Properties

Label 14.16.c.a
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: \(x^{10} - 2 x^{9} + 11041039 x^{8} + 10251836788 x^{7} + 100086056086567 x^{6} + 67517626050179350 x^{5} + 266968630608932668831 x^{4} + 130403899669863282233290 x^{3} + 531639154391012701500584575 x^{2} + 238948900825518555440992768500 x + 120052637231856930389048520650625\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{3}\cdot 5^{2}\cdot 7^{7} \)
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.

\(f(q)\) \(=\) \( q + ( -128 - 128 \beta_{1} ) q^{2} + ( -396 \beta_{1} - \beta_{2} + \beta_{3} ) q^{3} + 16384 \beta_{1} q^{4} + ( 60192 + 60192 \beta_{1} + 7 \beta_{2} - \beta_{4} ) q^{5} + ( -50688 - 128 \beta_{3} ) q^{6} + ( 116059 - 578688 \beta_{1} + 32 \beta_{2} - 158 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} ) q^{7} + 2097152 q^{8} + ( -3473847 - 3473842 \beta_{1} - 2194 \beta_{2} + 4 \beta_{3} + 42 \beta_{4} - 5 \beta_{5} - 10 \beta_{6} - 4 \beta_{7} - 3 \beta_{8} - 2 \beta_{9} ) q^{9} +O(q^{10})\) \( q + ( -128 - 128 \beta_{1} ) q^{2} + ( -396 \beta_{1} - \beta_{2} + \beta_{3} ) q^{3} + 16384 \beta_{1} q^{4} + ( 60192 + 60192 \beta_{1} + 7 \beta_{2} - \beta_{4} ) q^{5} + ( -50688 - 128 \beta_{3} ) q^{6} + ( 116059 - 578688 \beta_{1} + 32 \beta_{2} - 158 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} ) q^{7} + 2097152 q^{8} + ( -3473847 - 3473842 \beta_{1} - 2194 \beta_{2} + 4 \beta_{3} + 42 \beta_{4} - 5 \beta_{5} - 10 \beta_{6} - 4 \beta_{7} - 3 \beta_{8} - 2 \beta_{9} ) q^{9} + ( -7704576 \beta_{1} - 896 \beta_{2} + 896 \beta_{3} + 128 \beta_{5} ) q^{10} + ( 30 - 2838243 \beta_{1} - 1443 \beta_{2} + 1503 \beta_{3} - 30 \beta_{4} - 142 \beta_{5} - 17 \beta_{6} - 60 \beta_{7} + 13 \beta_{8} + 39 \beta_{9} ) q^{11} + ( 6488064 + 6488064 \beta_{1} + 16384 \beta_{2} ) q^{12} + ( -77266326 - 15 \beta_{1} - 30144 \beta_{3} - 641 \beta_{4} + 656 \beta_{5} - 94 \beta_{6} + 124 \beta_{7} - 30 \beta_{8} + 15 \beta_{9} ) q^{13} + ( -88927744 - 14855424 \beta_{1} + 16384 \beta_{2} + 3968 \beta_{3} + 256 \beta_{4} - 640 \beta_{5} - 128 \beta_{6} + 128 \beta_{7} - 128 \beta_{8} - 128 \beta_{9} ) q^{14} + ( 142711377 - 92 \beta_{1} + 167887 \beta_{3} + 1155 \beta_{4} - 1063 \beta_{5} + 453 \beta_{6} - 269 \beta_{7} - 184 \beta_{8} + 92 \beta_{9} ) q^{15} + ( -268435456 - 268435456 \beta_{1} ) q^{16} + ( -375 + 29093100 \beta_{1} - 96750 \beta_{2} + 96000 \beta_{3} + 375 \beta_{4} + 6749 \beta_{5} + 334 \beta_{6} + 750 \beta_{7} - 41 \beta_{8} - 123 \beta_{9} ) q^{17} + ( -640 + 444651648 \beta_{1} + 280832 \beta_{2} - 282112 \beta_{3} + 640 \beta_{4} - 5248 \beta_{5} + 768 \beta_{6} + 1280 \beta_{7} + 128 \beta_{8} + 384 \beta_{9} ) q^{18} + ( 709937924 + 709938932 \beta_{1} - 443017 \beta_{2} + 1981 \beta_{3} - 4543 \beta_{4} - 1008 \beta_{5} - 2016 \beta_{6} - 1981 \beta_{7} + 2919 \beta_{8} + 1946 \beta_{9} ) q^{19} + ( -986185728 - 114688 \beta_{3} + 16384 \beta_{4} - 16384 \beta_{5} ) q^{20} + ( 343487471 + 2552587498 \beta_{1} - 1367203 \beta_{2} - 83431 \beta_{3} + 17800 \beta_{4} - 17660 \beta_{5} - 2530 \beta_{6} + 2046 \beta_{7} + 3003 \beta_{8} + 1388 \beta_{9} ) q^{21} + ( -363301120 + 1664 \beta_{1} - 186880 \beta_{3} - 16000 \beta_{4} + 14336 \beta_{5} - 5504 \beta_{6} + 2176 \beta_{7} + 3328 \beta_{8} - 1664 \beta_{9} ) q^{22} + ( 1973844450 + 1973836463 \beta_{1} - 294300 \beta_{2} - 4490 \beta_{3} + 44599 \beta_{4} + 7987 \beta_{5} + 15974 \beta_{6} + 4490 \beta_{7} + 10491 \beta_{8} + 6994 \beta_{9} ) q^{23} + ( -830472192 \beta_{1} - 2097152 \beta_{2} + 2097152 \beta_{3} ) q^{24} + ( -12138 + 10367508466 \beta_{1} - 1922776 \beta_{2} + 1898500 \beta_{3} + 12138 \beta_{4} - 120538 \beta_{5} + 15392 \beta_{6} + 24276 \beta_{7} + 3254 \beta_{8} + 9762 \beta_{9} ) q^{25} + ( 9890075776 + 9890061824 \beta_{1} + 3842560 \beta_{2} - 12032 \beta_{3} + 96000 \beta_{4} + 13952 \beta_{5} + 27904 \beta_{6} + 12032 \beta_{7} + 5760 \beta_{8} + 3840 \beta_{9} ) q^{26} + ( -34295002066 + 5587 \beta_{1} - 1816664 \beta_{3} + 127921 \beta_{4} - 133508 \beta_{5} - 33317 \beta_{6} + 22143 \beta_{7} + 11174 \beta_{8} - 5587 \beta_{9} ) q^{27} + ( 9481240576 + 11382718464 \beta_{1} - 2621440 \beta_{2} + 2080768 \beta_{3} - 65536 \beta_{4} + 65536 \beta_{5} + 32768 \beta_{6} + 16384 \beta_{7} + 16384 \beta_{9} ) q^{28} + ( -16522003972 + 1225 \beta_{1} + 18049966 \beta_{3} + 22373 \beta_{4} - 23598 \beta_{5} - 31394 \beta_{6} + 28944 \beta_{7} + 2450 \beta_{8} - 1225 \beta_{9} ) q^{29} + ( -18267010048 - 18266963840 \beta_{1} - 21455104 \beta_{2} + 57984 \beta_{3} - 194048 \beta_{4} - 46208 \beta_{5} - 92416 \beta_{6} - 57984 \beta_{7} + 35328 \beta_{8} + 23552 \beta_{9} ) q^{30} + ( -37285 + 21368844101 \beta_{1} + 18953688 \beta_{2} - 19028258 \beta_{3} + 37285 \beta_{4} - 9644 \beta_{5} + 72515 \beta_{6} + 74570 \beta_{7} + 35230 \beta_{8} + 105690 \beta_{9} ) q^{31} + 34359738368 \beta_{1} q^{32} + ( -27966754345 - 27966719459 \beta_{1} - 7664962 \beta_{2} + 46524 \beta_{3} + 1541542 \beta_{4} - 34886 \beta_{5} - 69772 \beta_{6} - 46524 \beta_{7} + 34914 \beta_{8} + 23276 \beta_{9} ) q^{33} + ( 3724007552 - 5248 \beta_{1} - 12341248 \beta_{3} + 821120 \beta_{4} - 815872 \beta_{5} + 53248 \beta_{6} - 42752 \beta_{7} - 10496 \beta_{8} + 5248 \beta_{9} ) q^{34} + ( 97757717774 - 99664244369 \beta_{1} - 40539699 \beta_{2} + 38304694 \beta_{3} - 2062162 \beta_{4} + 1278616 \beta_{5} - 148545 \beta_{6} - 103434 \beta_{7} + 68245 \beta_{8} + 5567 \beta_{9} ) q^{35} + ( 56915591168 + 16384 \beta_{1} + 36044800 \beta_{3} - 770048 \beta_{4} + 753664 \beta_{5} + 65536 \beta_{6} - 98304 \beta_{7} + 32768 \beta_{8} - 16384 \beta_{9} ) q^{36} + ( -119375667475 - 119375697432 \beta_{1} - 78905903 \beta_{2} - 70230 \beta_{3} + 1326647 \beta_{4} + 29957 \beta_{5} + 59914 \beta_{6} + 70230 \beta_{7} - 120819 \beta_{8} - 80546 \beta_{9} ) q^{37} + ( -129024 - 90872058752 \beta_{1} + 56706176 \beta_{2} - 56964224 \beta_{3} + 129024 \beta_{4} + 456960 \beta_{5} + 4480 \beta_{6} + 258048 \beta_{7} - 124544 \beta_{8} - 373632 \beta_{9} ) q^{38} + ( 6118 + 565590967365 \beta_{1} + 159354926 \beta_{2} - 159342690 \beta_{3} - 6118 \beta_{4} - 4677116 \beta_{5} + 16777 \beta_{6} - 12236 \beta_{7} + 22895 \beta_{8} + 68685 \beta_{9} ) q^{39} + ( 126231773184 + 126231773184 \beta_{1} + 14680064 \beta_{2} - 2097152 \beta_{4} ) q^{40} + ( 175739858228 - 29647 \beta_{1} + 207985008 \beta_{3} - 1387139 \beta_{4} + 1416786 \beta_{5} + 256852 \beta_{6} - 197558 \beta_{7} - 59294 \beta_{8} + 29647 \beta_{9} ) q^{41} + ( 282764479616 - 43966982016 \beta_{1} + 185419264 \beta_{2} - 175325824 \beta_{3} - 1936640 \beta_{4} + 243968 \beta_{5} + 585728 \beta_{6} + 323840 \beta_{7} - 206720 \beta_{8} - 384384 \beta_{9} ) q^{42} + ( -530514794174 - 233986 \beta_{1} - 16019430 \beta_{3} - 4941572 \beta_{4} + 5175558 \beta_{5} + 267568 \beta_{6} + 200404 \beta_{7} - 467972 \beta_{8} + 233986 \beta_{9} ) q^{43} + ( 46502051840 + 46501560320 \beta_{1} + 23642112 \beta_{2} - 704512 \beta_{3} + 2539520 \beta_{4} + 491520 \beta_{5} + 983040 \beta_{6} + 704512 \beta_{7} - 638976 \beta_{8} - 425984 \beta_{9} ) q^{44} + ( 1082121 - 2167355633283 \beta_{1} - 442676694 \beta_{2} + 444840936 \beta_{3} - 1082121 \beta_{4} + 7521009 \beta_{5} - 1080444 \beta_{6} - 2164242 \beta_{7} + 1677 \beta_{8} + 5031 \beta_{9} ) q^{45} + ( 1022336 - 252650619648 \beta_{1} + 37670400 \beta_{2} - 35625728 \beta_{3} - 1022336 \beta_{4} - 6156288 \beta_{5} - 1469952 \beta_{6} - 2044672 \beta_{7} - 447616 \beta_{8} - 1342848 \beta_{9} ) q^{46} + ( 856943366208 + 856943867335 \beta_{1} + 324478388 \beta_{2} + 60077 \beta_{3} + 9519606 \beta_{4} - 501127 \beta_{5} - 1002254 \beta_{6} - 60077 \beta_{7} - 1323150 \beta_{8} - 882100 \beta_{9} ) q^{47} + ( -106300440576 - 268435456 \beta_{3} ) q^{48} + ( -277689268213 - 872715997153 \beta_{1} + 696378914 \beta_{2} - 1054091514 \beta_{3} + 2962337 \beta_{4} - 713440 \beta_{5} - 1363348 \beta_{6} - 626374 \beta_{7} - 1200094 \beta_{8} + 453859 \beta_{9} ) q^{49} + ( 1327044607488 + 416512 \beta_{1} - 244145152 \beta_{3} - 17399040 \beta_{4} + 16982528 \beta_{5} + 1137152 \beta_{6} - 1970176 \beta_{7} + 833024 \beta_{8} - 416512 \beta_{9} ) q^{50} + ( -1659772160044 - 1659770057906 \beta_{1} - 503526535 \beta_{2} + 2365041 \beta_{3} - 18780437 \beta_{4} - 2102138 \beta_{5} - 4204276 \beta_{6} - 2365041 \beta_{7} + 788709 \beta_{8} + 525806 \beta_{9} ) q^{51} + ( 1785856 - 1265927667712 \beta_{1} - 491847680 \beta_{2} + 495419392 \beta_{3} - 1785856 \beta_{4} - 12533760 \beta_{5} - 2031616 \beta_{6} - 3571712 \beta_{7} - 245760 \beta_{8} - 737280 \beta_{9} ) q^{52} + ( -1324409 + 807735827337 \beta_{1} + 891111091 \beta_{2} - 893759909 \beta_{3} + 1324409 \beta_{4} + 30320920 \beta_{5} + 2405764 \beta_{6} + 2648818 \beta_{7} + 1081355 \beta_{8} + 3244065 \beta_{9} ) q^{53} + ( 4389756715008 + 4389753165568 \beta_{1} + 229698688 \beta_{2} - 4264576 \beta_{3} - 12824448 \beta_{4} + 3549440 \beta_{5} + 7098880 \beta_{6} + 