + 1080368 \beta_{8} - 873588 \beta_{9} ) q^{43} + ( -11159175168 - 11159355392 \beta_{1} - 67567616 \beta_{2} - 67829760 \beta_{3} - 262144 \beta_{4} + 2883584 \beta_{5} + 344064 \beta_{6} - 1048576 \beta_{7} + 770048 \beta_{8} + 98304 \beta_{9} ) q^{44} + ( -685462 + 1523279057286 \beta_{1} - 342731 \beta_{2} + 44647392 \beta_{3} + 16132565 \beta_{4} + 15789834 \beta_{5} + 207804 \beta_{6} + 893266 \beta_{7} - 1370924 \beta_{8} - 273463 \beta_{9} ) q^{45} + ( 1165568 - 811017254656 \beta_{1} + 582784 \beta_{2} + 137420672 \beta_{3} - 6891008 \beta_{4} - 6308224 \beta_{5} - 222720 \beta_{6} - 1388288 \beta_{7} + 2331136 \beta_{8} + 508544 \beta_{9} ) q^{46} + ( -2521394893487 - 2521394071729 \beta_{1} + 266782550 \beta_{2} + 266918233 \beta_{3} + 135683 \beta_{4} - 5496890 \beta_{5} + 550392 \beta_{6} + 542732 \beta_{7} + 1786859 \beta_{8} - 1507833 \beta_{9} ) q^{47} + ( -128580583424 + 268435456 \beta_{2} ) q^{48} + ( -409526892227 + 794531536287 \beta_{1} - 776587317 \beta_{2} - 184719727 \beta_{3} + 5103952 \beta_{4} + 5371863 \beta_{5} + 789827 \beta_{6} + 2809928 \beta_{7} + 2107788 \beta_{8} + 438670 \beta_{9} ) q^{49} + ( 998719052800 - 907520 \beta_{1} + 347377920 \beta_{2} - 588032 \beta_{3} - 2511360 \beta_{4} - 588032 \beta_{5} + 268544 \beta_{6} - 1655296 \beta_{7} - 696832 \beta_{8} - 479232 \beta_{9} ) q^{50} + ( 1371430433172 + 1371431547429 \beta_{1} - 2092538970 \beta_{2} - 2092022378 \beta_{3} + 516592 \beta_{4} - 17479268 \beta_{5} + 81073 \beta_{6} + 2066368 \beta_{7} + 759811 \beta_{8} - 1711922 \beta_{9} ) q^{51} + ( -753664 + 605238984704 \beta_{1} - 376832 \beta_{2} + 157286400 \beta_{3} - 14303232 \beta_{4} - 14680064 \beta_{5} - 2162688 \beta_{6} - 1409024 \beta_{7} - 1507328 \beta_{8} - 1097728 \beta_{9} ) q^{52} + ( -2290410 - 2123337389334 \beta_{1} - 1145205 \beta_{2} + 1418090302 \beta_{3} + 8364164 \beta_{4} + 7218959 \beta_{5} + 2011260 \beta_{6} + 4301670 \beta_{7} - 4580820 \beta_{8} - 474785 \beta_{9} ) q^{53} + ( 850274525952 + 850272980352 \beta_{1} + 1121723904 \beta_{2} + 1121768960 \beta_{3} + 45056 \beta_{4} - 7120384 \beta_{5} - 1635712 \beta_{6} + 180224 \beta_{7} - 4862080 \beta_{8} + 3136256 \beta_{9} ) q^{54} + ( -4085141880930 - 712465 \beta_{1} + 1322483100 \beta_{2} - 593233 \beta_{3} - 14592345 \beta_{4} - 593233 \beta_{5} + 474001 \beta_{6} - 1365314 \beta_{7} - 1007618 \beta_{8} - 178848 \beta_{9} ) q^{55} + ( -214651895808 + 566950363136 \beta_{1} + 79691776 \beta_{2} + 306184192 \beta_{3} + 18874368 \beta_{4} + 6291456 \beta_{5} - 2097152 \beta_{7} + 2097152 \beta_{9} ) q^{56} + ( 4466561087003 + 4068870 \beta_{1} + 2502620126 \beta_{2} + 409014 \beta_{3} + 38519124 \beta_{4} + 409014 \beta_{5} + 3250842 \beta_{6} + 6307812 \beta_{7} - 4671756 \beta_{8} + 5489784 \beta_{9} ) q^{57} + ( 3775166514432 + 3775162824320 \beta_{1} - 1387815424 \beta_{2} - 1390078080 \beta_{3} - 2262656 \beta_{4} + 35674240 \beta_{5} + 835200 \beta_{6} - 9050624 \beta_{7} + 242944 \beta_{8} + 5117568 \beta_{9} ) q^{58} + ( 4015396 - 7998750067181 \beta_{1} + 2007698 \beta_{2} - 2477011375 \beta_{3} - 1684208 \beta_{4} + 323490 \beta_{5} + 5336604 \beta_{6} + 1321208 \beta_{7} + 8030792 \beta_{8} + 3786566 \beta_{9} ) q^{59} + ( -1015808 + 743297974272 \beta_{1} - 507904 \beta_{2} + 1057849344 \beta_{3} - 58621952 \beta_{4} - 59129856 \beta_{5} - 6242304 \beta_{6} - 5226496 \beta_{7} - 2031616 \beta_{8} - 2588672 \beta_{9} ) q^{60} + ( 1029751292992 + 1029751252567 \beta_{1} + 1038947454 \beta_{2} + 1039991077 \beta_{3} + 1043623 \beta_{4} + 101475138 \beta_{5} - 2127671 \beta_{6} + 4174492 \beta_{7} - 5339390 \beta_{8} + 1124473 \beta_{9} ) q^{61} + ( 773141402752 + 510848 \beta_{1} + 3292160896 \beta_{2} - 877440 \beta_{3} - 