Properties

Label 273.12.a.c
Level $273$
Weight $12$
Character orbit 273.a
Self dual yes
Analytic conductor $209.758$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 273.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(209.757688293\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - x^{15} - 26640 x^{14} - 38138 x^{13} + 279954048 x^{12} + 940095168 x^{11} - 1473489774048 x^{10} - 6951601887328 x^{9} + 4072481535854976 x^{8} + 21355761647276288 x^{7} - 5703452025696599040 x^{6} - 25041700255043457024 x^{5} + 3737994671267475603456 x^{4} + 6711995200063092490240 x^{3} - 963518272786595267739648 x^{2} + 1287697466669064032092160 x + 45555354651594891689197568\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{12}\cdot 5\cdot 7^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -4 + \beta_{1} ) q^{2} -243 q^{3} + ( 1298 - 4 \beta_{1} + \beta_{2} ) q^{4} + ( -199 + 20 \beta_{1} + \beta_{3} ) q^{5} + ( 972 - 243 \beta_{1} ) q^{6} + 16807 q^{7} + ( -11845 + 1533 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{8} + 59049 q^{9} +O(q^{10})\) \( q + ( -4 + \beta_{1} ) q^{2} -243 q^{3} + ( 1298 - 4 \beta_{1} + \beta_{2} ) q^{4} + ( -199 + 20 \beta_{1} + \beta_{3} ) q^{5} + ( 972 - 243 \beta_{1} ) q^{6} + 16807 q^{7} + ( -11845 + 1533 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{8} + 59049 q^{9} + ( 66068 - 439 \beta_{1} + 5 \beta_{2} - 20 \beta_{3} - \beta_{4} + \beta_{5} ) q^{10} + ( -29142 - 758 \beta_{1} - 38 \beta_{2} - 9 \beta_{3} - \beta_{4} + \beta_{9} ) q^{11} + ( -315414 + 972 \beta_{1} - 243 \beta_{2} ) q^{12} -371293 q^{13} + ( -67228 + 16807 \beta_{1} ) q^{14} + ( 48357 - 4860 \beta_{1} - 243 \beta_{3} ) q^{15} + ( 2504810 - 15213 \beta_{1} + 1346 \beta_{2} - 135 \beta_{3} - 9 \beta_{4} - 4 \beta_{5} + \beta_{6} + 7 \beta_{7} + 3 \beta_{8} - \beta_{9} + \beta_{11} + 5 \beta_{12} - 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{16} + ( 109068 + 8475 \beta_{1} + 46 \beta_{2} - 26 \beta_{3} + 21 \beta_{4} + 5 \beta_{5} - 7 \beta_{7} - 7 \beta_{8} - 2 \beta_{9} + \beta_{10} - 3 \beta_{11} - 2 \beta_{12} + \beta_{13} + 5 \beta_{14} + \beta_{15} ) q^{17} + ( -236196 + 59049 \beta_{1} ) q^{18} + ( 270037 - 81702 \beta_{1} + 458 \beta_{2} - 53 \beta_{3} + \beta_{4} - 7 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 3 \beta_{8} + 4 \beta_{9} + 5 \beta_{10} - 3 \beta_{11} - 3 \beta_{12} - \beta_{13} - 5 \beta_{14} + 2 \beta_{15} ) q^{19} + ( -1296634 + 40535 \beta_{1} - 1296 \beta_{2} + 2209 \beta_{3} + 27 \beta_{4} - 14 \beta_{5} - 6 \beta_{6} + 4 \beta_{7} + 4 \beta_{8} - 11 \beta_{9} - 11 \beta_{10} + 5 \beta_{11} - 5 \beta_{13} - 5 \beta_{14} - 3 \beta_{15} ) q^{20} -4084101 q^{21} + ( -2335338 - 114324 \beta_{1} - 1966 \beta_{2} + 1605 \beta_{3} - 26 \beta_{4} - 28 \beta_{5} + 4 \beta_{6} - 21 \beta_{7} + 8 \beta_{8} - 18 \beta_{9} + 5 \beta_{10} + 17 \beta_{11} - 26 \beta_{12} + 14 \beta_{13} - 4 \beta_{14} + 5 \beta_{15} ) q^{22} + ( -9400775 - 71794 \beta_{1} - 2458 \beta_{2} + 1107 \beta_{3} + 51 \beta_{4} + \beta_{5} - 9 \beta_{6} - 33 \beta_{7} - 2 \beta_{8} - 6 \beta_{9} - 4 \beta_{10} + 3 \beta_{11} + 19 \beta_{13} + 12 \beta_{14} + 4 \beta_{15} ) q^{23} + ( 2878335 - 372519 \beta_{1} + 1458 \beta_{2} + 486 \beta_{3} - 243 \beta_{4} ) q^{24} + ( 9455476 - 285101 \beta_{1} + 6711 \beta_{2} + 45 \beta_{3} - 192 \beta_{4} + 49 \beta_{5} - 5 \beta_{6} + 15 \beta_{7} - 22 \beta_{8} - \beta_{9} - 11 \beta_{10} - 33 \beta_{11} - 10 \beta_{12} + 17 \beta_{13} - 10 \beta_{14} - 18 \beta_{15} ) q^{25} + ( 1485172 - 371293 \beta_{1} ) q^{26} -14348907 q^{27} + ( 21815486 - 67228 \beta_{1} + 16807 \beta_{2} ) q^{28} + ( -6954128 - 157359 \beta_{1} - 5841 \beta_{2} + 1582 \beta_{3} - 50 \beta_{4} - 59 \beta_{5} - 24 \beta_{6} - 36 \beta_{7} - 62 \beta_{8} - 20 \beta_{9} + 41 \beta_{10} - 12 \beta_{11} + 6 \beta_{12} - 44 \beta_{13} + 45 \beta_{14} - 17 \beta_{15} ) q^{29} + ( -16054524 + 106677 \beta_{1} - 1215 \beta_{2} + 4860 \beta_{3} + 243 \beta_{4} - 243 \beta_{5} ) q^{30} + ( -5661422 + 51704 \beta_{1} + 12366 \beta_{2} - 7033 \beta_{3} - 685 \beta_{4} - 152 \beta_{5} - 29 \beta_{6} + 57 \beta_{7} + 120 \beta_{8} - 36 \beta_{9} + 2 \beta_{10} + 47 \beta_{11} + 36 \beta_{12} - 33 \beta_{13} - 115 \beta_{14} + 11 \beta_{15} ) q^{31} + ( -38276872 + 2447853 \beta_{1} - 15858 \beta_{2} - 227 \beta_{3} + 1737 \beta_{4} - 116 \beta_{5} + 75 \beta_{6} - 31 \beta_{7} - 35 \beta_{8} - 49 \beta_{9} - 10 \beta_{10} - 87 \beta_{11} - 101 \beta_{12} - 36 \beta_{13} + 251 \beta_{14} + 243 \beta_{15} ) q^{32} + ( 7081506 + 184194 \beta_{1} + 9234 \beta_{2} + 2187 \beta_{3} + 243 \beta_{4} - 243 \beta_{9} ) q^{33} + ( 27725624 + 267085 \beta_{1} + 41673 \beta_{2} + 3514 \beta_{3} - 1042 \beta_{4} - 310 \beta_{5} + 178 \beta_{6} + 338 \beta_{7} + 207 \beta_{8} - 270 \beta_{9} - 58 \beta_{10} + 218 \beta_{11} + 355 \beta_{12} - 64 \beta_{13} - 394 \beta_{14} - 320 \beta_{15} ) q^{34} + ( -3344593 + 336140 \beta_{1} + 16807 \beta_{3} ) q^{35} + ( 76645602 - 236196 \beta_{1} + 59049 \beta_{2} ) q^{36} + ( -109829746 + 850542 \beta_{1} - 18748 \beta_{2} - 6901 \beta_{3} - 210 \beta_{4} - 357 \beta_{5} - 4 \beta_{6} + 182 \beta_{7} - 103 \beta_{8} - 113 \beta_{9} + 59 \beta_{10} + 75 \beta_{11} + 25 \beta_{12} - 95 \beta_{13} + \beta_{14} - 16 \beta_{15} ) q^{37} + ( -273783819 + 1324189 \beta_{1} - 83221 \beta_{2} - 23981 \beta_{3} + 374 \beta_{4} - 369 \beta_{5} - 61 \beta_{6} - 206 \beta_{7} - 294 \beta_{8} + 168 \beta_{9} - 6 \beta_{10} - 57 \beta_{11} - 72 \beta_{12} + 161 \beta_{13} + 112 \beta_{14} - 143 \beta_{15} ) q^{38} + 90224199 q^{39} + ( 3826808 - 3886003 \beta_{1} + 46206 \beta_{2} - 83189 \beta_{3} - 4027 \beta_{4} + 1694 \beta_{5} - 217 \beta_{6} + 183 \beta_{7} - 143 \beta_{8} + 705 \beta_{9} + 290 \beta_{10} - 669 \beta_{11} + 15 \beta_{12} - 194 \beta_{13} + 265 \beta_{14} + 465 \beta_{15} ) q^{40} + ( 40667506 - 1105331 \beta_{1} + 17958 \beta_{2} - 12645 \beta_{3} - 1734 \beta_{4} - 297 \beta_{5} + 