Newspace parameters
| Level: | \( N \) | \(=\) | \( 65 = 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 65.o (of order \(12\), degree \(4\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(0.519027613138\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(5\) over \(\Q(\zeta_{12})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} + \cdots)\) |
| Defining polynomial: | \(x^{20} + 26 x^{18} + 279 x^{16} + 1604 x^{14} + 5353 x^{12} + 10466 x^{10} + 11441 x^{8} + 6176 x^{6} + 1263 x^{4} + 78 x^{2} + 1\) |
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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.
Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} + 26 x^{18} + 279 x^{16} + 1604 x^{14} + 5353 x^{12} + 10466 x^{10} + 11441 x^{8} + 6176 x^{6} + 1263 x^{4} + 78 x^{2} + 1\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \((\)\( -20 \nu^{18} - 389 \nu^{16} - 2695 \nu^{14} - 7125 \nu^{12} + 1214 \nu^{10} + 39860 \nu^{8} + 68102 \nu^{6} + 46015 \nu^{4} + 16571 \nu^{2} + 2996 \nu - 409 \)\()/5992\) |
| \(\beta_{2}\) | \(=\) | \((\)\( 20 \nu^{18} + 389 \nu^{16} + 2695 \nu^{14} + 7125 \nu^{12} - 1214 \nu^{10} - 39860 \nu^{8} - 68102 \nu^{6} - 46015 \nu^{4} - 16571 \nu^{2} + 2996 \nu + 409 \)\()/5992\) |
| \(\beta_{3}\) | \(=\) | \((\)\( -69 \nu^{18} - 1647 \nu^{16} - 15798 \nu^{14} - 78322 \nu^{12} - 214723 \nu^{10} - 324081 \nu^{8} - 257858 \nu^{6} - 105030 \nu^{4} - 19961 \nu^{2} - 539 \)\()/1712\) |
| \(\beta_{4}\) | \(=\) | \((\)\(321 \nu^{19} - 483 \nu^{18} + 9951 \nu^{17} - 11529 \nu^{16} + 127330 \nu^{15} - 110586 \nu^{14} + 869910 \nu^{13} - 548254 \nu^{12} + 3425391 \nu^{11} - 1503061 \nu^{10} + 7807469 \nu^{9} - 2268567 \nu^{8} + 9753906 \nu^{7} - 1805006 \nu^{6} + 5800042 \nu^{5} - 735210 \nu^{4} + 1183313 \nu^{3} - 139727 \nu^{2} + 39483 \nu - 3773\)\()/23968\) |
| \(\beta_{5}\) | \(=\) | \((\)\(700 \nu^{19} + 617 \nu^{18} + 17360 \nu^{17} + 16607 \nu^{16} + 174468 \nu^{15} + 185192 \nu^{14} + 912240 \nu^{13} + 1108682 \nu^{12} + 2625448 \nu^{11} + 3846937 \nu^{10} + 3934784 \nu^{9} + 7752327 \nu^{8} + 2192820 \nu^{7} + 8524592 \nu^{6} - 974624 \nu^{5} + 4331072 \nu^{4} - 1268316 \nu^{3} + 669321 \nu^{2} - 170688 \nu + 26961\)\()/11984\) |
| \(\beta_{6}\) | \(=\) | \((\)\( -409 \nu^{19} - 10614 \nu^{17} - 113722 \nu^{15} - 653341 \nu^{13} - 2182252 \nu^{11} - 4281808 \nu^{9} - 4719229 \nu^{7} - 2594086 \nu^{5} - 562582 \nu^{3} - 48473 \nu - 2996 \)\()/5992\) |
| \(\beta_{7}\) | \(=\) | \((\)\(1635 \nu^{19} - 309 \nu^{18} + 41725 \nu^{17} - 7171 \nu^{16} + 436590 \nu^{15} - 66542 \nu^{14} + 2423698 \nu^{13} - 319614 \nu^{12} + 7689981 \nu^{11} - 870831 \nu^{10} + 13910023 \nu^{9} - 1440917 \nu^{8} + 13309334 \nu^{7} - 1613590 \nu^{6} + 5445918 \nu^{5} - 1272982 \nu^{4} + 481307 \nu^{3} - 501629 \nu^{2} + 71073 \nu - 30699\)\()/23968\) |
| \(\beta_{8}\) | \(=\) | \((\)\(700 \nu^{19} - 617 \nu^{18} + 17360 \nu^{17} - 16607 \nu^{16} + 174468 \nu^{15} - 185192 \nu^{14} + 912240 \nu^{13} - 1108682 \nu^{12} + 2625448 \nu^{11} - 3846937 \nu^{10} + 3934784 \nu^{9} - 7752327 \nu^{8} + 2192820 \nu^{7} - 8524592 \nu^{6} - 974624 \nu^{5} - 4331072 \nu^{4} - 1268316 \nu^{3} - 669321 \nu^{2} - 170688 \nu - 26961\)\()/11984\) |
| \(\beta_{9}\) | \(=\) | \((\)\(-419 \nu^{19} + 4593 \nu^{18} - 12681 \nu^{17} + 118807 \nu^{16} - 158886 \nu^{15} + 1266230 \nu^{14} - 1064734 \nu^{13} + 7212374 \nu^{12} - 4108073 \nu^{11} + 23752795 \nu^{10} - 9110155 \nu^{9} + 45500681 \nu^{8} - 10820986 \nu^{7} + 47979462 \nu^{6} - 5625126 \nu^{5} + 23917926 \nu^{4} - 518719 \nu^{3} + 3775361 \nu^{2} + 159919 \nu + 83703\)\()/23968\) |
| \(\beta_{10}\) | \(=\) | \((\)\(-3773 \nu^{19} - 565 \nu^{18} - 97615 \nu^{17} - 11551 \nu^{16} - 1041138 \nu^{15} - 83062 \nu^{14} - 5941306 \nu^{13} - 198098 \nu^{12} - 19648615 \nu^{11} + 414413 \nu^{10} - 37985157 \nu^{9} + 3110895 \nu^{8} - 40898326 \nu^{7} + 5871486 \nu^{6} - 21497042 \nu^{5} + 4145562 \nu^{4} - 4030089 \nu^{3} + 861543 \nu^{2} - 166551 \nu + 81509\)\()/23968\) |
