Properties

Label 14.15.d.a
Level $14$
Weight $15$
Character orbit 14.d
Analytic conductor $17.406$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 15 \)
Character orbit: \([\chi]\) \(=\) 14.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.4060555413\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 4 x^{19} + 5366534 x^{18} + 786490608 x^{17} + 19826027932115 x^{16} + 3116508483645696 x^{15} + 38794655189396647126 x^{14} + \)\(69\!\cdots\!60\)\( x^{13} + \)\(54\!\cdots\!57\)\( x^{12} + \)\(66\!\cdots\!72\)\( x^{11} + \)\(34\!\cdots\!28\)\( x^{10} - \)\(50\!\cdots\!72\)\( x^{9} + \)\(15\!\cdots\!96\)\( x^{8} - \)\(88\!\cdots\!84\)\( x^{7} + \)\(37\!\cdots\!08\)\( x^{6} - \)\(63\!\cdots\!24\)\( x^{5} + \)\(59\!\cdots\!84\)\( x^{4} - \)\(42\!\cdots\!56\)\( x^{3} + \)\(11\!\cdots\!48\)\( x^{2} + \)\(52\!\cdots\!00\)\( x + \)\(13\!\cdots\!24\)\(\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{76}\cdot 3^{12}\cdot 7^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} + ( -146 - 146 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} ) q^{3} + ( -8192 + 8192 \beta_{1} ) q^{4} + ( 222 - 111 \beta_{1} - 17 \beta_{2} - 4 \beta_{3} - 33 \beta_{4} + 4 \beta_{5} + \beta_{10} ) q^{5} + ( -9650 + 19300 \beta_{1} + 294 \beta_{2} + 5 \beta_{3} + 146 \beta_{4} + \beta_{8} - \beta_{10} + \beta_{11} ) q^{6} + ( 26479 + 92582 \beta_{1} - 105 \beta_{2} - 92 \beta_{3} - 1344 \beta_{4} + 34 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} ) q^{7} -8192 \beta_{4} q^{8} + ( 1 + 1754834 \beta_{1} - 3150 \beta_{2} - 154 \beta_{3} - 2 \beta_{4} - 156 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} + 9 \beta_{8} + \beta_{9} - 4 \beta_{10} + 4 \beta_{11} + \beta_{12} + \beta_{13} - \beta_{15} - \beta_{16} - \beta_{17} - \beta_{18} - \beta_{19} ) q^{9} +O(q^{10})\) \( q -\beta_{2} q^{2} + ( -146 - 146 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} ) q^{3} + ( -8192 + 8192 \beta_{1} ) q^{4} + ( 222 - 111 \beta_{1} - 17 \beta_{2} - 4 \beta_{3} - 33 \beta_{4} + 4 \beta_{5} + \beta_{10} ) q^{5} + ( -9650 + 19300 \beta_{1} + 294 \beta_{2} + 5 \beta_{3} + 146 \beta_{4} + \beta_{8} - \beta_{10} + \beta_{11} ) q^{6} + ( 26479 + 92582 \beta_{1} - 105 \beta_{2} - 92 \beta_{3} - 1344 \beta_{4} + 34 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} ) q^{7} -8192 \beta_{4} q^{8} + ( 1 + 1754834 \beta_{1} - 3150 \beta_{2} - 154 \beta_{3} - 2 \beta_{4} - 156 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} + 9 \beta_{8} + \beta_{9} - 4 \beta_{10} + 4 \beta_{11} + \beta_{12} + \beta_{13} - \beta_{15} - \beta_{16} - \beta_{17} - \beta_{18} - \beta_{19} ) q^{9} + ( -131293 - 131285 \beta_{1} + 34 \beta_{2} - 5 \beta_{3} - 36 \beta_{4} + 827 \beta_{5} + 3 \beta_{6} + 5 \beta_{7} + 8 \beta_{8} + 4 \beta_{9} - 3 \beta_{10} + 3 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} - \beta_{14} + 2 \beta_{15} - \beta_{17} + \beta_{18} + 2 \beta_{19} ) q^{10} + ( 840194 - 840199 \beta_{1} + 33020 \beta_{2} + 486 \beta_{3} + 33106 \beta_{4} - 240 \beta_{5} - 11 \beta_{6} - 5 \beta_{7} - 43 \beta_{8} - 3 \beta_{9} + 88 \beta_{10} + 24 \beta_{11} - 3 \beta_{12} - 5 \beta_{13} - 3 \beta_{14} - 4 \beta_{16} - 3 \beta_{17} + 5 \beta_{18} + 4 \beta_{19} ) q^{11} + ( 2392064 - 1196032 \beta_{1} - 8192 \beta_{2} + 8192 \beta_{3} - 16384 \beta_{4} - 8192 \beta_{5} ) q^{12} + ( 5348459 - 10696890 \beta_{1} + 103626 \beta_{2} + 1786 \beta_{3} + 51751 \beta_{4} + \beta_{5} + \beta_{6} + 17 \beta_{7} + 55 \beta_{8} - 28 \beta_{9} - 71 \beta_{10} - 107 \beta_{11} - 9 \beta_{12} - \beta_{13} + \beta_{14} - 6 \beta_{15} + \beta_{16} + 7 \beta_{17} - 12 \beta_{18} + 7 \beta_{19} ) q^{13} + ( -642999 - 10184488 \beta_{1} - 116563 \beta_{2} + 5890 \beta_{3} - 91915 \beta_{4} + 1708 \beta_{5} + 80 \beta_{6} + 23 \beta_{7} + 223 \beta_{8} + 16 \beta_{9} - 142 \beta_{10} - 54 \beta_{11} + 6 \beta_{12} - 21 \beta_{13} - 3 \beta_{14} + 5 \beta_{15} - 11 \beta_{16} - 8 \beta_{17} + 18 \beta_{18} + 16 \beta_{19} ) q^{14} + ( 28134507 - 32 \beta_{1} + 25 \beta_{2} - 2733 \beta_{3} + 658832 \beta_{4} + 5496 \beta_{5} - 348 \beta_{6} - 48 \beta_{7} - 1018 \beta_{8} - 2 \beta_{9} - 1000 \beta_{10} + 173 \beta_{11} - 3 \beta_{12} + 28 \beta_{13} + 10 \beta_{14} + 40 \beta_{15} - 34 \beta_{16} - 3 \beta_{17} - 8 \beta_{19} ) q^{15} -67108864 \beta_{1} q^{16} + ( 72689420 + 72689048 \beta_{1} + 292472 \beta_{2} + 103 \beta_{3} - 292379 \beta_{4} - 32501 \beta_{5} - 198 \beta_{6} - 138 \beta_{7} + 2820 \beta_{8} - 228 \beta_{9} + 17 \beta_{10} - 24 \beta_{11} + 51 \beta_{12} + 114 \beta_{13} - 20 \beta_{14} + 26 \beta_{15} - 56 \beta_{16} - 13 \beta_{17} + 13 \beta_{18} - 72 \beta_{19} ) q^{17} + ( -25156103 + 25156183 \beta_{1} - 1738867 \beta_{2} + 65425 \beta_{3} - 1734891 \beta_{4} - 33219 \beta_{5} - 123 \beta_{6} + 367 \beta_{7} - 1578 \beta_{8} + 404 \beta_{9} + 3463 \beta_{10} + 241 \beta_{11} + 2 \beta_{12} - 240 \beta_{13} - 179 \beta_{14} - 66 \beta_{16} - 129 \beta_{17} + 157 \beta_{18} + 66 \beta_{19} ) q^{18} + ( -261360557 + 130680793 \beta_{1} - 1460890 \beta_{2} - 113001 \beta_{3} - 2920781 \beta_{4} + 111727 \beta_{5} - 821 \beta_{6} + 882 \beta_{7} + 238 \beta_{8} + 630 \beta_{9} - 562 \beta_{10} + 919 \beta_{11} - 147 \beta_{12} - 574 \beta_{13} - 574 \beta_{14} - 91 \beta_{15} - 105 \beta_{16} - 245 \beta_{17} + 91 \beta_{18} + 203 \beta_{19} ) q^{19} + ( -909312 + 1818624 \beta_{1} + 278528 \beta_{2} + 32768 \beta_{3} + 131072 \beta_{4} + 8192 \beta_{8} - 8192 \beta_{10} ) q^{20} + ( 582387505 - 406332798 \beta_{1} - 5005296 \beta_{2} + 264401 \beta_{3} + 1503012 \beta_{4} + 182958 \beta_{5} - 2946 \beta_{6} + 82 \beta_{7} - 1845 \beta_{8} + 424 \beta_{9} - 3179 \beta_{10} + 1628 \beta_{11} + 35 \beta_{12} - 163 \beta_{13} - 262 \beta_{14} + 194 \beta_{15} - 422 \beta_{16} - 233 \beta_{17} + 466 \beta_{18} - 220 \beta_{19} ) q^{21} + ( 269751549 - 632 \beta_{1} + 500 \beta_{2} + 37942 \beta_{3} + 859370 \beta_{4} - 74839 \beta_{5} - 463 \beta_{6} - 483 \beta_{7} - 3128 \beta_{8} - 968 \beta_{9} - 2922 \beta_{10} + 54 \beta_{11} + 10 \beta_{12} + 239 \beta_{13} - 129 \beta_{14} + 561 \beta_{15} - 219 \beta_{16} + 10 \beta_{17} - 84 \beta_{19} ) q^{22} + ( 1321 - 690265046 \beta_{1} - 3715059 \beta_{2} - 405720 \beta_{3} - 999 \beta_{4} - 405414 \beta_{5} - 5455 \beta_{6} - 1150 \beta_{7} + 595 \beta_{8} - 1614 \beta_{9} - 291 \beta_{10} - 3537 \beta_{11} + 823 \beta_{12} + 647 \beta_{13} + 578 \beta_{15} - 891 \beta_{16} - 41 \beta_{17} + 578 \beta_{18} + 27 \beta_{19} ) q^{23} + ( -79052800 - 79052800 \beta_{1} - 1204224 \beta_{2} + 1204224 \beta_{4} - 40960 \beta_{5} - 8192 \beta_{6} - 8192 \beta_{8} ) q^{24} + ( 1417065640 - 1417065190 \beta_{1} + 7602590 \beta_{2} + 1869090 \beta_{3} + 7566212 \beta_{4} - 936978 \beta_{5} + 2008 \beta_{6} + 2074 \beta_{7} + 19938 \beta_{8} + 1056 \beta_{9} - 39838 \beta_{10} - 6280 \beta_{11} - 464 \beta_{12} - 2200 \beta_{13} - 2412 \beta_{14} - 1494 \beta_{16} - 818 \beta_{17} - 420 \beta_{18} + 1494 \beta_{19} ) q^{25} + ( 843202262 - 421601549 \beta_{1} + 5224831 \beta_{2} - 694590 \beta_{3} + 10448615 \beta_{4} + 693634 \beta_{5} - 2342 \beta_{6} - 194 \beta_{7} - 153 \beta_{8} + 844 \beta_{9} - 2746 \beta_{10} + 2790 \beta_{11} - 1344 \beta_{12} - 1639 \beta_{13} - 1486 \beta_{14} - 489 \beta_{15} - 1079 \beta_{16} - 631 \beta_{17} + 489 \beta_{18} + 366 \beta_{19} ) q^{26} + ( 510865423 - 1021730468 \beta_{1} - 41420340 \beta_{2} + 2832675 \beta_{3} - 20702935 \beta_{4} + 383 \beta_{5} - 3936 \beta_{6} + 5643 \beta_{7} - 4547 \beta_{8} - 357 \beta_{9} + 2528 \beta_{10} + 8253 \beta_{11} - 1564 \beta_{12} - 3603 \beta_{13} - 1045 \beta_{14} + 2987 \beta_{15} - 2487 \beta_{16} - 2436 \beta_{17} + 5974 \beta_{18} + 2051 \beta_{19} ) q^{27} + ( -975339520 + 216915968 \beta_{1} + 11010048 \beta_{2} + 286720 \beta_{3} + 10149888 \beta_{4} + 466944 \beta_{5} - 8192 \beta_{6} - 8192 \beta_{8} - 8192 \beta_{9} + 8192 \beta_{11} ) q^{28} + ( 1394598239 + 18308 \beta_{1} - 8726 \beta_{2} + 591042 \beta_{3} - 28535917 \beta_{4} - 1220667 \beta_{5} - 21605 \beta_{6} + 17197 \beta_{7} + 49187 \beta_{8} + 20250 \beta_{9} + 38753 \beta_{10} + 16575 \beta_{11} - 4505 \beta_{12} - 7043 \beta_{13} - 7899 \beta_{14} - 550 \beta_{15} - 1967 \beta_{16} - 4505 \beta_{17} + 5357 \beta_{19} ) q^{29} + ( 1364 + 5411531901 \beta_{1} - 29162576 \beta_{2} - 1967441 \beta_{3} - 19988 \beta_{4} - 1972025 \beta_{5} - 2685 \beta_{6} - 616 \beta_{7} + 48259 \beta_{8} + 3116 \beta_{9} - 20123 \beta_{10} - 17 \beta_{11} + 4116 \beta_{12} + 2479 \beta_{13} - 2031 \beta_{15} - 2725 \beta_{16} - 2203 \beta_{17} - 2031 \beta_{18} - 3594 \beta_{19} ) q^{30} + ( 1521502766 + 1521491802 \beta_{1} + 53486900 \beta_{2} + 1662 \beta_{3} - 53479805 \beta_{4} - 1084396 \beta_{5} + 49610 \beta_{6} + 81 \beta_{7} - 48522 \beta_{8} - 2395 \beta_{9} + 1084 \beta_{10} + 855 \beta_{11} + 4575 \beta_{12} + 7319 \beta_{13} + 2116 \beta_{14} + 332 \beta_{15} - 4683 \beta_{16} + 177 \beta_{17} + 166 \beta_{18} - 4113 \beta_{19} ) q^{31} + ( 67108864 \beta_{2} + 67108864 \beta_{4} ) q^{32} + ( -2472964868 + 1236481947 \beta_{1} - 36049134 \beta_{2} - 8926024 \beta_{3} - 72197416 \beta_{4} + 8938506 \beta_{5} - 17702 \beta_{6} - 5576 \beta_{7} - 2200 \beta_{8} - 5048 \beta_{9} - 83318 \beta_{10} + 12522 \beta_{11} + 6328 \beta_{12} + 10754 \beta_{13} + 6696 \beta_{14} + 6802 \beta_{15} + 2116 \beta_{16} + 3194 \beta_{17} - 6802 \beta_{18} + 1018 \beta_{19} ) q^{33} + ( 2435630231 - 4871274238 \beta_{1} - 144854812 \beta_{2} + 1622903 \beta_{3} - 72325239 \beta_{4} - 10305 \beta_{5} + 10079 \beta_{6} + 20605 \beta_{7} - 83239 \beta_{8} + 25536 \beta_{9} + 81337 \beta_{10} - 23861 \beta_{11} - 6718 \beta_{12} - 6943 \beta_{13} + 3023 \beta_{14} + 1567 \beta_{15} + 8511 \beta_{16} + 2240 \beta_{17} + 3134 \beta_{18} - 4032 \beta_{19} ) q^{34} + ( -2685866328 + 3544740789 \beta_{1} + 48296320 \beta_{2} + 4848315 \beta_{3} + 204458263 \beta_{4} - 773154 \beta_{5} + 60655 \beta_{6} - 19991 \beta_{7} + 280429 \beta_{8} - 5151 \beta_{9} - 