4264576 \beta_{7} - 2145408 \beta_{8} - 1430272 \beta_{9} ) q^{54} + ( -4722595285775 - 979251 \beta_{1} + 3470450470 \beta_{3} + 6626748 \beta_{4} - 5647497 \beta_{5} - 919786 \beta_{6} + 2878288 \beta_{7} - 1958502 \beta_{8} + 979251 \beta_{9} ) q^{55} + ( 243393363968 - 1213596696576 \beta_{1} + 67108864 \beta_{2} - 331350016 \beta_{3} + 4194304 \beta_{4} + 2097152 \beta_{5} - 2097152 \beta_{6} - 4194304 \beta_{7} + 2097152 \beta_{8} ) q^{56} + ( -7597210874781 + 251082 \beta_{1} + 95534332 \beta_{3} + 91853634 \beta_{4} - 92104716 \beta_{5} - 5300124 \beta_{6} + 4797960 \beta_{7} + 502164 \beta_{8} - 251082 \beta_{9} ) q^{57} + ( 2114812646784 + 2114808785152 \beta_{1} - 2314100480 \beta_{2} - 4018432 \beta_{3} + 997888 \beta_{4} + 3861632 \beta_{5} + 7723264 \beta_{6} + 4018432 \beta_{7} - 470400 \beta_{8} - 313600 \beta_{9} ) q^{58} + ( -11017294 - 1216968067188 \beta_{1} - 2833557183 \beta_{2} + 2811522595 \beta_{3} + 11017294 \beta_{4} + 6621264 \beta_{5} + 13102596 \beta_{6} + 22034588 \beta_{7} + 2085302 \beta_{8} + 6255906 \beta_{9} ) q^{59} + ( -5914624 + 2338172878848 \beta_{1} + 2746253312 \beta_{2} - 2758082560 \beta_{3} + 5914624 \beta_{4} + 23330816 \beta_{5} + 4407296 \beta_{6} + 11829248 \beta_{7} - 1507328 \beta_{8} - 4521984 \beta_{9} ) q^{60} + ( 5896858308305 + 5896864836550 \beta_{1} - 53917909 \beta_{2} + 12748250 \beta_{3} - 40898319 \beta_{4} - 6528245 \beta_{5} - 13056490 \beta_{6} - 12748250 \beta_{7} + 18660015 \beta_{8} + 12440010 \beta_{9} ) q^{61} + ( 2735226099328 + 4509440 \beta_{1} + 2435353984 \beta_{3} - 10516352 \beta_{4} + 6006912 \beta_{5} + 263040 \beta_{6} - 9281920 \beta_{7} + 9018880 \beta_{8} - 4509440 \beta_{9} ) q^{62} + ( -13184249972492 + 3946791898038 \beta_{1} + 4120132306 \beta_{2} - 5608010351 \beta_{3} + 70499495 \beta_{4} - 75171800 \beta_{5} + 12399984 \beta_{6} + 15277981 \beta_{7} + 8015695 \beta_{8} + 4416456 \beta_{9} ) q^{63} + 4398046511104 q^{64} + ( 15605134208171 + 15605135272160 \beta_{1} - 13071246752 \beta_{2} + 6119612 \beta_{3} + 63082576 \beta_{4} - 1063989 \beta_{5} - 2127978 \beta_{6} - 6119612 \beta_{7} + 15166869 \beta_{8} + 10111246 \beta_{9} ) q^{65} + ( -4465408 + 3579741580416 \beta_{1} + 981115136 \beta_{2} - 990045952 \beta_{3} + 4465408 \beta_{4} - 198807040 \beta_{5} + 2975744 \beta_{6} + 8930816 \beta_{7} - 1489664 \beta_{8} - 4468992 \beta_{9} ) q^{66} + ( 9238370 + 569859942962 \beta_{1} - 1923621067 \beta_{2} + 1942097807 \beta_{3} - 9238370 \beta_{4} + 126273972 \beta_{5} - 4750478 \beta_{6} - 18476740 \beta_{7} + 4487892 \beta_{8} + 13463676 \beta_{9} ) q^{67} + ( -476666822656 - 476660678656 \beta_{1} + 1585152000 \beta_{2} + 6815744 \beta_{3} - 111247360 \beta_{4} - 6144000 \beta_{5} - 12288000 \beta_{6} - 6815744 \beta_{7} + 2015232 \beta_{8} + 1343488 \beta_{9} ) q^{68} + ( -4169356166988 - 997720 \beta_{1} + 5922304853 \beta_{3} - 115971687 \beta_{4} + 116969407 \beta_{5} - 16090272 \beta_{6} + 18085712 \beta_{7} - 1995440 \beta_{8} + 997720 \beta_{9} ) q^{69} + ( -25270030168064 - 12512993649280 \beta_{1} + 299320192 \beta_{2} - 5208095232 \beta_{3} + 182676608 \beta_{4} + 87054336 \beta_{5} + 5774208 \beta_{6} + 19013760 \beta_{7} - 8022784 \beta_{8} - 8735360 \beta_{9} ) q^{70} + ( 19018434989446 - 3149988 \beta_{1} + 7305688056 \beta_{3} + 144464986 \beta_{4} - 141314998 \beta_{5} + 3864074 \beta_{6} + 2435902 \beta_{7} - 6299976 \beta_{8} + 3149988 \beta_{9} ) q^{71} + ( -7285185183744 - 7285174697984 \beta_{1} - 4601151488 \beta_{2} + 8388608 \beta_{3} + 88080384 \beta_{4} - 10485760 \beta_{5} - 20971520 \beta_{6} - 8388608 \beta_{7} - 6291456 \beta_{8} - 4194304 \beta_{9} ) q^{72} + ( 33955558 - 56840775736135 \beta_{1} - 1197039064 \beta_{2} + 1264950180 \beta_{3} - 33955558 \beta_{4} + 305810774 \beta_{5} - 33578020 \beta_{6} - 67911116 \beta_{7} + 377538 \beta_{8} + 1132614 \beta_{9} ) q^{73} + ( 3834496 + 15280084116352 \beta_{1} + 10099955584 \beta_{2} - 10092286592 \beta_{3} - 3834496 \beta_{4} - 164655872 \beta_{5} + 1320448 \beta_{6} - 7668992 \beta_{7} + 5154944 \beta_{8} + 15464832 \beta_{9} ) q^{74} + ( -30364265324932 - 30364300914440 \beta_{1} + 30662153724 \beta_{2} - 58271994 \beta_{3} + 131021458 \beta_{4} + 35589508 \beta_{5} + 71179016 \beta_{6} + 58271994 \beta_{7} - 68047458 \beta_{8} - 45364972 \beta_{9} ) q^{75} + ( -11631606431744 - 15941632 \beta_{1} + 7258963968 \beta_{3} + 57917440 \beta_{4} - 41975808 \beta_{5} + 32456704 \beta_{6} - 573440 \beta_{7} - 31883264 \beta_{8} + 