121092224 \beta_{4} - 877440 \beta_{5} + 2265728 \beta_{6} + 327552 \beta_{7} - 3837312 \beta_{8} + 2082432 \beta_{9} ) q^{62} + ( -13017467519769 - 6501313785906 \beta_{1} - 2224834653 \beta_{2} + 4496682035 \beta_{3} + 7096886 \beta_{4} - 161521288 \beta_{5} - 999343 \beta_{6} - 15509692 \beta_{7} - 17930827 \beta_{8} - 6580378 \beta_{9} ) q^{63} + 4398046511104 q^{64} + ( 18260458528834 + 18260459658423 \beta_{1} - 7759855374 \beta_{2} - 7755603283 \beta_{3} + 4252091 \beta_{4} - 69931113 \beta_{5} - 7374593 \beta_{6} + 17008364 \beta_{7} - 17871688 \beta_{8} + 1992913 \beta_{9} ) q^{65} + ( 2350592 - 12353558256000 \beta_{1} + 1175296 \beta_{2} - 2491972608 \beta_{3} - 183840256 \beta_{4} - 182664960 \beta_{5} - 2070528 \beta_{6} - 4421120 \beta_{7} + 4701184 \beta_{8} + 485120 \beta_{9} ) q^{66} + ( 7662508 + 5853510838379 \beta_{1} + 3831254 \beta_{2} + 6141958011 \beta_{3} - 271943232 \beta_{4} - 268111978 \beta_{5} + 4853166 \beta_{6} - 2809342 \beta_{7} + 15325016 \beta_{8} + 5448976 \beta_{9} ) q^{67} + ( -10491356725248 - 10491364409344 \beta_{1} + 773292032 \beta_{2} + 760987648 \beta_{3} - 12304384 \beta_{4} - 29966336 \beta_{5} + 16924672 \beta_{6} - 49217536 \beta_{7} + 38469632 \beta_{8} + 3063808 \beta_{9} ) q^{68} + ( -27969464782250 - 2535192 \beta_{1} + 14165793902 \beta_{2} - 8319160 \beta_{3} + 327829695 \beta_{4} - 8319160 \beta_{5} + 14103128 \beta_{6} - 7962368 \beta_{7} - 25314272 \beta_{8} + 8675952 \beta_{9} ) q^{69} + ( -11922185738624 - 27509782670720 \beta_{1} - 3811602816 \beta_{2} + 2834869632 \beta_{3} - 38087168 \beta_{4} + 32985344 \beta_{5} + 10995712 \beta_{6} - 3633280 \beta_{7} + 20483456 \beta_{8} + 9009280 \beta_{9} ) q^{70} + ( -3776803189000 - 2956578 \beta_{1} - 6102589508 \beta_{2} - 23054 \beta_{3} - 185783222 \beta_{4} - 23054 \beta_{5} - 2910470 \beta_{6} - 4446394 \beta_{7} + 4354178 \beta_{8} - 4400286 \beta_{9} ) q^{71} + ( 19086314569728 + 19086320861184 \beta_{1} + 364904448 \beta_{2} + 375390208 \beta_{3} + 10485760 \beta_{4} + 148897792 \beta_{5} - 14680064 \beta_{6} + 41943040 \beta_{7} - 33554432 \beta_{8} - 2097152 \beta_{9} ) q^{72} + ( -19246324 - 48774034218161 \beta_{1} - 9623162 \beta_{2} - 13976989808 \beta_{3} + 460523798 \beta_{4} + 450900636 \beta_{5} - 660060 \beta_{6} + 18586264 \beta_{7} - 38492648 \beta_{8} - 9843182 \beta_{9} ) q^{73} + ( -22827264 + 35064023666432 \beta_{1} - 11413632 \beta_{2} - 4476824832 \beta_{3} + 166695168 \beta_{4} + 155281536 \beta_{5} + 9192960 \beta_{6} + 32020224 \beta_{7} - 45654528 \beta_{8} - 8349312 \beta_{9} ) q^{74} + ( 59161944809630 + 59161955768900 \beta_{1} + 2898714350 \beta_{2} + 2915021230 \beta_{3} + 16306880 \beta_{4} + 15406520 \beta_{5} - 21654490 \beta_{6} + 65227520 \beta_{7} - 48656590 \beta_{8} - 5611660 \beta_{9} ) q^{75} + ( 3484675768320 + 16252928 \beta_{1} + 3239051264 \beta_{2} + 17268736 \beta_{3} - 196378624 \beta_{4} + 17268736 \beta_{5} - 18284544 \beta_{6} + 33013760 \beta_{7} + 36061184 \beta_{8} - 1523712 \beta_{9} ) q^{76} + ( -18908085254352 - 45008105504117 \beta_{1} + 6193666073 \beta_{2} + 15343448659 \beta_{3} - 36703751 \beta_{4} + 246331148 \beta_{5} - 20296717 \beta_{6} + 116756682 \beta_{7} + 16892442 \beta_{8} + 14496692 \beta_{9} ) q^{77} + ( -26848450987904 - 5335296 \beta_{1} - 9992574336 \beta_{2} - 6792704 \beta_{3} + 1053609216 \beta_{4} - 6792704 \beta_{5} + 8250112 \beta_{6} - 11399296 \beta_{7} - 15771520 \beta_{8} + 2186112 \beta_{9} ) q^{78} + ( 27471374986204 + 27471359320981 \beta_{1} - 14227617465 \beta_{2} - 14276415224 \beta_{3} - 48797759 \beta_{4} - 48157152 \beta_{5} + 81930295 \beta_{6} - 195191036 \beta_{7} + 196993126 \beta_{8} - 17467313 \beta_{9} ) q^{79} + ( -588947390464 \beta_{1} + 