132408 \beta_{10} + 38364 \beta_{11} - 2101 \beta_{12} + 13483 \beta_{13} + 17117 \beta_{14} - 17578 \beta_{15} + 13472 \beta_{16} + 14553 \beta_{17} - 2091 \beta_{18} - 12936 \beta_{19} ) q^{35} + ( -14375583744 - 8192 \beta_{1} - 1253376 \beta_{3} - 25862144 \beta_{4} + 2531328 \beta_{5} - 65536 \beta_{6} - 8192 \beta_{7} - 40960 \beta_{8} - 24576 \beta_{9} - 40960 \beta_{10} + 32768 \beta_{11} - 8192 \beta_{13} + 8192 \beta_{15} + 8192 \beta_{16} + 8192 \beta_{19} ) q^{36} + ( 11788 - 2700739262 \beta_{1} - 8813110 \beta_{2} - 3063025 \beta_{3} + 275596 \beta_{4} - 3101934 \beta_{5} - 16876 \beta_{6} - 49628 \beta_{7} - 592086 \beta_{8} - 33454 \beta_{9} + 309366 \beta_{10} - 77858 \beta_{11} + 1963 \beta_{12} + 4017 \beta_{13} - 16596 \beta_{15} + 14264 \beta_{16} + 19981 \beta_{17} - 16596 \beta_{18} + 3754 \beta_{19} ) q^{37} + ( -11800693435 - 11800726083 \beta_{1} + 130067018 \beta_{2} + 30359 \beta_{3} - 130101668 \beta_{4} - 3909762 \beta_{5} + 116714 \beta_{6} - 74935 \beta_{7} + 376547 \beta_{8} - 35644 \beta_{9} + 19425 \beta_{10} - 24465 \beta_{11} - 6986 \beta_{12} - 22106 \beta_{13} - 21 \beta_{14} - 630 \beta_{15} + 20160 \beta_{16} + 5019 \beta_{17} - 315 \beta_{18} + 3850 \beta_{19} ) q^{38} + ( -12617869550 + 12617846803 \beta_{1} + 536090842 \beta_{2} - 17278186 \beta_{3} + 537121274 \beta_{4} + 8696711 \beta_{5} + 273670 \beta_{6} - 82716 \beta_{7} - 552560 \beta_{8} - 2649 \beta_{9} + 1126003 \beta_{10} - 534725 \beta_{11} + 25842 \beta_{12} + 55713 \beta_{13} + 31346 \beta_{14} + 13545 \beta_{16} + 12070 \beta_{17} - 17946 \beta_{18} - 13545 \beta_{19} ) q^{39} + ( 2151071744 - 1075552256 \beta_{1} + 311296 \beta_{2} + 6799360 \beta_{3} + 532480 \beta_{4} - 6766592 \beta_{5} - 16384 \beta_{6} - 16384 \beta_{7} - 8192 \beta_{8} - 32768 \beta_{9} - 49152 \beta_{10} + 16384 \beta_{11} + 8192 \beta_{13} + 16384 \beta_{14} - 8192 \beta_{15} + 8192 \beta_{16} + 8192 \beta_{17} + 8192 \beta_{18} - 16384 \beta_{19} ) q^{40} + ( 8441550084 - 16883045043 \beta_{1} - 360631658 \beta_{2} - 7786349 \beta_{3} - 180476507 \beta_{4} + 83249 \beta_{5} + 67548 \beta_{6} - 79407 \beta_{7} + 21609 \beta_{8} - 110481 \beta_{9} - 56451 \beta_{10} + 2208 \beta_{11} - 21290 \beta_{12} + 58893 \beta_{13} + 47262 \beta_{14} - 20743 \beta_{15} + 48179 \beta_{16} + 33838 \beta_{17} - 41486 \beta_{18} - 20615 \beta_{19} ) q^{41} + ( -42045407082 + 54250730699 \beta_{1} - 181449227 \beta_{2} - 8776966 \beta_{3} + 406730150 \beta_{4} - 12150698 \beta_{5} - 206530 \beta_{6} - 21946 \beta_{7} - 411221 \beta_{8} - 23508 \beta_{9} - 384786 \beta_{10} + 447166 \beta_{11} + 19344 \beta_{12} - 6655 \beta_{13} + 20186 \beta_{14} + 8527 \beta_{15} + 15841 \beta_{16} - 12119 \beta_{17} - 1479 \beta_{18} - 22802 \beta_{19} ) q^{42} + ( 36322845122 + 51850 \beta_{1} - 25698 \beta_{2} - 18946292 \beta_{3} + 18631832 \beta_{4} + 37773034 \beta_{5} - 212538 \beta_{6} + 36454 \beta_{7} - 948718 \beta_{8} + 117772 \beta_{9} - 1011852 \beta_{10} + 145534 \beta_{11} + 22980 \beta_{12} + 61810 \beta_{13} + 61356 \beta_{14} + 41136 \beta_{15} - 15850 \beta_{16} + 22980 \beta_{17} + 14002 \beta_{19} ) q^{43} + ( -57344 + 6882942976 \beta_{1} - 270532608 \beta_{2} - 2007040 \beta_{3} - 303104 \beta_{4} - 2056192 \beta_{5} - 40960 \beta_{6} + 32768 \beta_{7} + 720896 \beta_{8} + 98304 \beta_{9} - 327680 \beta_{10} - 122880 \beta_{11} - 8192 \beta_{12} - 16384 \beta_{13} - 40960 \beta_{15} + 24576 \beta_{16} - 8192 \beta_{17} - 40960 \beta_{18} - 24576 \beta_{19} ) q^{44} + ( 16622358847 + 16622653801 \beta_{1} + 1531561617 \beta_{2} - 69723 \beta_{3} - 1531487050 \beta_{4} + 29309789 \beta_{5} + 1827771 \beta_{6} + 323655 \beta_{7} + 2028109 \beta_{8} + 193362 \beta_{9} - 45730 \beta_{10} + 23925 \beta_{11} - 15208 \beta_{12} - 11442 \beta_{13} - 10999 \beta_{14} - 88840 \beta_{15} + 18039 \beta_{16} + 10806 \beta_{17} - 44420 \beta_{18} + 33989 \beta_{19} ) q^{45} + ( -28685872778 + 28685962778 \beta_{1} + 672499339 \beta_{2} - 64155498 \beta_{3} + 674447029 \beta_{4} + 32036265 \beta_{5} + 297989 \beta_{6} + 156584 \beta_{7} - 985869 \beta_{8} - 73488 \beta_{9} + 1847074 \beta_{10} - 650546 \beta_{11} - 20272 \beta_{12} - 20144 \beta_{13} + 12408 \beta_{14} + 29424 \beta_{16} + 17144 \beta_{17} - 28152 \beta_{18} - 29424 \beta_{19} ) q^{46} + ( -136262886975 + 68131497984 \beta_{1} - 879970373 \beta_{2} + 113645345 \beta_{3} - 1757281233 \beta_{4} - 113832332 \beta_{5} - 310830 \beta_{6} + 34221 \beta_{7} + 101058 \beta_{8} + 248349 \beta_{9} + 2467270 \beta_{10} + 397728 \beta_{11} + 8904 \beta_{12} - 15987 \beta_{13} - 10512 \beta_{14} - 26286 \beta_{15} - 8259 \beta_{16} - 27894 \beta_{17} + 26286 \beta_{18} + 45057 \beta_{19} ) q^{47} + ( -9797894144 + 19595788288 \beta_{1} + 134217728 \beta_{2} - 67108864 \beta_{3} + 67108864 \beta_{4} ) q^{48} + ( 113580294011 + 66863463901 \beta_{1} + 515150656 \beta_{2} + 5613705 \beta_{3} + 1509612465 \beta_{4} - 44561297 \beta_{5} - 92978 \beta_{6} + 90601 \beta_{7} + 2006331 \beta_{8} + 91667 \beta_{9} - 1112987 \beta_{10} + 2290270 \beta_{11} - 408 \beta_{12} + 17301 \beta_{13} - 76020 \beta_{14} + 129913 \beta_{15} - 130169 \beta_{16} - 20858 \beta_{17} + 52220 \beta_{18} + 119185 \beta_{19} ) q^{49} + ( 58051853752 - 345440 \beta_{1} + 142784 \beta_{2} - 7055044 \beta_{3} + 1412660398 \beta_{4} + 14808488 \beta_{5} - 2273656 \beta_{6} - 397792 \beta_{7} + 771284 \beta_{8} - 379200 \beta_{9} + 1096628 \beta_{10} + 1000332 \beta_{11} - 29248 \beta_{12} - 66016 \beta_{13} - 6144 \beta_{14} - 80864 \beta_{15} + 7520 \beta_{16} - 29248 \beta_{17} - 93440 \beta_{19} ) q^{50} + ( 106365 - 242043917521 \beta_{1} - 1579097624 \beta_{2} + 138939661 \beta_{3} + 2295329 \beta_{4} + 139229387 \beta_{5} - 1140007 \beta_{6} + 328046 \beta_{7} - 4628612 \beta_{8} - 111894 \beta_{9} + 2102106 \beta_{10} - 682321 \beta_{11} - 55205 \beta_{12} + 21218 \beta_{13} + 149225 \beta_{15} - 86819 \beta_{16} - 78095 \beta_{17} + 149225 \beta_{18} + 63929 \beta_{19} ) q^{51} + ( 43814494208 + 43814199296 \beta_{1} - 424321024 \beta_{2} + 40960 \beta_{3} + 424280064 \beta_{4} - 14245888 \beta_{5} + 778240 \beta_{6} - 385024 \beta_{7} - 630784 \beta_{8} - 147456 \beta_{9} + 81920 \beta_{10} - 24576 \beta_{11} + 16384 \beta_{12} + 16384 \beta_{13} + 8192 \beta_{14} + 98304 \beta_{15} - 57344 \beta_{16} - 49152 \beta_{17} + 49152 \beta_{18} - 73728 \beta_{19} ) q^{52} + ( 154643581176 - 154643763648 \beta_{1} + 437338620 \beta_{2} + 61250383 \beta_{3} + 432166259 \beta_{4} - 30723623 \beta_{5} + 2901603 \beta_{6} - 53427 \beta_{7} + 2756408 \beta_{8} + 139830 \beta_{9} - 5379166 \beta_{10} - 5650143 \beta_{11} - 150168 \beta_{12} - 260628 \beta_{13} - 209343 \beta_{14} - 170463 \beta_{16} - 71580 \beta_{17} + 173754 \beta_{18} + 170463 \beta_{19} ) q^{53} + ( -347538187418 + 173769101031 \beta_{1} + 516109632 \beta_{2} + 14587529 \beta_{3} + 1020956804 \beta_{4} - 14349501 \beta_{5} - 2186873 \beta_{6} + 160762 \beta_{7} - 220507 \beta_{8} - 646268 \beta_{9} - 11066453 \beta_{10} + 1929273 \beta_{11} - 4928 \beta_{12} - 111013 \beta_{13} - 84042 \beta_{14} + 74389 \beta_{15} + 49483 \beta_{16} - 14581 \beta_{17} - 74389 \beta_{18} - 39830 \beta_{19} ) q^{54} + ( 334950320257 - 669900880873 \beta_{1} - 5178401767 \beta_{2} + 15398280 \beta_{3} - 2587971312 \beta_{4} - 426959 \beta_{5} - 614671 \beta_{6} - 105465 \beta_{7} - 642317 \beta_{8} + 364994 \beta_{9} + 1067478 \beta_{10} - 3591405 \beta_{11} + 384176 \beta_{12} - 195061 \beta_{13} - 420288 \beta_{14} + 43000 \beta_{15} - 496707 \beta_{16} - 235368 \beta_{17} + 86000 \beta_{18} + 186935 \beta_{19} ) q^{55} + ( 88698544128 - 5267087360 \beta_{1} + 751771648 \beta_{2} + 14090240 \beta_{3} - 203153408 \beta_{4} - 62808064 \beta_{5} + 40960 \beta_{6} + 188416 \beta_{7} - 1122304 \beta_{8} + 229376 \beta_{9} - 557056 \beta_{10} + 688128 \beta_{11} - 180224 \beta_{12} - 245760 \beta_{13} - 221184 \beta_{14} - 147456 \beta_{15} + 65536 \beta_{16} - 24576 \beta_{17} - 106496 \beta_{18} + 49152 \beta_{19} ) q^{56} + ( 739626468945 + 77998 \beta_{1} + 151292 \beta_{2} + 215266978 \beta_{3} - 2706508236 \beta_{4} - 430628516 \beta_{5} - 10284986 \beta_{6} + 520016 \beta_{7} - 4987516 \beta_{8} - 180106 \beta_{9} - 5026624 \beta_{10} + 4941038 \beta_{11} + 103230 \beta_{12} + 145024 \beta_{13} - 235558 \beta_{14} - 393738 \beta_{15} + 61436 \beta_{16} + 103230 \beta_{17} - 293412 \beta_{19} ) q^{57} + ( -310532 - 236559250607 \beta_{1} - 1411972093 \beta_{2} - 44089276 \beta_{3} - 7560991 \beta_{4} - 43223156 \beta_{5} - 1382748 \beta_{6} - 469288 \beta_{7} + 14333687 \beta_{8} + 107732 \beta_{9} - 7295324 \beta_{10} - 1035320 \beta_{11} - 84068 \beta_{12} - 225643 \beta_{13} + 461403 \beta_{15} - 44823 \beta_{16} + 11863 \beta_{17} + 461403 \beta_{18} + 140754 \beta_{19} ) q^{58} + ( -226673260318 - 226674211642 \beta_{1} - 1635932525 \beta_{2} - 327760 \beta_{3} + 1635644299 \beta_{4} + 202855927 \beta_{5} + 8428650 \beta_{6} - 454980 \beta_{7} + 4289458 \beta_{8} - 702966 \beta_{9} - 419130 \beta_{10} + 510480 \beta_{11} + 31184 \beta_{12} + 254442 \beta_{13} - 21914 \beta_{14} + 826092 \beta_{15} - 314608 \beta_{16} - 113264 \beta_{17} + 413046 \beta_{18} + 25712 \beta_{19} ) q^{59} + ( -230478053376 + 230478299136 \beta_{1} - 5405753344 \beta_{2} + 44007424 \beta_{3} - 5388935168 \beta_{4} - 22724608 \beta_{5} + 1114112 \beta_{6} + 647168 \beta_{7} - 7864320 \beta_{8} + 434176 \beta_{9} + 16203776 \beta_{10} - 2269184 \beta_{11} + 90112 \beta_{12} - 270336 \beta_{13} - 81920 \beta_{14} + 24576 \beta_{16} - 253952 \beta_{17} + 327680 \beta_{18} - 24576 \beta_{19} ) q^{60} + ( -147771375074 + 73885490802 \beta_{1} + 5132713672 \beta_{2} - 355189187 \beta_{3} + 10271784724 \beta_{4} + 354996064 \beta_{5} - 3268654 \beta_{6} - 229012 \beta_{7} - 71540 \beta_{8} + 615664 \beta_{9} + 6006960 \beta_{10} + 3447532 \beta_{11} - 407715 \beta_{12} - 49525 \beta_{13} - 314594 \beta_{14} - 92918 \beta_{15} - 404236 \beta_{16} + 111251 \beta_{17} + 92918 \beta_{18} - 114730 \beta_{19} ) q^{61} + ( 439860940327 - 879721972942 \beta_{1} - 2916376808 \beta_{2} + 370956996 \beta_{3} - 1446676594 \beta_{4} - 973227 \beta_{5} - 548971 \beta_{6} + 1046015 \beta_{7} - 10168706 \beta_{8} + 1002176 \beta_{9} + 10428440 \beta_{10} + 128724 \beta_{11} + 226582 \beta_{12} - 395861 \beta_{13} - 326315 \beta_{14} + 7445 \beta_{15} - 285995 \beta_{16} - 255808 \beta_{17} + 14890 \beta_{18} + 44800 \beta_{19} ) q^{62} + ( -1450755182996 + 2459184639190 \beta_{1} + 2971231332 \beta_{2} - 250945705 \beta_{3} - 7979316522 \beta_{4} + 923741490 \beta_{5} - 3909531 \beta_{6} + 894815 \beta_{7} + 39406837 \beta_{8} + 698311 \beta_{9} - 26224497 \beta_{10} + 13191491 \beta_{11} + 155401 \beta_{12} - 328482 \beta_{13} - 474954 \beta_{14} - 162342 \beta_{15} - 68190 \beta_{16} - 525055 \beta_{17} - 384178 \beta_{18} + 316776 \beta_{19} ) q^{63} + 549755813888 q^{64} + ( -30167 + 751995543838 \beta_{1} - 295985166 \beta_{2} - 537944624 \beta_{3} + 21955328 \beta_{4} - 538414646 \beta_{5} - 3659680 \beta_{6} + 1900505 \beta_{7} - 41568777 \beta_{8} + 776907 \beta_{9} + 20659688 \beta_{10} - 2679510 \beta_{11} + 98809 \beta_{12} + 12899 \beta_{13} - 213589 \beta_{15} - 294447 \beta_{16} - 251603 \beta_{17} - 213589 \beta_{18} - 55965 \beta_{19} ) q^{65} + ( -282508221652 - 282506638900 \beta_{1} + 1217465763 \beta_{2} - 313692 \beta_{3} - 1216680635 \beta_{4} - 146398040 \beta_{5} + 6627960 \beta_{6} + 2082396 \beta_{7} + 31752980 \beta_{8} + 1171056 \beta_{9} - 190724 \beta_{10} + 75588 \beta_{11} + 15016 \beta_{12} + 122088 \beta_{13} - 66860 \beta_{14} - 686120 \beta_{15} + 8064 \beta_{16} + 48276 \beta_{17} - 343060 \beta_{18} + 58456 \beta_{19} ) q^{66} + ( -465249275246 + 465249549806 \beta_{1} - 26691618625 \beta_{2} + 1125928858 \beta_{3} - 26673304485 \beta_{4} - 562049415 \beta_{5} + 10033098 \beta_{6} - 105932 \beta_{7} - 10089366 \beta_{8} - 1112762 \beta_{9} + 19174624 \beta_{10} - 20010132 \beta_{11} + 22000 \beta_{12} + 633334 \beta_{13} + 732834 \beta_{14} + 492838 \beta_{16} + 371338 \beta_{17} - 293556 \beta_{18} - 492838 \beta_{19} ) q^{67} + ( -1190940729344 + 595470180352 \beta_{1} + 2394406912 \beta_{2} - 267190272 \beta_{3} + 4765106176 \beta_{4} + 267386880 \beta_{5} + 983040 \beta_{6} - 614400 \beta_{7} + 352256 \beta_{8} + 319488 \beta_{9} - 23347200 \beta_{10} - 638976 \beta_{11} + 172032 \beta_{12} - 122880 \beta_{13} + 327680 \beta_{14} - 106496 \beta_{15} + 106496 \beta_{16} - 352256 \beta_{17} + 106496 \beta_{18} + 417792 \beta_{19} ) q^{68} + ( 2474408104319 - 4948815611888 \beta_{1} + 53370207996 \beta_{2} - 688146184 \beta_{3} + 26664581788 \beta_{4} + 2440538 \beta_{5} + 2347270 \beta_{6} - 1917686 \beta_{7} + 16974939 \beta_{8} - 2603384 \beta_{9} - 18198333 \beta_{10} + 4960230 \beta_{11} - 1122174 \beta_{12} + 1134446 \beta_{13} + 1294652 \beta_{14} - 308922 \beta_{15} + 1486522 \beta_{16} + 1153838 \beta_{17} - 617844 \beta_{18} - 348516 \beta_{19} ) q^{69} + ( 381990914587 + 1286109330562 \beta_{1} - 649825633 \beta_{2} + 918824019 \beta_{3} - 3265461129 \beta_{4} - 573959658 \beta_{5} + 346450 \beta_{6} - 1385705 \beta_{7} - 53391092 \beta_{8} - 458008 \beta_{9} + 2220613 \beta_{10} + 5040863 \beta_{11} + 216510 \beta_{12} + 1120341 \beta_{13} + 855901 \beta_{14} + 1053995 \beta_{15} - 735625 \beta_{16} + 584694 \beta_{17} + 965752 \beta_{18} + 19156 \beta_{19} ) q^{70} + ( 4830508817338 - 1673432 \beta_{1} - 160684 \beta_{2} + 335159648 \beta_{3} + 26726076704 \beta_{4} - 666977074 \beta_{5} - 1247344 \beta_{6} - 3672874 \beta_{7} - 23059804 \beta_{8} - 510498 \beta_{9} - 21755710 \beta_{10} + 971346 \beta_{11} - 235906 \beta_{12} - 467170 \beta_{13} + 1527630 \beta_{14} + 1139908 \beta_{15} - 4642 \beta_{16} - 235906 \beta_{17} + 765928 \beta_{19} ) q^{71} + ( 753664 - 206076895232 \beta_{1} + 14243454976 \beta_{2} - 267452416 \beta_{3} - 14508032 \beta_{4} - 269484032 \beta_{5} + 1376256 \beta_{6} - 65536 \beta_{7} + 28368896 \beta_{8} - 1409024 \beta_{9} - 14155776 \beta_{10} - 1736704 \beta_{11} - 557056 \beta_{12} + 368640 \beta_{13} - 1286144 \beta_{15} + 1056768 \beta_{16} + 516096 \beta_{17} - 1286144 \beta_{18} + 16384 \beta_{19} ) q^{72} + ( -2178169435911 - 2178166003229 \beta_{1} - 11063939276 \beta_{2} + 1827436 \beta_{3} + 11062873328 \beta_{4} - 566317744 \beta_{5} + 9625698 \beta_{6} - 962824 \beta_{7} - 2513468 \beta_{8} + 1729172 \beta_{9} + 1976146 \beta_{10} - 2504142 \beta_{11} - 668152 \beta_{12} - 2308392 \beta_{13} + 137374 \beta_{14} - 2742528 \beta_{15} + 2168236 \beta_{16} + 665370 \beta_{17} - 1371264 \beta_{18} + 498808 \beta_{19} ) q^{73} + ( -57002589948 + 57000537996 \beta_{1} + 2374477315 \beta_{2} - 1163642908 \beta_{3} + 2415486995 \beta_{4} + 586029140 \beta_{5} + 1801092 \beta_{6} - 4901316 \beta_{7} - 23339240 \beta_{8} - 1659360 \beta_{9} + 45248284 \beta_{10} - 3301068 \beta_{11} - 176664 \beta_{12} + 1678032 \beta_{13} + 217908 \beta_{14} - 476712 \beta_{16} + 1160076 \beta_{17} - 1739580 \beta_{18} + 476712 \beta_{19} ) q^{74} + ( -12444498974072 + 6222250486004 \beta_{1} + 6726545600 \beta_{2} - 703600912 \beta_{3} + 13542072910 \beta_{4} + 704938126 \beta_{5} + 3134072 \beta_{6} + 1407320 \beta_{7} + 263822 \beta_{8} - 3302884 \beta_{9} + 91560340 \beta_{10} - 4049028 \beta_{11} + 2420194 \beta_{12} + 1660660 \beta_{13} + 2482832 \beta_{14} + 326794 \beta_{15} + 2550878 \beta_{16} + 549928 \beta_{17} - 326794 \beta_{18} - 680612 \beta_{19} ) q^{75} + ( 1070534533120 - 2141066428416 \beta_{1} + 23931895808 \beta_{2} + 926162944 \beta_{3} + 11963441152 \beta_{4} + 4874240 \beta_{5} + 2752512 \beta_{6} - 5447680 \beta_{7} - 5349376 \beta_{8} - 5906432 \beta_{9} + 4145152 \beta_{10} - 8560640 \beta_{11} - 458752 \beta_{12} + 3383296 \beta_{13} + 2351104 \beta_{14} - 745472 \beta_{15} + 2007040 \beta_{16} + 1146880 \beta_{17} - 1490944 \beta_{18} - 1204224 \beta_{19} ) q^{76} + ( 518391237200 + 6715836749813 \beta_{1} - 21714630679 \beta_{2} - 680136656 \beta_{3} - 43178312308 \beta_{4} - 157745451 \beta_{5} + 12108431 \beta_{6} - 5110495 \beta_{7} - 10640522 \beta_{8} - 5709666 \beta_{9} + 72979529 \beta_{10} - 22755909 \beta_{11} - 76091 \beta_{12} + 1220099 \beta_{13} + 3998231 \beta_{14} - 651360 \beta_{15} + 3056993 \beta_{16} + 2632901 \beta_{17} + 516676 \beta_{18} - 3666393 \beta_{19} ) q^{77} + ( 4431809524833 + 5030360 \beta_{1} - 1391524 \beta_{2} - 2305068457 \beta_{3} - 13835639874 \beta_{4} + 4599526001 \beta_{5} + 8740369 \beta_{6} + 6727479 \beta_{7} + 31932519 \beta_{8} + 6133160 \beta_{9} + 26351313 \beta_{10} - 2428141 \beta_{11} + 1400286 \beta_{12} + 3350765 \beta_{13} + 1103453 \beta_{14} + 1643123 \beta_{15} - 550193 \beta_{16} + 1400286 \beta_{17} + 542084 \beta_{19} ) q^{78} + ( -1806607 - 6052325523400 \beta_{1} - 11725956737 \beta_{2} + 951839980 \beta_{3} + 6013421 \beta_{4} + 947380284 \beta_{5} + 18691819 \beta_{6} - 14692108 \beta_{7} - 20295729 \beta_{8} - 497588 \beta_{9} + 15558033 \beta_{10} + 4524821 \beta_{11} + 893537 \beta_{12} + 143157 \beta_{13} - 2829540 \beta_{15} + 3094981 \beta_{16} + 1895597 \beta_{17} - 2829540 \beta_{18} - 2092921 \beta_{19} ) q^{79} + ( -7449083904 - 7449083904 \beta_{1} - 1140850688 \beta_{2} + 1140850688 \beta_{4} - 268435456 \beta_{5} - 67108864 \beta_{8} ) q^{80} + ( -10719062575206 + 10719066051606 \beta_{1} + 9995828424 \beta_{2} - 3529069671 \beta_{3} + 9794189823 \beta_{4} + 1759320489 \beta_{5} - 40865280 \beta_{6} + 4429758 \beta_{7} + 103772814 \beta_{8} + 4581942 \beta_{9} - 204682491 \beta_{10} + 78682974 \beta_{11} + 2984829 \beta_{12} + 1567836 \beta_{13} + 430662 \beta_{14} + 1475928 \beta_{16} - 371727 \beta_{17} - 1720653 \beta_{18} - 1475928 \beta_{19} ) q^{81} + ( -2931695142370 + 1465849561031 \beta_{1} + 8564742515 \beta_{2} + 625564278 \beta_{3} + 17094173703 \beta_{4} - 633248034 \beta_{5} + 15971686 \beta_{6} + 1154550 \beta_{7} + 3104723 \beta_{8} + 12855132 \beta_{9} - 43461086 \beta_{10} - 12800294 \beta_{11} - 532672 \beta_{12} + 2289197 \beta_{13} - 932006 \beta_{14} - 279581 \beta_{15} - 1838787 \beta_{16} + 1449085 \beta_{17} + 279581 \beta_{18} - 142970 \beta_{19} ) q^{82} + ( 7166911729506 - 14333827954944 \beta_{1} - 26298937312 \beta_{2} - 4172216290 \beta_{3} - 13130793286 \beta_{4} - 7865742 \beta_{5} + 43488 \beta_{6} + 16957134 \beta_{7} - 4398940 \beta_{8} + 14397138 \beta_{9} + 3125422 \beta_{10} + 11176902 \beta_{11} - 2781570 \beta_{12} - 5782410 \beta_{13} + 271338 \beta_{14} + 2230512 \beta_{15} + 1807614 \beta_{16} - 1735902 \beta_{17} + 4461024 \beta_{18} - 1284780 \beta_{19} ) q^{83} + ( -1442240618496 + 4770922373120 \beta_{1} - 12336291840 \beta_{2} + 1502183424 \beta_{3} - 53296021504 \beta_{4} - 3674988544 \beta_{5} + 11264000 \beta_{6} + 2596864 \beta_{7} - 24453120 \beta_{8} + 458752 \beta_{9} + 43384832 \beta_{10} - 22618112 \beta_{11} + 1515520 \beta_{12} + 204800 \beta_{13} - 1024000 \beta_{14} - 3817472 \beta_{15} + 1908736 \beta_{16} - 1548288 \beta_{17} - 2228224 \beta_{18} + 286720 \beta_{19} ) q^{84} + ( 20817092314550 + 3657932 \beta_{1} - 1767288 \beta_{2} + 1297057148 \beta_{3} - 71991177323 \beta_{4} - 2600736001 \beta_{5} + 61086495 \beta_{6} + 4475447 \beta_{7} + 83918872 \beta_{8} - 2842708 \beta_{9} + 82377936 \beta_{10} - 30181537 \beta_{11} - 3552981 \beta_{12} - 7800121 \beta_{13} - 7923477 \beta_{14} + 392118 \beta_{15} + 694159 \beta_{16} - 3552981 \beta_{17} + 3203273 \beta_{19} ) q^{85} + ( 7363568 + 237239019992 \beta_{1} - 37096616638 \beta_{2} - 1526336322 \beta_{3} - 48671778 \beta_{4} - 1531993122 \beta_{5} + 5316550 \beta_{6} + 8443456 \beta_{7} + 109548248 \beta_{8} - 3278928 \beta_{9} - 54893766 \beta_{10} + 20527254 \beta_{11} + 5502896 \beta_{12} + 3723844 \beta_{13} - 119012 \beta_{15} - 6554124 \beta_{16} - 1135348 \beta_{17} - 119012 \beta_{18} - 84120 \beta_{19} ) q^{86} + ( -4225919927042 - 4225945764875 \beta_{1} + 26019802434 \beta_{2} + 6036072 \beta_{3} - 26015605276 \beta_{4} + 1743683819 \beta_{5} - 147963504 \beta_{6} - 13856934 \beta_{7} - 144721094 \beta_{8} - 