# Properties

 Label 387.10.a.c Level 387 Weight 10 Character orbit 387.a Self dual yes Analytic conductor 199.319 Analytic rank 0 Dimension 15 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$387 = 3^{2} \cdot 43$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 387.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$199.318868595$$ Analytic rank: $$0$$ Dimension: $$15$$ Coefficient field: $$\mathbb{Q}[x]/(x^{15} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: multiple of $$2^{10}\cdot 3^{3}$$ Twist minimal: no (minimal twist has level 43) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{14}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 + \beta_{1} ) q^{2} + ( 216 + 3 \beta_{1} + \beta_{2} ) q^{4} + ( 315 - 4 \beta_{1} - \beta_{8} ) q^{5} + ( -644 - 37 \beta_{1} - 5 \beta_{2} - 4 \beta_{3} + \beta_{5} - \beta_{8} - 2 \beta_{9} ) q^{7} + ( 1335 + 242 \beta_{1} + 5 \beta_{2} + 12 \beta_{3} + 3 \beta_{5} - \beta_{8} + 2 \beta_{9} + \beta_{11} ) q^{8} +O(q^{10})$$ $$q + ( 2 + \beta_{1} ) q^{2} + ( 216 + 3 \beta_{1} + \beta_{2} ) q^{4} + ( 315 - 4 \beta_{1} - \beta_{8} ) q^{5} + ( -644 - 37 \beta_{1} - 5 \beta_{2} - 4 \beta_{3} + \beta_{5} - \beta_{8} - 2 \beta_{9} ) q^{7} + ( 1335 + 242 \beta_{1} + 5 \beta_{2} + 12 \beta_{3} + 3 \beta_{5} - \beta_{8} + 2 \beta_{9} + \beta_{11} ) q^{8} + ( -2497 + 511 \beta_{1} - 15 \beta_{2} - 34 \beta_{3} + \beta_{4} + 5 \beta_{5} - 4 \beta_{6} + 5 \beta_{7} - 9 \beta_{8} + \beta_{9} - 4 \beta_{10} + 3 \beta_{11} + 4 \beta_{12} + 4 \beta_{14} ) q^{10} + ( 7062 - 687 \beta_{1} + 10 \beta_{2} + 16 \beta_{3} + 5 \beta_{4} + 2 \beta_{5} + 6 \beta_{7} - 12 \beta_{8} - 10 \beta_{9} + \beta_{10} + 9 \beta_{11} - \beta_{13} ) q^{11} + ( -7905 + 1267 \beta_{1} + 26 \beta_{2} - 20 \beta_{3} + 16 \beta_{4} + 8 \beta_{5} + \beta_{7} - 11 \beta_{8} + 6 \beta_{9} - 14 \beta_{10} + 4 \beta_{11} - \beta_{12} + \beta_{13} - 7 \beta_{14} ) q^{13} + ( -27343 - 2956 \beta_{1} - 69 \beta_{2} + 60 \beta_{3} + 14 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + 14 \beta_{7} - 51 \beta_{8} - 7 \beta_{9} + 4 \beta_{10} - 14 \beta_{11} - 5 \beta_{12} - 25 \beta_{13} - 11 \beta_{14} ) q^{14} + ( 66169 + 2617 \beta_{1} + 351 \beta_{2} - 5 \beta_{3} - 10 \beta_{4} - \beta_{5} - 27 \beta_{6} - 18 \beta_{7} - 8 \beta_{8} - 29 \beta_{9} - 65 \beta_{10} + 13 \beta_{11} + 3 \beta_{12} - 51 \beta_{13} - 25 \beta_{14} ) q^{16} + ( 59338 - 2978 \beta_{1} + 2 \beta_{2} - 21 \beta_{3} - 23 \beta_{4} + 37 \beta_{5} - 37 \beta_{6} - 39 \beta_{7} + 6 \beta_{8} + 15 \beta_{9} - 33 \beta_{10} + 8 \beta_{11} - 34 \beta_{12} - 6 \beta_{13} + 13 \beta_{14} ) q^{17} + ( -46704 + 6066 \beta_{1} + 73 \beta_{2} + 167 \beta_{3} + 11 \beta_{4} + \beta_{5} + 20 \beta_{6} + 61 \beta_{7} - 96 \beta_{8} - 16 \beta_{9} + 46 \beta_{10} - 11 \beta_{11} - 128 \beta_{12} + 49 \beta_{13} + 27 \beta_{14} ) q^{19} + ( 205946 - 5736 \beta_{1} + 572 \beta_{2} - 531 \beta_{3} + 27 \beta_{4} - 11 \beta_{5} - 40 \beta_{6} - \beta_{7} - 163 \beta_{8} + 56 \beta_{9} - 207 \beta_{10} + 45 \beta_{11} + 68 \beta_{12} - 60 \beta_{13} + 17 \beta_{14} ) q^{20} + ( -486765 + 12702 \beta_{1} - 604 \beta_{2} + 487 \beta_{3} + 35 \beta_{4} + 173 \beta_{5} + 7 \beta_{6} + 127 \beta_{7} - 342 \beta_{8} - 49 \beta_{9} - 19 \beta_{10} + 5 \beta_{11} + 135 \beta_{12} - 269 \beta_{13} + 21 \beta_{14} ) q^{22} + ( 166938 - 782 \beta_{1} + 144 \beta_{2} - 390 \beta_{3} + 29 \beta_{4} + 69 \beta_{5} - 17 \beta_{6} + 321 \beta_{7} + 495 \beta_{8} - 269 \beta_{9} + 9 \beta_{10} + 34 \beta_{11} - 180 \beta_{12} + 150 \beta_{13} + 23 \beta_{14} ) q^{23} + ( 87004 + 3275 \beta_{1} - 624 \beta_{2} - 173 \beta_{3} - 69 \beta_{4} + 27 \beta_{5} - 141 \beta_{6} - 166 \beta_{7} - 495 \beta_{8} + 291 \beta_{9} + 377 \beta_{10} + 206 \beta_{11} - 295 \beta_{12} + 229 \beta_{13} + 136 \beta_{14} ) q^{25} + ( 888931 + 6816 \beta_{1} + 2060 \beta_{2} - 813 \beta_{3} + 119 \beta_{4} + 297 \beta_{5} + 99 \beta_{6} - 317 \beta_{7} - 990 \beta_{8} - 93 \beta_{9} - 599 \beta_{10} + 9 \beta_{11} + 491 \beta_{12} - 241 \beta_{13} + 33 \beta_{14} ) q^{26} + ( -1864062 - 54068 \beta_{1} - 2022 \beta_{2} + 444 \beta_{3} + 540 \beta_{4} + 258 \beta_{5} + 296 \beta_{6} - 372 \beta_{7} + 46 \beta_{8} - 376 \beta_{9} + 124 \beta_{10} + 166 \beta_{11} + 476 \beta_{12} - 300 \beta_{13} - 168 \beta_{14} ) q^{28} + ( 1224774 + 22936 \beta_{1} - 1939 \beta_{2} + 2139 \beta_{3} + 400 \beta_{4} + 468 \beta_{5} - 543 \beta_{6} - 116 \beta_{7} - 625 \beta_{8} - 81 \beta_{9} + 87 \beta_{10} + 375 \beta_{11} + 255 \beta_{12} + 564 \beta_{13} - 130 \beta_{14} ) q^{29} + ( -800733 - 10806 \beta_{1} - 193 \beta_{2} - 943 \beta_{3} + 89 \beta_{4} + 265 \beta_{5} - 304 \beta_{6} + 1179 \beta_{7} - 659 \beta_{8} - 230 \beta_{9} + 308 \beta_{10} + 135 \beta_{11} - 109 \beta_{12} + 32 \beta_{13} - 1007 \beta_{14} ) q^{31} + ( 1248461 + 122091 \beta_{1} - 1211 \beta_{2} + 2331 \beta_{3} + 338 \beta_{4} + 551 \beta_{5} + 333 \beta_{6} + 1682 \beta_{7} - 2012 \beta_{8} - 185 \beta_{9} - 633 \beta_{10} + 253 \beta_{11} + 391 \beta_{12} - 639 \beta_{13} - 701 \beta_{14} ) q^{32} + ( -2032434 + 56292 \beta_{1} + 614 \beta_{2} - 928 \beta_{3} - 359 \beta_{4} + 415 \beta_{5} - 94 \beta_{6} + 1605 \beta_{7} - 998 \beta_{8} + 132 \beta_{9} - 774 \beta_{10} + 283 \beta_{11} - 229 \beta_{12} + 323 \beta_{13} + 315 \beta_{14} ) q^{34} + ( 1849557 + 50676 \beta_{1} - 5617 \beta_{2} + 3258 \beta_{3} - 894 \beta_{4} - 588 \beta_{5} - 1163 \beta_{6} - 35 \beta_{7} + 2685 \beta_{8} + 271 \beta_{9} + 2406 \beta_{10} - 749 \beta_{11} - 666 \beta_{12} + 195 \beta_{13} + 817 \beta_{14} ) q^{35} + ( -577454 - 63537 \beta_{1} + 1034 \beta_{2} - 10099 \beta_{3} + 746 \beta_{4} + 2291 \beta_{5} + 735 \beta_{6} - 1550 \beta_{7} - 5538 \beta_{8} - 439 \beta_{9} + 1501 \beta_{10} + 1335 \beta_{11} + 825 \beta_{12} + 418 \beta_{13} + 1638 \beta_{14} ) q^{37} + ( 4260610 - 12543 \beta_{1} + 6326 \beta_{2} - 12018 \beta_{3} - 829 \beta_{4} + 1485 \beta_{5} - 752 \beta_{6} - 1481 \beta_{7} + 2790 \beta_{8} - 1598 \beta_{9} + 188 \beta_{10} + 697 \beta_{11} + 949 \beta_{12} + 1501 \beta_{13} + 3549 \beta_{14} ) q^{38} + ( -2672478 + 261864 \beta_{1} - 3488 \beta_{2} + 13611 \beta_{3} + 213 \beta_{4} + 543 \beta_{5} + 1686 \beta_{6} - 239 \beta_{7} - 6881 \beta_{8} + 2714 \beta_{9} - 3421 \beta_{10} + 1143 \beta_{11} + 464 \beta_{12} + 804 \beta_{13} - 1107 \beta_{14} ) q^{40} + ( 1233270 + 83615 \beta_{1} - 5275 \beta_{2} + 1341 \beta_{3} + 1540 \beta_{4} + 2411 \beta_{5} + 3240 \beta_{6} - 347 \beta_{7} - 972 \beta_{8} - 2546 \beta_{9} + 411 \beta_{10} + 1600 \beta_{11} + 1042 \beta_{12} + 1440 \beta_{13} + 1139 \beta_{14} ) q^{41} -3418801 q^{43} + ( 4645389 - 482962 \beta_{1} + 20196 \beta_{2} - 8597 \beta_{3} + 1956 \beta_{4} + 1843 \beta_{5} + 197 \beta_{6} - 2604 \beta_{7} - 2088 \beta_{8} - 4637 \beta_{9} - 2345 \beta_{10} - 2015 \beta_{11} + 487 \beta_{12} - 4303 \beta_{13} - 4709 \beta_{14} ) q^{44} + ( -148891 + 106300 \beta_{1} + 2358 \beta_{2} + 17000 \beta_{3} - 1938 \beta_{4} - 252 \beta_{5} + 642 \beta_{6} - 9730 \beta_{7} - 1257 \beta_{8} + 648 \beta_{9} + 1266 \beta_{10} - 1914 \beta_{11} + 603 \beta_{12} - 1761 \beta_{13} + 2787 \beta_{14} ) q^{46} + ( 6943537 + 413295 \beta_{1} + 7375 \beta_{2} - 1917 \beta_{3} - 4314 \beta_{4} + 641 \beta_{5} + 1306 \beta_{6} + 5183 \beta_{7} - 7919 \beta_{8} + 1628 \beta_{9} + 2615 \beta_{10} - 2710 \beta_{11} - 3673 \beta_{12} - 2471 \beta_{13} + 113 \beta_{14} ) q^{47} + ( 6109224 + 540200 \beta_{1} + 77 \beta_{2} - 2024 \beta_{3} - 6962 \beta_{4} - 278 \beta_{5} - 8177 \beta_{6} + 6751 \beta_{7} + 10451 \beta_{8} + 905 \beta_{9} + 1766 \beta_{10} - 273 \beta_{11} - 4750 \beta_{12} + 7663 \beta_{13} + 3677 \beta_{14} ) q^{49} + ( 2814095 + 15193 \beta_{1} + 13070 \beta_{2} - 50351 \beta_{3} - 6858 \beta_{4} - 2850 \beta_{5} - 3065 \beta_{6} + 10070 \beta_{7} + 4042 \beta_{8} + 4133 \beta_{9} + 3337 \beta_{10} - 642 \beta_{11} - 430 \beta_{12} + 1718 \beta_{13} + 8028 \beta_{14} ) q^{50} + ( 9766137 + 1529876 \beta_{1} + 6726 \beta_{2} + 28863 \beta_{3} + 1340 \beta_{4} + 7951 \beta_{5} - 3127 \beta_{6} + 11468 \beta_{7} - 29576 \beta_{8} + 1327 \beta_{9} - 11429 \beta_{10} + 3941 \beta_{11} + 4243 \beta_{12} - 3675 \beta_{13} - 1969 \beta_{14} ) q^{52} + ( 14436444 - 196345 \beta_{1} + 34069 \beta_{2} + 15140 \beta_{3} - 1040 \beta_{4} - 3140 \beta_{5} - 5553 \beta_{6} - 2452 \beta_{7} + 2900 \beta_{8} - 1465 \beta_{9} - 2718 \beta_{10} + 3653 \beta_{11} - 8383 \beta_{12} + 9990 \beta_{13} - 3490 \beta_{14} ) q^{53} + ( 25659851 - 374031 \beta_{1} - 12603 \beta_{2} - 8138 \beta_{3} - 1127 \beta_{4} - 6833 \beta_{5} + 4866 \beta_{6} - 10715 \beta_{7} - 3626 \beta_{8} + 4752 \beta_{9} + 21093 \beta_{10} - 481 \beta_{11} - 15401 \beta_{12} + 5838 \beta_{13} - 3389 \beta_{14} ) q^{55} + ( -28504778 - 1469346 \beta_{1} - 11226 \beta_{2} + 16098 \beta_{3} + 6056 \beta_{4} - 3226 \beta_{5} + 10858 \beta_{6} - 1432 \beta_{7} + 18408 \beta_{8} - 15002 \beta_{9} + 1786 \beta_{10} - 2310 \beta_{11} + 3270 \beta_{12} - 4390 \beta_{13} - 12038 \beta_{14} ) q^{56} + ( 19685556 + 457083 \beta_{1} + 66389 \beta_{2} + 2158 \beta_{3} - 220 \beta_{4} - 5282 \beta_{5} + 3882 \beta_{6} + 660 \beta_{7} - 25634 \beta_{8} - 695 \beta_{9} - 8160 \beta_{10} - 15202 \beta_{11} + 10370 \beta_{12} - 20482 \beta_{13} - 6116 \beta_{14} ) q^{58} + ( -12296489 - 521718 \beta_{1} + 43251 \beta_{2} + 65198 \beta_{3} + 4802 \beta_{4} + 9324 \beta_{5} - 2801 \beta_{6} - 15937 \beta_{7} + 11591 \beta_{8} - 18693 \beta_{9} - 3376 \beta_{10} - 5983 \beta_{11} - 11018 \beta_{12} + 4153 \beta_{13} - 6447 \beta_{14} ) q^{59} + ( 16692599 - 1267897 \beta_{1} + 24719 \beta_{2} + 10432 \beta_{3} + 689 \beta_{4} - 7677 \beta_{5} - 306 \beta_{6} - 16615 \beta_{7} - 21614 \beta_{8} + 14710 \beta_{9} - 16593 \beta_{10} - 14441 \beta_{11} + 9311 \beta_{12} + 834 \beta_{13} - 8983 \beta_{14} ) q^{61} + ( -9221566 - 690422 \beta_{1} + 26298 \beta_{2} + 35462 \beta_{3} - 3999 \beta_{4} + 4305 \beta_{5} - 3178 \beta_{6} - 38643 \beta_{7} - 41860 \beta_{8} + 11798 \beta_{9} - 2544 \beta_{10} - 5599 \beta_{11} + 18375 \beta_{12} - 39957 \beta_{13} - 16459 \beta_{14} ) q^{62} + ( 56659711 - 735489 \beta_{1} + 51379 \beta_{2} - 39845 \beta_{3} + 13866 \beta_{4} + 11025 \beta_{5} + 101 \beta_{6} - 31894 \beta_{7} - 52242 \beta_{8} - 1605 \beta_{9} - 3441 \beta_{10} + 7067 \beta_{11} + 26479 \beta_{12} - 11095 \beta_{13} + 827 \beta_{14} ) q^{64} + ( -6581693 + 2503589 \beta_{1} + 88955 \beta_{2} + 20386 \beta_{3} + 2349 \beta_{4} - 6989 \beta_{5} + 12982 \beta_{6} + 4745 \beta_{7} + 19678 \beta_{8} + 346 \beta_{9} - 8063 \beta_{10} + 5891 \beta_{11} - 4775 \beta_{12} + 20576 \beta_{13} + 2661 \beta_{14} ) q^{65} + ( 31466364 - 2031243 \beta_{1} + 41092 \beta_{2} - 27230 \beta_{3} + 9897 \beta_{4} - 1458 \beta_{5} - 8188 \beta_{6} + 24470 \beta_{7} - 22008 \beta_{8} - 7644 \beta_{9} - 21447 \beta_{10} + 16709 \beta_{11} - 21014 \beta_{12} + 3113 \beta_{13} + 4634 \beta_{14} ) q^{67} + ( 5914826 - 195013 \beta_{1} + 144623 \beta_{2} - 92833 \beta_{3} - 325 \beta_{4} - 20627 \beta_{5} - 5638 \beta_{6} - 9425 \beta_{7} - 48625 \beta_{8} + 9550 \beta_{9} - 22349 \beta_{10} + 14413 \beta_{11} + 30850 \beta_{12} + 3998 \beta_{13} + 15611 \beta_{14} ) q^{68} + ( 43332180 - 1562805 \beta_{1} - 69940 \beta_{2} + 29479 \beta_{3} - 12272 \beta_{4} - 27526 \beta_{5} + 22909 \beta_{6} + 4944 \beta_{7} + 