91 \beta_{6} - 8 \beta_{7} + 378 \beta_{8} + 9 \beta_{9} - 204 \beta_{10} + 123 \beta_{11} + 229 \beta_{12} - 231 \beta_{13} + 172 \beta_{14} - 233 \beta_{15} ) q^{41} + ( 16336404 - 4084101 \beta_{1} ) q^{42} + ( -118087957 - 569963 \beta_{1} - 79465 \beta_{2} - 45635 \beta_{3} - 716 \beta_{4} - 953 \beta_{5} + 533 \beta_{6} + 13 \beta_{7} + 199 \beta_{8} + 841 \beta_{9} - 141 \beta_{10} - 91 \beta_{11} - 407 \beta_{12} + 344 \beta_{13} + 304 \beta_{14} + 378 \beta_{15} ) q^{43} + ( -310922347 - 5710691 \beta_{1} - 146540 \beta_{2} - 105340 \beta_{3} - 9487 \beta_{4} + 2266 \beta_{5} - 1210 \beta_{6} - 982 \beta_{7} - 1214 \beta_{8} + 972 \beta_{9} + 286 \beta_{10} - 790 \beta_{11} + 366 \beta_{12} + 90 \beta_{13} + 380 \beta_{14} + 94 \beta_{15} ) q^{44} + ( -11750751 + 1180980 \beta_{1} + 59049 \beta_{3} ) q^{45} + ( -199319187 - 15195132 \beta_{1} - 32726 \beta_{2} - 42730 \beta_{3} - 6245 \beta_{4} + 61 \beta_{5} - 53 \beta_{6} - 226 \beta_{7} + 472 \beta_{8} - 173 \beta_{9} + 401 \beta_{10} + 608 \beta_{11} - 404 \beta_{12} + 456 \beta_{13} - 115 \beta_{14} - 800 \beta_{15} ) q^{46} + ( -252462946 + 3128996 \beta_{1} - 28244 \beta_{2} - 30615 \beta_{3} - 272 \beta_{4} - 357 \beta_{5} + 804 \beta_{6} + 1072 \beta_{7} + 174 \beta_{8} + 230 \beta_{9} - 566 \beta_{10} + 1318 \beta_{11} + 466 \beta_{12} + 518 \beta_{13} - 293 \beta_{14} + 363 \beta_{15} ) q^{47} + ( -608668830 + 3696759 \beta_{1} - 327078 \beta_{2} + 32805 \beta_{3} + 2187 \beta_{4} + 972 \beta_{5} - 243 \beta_{6} - 1701 \beta_{7} - 729 \beta_{8} + 243 \beta_{9} - 243 \beta_{11} - 1215 \beta_{12} + 486 \beta_{13} + 243 \beta_{14} + 243 \beta_{15} ) q^{48} + 282475249 q^{49} + ( -997161541 + 24427742 \beta_{1} - 627375 \beta_{2} + 157563 \beta_{3} + 4968 \beta_{4} + 1067 \beta_{5} - 35 \beta_{6} - 880 \beta_{7} + 620 \beta_{8} - 2460 \beta_{9} - 400 \beta_{10} - 65 \beta_{11} - 790 \beta_{12} + 115 \beta_{13} - 1480 \beta_{14} - 1695 \beta_{15} ) q^{50} + ( -26503524 - 2059425 \beta_{1} - 11178 \beta_{2} + 6318 \beta_{3} - 5103 \beta_{4} - 1215 \beta_{5} + 1701 \beta_{7} + 1701 \beta_{8} + 486 \beta_{9} - 243 \beta_{10} + 729 \beta_{11} + 486 \beta_{12} - 243 \beta_{13} - 1215 \beta_{14} - 243 \beta_{15} ) q^{51} + ( -481938314 + 1485172 \beta_{1} - 371293 \beta_{2} ) q^{52} + ( 31412868 + 8632700 \beta_{1} - 31278 \beta_{2} + 39511 \beta_{3} - 3995 \beta_{4} + 2240 \beta_{5} + 415 \beta_{6} + 2527 \beta_{7} + 1742 \beta_{8} + 1018 \beta_{9} - 1750 \beta_{10} + 799 \beta_{11} + 1530 \beta_{12} - 2091 \beta_{13} - 2273 \beta_{14} + 575 \beta_{15} ) q^{53} + ( 57395628 - 14348907 \beta_{1} ) q^{54} + ( -534120328 + 24549392 \beta_{1} - 694986 \beta_{2} + 10883 \beta_{3} + 14914 \beta_{4} - 1781 \beta_{5} + 938 \beta_{6} - 1232 \beta_{7} - 99 \beta_{8} - 393 \beta_{9} + 1267 \beta_{10} - 563 \beta_{11} - 3775 \beta_{12} + 1357 \beta_{13} + 1645 \beta_{14} + 2726 \beta_{15} ) q^{55} + ( -199078915 + 25765131 \beta_{1} - 100842 \beta_{2} - 33614 \beta_{3} + 16807 \beta_{4} ) q^{56} + ( -65618991 + 19853586 \beta_{1} - 111294 \beta_{2} + 12879 \beta_{3} - 243 \beta_{4} + 1701 \beta_{5} - 486 \beta_{6} - 486 \beta_{7} - 729 \beta_{8} - 972 \beta_{9} - 1215 \beta_{10} + 729 \beta_{11} + 729 \beta_{12} + 243 \beta_{13} + 1215 \beta_{14} - 486 \beta_{15} ) q^{57} + ( -489536814 - 20658239 \beta_{1} - 247237 \beta_{2} - 331370 \beta_{3} - 6822 \beta_{4} + 777 \beta_{5} + 1144 \beta_{6} - 755 \beta_{7} - 1269 \beta_{8} - 223 \beta_{9} + 760 \beta_{10} + 60 \beta_{11} + 843 \beta_{12} + 1249 \beta_{13} - 1159 \beta_{14} + 472 \beta_{15} ) q^{58} + ( 232522531 + 6287539 \beta_{1} - 660525 \beta_{2} + 107873 \beta_{3} - 10606 \beta_{4} - 5049 \beta_{5} - 953 \beta_{6} - 3513 \beta_{7} + 234 \beta_{8} - 1738 \beta_{9} + 362 \beta_{10} + 3092 \beta_{11} + 1560 \beta_{12} - 605 \beta_{13} - 1392 \beta_{14} - 2687 \beta_{15} ) q^{59} + ( 315082062 - 9850005 \beta_{1} + 314928 \beta_{2} - 536787 \beta_{3} - 6561 \beta_{4} + 3402 \beta_{5} + 1458 \beta_{6} - 972 \beta_{7} - 972 \beta_{8} + 2673 \beta_{9} + 2673 \beta_{10} - 1215 \beta_{11} + 1215 \beta_{13} + 1215 \beta_{14} + 729 \beta_{15} ) q^{60} + ( 1638380044 + 11203266 \beta_{1} - 457491 \beta_{2} - 159846 \beta_{3} + 10776 \beta_{4} + 3971 \beta_{5} - 4820 \beta_{6} + 603 \beta_{7} + 834 \beta_{8} + 879 \beta_{9} + 1300 \beta_{10} - 3261 \beta_{11} - 1591 \beta_{12} - 4056 \beta_{13} + 4619 \beta_{14} + 825 \beta_{15} ) q^{61} + ( 185567917 + 22740755 \beta_{1} - 1115211 \beta_{2} - 384892 \beta_{3} + 26691 \beta_{4} - 1725 \beta_{5} - 4201 \beta_{6} - 10036 \beta_{7} - 7977 \beta_{8} + 8567 \beta_{9} + 2461 \beta_{10} - 5282 \beta_{11} - 9113 \beta_{12} + 3280 \beta_{13} + 8969 \beta_{14} + 4276 \beta_{15} ) q^{62} + 992436543 q^{63} + ( 3196870544 - 40385877 \beta_{1} + 2403122 \beta_{2} - 573017 \beta_{3} - 2505 \beta_{4} - 1944 \beta_{5} + 5229 \beta_{6} + 17363 \beta_{7} + 10987 \beta_{8} + 4293 \beta_{9} - 4294 \beta_{10} + 5923 \beta_{11} + 15485 \beta_{12} - 11400 \beta_{13} - 7455 \beta_{14} + 129 \beta_{15} ) q^{64} + ( 73887307 - 7425860 \beta_{1} - 371293 \beta_{3} ) q^{65} + ( 567487134 + 27780732 \beta_{1} + 477738 \beta_{2} - 390015 \beta_{3} + 6318 \beta_{4} + 6804 \beta_{5} - 972 \beta_{6} + 5103 \beta_{7} - 1944 \beta_{8} + 4374 \beta_{9} - 1215 \beta_{10} - 4131 \beta_{11} + 6318 \beta_{12} - 3402 \beta_{13} + 972 \beta_{14} - 1215 \beta_{15} ) q^{66} + ( -331594140 + 5189100 \beta_{1} - 1701177 \beta_{2} + 163249 \beta_{3} - 234 \beta_{4} + 7702 \beta_{5} + 1282 \beta_{6} + 1453 \beta_{7} + 320 \beta_{8} - 18084 \beta_{9} + 2047 \beta_{10} + 1194 \beta_{11} + 1133 \beta_{12} + 1722 \beta_{13} - 9412 \beta_{14} - 6611 \beta_{15} ) q^{67} + ( 488209899 + 102134017 \beta_{1} - 1522494 \beta_{2} - 442770 \beta_{3} + 89155 \beta_{4} + 7418 \beta_{5} + 3476 \beta_{6} - 8874 \beta_{7} - 14518 \beta_{8} + 3476 \beta_{9} + 42 \beta_{10} - 15088 \beta_{11} - 12610 \beta_{12} + 11012 \beta_{13} + 11412 \beta_{14} + 7980 \beta_{15} ) q^{68} + ( 2284388325 + 17445942 \beta_{1} + 597294 \beta_{2} - 269001 \beta_{3} - 12393 \beta_{4} - 243 \beta_{5} + 2187 \beta_{6} + 8019 \beta_{7} + 486 \beta_{8} + 1458 \beta_{9} + 972 \beta_{10} - 729 \beta_{11} - 4617 \beta_{13} - 2916 \beta_{14} - 972 \beta_{15} ) q^{69} + ( 1110404876 - 7378273 \beta_{1} + 84035 \beta_{2} - 336140 \beta_{3} - 16807 \beta_{4} + 16807 \beta_{5} ) q^{70} + ( 1066178697 + 21303921 \beta_{1} - 1308991 \beta_{2} + 425629 \beta_{3} - 21446 \beta_{4} - 3749 \beta_{5} + 1822 \beta_{6} + 4386 \beta_{7} + 4609 \beta_{8} - 21370 \beta_{9} - 2339 \beta_{10} + 5576 \beta_{11} - 873 \beta_{12} + 6911 \beta_{13} - 8955 \beta_{14} - 16363 \beta_{15} ) q^{71} + ( -699435405 + 90522117 \beta_{1} - 354294 \beta_{2} - 118098 \beta_{3} + 59049 \beta_{4} ) q^{72} + ( 1318718484 - 22521913 \beta_{1} + 474162 \beta_{2} - 176543 \beta_{3} + 6710 \beta_{4} - 9081 \beta_{5} - 4322 \beta_{6} + 6227 \beta_{7} - 2844 \beta_{8} - 11312 \beta_{9} - 5878 \beta_{10} + 5960 \beta_{11} + 12203 \beta_{12} + 5648 \beta_{13} - 283 \beta_{14} - 6056 \beta_{15} ) q^{73} + ( 3308982835 - 151480994 \beta_{1} + 723974 \beta_{2} - 1136062 \beta_{3} + 9537 \beta_{4} - 776 \beta_{5} - 1213 \beta_{6} - 923 \beta_{7} - 1052 \beta_{8} + 15264 \beta_{9} + 3621 \beta_{10} - 3782 \beta_{11} + 4700 \beta_{12} - 7403 \beta_{13} + 4378 \beta_{14} + 12160 \beta_{15} ) q^{74} + ( -2297680668 + 69279543 \beta_{1} - 1630773 \beta_{2} - 10935 \beta_{3} + 46656 \beta_{4} - 11907 \beta_{5} + 1215 \beta_{6} - 3645 \beta_{7} + 5346 \beta_{8} + 243 \beta_{9} + 2673 \beta_{10} + 8019 \beta_{11} + 2430 \beta_{12} - 4131 \beta_{13} + 2430 \beta_{14} + 4374 \beta_{15} ) q^{75} + ( 5109363449 - 289893173 \beta_{1} + 1306328 \beta_{2} - 619220 \beta_{3} - 102107 \beta_{4} - 19920 \beta_{5} + 1359 \beta_{6} - 14061 \beta_{7} - 1095 \beta_{8} - 16066 \beta_{9} - 1321 \beta_{10} + 11650 \beta_{11} + 2443 \beta_{12} + 10131 \beta_{13} - 348 \beta_{14} - 11024 \beta_{15} ) q^{76} + ( -489789594 - 12739706 \beta_{1} - 638666 \beta_{2} - 151263 \beta_{3} - 16807 \beta_{4} + 16807 \beta_{9} ) q^{77} + ( -360896796 + 90224199 \beta_{1} ) q^{78} + ( -5228324044 - 41212575 \beta_{1} - 4640952 \beta_{2} - 597540 \beta_{3} - 40538 \beta_{4} + 4454 \beta_{5} - 14794 \beta_{6} - 6985 \beta_{7} + 13111 \beta_{8} - 4297 \beta_{9} + 1273 \beta_{10} - 4169 \beta_{11} - 5080 \beta_{12} - 5139 \beta_{13} + 7132 \beta_{14} + 7374 \beta_{15} ) q^{79} + ( -10252222518 + 41202179 \beta_{1} - 6378890 \beta_{2} + 3325089 \beta_{3} + 181735 \beta_{4} - 88700 \beta_{5} + 11549 \beta_{6} - 821 \beta_{7} + 26287 \beta_{8} - 23857 \beta_{9} + 2028 \beta_{10} + 29137 \beta_{11} - 15775 \beta_{12} - 638 \beta_{13} - 7625 \beta_{14} - 9041 \beta_{15} ) q^{80} + 3486784401 q^{81} + ( -3852300328 + 81688418 \beta_{1} - 4289300 \beta_{2} + 333788 \beta_{3} + 116289 \beta_{4} + 7930 \beta_{5} + 11740 \beta_{6} - 10619 \beta_{7} - 20695 \beta_{8} - 19441 \beta_{9} - 3406 \beta_{10} - 17890 \beta_{11} - 13243 \beta_{12} + 23235 \beta_{13} + 7211 \beta_{14} + 11034 \beta_{15} ) q^{82} + ( -4557373084 + 46686629 \beta_{1} - 3588050 \beta_{2} - 924532 \beta_{3} - 21053 \beta_{4} + 24197 \beta_{5} - 4611 \beta_{6} + 4486 \beta_{7} - 4273 \beta_{8} - 2337 \beta_{9} + 9705 \beta_{10} + 1160 \beta_{11} + 6876 \beta_{12} - 3268 \beta_{13} + 1930 \beta_{14} + 22084 \beta_{15} ) q^{83} + ( -5301163098 + 16336404 \beta_{1} - 4084101 \beta_{2} ) q^{84} + ( -791758367 + 52949282 \beta_{1} - 1902241 \beta_{2} - 256291 \beta_{3} - 54536 \beta_{4} + 43679 \beta_{5} - 13527 \beta_{6} + 3988 \beta_{7} - 32424 \beta_{8} + 8432 \beta_{9} - 21928 \beta_{10} - 31538 \beta_{11} + 7815 \beta_{12} + 1957 \beta_{13} + 8310 \beta_{14} + 15776 \beta_{15} ) q^{85} + ( -1243579372 - 289561899 \beta_{1} - 2609671 \beta_{2} - 151974 \beta_{3} - 34000 \beta_{4} - 82613 \beta_{5} + 16926 \beta_{6} + 7797 \beta_{7} + 18661 \beta_{8} - 573 \beta_{9} - 2770 \beta_{10} + 15756 \beta_{11} + 12141 \beta_{12} - 2291 \beta_{13} - 16377 \beta_{14} - 4700 \beta_{15} ) q^{86} + ( 1689853104 + 38238237 \beta_{1} + 1419363 \beta_{2} - 384426 \beta_{3} + 12150 \beta_{4} + 14337 \beta_{5} + 5832 \beta_{6} + 8748 \beta_{7} + 15066 \beta_{8} + 4860 \beta_{9} - 9963 \beta_{10} + 2916 \beta_{11} - 1458 \beta_{12} + 10692 \beta_{13} - 10935 \beta_{14} + 4131 \beta_{15} ) q^{87} + ( -12614919792 - 399285109 \beta_{1} - 15389254 \beta_{2} + 5894431 \beta_{3} - 20569 \beta_{4} - 94436 \beta_{5} - 16639 \beta_{6} - 41121 \beta_{7} + 18751 \beta_{8} - 42919 \beta_{9} + 4682 \beta_{10} + 32739 \beta_{11} - 51439 \beta_{12} + 4960 \beta_{13} - 18371 \beta_{14} - 45999 \beta_{15} ) q^{88} + ( 2012719387 + 33129976 \beta_{1} - 249676 \beta_{2} - 559722 \beta_{3} + 111998 \beta_{4} - 11475 \beta_{5} - 446 \beta_{6} + 35512 \beta_{7} + 35676 \beta_{8} + 45683 \beta_{9} + 7635 \beta_{10} - 29235 \beta_{11} + 18274 \beta_{12} - 46164 \beta_{13} + 8917 \beta_{14} + 19960 \beta_{15} ) q^{89} + ( 3901249332 - 25922511 \beta_{1} + 295245 \beta_{2} - 1180980 \beta_{3} - 59049 \beta_{4} + 59049 \beta_{5} ) q^{90} -6240321451 q^{91} + ( -30436459956 - 125523585 \beta_{1} - 20133078 \beta_{2} + 2371815 \beta_{3} - 72651 \beta_{4} - 10190 \beta_{5} - 31264 \beta_{6} - 33724 \beta_{7} - 60896 \beta_{8} + 54885 \beta_{9} + 9477 \beta_{10} - 44129 \beta_{11} - 46364 \beta_{12} + 7829 \beta_{13} + 2003 \beta_{14} + 10127 \beta_{15} ) q^{92} + ( 1375725546 - 12564072 \beta_{1} - 3004938 \beta_{2} + 1709019 \beta_{3} + 166455 \beta_{4} + 36936 \beta_{5} + 7047 \beta_{6} - 13851 \beta_{7} - 29160 \beta_{8} + 8748 \beta_{9} - 486 \beta_{10} - 11421 \beta_{11} - 8748 \beta_{12} + 8019 \beta_{13} + 27945 \beta_{14} - 2673 \beta_{15} ) q^{93} + ( 11515043604 - 309833988 \beta_{1} + 3718650 \beta_{2} + 3164456 \beta_{3} + 165057 \beta_{4} - 29447 \beta_{5} - 3240 \beta_{6} + 14472 \beta_{7} + 40583 \beta_{8} + 72236 \beta_{9} + 6086 \beta_{10} + 17044 \beta_{11} + 13271 \beta_{12} - 