14017521 \beta_{9} + 2386097 \beta_{10} - 951153 \beta_{11} + 4352516 \beta_{12} + 7583919 \beta_{13} + 804062 \beta_{14} + 3700592 \beta_{15} - 4666347 \beta_{16} - 630882 \beta_{17} + 1850296 \beta_{18} - 5300449 \beta_{19} ) q^{87} + ( -2209801641984 + 2209798889472 \beta_{1} - 7066107904 \beta_{2} - 621682688 \beta_{3} - 7024508928 \beta_{4} + 311328768 \beta_{5} + 786432 \beta_{6} - 3842048 \beta_{7} - 19341312 \beta_{8} + 2129920 \beta_{9} + 44965888 \beta_{10} + 2678784 \beta_{11} + 606208 \beta_{12} - 131072 \beta_{13} + 1056768 \beta_{14} - 81920 \beta_{16} - 1875968 \beta_{17} + 4595712 \beta_{18} + 81920 \beta_{19} ) q^{88} + ( -15806668580422 + 7903323973181 \beta_{1} + 83207491920 \beta_{2} + 7810857186 \beta_{3} + 166298437588 \beta_{4} - 7789389356 \beta_{5} - 13104 \beta_{6} - 11009484 \beta_{7} - 11939380 \beta_{8} - 24459756 \beta_{9} - 97834688 \beta_{10} - 7751604 \beta_{11} - 6930938 \beta_{12} - 7574562 \beta_{13} - 5512532 \beta_{14} + 2314804 \beta_{15} - 5363388 \beta_{16} - 3250746 \beta_{17} - 2314804 \beta_{18} + 1683196 \beta_{19} ) q^{89} + ( 12502378902613 - 25004767792602 \beta_{1} - 27934382946 \beta_{2} + 14954344565 \beta_{3} - 13956346012 \beta_{4} - 1346931 \beta_{5} - 4899347 \beta_{6} + 1367911 \beta_{7} - 2905669 \beta_{8} + 2395456 \beta_{9} + 3116827 \beta_{10} + 35436081 \beta_{11} - 2200538 \beta_{12} - 9170989 \beta_{13} - 6748803 \beta_{14} + 4133677 \beta_{15} - 7089395 \beta_{16} - 562240 \beta_{17} + 8267354 \beta_{18} + 7908544 \beta_{19} ) q^{90} + ( 545404950795 + 19257886432998 \beta_{1} - 70044006482 \beta_{2} - 8370986708 \beta_{3} + 48118649266 \beta_{4} + 3212759077 \beta_{5} - 37524494 \beta_{6} + 2567013 \beta_{7} - 6232843 \beta_{8} + 3217991 \beta_{9} - 112911066 \beta_{10} - 117382779 \beta_{11} - 3492504 \beta_{12} - 2519485 \beta_{13} - 6806671 \beta_{14} - 3080663 \beta_{15} - 6655495 \beta_{16} - 2743374 \beta_{17} + 508280 \beta_{18} + 3591379 \beta_{19} ) q^{91} + ( 5654622052352 + 15982592 \beta_{1} - 10919936 \beta_{2} - 3338092544 \beta_{3} - 30435049472 \beta_{4} + 6644555776 \beta_{5} + 71655424 \beta_{6} + 10461184 \beta_{7} + 2351104 \beta_{8} + 18382848 \beta_{9} - 2490368 \beta_{10} - 30646272 \beta_{11} - 6963200 \beta_{12} - 14262272 \beta_{13} - 8404992 \beta_{14} - 4734976 \beta_{15} + 335872 \beta_{16} - 6963200 \beta_{17} + 6742016 \beta_{19} ) q^{92} + ( -2006952 - 9423360238278 \beta_{1} - 312143222934 \beta_{2} + 9759497993 \beta_{3} - 9095496 \beta_{4} + 9780049934 \beta_{5} - 23288604 \beta_{6} + 18439320 \beta_{7} + 24156426 \beta_{8} + 4555650 \beta_{9} - 22287138 \beta_{10} + 10139490 \beta_{11} - 1004823 \beta_{12} - 3701109 \beta_{13} + 12471192 \beta_{15} - 7854612 \beta_{16} - 3464169 \beta_{17} + 12471192 \beta_{18} + 5395266 \beta_{19} ) q^{93} + ( -7368350171053 - 7368353846165 \beta_{1} + 67907876296 \beta_{2} - 9294803 \beta_{3} - 67914967254 \beta_{4} - 1263266968 \beta_{5} - 128066748 \beta_{6} + 8625491 \beta_{7} - 52114825 \beta_{8} - 5594372 \beta_{9} - 12666213 \beta_{10} + 12657477 \beta_{11} + 858130 \beta_{12} + 4031986 \beta_{13} + 1903433 \beta_{14} + 13429934 \beta_{15} - 3165120 \beta_{16} + 1912169 \beta_{17} + 6714967 \beta_{18} + 5273198 \beta_{19} ) q^{94} + ( 2523515746274 - 2523517983949 \beta_{1} + 341660122544 \beta_{2} + 10642824698 \beta_{3} + 341656316471 \beta_{4} - 5325618049 \beta_{5} - 125701238 \beta_{6} + 10969729 \beta_{7} + 5782056 \beta_{8} - 11139072 \beta_{9} - 22795549 \beta_{10} + 246402774 \beta_{11} - 13076853 \beta_{12} - 21657596 \beta_{13} - 14232264 \beta_{14} - 11341228 \beta_{16} - 5689707 \beta_{17} + 6902522 \beta_{18} + 11341228 \beta_{19} ) q^{95} + ( 1295201075200 - 647600537600 \beta_{1} - 9865003008 \beta_{2} - 335544320 \beta_{3} - 19662897152 \beta_{4} + 335544320 \beta_{5} + 67108864 \beta_{6} + 67108864 \beta_{10} - 67108864 \beta_{11} ) q^{96} + ( 13530097747164 - 27060174571895 \beta_{1} - 78098423956 \beta_{2} - 16685817567 \beta_{3} - 39205038363 \beta_{4} - 11764261 \beta_{5} - 30357416 \beta_{6} + 2496339 \beta_{7} + 164888221 \beta_{8} - 7947163 \beta_{9} - 155430217 \beta_{10} + 68501992 \beta_{11} + 16598466 \beta_{12} + 1020927 \beta_{13} - 10753496 \beta_{14} - 1568551 \beta_{15} - 18392911 \beta_{16} - 12463458 \beta_{17} - 3137102 \beta_{18} + 3861949 \beta_{19} ) q^{97} + ( 4267165948708 + 8036231827131 \beta_{1} - 177105473810 \beta_{2} + 17276827520 \beta_{3} - 60958556369 \beta_{4} - 16411368564 \beta_{5} - 26902956 \beta_{6} + 16592352 \beta_{7} + 276120721 \beta_{8} + 1867612 \beta_{9} - 72285464 \beta_{10} - 54923172 \beta_{11} - 2816476 \beta_{12} - 3746369 \beta_{13} - 1047032 \beta_{14} + 6322001 \beta_{15} - 3638869 \beta_{16} + 204125 \beta_{17} - 1434727 \beta_{18} + 143686 \beta_{19} ) q^{98} + ( 52842154867155 - 20065368 \beta_{1} + 19653990 \beta_{2} + 13149632280 \beta_{3} - 243611313618 \beta_{4} - 26265214185 \beta_{5} + 219307992 \beta_{6} - 6903117 \beta_{7} + 3961275 \beta_{8} - 13586547 \beta_{9} + 13661034 \beta_{10} - 121085001 \beta_{11} + 11963160 \beta_{12} + 30006681 \beta_{13} + 10764069 \beta_{14} - 4805571 \beta_{15} - 6080361 \beta_{16} + 11963160 \beta_{17} - 21251541 \beta_{19} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 4374q^{3} - 81920q^{4} + 3354q^{5} + 1455616q^{7} + 17547432q^{9} + O(q^{10}) \) \( 20q - 4374q^{3} - 81920q^{4} + 3354q^{5} + 1455616q^{7} + 17547432q^{9} - 3933696q^{10} + 8400426q^{11} + 35831808q^{12} - 114693888q^{14} + 562720524q^{15} - 671088640q^{16} + 2180481042q^{17} - 251755008q^{18} - 3919727442q^{19} + 7585509006q^{21} + 5394565632q^{22} - 6905098386q^{23} - 2371878912q^{24} + 14165082644q^{25} + 12652202496q^{26} - 17334943744q^{28} + 27884908704q^{29} + 54103511808q^{30} + 45638710782q^{31} - 37041090498q^{33} - 18274367202q^{35} - 287497125888q^{36} - 27026027926q^{37} - 354043974912q^{38} - 126125404380q^{39} + 32224837632q^{40} - 298475364864q^{42} + 726682953656q^{43} + 68816289792q^{44} + 498861631944q^{45} - 286664984832q^{46} - 2044625353338q^{47} + 2939974016204q^{49} + 1161106642944q^{50} - 2419609945602q^{51} + 1314350333952q^{52} + 1546271487546q^{53} - 5213176950528q^{54} + 1720927125504q^{56} + 14789884876092q^{57} - 2365863040512q^{58} - 6798944731566q^{59} - 2304903266304q^{60} - 2214453865554q^{61} - 4417730390688q^{63} + 10995116277760q^{64} + 7516703932836q^{65} - 8476063570944q^{66} - 4655820763226q^{67} - 17862500696064q^{68} + 20497461621504q^{70} + 96606137494152q^{71} - 2062377025536q^{72} - 65348368908666q^{73} - 566532483072q^{74} - 186663280957308q^{75} + 77525241691422q^{77} + 88663911671808q^{78} - 60517474082978q^{79} - 225083129856q^{80} - 107180264511342q^{81} - 43979002397184q^{82} + 18842436550656q^{84} + 416326699526124q^{85} + 2363335174656q^{86} - 126768392660088q^{87} - 22096140828672q^{88} - 237147002561826q^{89} + 203506111374408q^{91} + 113133131956224q^{92} - 94175068190130q^{93} - 221058962902272q^{94} + 25202514515490q^{95} + 19430432047104q^{96} + 165606984015360q^{98} + 1056686373672768q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 4 x^{19} + 5366534 x^{18} + 786490608 x^{17} + 19826027932115 x^{16} + 3116508483645696 x^{15} + 38794655189396647126 x^{14} + \)\(69\!\cdots\!60\)\( x^{13} + \)\(54\!\cdots\!57\)\( x^{12} + \)\(66\!\cdots\!72\)\( x^{11} + \)\(34\!\cdots\!28\)\( x^{10} - \)\(50\!\cdots\!72\)\( x^{9} + \)\(15\!\cdots\!96\)\( x^{8} - \)\(88\!\cdots\!84\)\( x^{7} + \)\(37\!\cdots\!08\)\( x^{6} - \)\(63\!\cdots\!24\)\( x^{5} + \)\(59\!\cdots\!84\)\( x^{4} - \)\(42\!\cdots\!56\)\( x^{3} + \)\(11\!\cdots\!48\)\( x^{2} + \)\(52\!\cdots\!00\)\( x + \)\(13\!\cdots\!24\)\(\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-\)\(39\!\cdots\!16\)\( \nu^{19} + \)\(10\!\cdots\!39\)\( \nu^{18} - \)\(21\!\cdots\!28\)\( \nu^{17} + \)\(22\!\cdots\!26\)\( \nu^{16} - \)\(77\!\cdots\!60\)\( \nu^{15} + \)\(76\!\cdots\!73\)\( \nu^{14} - \)\(14\!\cdots\!64\)\( \nu^{13} + \)\(11\!\cdots\!38\)\( \nu^{12} - \)\(20\!\cdots\!84\)\( \nu^{11} + \)\(28\!\cdots\!79\)\( \nu^{10} - \)\(12\!\cdots\!16\)\( \nu^{9} + \)\(36\!\cdots\!24\)\( \nu^{8} - \)\(60\!\cdots\!64\)\( \nu^{7} + \)\(18\!\cdots\!36\)\( \nu^{6} - \)\(15\!\cdots\!28\)\( \nu^{5} + \)\(60\!\cdots\!48\)\( \nu^{4} - \)\(27\!\cdots\!24\)\( \nu^{3} + \)\(72\!\cdots\!08\)\( \nu^{2} - \)\(46\!\cdots\!88\)\( \nu + \)\(90\!\cdots\!32\)\(\)\()/ \)\(10\!\cdots\!80\)\( \)
\(\beta_{2}\)\(=\)\((\)\(\)\(28\!\cdots\!43\)\( \nu^{19} - \)\(26\!\cdots\!11\)\( \nu^{18} + \)\(16\!\cdots\!32\)\( \nu^{17} - \)\(65\!\cdots\!34\)\( \nu^{16} + \)\(65\!\cdots\!61\)\( \nu^{15} - \)\(32\!\cdots\!09\)\( \nu^{14} + \)\(14\!\cdots\!04\)\( \nu^{13} - \)\(14\!\cdots\!06\)\( \nu^{12} + \)\(22\!\cdots\!51\)\( \nu^{11} - \)\(40\!\cdots\!07\)\( \nu^{10} + \)\(19\!\cdots\!82\)\( \nu^{9} - \)\(73\!\cdots\!52\)\( \nu^{8} + \)\(11\!\cdots\!24\)\( \nu^{7} - \)\(47\!\cdots\!20\)\( \nu^{6} + \)\(47\!\cdots\!12\)\( \nu^{5} - \)\(17\!\cdots\!12\)\( \nu^{4} + \)\(11\!\cdots\!04\)\( \nu^{3} - \)\(31\!\cdots\!64\)\( \nu^{2} + \)\(14\!\cdots\!36\)\( \nu - \)\(39\!\cdots\!36\)\(\)\()/ \)\(18\!\cdots\!00\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(17\!\cdots\!51\)\( \nu^{19} - \)\(15\!\cdots\!84\)\( \nu^{18} + \)\(10\!\cdots\!90\)\( \nu^{17} - \)\(74\!\cdots\!52\)\( \nu^{16} + \)\(40\!\cdots\!21\)\( \nu^{15} - \)\(27\!\cdots\!04\)\( \nu^{14} + \)\(86\!\cdots\!26\)\( \nu^{13} - \)\(57\!\cdots\!80\)\( \nu^{12} + \)\(13\!\cdots\!51\)\( \nu^{11} - \)\(89\!\cdots\!32\)\( \nu^{10} + \)\(11\!\cdots\!64\)\( \nu^{9} - \)\(78\!\cdots\!76\)\( \nu^{8} + \)\(69\!\cdots\!36\)\( \nu^{7} - \)\(41\!\cdots\!72\)\( \nu^{6} + \)\(22\!\cdots\!84\)\( \nu^{5} - \)\(13\!\cdots\!72\)\( \nu^{4} + \)\(49\!\cdots\!20\)\( \nu^{3} - \)\(18\!\cdots\!60\)\( \nu^{2} + \)\(33\!\cdots\!76\)\( \nu - \)\(11\!\cdots\!24\)\(\)\()/ \)\(79\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(21\!\cdots\!05\)\( \nu^{19} + \)\(58\!\cdots\!44\)\( \nu^{18} + \)\(11\!\cdots\!22\)\( \nu^{17} + \)\(20\!\cdots\!56\)\( \nu^{16} + \)\(43\!\cdots\!11\)\( \nu^{15} + \)\(81\!