| \(\beta_{11}\) | \(=\) | \((\)\(3773 \nu^{19} + 321 \nu^{18} + 97615 \nu^{17} + 9951 \nu^{16} + 1041138 \nu^{15} + 127330 \nu^{14} + 5941306 \nu^{13} + 869910 \nu^{12} + 19648615 \nu^{11} + 3425391 \nu^{10} + 37985157 \nu^{9} + 7807469 \nu^{8} + 40898326 \nu^{7} + 9753906 \nu^{6} + 21497042 \nu^{5} + 5800042 \nu^{4} + 4030089 \nu^{3} + 1183313 \nu^{2} + 154567 \nu + 39483\)\()/23968\) |
| \(\beta_{12}\) | \(=\) | \((\)\(3773 \nu^{19} - 321 \nu^{18} + 97615 \nu^{17} - 9951 \nu^{16} + 1041138 \nu^{15} - 127330 \nu^{14} + 5941306 \nu^{13} - 869910 \nu^{12} + 19648615 \nu^{11} - 3425391 \nu^{10} + 37985157 \nu^{9} - 7807469 \nu^{8} + 40898326 \nu^{7} - 9753906 \nu^{6} + 21497042 \nu^{5} - 5800042 \nu^{4} + 4030089 \nu^{3} - 1183313 \nu^{2} + 154567 \nu - 39483\)\()/23968\) |
| \(\beta_{13}\) | \(=\) | \((\)\(-2753 \nu^{19} - 867 \nu^{18} - 71035 \nu^{17} - 22593 \nu^{16} - 754642 \nu^{15} - 243222 \nu^{14} - 4279914 \nu^{13} - 1404094 \nu^{12} - 14010639 \nu^{11} - 4705845 \nu^{10} - 26598933 \nu^{9} - 9214959 \nu^{8} - 27637370 \nu^{7} - 9971998 \nu^{6} - 13384022 \nu^{5} - 5100714 \nu^{4} - 1923401 \nu^{3} - 798859 \nu^{2} - 7127 \nu - 15221\)\()/11984\) |
| \(\beta_{14}\) | \(=\) | \((\)\(2753 \nu^{19} - 867 \nu^{18} + 71035 \nu^{17} - 22593 \nu^{16} + 754642 \nu^{15} - 243222 \nu^{14} + 4279914 \nu^{13} - 1404094 \nu^{12} + 14010639 \nu^{11} - 4705845 \nu^{10} + 26598933 \nu^{9} - 9214959 \nu^{8} + 27637370 \nu^{7} - 9971998 \nu^{6} + 13384022 \nu^{5} - 5100714 \nu^{4} + 1923401 \nu^{3} - 798859 \nu^{2} + 7127 \nu - 15221\)\()/11984\) |
| \(\beta_{15}\) | \(=\) | \((\)\(-6965 \nu^{19} + 1863 \nu^{18} - 180971 \nu^{17} + 49605 \nu^{16} - 1940134 \nu^{15} + 545958 \nu^{14} - 11139534 \nu^{13} + 3217222 \nu^{12} - 37111487 \nu^{11} + 10951069 \nu^{10} - 72399061 \nu^{9} + 21535831 \nu^{8} - 78924566 \nu^{7} + 22849098 \nu^{6} - 42433790 \nu^{5} + 10804742 \nu^{4} - 8585045 \nu^{3} + 1254563 \nu^{2} - 504763 \nu + 1285\)\()/23968\) |
| \(\beta_{16}\) | \(=\) | \((\)\(8297 \nu^{19} - 175 \nu^{18} + 216915 \nu^{17} - 2093 \nu^{16} + 2342074 \nu^{15} + 9562 \nu^{14} + 13551722 \nu^{13} + 273770 \nu^{12} + 45488043 \nu^{11} + 1753171 \nu^{10} + 89206401 \nu^{9} + 5197801 \nu^{8} + 97040830 \nu^{7} + 7462350 \nu^{6} + 50921018 \nu^{5} + 4439554 \nu^{4} + 9271077 \nu^{3} + 591213 \nu^{2} + 419203 \nu + 23947\)\()/23968\) |
| \(\beta_{17}\) | \(=\) | \((\)\(-4979 \nu^{19} - 129835 \nu^{17} - 1398012 \nu^{15} - 8068552 \nu^{13} - 27041293 \nu^{11} - 53105203 \nu^{9} - 58320582 \nu^{7} - 31671424 \nu^{5} - 6571905 \nu^{3} + 5992 \nu^{2} - 381235 \nu + 11984\)\()/11984\) |
| \(\beta_{18}\) | \(=\) | \((\)\(10417 \nu^{19} - 483 \nu^{18} + 268635 \nu^{17} - 11529 \nu^{16} + 2853942 \nu^{15} - 110586 \nu^{14} + 16210930 \nu^{13} - 548254 \nu^{12} + 53334711 \nu^{11} - 1503061 \nu^{10} + 102576749 \nu^{9} - 2268567 \nu^{8} + 110068986 \nu^{7} - 1805006 \nu^{6} + 58130790 \nu^{5} - 735210 \nu^{4} + 11443805 \nu^{3} - 127743 \nu^{2} + 679767 \nu + 20195\)\()/23968\) |
| \(\beta_{19}\) | \(=\) | \((\)\(-16083 \nu^{19} - 483 \nu^{18} - 419809 \nu^{17} - 11529 \nu^{16} - 4525598 \nu^{15} - 110586 \nu^{14} - 26151990 \nu^{13} - 548254 \nu^{12} - 87742737 \nu^{11} - 1503061 \nu^{10} - 172338195 \nu^{9} - 2268567 \nu^{8} - 188614114 \nu^{7} - 1805006 \nu^{6} - 100694134 \nu^{5} - 735210 \nu^{4} - 19274543 \nu^{3} - 139727 \nu^{2} - 820129 \nu - 15757\)\()/23968\) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \(\beta_{2} + \beta_{1}\) |
| \(\nu^{2}\) | \(=\) | \(\beta_{12} + \beta_{10} - \beta_{3} + \beta_{1} - 2\) |