536870912 \beta_{3} + 268435456 \beta_{4} + 268435456 \beta_{5} ) q^{80} + ( 35424722 + 70411172327571 \beta_{1} + 17712361 \beta_{2} + 24317179410 \beta_{3} + 1043264015 \beta_{4} + 1060976376 \beta_{5} - 17730690 \beta_{6} - 53155412 \beta_{7} + 70849444 \beta_{8} + 11802131 \beta_{9} ) q^{81} + ( 46525984951808 + 46526036580224 \beta_{1} + 6983861504 \beta_{2} + 7032608640 \beta_{3} + 48747136 \beta_{4} - 521830016 \beta_{5} - 45865856 \beta_{6} + 194988544 \beta_{7} - 88850432 \beta_{8} - 54509696 \beta_{9} ) q^{82} + ( -91460545784342 - 22551748 \beta_{1} - 22438500006 \beta_{2} + 41189376 \beta_{3} - 778553328 \beta_{4} + 41189376 \beta_{5} - 104930500 \beta_{6} - 13232934 \beta_{7} + 177990438 \beta_{8} - 95611686 \beta_{9} ) q^{83} + ( -37883711782912 + 15560677703680 \beta_{1} - 13658144768 \beta_{2} - 5237850112 \beta_{3} + 437944320 \beta_{4} + 919207936 \beta_{5} - 25444352 \beta_{6} - 69369856 \beta_{7} + 3768320 \beta_{8} - 70877184 \beta_{9} ) q^{84} + ( 26497758994086 - 44597875 \beta_{1} + 23704560867 \beta_{2} + 9515285 \beta_{3} - 1962276831 \beta_{4} + 9515285 \beta_{5} - 63628445 \beta_{6} - 62139170 \beta_{7} + 100200310 \beta_{8} - 81169740 \beta_{9} ) q^{85} + ( 5319672629504 + 5319723136512 \beta_{1} - 6192352768 \beta_{2} - 6179118848 \beta_{3} + 13233920 \beta_{4} + 103889920 \beta_{5} + 24039168 \beta_{6} + 52935680 \beta_{7} + 85351424 \beta_{8} - 87780096 \beta_{9} ) q^{86} + ( 48326260 - 266505780956484 \beta_{1} + 24163130 \beta_{2} + 33816290913 \beta_{3} - 691873646 \beta_{4} - 667710516 \beta_{5} - 107700627 \beta_{6} - 156026887 \beta_{7} + 96652520 \beta_{8} - 11737079 \beta_{9} ) q^{87} + ( 67108864 - 1428342964224 \beta_{1} + 33554432 \beta_{2} - 8648654848 \beta_{3} + 369098752 \beta_{4} + 402653184 \beta_{5} + 31457280 \beta_{6} - 35651584 \beta_{7} + 134217728 \beta_{8} + 44040192 \beta_{9} ) q^{88} + ( 63066577386191 + 63066497364977 \beta_{1} - 37583574124 \beta_{2} - 37701539538 \beta_{3} - 117965414 \beta_{4} + 425101600 \beta_{5} + 155909614 \beta_{6} - 471861656 \beta_{7} + 349763428 \beta_{8} + 42077014 \beta_{9} ) q^{89} + ( -194979780472832 - 26136960 \beta_{1} - 5758735744 \beta_{2} - 43869568 \beta_{3} + 2021098752 \beta_{4} - 43869568 \beta_{5} + 61602176 \beta_{6} - 61140224 \beta_{7} - 114338048 \beta_{8} + 26598912 \beta_{9} ) q^{90} + ( -354181326543779 - 180616416261976 \beta_{1} + 41049062101 \beta_{2} - 39919611484 \beta_{3} - 1322554540 \beta_{4} - 960773298 \beta_{5} + 33420814 \beta_{6} - 315422503 \beta_{7} + 73142475 \beta_{8} + 12693223 \beta_{9} ) q^{91} + ( 103810329280512 + 55590912 \beta_{1} - 17515249664 \beta_{2} + 74596352 \beta_{3} - 807452672 \beta_{4} + 74596352 \beta_{5} - 93601792 \beta_{6} + 120684544 \beta_{7} + 177700864 \beta_{8} - 28508160 \beta_{9} ) q^{92} + ( 592544164207714 + 592544144850499 \beta_{1} + 83219397562 \beta_{2} + 83378167719 \beta_{3} + 158770157 \beta_{4} - 1125876196 \beta_{5} - 336897529 \beta_{6} + 635080628 \beta_{7} - 851922430 \beta_{8} + 197484587 \beta_{9} ) q^{93} + ( -34734848 - 322738282913536 \beta_{1} - 17367424 \beta_{2} + 34148166400 \beta_{3} - 703601920 \beta_{4} - 720969344 \beta_{5} + 263452800 \beta_{6} + 298187648 \beta_{7} - 69469696 \beta_{8} + 70450176 \beta_{9} ) q^{94} + ( -154242414 + 269251168287766 \beta_{1} - 77121207 \beta_{2} - 24528256283 \beta_{3} + 952427564 \beta_{4} + 875306357 \beta_{5} - 117873702 \beta_{6} + 36368712 \beta_{7} - 308484828 \beta_{8} - 116412441 \beta_{9} ) q^{95} + ( -16458314678272 - 16458314678272 \beta_{1} + 34359738368 \beta_{2} + 34359738368 \beta_{3} ) q^{96} + ( 5832188922336 + 109678117 \beta_{1} - 97625807485 \beta_{2} - 113285199 \beta_{3} + 1266412796 \beta_{4} - 113285199 \beta_{5} + 