\cdots\!12\)\( \nu^{14} + \)\(85\!\cdots\!90\)\( \nu^{13} + \)\(18\!\cdots\!24\)\( \nu^{12} + \)\(11\!\cdots\!01\)\( \nu^{11} + \)\(18\!\cdots\!68\)\( \nu^{10} + \)\(76\!\cdots\!92\)\( \nu^{9} + \)\(14\!\cdots\!08\)\( \nu^{8} + \)\(33\!\cdots\!40\)\( \nu^{7} - \)\(50\!\cdots\!84\)\( \nu^{6} + \)\(78\!\cdots\!56\)\( \nu^{5} - \)\(98\!\cdots\!44\)\( \nu^{4} + \)\(11\!\cdots\!96\)\( \nu^{3} - \)\(15\!\cdots\!96\)\( \nu^{2} + \)\(13\!\cdots\!00\)\( \nu + \)\(18\!\cdots\!00\)\(\)\()/ \)\(31\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(60\!\cdots\!95\)\( \nu^{19} + \)\(16\!\cdots\!02\)\( \nu^{18} + \)\(31\!\cdots\!14\)\( \nu^{17} + \)\(57\!\cdots\!08\)\( \nu^{16} + \)\(11\!\cdots\!13\)\( \nu^{15} + \)\(21\!\cdots\!62\)\( \nu^{14} + \)\(22\!\cdots\!78\)\( \nu^{13} + \)\(43\!\cdots\!80\)\( \nu^{12} + \)\(31\!\cdots\!31\)\( \nu^{11} + \)\(35\!\cdots\!62\)\( \nu^{10} + \)\(18\!\cdots\!96\)\( \nu^{9} - \)\(12\!\cdots\!60\)\( \nu^{8} + \)\(97\!\cdots\!44\)\( \nu^{7} - \)\(11\!\cdots\!40\)\( \nu^{6} + \)\(24\!\cdots\!40\)\( \nu^{5} - \)\(73\!\cdots\!16\)\( \nu^{4} + \)\(55\!\cdots\!64\)\( \nu^{3} - \)\(90\!\cdots\!72\)\( \nu^{2} + \)\(20\!\cdots\!52\)\( \nu - \)\(22\!\cdots\!24\)\(\)\()/ \)\(79\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(57\!\cdots\!13\)\( \nu^{19} - \)\(97\!\cdots\!34\)\( \nu^{18} - \)\(31\!\cdots\!26\)\( \nu^{17} - \)\(58\!\cdots\!56\)\( \nu^{16} - \)\(12\!\cdots\!51\)\( \nu^{15} - \)\(21\!\cdots\!74\)\( \nu^{14} - \)\(26\!\cdots\!58\)\( \nu^{13} - \)\(44\!\cdots\!08\)\( \nu^{12} - \)\(39\!\cdots\!65\)\( \nu^{11} - \)\(62\!\cdots\!22\)\( \nu^{10} - \)\(29\!\cdots\!32\)\( \nu^{9} - \)\(40\!\cdots\!36\)\( \nu^{8} - \)\(10\!\cdots\!36\)\( \nu^{7} - \)\(18\!\cdots\!96\)\( \nu^{6} - \)\(20\!\cdots\!88\)\( \nu^{5} - \)\(48\!\cdots\!76\)\( \nu^{4} + \)\(46\!\cdots\!08\)\( \nu^{3} - \)\(84\!\cdots\!04\)\( \nu^{2} + \)\(11\!\cdots\!68\)\( \nu - \)\(26\!\cdots\!12\)\(\)\()/ \)\(21\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(43\!\cdots\!83\)\( \nu^{19} - \)\(18\!\cdots\!80\)\( \nu^{18} + \)\(16\!\cdots\!30\)\( \nu^{17} - \)\(98\!\cdots\!52\)\( \nu^{16} - \)\(10\!\cdots\!51\)\( \nu^{15} - \)\(36\!\cdots\!64\)\( \nu^{14} - \)\(58\!\cdots\!54\)\( \nu^{13} - \)\(71\!\cdots\!16\)\( \nu^{12} - \)\(14\!\cdots\!45\)\( \nu^{11} - \)\(99\!\cdots\!12\)\( \nu^{10} - \)\(16\!\cdots\!08\)\( \nu^{9} - \)\(62\!\cdots\!04\)\( \nu^{8} - \)\(32\!\cdots\!52\)\( \nu^{7} - \)\(26\!\cdots\!20\)\( \nu^{6} - \)\(32\!\cdots\!44\)\( \nu^{5} - \)\(61\!\cdots\!36\)\( \nu^{4} + \)\(55\!\cdots\!80\)\( \nu^{3} - \)\(84\!\cdots\!16\)\( \nu^{2} + \)\(13\!\cdots\!08\)\( \nu - \)\(96\!\cdots\!28\)\(\)\()/ \)\(15\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(10\!\cdots\!65\)\( \nu^{19} + \)\(42\!\cdots\!18\)\( \nu^{18} - \)\(57\!\cdots\!74\)\( \nu^{17} + \)\(14\!\cdots\!72\)\( \nu^{16} - \)\(20\!\cdots\!03\)\( \nu^{15} + \)\(54\!\cdots\!78\)\( \nu^{14} - \)\(39\!\cdots\!78\)\( \nu^{13} + \)\(10\!\cdots\!60\)\( \nu^{12} - \)\(53\!\cdots\!01\)\( \nu^{11} + \)\(18\!\cdots\!18\)\( \nu^{10} - \)\(31\!\cdots\!16\)\( \nu^{9} + \)\(18\!\cdots\!80\)\( \nu^{8} - \)\(14\!\cdots\!04\)\( \nu^{7} + \)\(95\!\cdots\!60\)\( \nu^{6} - \)\(37\!\cdots\!80\)\( \nu^{5} + \)\(30\!\cdots\!36\)\( \nu^{4} - \)\(78\!\cdots\!84\)\( \nu^{3} + \)\(42\!\cdots\!92\)\( \nu^{2} - \)\(43\!\cdots\!92\)\( \nu + \)\(12\!\cdots\!64\)\(\)\()/ \)\(10\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(64\!\cdots\!27\)\( \nu^{19} + \)\(14\!\cdots\!14\)\( \nu^{18} + \)\(35\!\cdots\!22\)\( \nu^{17} + \)\(83\!\cdots\!24\)\( \nu^{16} + \)\(14\!\cdots\!05\)\( \nu^{15} + \)\(30\!\cdots\!10\)\( \nu^{14} + \)\(31\!\cdots\!38\)\( \nu^{13} + \)\(59\!\cdots\!28\)\( \nu^{12} + \)\(48\!\cdots\!51\)\( \nu^{11} + \)\(80\!\cdots\!66\)\( \nu^{10} + \)\(35\!\cdots\!64\)\( \nu^{9} + \)\(44\!\cdots\!04\)\( \nu^{8} + \)\(11\!\cdots\!40\)\( \nu^{7} + \)\(18\!\cdots\!28\)\( \nu^{6} + \)\(19\!\cdots\!80\)\( \nu^{5} + \)\(36\!\cdots\!76\)\( \nu^{4} - \)\(37\!\cdots\!20\)\( \nu^{3} + \)\(58\!\cdots\!64\)\( \nu^{2} - \)\(50\!\cdots\!84\)\( \nu - \)\(28\!\cdots\!16\)\(\)\()/ \)\(15\!\cdots\!00\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(27\!\cdots\!58\)\( \nu^{19} - \)\(38\!\cdots\!65\)\( \nu^{18} + \)\(14\!\cdots\!82\)\( \nu^{17} + \)\(14\!\cdots\!82\)\( \nu^{16} + \)\(53\!\cdots\!06\)\( \nu^{15} + \)\(11\!\cdots\!61\)\( \nu^{14} + \)\(10\!\cdots\!74\)\( \nu^{13} + \)\(46\!\cdots\!18\)\( \nu^{12} + \)\(14\!\cdots\!22\)\( \nu^{11} - \)\(19\!\cdots\!21\)\( \nu^{10} + \)\(89\!\cdots\!78\)\( \nu^{9} - \)\(13\!\cdots\!72\)\( \nu^{8} + \)\(40\!\cdots\!16\)\( \nu^{7} - \)\(73\!\cdots\!24\)\( \nu^{6} + \)\(99\!\cdots\!24\)\( \nu^{5} - \)\(27\!\cdots\!56\)\( \nu^{4} + \)\(17\!\cdots\!00\)\( \nu^{3} - \)\(22\!\cdots\!76\)\( \nu^{2} + \)\(22\!\cdots\!32\)\( \nu + \)\(43\!\cdots\!20\)\(\)\()/ \)\(54\!\cdots\!00\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(15\!\cdots\!11\)\( \nu^{19} - \)\(22\!\cdots\!72\)\( \nu^{18} + \)\(83\!\cdots\!38\)\( \nu^{17} + \)\(45\!\cdots\!20\)\( \nu^{16} + \)\(30\!\cdots\!41\)\( \nu^{15} + \)\(46\!\cdots\!32\)\( \nu^{14} + \)\(59\!\cdots\!94\)\( \nu^{13} + \)\(22\!\cdots\!56\)\( \nu^{12} + \)\(82\!\cdots\!51\)\( \nu^{11} - \)\(20\!\cdots\!52\)\( \nu^{10} + \)\(51\!\cdots\!96\)\( \nu^{9} - \)\(88\!\cdots\!60\)\( \nu^{8} + \)\(23\!\cdots\!44\)\( \nu^{7} - \)\(48\!\cdots\!48\)\( \nu^{6} + \)\(56\!\cdots\!00\)\( \nu^{5} - \)\(18\!\cdots\!64\)\( \nu^{4} + \)\(99\!\cdots\!88\)\( \nu^{3} - \)\(17\!\cdots\!28\)\( \nu^{2} + \)\(23\!\cdots\!80\)\( \nu - \)\(60\!\cdots\!16\)\(\)\()/ \)\(21\!\cdots\!00\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(18\!\cdots\!77\)\( \nu^{19} - \)\(28\!\cdots\!98\)\( \nu^{18} - \)\(79\!\cdots\!42\)\( \nu^{17} - \)\(16\!\cdots\!52\)\( \nu^{16} - \)\(28\!\cdots\!27\)\( \nu^{15} - \)\(61\!\cdots\!06\)\( \nu^{14} - \)\(44\!\cdots\!34\)\( \nu^{13} - \)\(11\!\cdots\!76\)\( \nu^{12} - \)\(49\!\cdots\!85\)\( \nu^{11} - \)\(15\!\cdots\!74\)\( \nu^{10} + \)\(15\!\cdots\!56\)\( \nu^{9} - \)\(82\!\cdots\!32\)\( \nu^{8} + \)\(34\!\cdots\!60\)\( \nu^{7} - \)\(38\!\cdots\!60\)\( \nu^{6} + \)\(20\!\cdots\!92\)\( \nu^{5} - \)\(91\!\cdots\!36\)\( \nu^{4} + \)\(60\!\cdots\!00\)\( \nu^{3} - \)\(21\!\cdots\!12\)\( \nu^{2} + \)\(64\!\cdots\!84\)\( \nu - \)\(15\!\cdots\!52\)\(\)\()/ \)\(21\!\cdots\!00\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(15\!\cdots\!11\)\( \nu^{19} + \)\(13\!\cdots\!78\)\( \nu^{18} - \)\(81\!\cdots\!00\)\( \nu^{17} + \)\(62\!\cdots\!60\)\( \nu^{16} - \)\(29\!\cdots\!69\)\( \nu^{15} + \)\(22\!\cdots\!90\)\( \nu^{14} - \)\(54\!\cdots\!20\)\( \nu^{13} + \)\(43\!\cdots\!08\)\( \nu^{12} - \)\(74\!\cdots\!27\)\( \nu^{11} + \)\(65\!\cdots\!70\)\( \nu^{10} - \)\(46\!\cdots\!74\)\( \nu^{9} + \)\(49\!\cdots\!64\)\( \nu^{8} - \)\(28\!\cdots\!36\)\( \nu^{7} + \)\(23\!\cdots\!64\)\( \nu^{6} - \)\(90\!\cdots\!96\)\( \nu^{5} + \)\(63\!\cdots\!72\)\( \nu^{4} - \)\(24\!\cdots\!80\)\( \nu^{3} + \)\(92\!\cdots\!28\)\( \nu^{2} - \)\(10\!\cdots\!76\)\( \nu + \)\(13\!\cdots\!52\)\(\)\()/ \)\(54\!\cdots\!00\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(15\!\cdots\!86\)\( \nu^{19} - \)\(74\!\cdots\!41\)\( \nu^{18} - \)\(85\!\cdots\!60\)\( \nu^{17} - \)\(52\!\cdots\!30\)\( \nu^{16} - \)\(32\!\cdots\!82\)\( \nu^{15} - \)\(19\!\cdots\!99\)\( \nu^{14} - \)\(66\!\cdots\!64\)\( \nu^{13} - \)\(40\!\cdots\!46\)\( \nu^{12} - \)\(96\!\cdots\!26\)\( \nu^{11} - \)\(52\!\cdots\!65\)\( \nu^{10} - \)\(67\!\cdots\!52\)\( \nu^{9} - \)\(26\!\cdots\!08\)\( \nu^{8} - \)\(27\!\cdots\!92\)\( \nu^{7} - \)\(10\!\cdots\!12\)\( \nu^{6} - \)\(65\!\cdots\!64\)\( \nu^{5} - \)\(19\!\cdots\!16\)\( \nu^{4} - \)\(50\!\cdots\!88\)\( \nu^{3} - \)\(40\!\cdots\!88\)\( \nu^{2} + \)\(43\!\cdots\!16\)\( \nu - \)\(11\!\cdots\!68\)\(\)\()/ \)\(37\!\cdots\!00\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(80\!\cdots\!75\)\( \nu^{19} - \)\(45\!\cdots\!08\)\( \nu^{18} - \)\(42\!\cdots\!54\)\( \nu^{17} - \)\(91\!\cdots\!24\)\( \nu^{16} - \)\(15\!\cdots\!53\)\( \nu^{15} - \)\(35\!\cdots\!72\)\( \nu^{14} - \)\(30\!\cdots\!98\)\( \nu^{13} - \)\(78\!\cdots\!04\)\( \nu^{12} - \)\(43\!\cdots\!67\)\( \nu^{11} - \)\(86\!\cdots\!28\)\( \nu^{10} - \)\(27\!\cdots\!12\)\( \nu^{9} - \)\(21\!\cdots\!40\)\( \nu^{8} - \)\(12\!\cdots\!96\)\( \nu^{7} - \)\(53\!\cdots\!36\)\( \nu^{6} - \)\(28\!\cdots\!32\)\( \nu^{5} + \)\(20\!\cdots\!40\)\( \nu^{4} - \)\(44\!\cdots\!20\)\( \nu^{3} + \)\(41\!\cdots\!84\)\( \nu^{2} - \)\(12\!\cdots\!48\)\( \nu + \)\(48\!\cdots\!08\)\(\)\()/ \)\(15\!\cdots\!00\)\( \)
\(\beta_{16}\)\(=\)\((\)\(-\)\(37\!\cdots\!17\)\( \nu^{19} + \)\(91\!\cdots\!41\)\( \nu^{18} - \)\(20\!\cdots\!60\)\( \nu^{17} + \)\(17\!\cdots\!22\)\( \nu^{16} - \)\(74\!\cdots\!87\)\( \nu^{15} + \)\(56\!\cdots\!43\)\( \nu^{14} - \)\(14\!\cdots\!12\)\( \nu^{13} + \)\(70\!\cdots\!22\)\( \nu^{12} - \)\(20\!\cdots\!09\)\( \nu^{11} + \)\(21\!\cdots\!17\)\( \nu^{10} - \)\(12\!\cdots\!90\)\( \nu^{9} + \)\(30\!\cdots\!32\)\( \nu^{8} - \)\(62\!\cdots\!36\)\( \nu^{7} + \)\(16\!\cdots\!32\)\( \nu^{6} - \)\(16\!\cdots\!92\)\( \nu^{5} + \)\(55\!\cdots\!32\)\( \nu^{4} - \)\(32\!\cdots\!32\)\( \nu^{3} + \)\(73\!\cdots\!84\)\( \nu^{2} - \)\(12\!\cdots\!84\)\( \nu + \)\(95\!\cdots\!56\)\(\)\()/ \)\(37\!\cdots\!00\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-\)\(97\!\cdots\!45\)\( \nu^{19} - \)\(17\!\cdots\!23\)\( \nu^{18} - \)\(51\!\cdots\!60\)\( \nu^{17} - \)\(17\!\cdots\!30\)\( \nu^{16} - \)\(19\!\cdots\!11\)\( \nu^{15} - \)\(64\!\cdots\!53\)\( \nu^{14} - \)\(37\!\cdots\!68\)\( \nu^{13} - \)\(13\!\cdots\!58\)\( \nu^{12} - \)\(52\!\cdots\!53\)\( \nu^{11} - \)\(15\!\cdots\!35\)\( \nu^{10} - \)\(32\!\cdots\!26\)\( \nu^{9} - \)\(48\!\cdots\!04\)\( \nu^{8} - \)\(13\!\cdots\!12\)\( \nu^{7} - \)\(15\!\cdots\!92\)\( \nu^{6} - \)\(27\!\cdots\!16\)\( \nu^{5} + \)\(44\!\cdots\!04\)\( \nu^{4} - \)\(31\!\cdots\!96\)\( \nu^{3} - \)\(55\!\cdots\!72\)\( \nu^{2} + \)\(12\!\cdots\!24\)\( \nu - \)\(13\!\cdots\!00\)\(\)\()/ \)\(75\!\cdots\!00\)\( \)