110371 \beta_{8} - 34875 \beta_{9} + 87493 \beta_{10} - 19036 \beta_{11} - 42209 \beta_{12} + 27191 \beta_{13} - 1773 \beta_{14} ) q^{70} + ( 402709 + 891494 \beta_{1} + 62584 \beta_{2} - 44922 \beta_{3} + 15225 \beta_{4} - 14308 \beta_{5} + 30932 \beta_{6} - 8823 \beta_{7} + 17091 \beta_{8} - 21428 \beta_{9} + 54755 \beta_{10} - 16825 \beta_{11} - 42649 \beta_{12} + 22558 \beta_{13} - 4779 \beta_{14} ) q^{71} + ( -47462648 - 1913733 \beta_{1} - 146775 \beta_{2} + 64046 \beta_{3} + 6468 \beta_{4} + 26405 \beta_{5} + 13416 \beta_{6} + 26456 \beta_{7} - 30649 \beta_{8} - 26636 \beta_{9} + 8084 \beta_{10} + 5796 \beta_{11} + 11354 \beta_{12} + 21498 \beta_{13} + 40602 \beta_{14} ) q^{73} + ( -48640289 + 2236729 \beta_{1} + 44918 \beta_{2} - 100294 \beta_{3} + 19410 \beta_{4} + 15246 \beta_{5} - 21568 \beta_{6} + 83346 \beta_{7} - 123743 \beta_{8} + 11416 \beta_{9} - 16092 \beta_{10} - 552 \beta_{11} - 13293 \beta_{12} - 27501 \beta_{13} + 3407 \beta_{14} ) q^{74} + ( 22500072 + 5048134 \beta_{1} - 84268 \beta_{2} + 133929 \beta_{3} - 7101 \beta_{4} - 17265 \beta_{5} - 11392 \beta_{6} + 57463 \beta_{7} - 11665 \beta_{8} + 65080 \beta_{9} - 6315 \beta_{10} - 5357 \beta_{11} - 6308 \beta_{12} + 21252 \beta_{13} + 5853 \beta_{14} ) q^{76} + ( 81687577 + 5097981 \beta_{1} + 88213 \beta_{2} - 279780 \beta_{3} - 17103 \beta_{4} - 9855 \beta_{5} + 5624 \beta_{6} - 17761 \beta_{7} + 81016 \beta_{8} + 33680 \beta_{9} + 66129 \beta_{10} + 8375 \beta_{11} - 5439 \beta_{12} + 41832 \beta_{13} + 23753 \beta_{14} ) q^{77} + ( 37285066 + 1469461 \beta_{1} + 174610 \beta_{2} + 119572 \beta_{3} - 24166 \beta_{4} - 36295 \beta_{5} - 37899 \beta_{6} + 64683 \beta_{7} - 115787 \beta_{8} + 31219 \beta_{9} - 26528 \beta_{10} + 2581 \beta_{11} + 45144 \beta_{12} - 88833 \beta_{13} - 12757 \beta_{14} ) q^{79} + ( 77261982 - 510888 \beta_{1} + 437680 \beta_{2} - 228955 \beta_{3} - 35053 \beta_{4} - 21111 \beta_{5} - 59738 \beta_{6} - 7065 \beta_{7} + 66289 \beta_{8} - 142 \beta_{9} - 65435 \beta_{10} + 15009 \beta_{11} + 34160 \beta_{12} + 23060 \beta_{13} + 32071 \beta_{14} ) q^{80} + ( 62968505 - 877529 \beta_{1} + 433624 \beta_{2} + 13467 \beta_{3} + 20818 \beta_{4} - 41762 \beta_{5} - 32403 \beta_{6} + 2178 \beta_{7} + 3610 \beta_{8} - 24063 \beta_{9} - 42945 \beta_{10} - 26050 \beta_{11} - 12614 \beta_{12} - 43098 \beta_{13} + 1672 \beta_{14} ) q^{82} + ( 95761721 + 2750841 \beta_{1} + 95733 \beta_{2} - 181508 \beta_{3} + 24903 \beta_{4} - 25832 \beta_{5} + 45317 \beta_{6} + 66643 \beta_{7} - 79257 \beta_{8} - 17317 \beta_{9} - 33255 \beta_{10} + 27420 \beta_{11} + 3218 \beta_{12} - 70158 \beta_{13} - 65569 \beta_{14} ) q^{83} + ( 46614231 + 1575075 \beta_{1} + 258692 \beta_{2} + 336227 \beta_{3} - 22030 \beta_{4} - 30735 \beta_{5} + 12277 \beta_{6} - 29574 \beta_{7} - 114961 \beta_{8} - 28413 \beta_{9} - 46369 \beta_{10} + 14471 \beta_{11} + 21315 \beta_{12} + 40792 \beta_{13} - 19728 \beta_{14} ) q^{85} + ( -6837602 - 3418801 \beta_{1} ) q^{86} + ( -98615654 + 8455177 \beta_{1} - 376372 \beta_{2} + 539871 \beta_{3} + 89234 \beta_{4} + 107388 \beta_{5} + 68273 \beta_{6} - 46782 \beta_{7} - 33623 \beta_{8} - 15863 \beta_{9} + 17735 \beta_{10} - 11764 \beta_{11} - 23557 \beta_{12} + 10173 \beta_{13} - 150901 \beta_{14} ) q^{88} + ( 26326865 + 67477 \beta_{1} - 20813 \beta_{2} - 454930 \beta_{3} + 25405 \beta_{4} + 9373 \beta_{5} - 49610 \beta_{6} + 17133 \beta_{7} - 58560 \beta_{8} + 104674 \beta_{9} - 61983 \beta_{10} - 25855 \beta_{11} + 107279 \beta_{12} - 37656 \beta_{13} + 89595 \beta_{14} ) q^{89} + ( -218605109 + 560803 \beta_{1} + 191729 \beta_{2} + 167612 \beta_{3} + 29077 \beta_{4} - 69001 \beta_{5} + 139820 \beta_{6} - 90597 \beta_{7} + 301916 \beta_{8} - 47566 \beta_{9} + 126661 \beta_{10} - 37865 \beta_{11} - 53217 \beta_{12} + 18890 \beta_{13} - 2925 \beta_{14} ) q^{91} + ( -10502363 + 2169178 \beta_{1} - 57990 \beta_{2} + 322622 \beta_{3} + 25585 \beta_{4} - 15224 \beta_{5} + 30581 \beta_{6} + 130853 \beta_{7} + 47621 \beta_{8} + 49311 \beta_{9} + 93134 \beta_{10} - 13966 \beta_{11} - 39405 \beta_{12} + 55045 \beta_{13} - 23542 \beta_{14} ) q^{92} + ( 309481839 + 11774278 \beta_{1} + 612092 \beta_{2} - 454165 \beta_{3} - 32792 \beta_{4} + 61954 \beta_{5} - 109289 \beta_{6} - 92720 \beta_{7} + 73758 \beta_{8} - 24251 \beta_{9} + 14751 \beta_{10} + 121898 \beta_{11} - 24918 \beta_{12} + 42554 \beta_{13} + 58230 \beta_{14} ) q^{94} + ( 256898810 + 374965 \beta_{1} - 306388 \beta_{2} + 41743 \beta_{3} + 84808 \beta_{4} + 68239 \beta_{5} - 11293 \beta_{6} + 41664 \beta_{7} - 256534 \beta_{8} + 37491 \beta_{9} - 1531 \beta_{10} + 87245 \beta_{11} + 4785 \beta_{12} + 68968 \beta_{13} + 1034 \beta_{14} ) q^{95} + ( -204322073 + 1038679 \beta_{1} + 498290 \beta_{2} - 109903 \beta_{3} + 110556 \beta_{4} + 62061 \beta_{5} - 13215 \beta_{6} - 15464 \beta_{7} - 30771 \beta_{8} - 200193 \beta_{9} - 172109 \beta_{10} + 159331 \beta_{11} + 146140 \beta_{12} + 57059 \beta_{13} + 59510 \beta_{14} ) q^{97} + ( 410161410 + 5504792 \beta_{1} + 596368 \beta_{2} - 46943 \beta_{3} - 196562 \beta_{4} - 138284 \beta_{5} - 63193 \beta_{6} - 125434 \beta_{7} + 81565 \beta_{8} + 223351 \beta_{9} + 61599 \beta_{10} - 25886 \beta_{11} - 77461 \beta_{12} + 110811 \beta_{13} + 143911 \beta_{14} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$15q + 32q^{2} + 3242q^{4} + 4717q^{5} - 9680q^{7} + 20394q^{8} + O(q^{10})$$ $$15q + 32q^{2} + 3242q^{4} + 4717q^{5} - 9680q^{7} + 20394q^{8} - 36237q^{10} + 104484q^{11} - 116174q^{13} - 416064q^{14} + 996762q^{16} + 884265q^{17} - 689535q^{19} + 3077879q^{20} - 7276218q^{22} + 2504077q^{23} + 1315350q^{25} + 13343414q^{26} - 28059568q^{28} + 18406221q^{29} - 12033699q^{31} + 18952630q^{32} - 