15941632 \beta_{9} ) q^{76} + ( -70040368405692 - 8743569427897 \beta_{1} - 8185111033 \beta_{2} - 11106683169 \beta_{3} - 450475497 \beta_{4} - 247678519 \beta_{5} - 57754230 \beta_{6} - 88742200 \beta_{7} - 17933006 \beta_{8} + 1058413 \beta_{9} ) q^{77} + ( 72395645187072 + 2930560 \beta_{1} + 20399577984 \beta_{3} - 600818304 \beta_{4} + 597887744 \beta_{5} - 3713664 \beta_{6} - 2147456 \beta_{7} + 5861120 \beta_{8} - 2930560 \beta_{9} ) q^{78} + ( 107305337695114 + 107305319782829 \beta_{1} - 2617101050 \beta_{2} - 46477042 \beta_{3} - 640246751 \beta_{4} + 17912285 \beta_{5} + 35824570 \beta_{6} + 46477042 \beta_{7} - 85694271 \beta_{8} - 57129514 \beta_{9} ) q^{79} + ( -16157666967552 \beta_{1} - 1879048192 \beta_{2} + 1879048192 \beta_{3} + 268435456 \beta_{5} ) q^{80} + ( -21941015 - 4597694355624 \beta_{1} + 35090343748 \beta_{2} - 35134225778 \beta_{3} + 21941015 \beta_{4} - 211892801 \beta_{5} - 24544254 \beta_{6} + 43882030 \beta_{7} - 46485269 \beta_{8} - 139455807 \beta_{9} ) q^{81} + ( -22494672770944 - 22494643688704 \beta_{1} - 26596793600 \beta_{2} + 32877056 \beta_{3} + 148471552 \beta_{4} - 29082240 \beta_{5} - 58164480 \beta_{6} - 32877056 \beta_{7} + 11384448 \beta_{8} + 7589632 \beta_{9} ) q^{82} + ( -94965386519904 + 13727854 \beta_{1} - 6137357146 \beta_{3} + 89787170 \beta_{4} - 103515024 \beta_{5} + 50788542 \beta_{6} - 78244250 \beta_{7} + 27455708 \beta_{8} - 13727854 \beta_{9} ) q^{83} + ( -41821552115712 - 36193819869184 \beta_{1} - 1333411840 \beta_{2} + 23808638976 \beta_{3} - 43745280 \beta_{4} + 258113536 \beta_{5} - 33521664 \beta_{6} - 74973184 \beta_{7} - 22740992 \beta_{8} + 26460160 \beta_{9} ) q^{84} + ( 247135340365252 + 17647035 \beta_{1} - 32012416553 \beta_{3} - 183247116 \beta_{4} + 165600081 \beta_{5} + 92575550 \beta_{6} - 127869620 \beta_{7} + 35294070 \beta_{8} - 17647035 \beta_{9} ) q^{85} + ( 67905897952768 + 67905902251264 \beta_{1} + 2024835328 \beta_{2} + 34248704 \beta_{3} + 628222720 \beta_{4} - 4298496 \beta_{5} - 8596992 \beta_{6} - 34248704 \beta_{7} + 89850624 \beta_{8} + 59900416 \beta_{9} ) q^{86} + ( 51580904 - 312405521149425 \beta_{1} + 7486111832 \beta_{2} - 7382950024 \beta_{3} - 51580904 \beta_{4} - 69642892 \beta_{5} - 93303565 \beta_{6} - 103161808 \beta_{7} - 41722661 \beta_{8} - 125167983 \beta_{9} ) q^{87} + ( 62914560 - 5952226983936 \beta_{1} - 3026190336 \beta_{2} + 3152019456 \beta_{3} - 62914560 \beta_{4} - 297795584 \beta_{5} - 35651584 \beta_{6} - 125829120 \beta_{7} + 27262976 \beta_{8} + 81788928 \beta_{9} ) q^{88} + ( 32018311454821 + 32018326290883 \beta_{1} - 56500283290 \beta_{2} + 25415716 \beta_{3} + 1067792642 \beta_{4} - 14836062 \beta_{5} - 29672124 \beta_{6} - 25415716 \beta_{7} + 31738962 \beta_{8} + 21159308 \beta_{9} ) q^{89} + ( -277421797868544 + 214656 \beta_{1} - 56800913664 \beta_{3} + 1100985984 \beta_{4} - 1101200640 \beta_{5} - 138726144 \beta_{6} + 138296832 \beta_{7} + 429312 \beta_{8} - 214656 \beta_{9} ) q^{90} + ( -70233141341216 + 120932674812131 \beta_{1} + 51254720454 \beta_{2} - 15593222533 \beta_{3} - 1524934089 \beta_{4} + 2003683948 \beta_{5} + 218982687 \beta_{6} + 339562335 \beta_{7} + 6211408 \beta_{8} - 150649793 \beta_{9} ) q^{91} + ( -32339598327808 - 57294848 \beta_{1} + 4633657344 \beta_{3} - 599851008 \beta_{4} + 657145856 \beta_{5} - 73564160 \beta_{6} + 188153856 \beta_{7} - 114589696 \beta_{8} + 57294848 \beta_{9} ) q^{92} + ( 342897223080077 + 342897178206268 \beta_{1} + 111894940855 \beta_{2} - 47603262 \beta_{3} - 324310211 \beta_{4} + 44873809 \beta_{5} + 89747618 \beta_{6} + 47603262 \beta_{7} - 8188359 \beta_{8} - 5458906 \beta_{9} ) q^{93} + ( -64144256 - 109688871473280 \beta_{1} - 41533233664 \beta_{2} + 41404945152 \beta_{3} + 64144256 \beta_{4} - 1162055168 \beta_{5} + 120598656 \beta_{6} + 128288512 \beta_{7} + 56454400 \beta_{8} + 169363200 \beta_{9} ) q^{94} + ( 77988257 + 147954666468696 \beta_{1} - 214747752516 \beta_{2} + 214903729030 \beta_{3} - 77988257 \beta_{4} - 2234427548 \beta_{5} + 1630262 \beta_{6} - 155976514 \beta_{7} + 79618519 \beta_{8} + 238855557 \beta_{9} ) q^{95} + ( 13606456393728 + 13606456393728 \beta_{1} + 34359738368 \beta_{2} ) q^{96} + ( -405706594201042 + 24015997 \beta_{1} + 32254997610 \beta_{3} - 636931625 \beta_{4} + 612915628 \beta_{5} + 172297140 \beta_{6} - 220329134 \beta_{7} + 48031994 \beta_{8} - 24015997 \beta_{9} ) q^{97} + ( -76163595812864 + 35544131998592 \beta_{1} + 45867388672 \beta_{2} + 88961992448 \beta_{3} + 