336248515 \beta_{6} + 107874576 \beta_{7} - 561015372 \beta_{8} + 334444974 \beta_{9} ) q^{97} + ( -154119377751936 - 52419397256960 \beta_{1} - 75759051520 \beta_{2} - 99403176576 \beta_{3} + 687598464 \beta_{4} + 34292608 \beta_{5} + 44948096 \beta_{6} + 629467648 \beta_{7} - 359670784 \beta_{8} + 101097856 \beta_{9} ) q^{98} + ( 393045690928277 + 150767736 \beta_{1} + 148065798679 \beta_{2} - 114270650 \beta_{3} + 1860170994 \beta_{4} - 114270650 \beta_{5} + 379309036 \beta_{6} + 169016279 \beta_{7} - 626098879 \beta_{8} + 397557579 \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 640q^{2} + 2395q^{3} - 81920q^{4} + 10969q^{5} + 613120q^{6} + 2375276q^{7} - 20971520q^{8} - 45505414q^{9} + O(q^{10}) \) \( 10q + 640q^{2} + 2395q^{3} - 81920q^{4} + 10969q^{5} + 613120q^{6} + 2375276q^{7} - 20971520q^{8} - 45505414q^{9} - 1404032q^{10} - 3405409q^{11} + 39239680q^{12} + 369408276q^{13} + 411547520q^{14} + 453669790q^{15} - 1342177280q^{16} - 3201713287q^{17} + 5824692992q^{18} - 1063446451q^{19} - 359432192q^{20} + 21058786691q^{21} - 871784704q^{22} - 31680429991q^{23} - 5022679040q^{24} + 39012502244q^{25} + 23642129664q^{26} + 66427564750q^{27} + 13761560576q^{28} + 294935242276q^{29} + 29034866560q^{30} + 30201804075q^{31} + 171798691840q^{32} - 482562276049q^{33} - 819638601472q^{34} - 1540311474205q^{35} + 1491121405952q^{36} + 1369689837461q^{37} + 136121145728q^{38} - 1048775641458q^{39} - 23003660288q^{40} + 3634838274156q^{41} - 871978643968q^{42} + 415602663752q^{43} - 55794221056q^{44} - 7616412162898q^{45} + 4055095038848q^{46} - 12606978599835q^{47} - 1285805834240q^{48} - 8067928630838q^{49} + 9987200574464q^{50} + 6857137867217q^{51} - 3026192596992q^{52} + 10616682242361q^{53} + 4251364144000q^{54} - 40851384759514q^{55} - 4981314813952q^{56} + 44665526899814q^{57} + 18875855505664q^{58} + 39993786962755q^{59} - 3716462919680q^{60} + 5148862074165q^{61} + 7731661843200q^{62} - 97668254816810q^{63} + 43980465111040q^{64} + 91302240851010q^{65} + 61767971334272q^{66} - 29267240444285q^{67} - 52456870494208q^{68} - 279695242712578q^{69} + 18327124839296q^{70} - 37767657205600q^{71} + 95431769980928q^{72} + 243869659370917q^{73} - 175320299195008q^{74} + 295809815641460q^{75} + 34847013306368q^{76} + 35960055270559q^{77} - 268486564213248q^{78} + 137356615917609q^{79} + 2944468516864q^{80} - 352056899083285q^{81} + 232629649545984q^{82} - 914604166510648q^{83} - 456640427573248q^{84} + 264981492485402q^{85} + 26598570480128q^{86} + 1332529214868790q^{87} + 7141660295168q^{88} + 315332760149685q^{89} - 1949801513701888q^{90} - 2638730194011720q^{91} + 1038104329945088q^{92} + 2962720310885787q^{93} + 1613693260778880q^{94} - 1346257729549705q^{95} - 82291573391360q^{96} + 58320039716044q^{97} - 1279096378461056q^{98} + 3930453838067596q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - x^{9} + 7979102 x^{8} - 3342530557 x^{7} + 48610066963550 x^{6} - 20531463882664153 x^{5} + 122945882607722292337 x^{4} - 89586690298994071428648 x^{3} + 238785864099387585455090736 x^{2} - 108333121326673799746890760320 x + 51761574501460447109413728819456\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(\)\(74\!\cdots\!37\)\( \nu^{9} - \)\(10\!\cdots\!43\)\( \nu^{8} + \)\(58\!\cdots\!88\)\( \nu^{7} - \)\(24\!\cdots\!93\)\( \nu^{6} + \)\(36\!\cdots\!24\)\( \nu^{5} - \)\(15\!\cdots\!93\)\( \nu^{4} + \)\(89\!\cdots\!23\)\( \nu^{3} - \)\(36\!\cdots\!90\)\( \nu^{2} + \)\(17\!\cdots\!32\)\( \nu - \)\(78\!\cdots\!36\)\(\)\()/ \)\(79\!\cdots\!80\)\( \)