| \(\nu^{3}\) | \(=\) | \(\beta_{19} + 2 \beta_{18} - \beta_{13} - 3 \beta_{12} - 2 \beta_{11} - 2 \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} - 6 \beta_{2} - 7 \beta_{1}\) |
| \(\nu^{4}\) | \(=\) | \(\beta_{19} - \beta_{18} + \beta_{17} + 2 \beta_{16} - 2 \beta_{15} + 3 \beta_{14} - 6 \beta_{12} - 2 \beta_{11} - 7 \beta_{10} + \beta_{9} - 3 \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + 7 \beta_{3} - \beta_{2} - 7 \beta_{1} + 8\) |
| \(\nu^{5}\) | \(=\) | \(-9 \beta_{19} - 17 \beta_{18} - \beta_{17} - 3 \beta_{14} + 11 \beta_{13} + 26 \beta_{12} + 17 \beta_{11} + 17 \beta_{10} + 8 \beta_{9} + 10 \beta_{8} + 8 \beta_{7} + 5 \beta_{6} - 6 \beta_{5} - 18 \beta_{4} + 9 \beta_{3} + 37 \beta_{2} + 46 \beta_{1} - 2\) |
| \(\nu^{6}\) | \(=\) | \(-13 \beta_{19} + 13 \beta_{18} - 13 \beta_{17} - 26 \beta_{16} + 26 \beta_{15} - 32 \beta_{14} + 4 \beta_{13} + 38 \beta_{12} + 25 \beta_{11} + 53 \beta_{10} - 10 \beta_{9} + 37 \beta_{8} + 10 \beta_{7} - 13 \beta_{6} - 11 \beta_{5} + 16 \beta_{4} - 53 \beta_{3} + 18 \beta_{2} + 51 \beta_{1} - 39\) |
| \(\nu^{7}\) | \(=\) | \(66 \beta_{19} + 122 \beta_{18} + 14 \beta_{17} + 37 \beta_{14} - 89 \beta_{13} - 189 \beta_{12} - 121 \beta_{11} - 120 \beta_{10} - 52 \beta_{9} - 76 \beta_{8} - 52 \beta_{7} - 24 \beta_{6} + 28 \beta_{5} + 138 \beta_{4} - 69 \beta_{3} - 236 \beta_{2} - 304 \beta_{1} + 21\) |
| \(\nu^{8}\) | \(=\) | \(120 \beta_{19} - 116 \beta_{18} + 116 \beta_{17} + 232 \beta_{16} - 240 \beta_{15} + 260 \beta_{14} - 56 \beta_{13} - 252 \beta_{12} - 229 \beta_{11} - 397 \beta_{10} + 76 \beta_{9} - 332 \beta_{8} - 84 \beta_{7} + 112 \beta_{6} + 100 \beta_{5} - 152 \beta_{4} + 398 \beta_{3} - 185 \beta_{2} - 368 \beta_{1} + 213\) |
| \(\nu^{9}\) | \(=\) | \(-462 \beta_{19} - 840 \beta_{18} - 136 \beta_{17} - 332 \beta_{14} + 657 \beta_{13} + 1306 \beta_{12} + 818 \beta_{11} + 813 \beta_{10} + 325 \beta_{9} + 537 \beta_{8} + 325 \beta_{7} + 122 \beta_{6} - 113 \beta_{5} - 1011 \beta_{4} + 506 \beta_{3} + 1544 \beta_{2} + 2032 \beta_{1} - 170\) |
| \(\nu^{10}\) | \(=\) | \(-974 \beta_{19} + 914 \beta_{18} - 914 \beta_{17} - 1828 \beta_{16} + 1948 \beta_{15} - 1941 \beta_{14} + 546 \beta_{13} + 1707 \beta_{12} + 1862 \beta_{11} + 2910 \beta_{10} - 539 \beta_{9} + 2648 \beta_{8} + 659 \beta_{7} - 854 \beta_{6} - 820 \beta_{5} + 1229 \beta_{4} - 2933 \beta_{3} + 1578 \beta_{2} + 2621 \beta_{1} - 1255\) |
| \(\nu^{11}\) | \(=\) | \(3210 \beta_{19} + 5733 \beta_{18} + 1135 \beta_{17} + 2648 \beta_{14} - 4690 \beta_{13} - 8890 \beta_{12} - 5456 \beta_{11} - 5476 \beta_{10} - 2042 \beta_{9} - 3716 \beta_{8} - 2042 \beta_{7} - 676 \beta_{6} + 368 \beta_{5} + 7277 \beta_{4} - 3655 \beta_{3} - 10296 \beta_{2} - 13730 \beta_{1} + 1267\) |
| \(\nu^{12}\) | \(=\) | \(7438 \beta_{19} - 6832 \beta_{18} + 6832 \beta_{17} + 13664 \beta_{16} - 14876 \beta_{15} + 14032 \beta_{14} - 4610 \beta_{13} - 11688 \beta_{12} - 14278 \beta_{11} - 20988 \beta_{10} + 3766 \beta_{9} - 19984 \beta_{8} - 4978 \beta_{7} + 6226 \beta_{6} + 6320 \beta_{5} - 9292 \beta_{4} + 21288 \beta_{3} - 12378 \beta_{2} - 18508 \beta_{1} + 7797\) |
| \(\nu^{13}\) | \(=\) | \(-22356 \beta_{19} - 39188 \beta_{18} - 8780 \beta_{17} - 19984 \beta_{14} + 33058 \beta_{13} + 60422 \beta_{12} + 36438 \beta_{11} + 37058 \beta_{10} + 13074 \beta_{9} + 25620 \beta_{8} + 13074 \beta_{7} + 4096 \beta_{6} - 528 \beta_{5} - 51934 \beta_{4} + 26218 \beta_{3} + 69645 \beta_{2} + 93629 \beta_{1} - 9130\) |
| \(\nu^{14}\) | \(=\) | \(-54982 \beta_{19} + 49778 \beta_{18} - 49778 \beta_{17} - 99556 \beta_{16} + 109964 \beta_{15} - 100084 \beta_{14} + 36202 \beta_{13} + 80555 \beta_{12} + 105990 \beta_{11} + 149815 \beta_{10} - 26322 \beta_{9} + 146494 \beta_{8} + 36730 \beta_{7} - 44574 \beta_{6} - 46938 \beta_{5} + 68030 \beta_{4} - 152877 \beta_{3} + 92972 \beta_{2} + 130077 \beta_{1} - 50290\) |