83188224 \beta_{4} - 368034688 \beta_{5} + 94332672 \beta_{6} + 174508544 \beta_{7} + 211705984 \beta_{8} + 153612032 \beta_{9} ) q^{98} + ( -99023482118724 - 9757149 \beta_{1} - 189241883439 \beta_{3} + 1214642715 \beta_{4} - 1204885566 \beta_{5} - 180631227 \beta_{6} + 200145525 \beta_{7} - 19514298 \beta_{8} + 9757149 \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 640q^{2} + 1979q^{3} - 81920q^{4} + 300953q^{5} - 506624q^{6} + 4054316q^{7} + 20971520q^{8} - 17367046q^{9} + O(q^{10}) \) \( 10q - 640q^{2} + 1979q^{3} - 81920q^{4} + 300953q^{5} - 506624q^{6} + 4054316q^{7} + 20971520q^{8} - 17367046q^{9} + 38521984q^{10} + 14189887q^{11} + 32423936q^{12} - 772602348q^{13} - 815024000q^{14} + 1426775646q^{15} - 1342177280q^{16} - 145564295q^{17} - 2222981888q^{18} + 3550131629q^{19} - 9861627904q^{20} - 9326509885q^{21} - 3632611072q^{22} + 9869524537q^{23} + 4150263808q^{24} - 51839554204q^{25} + 49446550272q^{26} - 342946209778q^{27} + 37897158656q^{28} - 165255956188q^{29} - 91313641344q^{30} - 106825666677q^{31} - 171798691840q^{32} - 139826141649q^{33} + 37264459520q^{34} + 1475862848131q^{35} + 569083363328q^{36} - 596799401515q^{37} + 454416848512q^{38} - 2827795559666q^{39} + 631144185856q^{40} + 1756981189740q^{41} + 3047643214336q^{42} - 5305116572344q^{43} + 232487108608q^{44} + 10836341974062q^{45} + 1263299140736q^{46} + 4284391851525q^{47} - 1062467534848q^{48} + 1588100454346q^{49} + 13270925876224q^{50} - 8298359375823q^{51} + 6329158434816q^{52} - 4037801378823q^{53} + 21948557425792q^{54} - 47232884322970q^{55} + 8502516908032q^{56} - 75972269020058q^{57} + 10576381196032q^{58} + 6081930248483q^{59} - 11688146092032q^{60} + 29484338931189q^{61} + 27347370669312q^{62} - 151569370755754q^{63} + 43980465111040q^{64} + 78038741223618q^{65} - 17897746131072q^{66} - 2851190345117q^{67} - 2384925409280q^{68} - 41705305747074q^{69} - 190125208943488q^{70} + 190169727933856q^{71} - 36421335252992q^{72} + 284202883487269q^{73} - 76390323393920q^{74} - 151851953188876q^{75} - 116330713219072q^{76} - 656655421831969q^{77} + 723915663274496q^{78} + 536529323488905q^{79} + 80786455789568q^{80} + 23023662902123q^{81} - 112446796143360q^{82} - 949641950127416q^{83} - 237292793479168q^{84} + 2471416802444186q^{85} + 339527460630016q^{86} + 1562035609957686q^{87} + 29758349901824q^{88} + 160148042721333q^{89} - 2774103545359872q^{90} - 1307014889481672q^{91} - 323404580028416q^{92} + 1714374265333339q^{93} + 548402156995200q^{94} - 739988010259049q^{95} + 67997922230272q^{96} - 4057131581852468q^{97} - 939580168008832q^{98} - 989855214604404q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 2 x^{9} + 11041039 x^{8} + 10251836788 x^{7} + 100086056086567 x^{6} + 67517626050179350 x^{5} + 266968630608932668831 x^{4} + 130403899669863282233290 x^{3} + 531639154391012701500584575 x^{2} + 238948900825518555440992768500 x + 120052637231856930389048520650625\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-\)\(31\!\cdots\!51\)\( \nu^{9} - \)\(16\!\cdots\!13\)\( \nu^{8} - \)\(34\!\cdots\!34\)\( \nu^{7} - \)\(30\!\cdots\!48\)\( \nu^{6} - \)\(31\!\cdots\!37\)\( \nu^{5} - \)\(19\!\cdots\!05\)\( \nu^{4} - \)\(76\!\cdots\!56\)\( \nu^{3} - \)\(73\!\cdots\!80\)\( \nu^{2} - \)\(14\!\cdots\!75\)\( \nu - \)\(66\!\cdots\!75\)\(\)\()/ \)\(73\!\cdots\!00\)\( \)
\(\beta_{2}\)\(=\)\((\)\(\)\(31\!\cdots\!51\)\( \nu^{9} + \)\(16\!\cdots\!13\)\( \nu^{8} + \)\(34\!\cdots\!34\)\( \nu^{7} + \)\(30\!\cdots\!48\)\( \nu^{6} + \)\(31\!\cdots\!37\)\( \nu^{5} + \)\(19\!\cdots\!05\)\( \nu^{4} + \)\(76\!\cdots\!56\)\( \nu^{3} + \)\(73\!\cdots\!80\)\( \nu^{2} + \)\(14\!\cdots\!75\)\( \nu - \)\(70\!\cdots\!25\)\(\)\()/ \)\(73\!\cdots\!00\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(22\!\cdots\!69\)\( \nu^{9} - \)\(10\!\cdots\!73\)\( \nu^{8} - \)\(22\!\cdots\!64\)\( \nu^{7} - \)\(75\!\cdots\!68\)\( \nu^{6} - \)\(30\!\cdots\!07\)\( \nu^{5} - \)\(11\!\cdots\!95\)\( \nu^{4} - \)\(46\!\cdots\!86\)\( \nu^{3} - \)\(28\!\cdots\!50\)\( \nu^{2} - \)\(12\!\cdots\!75\)\( \nu - \)\(52\!\cdots\!75\)\(\)\()/ \)\(50\!\cdots\!50\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(34\!\cdots\!78\)\( \nu^{9} - \)\(34\!\cdots\!64\)\( \nu^{8} - \)\(74\!\cdots\!02\)\( \nu^{7} - \)\(34\!\cdots\!94\)\( \nu^{6} - \)\(54\!\cdots\!86\)\( \nu^{5} - \)\(22\!\cdots\!90\)\( \nu^{4} - \)\(12\!\cdots\!68\)\( \nu^{3} - \)\(17\!\cdots\!90\)\( \nu^{2} - \)\(36\!\cdots\!50\)\( \nu - \)\(29\!\cdots\!00\)\(\)\()/ \)\(44\!