69316 \beta_{13} + 37200 \beta_{14} + 73954 \beta_{15} ) q^{94} + ( -8917311519 - 357367221 \beta_{1} - 6571203 \beta_{2} + 1943167 \beta_{3} + 186616 \beta_{4} - 109617 \beta_{5} + 7391 \beta_{6} + 64151 \beta_{7} + 20364 \beta_{8} + 30305 \beta_{9} + 5485 \beta_{10} + 27937 \beta_{11} + 26050 \beta_{12} - 21353 \beta_{13} - 28750 \beta_{14} - 12860 \beta_{15} ) q^{95} + ( 9301279896 - 594828279 \beta_{1} + 3853494 \beta_{2} + 55161 \beta_{3} - 422091 \beta_{4} + 28188 \beta_{5} - 18225 \beta_{6} + 7533 \beta_{7} + 8505 \beta_{8} + 11907 \beta_{9} + 2430 \beta_{10} + 21141 \beta_{11} + 24543 \beta_{12} + 8748 \beta_{13} - 60993 \beta_{14} - 59049 \beta_{15} ) q^{96} + ( 455290903 - 327830240 \beta_{1} - 5542011 \beta_{2} + 136449 \beta_{3} - 182617 \beta_{4} + 19554 \beta_{5} + 19296 \beta_{6} - 8935 \beta_{7} + 10863 \beta_{8} - 35807 \beta_{9} + 9597 \beta_{10} + 42600 \beta_{11} + 4166 \beta_{12} - 45853 \beta_{13} - 7314 \beta_{14} - 24949 \beta_{15} ) q^{97} + ( -1129900996 + 282475249 \beta_{1} ) q^{98} + ( -1720805958 - 44759142 \beta_{1} - 2243862 \beta_{2} - 531441 \beta_{3} - 59049 \beta_{4} + 59049 \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 63q^{2} - 3888q^{3} + 20761q^{4} - 3168q^{5} + 15309q^{6} + 268912q^{7} - 187965q^{8} + 944784q^{9} + O(q^{10}) \) \( 16q - 63q^{2} - 3888q^{3} + 20761q^{4} - 3168q^{5} + 15309q^{6} + 268912q^{7} - 187965q^{8} + 944784q^{9} + 1056711q^{10} - 466884q^{11} - 5044923q^{12} - 5940688q^{13} - 1058841q^{14} + 769824q^{15} + 40058265q^{16} + 1753452q^{17} - 3720087q^{18} + 4237800q^{19} - 20710503q^{20} - 65345616q^{21} - 37479661q^{22} - 150481440q^{23} + 45675495q^{24} + 150983176q^{25} + 23391459q^{26} - 229582512q^{27} + 348930127q^{28} - 111411432q^{29} - 256780773q^{30} - 90536236q^{31} - 609941313q^{32} + 113452812q^{33} + 443745577q^{34} - 53244576q^{35} + 1225916289q^{36} - 1756337900q^{37} - 4378868973q^{38} + 1443587184q^{39} + 57535383q^{40} + 649575720q^{41} + 257298363q^{42} - 1889554520q^{43} - 4979587689q^{44} - 187067232q^{45} - 3204000867q^{46} - 4036082940q^{47} - 9734158395q^{48} + 4519603984q^{49} - 15928886862q^{50} - 426088836q^{51} - 7708413973q^{52} + 511144020q^{53} + 903981141q^{54} - 8519365572q^{55} - 3159127755q^{56} - 1029785400q^{57} - 7851149577q^{58} + 3728287296q^{59} + 5032652229q^{60} + 26227205052q^{61} + 2996618490q^{62} + 15878984688q^{63} + 51104408465q^{64} + 1176256224q^{65} + 9107557623q^{66} - 5295680024q^{67} + 7919540325q^{68} + 36566989920q^{69} + 17760141777q^{70} + 17082800928q^{71} - 11099145285q^{72} + 21076301488q^{73} + 52794333675q^{74} - 36688911768q^{75} + 81459179177q^{76} - 7846919388q^{77} - 5684124537q^{78} - 83677977852q^{79} - 163988465427q^{80} + 55788550416q^{81} - 61543728760q^{82} - 72857072340q^{83} - 84790020861q^{84} - 12608565060q^{85} - 20177818011q^{86} + 27072977976q^{87} - 202213679643q^{88} + 32238476676q^{89} + 62397727839q^{90} - 99845143216q^{91} - 487057197033q^{92} + 22000305348q^{93} + 183904776602q^{94} - 143022915504q^{95} + 148215739059q^{96} + 6973535140q^{97} - 17795940687q^{98} - 27569033316q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - x^{15} - 26640 x^{14} - 38138 x^{13} + 279954048 x^{12} + 940095168 x^{11} - 1473489774048 x^{10} - 6951601887328 x^{9} + 4072481535854976 x^{8} + 21355761647276288 x^{7} - 5703452025696599040 x^{6} - 25041700255043457024 x^{5} + 3737994671267475603456 x^{4} + 6711995200063092490240 x^{3} - 963518272786595267739648 x^{2} + 1287697466669064032092160 x + 45555354651594891689197568\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \nu - 3330 \)
\(\beta_{3}\)\(=\)\((\)\(\)\(14\!\cdots\!61\)\( \nu^{15} - \)\(55\!\cdots\!69\)\( \nu^{14} - \)\(36\!\cdots\!08\)\( \nu^{13} + \)\(13\!\cdots\!06\)\( \nu^{12} + \)\(35\!\cdots\!60\)\( \nu^{11} - \)\(11\!\cdots\!32\)\( \nu^{10} - \)\(16\!\cdots\!32\)\( \nu^{9} + \)\(53\!\cdots\!88\)\( \nu^{8} + \)\(39\!\cdots\!72\)\( \nu^{7} - \)\(11\!\cdots\!48\)\( \nu^{6} - \)\(39\!\cdots\!96\)\( \nu^{5} + \)\(11\!\cdots\!24\)\( \nu^{4} + \)\(12\!\cdots\!44\)\( \nu^{3} - \)\(37\!\cdots\!92\)\( \nu^{2} + \)\(24\!\cdots\!48\)\( \nu + \)\(17\!\cdots\!16\)\(\)\()/ \)\(24\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(14\!\cdots\!61\)\( \nu^{15} - \)\(55\!\cdots\!69\)\( \nu^{14} - \)\(36\!\cdots\!08\)\( \nu^{13} + \)\(13\!\cdots\!06\)\( \nu^{12} + \)\(35\!\cdots\!60\)\( \nu^{11} - \)\(11\!\cdots\!32\)\( \nu^{10} - \)\(16\!\cdots\!32\)\( \nu^{9} + \)\(53\!\cdots\!88\)\( \nu^{8} + \)\(39\!\cdots\!72\)\( \nu^{7} - \)\(11\!\cdots\!48\)\( \nu^{6} - \)\(39\!\cdots\!96\)\( \nu^{5} + \)\(11\!\cdots\!24\)\( \nu^{4} + \)\(12\!\cdots\!44\)\( \nu^{3} - \)\(37\!\cdots\!92\)\( \nu^{2} + \)\(17\!\cdots\!48\)\( \nu + \)\(17\!\cdots\!16\)\(\)\()/ \)\(12\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(28\!\cdots\!61\)\( \nu^{15} + \)\(10\!\cdots\!49\)\( \nu^{14} + \)\(70\!\cdots\!08\)\( \nu^{13} - \)\(25\!\cdots\!86\)\( \nu^{12} - \)\(69\!\cdots\!60\)\( \nu^{11} + \)\(23\!\cdots\!72\)\( \nu^{10} + \)\(32\!\cdots\!12\)\( \nu^{9} - \)\(10\!\cdots\!28\)\( \nu^{8} - \)\(75\!\cdots\!12\)\( \nu^{7} + \)\(22\!\cdots\!88\)\( \nu^{6} + \)\(76\!\cdots\!96\)\( \nu^{5} - \)\(21\!\cdots\!64\)\( \nu^{4} - \)\(24\!\cdots\!84\)\( \nu^{3} + \)\(72\!\cdots\!92\)\( \nu^{2} - \)\(58\!\cdots\!68\)\( \nu - \)\(34\!\cdots\!16\)\(\)\()/ \)\(24\!\cdots\!80\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(51\!\cdots\!41\)\( \nu^{15} + \)\(19\!\cdots\!89\)\( \nu^{14} + \)\(12\!\cdots\!48\)\( \nu^{13} - \)\(46\!\cdots\!86\)\( \nu^{12} - \)\(12\!\cdots\!60\)\( \nu^{11} + \)\(42\!\cdots\!92\)\( \nu^{10} + \)\(59\!\cdots\!92\)\( \nu^{9} - \)\(18\!\cdots\!28\)\( \nu^{8} - \)\(13\!\cdots\!32\)\( \nu^{7} + \)\(40\!\cdots\!88\)\( \nu^{6} + \)\(13\!\cdots\!76\)\( \nu^{5} - \)\(39\!\cdots\!44\)\( \nu^{4} - \)\(44\!\cdots\!64\)\( \nu^{3} + \)\(13\!\cdots\!52\)\( \nu^{2} - \)\(10\!\cdots\!88\)\( \nu - \)\(62\!\cdots\!96\)\(\)\()/ \)\(62\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(88\!