| \(\nu^{15}\) | \(=\) | \(156271 \beta_{19} + 269080 \beta_{18} + 65190 \beta_{17} + 146494 \beta_{14} - 231975 \beta_{13} - 412141 \beta_{12} - 245006 \beta_{11} - 252616 \beta_{10} - 85481 \beta_{9} - 176993 \beta_{8} - 85481 \beta_{7} - 26707 \beta_{6} - 6031 \beta_{5} + 368826 \beta_{4} - 187213 \beta_{3} - 476096 \beta_{2} - 643231 \beta_{1} + 64782\) |
| \(\nu^{16}\) | \(=\) | \(398897 \beta_{19} - 357747 \beta_{18} + 357747 \beta_{17} + 715494 \beta_{16} - 797794 \beta_{15} + 709463 \beta_{14} - 272912 \beta_{13} - 557750 \beta_{12} - 771768 \beta_{11} - 1062637 \beta_{10} + 184581 \beta_{9} - 1056441 \beta_{8} - 266881 \beta_{7} + 316597 \beta_{6} + 340947 \beta_{5} - 489763 \beta_{4} + 1090029 \beta_{3} - 681429 \beta_{2} - 912121 \beta_{1} + 333034\) |
| \(\nu^{17}\) | \(=\) | \(-1095413 \beta_{19} - 1856511 \beta_{18} - 472963 \beta_{17} - 1056441 \beta_{14} + 1625819 \beta_{13} + 2825568 \beta_{12} + 1660831 \beta_{11} + 1734115 \beta_{10} + 569378 \beta_{9} + 1226922 \beta_{8} + 569378 \beta_{7} + 182897 \beta_{6} + 88166 \beta_{5} - 2610994 \beta_{4} + 1332033 \beta_{3} + 3279739 \beta_{2} + 4444476 \beta_{1} - 456258\) |
| \(\nu^{18}\) | \(=\) | \(-2861437 \beta_{19} + 2550751 \beta_{18} - 2550751 \beta_{17} - 5101502 \beta_{16} + 5722874 \beta_{15} - 5013336 \beta_{14} + 2007386 \beta_{13} + 3875098 \beta_{12} + 5552425 \beta_{11} + 7508303 \beta_{10} - 1297848 \beta_{9} + 7545155 \beta_{8} + 1919220 \beta_{7} - 2240065 \beta_{6} - 2443653 \beta_{5} + 3492968 \beta_{4} - 7734751 \beta_{3} + 4921700 \beta_{2} + 6390257 \beta_{1} - 2246355\) |
| \(\nu^{19}\) | \(=\) | \(7691530 \beta_{19} + 12862874 \beta_{18} + 3385870 \beta_{17} + 7545155 \beta_{14} - 11394317 \beta_{13} - 19469181 \beta_{12} - 11344809 \beta_{11} - 11973534 \beta_{10} - 3849162 \beta_{9} - 8532880 \beta_{8} - 3849162 \beta_{7} - 1287312 \beta_{6} - 834556 \beta_{5} + 18442656 \beta_{4} - 9449577 \beta_{3} - 22718962 \beta_{2} - 30843334 \beta_{1} + 3202109\) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/65\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) | \(41\) |
| \(\chi(n)\) | \(-\beta_{11} - \beta_{12}\) | \(-\beta_{11}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2.1 |
|
−1.32488 | − | 2.29475i | −1.25278 | − | 0.335680i | −2.51060 | + | 4.34849i | −1.30391 | − | 1.81654i | 0.889471 | + | 3.31955i | 0.0972962 | + | 0.0561740i | 8.00544 | −1.14131 | − | 0.658935i | −2.44100 | + | 5.39885i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2.2 | −0.915816 | − | 1.58624i | 1.91432 | + | 0.512942i | −0.677439 | + | 1.17336i | 1.45480 | + | 1.69810i | −0.939520 | − | 3.50634i | −3.06478 | − | 1.76945i | −1.18163 | 0.803451 | + | 0.463873i | 1.36126 | − | 3.86282i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2.3 | −0.246951 | − | 0.427732i | 0.908353 | + | 0.243392i | 0.878030 | − | 1.52079i | −2.21791 | + | 0.284413i | −0.120212 | − | 0.448637i | 3.18307 | + | 1.83775i | −1.85513 | −1.83221 | − | 1.05783i | 0.669366 | + | 0.878433i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2.4 | −0.137404 | − | 0.237991i | −2.28256 | − | 0.611610i | 0.962240 | − | 1.66665i | 1.69883 | − | 1.45395i | 0.168076 | + | 0.627267i | −0.334376 | − | 0.193052i | −1.07848 | 2.23793 | + | 1.29207i | −0.579454 | − | 0.204528i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2.5 | 0.759023 | + | 1.31467i | −0.653367 | − | 0.175069i | −0.152233 | + | 0.263675i | 0.600231 | + | 2.15400i | −0.265763 | − | 0.991842i | −2.24723 | − | 1.29744i | 2.57390 | −2.20184 | − | 1.27123i | −2.37621 | + | 2.42404i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 32.1 | −1.04397 | + | 1.80821i | 0.713171 | + | 2.66159i | −1.17974 | − | 2.04338i | −0.194361 | − | 2.22760i | −5.55724 | − | 1.48906i | 2.52122 | − | 1.45563i | 0.750585 | −3.97738 | + | 2.29634i | 4.23088 | + | 