30383125q^{34} + 27855546q^{35} - 8722847q^{37} + 63941843q^{38} - 39665611q^{40} + 18689389q^{41} - 51282015q^{43} + 68723220q^{44} - 2067521q^{46} + 104960741q^{47} + 92663095q^{49} + 42446347q^{50} + 149226080q^{52} + 215907800q^{53} + 384379852q^{55} - 430441344q^{56} + 295963139q^{58} - 185924544q^{59} + 247538102q^{61} - 139798853q^{62} + 848556290q^{64} - 94294394q^{65} + 467904656q^{67} + 88234341q^{68} + 647526126q^{70} + 8252944q^{71} - 715627902q^{73} - 725122989q^{74} + 346300359q^{76} + 1236779964q^{77} + 560681783q^{79} + 1157214179q^{80} + 941346367q^{82} + 1442854698q^{83} + 699302088q^{85} - 109401632q^{86} - 1464507256q^{88} + 396710008q^{89} - 3278076852q^{91} - 155864647q^{92} + 4666638949q^{94} + 3854114395q^{95} - 3063837815q^{97} + 6161086984q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{15} - 2 x^{14} - 5425 x^{13} + 14888 x^{12} + 11288030 x^{11} - 37600244 x^{10} - 11474166224 x^{9} + 47465836576 x^{8} + 5986976782464 x^{7} - 32493903147264 x^{6} - 1516975415483904 x^{5} + 10892588268404224 x^{4} + 139803541742443008 x^{3} - 1349125586394823680 x^{2} + 2103623681144094720 x + 529838441422848000$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu - 724$$ $$\beta_{3}$$ $$=$$ $$($$$$-$$$$19\!\cdots\!31$$$$\nu^{14} -$$$$19\!\cdots\!96$$$$\nu^{13} +$$$$99\!\cdots\!55$$$$\nu^{12} +$$$$92\!\cdots\!54$$$$\nu^{11} -$$$$19\!\cdots\!26$$$$\nu^{10} -$$$$16\!\cdots\!32$$$$\nu^{9} +$$$$18\!\cdots\!24$$$$\nu^{8} +$$$$11\!\cdots\!36$$$$\nu^{7} -$$$$89\!\cdots\!60$$$$\nu^{6} -$$$$27\!\cdots\!04$$$$\nu^{5} +$$$$20\!\cdots\!00$$$$\nu^{4} -$$$$37\!\cdots\!44$$$$\nu^{3} -$$$$18\!\cdots\!40$$$$\nu^{2} +$$$$13\!\cdots\!24$$$$\nu -$$$$11\!\cdots\!12$$$$)/$$$$25\!\cdots\!24$$ $$\beta_{4}$$ $$=$$ $$($$$$55\!\cdots\!29$$$$\nu^{14} -$$$$34\!\cdots\!28$$$$\nu^{13} -$$$$34\!\cdots\!25$$$$\nu^{12} +$$$$17\!\cdots\!22$$$$\nu^{11} +$$$$84\!\cdots\!10$$$$\nu^{10} -$$$$32\!\cdots\!56$$$$\nu^{9} -$$$$10\!\cdots\!16$$$$\nu^{8} +$$$$28\!\cdots\!24$$$$\nu^{7} +$$$$68\!\cdots\!96$$$$\nu^{6} -$$$$11\!\cdots\!96$$$$\nu^{5} -$$$$21\!\cdots\!56$$$$\nu^{4} +$$$$24\!\cdots\!16$$$$\nu^{3} +$$$$26\!\cdots\!32$$$$\nu^{2} -$$$$20\!\cdots\!40$$$$\nu -$$$$11\!\cdots\!00$$$$)/$$$$48\!\cdots\!20$$ $$\beta_{5}$$ $$=$$ $$($$$$-$$$$30\!\cdots\!41$$$$\nu^{14} +$$$$50\!\cdots\!48$$$$\nu^{13} +$$$$16\!\cdots\!53$$$$\nu^{12} -$$$$26\!\cdots\!14$$$$\nu^{11} -$$$$35\!\cdots\!30$$$$\nu^{10} +$$$$50\!\cdots\!88$$$$\nu^{9} +$$$$37\!\cdots\!32$$$$\nu^{8} -$$$$44\!\cdots\!72$$$$\nu^{7} -$$$$20\!\cdots\!88$$$$\nu^{6} +$$$$18\!\cdots\!00$$$$\nu^{5} +$$$$54\!\cdots\!88$$$$\nu^{4} -$$$$31\!\cdots\!60$$$$\nu^{3} -$$$$57\!\cdots\!64$$$$\nu^{2} +$$$$10\!\cdots\!24$$$$\nu +$$$$54\!\cdots\!28$$$$)/$$$$25\!\cdots\!24$$ $$\beta_{6}$$ $$=$$ $$($$$$14\!\cdots\!33$$$$\nu^{14} -$$$$26\!\cdots\!36$$$$\nu^{13} -$$$$85\!\cdots\!05$$$$\nu^{12} +$$$$14\!\cdots\!74$$$$\nu^{11} +$$$$19\!\cdots\!30$$$$\nu^{10} -$$$$29\!\cdots\!32$$$$\nu^{9} -$$$$22\!\cdots\!12$$$$\nu^{8} +$$$$29\!\cdots\!48$$$$\nu^{7} +$$$$13\!\cdots\!72$$$$\nu^{6} -$$$$14\!\cdots\!72$$$$\nu^{5} -$$$$37\!\cdots\!12$$$$\nu^{4} +$$$$35\!\cdots\!72$$$$\nu^{3} +$$$$37\!\cdots\!04$$$$\nu^{2} -$$$$31\!\cdots\!40$$$$\nu +$$$$15\!\cdots\!20$$$$)/$$$$48\!\cdots\!20$$ $$\beta_{7}$$ $$=$$ $$($$$$99\!\cdots\!51$$$$\nu^{14} +$$$$19\!\cdots\!88$$$$\nu^{13} -$$$$51\!\cdots\!55$$$$\nu^{12} -$$$$95\!\cdots\!22$$$$\nu^{11} +$$$$10\!\cdots\!10$$$$\nu^{10} +$$$$17\!\cdots\!16$$$$\nu^{9} -$$$$95\!\cdots\!04$$$$\nu^{8} -$$$$14\!\cdots\!64$$$$\nu^{7} +$$$$44\!\cdots\!84$$$$\nu^{6} +$$$$54\!\cdots\!16$$$$\nu^{5} -$$$$99\!\cdots\!44$$$$\nu^{4} -$$$$68\!\cdots\!96$$$$\nu^{3} +$$$$87\!\cdots\!88$$$$\nu^{2} -$$$$20\!\cdots\!60$$$$\nu -$$$$28\!\cdots\!00$$$$)/$$$$32\!\cdots\!80$$ $$\beta_{8}$$ $$=$$ $$($$$$-$$$$18\!\cdots\!43$$$$\nu^{14} -$$$$30\!\cdots\!04$$$$\nu^{13} +$$$$95\!\cdots\!75$$$$\nu^{12} +$$$$14\!\cdots\!06$$$$\nu^{11} -$$$$18\!\cdots\!10$$$$\nu^{10} -$$$$27\!\cdots\!28$$$$\nu^{9} +$$$$16\!\cdots\!92$$$$\nu^{8} +$$$$23\!\cdots\!72$$$$\nu^{7} -$$$$67\!\cdots\!32$$$$\nu^{6} -$$$$85\!\cdots\!88$$$$\nu^{5} +$$$$12\!\cdots\!92$$$$\nu^{4} +$$$$11\!\cdots\!48$$$$\nu^{3} -$$$$77\!\cdots\!44$$$$\nu^{2} -$$$$26\!\cdots\!40$$$$\nu +$$$$51\!\cdots\!00$$$$)/$$$$38\!\cdots\!60$$ $$\beta_{9}$$ $$=$$ $$($$$$-$$$$32\!\cdots\!87$$$$\nu^{14} -$$$$52\!\cdots\!76$$$$\nu^{13} +$$$$16\!\cdots\!55$$$$\nu^{12} +$$$$25\!\cdots\!74$$$$\nu^{11} -$$$$31\!\cdots\!30$$$$\nu^{10} -$$$$48\!\cdots\!32$$$$\nu^{9} +$$$$27\!\cdots\!28$$$$\nu^{8} +$$$$42\!\cdots\!28$$$$\nu^{7} -$$$$11\!\cdots\!28$$$$\nu^{6} -$$$$16\!\cdots\!32$$$$\nu^{5} +$$$$20\!\cdots\!48$$$$\nu^{4} +$$$$24\!\cdots\!32$$$$\nu^{3} -$$$$11\!\cdots\!16$$$$\nu^{2} -$$$$83\!\cdots\!40$$$$\nu +$$$$16\!\cdots\!00$$$$)/$$$$48\!\cdots\!20$$ $$\beta_{10}$$ $$=$$ $$($$$$15\!\cdots\!89$$$$\nu^{14} +$$$$25\!\cdots\!92$$$$\nu^{13} -$$$$77\!\cdots\!25$$$$\nu^{12} -$$$$12\!\cdots\!78$$$$\nu^{11} +$$$$14\!\cdots\!70$$$$\nu^{10} +$$$$22\!\cdots\!24$$$$\nu^{9} -$$$$13\!\cdots\!56$$$$\nu^{8} -$$$$18\!\cdots\!36$$$$\nu^{7} +$$$$58\!\cdots\!16$$$$\nu^{6} +$$$$68\!\cdots\!24$$$$\nu^{5} -$$$$11\!\cdots\!16$$$$\nu^{4} -$$$$86\!\cdots\!24$$$$\nu^{3} +$$$$95\!\cdots\!32$$$$\nu^{2} +$$$$32\!\cdots\!60$$$$\nu -$$$$66\!\cdots\!80$$$$)/$$$$12\!\cdots\!20$$ $$\beta_{11}$$ $$=$$ $$($$$$40\!\cdots\!87$$$$\nu^{14} +$$$$33\!\cdots\!16$$$$\nu^{13} -$$$$20\!\cdots\!95$$$$\nu^{12} -$$$$15\!\cdots\!34$$$$\nu^{11} +$$$$41\!\cdots\!50$$$$\nu^{10} +$$$$28\!\cdots\!92$$$$\nu^{9} -$$$$38\!\cdots\!08$$$$\nu^{8} -$$$$22\!\cdots\!08$$$$\nu^{7} +$$$$18\!\cdots\!88$$$$\nu^{6} +$$$$70\!\cdots\!72$$$$\nu^{5} -$$$$40\!\cdots\!68$$$$\nu^{4} -$$$$29\!\cdots\!92$$$$\nu^{3} +$$$$35\!\cdots\!76$$$$\nu^{2} -$$$$11\!