\cdots\!71\)\( \nu^{15} + \)\(33\!\cdots\!79\)\( \nu^{14} + \)\(22\!\cdots\!48\)\( \nu^{13} - \)\(80\!\cdots\!26\)\( \nu^{12} - \)\(21\!\cdots\!00\)\( \nu^{11} + \)\(73\!\cdots\!72\)\( \nu^{10} + \)\(10\!\cdots\!92\)\( \nu^{9} - \)\(32\!\cdots\!88\)\( \nu^{8} - \)\(23\!\cdots\!52\)\( \nu^{7} + \)\(70\!\cdots\!48\)\( \nu^{6} + \)\(24\!\cdots\!96\)\( \nu^{5} - \)\(67\!\cdots\!84\)\( \nu^{4} - \)\(77\!\cdots\!04\)\( \nu^{3} + \)\(22\!\cdots\!32\)\( \nu^{2} - \)\(16\!\cdots\!88\)\( \nu - \)\(10\!\cdots\!96\)\(\)\()/ \)\(77\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(14\!\cdots\!17\)\( \nu^{15} - \)\(56\!\cdots\!33\)\( \nu^{14} - \)\(37\!\cdots\!76\)\( \nu^{13} + \)\(13\!\cdots\!42\)\( \nu^{12} + \)\(36\!\cdots\!00\)\( \nu^{11} - \)\(12\!\cdots\!84\)\( \nu^{10} - \)\(17\!\cdots\!24\)\( \nu^{9} + \)\(53\!\cdots\!56\)\( \nu^{8} + \)\(39\!\cdots\!04\)\( \nu^{7} - \)\(11\!\cdots\!56\)\( \nu^{6} - \)\(40\!\cdots\!12\)\( \nu^{5} + \)\(11\!\cdots\!28\)\( \nu^{4} + \)\(12\!\cdots\!28\)\( \nu^{3} - \)\(37\!\cdots\!04\)\( \nu^{2} + \)\(25\!\cdots\!36\)\( \nu + \)\(17\!\cdots\!72\)\(\)\()/ \)\(12\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(81\!\cdots\!79\)\( \nu^{15} + \)\(31\!\cdots\!91\)\( \nu^{14} + \)\(20\!\cdots\!12\)\( \nu^{13} - \)\(73\!\cdots\!34\)\( \nu^{12} - \)\(20\!\cdots\!40\)\( \nu^{11} + \)\(67\!\cdots\!48\)\( \nu^{10} + \)\(95\!\cdots\!48\)\( \nu^{9} - \)\(29\!\cdots\!32\)\( \nu^{8} - \)\(22\!\cdots\!08\)\( \nu^{7} + \)\(65\!\cdots\!72\)\( \nu^{6} + \)\(22\!\cdots\!44\)\( \nu^{5} - \)\(62\!\cdots\!36\)\( \nu^{4} - \)\(71\!\cdots\!16\)\( \nu^{3} + \)\(21\!\cdots\!88\)\( \nu^{2} - \)\(16\!\cdots\!72\)\( \nu - \)\(99\!\cdots\!24\)\(\)\()/ \)\(62\!\cdots\!00\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(61\!\cdots\!71\)\( \nu^{15} - \)\(23\!\cdots\!39\)\( \nu^{14} - \)\(15\!\cdots\!08\)\( \nu^{13} + \)\(55\!\cdots\!46\)\( \nu^{12} + \)\(15\!\cdots\!20\)\( \nu^{11} - \)\(51\!\cdots\!72\)\( \nu^{10} - \)\(72\!\cdots\!52\)\( \nu^{9} + \)\(22\!\cdots\!28\)\( \nu^{8} + \)\(16\!\cdots\!32\)\( \nu^{7} - \)\(49\!\cdots\!68\)\( \nu^{6} - \)\(16\!\cdots\!36\)\( \nu^{5} + \)\(47\!\cdots\!04\)\( \nu^{4} + \)\(53\!\cdots\!84\)\( \nu^{3} - \)\(15\!\cdots\!32\)\( \nu^{2} + \)\(11\!\cdots\!28\)\( \nu + \)\(75\!\cdots\!36\)\(\)\()/ \)\(36\!\cdots\!00\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(45\!\cdots\!39\)\( \nu^{15} + \)\(17\!\cdots\!19\)\( \nu^{14} + \)\(11\!\cdots\!88\)\( \nu^{13} - \)\(40\!\cdots\!14\)\( \nu^{12} - \)\(11\!\cdots\!56\)\( \nu^{11} + \)\(37\!\cdots\!12\)\( \nu^{10} + \)\(52\!\cdots\!60\)\( \nu^{9} - \)\(16\!\cdots\!92\)\( \nu^{8} - \)\(12\!\cdots\!72\)\( \nu^{7} + \)\(36\!\cdots\!24\)\( \nu^{6} + \)\(12\!\cdots\!68\)\( \nu^{5} - \)\(34\!\cdots\!88\)\( \nu^{4} - \)\(39\!\cdots\!52\)\( \nu^{3} + \)\(11\!\cdots\!12\)\( \nu^{2} - \)\(81\!\cdots\!80\)\( \nu - \)\(55\!\cdots\!64\)\(\)\()/ \)\(24\!\cdots\!80\)\( \)
\(\beta_{12}\)\(=\)\((\)\(\)\(78\!\cdots\!29\)\( \nu^{15} - \)\(30\!\cdots\!21\)\( \nu^{14} - \)\(19\!\cdots\!52\)\( \nu^{13} + \)\(71\!\cdots\!74\)\( \nu^{12} + \)\(19\!\cdots\!00\)\( \nu^{11} - \)\(65\!\cdots\!28\)\( \nu^{10} - \)\(91\!\cdots\!08\)\( \nu^{9} + \)\(28\!\cdots\!12\)\( \nu^{8} + \)\(21\!\cdots\!48\)\( \nu^{7} - \)\(62\!\cdots\!52\)\( \nu^{6} - \)\(21\!\cdots\!04\)\( \nu^{5} + \)\(60\!\cdots\!16\)\( \nu^{4} + \)\(68\!\cdots\!96\)\( \nu^{3} - \)\(20\!\cdots\!68\)\( \nu^{2} + \)\(14\!\cdots\!12\)\( \nu + \)\(96\!\cdots\!04\)\(\)\()/ \)\(31\!\cdots\!00\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(17\!\cdots\!93\)\( \nu^{15} - \)\(66\!\cdots\!57\)\( \nu^{14} - \)\(43\!\cdots\!84\)\( \nu^{13} + \)\(15\!\cdots\!58\)\( \nu^{12} + \)\(42\!\cdots\!00\)\( \nu^{11} - \)\(14\!\cdots\!76\)\( \nu^{10} - \)\(20\!\cdots\!36\)\( \nu^{9} + \)\(63\!\cdots\!04\)\( \nu^{8} + \)\(46\!\cdots\!16\)\( \nu^{7} - \)\(13\!\cdots\!84\)\( \nu^{6} - \)\(47\!\cdots\!68\)\( \nu^{5} + \)\(13\!\cdots\!72\)\( \nu^{4} + \)\(15\!\cdots\!32\)\( \nu^{3} - \)\(44\!\cdots\!56\)\( \nu^{2} + \)\(30\!\cdots\!04\)\( \nu + \)\(21\!\cdots\!68\)\(\)\()/ \)\(62\!\cdots\!00\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(49\!\cdots\!03\)\( \nu^{15} + \)\(19\!\cdots\!07\)\( \nu^{14} + \)\(12\!\cdots\!64\)\( \nu^{13} - \)\(44\!\cdots\!58\)\( \nu^{12} - \)\(12\!\cdots\!20\)\( \nu^{11} + \)\(41\!\cdots\!16\)\( \nu^{10} + \)\(57\!\cdots\!36\)\( \nu^{9} - \)\(18\!\cdots\!64\)\( \nu^{8} - \)\(13\!\cdots\!16\)\( \nu^{7} + \)\(39\!\cdots\!64\)\( \nu^{6} + \)\(13\!\cdots\!28\)\( \nu^{5} - \)\(38\!\cdots\!12\)\( \nu^{4} - \)\(43\!\cdots\!12\)\( \nu^{3} + \)\(12\!\cdots\!96\)\( \nu^{2} - \)\(91\!\cdots\!04\)\( \nu - \)\(60\!\cdots\!08\)\(\)\()/ \)\(12\!\cdots\!00\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(12\!\cdots\!89\)\( \nu^{15} - \)\(46\!\cdots\!21\)\( \nu^{14} - \)\(30\!\cdots\!12\)\( \nu^{13} + \)\(11\!\cdots\!14\)\( \nu^{12} + \)\(29\!\cdots\!20\)\( \nu^{11} - \)\(10\!\cdots\!08\)\( \nu^{10} - \)\(14\!\cdots\!48\)\( \nu^{9} + \)\(44\!\cdots\!52\)\( \nu^{8} + \)\(32\!\cdots\!48\)\( \nu^{7} - \)\(97\!\cdots\!92\)\( \nu^{6} - \)\(33\!\cdots\!84\)\( \nu^{5} + \)\(93\!\cdots\!16\)\( \nu^{4} + \)\(10\!\cdots\!96\)\( \nu^{3} - \)\(31\!\cdots\!48\)\( \nu^{2} + \)\(22\!\cdots\!72\)\( \nu + \)\(14\!\cdots\!24\)\(\)\()/ \)\(24\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4 \beta_{1} + 3330\)
\(\nu^{3}\)\(=\)\(\beta_{4} - 2 \beta_{3} + 6 \beta_{2} + 5629 \beta_{1} + 11795\)
\(\nu^{4}\)\(=\)\(-\beta_{15} - \beta_{14} - 2 \beta_{13} + 5 \beta_{12} + \beta_{11} - \beta_{9} + 3 \beta_{8} + 7 \beta_{7} + \beta_{6} - 4 \beta_{5} + 7 \beta_{4} - 167 \beta_{3} + 7490 \beta_{2} + 50147 \beta_{1} + 18737114\)
\(\nu^{5}\)\(=\)\(223 \beta_{15} + 231 \beta_{14} - 76 \beta_{13} - \beta_{12} - 67 \beta_{11} - 10 \beta_{10} - 69 \beta_{9} + 25 \beta_{8} + 109 \beta_{7} + 95 \beta_{6} - 196 \beta_{5} + 9909 \beta_{4} - 19631 \beta_{3} + 84470 \beta_{2} + 36081289 \beta_{1} + 155790112\)