\(\beta_{2}\)\(=\)\((\)\(\)\(56\!\cdots\!43\)\( \nu^{9} + \)\(30\!\cdots\!77\)\( \nu^{8} - \)\(12\!\cdots\!62\)\( \nu^{7} + \)\(32\!\cdots\!21\)\( \nu^{6} - \)\(14\!\cdots\!26\)\( \nu^{5} + \)\(14\!\cdots\!97\)\( \nu^{4} - \)\(79\!\cdots\!45\)\( \nu^{3} + \)\(36\!\cdots\!00\)\( \nu^{2} - \)\(17\!\cdots\!28\)\( \nu + \)\(72\!\cdots\!80\)\(\)\()/ \)\(75\!\cdots\!96\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(20\!\cdots\!67\)\( \nu^{9} + \)\(77\!\cdots\!97\)\( \nu^{8} + \)\(35\!\cdots\!58\)\( \nu^{7} - \)\(14\!\cdots\!63\)\( \nu^{6} + \)\(24\!\cdots\!22\)\( \nu^{5} - \)\(13\!\cdots\!63\)\( \nu^{4} + \)\(11\!\cdots\!43\)\( \nu^{3} - \)\(76\!\cdots\!84\)\( \nu^{2} + \)\(28\!\cdots\!12\)\( \nu - \)\(13\!\cdots\!76\)\(\)\()/ \)\(26\!\cdots\!92\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(11\!\cdots\!47\)\( \nu^{9} - \)\(14\!\cdots\!71\)\( \nu^{8} + \)\(61\!\cdots\!26\)\( \nu^{7} - \)\(15\!\cdots\!91\)\( \nu^{6} + \)\(73\!\cdots\!98\)\( \nu^{5} - \)\(73\!\cdots\!31\)\( \nu^{4} - \)\(12\!\cdots\!93\)\( \nu^{3} - \)\(18\!\cdots\!00\)\( \nu^{2} + \)\(85\!\cdots\!44\)\( \nu - \)\(11\!\cdots\!60\)\(\)\()/ \)\(28\!\cdots\!36\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(51\!\cdots\!71\)\( \nu^{9} + \)\(32\!\cdots\!91\)\( \nu^{8} + \)\(45\!\cdots\!84\)\( \nu^{7} + \)\(73\!\cdots\!01\)\( \nu^{6} + \)\(24\!\cdots\!72\)\( \nu^{5} + \)\(34\!\cdots\!81\)\( \nu^{4} + \)\(57\!\cdots\!29\)\( \nu^{3} - \)\(13\!\cdots\!50\)\( \nu^{2} + \)\(23\!\cdots\!36\)\( \nu + \)\(34\!\cdots\!12\)\(\)\()/ \)\(49\!\cdots\!60\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(92\!\cdots\!01\)\( \nu^{9} + \)\(27\!\cdots\!01\)\( \nu^{8} + \)\(74\!\cdots\!64\)\( \nu^{7} + \)\(16\!\cdots\!31\)\( \nu^{6} + \)\(20\!\cdots\!92\)\( \nu^{5} + \)\(95\!\cdots\!31\)\( \nu^{4} + \)\(15\!\cdots\!59\)\( \nu^{3} + \)\(19\!\cdots\!50\)\( \nu^{2} - \)\(17\!\cdots\!84\)\( \nu + \)\(34\!\cdots\!52\)\(\)\()/ \)\(19\!\cdots\!40\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(11\!\cdots\!89\)\( \nu^{9} - \)\(19\!\cdots\!99\)\( \nu^{8} - \)\(10\!\cdots\!86\)\( \nu^{7} - \)\(12\!\cdots\!59\)\( \nu^{6} - \)\(61\!\cdots\!18\)\( \nu^{5} - \)\(67\!\cdots\!19\)\( \nu^{4} - \)\(16\!\cdots\!21\)\( \nu^{3} - \)\(10\!\cdots\!20\)\( \nu^{2} - \)\(23\!\cdots\!84\)\( \nu - \)\(16\!\cdots\!08\)\(\)\()/ \)\(19\!\cdots\!40\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(19\!\cdots\!77\)\( \nu^{9} - \)\(56\!\cdots\!53\)\( \nu^{8} + \)\(14\!\cdots\!78\)\( \nu^{7} - \)\(12\!\cdots\!53\)\( \nu^{6} + \)\(86\!\cdots\!34\)\( \nu^{5} - \)\(75\!\cdots\!73\)\( \nu^{4} + \)\(19\!\cdots\!33\)\( \nu^{3} - \)\(27\!\cdots\!80\)\( \nu^{2} + \)\(36\!\cdots\!12\)\( \nu - \)\(31\!\cdots\!16\)\(\)\()/ \)\(19\!\cdots\!40\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(64\!\cdots\!31\)\( \nu^{9} - \)\(17\!\cdots\!39\)\( \nu^{8} + \)\(40\!\cdots\!34\)\( \nu^{7} - \)\(36\!\cdots\!79\)\( \nu^{6} + \)\(27\!\cdots\!82\)\( \nu^{5} - \)\(18\!\cdots\!79\)\( \nu^{4} + \)\(61\!\cdots\!39\)\( \nu^{3} - \)\(70\!\cdots\!60\)\( \nu^{2} + \)\(14\!\cdots\!16\)\( \nu - \)\(60\!\cdots\!68\)\(\)\()/ \)\(39\!\cdots\!80\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{9} + 5 \beta_{8} - 4 \beta_{7} + 2 \beta_{6} - 61 \beta_{5} - \beta_{4} - 264 \beta_{3} - 263 \beta_{2} + 1189 \beta_{1} + 1189\)\()/5880\)
\(\nu^{2}\)\(=\)\((\)\(-2686 \beta_{9} - 12372 \beta_{8} + 7407 \beta_{7} + 1221 \beta_{6} + 15760 \beta_{5} + 18853 \beta_{4} + 1134479 \beta_{3} - 3093 \beta_{2} + 18766841004 \beta_{1} - 6186\)\()/5880\)
\(\nu^{3}\)\(=\)\((\)\(2680977 \beta_{9} - 1678795 \beta_{8} + 3683159 \beta_{7} + 1286227 \beta_{6} + 501091 \beta_{5} - 31771308 \beta_{4} + 501091 \beta_{3} + 212126208 \beta_{2} + 2288409 \beta_{1} + 840301192087\)\()/840\)
\(\nu^{4}\)\(=\)\((\)\(18574714003 \beta_{9} + 20945747373 \beta_{8} - 55780363860 \beta_{7} + 11630279446 \beta_{6} - 116249266621 \beta_{5} - 13945090965 \beta_{4} - 8807186946668 \beta_{3} - 8793241855703 \beta_{2} - 78903454223987387 \beta_{1} - 78903437964084903\)\()/5880\)