\cdots\!25\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(83\!\cdots\!97\)\( \nu^{9} - \)\(29\!\cdots\!29\)\( \nu^{8} + \)\(65\!\cdots\!28\)\( \nu^{7} + \)\(29\!\cdots\!96\)\( \nu^{6} - \)\(96\!\cdots\!31\)\( \nu^{5} - \)\(20\!\cdots\!95\)\( \nu^{4} - \)\(78\!\cdots\!18\)\( \nu^{3} + \)\(67\!\cdots\!70\)\( \nu^{2} - \)\(27\!\cdots\!75\)\( \nu - \)\(12\!\cdots\!25\)\(\)\()/ \)\(88\!\cdots\!50\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(10\!\cdots\!28\)\( \nu^{9} - \)\(14\!\cdots\!36\)\( \nu^{8} + \)\(14\!\cdots\!27\)\( \nu^{7} - \)\(56\!\cdots\!06\)\( \nu^{6} + \)\(11\!\cdots\!11\)\( \nu^{5} - \)\(46\!\cdots\!85\)\( \nu^{4} + \)\(35\!\cdots\!43\)\( \nu^{3} - \)\(17\!\cdots\!35\)\( \nu^{2} + \)\(39\!\cdots\!50\)\( \nu - \)\(31\!\cdots\!00\)\(\)\()/ \)\(44\!\cdots\!25\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(40\!\cdots\!33\)\( \nu^{9} + \)\(60\!\cdots\!19\)\( \nu^{8} + \)\(36\!\cdots\!42\)\( \nu^{7} + \)\(52\!\cdots\!44\)\( \nu^{6} + \)\(33\!\cdots\!91\)\( \nu^{5} + \)\(29\!\cdots\!95\)\( \nu^{4} + \)\(53\!\cdots\!48\)\( \nu^{3} + \)\(55\!\cdots\!80\)\( \nu^{2} + \)\(12\!\cdots\!25\)\( \nu + \)\(10\!\cdots\!25\)\(\)\()/ \)\(88\!\cdots\!50\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(30\!\cdots\!71\)\( \nu^{9} - \)\(49\!\cdots\!27\)\( \nu^{8} + \)\(31\!\cdots\!64\)\( \nu^{7} - \)\(19\!\cdots\!92\)\( \nu^{6} + \)\(23\!\cdots\!27\)\( \nu^{5} - \)\(27\!\cdots\!45\)\( \nu^{4} + \)\(40\!\cdots\!26\)\( \nu^{3} - \)\(71\!\cdots\!70\)\( \nu^{2} + \)\(11\!\cdots\!75\)\( \nu - \)\(10\!\cdots\!25\)\(\)\()/ \)\(62\!\cdots\!50\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(68\!\cdots\!59\)\( \nu^{9} + \)\(47\!\cdots\!97\)\( \nu^{8} + \)\(75\!\cdots\!46\)\( \nu^{7} + \)\(12\!\cdots\!52\)\( \nu^{6} + \)\(74\!\cdots\!73\)\( \nu^{5} + \)\(93\!\cdots\!05\)\( \nu^{4} + \)\(21\!\cdots\!04\)\( \nu^{3} + \)\(22\!\cdots\!00\)\( \nu^{2} + \)\(44\!\cdots\!75\)\( \nu + \)\(35\!\cdots\!75\)\(\)\()/ \)\(12\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(3 \beta_{9} + \beta_{8} + 10 \beta_{7} + 6 \beta_{6} - 41 \beta_{5} + 5 \beta_{4} - 1414 \beta_{3} + 1404 \beta_{2} + 17665933 \beta_{1} - 5\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-6772 \beta_{9} + 13544 \beta_{8} + 15033 \beta_{7} - 28577 \beta_{6} - 78998 \beta_{5} + 72226 \beta_{4} - 28375553 \beta_{3} + 6772 \beta_{1} - 24663094820\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(-6325062 \beta_{9} - 9487593 \beta_{8} - 95858454 \beta_{7} - 198041970 \beta_{6} - 99020985 \beta_{5} + 927193167 \beta_{4} + 95858454 \beta_{3} - 41582176961 \beta_{2} - 250595927687231 \beta_{1} - 250596026708216\)\()/8\)
\(\nu^{5}\)\(=\)\((\)\(193601427720 \beta_{9} + 64533809240 \beta_{8} - 584193167470 \beta_{7} - 227562774495 \beta_{6} + 1052737919465 \beta_{5} - 292096583735 \beta_{4} + 240926754963904 \beta_{3} - 240342561796434 \beta_{2} - 365782979524778704 \beta_{1} + 292096583735\)\()/8\)
\(\nu^{6}\)\(=\)\((\)\(43465737693923 \beta_{9} - 86931475387846 \beta_{8} - 943860837106617 \beta_{7} + 1030792312494463 \beta_{6} + 9695952046125937 \beta_{5} - 9652486308432014 \beta_{4} + 500530649226157180 \beta_{3} - 43465737693923 \beta_{1} + 2123153289603797958463\)\()/8\)
\(\nu^{7}\)\(=\)\((\)\(-1141319625486055974 \beta_{9} - 1711979438229083961 \beta_{8} + 3719067626274401457 \beta_{7} + 6296815627062746940 \beta_{6} + 3148407813531373470 \beta_{5} - 13650640495998372936 \beta_{4} - 3719067626274401457 \beta_{3} + 2218059715810079593315 \beta_{2} + 4399653641995079471004835 \beta_{1} + 4399656790402893002378305\)\()/8\)
\(\nu^{8}\)\(=\)\((\)\(-2417699299654876548837 \beta_{9} - 805899766551625516279 \beta_{8} + 20006123637322781321360 \beta_{7} + 9197162052109765144401 \beta_{6} - 80787235887343303520011 \beta_{5} + 10003061818661390660680 \beta_{4} - 5552147023703794992513161 \beta_{3} + 5532140900066472211191801 \beta_{2} + 19586255306016247934683796540 \beta_{1} - 10003061818661390660680\)\()/8\)
\(\nu^{9}\)\(=\)\((\)\(-5151895915882994794609135 \beta_{9} + 10303791831765989589218270 \beta_{8} + 27224795615753641637972010 \beta_{7} - 37528587447519631227190280 \beta_{6} - 201016938020235950198346995 \beta_{5} + 195865042104352955403737860 \beta_{4} - 21386248393611021318167968769 \beta_{3} + 5151895915882994794609135 \beta_{1} - 48752827578010188957322194162599\)\()/8\)

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
−1239.01 + 2146.02i
−879.415 + 1523.19i
−253.251 + 438.643i
778.235 1347.94i
1594.44 2761.64i
−1239.01 2146.02i