1.97411i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 32.2 | −0.792369 | + | 1.37242i | −0.0510678 | − | 0.190588i | −0.255697 | − | 0.442881i | 0.0672627 | + | 2.23506i | 0.302032 | + | 0.0809291i | −0.474866 | + | 0.274164i | −2.35905 | 2.56436 | − | 1.48053i | −3.12074 | − | 1.67868i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 32.3 | 0.0656513 | − | 0.113711i | −0.0890070 | − | 0.332179i | 0.991380 | + | 1.71712i | 0.813169 | − | 2.08297i | −0.0436159 | − | 0.0116869i | −2.40874 | + | 1.39069i | 0.522947 | 2.49566 | − | 1.44087i | −0.183472 | − | 0.229216i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 32.4 | 0.511309 | − | 0.885613i | −0.721300 | − | 2.69193i | 0.477126 | + | 0.826407i | −1.69584 | + | 1.45744i | −2.75281 | − | 0.737614i | 0.834479 | − | 0.481787i | 3.02107 | −4.12812 | + | 2.38337i | 0.423625 | + | 2.24706i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 32.5 | 1.12540 | − | 1.94926i | 0.514229 | + | 1.91913i | −1.53307 | − | 2.65535i | −2.22228 | − | 0.247944i | 4.31958 | + | 1.15743i | −1.10607 | + | 0.638592i | −2.39966 | −0.820542 | + | 0.473740i | −2.98427 | + | 4.05275i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.1 | −1.32488 | + | 2.29475i | −1.25278 | + | 0.335680i | −2.51060 | − | 4.34849i | −1.30391 | + | 1.81654i | 0.889471 | − | 3.31955i | 0.0972962 | − | 0.0561740i | 8.00544 | −1.14131 | + | 0.658935i | −2.44100 | − | 5.39885i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.2 | −0.915816 | + | 1.58624i | 1.91432 | − | 0.512942i | −0.677439 | − | 1.17336i | 1.45480 | − | 1.69810i | −0.939520 | + | 3.50634i | −3.06478 | + | 1.76945i | −1.18163 | 0.803451 | − | 0.463873i | 1.36126 | + | 3.86282i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.3 | −0.246951 | + | 0.427732i | 0.908353 | − | 0.243392i | 0.878030 | + | 1.52079i | −2.21791 | − | 0.284413i | −0.120212 | + | 0.448637i | 3.18307 | − | 1.83775i | −1.85513 | −1.83221 | + | 1.05783i | 0.669366 | − | 0.878433i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.4 | −0.137404 | + | 0.237991i | −2.28256 | + | 0.611610i | 0.962240 | + | 1.66665i | 1.69883 | + | 1.45395i | 0.168076 | − | 0.627267i | −0.334376 | + | 0.193052i | −1.07848 | 2.23793 | − | 1.29207i | −0.579454 | + | 0.204528i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.5 | 0.759023 | − | 1.31467i | −0.653367 | + | 0.175069i | −0.152233 | − | 0.263675i | 0.600231 | − | 2.15400i | −0.265763 | + | 0.991842i | −2.24723 | + | 1.29744i | 2.57390 | −2.20184 | + | 1.27123i | −2.37621 | − | 2.42404i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 63.1 | −1.04397 | − | 1.80821i | 0.713171 | − | 2.66159i | −1.17974 | + | 2.04338i | −0.194361 | + | 2.22760i | −5.55724 | + | 1.48906i | 2.52122 | + | 1.45563i | 0.750585 | −3.97738 | − | 2.29634i | 4.23088 | − | 1.97411i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 63.2 | −0.792369 | − | 1.37242i | −0.0510678 | + | 0.190588i | −0.255697 | + | 0.442881i | 0.0672627 | − | 2.23506i | 0.302032 | − | 0.0809291i | −0.474866 | − | 0.274164i | −2.35905 | 2.56436 | + | 1.48053i | −3.12074 | + | 1.67868i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 63.3 | 0.0656513 | + | 0.113711i | −0.0890070 | + | 0.332179i | 0.991380 | − | 1.71712i | 0.813169 | + | 2.08297i | −0.0436159 | + | 0.0116869i | −2.40874 | − | 1.39069i | 0.522947 | 2.49566 | + | 1.44087i | −0.183472 | + | 0.229216i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 63.4 | 0.511309 | + | 0.885613i | −0.721300 | + | 2.69193i | 0.477126 | − | 0.826407i | −1.69584 | − | 1.45744i | −2.75281 | + | 0.737614i | 0.834479 | + | 0.481787i | 3.02107 | −4.12812 | − | 2.38337i | 0.423625 | − | 2.24706i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 63.5 | 1.12540 | + | 1.94926i | 0.514229 | − | 1.91913i | −1.53307 | + | 2.65535i | −2.22228 | + | 0.247944i | 4.31958 | − | 1.15743i | −1.10607 | − | 0.638592i | −2.39966 | −0.820542 | − | 0.473740i | −2.98427 | − | 4.05275i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 65.o | even | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 65.2.o.a | ✓ | 20 |