\(\nu^{5}\)\(=\)\((\)\(-52274739910538 \beta_{9} - 72390987719956 \beta_{8} - 66335485081669 \beta_{7} - 102530978941647 \beta_{6} + 897385376399340 \beta_{5} + 915483123329329 \beta_{4} + 9751630601408867 \beta_{3} - 18097746929989 \beta_{2} + 36008053026683207192 \beta_{1} - 36195493859978\)\()/5880\)
\(\nu^{6}\)\(=\)\((\)\(-779196830115927 \beta_{9} + 4399738542069957 \beta_{8} + 2841344881838103 \beta_{7} - 2329735409387633 \beta_{6} + 1810270855977015 \beta_{5} - 14828325235840640 \beta_{4} + 1810270855977015 \beta_{3} + 1534933924669049020 \beta_{2} + 1290806302566397 \beta_{1} + 10194870848420738340939\)\()/168\)
\(\nu^{7}\)\(=\)\((\)\(-262499890620543554245 \beta_{9} + 717963286962522721937 \beta_{8} - 370529658321951303508 \beta_{7} + 270198567181003515938 \beta_{6} - 3961630295581102948921 \beta_{5} - 92632414580487825877 \beta_{4} - 62111016622476095514000 \beta_{3} - 62018384207895607688123 \beta_{2} - 195549482566400865487949855 \beta_{1} - 195549567500138885515814039\)\()/5880\)
\(\nu^{8}\)\(=\)\((\)\(-274952936357936668506694 \beta_{9} - 1202244265630444965340884 \beta_{8} + 677946522964246201156023 \beta_{7} + 76824390149023718485581 \beta_{6} + 2500041019430567988433432 \beta_{5} + 2800602085838179229768653 \beta_{4} + 301364154216771449855988359 \beta_{3} - 300561066407611241335221 \beta_{2} + 1676812691440135987109387188308 \beta_{1} - 601122132815222482670442\)\()/5880\)
\(\nu^{9}\)\(=\)\((\)\(387275660358521096104237641 \beta_{9} - 252382035198018120424147555 \beta_{8} + 522169285519024071784327727 \beta_{7} + 190736960992095909562780051 \beta_{6} + 67446812580251487840045043 \beta_{5} - 2498501445777031436328688644 \beta_{4} + 67446812580251487840045043 \beta_{3} + 53544940203162646538421536904 \beta_{2} + 325630586152598885242870137 \beta_{1} + 149503633326218807079452993284711\)\()/840\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/14\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1 - \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
−1143.06 + 1979.84i
1086.67 1882.17i
259.792 449.973i
739.390 1280.66i
−942.293 + 1632.10i
−1143.06 1979.84i
1086.67 + 1882.17i
259.792 + 449.973i
739.390 + 1280.66i
−942.293 1632.10i
64.0000 110.851i −3401.15 5890.97i −8192.00 14189.0i −55365.7 + 95896.2i −870695. 550858. + 2.10811e6i −2.09715e6 −1.59612e7 + 2.76456e7i 7.08681e6 + 1.22747e7i
9.2 64.0000 110.851i −1325.39 2295.64i −8192.00 14189.0i 56213.5 97364.6i −339300. 875396. 1.99531e6i −2.09715e6 3.66114e6 6.34128e6i −7.19532e6 1.24627e7i
9.3 64.0000 110.851i 350.525 + 607.127i −8192.00 14189.0i −6451.18 + 11173.8i 89734.4 −117963. + 2.17569e6i −2.09715e6 6.92872e6 1.20009e7i 825752. + 1.43024e6i
9.4 64.0000 110.851i 2378.65 + 4119.95i −8192.00 14189.0i 110581. 191531.i 608935. −2.15329e6 333042.i −2.09715e6 −4.14154e6 + 7.17335e6i −1.41543e7 2.45160e7i
9.5 64.0000 110.851i 3194.86 + 5533.66i −8192.00 14189.0i −99492.8 + 172327.i 817884. 2.03263e6 784833.i −2.09715e6 −1.32398e7 + 2.29320e7i 1.27351e7 + 2.20578e7i
11.1 64.0000 + 110.851i −3401.15 + 5890.97i −8192.00 + 14189.0i −55365.7 95896.2i −870695. 550858. 2.10811e6i −2.09715e6 −1.59612e7 2.76456e7i 7.08681e6 1.22747e7i
11.2 64.0000 + 110.851i −1325.39 + 2295.64i −8192.00 + 14189.0i 56213.5 + 97364.6i −339300. 875396. + 1.99531e6i −2.09715e6 3.66114e6 + 6.34128e6i −7.19532e6 + 1.24627e7i
11.3 64.0000 + 110.851i 350.525 607.127i −8192.00 + 14189.0i −6451.18 11173.8i 89734.4 −117963. 2.17569e6i −2.09715e6 6.92872e6 + 1.20009e7i 825752. 1.43024e6i
11.4 64.0000 + 110.851i 2378.65 4119.95i −8192.00 + 14189.0i 110581. + 191531.i 608935. −2.15329e6 + 333042.i −2.09715e6 −4.14154e6 7.17335e6i −1.41543e7 + 2.45160e7i