\cdots\!20$$$$\nu -$$$$50\!\cdots\!00$$$$)/$$$$19\!\cdots\!80$$ $$\beta_{12}$$ $$=$$ $$($$$$-$$$$87\!\cdots\!79$$$$\nu^{14} -$$$$90\!\cdots\!95$$$$\nu^{13} +$$$$45\!\cdots\!17$$$$\nu^{12} +$$$$43\!\cdots\!97$$$$\nu^{11} -$$$$90\!\cdots\!30$$$$\nu^{10} -$$$$78\!\cdots\!34$$$$\nu^{9} +$$$$84\!\cdots\!84$$$$\nu^{8} +$$$$63\!\cdots\!20$$$$\nu^{7} -$$$$39\!\cdots\!28$$$$\nu^{6} -$$$$20\!\cdots\!52$$$$\nu^{5} +$$$$86\!\cdots\!04$$$$\nu^{4} +$$$$12\!\cdots\!96$$$$\nu^{3} -$$$$70\!\cdots\!84$$$$\nu^{2} +$$$$27\!\cdots\!12$$$$\nu -$$$$26\!\cdots\!76$$$$)/$$$$40\!\cdots\!16$$ $$\beta_{13}$$ $$=$$ $$($$$$-$$$$86\!\cdots\!09$$$$\nu^{14} -$$$$11\!\cdots\!72$$$$\nu^{13} +$$$$45\!\cdots\!45$$$$\nu^{12} +$$$$55\!\cdots\!38$$$$\nu^{11} -$$$$91\!\cdots\!50$$$$\nu^{10} -$$$$99\!\cdots\!44$$$$\nu^{9} +$$$$87\!\cdots\!56$$$$\nu^{8} +$$$$79\!\cdots\!56$$$$\nu^{7} -$$$$41\!\cdots\!76$$$$\nu^{6} -$$$$25\!\cdots\!64$$$$\nu^{5} +$$$$97\!\cdots\!96$$$$\nu^{4} +$$$$17\!\cdots\!84$$$$\nu^{3} -$$$$87\!\cdots\!12$$$$\nu^{2} +$$$$29\!\cdots\!80$$$$\nu -$$$$15\!\cdots\!80$$$$)/$$$$38\!\cdots\!60$$ $$\beta_{14}$$ $$=$$ $$($$$$34\!\cdots\!11$$$$\nu^{14} +$$$$45\!\cdots\!36$$$$\nu^{13} -$$$$18\!\cdots\!55$$$$\nu^{12} -$$$$22\!\cdots\!42$$$$\nu^{11} +$$$$36\!\cdots\!50$$$$\nu^{10} +$$$$40\!\cdots\!52$$$$\nu^{9} -$$$$34\!\cdots\!84$$$$\nu^{8} -$$$$32\!\cdots\!24$$$$\nu^{7} +$$$$16\!\cdots\!56$$$$\nu^{6} +$$$$11\!\cdots\!96$$$$\nu^{5} -$$$$37\!\cdots\!88$$$$\nu^{4} -$$$$95\!\cdots\!80$$$$\nu^{3} +$$$$33\!\cdots\!52$$$$\nu^{2} -$$$$94\!\cdots\!96$$$$\nu -$$$$60\!\cdots\!84$$$$)/$$$$12\!\cdots\!12$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - \beta_{1} + 724$$ $$\nu^{3}$$ $$=$$ $$\beta_{11} + 2 \beta_{9} - \beta_{8} + 3 \beta_{5} + 12 \beta_{3} - \beta_{2} + 1260 \beta_{1} - 969$$ $$\nu^{4}$$ $$=$$ $$-25 \beta_{14} - 51 \beta_{13} + 3 \beta_{12} + 5 \beta_{11} - 65 \beta_{10} - 45 \beta_{9} - 18 \beta_{7} - 27 \beta_{6} - 25 \beta_{5} - 10 \beta_{4} - 101 \beta_{3} + 1871 \beta_{2} - 2863 \beta_{1} + 912593$$ $$\nu^{5}$$ $$=$$ $$-451 \beta_{14} - 129 \beta_{13} + 361 \beta_{12} + 2211 \beta_{11} + 17 \beta_{10} + 4281 \beta_{9} - 4020 \beta_{8} + 1862 \beta_{7} + 603 \beta_{6} + 6825 \beta_{5} + 438 \beta_{4} + 27437 \beta_{3} - 9721 \beta_{2} + 1906657 \beta_{1} - 2541141$$ $$\nu^{6}$$ $$=$$ $$-56261 \beta_{14} - 137047 \beta_{13} + 29647 \beta_{12} + 13355 \beta_{11} - 166145 \beta_{10} - 124837 \beta_{9} - 24322 \beta_{8} - 99238 \beta_{7} - 74635 \beta_{6} - 72415 \beta_{5} - 16390 \beta_{4} - 377749 \beta_{3} + 3313547 \beta_{2} - 9867737 \beta_{1} + 1382468255$$ $$\nu^{7}$$ $$=$$ $$-1476377 \beta_{14} - 101683 \beta_{13} + 624443 \beta_{12} + 3891795 \beta_{11} + 499011 \beta_{10} + 8280255 \beta_{9} - 9578646 \beta_{8} + 6641514 \beta_{7} + 2022465 \beta_{6} + 13341257 \beta_{5} + 1574810 \beta_{4} + 54263775 \beta_{3} - 32063849 \beta_{2} + 3159193055 \beta_{1} - 7989903641$$ $$\nu^{8}$$ $$=$$ $$-99010825 \beta_{14} - 289235299 \beta_{13} + 92490507 \beta_{12} + 24459623 \beta_{11} - 343317101 \beta_{10} - 293492905 \beta_{9} - 55318490 \beta_{8} - 304193382 \beta_{7} - 161180207 \beta_{6} - 172377627 \beta_{5} - 28019062 \beta_{4} - 1021357761 \beta_{3} + 5940685531 \beta_{2} - 28988601017 \beta_{1} + 2293879739123$$ $$\nu^{9}$$ $$=$$ $$-3505734621 \beta_{14} + 504553057 \beta_{13} + 400597703 \beta_{12} + 6498434995 \beta_{11} + 2257945439 \beta_{10} + 16121418211 \beta_{9} - 19514435778 \beta_{8} + 17240539538 \beta_{7} + 5036973621 \beta_{6} + 25209696809 \beta_{5} + 3950278306 \beta_{4} + 106123526619 \beta_{3} - 84361806929 \beta_{2} + 5517998843179 \beta_{1} - 22522119936385$$ $$\nu^{10}$$ $$=$$ $$-159606282881 \beta_{14} - 567035896587 \beta_{13} + 222487096275 \beta_{12} + 36914590119 \beta_{11} - 667007471669 \beta_{10} - 649593397041 \beta_{9} - 62970416226 \beta_{8} - 761822926566 \beta_{7} - 326638116615 \beta_{6} - 394538691451 \beta_{5} - 58080805974 \beta_{4} - 2452820561641 \beta_{3} + 10877476562787 \beta_{2} - 75178497365145 \beta_{1} + 4012983344773363$$ $$\nu^{11}$$ $$=$$ $$-7366410638861 \beta_{14} + 2712863734673 \beta_{13} - 1119132304553 \beta_{12} + 10759881290443 \beta_{11} + 7161703054575 \beta_{10} + 31937223329091 \beta_{9} - 37451118980778 \beta_{8} + 39697676628498 \beta_{7} + 11373902175909 \beta_{6} + 47419492364081 \beta_{5} + 8660440017570 \beta_{4} + 208551500150859 \beta_{3} - 202075889447449 \beta_{2} + 9982495692381739 \beta_{1} - 57250960348232121$$ $$\nu^{12}$$ $$=$$ $$-245248710611449 \beta_{14} - 1083049183240819 \beta_{13} + 481111511396539 \beta_{12} + 44325492776047 \beta_{11} - 1270622059987117 \beta_{10} - 1390238797362409 \beta_{9} + 21040233310126 \beta_{8} - 1740214024685526 \beta_{7} - 651653446777327 \beta_{6} - 888233290791267 \beta_{5} - 133596085914118 \beta_{4} - 5559703124423169 \beta_{3} + 20314753348708091 \beta_{2} - 180468425483910945 \beta_{1} + 7271772332801227259$$ $$\nu^{13}$$ $$=$$ $$-14571618375601765 \beta_{14} + 8905237902902281 \beta_{13} - 5718447457129825 \beta_{12} + 17961495859399203 \beta_{11} + 19293763112871287 \beta_{10} + 64070228644553227 \beta_{9} - 70251027192678154 \beta_{8} + 86357267464454082 \beta_{7} + 24602114347149757 \beta_{6} + 89692877074905401 \beta_{5} + 17865031729309778 \beta_{4} + 411871543490528051 \beta_{3} - 460231065324273793 \beta_{2} + 18525240229224447251 \beta_{1} - 135923840841627039137$$ $$\nu^{14}$$ $$=$$ $$-361913292449488609 \beta_{14} - 2053445921854982699 \beta_{13} + 989174116651597619 \beta_{12} + 28336960724061799 \beta_{11} - 2413405825177909877 \beta_{10} - 2914782746773674961 \beta_{9} + 354219391839938942 \beta_{8} - 3789566164596080582 \beta_{7} - 1298720351038661703 \beta_{6} - 1970212909654523387 \beta_{5} - 312873342030878966 \beta_{4} - 12191099478474904041 \beta_{3} + 38579726935375072371 \beta_{2} - 412635637684278462185 \beta_{1} + 13516338533949787247379$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −45.0936 −35.4490 −28.3075 −25.8680 −22.4329 −17.0644 −0.220103 2.38595 7.62988 15.9205 16.4876 21.7142 32.6728 39.2075 40.4171