| 3.b | odd | 2 | 1 | 585.2.cf.a | 20 | ||
| 5.b | even | 2 | 1 | 325.2.s.b | 20 | ||
| 5.c | odd | 4 | 1 | 65.2.t.a | yes | 20 | |
| 5.c | odd | 4 | 1 | 325.2.x.b | 20 | ||
| 13.b | even | 2 | 1 | 845.2.o.g | 20 | ||
| 13.c | even | 3 | 1 | 845.2.k.e | 20 | ||
| 13.c | even | 3 | 1 | 845.2.o.e | 20 | ||
| 13.d | odd | 4 | 1 | 845.2.t.e | 20 | ||
| 13.d | odd | 4 | 1 | 845.2.t.f | 20 | ||
| 13.e | even | 6 | 1 | 845.2.k.d | 20 | ||
| 13.e | even | 6 | 1 | 845.2.o.f | 20 | ||
| 13.f | odd | 12 | 1 | 65.2.t.a | yes | 20 | |
| 13.f | odd | 12 | 1 | 845.2.f.d | 20 | ||
| 13.f | odd | 12 | 1 | 845.2.f.e | 20 | ||
| 13.f | odd | 12 | 1 | 845.2.t.g | 20 | ||
| 15.e | even | 4 | 1 | 585.2.dp.a | 20 | ||
| 39.k | even | 12 | 1 | 585.2.dp.a | 20 | ||
| 65.f | even | 4 | 1 | 845.2.o.f | 20 | ||
| 65.h | odd | 4 | 1 | 845.2.t.g | 20 | ||
| 65.k | even | 4 | 1 | 845.2.o.e | 20 | ||
| 65.o | even | 12 | 1 | inner | 65.2.o.a | ✓ | 20 |
| 65.o | even | 12 | 1 | 845.2.k.e | 20 | ||
| 65.q | odd | 12 | 1 | 845.2.f.e | 20 | ||
| 65.q | odd | 12 | 1 | 845.2.t.f | 20 | ||
| 65.r | odd | 12 | 1 | 845.2.f.d | 20 | ||
| 65.r | odd | 12 | 1 | 845.2.t.e | 20 | ||
| 65.s | odd | 12 | 1 | 325.2.x.b | 20 | ||
| 65.t | even | 12 | 1 | 325.2.s.b | 20 | ||
| 65.t | even | 12 | 1 | 845.2.k.d | 20 | ||
| 65.t | even | 12 | 1 | 845.2.o.g | 20 | ||
| 195.bn | odd | 12 | 1 | 585.2.cf.a | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 65.2.o.a | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 65.2.o.a | ✓ | 20 | 65.o | even | 12 | 1 | inner |
| 65.2.t.a | yes | 20 | 5.c | odd | 4 | 1 | |
| 65.2.t.a | yes | 20 | 13.f | odd | 12 | 1 | |
| 325.2.s.b | 20 | 5.b | even | 2 | 1 | ||
| 325.2.s.b | 20 | 65.t | even | 12 | 1 | ||
| 325.2.x.b | 20 | 5.c | odd | 4 | 1 | ||
| 325.2.x.b | 20 | 65.s | odd | 12 | 1 | ||
| 585.2.cf.a | 20 | 3.b | odd | 2 | 1 | ||
| 585.2.cf.a | 20 | 195.bn | odd | 12 | 1 | ||
| 585.2.dp.a | 20 | 15.e | even | 4 | 1 | ||
| 585.2.dp.a | 20 | 39.k | even | 12 | 1 | ||
| 845.2.f.d | 20 | 13.f | odd | 12 | 1 | ||
| 845.2.f.d | 20 | 65.r | odd | 12 | 1 | ||
| 845.2.f.e | 20 | 13.f | odd | 12 | 1 | ||
| 845.2.f.e | 20 | 65.q | odd | 12 | 1 | ||
| 845.2.k.d | 20 | 13.e | even | 6 | 1 | ||
| 845.2.k.d | 20 | 65.t | even | 12 | 1 | ||
| 845.2.k.e | 20 | 13.c | even | 3 | 1 | ||
| 845.2.k.e | 20 | 65.o | even | 12 | 1 | ||
| 845.2.o.e | 20 | 13.c | even | 3 | 1 | ||
| 845.2.o.e | 20 | 65.k | even | 4 | 1 | ||
| 845.2.o.f | 20 | 13.e | even | 6 | 1 | ||
| 845.2.o.f | 20 | 65.f | even | 4 | 1 | ||
| 845.2.o.g | 20 | 13.b | even | 2 | 1 | ||
| 845.2.o.g | 20 | 65.t | even | 12 | 1 | ||
| 845.2.t.e | 20 | 13.d | odd | 4 | 1 | ||
| 845.2.t.e | 20 | 65.r | odd | 12 | 1 | ||
| 845.2.t.f | 20 | 13.d | odd | 4 | 1 | ||
| 845.2.t.f | 20 | 65.q | odd | 12 | 1 | ||
| 845.2.t.g | 20 | 13.f | odd | 12 | 1 | ||
| 845.2.t.g | 20 | 65.h | odd | 4 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(65, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ | \( 1 - 2 T + 41 T^{2} + 178 T^{3} + 1214 T^{4} + 2208 T^{5} + 4733 T^{6} + 3252 T^{7} + 6569 T^{8} + 4404 T^{9} + 5468 T^{10} + 2864 T^{11} + 2677 T^{12} + 1258 T^{13} + 893 T^{14} + 322 T^{15} + 170 T^{16} + 48 T^{17} + 21 T^{18} + 4 T^{19} + T^{20} \) |
| $3$ | \( 16 + 96 T + 708 T^{2} + 1948 T^{3} + 4097 T^{4} + 2690 T^{5} - 8584 T^{6} - 11256 T^{7} + 5374 T^{8} + 8882 T^{9} + 832 T^{10} + 418 T^{11} + 595 T^{12} - 618 T^{13} - 188 T^{14} + 2 T^{15} - 2 T^{16} + 20 T^{17} + 8 T^{18} + 2 T^{19} + T^{20} \) |