\(\nu^{6}\)\(=\)\(-4519 \beta_{15} - 11911 \beta_{14} - 33224 \beta_{13} + 65461 \beta_{12} + 14315 \beta_{11} - 4534 \beta_{10} - 7363 \beta_{9} + 41587 \beta_{8} + 89979 \beta_{7} + 17509 \beta_{6} - 46648 \beta_{5} + 142751 \beta_{4} - 2389041 \beta_{3} + 54168418 \beta_{2} + 513949619 \beta_{1} + 120038018864\)
\(\nu^{7}\)\(=\)\(3016319 \beta_{15} + 2826975 \beta_{14} - 1359700 \beta_{13} + 515327 \beta_{12} - 667675 \beta_{11} - 327582 \beta_{10} - 947725 \beta_{9} + 610633 \beta_{8} + 2370185 \beta_{7} + 1571303 \beta_{6} - 2737056 \beta_{5} + 84674381 \beta_{4} - 156690951 \beta_{3} + 969639486 \beta_{2} + 246474907289 \beta_{1} + 1631988243260\)
\(\nu^{8}\)\(=\)\(19632589 \beta_{15} - 90806243 \beta_{14} - 371302268 \beta_{13} + 643084029 \beta_{12} + 137173839 \beta_{11} - 67455826 \beta_{10} - 51131783 \beta_{9} + 417481275 \beta_{8} + 891517779 \beta_{7} + 200153893 \beta_{6} - 408398104 \beta_{5} + 1879596095 \beta_{4} - 23824279781 \beta_{3} + 400194063386 \beta_{2} + 5006848471091 \beta_{1} + 819405787445428\)
\(\nu^{9}\)\(=\)\(30318886279 \beta_{15} + 26065139479 \beta_{14} - 17434683404 \beta_{13} + 10583726927 \beta_{12} - 4614326403 \beta_{11} - 5009898798 \beta_{10} - 9940470501 \beta_{9} + 9662640057 \beta_{8} + 32072797345 \beta_{7} + 17765929239 \beta_{6} - 27720787080 \beta_{5} + 697606312053 \beta_{4} - 1198188533263 \beta_{3} + 10080630775054 \beta_{2} + 1761018033125297 \beta_{1} + 16093470442367188\)
\(\nu^{10}\)\(=\)\(658708717629 \beta_{15} - 531994143347 \beta_{14} - 3594417665428 \beta_{13} + 5713230246213 \beta_{12} + 1143387161903 \beta_{11} - 744270717722 \beta_{10} - 395361234983 \beta_{9} + 3756001736851 \beta_{8} + 8080896495355 \beta_{7} + 1962729259677 \beta_{6} - 3251877274056 \beta_{5} + 20887698307799 \beta_{4} - 210112956052949 \beta_{3} + 3030108216926010 \beta_{2} + 47561726449856651 \beta_{1} + 5849717499791410444\)
\(\nu^{11}\)\(=\)\(275044655337567 \beta_{15} + 218121123242959 \beta_{14} - 192481955056812 \beta_{13} + 143115322007559 \beta_{12} - 24971499305435 \beta_{11} - 59067996957518 \beta_{10} - 93608051374925 \beta_{9} + 121623499887873 \beta_{8} + 360276502713433 \beta_{7} + 174640637963119 \beta_{6} - 250507286716088 \beta_{5} + 5685102805810797 \beta_{4} - 9206741129929703 \beta_{3} + 99069319901308158 \beta_{2} + 13035088967826978089 \beta_{1} + 154029265675351800148\)
\(\nu^{12}\)\(=\)\(9271079965215813 \beta_{15} - 2131935527077291 \beta_{14} - 32636943047759140 \beta_{13} + 48616484640565565 \beta_{12} + 8992989169154023 \beta_{11} - 7341298170305882 \beta_{10} - 3341016192294431 \beta_{9} + 32288456434353435 \beta_{8} + 70438692555043331 \beta_{7} + 17954462580982133 \beta_{6} - 25038106490343560 \beta_{5} + 212777306515936095 \beta_{4} - 1758423639541920029 \beta_{3} + 23439690891514719178 \beta_{2} + 443613991941971870195 \beta_{1} + 43262484296604959313436\)
\(\nu^{13}\)\(=\)\(2388657324717895303 \beta_{15} + 1754797531043935159 \beta_{14} - 1951971939580560364 \beta_{13} + 1622032146410458191 \beta_{12} - 87091950230352547 \beta_{11} - 614305640193829534 \beta_{10} - 833991386134105669 \beta_{9} + 1341736951725697913 \beta_{8} + 3676796861127852657 \beta_{7} + 1609255369199759863 \beta_{6} - 2154359507158022744 \beta_{5} + 46255029976921783269 \beta_{4} - 72438297223767266383 \beta_{3} + 939441114891038791118 \beta_{2} + 99260619904961143738113 \beta_{1} + 1443715443968935056215188\)
\(\nu^{14}\)\(=\)\(104827824961335943005 \beta_{15} + 1409955217071173357 \beta_{14} - 287096155021776296740 \beta_{13} + 405911744929578083509 \beta_{12} + 69167728295278119727 \beta_{11} - 68498648507027660010 \beta_{10} - 29524814055078451879 \beta_{9} + 272204117235497279587 \beta_{8} + 602635541654402110667 \beta_{7} + 158644633268400331117 \beta_{6} - 191715388217970030984 \beta_{5} + 2058399396769367143559 \beta_{4} - 14374316217219065192309 \beta_{3} + 184579594373257091197434 \beta_{2} + 4075952205154780947870075 \beta_{1} + 329157709469569942386589020\)
\(\nu^{15}\)\(=\)\(20343207496408982460527 \beta_{15} + 13912055686017851817631 \beta_{14} - 18774601349287886936332 \beta_{13} + 16722201213592025905015 \beta_{12} + 218529867206734145621 \beta_{11} - 5954284682400982800430 \beta_{10} - 7199022995207184707325 \beta_{9} + 13650719170022517569041 \beta_{8} + 35462321517509352586889 \beta_{7} + 14341452707112497164639 \beta_{6} - 18112396040050035171800 \beta_{5} + 377261595349433169741405 \beta_{4} - 586822336244933905096183 \beta_{3} + 8692016909621933873071262 \beta_{2} + 773191829260363607692866201 \beta_{1} + 13309379487939348254834094580\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−84.0758
−81.4415
−68.0202
−65.6788
−37.3711
−37.3707
−23.9940
−6.60872
9.73282
18.3601
25.8857
41.2629
67.4693
72.4378
78.0478
92.3644
−88.0758 −243.000 5709.35 −13401.9 21402.4 16807.0 −322476. 59049.0 1.18039e6
1.2 −85.4415 −243.000 5252.25 8053.09 20762.3 16807.0 −273776. 59049.0 −688068.
1.3 −72.0202 −243.000 3138.91 10386.5 17500.9 16807.0 −78567.7 59049.0 −748037.
1.4 −69.6788 −243.000 2807.14 −8533.64 16932.0 16807.0 −52895.7 59049.0 594614.
1.5 −41.3711 −243.000 −336.429 −4784.52 10053.2 16807.0 98646.5 59049.0 197941.
1.6 −41.3707 −243.000 −336.467 −335.043 10053.1 16807.0 98647.0 59049.0 13861.0
1.7 −27.9940 −243.000 −1264.33 −7765.11 6802.55 16807.0 92725.6 59049.0 217377.
1.8 −10.6087 −243.000 −1935.46 7850.01 2577.92 16807.0 42259.4 59049.0 −83278.5
1.9 5.73282 −243.000 −2015.13 −168.612 −1393.07 16807.0 −23293.2 59049.0 −966.623
1.10 14.3601 −243.000 −1841.79 4424.64 −3489.51 16807.0 −55857.8 59049.0 63538.3
1.11 21.8857 −243.000 −1569.02 −9779.77 −5318.23 16807.0 −79161.0 59049.0 −214037.
1.12 37.2629 −243.000 −659.475 3035.84 −9054.89 16807.0 −100888. 59049.0 113124.
1.13 63.4693 −243.000 1980.36 12941.9 −15423.1 16807.0 −4293.20 59049.0 821414.
1.14 68.4378 −243.000 2635.73 3481.49 −16630.4 16807.0 40223.3 59049.0 238265.
1.15 74.0478 −243.000 3435.08 −7551.16 −17993.6 16807.0 102710. 59049.0 −559147.