−43.0936 0 1345.05 1330.99 0 −743.545 −35899.3 0 −57356.9
1.2 −33.4490 0 606.833 752.925 0 −7164.62 −3172.07 0 −25184.6
1.3 −26.3075 0 180.084 −879.044 0 8431.97 8731.88 0 23125.4
1.4 −23.8680 0 57.6838 578.837 0 −413.035 10843.6 0 −13815.7
1.5 −20.4329 0 −94.4979 2583.58 0 −1769.99 12392.5 0 −52790.0
1.6 −15.0644 0 −285.063 −1876.98 0 −7041.18 12007.3 0 28275.7
1.7 1.77990 0 −508.832 −1139.29 0 4324.96 −1816.98 0 −2027.82
1.8 4.38595 0 −492.763 1237.15 0 12549.3 −4406.84 0 5426.06
1.9 9.62988 0 −419.265 −636.384 0 −3288.51 −8967.97 0 −6128.30
1.10 17.9205 0 −190.855 −238.484 0 −4524.82 −12595.5 0 −4273.75
1.11 18.4876 0 −170.210 2363.43 0 7025.62 −12612.4 0 43694.1
1.12 23.7142 0 50.3616 764.097 0 4079.63 −10947.4 0 18119.9
1.13 34.6728 0 690.205 −1855.36 0 −11539.9 6178.86 0 −64330.7
1.14 41.2075 0 1186.05 1998.59 0 −10690.1 27776.1 0 82356.9
1.15 42.4171 0 1287.21 −267.050 0 1084.23 32882.2 0 −11327.5
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.15 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 387.10.a.c 15
3.b odd 2 1 43.10.a.a 15

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.10.a.a 15 3.b odd 2 1
387.10.a.c 15 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$43$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{15} - \cdots$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(387))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 32 T + 2731 T^{2} - 77806 T^{3} + 3850358 T^{4} - 101839448 T^{5} + 3733988120 T^{6} - 89284085280 T^{7} + 2722683702592 T^{8} - 57999472061696 T^{9} + 1597215028061696 T^{10} - 30230480697280512 T^{11} + 801326892628557824 T^{12} - 13973639350763585536 T^{13} +$$$$38\!\cdots\!16$$$$T^{14} -$$$$67\!\cdots\!40$$$$T^{15} +$$$$19\!\cdots\!92$$$$T^{16} -$$$$36\!\cdots\!84$$$$T^{17} +$$$$10\!\cdots\!72$$$$T^{18} -$$$$20\!\cdots\!32$$$$T^{19} +$$$$56\!\cdots\!72$$$$T^{20} -$$$$10\!\cdots\!64$$$$T^{21} +$$$$25\!\cdots\!36$$$$T^{22} -$$$$42\!\cdots\!80$$$$T^{23} +$$$$90\!\cdots\!40$$$$T^{24} -$$$$12\!\cdots\!52$$$$T^{25} +$$$$24\!\cdots\!04$$$$T^{26} -$$$$25\!\cdots\!36$$$$T^{27} +$$$$45\!\cdots\!32$$$$T^{28} -$$$$27\!\cdots\!48$$$$T^{29} +$$$$43\!\cdots\!68$$$$T^{30}$$
$3$ 1
$5$ $$1 - 4717 T + 25115807 T^{2} - 83707229397 T^{3} + 275195357593257 T^{4} - 730256475016765590 T^{5} +$$$$18\!\cdots\!78$$$$T^{6} -$$$$41\!\cdots\!18$$$$T^{7} +$$$$89\!\cdots\!28$$$$T^{8} -$$$$17\!\cdots\!68$$$$T^{9} +$$$$32\!\cdots\!59$$$$T^{10} -$$$$57\!\cdots\!55$$$$T^{11} +$$$$96\!\cdots\!25$$$$T^{12} -$$$$15\!\cdots\!25$$$$T^{13} +$$$$22\!\cdots\!25$$$$T^{14} -$$$$32\!\cdots\!00$$$$T^{15} +$$$$44\!\cdots\!25$$$$T^{16} -$$$$57\!\cdots\!25$$$$T^{17} +$$$$71\!\cdots\!25$$$$T^{18} -$$$$83\!\cdots\!75$$$$T^{19} +$$$$93\!\cdots\!75$$$$T^{20} -$$$$97\!\cdots\!00$$$$T^{21} +$$$$97\!\cdots\!00$$$$T^{22} -$$$$88\!\cdots\!50$$$$T^{23} +$$$$77\!\cdots\!50$$$$T^{24} -$$$$58\!\cdots\!50$$$$T^{25} +$$$$43\!\cdots\!25$$$$T^{26} -$$$$25\!\cdots\!25$$$$T^{27} +$$$$15\!\cdots\!75$$$$T^{28} -$$$$55\!\cdots\!25$$$$T^{29} +$$$$22\!\cdots\!25$$$$T^{30}$$
$7$ $$1 + 9680 T + 303171705 T^{2} + 2420677168232 T^{3} + 41789850773745109 T^{4} +$$$$27\!\cdots\!60$$$$T^{5} +$$$$34\!\cdots\!17$$$$T^{6} +$$$$18\!\cdots\!84$$$$T^{7} +$$$$18\!\cdots\!61$$$$T^{8} +$$$$63\!\cdots\!36$$$$T^{9} +$$$$58\!\cdots\!33$$$$T^{10} -$$$$81\!\cdots\!12$$$$T^{11} +$$$$53\!\cdots\!53$$$$T^{12} -$$$$14\!\cdots\!36$$$$T^{13} -$$$$44\!\cdots\!71$$$$T^{14} -$$$$84\!\cdots\!16$$$$T^{15} -$$$$17\!\cdots\!97$$$$T^{16} -$$$$23\!\cdots\!64$$$$T^{17} +$$$$35\!\cdots\!79$$$$T^{18} -$$$$21\!\cdots\!12$$$$T^{19} +$$$$62\!\cdots\!31$$$$T^{20} +$$$$27\!\cdots\!64$$$$T^{21} +$$$$31\!\cdots\!23$$$$T^{22} +$$$$12\!\cdots\!84$$$$T^{23} +$$$$97\!\cdots\!19$$$$T^{24} +$$$$31\!\cdots\!40$$$$T^{25} +$$$$19\!\cdots\!87$$$$T^{26} +$$$$45\!\cdots\!32$$$$T^{27} +$$$$22\!\cdots\!35$$$$T^{28} +$$$$29\!\cdots\!20$$$$T^{29} +$$$$12\!\cdots\!43$$$$T^{30}$$
$11$ $$1 - 104484 T + 21671047097 T^{2} - 1826154580412208 T^{3} +$$$$22\!\cdots\!79$$$$T^{4} -$$$$16\!\cdots\!04$$$$T^{5} +$$$$15\!\cdots\!71$$$$T^{6} -$$$$10\!\cdots\!88$$$$T^{7} +$$$$81\!\cdots\!64$$$$T^{8} -$$$$46\!\cdots\!00$$$$T^{9} +$$$$32\!\cdots\!04$$$$T^{10} -$$$$17\!\cdots\!28$$$$T^{11} +$$$$10\!\cdots\!38$$$$T^{12} -$$$$52\!\cdots\!68$$$$T^{13} +$$$$30\!\cdots\!42$$$$T^{14} -$$$$13\!\cdots\!92$$$$T^{15} +$$$$72\!\cdots\!22$$$$T^{16} -$$$$29\!\cdots\!08$$$$T^{17} +$$$$14\!\cdots\!98$$$$T^{18} -$$$$53\!\cdots\!08$$$$T^{19} +$$$$24\!\cdots\!04$$$$T^{20} -$$$$79\!\cdots\!00$$$$T^{21} +$$$$32\!\cdots\!84$$$$T^{22} -$$$$96\!\cdots\!48$$$$T^{23} +$$$$35\!\cdots\!81$$$$T^{24} -$$$$87\!\cdots\!04$$$$T^{25} +$$$$28\!\cdots\!89$$$$T^{26} -$$$$53\!\cdots\!48$$$$T^{27} +$$$$15\!\cdots\!87$$$$T^{28} -$$$$17\!\cdots\!24$$$$T^{29} +$$$$38\!\cdots\!51$$$$T^{30}$$
$13$ $$1 + 116174 T + 93110646219 T^{2} + 9939551633003756 T^{3} +$$$$43\!\cdots\!83$$$$T^{4} +$$$$42\!\cdots\!54$$$$T^{5} +$$$$13\!\cdots\!41$$$$T^{6} +$$$$12\!\cdots\!40$$$$T^{7} +$$$$31\!\cdots\!88$$$$T^{8} +$$$$25\!\cdots\!00$$$$T^{9} +$$$$57\!\cdots\!96$$$$T^{10} +$$$$42\!\cdots\!04$$$$T^{11} +$$$$85\!\cdots\!78$$$$T^{12} +$$$$58\!\cdots\!52$$$$T^{13} +$$$$10\!\cdots\!26$$$$T^{14} +$$$$67\!\cdots\!12$$$$T^{15} +$$$$11\!\cdots\!98$$$$T^{16} +$$$$65\!\cdots\!08$$$$T^{17} +$$$$10\!\cdots\!26$$$$T^{18} +$$$$53\!\cdots\!64$$$$T^{19} +$$$$76\!\cdots\!28$$$$T^{20} +$$$$36\!\cdots\!00$$$$T^{21} +$$$$47\!\cdots\!36$$$$T^{22} +$$$$19\!\cdots\!40$$$$T^{23} +$$$$23\!\cdots\!33$$$$T^{24} +$$$$76\!\cdots\!46$$$$T^{25} +$$$$83\!\cdots\!91$$$$T^{26} +$$$$20\!\cdots\!76$$$$T^{27} +$$$$19\!\cdots\!27$$$$T^{28} +$$$$26\!\cdots\!66$$$$T^{29} +$$$$24\!\cdots\!57$$$$T^{30}$$