11.5 64.0000 + 110.851i 3194.86 5533.66i −8192.00 + 14189.0i −99492.8 172327.i 817884. 2.03263e6 + 784833.i −2.09715e6 −1.32398e7 2.29320e7i 1.27351e7 2.20578e7i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 14.16.c.b 10
3.b odd 2 1 126.16.g.c 10
7.b odd 2 1 98.16.c.n 10
7.c even 3 1 inner 14.16.c.b 10
7.c even 3 1 98.16.a.h 5
7.d odd 6 1 98.16.a.i 5
7.d odd 6 1 98.16.c.n 10
21.h odd 6 1 126.16.g.c 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.16.c.b 10 1.a even 1 1 trivial
14.16.c.b 10 7.c even 3 1 inner
98.16.a.h 5 7.c even 3 1
98.16.a.i 5 7.d odd 6 1
98.16.c.n 10 7.b odd 2 1
98.16.c.n 10 7.d odd 6 1
126.16.g.c 10 3.b odd 2 1
126.16.g.c 10 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(29\!\cdots\!33\)\( T_{3}^{6} - \)\(50\!\cdots\!05\)\( T_{3}^{5} + \)\(42\!\cdots\!57\)\( T_{3}^{4} + \)\(18\!\cdots\!90\)\( T_{3}^{3} + \)\(28\!\cdots\!71\)\( T_{3}^{2} - \)\(18\!\cdots\!35\)\( T_{3} + \)\(14\!\cdots\!25\)\( \)">\(T_{3}^{10} - \cdots\) acting on \(S_{16}^{\mathrm{new}}(14, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 16384 - 128 T + T^{2} )^{5} \)
$3$ \( \)\(14\!\cdots\!25\)\( - \)\(18\!\cdots\!35\)\( T + \)\(28\!\cdots\!71\)\( T^{2} + \)\(18\!\cdots\!90\)\( T^{3} + \)\(42\!\cdots\!57\)\( T^{4} - 5089648660179754905 T^{5} + 2923699083469533 T^{6} - 119018122470 T^{7} + 61492987 T^{8} - 2395 T^{9} + T^{10} \)
$5$ \( \)\(49\!\cdots\!25\)\( + \)\(39\!\cdots\!75\)\( T + \)\(30\!\cdots\!75\)\( T^{2} + \)\(59\!\cdots\!50\)\( T^{3} + \)\(31\!\cdots\!25\)\( T^{4} + \)\(40\!\cdots\!25\)\( T^{5} + \)\(26\!\cdots\!25\)\( T^{6} + 1376760484456490 T^{7} + 56847853671 T^{8} - 10969 T^{9} + T^{10} \)
$7$ \( \)\(24\!\cdots\!43\)\( - \)\(12\!\cdots\!76\)\( T + \)\(73\!\cdots\!49\)\( T^{2} + \)\(32\!\cdots\!60\)\( T^{3} - \)\(17\!\cdots\!06\)\( T^{4} + \)\(11\!\cdots\!88\)\( T^{5} - \)\(36\!\cdots\!42\)\( T^{6} + 1460543381538253040 T^{7} + 6854932353507 T^{8} - 2375276 T^{9} + T^{10} \)
$11$ \( \)\(53\!\cdots\!25\)\( + \)\(31\!\cdots\!25\)\( T + \)\(21\!\cdots\!75\)\( T^{2} + \)\(21\!\cdots\!70\)\( T^{3} + \)\(12\!\cdots\!69\)\( T^{4} + \)\(10\!\cdots\!59\)\( T^{5} + \)\(51\!\cdots\!01\)\( T^{6} + \)\(19\!\cdots\!94\)\( T^{7} + 8067224900610939 T^{8} + 3405409 T^{9} + T^{10} \)
$13$ \( ( \)\(13\!\cdots\!00\)\( + \)\(38\!\cdots\!00\)\( T + \)\(87\!\cdots\!60\)\( T^{2} - 126907824737012904 T^{3} - 184704138 T^{4} + T^{5} )^{2} \)
$17$ \( \)\(10\!\cdots\!01\)\( - \)\(16\!\cdots\!01\)\( T + \)\(10\!\cdots\!91\)\( T^{2} + \)\(68\!\cdots\!38\)\( T^{3} + \)\(86\!\cdots\!13\)\( T^{4} + \)\(32\!\cdots\!85\)\( T^{5} + \)\(17\!\cdots\!45\)\( T^{6} + \)\(37\!\cdots\!58\)\( T^{7} + 18334030904695777095 T^{8} + 3201713287 T^{9} + T^{10} \)
$19$ \( \)\(21\!\cdots\!25\)\( + \)\(17\!\cdots\!55\)\( T + \)\(21\!\cdots\!71\)\( T^{2} + \)\(27\!\cdots\!18\)\( T^{3} + \)\(59\!\cdots\!77\)\( T^{4} + \)\(14\!\cdots\!13\)\( T^{5} + \)\(81\!\cdots\!13\)\( T^{6} + \)\(71\!\cdots\!74\)\( T^{7} + 30835900979958832323 T^{8} + 1063446451 T^{9} + T^{10} \)
$23$ \( \)\(13\!\cdots\!01\)\( + \)\(66\!\cdots\!43\)\( T + \)\(23\!\cdots\!95\)\( T^{2} + \)\(45\!\cdots\!02\)\( T^{3} + \)\(66\!\cdots\!45\)\( T^{4} + \)\(53\!\cdots\!65\)\( T^{5} + \)\(37\!\cdots\!93\)\( T^{6} + \)\(10\!\cdots\!02\)\( T^{7} + \)\(12\!\cdots\!71\)\( T^{8} + 31680429991 T^{9} + T^{10} \)
$29$ \( ( \)\(49\!\cdots\!36\)\( - \)\(10\!\cdots\!48\)\( T + \)\(33\!\cdots\!52\)\( T^{2} - \)\(18\!\cdots\!28\)\( T^{3} - 147467621138 T^{4} + T^{5} )^{2} \)
$31$ \( \)\(19\!\cdots\!25\)\( - \)\(62\!\cdots\!75\)\( T + \)\(22\!\cdots\!75\)\( T^{2} - \)\(17\!\cdots\!90\)\( T^{3} + \)\(44\!\cdots\!49\)\( T^{4} - \)\(16\!\cdots\!29\)\( T^{5} + \)\(81\!\cdots\!49\)\( T^{6} - \)\(10\!\cdots\!66\)\( T^{7} + \)\(93\!\cdots\!63\)\( T^{8} - 30201804075 T^{9} + T^{10} \)