| $5$ | \( 9765625 + 11718750 T + 10546875 T^{2} + 8906250 T^{3} + 6140625 T^{4} + 3887500 T^{5} + 2295000 T^{6} + 1264500 T^{7} + 649950 T^{8} + 316320 T^{9} + 146954 T^{10} + 63264 T^{11} + 25998 T^{12} + 10116 T^{13} + 3672 T^{14} + 1244 T^{15} + 393 T^{16} + 114 T^{17} + 27 T^{18} + 6 T^{19} + T^{20} \) |
| $7$ | \( 64 - 480 T - 1128 T^{2} + 17460 T^{3} + 94417 T^{4} + 166374 T^{5} + 54704 T^{6} - 153888 T^{7} - 73546 T^{8} + 175722 T^{9} + 198624 T^{10} + 46566 T^{11} - 24333 T^{12} - 10674 T^{13} + 2792 T^{14} + 1950 T^{15} + 54 T^{16} - 120 T^{17} - 8 T^{18} + 6 T^{19} + T^{20} \) |
| $11$ | \( 256 + 15616 T + 147920 T^{2} - 3781400 T^{3} + 14856881 T^{4} - 2759960 T^{5} + 33036500 T^{6} - 15386688 T^{7} + 2684514 T^{8} - 1623624 T^{9} - 1360616 T^{10} + 30704 T^{11} + 266651 T^{12} + 196248 T^{13} + 81176 T^{14} + 23048 T^{15} + 5362 T^{16} + 976 T^{17} + 140 T^{18} + 16 T^{19} + T^{20} \) |
| $13$ | \( 137858491849 - 21208998746 T + 12235960815 T^{2} - 1631461442 T^{3} + 699887305 T^{4} - 350500592 T^{5} + 32102564 T^{6} + 456976 T^{7} - 1624090 T^{8} + 970580 T^{9} + 342346 T^{10} + 74660 T^{11} - 9610 T^{12} + 208 T^{13} + 1124 T^{14} - 944 T^{15} + 145 T^{16} - 26 T^{17} + 15 T^{18} - 2 T^{19} + T^{20} \) |
| $17$ | \( 1168561 - 7415660 T + 174841793 T^{2} - 5758158 T^{3} + 44759067 T^{4} + 26251528 T^{5} - 5096966 T^{6} + 2745932 T^{7} - 121343 T^{8} - 1056080 T^{9} + 96031 T^{10} - 38730 T^{11} - 2399 T^{12} + 17836 T^{13} - 446 T^{14} + 444 T^{15} + 107 T^{16} - 232 T^{17} - 7 T^{18} + 10 T^{19} + T^{20} \) |
| $19$ | \( 1583721616 + 3820734368 T + 1167831572 T^{2} + 463061396 T^{3} + 3184775849 T^{4} - 2773365368 T^{5} + 1830432156 T^{6} - 1254413560 T^{7} + 577282318 T^{8} - 205617120 T^{9} + 65790820 T^{10} - 15024876 T^{11} + 1989583 T^{12} + 74704 T^{13} - 116976 T^{14} + 25808 T^{15} - 1690 T^{16} - 440 T^{17} + 152 T^{18} - 20 T^{19} + T^{20} \) |
| $23$ | \( 144 - 1296 T + 2916 T^{2} - 600 T^{3} - 21887 T^{4} + 187570 T^{5} + 260036 T^{6} + 474444 T^{7} + 4070922 T^{8} + 2935886 T^{9} + 2860352 T^{10} + 2226574 T^{11} + 1100975 T^{12} + 380770 T^{13} + 87332 T^{14} + 10138 T^{15} - 326 T^{16} - 344 T^{17} - 52 T^{18} + 2 T^{19} + T^{20} \) |
| $29$ | \( 206213167449 - 1375693542936 T + 3313045860099 T^{2} - 1693561403784 T^{3} - 352134815045 T^{4} + 421814740512 T^{5} + 49257519182 T^{6} - 87906944184 T^{7} + 13003839581 T^{8} + 4237769376 T^{9} - 990563695 T^{10} - 156526392 T^{11} + 52332613 T^{12} + 1675704 T^{13} - 1236386 T^{14} - 14640 T^{15} + 20819 T^{16} - 173 T^{18} + T^{20} \) |
| $31$ | \( 2166784 + 14602240 T + 49203200 T^{2} - 45220864 T^{3} + 317008128 T^{4} + 1365233664 T^{5} + 2473766912 T^{6} + 2175176448 T^{7} + 1058401664 T^{8} + 187005312 T^{9} + 8238080 T^{10} - 315200 T^{11} + 3495856 T^{12} + 416544 T^{13} + 5408 T^{14} - 12992 T^{15} + 6072 T^{16} - 104 T^{17} + T^{20} \) |
| $37$ | \( 4508182449 - 23327492490 T + 49998796215 T^{2} - 50518059150 T^{3} + 19210271947 T^{4} + 5364502944 T^{5} - 2412054490 T^{6} - 9147396588 T^{7} + 11463108577 T^{8} - 6758027358 T^{9} + 2417790097 T^{10} - 546547590 T^{11} + 72292689 T^{12} - 3632340 T^{13} - 163690 T^{14} - 82248 T^{15} + 43003 T^{16} - 7854 T^{17} + 775 T^{18} - 42 T^{19} + T^{20} \) |
| $41$ | \( 3748255729 - 2192518076 T - 3998259115 T^{2} + 21282088462 T^{3} - 38294393429 T^{4} - 45649423224 T^{5} + 114452844834 T^{6} - 20978724348 T^{7} + 6465808597 T^{8} - 344134940 T^{9} - 181905193 T^{10} - 11505094 T^{11} - 5999131 T^{12} + 1385824 T^{13} + 318594 T^{14} + 7852 T^{15} + 5227 T^{16} - 1188 T^{17} + 53 T^{18} - 10 T^{19} + T^{20} \) |
| $43$ | \( 1370772640000 - 18203832560000 T + 57172378499600 T^{2} + 58430182109120 T^{3} + 12962355503081 T^{4} - 826155173166 T^{5} - 373100342360 T^{6} - 181959452144 T^{7} - 25100897594 T^{8} + 1062248466 T^{9} + 1284829516 T^{10} + 499552270 T^{11} + 114310435 T^{12} + 20569910 T^{13} + 3269428 T^{14} + 414538 T^{15} + 46734 T^{16} + 4500 T^{17} + 332 T^{18} + 22 T^{19} + T^{20} \) |