$17$ $$1 - 884265 T + 1384046763853 T^{2} - 1008991515705553461 T^{3} +$$$$92\!\cdots\!03$$$$T^{4} -$$$$56\!\cdots\!24$$$$T^{5} +$$$$38\!\cdots\!52$$$$T^{6} -$$$$20\!\cdots\!80$$$$T^{7} +$$$$11\!\cdots\!15$$$$T^{8} -$$$$54\!\cdots\!13$$$$T^{9} +$$$$26\!\cdots\!40$$$$T^{10} -$$$$11\!\cdots\!12$$$$T^{11} +$$$$47\!\cdots\!38$$$$T^{12} -$$$$17\!\cdots\!77$$$$T^{13} +$$$$69\!\cdots\!95$$$$T^{14} -$$$$23\!\cdots\!62$$$$T^{15} +$$$$81\!\cdots\!15$$$$T^{16} -$$$$25\!\cdots\!93$$$$T^{17} +$$$$79\!\cdots\!74$$$$T^{18} -$$$$21\!\cdots\!72$$$$T^{19} +$$$$62\!\cdots\!80$$$$T^{20} -$$$$15\!\cdots\!77$$$$T^{21} +$$$$38\!\cdots\!95$$$$T^{22} -$$$$80\!\cdots\!80$$$$T^{23} +$$$$18\!\cdots\!84$$$$T^{24} -$$$$31\!\cdots\!76$$$$T^{25} +$$$$60\!\cdots\!59$$$$T^{26} -$$$$78\!\cdots\!01$$$$T^{27} +$$$$12\!\cdots\!81$$$$T^{28} -$$$$96\!\cdots\!85$$$$T^{29} +$$$$12\!\cdots\!93$$$$T^{30}$$
$19$ $$1 + 689535 T + 2651913177639 T^{2} + 1907898195520184901 T^{3} +$$$$36\!\cdots\!69$$$$T^{4} +$$$$26\!\cdots\!68$$$$T^{5} +$$$$34\!\cdots\!48$$$$T^{6} +$$$$24\!\cdots\!24$$$$T^{7} +$$$$24\!\cdots\!78$$$$T^{8} +$$$$17\!\cdots\!10$$$$T^{9} +$$$$14\!\cdots\!57$$$$T^{10} +$$$$92\!\cdots\!71$$$$T^{11} +$$$$66\!\cdots\!35$$$$T^{12} +$$$$40\!\cdots\!15$$$$T^{13} +$$$$25\!\cdots\!19$$$$T^{14} +$$$$14\!\cdots\!88$$$$T^{15} +$$$$82\!\cdots\!01$$$$T^{16} +$$$$41\!\cdots\!15$$$$T^{17} +$$$$22\!\cdots\!65$$$$T^{18} +$$$$99\!\cdots\!51$$$$T^{19} +$$$$49\!\cdots\!43$$$$T^{20} +$$$$19\!\cdots\!10$$$$T^{21} +$$$$91\!\cdots\!02$$$$T^{22} +$$$$29\!\cdots\!64$$$$T^{23} +$$$$13\!\cdots\!12$$$$T^{24} +$$$$32\!\cdots\!68$$$$T^{25} +$$$$14\!\cdots\!51$$$$T^{26} +$$$$24\!\cdots\!41$$$$T^{27} +$$$$10\!\cdots\!21$$$$T^{28} +$$$$91\!\cdots\!35$$$$T^{29} +$$$$42\!\cdots\!99$$$$T^{30}$$
$23$ $$1 - 2504077 T + 19837371032721 T^{2} - 44904598844137902367 T^{3} +$$$$19\!\cdots\!31$$$$T^{4} -$$$$39\!\cdots\!74$$$$T^{5} +$$$$12\!\cdots\!34$$$$T^{6} -$$$$22\!\cdots\!78$$$$T^{7} +$$$$54\!\cdots\!85$$$$T^{8} -$$$$92\!\cdots\!11$$$$T^{9} +$$$$19\!\cdots\!50$$$$T^{10} -$$$$29\!\cdots\!96$$$$T^{11} +$$$$52\!\cdots\!24$$$$T^{12} -$$$$73\!\cdots\!07$$$$T^{13} +$$$$11\!\cdots\!35$$$$T^{14} -$$$$14\!\cdots\!00$$$$T^{15} +$$$$20\!\cdots\!05$$$$T^{16} -$$$$23\!\cdots\!83$$$$T^{17} +$$$$30\!\cdots\!28$$$$T^{18} -$$$$30\!\cdots\!56$$$$T^{19} +$$$$36\!\cdots\!50$$$$T^{20} -$$$$31\!\cdots\!99$$$$T^{21} +$$$$33\!\cdots\!95$$$$T^{22} -$$$$24\!\cdots\!38$$$$T^{23} +$$$$23\!\cdots\!82$$$$T^{24} -$$$$14\!\cdots\!26$$$$T^{25} +$$$$12\!\cdots\!97$$$$T^{26} -$$$$52\!\cdots\!27$$$$T^{27} +$$$$41\!\cdots\!63$$$$T^{28} -$$$$94\!\cdots\!53$$$$T^{29} +$$$$68\!\cdots\!07$$$$T^{30}$$
$29$ $$1 - 18406221 T + 231322256139583 T^{2} -$$$$21\!\cdots\!69$$$$T^{3} +$$$$16\!\cdots\!45$$$$T^{4} -$$$$10\!\cdots\!86$$$$T^{5} +$$$$64\!\cdots\!34$$$$T^{6} -$$$$34\!\cdots\!58$$$$T^{7} +$$$$16\!\cdots\!04$$$$T^{8} -$$$$73\!\cdots\!72$$$$T^{9} +$$$$30\!\cdots\!31$$$$T^{10} -$$$$11\!\cdots\!11$$$$T^{11} +$$$$44\!\cdots\!21$$$$T^{12} -$$$$16\!\cdots\!53$$$$T^{13} +$$$$60\!\cdots\!37$$$$T^{14} -$$$$22\!\cdots\!12$$$$T^{15} +$$$$87\!\cdots\!53$$$$T^{16} -$$$$34\!\cdots\!33$$$$T^{17} +$$$$13\!\cdots\!89$$$$T^{18} -$$$$52\!\cdots\!31$$$$T^{19} +$$$$19\!\cdots\!19$$$$T^{20} -$$$$68\!\cdots\!32$$$$T^{21} +$$$$22\!\cdots\!56$$$$T^{22} -$$$$66\!\cdots\!78$$$$T^{23} +$$$$18\!\cdots\!86$$$$T^{24} -$$$$45\!\cdots\!86$$$$T^{25} +$$$$99\!\cdots\!05$$$$T^{26} -$$$$18\!\cdots\!09$$$$T^{27} +$$$$29\!\cdots\!47$$$$T^{28} -$$$$33\!\cdots\!41$$$$T^{29} +$$$$26\!\cdots\!49$$$$T^{30}$$
$31$ $$1 + 12033699 T + 157221027222111 T^{2} +$$$$11\!\cdots\!53$$$$T^{3} +$$$$11\!\cdots\!51$$$$T^{4} +$$$$78\!\cdots\!70$$$$T^{5} +$$$$65\!\cdots\!70$$$$T^{6} +$$$$42\!\cdots\!02$$$$T^{7} +$$$$29\!\cdots\!41$$$$T^{8} +$$$$18\!\cdots\!53$$$$T^{9} +$$$$11\!\cdots\!80$$$$T^{10} +$$$$68\!\cdots\!12$$$$T^{11} +$$$$39\!\cdots\!74$$$$T^{12} +$$$$21\!\cdots\!69$$$$T^{13} +$$$$11\!\cdots\!43$$$$T^{14} +$$$$62\!\cdots\!76$$$$T^{15} +$$$$31\!\cdots\!53$$$$T^{16} +$$$$15\!\cdots\!29$$$$T^{17} +$$$$73\!\cdots\!14$$$$T^{18} +$$$$33\!\cdots\!72$$$$T^{19} +$$$$15\!\cdots\!80$$$$T^{20} +$$$$61\!\cdots\!13$$$$T^{21} +$$$$26\!\cdots\!31$$$$T^{22} +$$$$10\!\cdots\!22$$$$T^{23} +$$$$41\!\cdots\!70$$$$T^{24} +$$$$13\!\cdots\!70$$$$T^{25} +$$$$48\!\cdots\!21$$$$T^{26} +$$$$13\!\cdots\!73$$$$T^{27} +$$$$48\!\cdots\!21$$$$T^{28} +$$$$98\!\cdots\!19$$$$T^{29} +$$$$21\!\cdots\!51$$$$T^{30}$$
$37$ $$1 + 8722847 T + 568554278402447 T^{2} +$$$$60\!\cdots\!91$$$$T^{3} +$$$$19\!\cdots\!13$$$$T^{4} +$$$$18\!\cdots\!10$$$$T^{5} +$$$$47\!\cdots\!42$$$$T^{6} +$$$$39\!\cdots\!30$$$$T^{7} +$$$$80\!\cdots\!88$$$$T^{8} +$$$$52\!\cdots\!84$$$$T^{9} +$$$$99\!\cdots\!43$$$$T^{10} +$$$$39\!\cdots\!97$$$$T^{11} +$$$$88\!\cdots\!13$$$$T^{12} -$$$$31\!\cdots\!73$$$$T^{13} +$$$$63\!\cdots\!65$$$$T^{14} -$$$$36\!\cdots\!60$$$$T^{15} +$$$$82\!\cdots\!05$$$$T^{16} -$$$$52\!\cdots\!17$$$$T^{17} +$$$$19\!\cdots\!29$$$$T^{18} +$$$$11\!\cdots\!77$$$$T^{19} +$$$$36\!\cdots\!51$$$$T^{20} +$$$$25\!\cdots\!76$$$$T^{21} +$$$$50\!\cdots\!64$$$$T^{22} +$$$$31\!\cdots\!30$$$$T^{23} +$$$$50\!\cdots\!54$$$$T^{24} +$$$$26\!\cdots\!90$$$$T^{25} +$$$$35\!\cdots\!49$$$$T^{26} +$$$$13\!\cdots\!11$$$$T^{27} +$$$$17\!\cdots\!99$$$$T^{28} +$$$$34\!\cdots\!23$$$$T^{29} +$$$$50\!\cdots\!93$$$$T^{30}$$