$37$ \( \)\(31\!\cdots\!09\)\( - \)\(36\!\cdots\!57\)\( T + \)\(37\!\cdots\!11\)\( T^{2} - \)\(19\!\cdots\!06\)\( T^{3} + \)\(11\!\cdots\!77\)\( T^{4} - \)\(44\!\cdots\!15\)\( T^{5} + \)\(22\!\cdots\!45\)\( T^{6} - \)\(61\!\cdots\!86\)\( T^{7} + \)\(14\!\cdots\!95\)\( T^{8} - 1369689837461 T^{9} + T^{10} \)
$41$ \( ( -\)\(90\!\cdots\!28\)\( - \)\(12\!\cdots\!24\)\( T + \)\(61\!\cdots\!44\)\( T^{2} - \)\(28\!\cdots\!48\)\( T^{3} - 1817419137078 T^{4} + T^{5} )^{2} \)
$43$ \( ( \)\(28\!\cdots\!00\)\( + \)\(45\!\cdots\!00\)\( T - \)\(19\!\cdots\!00\)\( T^{2} - \)\(49\!\cdots\!40\)\( T^{3} - 207801331876 T^{4} + T^{5} )^{2} \)
$47$ \( \)\(23\!\cdots\!61\)\( + \)\(17\!\cdots\!75\)\( T + \)\(93\!\cdots\!35\)\( T^{2} + \)\(31\!\cdots\!90\)\( T^{3} + \)\(91\!\cdots\!65\)\( T^{4} + \)\(20\!\cdots\!81\)\( T^{5} + \)\(45\!\cdots\!25\)\( T^{6} + \)\(78\!\cdots\!70\)\( T^{7} + \)\(12\!\cdots\!95\)\( T^{8} + 12606978599835 T^{9} + T^{10} \)
$53$ \( \)\(14\!\cdots\!49\)\( - \)\(15\!\cdots\!17\)\( T + \)\(67\!\cdots\!51\)\( T^{2} - \)\(43\!\cdots\!06\)\( T^{3} + \)\(18\!\cdots\!77\)\( T^{4} - \)\(11\!\cdots\!55\)\( T^{5} + \)\(25\!\cdots\!45\)\( T^{6} - \)\(12\!\cdots\!66\)\( T^{7} + \)\(24\!\cdots\!55\)\( T^{8} - 10616682242361 T^{9} + T^{10} \)
$59$ \( \)\(29\!\cdots\!25\)\( + \)\(30\!\cdots\!25\)\( T + \)\(41\!\cdots\!75\)\( T^{2} - \)\(96\!\cdots\!10\)\( T^{3} + \)\(38\!\cdots\!41\)\( T^{4} - \)\(29\!\cdots\!17\)\( T^{5} + \)\(88\!\cdots\!89\)\( T^{6} - \)\(72\!\cdots\!02\)\( T^{7} + \)\(15\!\cdots\!07\)\( T^{8} - 39993786962755 T^{9} + T^{10} \)
$61$ \( \)\(34\!\cdots\!61\)\( + \)\(21\!\cdots\!75\)\( T + \)\(12\!\cdots\!35\)\( T^{2} + \)\(34\!\cdots\!90\)\( T^{3} + \)\(81\!\cdots\!65\)\( T^{4} + \)\(81\!\cdots\!81\)\( T^{5} + \)\(39\!\cdots\!25\)\( T^{6} + \)\(21\!\cdots\!70\)\( T^{7} + \)\(73\!\cdots\!95\)\( T^{8} - 5148862074165 T^{9} + T^{10} \)
$67$ \( \)\(11\!\cdots\!81\)\( + \)\(19\!\cdots\!97\)\( T + \)\(24\!\cdots\!55\)\( T^{2} + \)\(16\!\cdots\!34\)\( T^{3} + \)\(83\!\cdots\!97\)\( T^{4} + \)\(24\!\cdots\!19\)\( T^{5} + \)\(55\!\cdots\!17\)\( T^{6} + \)\(50\!\cdots\!22\)\( T^{7} + \)\(71\!\cdots\!43\)\( T^{8} + 29267240444285 T^{9} + T^{10} \)
$71$ \( ( -\)\(30\!\cdots\!76\)\( - \)\(22\!\cdots\!28\)\( T - \)\(18\!\cdots\!44\)\( T^{2} - \)\(40\!\cdots\!28\)\( T^{3} + 18883828602800 T^{4} + T^{5} )^{2} \)
$73$ \( \)\(13\!\cdots\!25\)\( - \)\(48\!\cdots\!25\)\( T + \)\(12\!\cdots\!75\)\( T^{2} - \)\(18\!\cdots\!70\)\( T^{3} + \)\(23\!\cdots\!69\)\( T^{4} - \)\(19\!\cdots\!67\)\( T^{5} + \)\(15\!\cdots\!37\)\( T^{6} - \)\(89\!\cdots\!18\)\( T^{7} + \)\(60\!\cdots\!23\)\( T^{8} - 243869659370917 T^{9} + T^{10} \)
$79$ \( \)\(13\!\cdots\!25\)\( - \)\(87\!\cdots\!25\)\( T + \)\(21\!\cdots\!75\)\( T^{2} - \)\(93\!\cdots\!50\)\( T^{3} + \)\(22\!\cdots\!25\)\( T^{4} - \)\(10\!\cdots\!75\)\( T^{5} + \)\(84\!\cdots\!25\)\( T^{6} - \)\(14\!\cdots\!70\)\( T^{7} + \)\(10\!\cdots\!51\)\( T^{8} - 137356615917609 T^{9} + T^{10} \)
$83$ \( ( \)\(36\!\cdots\!32\)\( - \)\(38\!\cdots\!20\)\( T - \)\(51\!\cdots\!44\)\( T^{2} - \)\(69\!\cdots\!44\)\( T^{3} + 457302083255324 T^{4} + T^{5} )^{2} \)
$89$ \( \)\(19\!\cdots\!29\)\( - \)\(65\!\cdots\!61\)\( T + \)\(24\!\cdots\!15\)\( T^{2} + \)\(45\!\cdots\!62\)\( T^{3} + \)\(73\!\cdots\!53\)\( T^{4} - \)\(81\!\cdots\!47\)\( T^{5} + \)\(11\!\cdots\!77\)\( T^{6} + \)\(43\!\cdots\!86\)\( T^{7} + \)\(43\!\cdots\!67\)\( T^{8} - 315332760149685 T^{9} + T^{10} \)
$97$ \( ( \)\(22\!\cdots\!00\)\( + \)\(44\!\cdots\!80\)\( T + \)\(17\!\cdots\!16\)\( T^{2} - \)\(15\!\cdots\!60\)\( T^{3} - 29160019858022 T^{4} + T^{5} )^{2} \)
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