\(\beta_{18}\)\(=\)\((\)\(-\)\(24\!\cdots\!91\)\( \nu^{19} + \)\(43\!\cdots\!74\)\( \nu^{18} - \)\(12\!\cdots\!34\)\( \nu^{17} + \)\(40\!\cdots\!28\)\( \nu^{16} - \)\(47\!\cdots\!33\)\( \nu^{15} + \)\(10\!\cdots\!70\)\( \nu^{14} - \)\(91\!\cdots\!86\)\( \nu^{13} + \)\(98\!\cdots\!84\)\( \nu^{12} - \)\(12\!\cdots\!75\)\( \nu^{11} + \)\(79\!\cdots\!54\)\( \nu^{10} - \)\(79\!\cdots\!80\)\( \nu^{9} + \)\(16\!\cdots\!00\)\( \nu^{8} - \)\(36\!\cdots\!60\)\( \nu^{7} + \)\(91\!\cdots\!08\)\( \nu^{6} - \)\(89\!\cdots\!60\)\( \nu^{5} + \)\(32\!\cdots\!28\)\( \nu^{4} - \)\(16\!\cdots\!44\)\( \nu^{3} + \)\(36\!\cdots\!76\)\( \nu^{2} - \)\(15\!\cdots\!88\)\( \nu - \)\(10\!\cdots\!48\)\(\)\()/ \)\(15\!\cdots\!00\)\( \)
\(\beta_{19}\)\(=\)\((\)\(\)\(12\!\cdots\!51\)\( \nu^{19} + \)\(15\!\cdots\!44\)\( \nu^{18} + \)\(65\!\cdots\!70\)\( \nu^{17} + \)\(17\!\cdots\!60\)\( \nu^{16} + \)\(24\!\cdots\!53\)\( \nu^{15} + \)\(69\!\cdots\!72\)\( \nu^{14} + \)\(47\!\cdots\!82\)\( \nu^{13} + \)\(14\!\cdots\!44\)\( \nu^{12} + \)\(67\!\cdots\!19\)\( \nu^{11} + \)\(16\!\cdots\!80\)\( \nu^{10} + \)\(43\!\cdots\!88\)\( \nu^{9} + \)\(51\!\cdots\!52\)\( \nu^{8} + \)\(18\!\cdots\!24\)\( \nu^{7} + \)\(14\!\cdots\!64\)\( \nu^{6} + \)\(43\!\cdots\!20\)\( \nu^{5} - \)\(18\!\cdots\!68\)\( \nu^{4} + \)\(59\!\cdots\!64\)\( \nu^{3} + \)\(24\!\cdots\!80\)\( \nu^{2} + \)\(10\!\cdots\!20\)\( \nu + \)\(33\!\cdots\!88\)\(\)\()/ \)\(72\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{11} - \beta_{10} + 2 \beta_{8} + \beta_{6} + 5 \beta_{5} - \beta_{4} + 5 \beta_{3} + 6 \beta_{2} + 150 \beta_{1}\)\()/384\)
\(\nu^{2}\)\(=\)\((\)\(-32 \beta_{18} - 32 \beta_{17} + 32 \beta_{12} + 106 \beta_{11} - 138 \beta_{10} - 64 \beta_{9} + 53 \beta_{8} + 64 \beta_{7} - 117 \beta_{6} - 4009 \beta_{5} - 129228 \beta_{4} + 7986 \beta_{3} - 129090 \beta_{2} + 206072174 \beta_{1} - 206072110\)\()/192\)
\(\nu^{3}\)\(=\)\((\)\(-13708 \beta_{19} - 8538 \beta_{17} + 18227 \beta_{16} - 11353 \beta_{15} - 59943 \beta_{14} - 35303 \beta_{13} - 8538 \beta_{12} + 1885783 \beta_{11} - 3025827 \beta_{10} - 159768 \beta_{9} - 3066261 \beta_{8} - 18149 \beta_{7} - 3854867 \beta_{6} - 31603123 \beta_{5} - 393072544 \beta_{4} + 15871691 \beta_{3} + 42828 \beta_{2} - 61016 \beta_{1} - 46527608787\)\()/384\)
\(\nu^{4}\)\(=\)\((\)\(14256582 \beta_{19} + 10104385 \beta_{18} + 15327269 \beta_{17} + 15110363 \beta_{16} + 10104385 \beta_{15} - 14087889 \beta_{13} - 14039676 \beta_{12} - 42547328 \beta_{11} + 64430404 \beta_{10} - 12534932 \beta_{9} - 140927803 \beta_{8} - 36689064 \beta_{7} - 10185228 \beta_{6} - 2403121124 \beta_{5} + 39987099 \beta_{4} - 2423112988 \beta_{3} + 68795799893 \beta_{2} - 69495056442301 \beta_{1} - 13919196\)\()/32\)
\(\nu^{5}\)\(=\)\((\)\(-34660198122 \beta_{19} - 49147590849 \beta_{18} + 110574842517 \beta_{17} + 34660198122 \beta_{16} + 183807036735 \beta_{14} + 311718586560 \beta_{13} + 51996905430 \beta_{12} - 8815296232937 \beta_{11} + 14391699060545 \beta_{10} - 127340849940 \beta_{9} - 7165660537888 \beta_{8} - 730030669035 \beta_{7} + 4533031143049 \beta_{6} + 54810825695273 \beta_{5} + 1455802488037199 \beta_{4} - 108504558190441 \beta_{3} + 1442135287880931 \beta_{2} - 218864745819308175 \beta_{1} + 218864441993866191\)\()/384\)
\(\nu^{6}\)\(=\)\((\)\(-199245358320764 \beta_{19} + 2198983401774 \beta_{17} - 230196384470633 \beta_{16} - 113813426083301 \beta_{15} + 14827896586725 \beta_{14} + 234594351274181 \beta_{13} + 2198983401774 \beta_{12} - 803578955709953 \beta_{11} + 1389699336801253 \beta_{10} + 504991342606344 \beta_{9} + 1360137493883899 \beta_{8} + 104772679532639 \beta_{7} + 1588025998285729 \beta_{6} + 99182552665482977 \beta_{5} + 1398582188136600620 \beta_{4} - 49821577134292621 \beta_{3} + 52281922685820 \beta_{2} + 115202609315816 \beta_{1} + 945484413357984726033\)\()/192\)
\(\nu^{7}\)\(=\)\((\)\(89784613020509038 \beta_{19} + 152900965431282085 \beta_{18} - 255969806473723511 \beta_{17} - 357710831769998153 \beta_{16} + 152900965431282085 \beta_{15} - 366815540363033317 \beta_{13} + 11956412275765604 \beta_{12} + 10846358216153333182 \beta_{11} - 16786002845332564186 \beta_{10} + 1471441680881030492 \beta_{9} + 33169484191294953157 \beta_{8} + 1994595573310347960 \beta_{7} + 9439427713624871778 \beta_{6} + 176853275335255070594 \beta_{5} - 15330015145579399239 \beta_{4} + 176649214429688781066 \beta_{3} - 4924301121185370692139 \beta_{2} + 739579947741219435785539 \beta_{1} - 643846467705557596\)\()/384\)
\(\nu^{8}\)\(=\)\((\)\(1964703311133960810 \beta_{19} - 16954922705751485671 \beta_{18} - 49278844579657540805 \beta_{17} - 1964703311133960810 \beta_{16} - 7774000518603417399 \beta_{14} - 18074252497447530272 \beta_{13} + 37013889289679467122 \beta_{12} + 441176951228965657825 \beta_{11} - 629805457332988088017 \beta_{10} - 49347413036760901004 \beta_{9} + 309059147230472907320 \beta_{8} + 102297459554371661179 \beta_{7} - 291972368889313911825 \beta_{6} - 13164320500379207498449 \beta_{5} - 377009835029391917961151 \beta_{4} + 26220602812453677239329 \beta_{3} - 376413567012867651812363 \beta_{2} + 183941945201419106998445311 \beta_{1} - 183941864822805946409810183\)\()/16\)
\(\nu^{9}\)\(=\)\((\)\(162892303044235968946500 \beta_{19} - 284503565386330592803842 \beta_{17} + 792021214155031630787103 \beta_{16} - 451306154262471399580701 \beta_{15} - 1235548589226261262411107 \beta_{14} - 1361028344927692816394787 \beta_{13} - 284503565386330592803842 \beta_{12} + 23174249301733964442155767 \beta_{11} - 37819289817465357642300019 \beta_{10} - 2929740354311886481676280 \beta_{9} - 38365814854699302706258285 \beta_{8} - 57806335628669249414745 \beta_{7} - 47561565099155474025073223 \beta_{6} - 1100263465664475480619027207 \beta_{5} - 15646387459887785848818658084 \beta_{4} + 551252091233397525451125323 \beta_{3} + 299434019190418886117244 \beta_{2} - 724347794082269326218168 \beta_{1} - 2253800490661545150350898880215\)\()/384\)
\(\nu^{10}\)\(=\)\((\)\(\)\(10\!\cdots\!54\)\( \beta_{19} + \)\(32\!\cdots\!63\)\( \beta_{18} + \)\(14\!\cdots\!39\)\( \beta_{17} + \)\(15\!\cdots\!37\)\( \beta_{16} + \)\(32\!\cdots\!63\)\( \beta_{15} - \)\(84\!\cdots\!27\)\( \beta_{13} - \)\(11\!\cdots\!52\)\( \beta_{12} - \)\(83\!\cdots\!82\)\( \beta_{11} + \)\(10\!\cdots\!38\)\( \beta_{10} - \)\(17\!\cdots\!12\)\( \beta_{9} - \)\(20\!\cdots\!85\)\( \beta_{8} - \)\(36\!\cdots\!24\)\( \beta_{7} - \)\(45\!\cdots\!38\)\( \beta_{6} - \)\(47\!\cdots\!30\)\( \beta_{5} + \)\(77\!\cdots\!47\)\( \beta_{4} - \)\(47\!\cdots\!54\)\( \beta_{3} + \)\(13\!\cdots\!83\)\( \beta_{2} - \)\(52\!\cdots\!79\)\( \beta_{1} - \)\(67\!\cdots\!00\)\(\)\()/192\)
\(\nu^{11}\)\(=\)\((\)\(-\)\(78\!\cdots\!26\)\( \beta_{19} - \)\(13\!\cdots\!37\)\( \beta_{18} + \)\(31\!\cdots\!05\)\( \beta_{17} + \)\(78\!\cdots\!26\)\( \beta_{16} + \)\(31\!\cdots\!51\)\( \beta_{14} + \)\(55\!\cdots\!24\)\( \beta_{13} + \)\(75\!\cdots\!94\)\( \beta_{12} - \)\(11\!\cdots\!89\)\( \beta_{11} + \)\(17\!\cdots\!49\)\( \beta_{10} - \)\(18\!\cdots\!04\)\( \beta_{9} - \)\(89\!\cdots\!04\)\( \beta_{8} - \)\(13\!\cdots\!63\)\( \beta_{7} + \)\(61\!\cdots\!49\)\( \beta_{6} + \)\(16\!\cdots\!41\)\( \beta_{5} + \)\(47\!\cdots\!79\)\( \beta_{4} - \)\(32\!\cdots\!85\)\( \beta_{3} + \)\(47\!\cdots\!03\)\( \beta_{2} - \)\(65\!\cdots\!35\)\( \beta_{1} + \)\(65\!\cdots\!95\)\(\)\()/384\)
\(\nu^{12}\)\(=\)\((\)\(-\)\(44\!\cdots\!32\)\( \beta_{19} + \)\(61\!\cdots\!46\)\( \beta_{17} - \)\(58\!\cdots\!53\)\( \beta_{16} - \)\(65\!\cdots\!05\)\( \beta_{15} + \)\(20\!\cdots\!73\)\( \beta_{14} + \)\(70\!\cdots\!45\)\( \beta_{13} + \)\(61\!\cdots\!46\)\( \beta_{12} - \)\(33\!\cdots\!49\)\( \beta_{11} + \)\(47\!\cdots\!49\)\( \beta_{10} + \)\(11\!\cdots\!04\)\( \beta_{9} + \)\(47\!\cdots\!75\)\( \beta_{8} + \)\(82\!\cdots\!71\)\( \beta_{7} + \)\(67\!\cdots\!05\)\( \beta_{6} + \)\(46\!\cdots\!37\)\( \beta_{5} + \)\(68\!\cdots\!12\)\( \beta_{4} - \)\(23\!\cdots\!97\)\( \beta_{3} + \)\(16\!\cdots\!60\)\( \beta_{2} + \)\(16\!\cdots\!52\)\( \beta_{1} + \)\(20\!\cdots\!21\)\(\)\()/32\)
\(\nu^{13}\)\(=\)\((\)\(-\)\(30\!\cdots\!46\)\( \beta_{19} + \)\(37\!\cdots\!13\)\( \beta_{18} - \)\(66\!\cdots\!55\)\( \beta_{17} - \)\(88\!\cdots\!93\)\( \beta_{16} + \)\(37\!\cdots\!13\)\( \beta_{15} - \)\(36\!\cdots\!49\)\( \beta_{13} + \)\(24\!\cdots\!84\)\( \beta_{12} + \)\(15\!\cdots\!74\)\( \beta_{11} - \)\(21\!\cdots\!74\)\( \beta_{10} + \)\(23\!\cdots\!28\)\( \beta_{9} + \)\(41\!\cdots\!09\)\( \beta_{8} + \)\(33\!\cdots\!48\)\( \beta_{7} + \)\(12\!\cdots\!70\)\( \beta_{6} + \)\(47\!\cdots\!90\)\( \beta_{5} - \)\(18\!\cdots\!31\)\( \beta_{4} + \)\(47\!\cdots\!02\)\( \beta_{3} - \)\(13\!\cdots\!71\)\( \beta_{2} + \)\(18\!\cdots\!47\)\( \beta_{1} - \)\(76\!\cdots\!44\)\(\)\()/384\)
\(\nu^{14}\)\(=\)\((\)\(\)\(11\!\cdots\!90\)\( \beta_{19} - \)\(17\!\cdots\!45\)\( \beta_{18} - \)\(10\!\cdots\!55\)\( \beta_{17} - \)\(11\!\cdots\!90\)\( \beta_{16} - \)\(38\!\cdots\!57\)\( \beta_{14} - \)\(75\!\cdots\!72\)\( \beta_{13} + \)\(51\!\cdots\!50\)\( \beta_{12} + \)\(12\!\cdots\!71\)\( \beta_{11} - \)\(15\!\cdots\!11\)\( \beta_{10} - \)\(41\!\cdots\!24\)\( \beta_{9} + \)\(76\!\cdots\!04\)\( \beta_{8} + \)\(24\!\cdots\!41\)\( \beta_{7} - \)\(73\!\cdots\!43\)\( \beta_{6} - \)\(39\!\cdots\!95\)\( \beta_{5} - \)\(11\!\cdots\!85\)\( \beta_{4} + \)\(79\!\cdots\!51\)\( \beta_{3} - \)\(11\!\cdots\!57\)\( \beta_{2} + \)\(30\!\cdots\!41\)\( \beta_{1} - \)\(30\!\cdots\!49\)\(\)\()/192\)
\(\nu^{15}\)\(=\)\((\)\(\)\(77\!\cdots\!96\)\( \beta_{19} - \)\(58\!\cdots\!34\)\( \beta_{17} + \)\(18\!\cdots\!03\)\( \beta_{16} - \)\(10\!\cdots\!73\)\( \beta_{15} - \)\(21\!\cdots\!15\)\( \beta_{14} - \)\(30\!\cdots\!71\)\( \beta_{13} - \)\(58\!\cdots\!34\)\( \beta_{12} + \)\(32\!\cdots\!23\)\( \beta_{11} - \)\(48\!\cdots\!39\)\( \beta_{10} - \)\(46\!\cdots\!52\)\( \beta_{9} - \)\(49\!\cdots\!05\)\( \beta_{8} + \)\(39\!\cdots\!47\)\( \beta_{7} - \)\(66\!\cdots\!59\)\( \beta_{6} - \)\(26\!\cdots\!87\)\( \beta_{5} - \)\(39\!\cdots\!68\)\( \beta_{4} + \)\(13\!\cdots\!95\)\( \beta_{3} - \)\(15\!\cdots\!28\)\( \beta_{2} - \)\(58\!\cdots\!00\)\( \beta_{1} - \)\(50\!\cdots\!39\)\(\)\()/384\)