−879.415 1523.19i
−253.251 438.643i
778.235 + 1347.94i
1594.44 + 2761.64i
−64.0000 + 110.851i −2280.51 3949.96i −8192.00 14189.0i 151908. 263112.i 583811. 1.80034e6 + 1.22733e6i 2.09715e6 −3.22701e6 + 5.58934e6i 1.94442e7 + 3.36783e7i
9.2 −64.0000 + 110.851i −1561.33 2704.30i −8192.00 14189.0i −101823. + 176362.i 399701. 1.89733e6 + 1.07131e6i 2.09715e6 2.29895e6 3.98189e6i −1.30333e7 2.25744e7i
9.3 −64.0000 + 110.851i −309.001 535.206i −8192.00 14189.0i 8985.53 15563.4i 79104.3 −2.17743e6 + 79880.9i 2.09715e6 6.98349e6 1.20958e7i 1.15015e6 + 1.99212e6i
9.4 −64.0000 + 110.851i 1753.97 + 3037.97i −8192.00 14189.0i −36165.1 + 62639.7i −449016. 780333. 2.03437e6i 2.09715e6 1.02163e6 1.76951e6i −4.62913e6 8.01788e6i
9.5 −64.0000 + 110.851i 3386.37 + 5865.37i −8192.00 14189.0i 127571. 220960.i −866911. −273416. + 2.16167e6i 2.09715e6 −1.57606e7 + 2.72981e7i 1.63291e7 + 2.82829e7i
11.1 −64.0000 110.851i −2280.51 + 3949.96i −8192.00 + 14189.0i 151908. + 263112.i 583811. 1.80034e6 1.22733e6i 2.09715e6 −3.22701e6 5.58934e6i 1.94442e7 3.36783e7i
11.2 −64.0000 110.851i −1561.33 + 2704.30i −8192.00 + 14189.0i −101823. 176362.i 399701. 1.89733e6 1.07131e6i 2.09715e6 2.29895e6 + 3.98189e6i −1.30333e7 + 2.25744e7i
11.3 −64.0000 110.851i −309.001 + 535.206i −8192.00 + 14189.0i 8985.53 + 15563.4i 79104.3 −2.17743e6 79880.9i 2.09715e6 6.98349e6 + 1.20958e7i 1.15015e6 1.99212e6i
11.4 −64.0000 110.851i 1753.97 3037.97i −8192.00 + 14189.0i −36165.1 62639.7i −449016. 780333. + 2.03437e6i 2.09715e6 1.02163e6 + 1.76951e6i −4.62913e6 + 8.01788e6i
11.5 −64.0000 110.851i 3386.37 5865.37i −8192.00 + 14189.0i 127571. + 220960.i −866911. −273416. 2.16167e6i 2.09715e6 −1.57606e7 2.72981e7i 1.63291e7 2.82829e7i
\(n\): 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.a 10
3.b odd 2 1 126.16.g.e 10
7.b odd 2 1 98.16.c.m 10
7.c even 3 1 inner 14.16.c.a 10
7.c even 3 1 98.16.a.j 5
7.d odd 6 1 98.16.a.k 5
7.d odd 6 1 98.16.c.m 10
21.h odd 6 1 126.16.g.e 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.16.c.a 10 1.a even 1 1 trivial
14.16.c.a 10 7.c even 3 1 inner
98.16.a.j 5 7.c even 3 1
98.16.a.k 5 7.d odd 6 1
98.16.c.m 10 7.b odd 2 1
98.16.c.m 10 7.d odd 6 1
126.16.g.e 10 3.b odd 2 1
126.16.g.e 10 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(14\!\cdots\!17\)\( T_{3}^{6} + \)\(11\!\cdots\!75\)\( T_{3}^{5} + \)\(15\!\cdots\!09\)\( T_{3}^{4} + \)\(21\!\cdots\!10\)\( T_{3}^{3} + \)\(12\!\cdots\!75\)\( T_{3}^{2} + \)\(75\!\cdots\!25\)\( T_{3} + \)\(43\!\cdots\!25\)\( \)">\(T_{3}^{10} - \cdots\) acting on \(S_{16}^{\mathrm{new}}(14, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 16384 + 128 T + T^{2} )^{5} \)
$3$ \( \)\(43\!\cdots\!25\)\( + \)\(75\!\cdots\!25\)\( T + \)\(12\!\cdots\!75\)\( T^{2} + \)\(21\!\cdots\!10\)\( T^{3} + \)\(15\!\cdots\!09\)\( T^{4} + 1176939196031463975 T^{5} + 1475209864305117 T^{6} + 62709703674 T^{7} + 46514011 T^{8} - 1979 T^{9} + T^{10} \)
$5$ \( \)\(42\!\cdots\!25\)\( - \)\(18\!\cdots\!25\)\( T + \)\(11\!\cdots\!75\)\( T^{2} + \)\(10\!\cdots\!50\)\( T^{3} + \)\(24\!\cdots\!25\)\( T^{4} - \)\(28\!\cdots\!75\)\( T^{5} + \)\(66\!\cdots\!25\)\( T^{6} - 11414991893674070 T^{7} + 147500076519 T^{8} - 300953 T^{9} + T^{10} \)
$7$ \( \)\(24\!\cdots\!43\)\( - \)\(20\!\cdots\!16\)\( T + \)\(79\!\cdots\!85\)\( T^{2} + \)\(87\!\cdots\!88\)\( T^{3} - \)\(24\!\cdots\!38\)\( T^{4} + \)\(16\!\cdots\!20\)\( T^{5} - \)\(51\!\cdots\!66\)\( T^{6} + 3902858949602838512 T^{7} + 7424688886755 T^{8} - 4054316 T^{9} + T^{10} \)
$11$ \( \)\(19\!\cdots\!25\)\( - \)\(66\!\cdots\!95\)\( T + \)\(26\!\cdots\!39\)\( T^{2} - \)\(27\!\cdots\!90\)\( T^{3} + \)\(71\!\cdots\!21\)\( T^{4} - \)\(38\!\cdots\!53\)\( T^{5} + \)\(15\!\cdots\!81\)\( T^{6} - \)\(33\!\cdots\!66\)\( T^{7} + 14319408792027291 T^{8} - 14189887 T^{9} + T^{10} \)
$13$ \( ( \)\(72\!\cdots\!00\)\( + \)\(30\!\cdots\!20\)\( T - \)\(37\!\cdots\!96\)\( T^{2} - 103800355267236264 T^{3} + 386301174 T^{4} + T^{5} )^{2} \)
$17$ \( \)\(42\!\cdots\!25\)\( + \)\(40\!\cdots\!75\)\( T + \)\(41\!\cdots\!39\)\( T^{2} - \)\(29\!\cdots\!54\)\( T^{3} + \)\(18\!\cdots\!61\)\( T^{4} - \)\(24\!\cdots\!43\)\( T^{5} + \)\(28\!\cdots\!49\)\( T^{6} - \)\(10\!\cdots\!46\)\( T^{7} + 5330661804775905351 T^{8} + 145564295 T^{9} + T^{10} \)
$19$ \( \)\(17\!\cdots\!25\)\( + \)\(22\!\cdots\!75\)\( T + \)\(26\!\cdots\!75\)\( T^{2} + \)\(69\!\cdots\!10\)\( T^{3} + \)\(28\!\cdots\!61\)\( T^{4} + \)\(16\!\cdots\!41\)\( T^{5} + \)\(16\!\cdots\!85\)\( T^{6} + \)\(36\!\cdots\!74\)\( T^{7} + 56641387259663817315 T^{8} - 3550131629 T^{9} + T^{10} \)