1.16 88.3644 −243.000 5760.27 −1021.65 −21472.6 16807.0 328033. 59049.0 −90277.5
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(13\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.12.a.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.12.a.c 16 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(11\!\cdots\!64\)\( T_{2}^{10} - \)\(61\!\cdots\!28\)\( T_{2}^{9} + \)\(28\!\cdots\!64\)\( T_{2}^{8} + \)\(13\!\cdots\!88\)\( T_{2}^{7} - \)\(33\!\cdots\!64\)\( T_{2}^{6} - \)\(14\!\cdots\!16\)\( T_{2}^{5} + \)\(19\!\cdots\!24\)\( T_{2}^{4} + \)\(55\!\cdots\!00\)\( T_{2}^{3} - \)\(56\!\cdots\!12\)\( T_{2}^{2} - \)\(52\!\cdots\!80\)\( T_{2} + \)\(36\!\cdots\!08\)\( \)">\(T_{2}^{16} + \cdots\) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(273))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( \)\(36\!\cdots\!08\)\( - \)\(52\!\cdots\!80\)\( T - \)\(56\!\cdots\!12\)\( T^{2} + \)\(55\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!24\)\( T^{4} - \)\(14\!\cdots\!16\)\( T^{5} - 3396107610598334464 T^{6} + 136659347406252288 T^{7} + 2806337209265664 T^{8} - 61183583231328 T^{9} - 1143989124064 T^{10} + 13713811200 T^{11} + 239619832 T^{12} - 1495818 T^{13} - 24780 T^{14} + 63 T^{15} + T^{16} \)
$3$ \( ( 243 + T )^{16} \)
$5$ \( -\)\(71\!\cdots\!00\)\( - \)\(67\!\cdots\!00\)\( T - \)\(15\!\cdots\!00\)\( T^{2} - \)\(24\!\cdots\!00\)\( T^{3} + \)\(72\!\cdots\!00\)\( T^{4} - \)\(22\!\cdots\!00\)\( T^{5} - \)\(73\!\cdots\!00\)\( T^{6} + \)\(36\!\cdots\!00\)\( T^{7} + \)\(31\!\cdots\!25\)\( T^{8} - \)\(13\!\cdots\!00\)\( T^{9} - \)\(70\!\cdots\!00\)\( T^{10} + \)\(19\!\cdots\!40\)\( T^{11} + 81240793211506326 T^{12} - 1297215167688 T^{13} - 461098476 T^{14} + 3168 T^{15} + T^{16} \)
$7$ \( ( -16807 + T )^{16} \)
$11$ \( \)\(26\!\cdots\!00\)\( - \)\(73\!\cdots\!72\)\( T - \)\(16\!\cdots\!84\)\( T^{2} + \)\(22\!\cdots\!92\)\( T^{3} + \)\(19\!\cdots\!12\)\( T^{4} - \)\(60\!\cdots\!88\)\( T^{5} - \)\(42\!\cdots\!08\)\( T^{6} + \)\(69\!\cdots\!24\)\( T^{7} + \)\(36\!\cdots\!00\)\( T^{8} - \)\(39\!\cdots\!00\)\( T^{9} - \)\(14\!\cdots\!44\)\( T^{10} + \)\(10\!\cdots\!80\)\( T^{11} + \)\(28\!\cdots\!93\)\( T^{12} - 1149148363875469452 T^{13} - 2745154787702 T^{14} + 466884 T^{15} + T^{16} \)
$13$ \( ( 371293 + T )^{16} \)
$17$ \( \)\(37\!\cdots\!00\)\( + \)\(74\!\cdots\!60\)\( T - \)\(17\!\cdots\!60\)\( T^{2} - \)\(23\!\cdots\!80\)\( T^{3} + \)\(52\!\cdots\!84\)\( T^{4} + \)\(28\!\cdots\!24\)\( T^{5} - \)\(66\!\cdots\!16\)\( T^{6} - \)\(17\!\cdots\!20\)\( T^{7} + \)\(43\!\cdots\!24\)\( T^{8} + \)\(58\!\cdots\!80\)\( T^{9} - \)\(15\!\cdots\!28\)\( T^{10} - \)\(99\!\cdots\!40\)\( T^{11} + \)\(28\!\cdots\!73\)\( T^{12} + \)\(76\!\cdots\!20\)\( T^{13} - 275571876515346 T^{14} - 1753452 T^{15} + T^{16} \)
$19$ \( \)\(19\!\cdots\!68\)\( - \)\(58\!\cdots\!16\)\( T - \)\(40\!\cdots\!64\)\( T^{2} + \)\(78\!\cdots\!40\)\( T^{3} + \)\(35\!\cdots\!04\)\( T^{4} - \)\(32\!\cdots\!16\)\( T^{5} - \)\(14\!\cdots\!20\)\( T^{6} + \)\(56\!\cdots\!80\)\( T^{7} + \)\(29\!\cdots\!45\)\( T^{8} - \)\(37\!\cdots\!92\)\( T^{9} - \)\(34\!\cdots\!00\)\( T^{10} - \)\(46\!\cdots\!44\)\( T^{11} + \)\(22\!\cdots\!46\)\( T^{12} + \)\(14\!\cdots\!20\)\( T^{13} - 744549958259692 T^{14} - 4237800 T^{15} + T^{16} \)
$23$ \( -\)\(12\!\cdots\!00\)\( - \)\(50\!\cdots\!40\)\( T - \)\(90\!\cdots\!60\)\( T^{2} + \)\(18\!\cdots\!20\)\( T^{3} + \)\(19\!\cdots\!56\)\( T^{4} - \)\(92\!\cdots\!60\)\( T^{5} - \)\(26\!\cdots\!00\)\( T^{6} - \)\(14\!\cdots\!36\)\( T^{7} - \)\(10\!\cdots\!79\)\( T^{8} + \)\(19\!\cdots\!48\)\( T^{9} + \)\(68\!\cdots\!44\)\( T^{10} - \)\(48\!\cdots\!00\)\( T^{11} - \)\(40\!\cdots\!22\)\( T^{12} - \)\(63\!\cdots\!04\)\( T^{13} + 3073440182328792 T^{14} + 150481440 T^{15} + T^{16} \)
$29$ \( -\)\(18\!\cdots\!12\)\( - \)\(67\!\cdots\!36\)\( T + \)\(40\!\cdots\!96\)\( T^{2} + \)\(15\!\cdots\!16\)\( T^{3} - \)\(33\!\cdots\!64\)\( T^{4} - \)\(13\!\cdots\!52\)\( T^{5} + \)\(11\!\cdots\!20\)\( T^{6} + \)\(58\!\cdots\!12\)\( T^{7} - \)\(91\!\cdots\!23\)\( T^{8} - \)\(13\!\cdots\!56\)\( T^{9} - \)\(33\!\cdots\!16\)\( T^{10} + \)\(15\!\cdots\!80\)\( T^{11} + \)\(78\!\cdots\!58\)\( T^{12} - \)\(78\!\cdots\!16\)\( T^{13} - 54300566909843880 T^{14} + 111411432 T^{15} + T^{16} \)
$31$ \( \)\(53\!\cdots\!20\)\( + \)\(11\!\cdots\!88\)\( T + \)\(67\!\cdots\!92\)\( T^{2} + \)\(10\!\cdots\!92\)\( T^{3} - \)\(69\!\cdots\!36\)\( T^{4} - \)\(29\!\cdots\!08\)\( T^{5} - \)\(80\!\cdots\!28\)\( T^{6} + \)\(27\!\cdots\!00\)\( T^{7} + \)\(14\!\cdots\!64\)\( T^{8} - \)\(12\!\cdots\!64\)\( T^{9} - \)\(82\!\cdots\!08\)\( T^{10} + \)\(25\!\cdots\!44\)\( T^{11} + \)\(20\!\cdots\!93\)\( T^{12} - \)\(25\!\cdots\!04\)\( T^{13} - 235548240933258394 T^{14} + 90536236 T^{15} + T^{16} \)
$37$ \( -\)\(99\!\cdots\!12\)\( - \)\(52\!\cdots\!36\)\( T + \)\(15\!\cdots\!80\)\( T^{2} + \)\(29\!\cdots\!00\)\( T^{3} - \)\(98\!\cdots\!04\)\( T^{4} - \)\(55\!\cdots\!40\)\( T^{5} + \)\(36\!\cdots\!36\)\( T^{6} + \)\(21\!\cdots\!12\)\( T^{7} - \)\(18\!\cdots\!92\)\( T^{8} - \)\(38\!\cdots\!60\)\( T^{9} + \)\(34\!\cdots\!76\)\( T^{10} + \)\(50\!\cdots\!28\)\( T^{11} - \)\(26\!\cdots\!23\)\( T^{12} - \)\(49\!\cdots\!44\)\( T^{13} + 581954642242000178 T^{14} + 1756337900 T^{15} + T^{16} \)
$41$ \( -\)\(16\!\cdots\!00\)\( + \)\(72\!\cdots\!00\)\( T - \)\(72\!\cdots\!60\)\( T^{2} - \)\(47\!\cdots\!40\)\( T^{3} + \)\(73\!\cdots\!16\)\( T^{4} + \)\(76\!\cdots\!44\)\( T^{5} - \)\(13\!\cdots\!72\)\( T^{6} - \)\(96\!\cdots\!96\)\( T^{7} + \)\(88\!\cdots\!96\)\( T^{8} + \)\(57\!\cdots\!04\)\( T^{9} - \)\(27\!\cdots\!64\)\( T^{10} - \)\(15\!\cdots\!16\)\( T^{11} + \)\(44\!\cdots\!48\)\( T^{12} + \)\(16\!\cdots\!60\)\( T^{13} - 3393263600912409328 T^{14} - 649575720 T^{15} + T^{16} \)