\(\nu^{16}\)\(=\)\((\)\(\)\(49\!\cdots\!46\)\( \beta_{19} - \)\(92\!\cdots\!33\)\( \beta_{18} + \)\(93\!\cdots\!23\)\( \beta_{17} + \)\(10\!\cdots\!33\)\( \beta_{16} - \)\(92\!\cdots\!33\)\( \beta_{15} - \)\(31\!\cdots\!67\)\( \beta_{13} - \)\(64\!\cdots\!56\)\( \beta_{12} - \)\(76\!\cdots\!94\)\( \beta_{11} + \)\(84\!\cdots\!66\)\( \beta_{10} - \)\(14\!\cdots\!68\)\( \beta_{9} - \)\(16\!\cdots\!13\)\( \beta_{8} - \)\(26\!\cdots\!52\)\( \beta_{7} - \)\(45\!\cdots\!74\)\( \beta_{6} - \)\(46\!\cdots\!70\)\( \beta_{5} + \)\(65\!\cdots\!35\)\( \beta_{4} - \)\(46\!\cdots\!14\)\( \beta_{3} + \)\(13\!\cdots\!91\)\( \beta_{2} - \)\(30\!\cdots\!59\)\( \beta_{1} - \)\(13\!\cdots\!88\)\(\)\()/8\)
\(\nu^{17}\)\(=\)\((\)\(-\)\(16\!\cdots\!58\)\( \beta_{19} - \)\(30\!\cdots\!65\)\( \beta_{18} + \)\(67\!\cdots\!61\)\( \beta_{17} + \)\(16\!\cdots\!58\)\( \beta_{16} + \)\(56\!\cdots\!59\)\( \beta_{14} + \)\(10\!\cdots\!48\)\( \beta_{13} - \)\(69\!\cdots\!14\)\( \beta_{12} - \)\(17\!\cdots\!53\)\( \beta_{11} + \)\(23\!\cdots\!69\)\( \beta_{10} - \)\(31\!\cdots\!08\)\( \beta_{9} - \)\(11\!\cdots\!68\)\( \beta_{8} - \)\(23\!\cdots\!19\)\( \beta_{7} + \)\(92\!\cdots\!17\)\( \beta_{6} + \)\(37\!\cdots\!05\)\( \beta_{5} + \)\(11\!\cdots\!91\)\( \beta_{4} - \)\(74\!\cdots\!21\)\( \beta_{3} + \)\(11\!\cdots\!15\)\( \beta_{2} - \)\(13\!\cdots\!75\)\( \beta_{1} + \)\(13\!\cdots\!47\)\(\)\()/384\)
\(\nu^{18}\)\(=\)\((\)\(-\)\(38\!\cdots\!32\)\( \beta_{19} + \)\(10\!\cdots\!82\)\( \beta_{17} - \)\(57\!\cdots\!19\)\( \beta_{16} + \)\(10\!\cdots\!77\)\( \beta_{15} + \)\(33\!\cdots\!35\)\( \beta_{14} + \)\(78\!\cdots\!83\)\( \beta_{13} + \)\(10\!\cdots\!82\)\( \beta_{12} - \)\(40\!\cdots\!19\)\( \beta_{11} + \)\(52\!\cdots\!19\)\( \beta_{10} + \)\(10\!\cdots\!32\)\( \beta_{9} + \)\(52\!\cdots\!77\)\( \beta_{8} - \)\(63\!\cdots\!03\)\( \beta_{7} + \)\(82\!\cdots\!67\)\( \beta_{6} + \)\(62\!\cdots\!91\)\( \beta_{5} + \)\(93\!\cdots\!20\)\( \beta_{4} - \)\(31\!\cdots\!23\)\( \beta_{3} + \)\(19\!\cdots\!40\)\( \beta_{2} + \)\(58\!\cdots\!68\)\( \beta_{1} + \)\(17\!\cdots\!15\)\(\)\()/192\)
\(\nu^{19}\)\(=\)\((\)\(-\)\(22\!\cdots\!66\)\( \beta_{19} + \)\(83\!\cdots\!05\)\( \beta_{18} - \)\(14\!\cdots\!23\)\( \beta_{17} - \)\(18\!\cdots\!29\)\( \beta_{16} + \)\(83\!\cdots\!05\)\( \beta_{15} - \)\(39\!\cdots\!81\)\( \beta_{13} + \)\(65\!\cdots\!72\)\( \beta_{12} + \)\(23\!\cdots\!94\)\( \beta_{11} - \)\(28\!\cdots\!86\)\( \beta_{10} + \)\(38\!\cdots\!56\)\( \beta_{9} + \)\(56\!\cdots\!77\)\( \beta_{8} + \)\(56\!\cdots\!80\)\( \beta_{7} + \)\(17\!\cdots\!22\)\( \beta_{6} + \)\(10\!\cdots\!62\)\( \beta_{5} - \)\(24\!\cdots\!15\)\( \beta_{4} + \)\(10\!\cdots\!58\)\( \beta_{3} - \)\(30\!\cdots\!19\)\( \beta_{2} + \)\(37\!\cdots\!67\)\( \beta_{1} - \)\(10\!\cdots\!28\)\(\)\()/384\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
753.046 + 1304.31i
222.426 + 385.253i
87.1998 + 151.035i
−332.618 576.111i
−727.640 1260.31i
−728.515 1261.82i
−354.747 614.440i
−48.0702 83.2600i
319.863 + 554.019i
811.055 + 1404.79i
753.046 1304.31i
222.426 385.253i
87.1998 151.035i
−332.618 + 576.111i
−727.640 + 1260.31i
−728.515 + 1261.82i
−354.747 + 614.440i
−48.0702 + 83.2600i
319.863 554.019i
811.055 1404.79i
−45.2548 78.3837i −3571.50 2062.01i −4096.00 + 7094.48i 81056.3 46797.9i 373263.i 587107. + 577519.i 741455. 6.11227e6 + 1.05868e7i −7.33638e6 4.23566e6i
3.2 −45.2548 78.3837i −1320.27 762.260i −4096.00 + 7094.48i −130389. + 75280.2i 137984.i 789127. + 235587.i 741455. −1.22940e6 2.12939e6i 1.18015e7 + 6.81358e6i
3.3 −45.2548 78.3837i −746.556 431.025i −4096.00 + 7094.48i 45757.7 26418.2i 78023.8i −602075. 561898.i 741455. −2.01992e6 3.49860e6i −4.14152e6 2.39111e6i
3.4 −45.2548 78.3837i 1034.58 + 597.314i −4096.00 + 7094.48i 5108.24 2949.25i 108125.i −382860. + 729137.i 741455. −1.67792e6 2.90624e6i −462345. 266935.i
3.5 −45.2548 78.3837i 2710.52 + 1564.92i −4096.00 + 7094.48i 10170.7 5872.03i 283280.i 552858. 610386.i 741455. 2.50645e6 + 4.34129e6i −920543. 531476.i
3.6 45.2548 + 78.3837i −3149.23 1818.21i −4096.00 + 7094.48i −112471. + 64935.4i 329131.i −629509. + 530982.i −741455. 4.22027e6 + 7.30973e6i −1.01797e7 5.87728e6i
3.7 45.2548 + 78.3837i −1563.47 902.668i −4096.00 + 7094.48i 25206.6 14553.0i 163400.i 823165. + 24943.0i −741455. −761866. 1.31959e6i 2.28144e6 + 1.31719e6i
3.8 45.2548 + 78.3837i −262.346 151.465i −4096.00 + 7094.48i 106192. 61309.7i 27418.1i −789690. + 233694.i −741455. −2.34560e6 4.06270e6i 9.61136e6 + 5.54912e6i
3.9 45.2548 + 78.3837i 1298.66 + 749.784i −4096.00 + 7094.48i −67507.4 + 38975.4i 135725.i −346065. 747303.i −741455. −1.26713e6 2.19474e6i −6.11007e6 3.52765e6i
3.10 45.2548 + 78.3837i 3382.61 + 1952.95i −4096.00 + 7094.48i 38553.6 22259.0i 353522.i 725749. + 389245.i −741455. 5.23657e6 + 9.07000e6i 3.48948e6 + 2.01465e6i
5.1 −45.2548 + 78.3837i −3571.50 + 2062.01i −4096.00 7094.48i 81056.3 + 46797.9i 373263.i 587107. 577519.i 741455. 6.11227e6 1.05868e7i −7.33638e6 + 4.23566e6i
5.2 −45.2548 + 78.3837i −1320.27 + 762.260i −4096.00 7094.48i −130389. 75280.2i 137984.i 789127. 235587.i 741455. −1.22940e6 + 2.12939e6i 1.18015e7 6.81358e6i
5.3 −45.2548 + 78.3837i −746.556 + 431.025i −4096.00 7094.48i 45757.7 + 26418.2i 78023.8i −602075. + 561898.i 741455. −2.01992e6 + 3.49860e6i −4.14152e6 + 2.39111e6i
5.4 −45.2548 + 78.3837i 1034.58 597.314i −4096.00 7094.48i 5108.24 + 2949.25i 108125.i −382860. 729137.i 741455. −1.67792e6 + 2.90624e6i −462345. + 266935.i
5.5 −45.2548 + 78.3837i 2710.52 1564.92i −4096.00 7094.48i 10170.7 + 5872.03i 283280.i 552858. + 610386.i 741455. 2.50645e6 4.34129e6i −920543. + 531476.i
5.6 45.2548 78.3837i −3149.23 + 1818.21i −4096.00 7094.48i −112471. 64935.4i 329131.i −629509. 530982.i −741455. 4.22027e6 7.30973e6i −1.01797e7 + 5.87728e6i
5.7 45.2548 78.3837i −1563.47 + 902.668i −4096.00 7094.48i 25206.6 + 14553.0i 163400.i 823165. 24943.0i −741455. −761866. + 1.31959e6i 2.28144e6 1.31719e6i
5.8 45.2548 78.3837i −262.346 + 151.465i −4096.00 7094.48i 106192. + 61309.7i 27418.1i −789690. 233694.i −741455. −2.34560e6 + 4.06270e6i 9.61136e6 5.54912e6i
5.9 45.2548 78.3837i 1298.66 749.784i −4096.00 7094.48i −67507.4 38975.4i 135725.i −346065. + 747303.i −741455. −1.26713e6 + 2.19474e6i −6.11007e6 + 3.52765e6i
5.10 45.2548 78.3837i 3382.61 1952.95i −4096.00 7094.48i 38553.6 + 22259.0i 353522.i 725749. 389245.i −741455. 5.23657e6 9.07000e6i 3.48948e6 2.01465e6i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 14.15.d.a 20
3.b odd 2 1 126.15.n.b 20
7.b odd 2 1 98.15.d.b 20
7.c even 3 1 98.15.b.c 20
7.c even 3 1 98.15.d.b 20
7.d odd 6 1 inner 14.15.d.a 20
7.d odd 6 1 98.15.b.c 20
21.g even 6 1 126.15.n.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.15.d.a 20 1.a even 1 1 trivial
14.15.d.a 20 7.d odd 6 1 inner
98.15.b.c 20 7.c even 3 1
98.15.b.c 20 7.d odd 6 1
98.15.d.b 20 7.b odd 2 1
98.15.d.b 20 7.c even 3 1
126.15.n.b 20 3.b odd 2 1
126.15.n.b 20 21.g even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{15}^{\mathrm{new}}(14, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 67108864 + 8192 T^{2} + T^{4} )^{5} \)
$3$ \( \)\(55\!\cdots\!49\)\( + \)\(39\!\cdots\!90\)\( T + \)\(10\!\cdots\!77\)\( T^{2} + \)\(48\!\cdots\!70\)\( T^{3} - \)\(10\!\cdots\!01\)\( T^{4} - \)\(90\!\cdots\!48\)\( T^{5} + \)\(98\!\cdots\!14\)\( T^{6} + \)\(10\!\cdots\!40\)\( T^{7} - \)\(30\!\cdots\!51\)\( T^{8} - \)\(49\!\cdots\!46\)\( T^{9} + \)\(68\!\cdots\!07\)\( T^{10} + \)\(17\!\cdots\!94\)\( T^{11} + \)\(15\!\cdots\!73\)\( T^{12} - \)\(25\!\cdots\!08\)\( T^{13} - \)\(29\!\cdots\!62\)\( T^{14} + 2566441826346788028 T^{15} + 440951103126963 T^{16} - 129032628210 T^{17} - 23122623 T^{18} + 4374 T^{19} + T^{20} \)
$5$ \( \)\(68\!\cdots\!25\)\( - \)\(38\!\cdots\!50\)\( T + \)\(93\!\cdots\!75\)\( T^{2} - \)\(12\!\cdots\!50\)\( T^{3} + \)\(99\!\cdots\!75\)\( T^{4} - \)\(47\!\cdots\!00\)\( T^{5} + \)\(13\!\cdots\!50\)\( T^{6} - \)\(17\!\cdots\!00\)\( T^{7} - \)\(53\!\cdots\!75\)\( T^{8} + \)\(44\!\cdots\!50\)\( T^{9} - \)\(12\!\cdots\!75\)\( T^{10} - \)\(10\!\cdots\!50\)\( T^{11} + \)\(11\!\cdots\!25\)\( T^{12} + \)\(63\!\cdots\!00\)\( T^{13} - \)\(12\!\cdots\!30\)\( T^{14} - \)\(31\!\cdots\!60\)\( T^{15} + \)\(99\!\cdots\!11\)\( T^{16} + 126104512257594 T^{17} - 37594494789 T^{18} - 3354 T^{19} + T^{20} \)
$7$ \( \)\(20\!\cdots\!01\)\( - \)\(44\!\cdots\!84\)\( T - \)\(18\!\cdots\!74\)\( T^{2} + \)\(41\!\cdots\!16\)\( T^{3} + \)\(18\!\cdots\!89\)\( T^{4} - \)\(19\!\cdots\!88\)\( T^{5} - \)\(28\!\cdots\!16\)\( T^{6} + \)\(10\!\cdots\!96\)\( T^{7} + \)\(77\!\cdots\!66\)\( T^{8} - \)\(17\!\cdots\!28\)\( T^{9} - \)\(10\!\cdots\!08\)\( T^{10} - \)\(26\!\cdots\!72\)\( T^{11} + \)\(16\!\cdots\!66\)\( T^{12} + \)\(34\!\cdots\!04\)\( T^{13} - \)\(13\!\cdots\!16\)\( T^{14} - \)\(13\!\cdots\!12\)\( T^{15} + \)\(18\!\cdots\!89\)\( T^{16} + 626100610013791584 T^{17} - 410578038374 T^{18} - 1455616 T^{19} + T^{20} \)
$11$ \( \)\(29\!\cdots\!89\)\( - \)\(23\!\cdots\!50\)\( T + \)\(16\!\cdots\!93\)\( T^{2} - \)\(25\!\cdots\!94\)\( T^{3} + \)\(54\!\cdots\!59\)\( T^{4} - \)\(27\!\cdots\!96\)\( T^{5} + \)\(86\!\cdots\!98\)\( T^{6} - \)\(39\!\cdots\!16\)\( T^{7} + \)\(66\!\cdots\!01\)\( T^{8} - \)\(26\!\cdots\!58\)\( T^{9} + \)\(34\!\cdots\!03\)\( T^{10} - \)\(12\!\cdots\!62\)\( T^{11} + \)\(10\!\cdots\!37\)\( T^{12} - \)\(26\!\cdots\!80\)\( T^{13} + \)\(19\!\cdots\!86\)\( T^{14} - \)\(35\!\cdots\!16\)\( T^{15} + \)\(25\!\cdots\!95\)\( T^{16} - \)\(26\!\cdots\!18\)\( T^{17} + 1943646070397229 T^{18} - 8400426 T^{19} + T^{20} \)
$13$ \( \)\(19\!\cdots\!16\)\( + \)\(28\!\cdots\!36\)\( T^{2} + \)\(53\!\cdots\!76\)\( T^{4} + \)\(32\!\cdots\!92\)\( T^{6} + \)\(57\!\cdots\!08\)\( T^{8} + \)\(45\!\cdots\!20\)\( T^{10} + \)\(18\!\cdots\!40\)\( T^{12} + \)\(42\!\cdots\!32\)\( T^{14} + \)\(54\!\cdots\!60\)\( T^{16} + 36752262711335448 T^{18} + T^{20} \)