| $47$ | \( 807469056 + 11345362944 T^{2} + 44521738240 T^{4} + 54003212288 T^{6} + 17345956864 T^{8} + 2242437632 T^{10} + 133025920 T^{12} + 3738752 T^{14} + 52960 T^{16} + 368 T^{18} + T^{20} \) |
| $53$ | \( 2978634160384 + 1638501455872 T + 450657394688 T^{2} + 451854140672 T^{3} + 938935450496 T^{4} + 662260749312 T^{5} + 256514576512 T^{6} + 53765212096 T^{7} + 8410884016 T^{8} + 2137889376 T^{9} + 822713344 T^{10} + 169978000 T^{11} + 19301176 T^{12} + 1286456 T^{13} + 475072 T^{14} + 100768 T^{15} + 10713 T^{16} + 78 T^{17} + 50 T^{18} + 10 T^{19} + T^{20} \) |
| $59$ | \( 33856 + 28392672 T + 6360946824 T^{2} - 9278523364 T^{3} + 8028346497 T^{4} - 6999681172 T^{5} + 4362869144 T^{6} - 1471550856 T^{7} + 112358362 T^{8} + 234546884 T^{9} - 132826648 T^{10} + 26431368 T^{11} + 5027435 T^{12} - 5604924 T^{13} + 1943264 T^{14} - 363348 T^{15} + 43194 T^{16} - 4136 T^{17} + 320 T^{18} - 16 T^{19} + T^{20} \) |
| $61$ | \( 826457355409 + 152931933728 T + 714855732255 T^{2} + 167561667840 T^{3} + 464934395195 T^{4} + 101940472768 T^{5} + 118218961526 T^{6} + 20885521424 T^{7} + 20383774133 T^{8} + 3839755696 T^{9} + 1784573189 T^{10} + 385162944 T^{11} + 121471973 T^{12} + 21057264 T^{13} + 3732310 T^{14} + 391232 T^{15} + 44619 T^{16} + 3184 T^{17} + 319 T^{18} + 16 T^{19} + T^{20} \) |
| $67$ | \( 15478905336976 - 87299958482384 T + 620793296665580 T^{2} + 855544406309724 T^{3} + 710415918999441 T^{4} + 375296027359966 T^{5} + 149347427286784 T^{6} + 45533093686732 T^{7} + 11395073107926 T^{8} + 2376642817662 T^{9} + 433822790124 T^{10} + 69587684462 T^{11} + 9990258843 T^{12} + 1247036682 T^{13} + 133985072 T^{14} + 11877582 T^{15} + 858582 T^{16} + 48068 T^{17} + 2044 T^{18} + 58 T^{19} + T^{20} \) |
| $71$ | \( 112163628352758544 + 124423487088552896 T + 74662563174729188 T^{2} + 31199257022438180 T^{3} + 9682913168509481 T^{4} + 2301254681694640 T^{5} + 419603164556892 T^{6} + 55401972752540 T^{7} + 4475729325790 T^{8} - 28579680612 T^{9} - 68391405476 T^{10} - 9818440548 T^{11} - 554987105 T^{12} + 39729340 T^{13} + 10077960 T^{14} + 615752 T^{15} - 10042 T^{16} - 5924 T^{17} - 256 T^{18} + 16 T^{19} + T^{20} \) |
| $73$ | \( ( 253113232 - 309584640 T + 145220576 T^{2} - 29843280 T^{3} + 1070440 T^{4} + 651712 T^{5} - 115932 T^{6} + 5932 T^{7} + 249 T^{8} - 36 T^{9} + T^{10} )^{2} \) |
| $79$ | \( 7586504765095936 + 5385471164325888 T^{2} + 1241871864168704 T^{4} + 134326970823168 T^{6} + 7680125227520 T^{8} + 243260666240 T^{10} + 4371796960 T^{12} + 45175136 T^{14} + 264160 T^{16} + 808 T^{18} + T^{20} \) |
| $83$ | \( 11512611864576 + 13783820697600 T^{2} + 5842988808448 T^{4} + 1204294956032 T^{6} + 135217417088 T^{8} + 8634255104 T^{10} + 316478512 T^{12} + 6614080 T^{14} + 76568 T^{16} + 448 T^{18} + T^{20} \) |
| $89$ | \( 329648222500 + 1177174003500 T + 2273701009050 T^{2} + 5083577201910 T^{3} + 8474675429921 T^{4} + 6925519621868 T^{5} + 3497464766988 T^{6} + 1358361684772 T^{7} + 446547141742 T^{8} + 120073964504 T^{9} + 25613241926 T^{10} + 4417984798 T^{11} + 621940219 T^{12} + 69631212 T^{13} + 6066956 T^{14} + 351592 T^{15} - 498 T^{16} - 1876 T^{17} - 150 T^{18} - 6 T^{19} + T^{20} \) |
| $97$ | \( 28761567213630736 - 55405419075392720 T + 143232059180324060 T^{2} + 79826789169984204 T^{3} + 36437734918301265 T^{4} + 8992046654782666 T^{5} + 2040319607503288 T^{6} + 320016998462884 T^{7} + 53902773927654 T^{8} + 6669078921834 T^{9} + 926859773148 T^{10} + 85742329586 T^{11} + 9510255507 T^{12} + 663004590 T^{13} + 67111568 T^{14} + 3595530 T^{15} + 288774 T^{16} + 10988 T^{17} + 796 T^{18} + 22 T^{19} + T^{20} \) |
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