$43$ \( \)\(73\!\cdots\!32\)\( - \)\(23\!\cdots\!40\)\( T - \)\(92\!\cdots\!88\)\( T^{2} + \)\(55\!\cdots\!32\)\( T^{3} + \)\(12\!\cdots\!12\)\( T^{4} - \)\(22\!\cdots\!24\)\( T^{5} - \)\(27\!\cdots\!92\)\( T^{6} + \)\(22\!\cdots\!04\)\( T^{7} - \)\(23\!\cdots\!67\)\( T^{8} - \)\(86\!\cdots\!24\)\( T^{9} - \)\(10\!\cdots\!40\)\( T^{10} + \)\(14\!\cdots\!04\)\( T^{11} + \)\(50\!\cdots\!62\)\( T^{12} - \)\(93\!\cdots\!32\)\( T^{13} - 4405323488821409560 T^{14} + 1889554520 T^{15} + T^{16} \)
$47$ \( -\)\(13\!\cdots\!00\)\( + \)\(12\!\cdots\!00\)\( T + \)\(36\!\cdots\!20\)\( T^{2} - \)\(25\!\cdots\!60\)\( T^{3} - \)\(58\!\cdots\!68\)\( T^{4} + \)\(39\!\cdots\!12\)\( T^{5} + \)\(15\!\cdots\!12\)\( T^{6} + \)\(35\!\cdots\!48\)\( T^{7} - \)\(13\!\cdots\!80\)\( T^{8} - \)\(79\!\cdots\!56\)\( T^{9} + \)\(39\!\cdots\!28\)\( T^{10} + \)\(40\!\cdots\!04\)\( T^{11} - \)\(10\!\cdots\!23\)\( T^{12} - \)\(72\!\cdots\!12\)\( T^{13} - 12231745397209532446 T^{14} + 4036082940 T^{15} + T^{16} \)
$53$ \( \)\(51\!\cdots\!64\)\( - \)\(44\!\cdots\!80\)\( T + \)\(13\!\cdots\!00\)\( T^{2} - \)\(16\!\cdots\!84\)\( T^{3} + \)\(82\!\cdots\!24\)\( T^{4} + \)\(11\!\cdots\!72\)\( T^{5} - \)\(47\!\cdots\!04\)\( T^{6} - \)\(31\!\cdots\!12\)\( T^{7} + \)\(17\!\cdots\!96\)\( T^{8} + \)\(39\!\cdots\!68\)\( T^{9} - \)\(26\!\cdots\!20\)\( T^{10} - \)\(24\!\cdots\!60\)\( T^{11} + \)\(19\!\cdots\!49\)\( T^{12} + \)\(66\!\cdots\!44\)\( T^{13} - 72865478781976582002 T^{14} - 511144020 T^{15} + T^{16} \)
$59$ \( -\)\(10\!\cdots\!00\)\( + \)\(56\!\cdots\!40\)\( T - \)\(52\!\cdots\!60\)\( T^{2} - \)\(35\!\cdots\!00\)\( T^{3} + \)\(48\!\cdots\!08\)\( T^{4} + \)\(79\!\cdots\!80\)\( T^{5} - \)\(12\!\cdots\!04\)\( T^{6} - \)\(82\!\cdots\!44\)\( T^{7} + \)\(13\!\cdots\!04\)\( T^{8} + \)\(41\!\cdots\!00\)\( T^{9} - \)\(75\!\cdots\!52\)\( T^{10} - \)\(98\!\cdots\!32\)\( T^{11} + \)\(19\!\cdots\!16\)\( T^{12} + \)\(10\!\cdots\!60\)\( T^{13} - \)\(23\!\cdots\!16\)\( T^{14} - 3728287296 T^{15} + T^{16} \)
$61$ \( -\)\(71\!\cdots\!00\)\( + \)\(44\!\cdots\!40\)\( T + \)\(10\!\cdots\!96\)\( T^{2} - \)\(57\!\cdots\!32\)\( T^{3} - \)\(10\!\cdots\!16\)\( T^{4} + \)\(14\!\cdots\!08\)\( T^{5} + \)\(28\!\cdots\!84\)\( T^{6} - \)\(23\!\cdots\!84\)\( T^{7} - \)\(30\!\cdots\!08\)\( T^{8} + \)\(28\!\cdots\!36\)\( T^{9} + \)\(14\!\cdots\!36\)\( T^{10} - \)\(16\!\cdots\!28\)\( T^{11} - \)\(24\!\cdots\!35\)\( T^{12} + \)\(40\!\cdots\!72\)\( T^{13} + 31516447958545087982 T^{14} - 26227205052 T^{15} + T^{16} \)
$67$ \( -\)\(53\!\cdots\!08\)\( + \)\(57\!\cdots\!40\)\( T - \)\(52\!\cdots\!52\)\( T^{2} - \)\(81\!\cdots\!44\)\( T^{3} + \)\(57\!\cdots\!92\)\( T^{4} + \)\(79\!\cdots\!76\)\( T^{5} - \)\(16\!\cdots\!32\)\( T^{6} + \)\(14\!\cdots\!04\)\( T^{7} + \)\(21\!\cdots\!24\)\( T^{8} - \)\(29\!\cdots\!08\)\( T^{9} - \)\(15\!\cdots\!92\)\( T^{10} + \)\(18\!\cdots\!04\)\( T^{11} + \)\(59\!\cdots\!56\)\( T^{12} - \)\(51\!\cdots\!32\)\( T^{13} - \)\(12\!\cdots\!00\)\( T^{14} + 5295680024 T^{15} + T^{16} \)
$71$ \( -\)\(62\!\cdots\!60\)\( + \)\(14\!\cdots\!20\)\( T + \)\(35\!\cdots\!04\)\( T^{2} - \)\(11\!\cdots\!68\)\( T^{3} - \)\(47\!\cdots\!84\)\( T^{4} + \)\(22\!\cdots\!44\)\( T^{5} - \)\(91\!\cdots\!32\)\( T^{6} - \)\(20\!\cdots\!84\)\( T^{7} + \)\(11\!\cdots\!64\)\( T^{8} + \)\(94\!\cdots\!36\)\( T^{9} - \)\(62\!\cdots\!56\)\( T^{10} - \)\(24\!\cdots\!04\)\( T^{11} + \)\(16\!\cdots\!40\)\( T^{12} + \)\(32\!\cdots\!84\)\( T^{13} - \)\(20\!\cdots\!52\)\( T^{14} - 17082800928 T^{15} + T^{16} \)
$73$ \( \)\(42\!\cdots\!00\)\( + \)\(29\!\cdots\!40\)\( T - \)\(60\!\cdots\!60\)\( T^{2} - \)\(92\!\cdots\!56\)\( T^{3} + \)\(29\!\cdots\!08\)\( T^{4} + \)\(31\!\cdots\!04\)\( T^{5} - \)\(58\!\cdots\!96\)\( T^{6} - \)\(45\!\cdots\!12\)\( T^{7} + \)\(58\!\cdots\!89\)\( T^{8} + \)\(33\!\cdots\!00\)\( T^{9} - \)\(32\!\cdots\!56\)\( T^{10} - \)\(13\!\cdots\!24\)\( T^{11} + \)\(98\!\cdots\!26\)\( T^{12} + \)\(26\!\cdots\!92\)\( T^{13} - \)\(15\!\cdots\!52\)\( T^{14} - 21076301488 T^{15} + T^{16} \)
$79$ \( -\)\(67\!\cdots\!00\)\( - \)\(57\!\cdots\!92\)\( T - \)\(16\!\cdots\!08\)\( T^{2} - \)\(16\!\cdots\!68\)\( T^{3} + \)\(32\!\cdots\!48\)\( T^{4} + \)\(18\!\cdots\!36\)\( T^{5} + \)\(89\!\cdots\!76\)\( T^{6} - \)\(44\!\cdots\!48\)\( T^{7} - \)\(49\!\cdots\!48\)\( T^{8} - \)\(36\!\cdots\!52\)\( T^{9} + \)\(89\!\cdots\!32\)\( T^{10} + \)\(25\!\cdots\!68\)\( T^{11} - \)\(50\!\cdots\!07\)\( T^{12} - \)\(28\!\cdots\!52\)\( T^{13} - \)\(12\!\cdots\!58\)\( T^{14} + 83677977852 T^{15} + T^{16} \)
$83$ \( -\)\(86\!\cdots\!92\)\( + \)\(85\!\cdots\!40\)\( T - \)\(11\!\cdots\!88\)\( T^{2} - \)\(52\!\cdots\!92\)\( T^{3} + \)\(82\!\cdots\!12\)\( T^{4} + \)\(24\!\cdots\!04\)\( T^{5} - \)\(79\!\cdots\!56\)\( T^{6} + \)\(15\!\cdots\!96\)\( T^{7} + \)\(21\!\cdots\!52\)\( T^{8} - \)\(68\!\cdots\!12\)\( T^{9} - \)\(24\!\cdots\!44\)\( T^{10} + \)\(83\!\cdots\!80\)\( T^{11} + \)\(15\!\cdots\!69\)\( T^{12} - \)\(41\!\cdots\!56\)\( T^{13} - \)\(59\!\cdots\!54\)\( T^{14} + 72857072340 T^{15} + T^{16} \)
$89$ \( -\)\(54\!\cdots\!00\)\( - \)\(10\!\cdots\!00\)\( T - \)\(33\!\cdots\!00\)\( T^{2} + \)\(34\!\cdots\!80\)\( T^{3} + \)\(18\!\cdots\!60\)\( T^{4} - \)\(32\!\cdots\!80\)\( T^{5} - \)\(29\!\cdots\!28\)\( T^{6} + \)\(64\!\cdots\!68\)\( T^{7} + \)\(22\!\cdots\!80\)\( T^{8} + \)\(48\!\cdots\!24\)\( T^{9} - \)\(88\!\cdots\!60\)\( T^{10} - \)\(27\!\cdots\!96\)\( T^{11} + \)\(19\!\cdots\!17\)\( T^{12} + \)\(50\!\cdots\!96\)\( T^{13} - \)\(22\!\cdots\!26\)\( T^{14} - 32238476676 T^{15} + T^{16} \)
$97$ \( \)\(37\!\cdots\!40\)\( + \)\(52\!\cdots\!80\)\( T + \)\(22\!\cdots\!52\)\( T^{2} - \)\(11\!\cdots\!32\)\( T^{3} - \)\(37\!\cdots\!52\)\( T^{4} - \)\(94\!\cdots\!88\)\( T^{5} + \)\(37\!\cdots\!48\)\( T^{6} + \)\(45\!\cdots\!20\)\( T^{7} + \)\(37\!\cdots\!20\)\( T^{8} - \)\(71\!\cdots\!76\)\( T^{9} - \)\(10\!\cdots\!08\)\( T^{10} + \)\(44\!\cdots\!24\)\( T^{11} + \)\(11\!\cdots\!73\)\( T^{12} - \)\(82\!\cdots\!88\)\( T^{13} - \)\(54\!\cdots\!74\)\( T^{14} - 6973535140 T^{15} + T^{16} \)
show more
show less