$17$ \( \)\(57\!\cdots\!21\)\( - \)\(28\!\cdots\!18\)\( T - \)\(72\!\cdots\!49\)\( T^{2} + \)\(38\!\cdots\!26\)\( T^{3} + \)\(76\!\cdots\!31\)\( T^{4} - \)\(62\!\cdots\!08\)\( T^{5} - \)\(58\!\cdots\!54\)\( T^{6} + \)\(15\!\cdots\!92\)\( T^{7} + \)\(54\!\cdots\!13\)\( T^{8} - \)\(25\!\cdots\!34\)\( T^{9} + \)\(22\!\cdots\!49\)\( T^{10} + \)\(19\!\cdots\!54\)\( T^{11} - \)\(29\!\cdots\!47\)\( T^{12} - \)\(10\!\cdots\!72\)\( T^{13} + \)\(30\!\cdots\!26\)\( T^{14} - \)\(97\!\cdots\!32\)\( T^{15} - \)\(69\!\cdots\!89\)\( T^{16} + \)\(31\!\cdots\!14\)\( T^{17} + 1439500397616172371 T^{18} - 2180481042 T^{19} + T^{20} \)
$19$ \( \)\(34\!\cdots\!25\)\( + \)\(18\!\cdots\!50\)\( T - \)\(52\!\cdots\!75\)\( T^{2} - \)\(21\!\cdots\!50\)\( T^{3} + \)\(10\!\cdots\!75\)\( T^{4} + \)\(12\!\cdots\!00\)\( T^{5} + \)\(13\!\cdots\!30\)\( T^{6} - \)\(34\!\cdots\!80\)\( T^{7} - \)\(13\!\cdots\!11\)\( T^{8} + \)\(69\!\cdots\!14\)\( T^{9} + \)\(69\!\cdots\!55\)\( T^{10} - \)\(40\!\cdots\!98\)\( T^{11} - \)\(79\!\cdots\!99\)\( T^{12} + \)\(10\!\cdots\!48\)\( T^{13} + \)\(72\!\cdots\!02\)\( T^{14} + \)\(33\!\cdots\!84\)\( T^{15} - \)\(12\!\cdots\!41\)\( T^{16} - \)\(11\!\cdots\!86\)\( T^{17} + 2086580222205143505 T^{18} + 3919727442 T^{19} + T^{20} \)
$23$ \( \)\(33\!\cdots\!49\)\( + \)\(88\!\cdots\!94\)\( T + \)\(15\!\cdots\!41\)\( T^{2} + \)\(16\!\cdots\!42\)\( T^{3} + \)\(13\!\cdots\!67\)\( T^{4} + \)\(75\!\cdots\!56\)\( T^{5} + \)\(37\!\cdots\!30\)\( T^{6} + \)\(14\!\cdots\!48\)\( T^{7} + \)\(53\!\cdots\!37\)\( T^{8} + \)\(16\!\cdots\!18\)\( T^{9} + \)\(51\!\cdots\!91\)\( T^{10} + \)\(12\!\cdots\!82\)\( T^{11} + \)\(32\!\cdots\!33\)\( T^{12} + \)\(66\!\cdots\!92\)\( T^{13} + \)\(14\!\cdots\!50\)\( T^{14} + \)\(23\!\cdots\!92\)\( T^{15} + \)\(45\!\cdots\!03\)\( T^{16} + \)\(52\!\cdots\!50\)\( T^{17} + 89184082730661708381 T^{18} + 6905098386 T^{19} + T^{20} \)
$29$ \( ( -\)\(12\!\cdots\!48\)\( - \)\(13\!\cdots\!92\)\( T + \)\(29\!\cdots\!24\)\( T^{2} + \)\(23\!\cdots\!60\)\( T^{3} - \)\(27\!\cdots\!04\)\( T^{4} - \)\(12\!\cdots\!84\)\( T^{5} + \)\(11\!\cdots\!68\)\( T^{6} + \)\(22\!\cdots\!60\)\( T^{7} - \)\(18\!\cdots\!32\)\( T^{8} - 13942454352 T^{9} + T^{10} )^{2} \)
$31$ \( \)\(17\!\cdots\!29\)\( - \)\(72\!\cdots\!82\)\( T + \)\(43\!\cdots\!01\)\( T^{2} + \)\(24\!\cdots\!58\)\( T^{3} - \)\(83\!\cdots\!85\)\( T^{4} - \)\(55\!\cdots\!40\)\( T^{5} + \)\(53\!\cdots\!98\)\( T^{6} + \)\(11\!\cdots\!60\)\( T^{7} - \)\(30\!\cdots\!71\)\( T^{8} + \)\(37\!\cdots\!86\)\( T^{9} + \)\(13\!\cdots\!43\)\( T^{10} - \)\(39\!\cdots\!54\)\( T^{11} - \)\(23\!\cdots\!31\)\( T^{12} + \)\(11\!\cdots\!84\)\( T^{13} + \)\(29\!\cdots\!98\)\( T^{14} - \)\(25\!\cdots\!60\)\( T^{15} + \)\(27\!\cdots\!07\)\( T^{16} + \)\(10\!\cdots\!06\)\( T^{17} - \)\(17\!\cdots\!75\)\( T^{18} - 45638710782 T^{19} + T^{20} \)
$37$ \( \)\(65\!\cdots\!81\)\( + \)\(22\!\cdots\!42\)\( T + \)\(66\!\cdots\!83\)\( T^{2} + \)\(28\!\cdots\!50\)\( T^{3} + \)\(10\!\cdots\!63\)\( T^{4} + \)\(50\!\cdots\!60\)\( T^{5} + \)\(60\!\cdots\!22\)\( T^{6} - \)\(14\!\cdots\!68\)\( T^{7} + \)\(22\!\cdots\!97\)\( T^{8} - \)\(37\!\cdots\!18\)\( T^{9} + \)\(25\!\cdots\!85\)\( T^{10} - \)\(28\!\cdots\!30\)\( T^{11} + \)\(20\!\cdots\!45\)\( T^{12} - \)\(12\!\cdots\!64\)\( T^{13} + \)\(67\!\cdots\!10\)\( T^{14} + \)\(32\!\cdots\!92\)\( T^{15} + \)\(12\!\cdots\!87\)\( T^{16} + \)\(12\!\cdots\!50\)\( T^{17} + \)\(37\!\cdots\!99\)\( T^{18} + 27026027926 T^{19} + T^{20} \)
$41$ \( \)\(13\!\cdots\!24\)\( + \)\(22\!\cdots\!00\)\( T^{2} + \)\(12\!\cdots\!36\)\( T^{4} + \)\(27\!\cdots\!84\)\( T^{6} + \)\(32\!\cdots\!08\)\( T^{8} + \)\(21\!\cdots\!64\)\( T^{10} + \)\(89\!\cdots\!80\)\( T^{12} + \)\(22\!\cdots\!28\)\( T^{14} + \)\(34\!\cdots\!32\)\( T^{16} + \)\(28\!\cdots\!44\)\( T^{18} + T^{20} \)
$43$ \( ( \)\(15\!\cdots\!28\)\( - \)\(20\!\cdots\!60\)\( T - \)\(18\!\cdots\!88\)\( T^{2} + \)\(28\!\cdots\!84\)\( T^{3} + \)\(33\!\cdots\!72\)\( T^{4} - \)\(11\!\cdots\!44\)\( T^{5} + \)\(17\!\cdots\!04\)\( T^{6} + \)\(11\!\cdots\!12\)\( T^{7} - \)\(27\!\cdots\!20\)\( T^{8} - 363341476828 T^{9} + T^{10} )^{2} \)
$47$ \( \)\(68\!\cdots\!89\)\( + \)\(39\!\cdots\!86\)\( T + \)\(88\!\cdots\!57\)\( T^{2} + \)\(78\!\cdots\!06\)\( T^{3} + \)\(19\!\cdots\!71\)\( T^{4} - \)\(12\!\cdots\!04\)\( T^{5} - \)\(51\!\cdots\!42\)\( T^{6} + \)\(51\!\cdots\!24\)\( T^{7} + \)\(49\!\cdots\!69\)\( T^{8} + \)\(15\!\cdots\!66\)\( T^{9} + \)\(13\!\cdots\!99\)\( T^{10} - \)\(49\!\cdots\!82\)\( T^{11} - \)\(90\!\cdots\!99\)\( T^{12} + \)\(21\!\cdots\!68\)\( T^{13} + \)\(80\!\cdots\!38\)\( T^{14} + \)\(57\!\cdots\!68\)\( T^{15} - \)\(75\!\cdots\!01\)\( T^{16} - \)\(98\!\cdots\!98\)\( T^{17} + \)\(91\!\cdots\!77\)\( T^{18} + 2044625353338 T^{19} + T^{20} \)
$53$ \( \)\(68\!\cdots\!69\)\( + \)\(85\!\cdots\!50\)\( T + \)\(38\!\cdots\!91\)\( T^{2} + \)\(20\!\cdots\!42\)\( T^{3} + \)\(11\!\cdots\!83\)\( T^{4} + \)\(43\!\cdots\!08\)\( T^{5} + \)\(19\!\cdots\!94\)\( T^{6} + \)\(22\!\cdots\!80\)\( T^{7} + \)\(20\!\cdots\!97\)\( T^{8} + \)\(16\!\cdots\!06\)\( T^{9} + \)\(15\!\cdots\!33\)\( T^{10} - \)\(15\!\cdots\!30\)\( T^{11} + \)\(76\!\cdots\!93\)\( T^{12} - \)\(11\!\cdots\!92\)\( T^{13} + \)\(28\!\cdots\!54\)\( T^{14} - \)\(51\!\cdots\!20\)\( T^{15} + \)\(71\!\cdots\!87\)\( T^{16} - \)\(10\!\cdots\!82\)\( T^{17} + \)\(11\!\cdots\!31\)\( T^{18} - 1546271487546 T^{19} + T^{20} \)
$59$ \( \)\(22\!\cdots\!49\)\( + \)\(92\!\cdots\!50\)\( T + \)\(10\!\cdots\!13\)\( T^{2} - \)\(82\!\cdots\!50\)\( T^{3} - \)\(14\!\cdots\!85\)\( T^{4} + \)\(73\!\cdots\!36\)\( T^{5} + \)\(17\!\cdots\!94\)\( T^{6} + \)\(21\!\cdots\!88\)\( T^{7} - \)\(51\!\cdots\!99\)\( T^{8} - \)\(11\!\cdots\!18\)\( T^{9} + \)\(11\!\cdots\!15\)\( T^{10} + \)\(40\!\cdots\!58\)\( T^{11} - \)\(96\!\cdots\!07\)\( T^{12} - \)\(52\!\cdots\!48\)\( T^{13} + \)\(56\!\cdots\!54\)\( T^{14} + \)\(51\!\cdots\!88\)\( T^{15} + \)\(21\!\cdots\!27\)\( T^{16} - \)\(21\!\cdots\!74\)\( T^{17} - \)\(16\!\cdots\!87\)\( T^{18} + 6798944731566 T^{19} + T^{20} \)
$61$ \( \)\(14\!\cdots\!29\)\( + \)\(54\!\cdots\!34\)\( T + \)\(70\!\cdots\!71\)\( T^{2} + \)\(58\!\cdots\!18\)\( T^{3} - \)\(42\!\cdots\!25\)\( T^{4} - \)\(11\!\cdots\!88\)\( T^{5} + \)\(24\!\cdots\!34\)\( T^{6} + \)\(16\!\cdots\!88\)\( T^{7} - \)\(81\!\cdots\!35\)\( T^{8} - \)\(34\!\cdots\!38\)\( T^{9} - \)\(33\!\cdots\!43\)\( T^{10} + \)\(51\!\cdots\!66\)\( T^{11} + \)\(11\!\cdots\!97\)\( T^{12} - \)\(41\!\cdots\!76\)\( T^{13} - \)\(12\!\cdots\!74\)\( T^{14} + \)\(25\!\cdots\!80\)\( T^{15} + \)\(93\!\cdots\!99\)\( T^{16} - \)\(86\!\cdots\!50\)\( T^{17} - \)\(37\!\cdots\!53\)\( T^{18} + 2214453865554 T^{19} + T^{20} \)
$67$ \( \)\(13\!\cdots\!21\)\( + \)\(36\!\cdots\!62\)\( T + \)\(63\!\cdots\!33\)\( T^{2} + \)\(65\!\cdots\!58\)\( T^{3} + \)\(50\!\cdots\!87\)\( T^{4} + \)\(27\!\cdots\!56\)\( T^{5} + \)\(12\!\cdots\!30\)\( T^{6} + \)\(46\!\cdots\!04\)\( T^{7} + \)\(16\!\cdots\!45\)\( T^{8} + \)\(46\!\cdots\!26\)\( T^{9} + \)\(12\!\cdots\!67\)\( T^{10} + \)\(26\!\cdots\!30\)\( T^{11} + \)\(53\!\cdots\!69\)\( T^{12} + \)\(75\!\cdots\!68\)\( T^{13} + \)\(12\!\cdots\!14\)\( T^{14} + \)\(13\!\cdots\!76\)\( T^{15} + \)\(19\!\cdots\!39\)\( T^{16} + \)\(11\!\cdots\!74\)\( T^{17} + \)\(15\!\cdots\!73\)\( T^{18} + 4655820763226 T^{19} + T^{20} \)
$71$ \( ( \)\(14\!\cdots\!36\)\( - \)\(21\!\cdots\!76\)\( T + \)\(10\!\cdots\!16\)\( T^{2} - \)\(18\!\cdots\!24\)\( T^{3} - \)\(14\!\cdots\!96\)\( T^{4} + \)\(89\!\cdots\!72\)\( T^{5} - \)\(10\!\cdots\!96\)\( T^{6} + \)\(19\!\cdots\!16\)\( T^{7} + \)\(71\!\cdots\!76\)\( T^{8} - 48303068747076 T^{9} + T^{10} )^{2} \)
$73$ \( \)\(14\!\cdots\!21\)\( + \)\(23\!\cdots\!94\)\( T + \)\(12\!\cdots\!83\)\( T^{2} - \)\(35\!\cdots\!46\)\( T^{3} - \)\(15\!\cdots\!29\)\( T^{4} + \)\(32\!\cdots\!16\)\( T^{5} + \)\(28\!\cdots\!38\)\( T^{6} + \)\(88\!\cdots\!80\)\( T^{7} + \)\(91\!\cdots\!73\)\( T^{8} - \)\(56\!\cdots\!98\)\( T^{9} - \)\(17\!\cdots\!35\)\( T^{10} + \)\(49\!\cdots\!46\)\( T^{11} + \)\(40\!\cdots\!81\)\( T^{12} + \)\(34\!\cdots\!20\)\( T^{13} - \)\(52\!\cdots\!42\)\( T^{14} - \)\(21\!\cdots\!56\)\( T^{15} - \)\(46\!\cdots\!01\)\( T^{16} + \)\(19\!\cdots\!78\)\( T^{17} + \)\(17\!\cdots\!35\)\( T^{18} + 65348368908666 T^{19} + T^{20} \)
$79$ \( \)\(24\!\cdots\!25\)\( + \)\(28\!\cdots\!50\)\( T + \)\(20\!\cdots\!25\)\( T^{2} + \)\(96\!\cdots\!50\)\( T^{3} + \)\(33\!\cdots\!75\)\( T^{4} + \)\(89\!\cdots\!00\)\( T^{5} + \)\(18\!\cdots\!50\)\( T^{6} + \)\(31\!\cdots\!00\)\( T^{7} + \)\(43\!\cdots\!25\)\( T^{8} + \)\(50\!\cdots\!50\)\( T^{9} + \)\(51\!\cdots\!75\)\( T^{10} + \)\(45\!\cdots\!90\)\( T^{11} + \)\(38\!\cdots\!81\)\( T^{12} + \)\(24\!\cdots\!24\)\( T^{13} + \)\(14\!\cdots\!26\)\( T^{14} + \)\(61\!\cdots\!56\)\( T^{15} + \)\(26\!\cdots\!03\)\( T^{16} + \)\(88\!\cdots\!98\)\( T^{17} + \)\(31\!\cdots\!89\)\( T^{18} + 60517474082978 T^{19} + T^{20} \)
$83$ \( \)\(21\!\cdots\!36\)\( + \)\(31\!\cdots\!16\)\( T^{2} + \)\(18\!\cdots\!04\)\( T^{4} + \)\(54\!\cdots\!84\)\( T^{6} + \)\(90\!\cdots\!08\)\( T^{8} + \)\(87\!\cdots\!24\)\( T^{10} + \)\(47\!\cdots\!24\)\( T^{12} + \)\(12\!\cdots\!04\)\( T^{14} + \)\(14\!\cdots\!76\)\( T^{16} + \)\(66\!\cdots\!60\)\( T^{18} + T^{20} \)
$89$ \( \)\(70\!\cdots\!81\)\( + \)\(13\!\cdots\!22\)\( T - \)\(56\!\cdots\!01\)\( T^{2} - \)\(11\!\cdots\!18\)\( T^{3} + \)\(47\!\cdots\!95\)\( T^{4} + \)\(12\!\cdots\!12\)\( T^{5} + \)\(13\!\cdots\!94\)\( T^{6} + \)\(80\!\cdots\!16\)\( T^{7} + \)\(26\!\cdots\!81\)\( T^{8} + \)\(31\!\cdots\!38\)\( T^{9} - \)\(74\!\cdots\!71\)\( T^{10} - \)\(16\!\cdots\!26\)\( T^{11} + \)\(89\!\cdots\!65\)\( T^{12} + \)\(43\!\cdots\!04\)\( T^{13} + \)\(67\!\cdots\!14\)\( T^{14} + \)\(10\!\cdots\!04\)\( T^{15} - \)\(82\!\cdots\!73\)\( T^{16} - \)\(23\!\cdots\!62\)\( T^{17} + \)\(17\!\cdots\!55\)\( T^{18} + 237147002561826 T^{19} + T^{20} \)
$97$ \( \)\(18\!\cdots\!56\)\( + \)\(50\!\cdots\!20\)\( T^{2} + \)\(27\!\cdots\!80\)\( T^{4} + \)\(46\!\cdots\!00\)\( T^{6} + \)\(30\!\cdots\!00\)\( T^{8} + \)\(10\!\cdots\!16\)\( T^{10} + \)\(21\!\cdots\!20\)\( T^{12} + \)\(24\!\cdots\!40\)\( T^{14} + \)\(17\!\cdots\!40\)\( T^{16} + \)\(63\!\cdots\!60\)\( T^{18} + T^{20} \)
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