$23$ \( \)\(12\!\cdots\!01\)\( + \)\(18\!\cdots\!83\)\( T + \)\(37\!\cdots\!15\)\( T^{2} - \)\(16\!\cdots\!66\)\( T^{3} + \)\(58\!\cdots\!81\)\( T^{4} - \)\(73\!\cdots\!11\)\( T^{5} + \)\(78\!\cdots\!21\)\( T^{6} - \)\(89\!\cdots\!46\)\( T^{7} + \)\(92\!\cdots\!15\)\( T^{8} - 9869524537 T^{9} + T^{10} \)
$29$ \( ( -\)\(36\!\cdots\!72\)\( + \)\(71\!\cdots\!00\)\( T - \)\(85\!\cdots\!92\)\( T^{2} - \)\(18\!\cdots\!96\)\( T^{3} + 82627978094 T^{4} + T^{5} )^{2} \)
$31$ \( \)\(69\!\cdots\!25\)\( + \)\(83\!\cdots\!65\)\( T + \)\(76\!\cdots\!31\)\( T^{2} + \)\(27\!\cdots\!98\)\( T^{3} + \)\(73\!\cdots\!85\)\( T^{4} + \)\(60\!\cdots\!23\)\( T^{5} + \)\(45\!\cdots\!69\)\( T^{6} + \)\(13\!\cdots\!82\)\( T^{7} + \)\(67\!\cdots\!95\)\( T^{8} + 106825666677 T^{9} + T^{10} \)
$37$ \( \)\(39\!\cdots\!09\)\( + \)\(67\!\cdots\!51\)\( T + \)\(12\!\cdots\!35\)\( T^{2} + \)\(23\!\cdots\!98\)\( T^{3} + \)\(33\!\cdots\!73\)\( T^{4} + \)\(57\!\cdots\!17\)\( T^{5} + \)\(22\!\cdots\!37\)\( T^{6} + \)\(15\!\cdots\!34\)\( T^{7} + \)\(70\!\cdots\!07\)\( T^{8} + 596799401515 T^{9} + T^{10} \)
$41$ \( ( \)\(58\!\cdots\!44\)\( - \)\(90\!\cdots\!72\)\( T - \)\(91\!\cdots\!76\)\( T^{2} - \)\(21\!\cdots\!96\)\( T^{3} - 878490594870 T^{4} + T^{5} )^{2} \)
$43$ \( ( -\)\(29\!\cdots\!40\)\( + \)\(18\!\cdots\!44\)\( T - \)\(28\!\cdots\!72\)\( T^{2} - \)\(10\!\cdots\!60\)\( T^{3} + 2652558286172 T^{4} + T^{5} )^{2} \)
$47$ \( \)\(29\!\cdots\!25\)\( - \)\(66\!\cdots\!25\)\( T + \)\(10\!\cdots\!79\)\( T^{2} - \)\(90\!\cdots\!54\)\( T^{3} + \)\(59\!\cdots\!81\)\( T^{4} - \)\(22\!\cdots\!23\)\( T^{5} + \)\(68\!\cdots\!49\)\( T^{6} - \)\(11\!\cdots\!46\)\( T^{7} + \)\(32\!\cdots\!51\)\( T^{8} - 4284391851525 T^{9} + T^{10} \)
$53$ \( \)\(24\!\cdots\!29\)\( + \)\(75\!\cdots\!23\)\( T + \)\(30\!\cdots\!43\)\( T^{2} + \)\(21\!\cdots\!54\)\( T^{3} + \)\(86\!\cdots\!85\)\( T^{4} + \)\(48\!\cdots\!69\)\( T^{5} + \)\(17\!\cdots\!45\)\( T^{6} + \)\(42\!\cdots\!54\)\( T^{7} + \)\(15\!\cdots\!27\)\( T^{8} + 4037801378823 T^{9} + T^{10} \)
$59$ \( \)\(76\!\cdots\!25\)\( - \)\(26\!\cdots\!75\)\( T + \)\(99\!\cdots\!59\)\( T^{2} + \)\(26\!\cdots\!54\)\( T^{3} + \)\(43\!\cdots\!41\)\( T^{4} - \)\(72\!\cdots\!29\)\( T^{5} + \)\(12\!\cdots\!37\)\( T^{6} - \)\(85\!\cdots\!74\)\( T^{7} + \)\(12\!\cdots\!03\)\( T^{8} - 6081930248483 T^{9} + T^{10} \)
$61$ \( \)\(12\!\cdots\!25\)\( + \)\(24\!\cdots\!95\)\( T + \)\(41\!\cdots\!79\)\( T^{2} + \)\(22\!\cdots\!70\)\( T^{3} + \)\(14\!\cdots\!09\)\( T^{4} + \)\(19\!\cdots\!29\)\( T^{5} + \)\(17\!\cdots\!61\)\( T^{6} + \)\(36\!\cdots\!62\)\( T^{7} + \)\(22\!\cdots\!59\)\( T^{8} - 29484338931189 T^{9} + T^{10} \)
$67$ \( \)\(61\!\cdots\!25\)\( - \)\(15\!\cdots\!15\)\( T + \)\(36\!\cdots\!11\)\( T^{2} - \)\(17\!\cdots\!58\)\( T^{3} + \)\(18\!\cdots\!49\)\( T^{4} + \)\(74\!\cdots\!39\)\( T^{5} + \)\(79\!\cdots\!45\)\( T^{6} + \)\(91\!\cdots\!94\)\( T^{7} + \)\(29\!\cdots\!79\)\( T^{8} + 2851190345117 T^{9} + T^{10} \)
$71$ \( ( \)\(15\!\cdots\!00\)\( - \)\(14\!\cdots\!96\)\( T + \)\(38\!\cdots\!28\)\( T^{2} - \)\(10\!\cdots\!80\)\( T^{3} - 95084863966928 T^{4} + T^{5} )^{2} \)
$73$ \( \)\(16\!\cdots\!25\)\( + \)\(21\!\cdots\!35\)\( T + \)\(33\!\cdots\!71\)\( T^{2} - \)\(52\!\cdots\!46\)\( T^{3} + \)\(20\!\cdots\!01\)\( T^{4} - \)\(13\!\cdots\!03\)\( T^{5} + \)\(59\!\cdots\!85\)\( T^{6} - \)\(64\!\cdots\!22\)\( T^{7} + \)\(66\!\cdots\!51\)\( T^{8} - 284202883487269 T^{9} + T^{10} \)
$79$ \( \)\(18\!\cdots\!25\)\( - \)\(88\!\cdots\!25\)\( T + \)\(42\!\cdots\!75\)\( T^{2} - \)\(29\!\cdots\!10\)\( T^{3} + \)\(29\!\cdots\!29\)\( T^{4} - \)\(15\!\cdots\!99\)\( T^{5} + \)\(11\!\cdots\!69\)\( T^{6} - \)\(54\!\cdots\!26\)\( T^{7} + \)\(23\!\cdots\!03\)\( T^{8} - 536529323488905 T^{9} + T^{10} \)
$83$ \( ( -\)\(17\!\cdots\!00\)\( - \)\(49\!\cdots\!48\)\( T - \)\(15\!\cdots\!32\)\( T^{2} + \)\(55\!\cdots\!08\)\( T^{3} + 474820975063708 T^{4} + T^{5} )^{2} \)
$89$ \( \)\(85\!\cdots\!25\)\( - \)\(38\!\cdots\!65\)\( T + \)\(11\!\cdots\!39\)\( T^{2} - \)\(18\!\cdots\!66\)\( T^{3} + \)\(22\!\cdots\!93\)\( T^{4} - \)\(14\!\cdots\!47\)\( T^{5} + \)\(70\!\cdots\!33\)\( T^{6} - \)\(74\!\cdots\!98\)\( T^{7} + \)\(26\!\cdots\!71\)\( T^{8} - 160148042721333 T^{9} + T^{10} \)
$97$ \( ( -\)\(10\!\cdots\!00\)\( - \)\(42\!\cdots\!00\)\( T + \)\(22\!\cdots\!32\)\( T^{2} + \)\(12\!\cdots\!40\)\( T^{3} + 2028565790926234 T^{4} + T^{5} )^{2} \)
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