$41$ $$1 - 18689389 T + 2776615386417537 T^{2} -$$$$46\!\cdots\!33$$$$T^{3} +$$$$37\!\cdots\!99$$$$T^{4} -$$$$58\!\cdots\!96$$$$T^{5} +$$$$33\!\cdots\!84$$$$T^{6} -$$$$48\!\cdots\!36$$$$T^{7} +$$$$21\!\cdots\!31$$$$T^{8} -$$$$30\!\cdots\!37$$$$T^{9} +$$$$11\!\cdots\!84$$$$T^{10} -$$$$15\!\cdots\!00$$$$T^{11} +$$$$48\!\cdots\!18$$$$T^{12} -$$$$63\!\cdots\!65$$$$T^{13} +$$$$18\!\cdots\!43$$$$T^{14} -$$$$22\!\cdots\!26$$$$T^{15} +$$$$59\!\cdots\!23$$$$T^{16} -$$$$68\!\cdots\!65$$$$T^{17} +$$$$17\!\cdots\!58$$$$T^{18} -$$$$17\!\cdots\!00$$$$T^{19} +$$$$42\!\cdots\!84$$$$T^{20} -$$$$37\!\cdots\!57$$$$T^{21} +$$$$87\!\cdots\!51$$$$T^{22} -$$$$63\!\cdots\!16$$$$T^{23} +$$$$14\!\cdots\!44$$$$T^{24} -$$$$82\!\cdots\!96$$$$T^{25} +$$$$17\!\cdots\!39$$$$T^{26} -$$$$71\!\cdots\!93$$$$T^{27} +$$$$13\!\cdots\!97$$$$T^{28} -$$$$30\!\cdots\!49$$$$T^{29} +$$$$53\!\cdots\!01$$$$T^{30}$$
$43$ $$( 1 + 3418801 T )^{15}$$
$47$ $$1 - 104960741 T + 13310159754260471 T^{2} -$$$$95\!\cdots\!91$$$$T^{3} +$$$$75\!\cdots\!21$$$$T^{4} -$$$$43\!\cdots\!16$$$$T^{5} +$$$$26\!\cdots\!08$$$$T^{6} -$$$$13\!\cdots\!76$$$$T^{7} +$$$$66\!\cdots\!90$$$$T^{8} -$$$$28\!\cdots\!38$$$$T^{9} +$$$$12\!\cdots\!89$$$$T^{10} -$$$$49\!\cdots\!97$$$$T^{11} +$$$$20\!\cdots\!91$$$$T^{12} -$$$$70\!\cdots\!13$$$$T^{13} +$$$$26\!\cdots\!11$$$$T^{14} -$$$$85\!\cdots\!24$$$$T^{15} +$$$$29\!\cdots\!37$$$$T^{16} -$$$$88\!\cdots\!57$$$$T^{17} +$$$$28\!\cdots\!33$$$$T^{18} -$$$$77\!\cdots\!37$$$$T^{19} +$$$$22\!\cdots\!23$$$$T^{20} -$$$$56\!\cdots\!22$$$$T^{21} +$$$$14\!\cdots\!70$$$$T^{22} -$$$$31\!\cdots\!16$$$$T^{23} +$$$$73\!\cdots\!76$$$$T^{24} -$$$$13\!\cdots\!84$$$$T^{25} +$$$$26\!\cdots\!43$$$$T^{26} -$$$$36\!\cdots\!51$$$$T^{27} +$$$$57\!\cdots\!77$$$$T^{28} -$$$$50\!\cdots\!89$$$$T^{29} +$$$$54\!\cdots\!43$$$$T^{30}$$
$53$ $$1 - 215907800 T + 45601589020862355 T^{2} -$$$$63\!\cdots\!44$$$$T^{3} +$$$$82\!\cdots\!07$$$$T^{4} -$$$$85\!\cdots\!20$$$$T^{5} +$$$$82\!\cdots\!01$$$$T^{6} -$$$$67\!\cdots\!72$$$$T^{7} +$$$$51\!\cdots\!60$$$$T^{8} -$$$$34\!\cdots\!76$$$$T^{9} +$$$$21\!\cdots\!24$$$$T^{10} -$$$$11\!\cdots\!24$$$$T^{11} +$$$$58\!\cdots\!90$$$$T^{12} -$$$$26\!\cdots\!88$$$$T^{13} +$$$$12\!\cdots\!62$$$$T^{14} -$$$$61\!\cdots\!84$$$$T^{15} +$$$$40\!\cdots\!46$$$$T^{16} -$$$$28\!\cdots\!32$$$$T^{17} +$$$$20\!\cdots\!30$$$$T^{18} -$$$$13\!\cdots\!04$$$$T^{19} +$$$$83\!\cdots\!32$$$$T^{20} -$$$$44\!\cdots\!44$$$$T^{21} +$$$$22\!\cdots\!20$$$$T^{22} -$$$$95\!\cdots\!52$$$$T^{23} +$$$$38\!\cdots\!53$$$$T^{24} -$$$$13\!\cdots\!80$$$$T^{25} +$$$$41\!\cdots\!19$$$$T^{26} -$$$$10\!\cdots\!84$$$$T^{27} +$$$$25\!\cdots\!15$$$$T^{28} -$$$$39\!\cdots\!00$$$$T^{29} +$$$$59\!\cdots\!57$$$$T^{30}$$
$59$ $$1 + 185924544 T + 72641643327616985 T^{2} +$$$$11\!\cdots\!56$$$$T^{3} +$$$$26\!\cdots\!25$$$$T^{4} +$$$$37\!\cdots\!76$$$$T^{5} +$$$$67\!\cdots\!65$$$$T^{6} +$$$$85\!\cdots\!64$$$$T^{7} +$$$$12\!\cdots\!49$$$$T^{8} +$$$$14\!\cdots\!84$$$$T^{9} +$$$$19\!\cdots\!45$$$$T^{10} +$$$$20\!\cdots\!16$$$$T^{11} +$$$$24\!\cdots\!49$$$$T^{12} +$$$$23\!\cdots\!92$$$$T^{13} +$$$$25\!\cdots\!69$$$$T^{14} +$$$$22\!\cdots\!84$$$$T^{15} +$$$$21\!\cdots\!91$$$$T^{16} +$$$$17\!\cdots\!32$$$$T^{17} +$$$$15\!\cdots\!31$$$$T^{18} +$$$$11\!\cdots\!56$$$$T^{19} +$$$$95\!\cdots\!55$$$$T^{20} +$$$$63\!\cdots\!24$$$$T^{21} +$$$$47\!\cdots\!71$$$$T^{22} +$$$$27\!\cdots\!84$$$$T^{23} +$$$$18\!\cdots\!35$$$$T^{24} +$$$$89\!\cdots\!76$$$$T^{25} +$$$$55\!\cdots\!75$$$$T^{26} +$$$$20\!\cdots\!76$$$$T^{27} +$$$$11\!\cdots\!15$$$$T^{28} +$$$$24\!\cdots\!04$$$$T^{29} +$$$$11\!\cdots\!99$$$$T^{30}$$
$61$ $$1 - 247538102 T + 123939924002267915 T^{2} -$$$$25\!\cdots\!76$$$$T^{3} +$$$$72\!\cdots\!53$$$$T^{4} -$$$$12\!\cdots\!30$$$$T^{5} +$$$$27\!\cdots\!23$$$$T^{6} -$$$$42\!\cdots\!04$$$$T^{7} +$$$$73\!\cdots\!05$$$$T^{8} -$$$$10\!\cdots\!70$$$$T^{9} +$$$$15\!\cdots\!63$$$$T^{10} -$$$$19\!\cdots\!64$$$$T^{11} +$$$$26\!\cdots\!33$$$$T^{12} -$$$$30\!\cdots\!38$$$$T^{13} +$$$$37\!\cdots\!95$$$$T^{14} -$$$$39\!\cdots\!08$$$$T^{15} +$$$$43\!\cdots\!95$$$$T^{16} -$$$$41\!\cdots\!78$$$$T^{17} +$$$$42\!\cdots\!93$$$$T^{18} -$$$$36\!\cdots\!04$$$$T^{19} +$$$$34\!\cdots\!63$$$$T^{20} -$$$$26\!\cdots\!70$$$$T^{21} +$$$$22\!\cdots\!05$$$$T^{22} -$$$$14\!\cdots\!84$$$$T^{23} +$$$$11\!\cdots\!03$$$$T^{24} -$$$$60\!\cdots\!30$$$$T^{25} +$$$$40\!\cdots\!73$$$$T^{26} -$$$$16\!\cdots\!56$$$$T^{27} +$$$$94\!\cdots\!15$$$$T^{28} -$$$$22\!\cdots\!22$$$$T^{29} +$$$$10\!\cdots\!01$$$$T^{30}$$
$67$ $$1 - 467904656 T + 289992155760016797 T^{2} -$$$$96\!\cdots\!32$$$$T^{3} +$$$$36\!\cdots\!55$$$$T^{4} -$$$$99\!\cdots\!68$$$$T^{5} +$$$$29\!\cdots\!91$$$$T^{6} -$$$$69\!\cdots\!20$$$$T^{7} +$$$$17\!\cdots\!04$$$$T^{8} -$$$$36\!\cdots\!76$$$$T^{9} +$$$$81\!\cdots\!48$$$$T^{10} -$$$$15\!\cdots\!04$$$$T^{11} +$$$$31\!\cdots\!82$$$$T^{12} -$$$$54\!\cdots\!16$$$$T^{13} +$$$$10\!\cdots\!46$$$$T^{14} -$$$$16\!\cdots\!44$$$$T^{15} +$$$$27\!\cdots\!62$$$$T^{16} -$$$$40\!\cdots\!44$$$$T^{17} +$$$$63\!\cdots\!86$$$$T^{18} -$$$$85\!\cdots\!24$$$$T^{19} +$$$$12\!\cdots\!36$$$$T^{20} -$$$$14\!\cdots\!04$$$$T^{21} +$$$$19\!\cdots\!52$$$$T^{22} -$$$$20\!\cdots\!20$$$$T^{23} +$$$$24\!\cdots\!97$$$$T^{24} -$$$$22\!\cdots\!32$$$$T^{25} +$$$$22\!\cdots\!65$$$$T^{26} -$$$$15\!\cdots\!12$$$$T^{27} +$$$$12\!\cdots\!19$$$$T^{28} -$$$$56\!\cdots\!64$$$$T^{29} +$$$$33\!\cdots\!43$$$$T^{30}$$
$71$ $$1 - 8252944 T + 295175699032282861 T^{2} -$$$$11\!\cdots\!80$$$$T^{3} +$$$$45\!\cdots\!81$$$$T^{4} -$$$$31\!\cdots\!80$$$$T^{5} +$$$$48\!\cdots\!57$$$$T^{6} -$$$$47\!\cdots\!60$$$$T^{7} +$$$$40\!\cdots\!77$$$$T^{8} -$$$$47\!\cdots\!80$$$$T^{9} +$$$$28\!\cdots\!97$$$$T^{10} -$$$$35\!\cdots\!60$$$$T^{11} +$$$$17\!\cdots\!41$$$$T^{12} -$$$$20\!\cdots\!76$$$$T^{13} +$$$$94\!\cdots\!13$$$$T^{14} -$$$$10\!\cdots\!84$$$$T^{15} +$$$$43\!\cdots\!03$$$$T^{16} -$$$$43\!\cdots\!36$$$$T^{17} +$$$$17\!\cdots\!31$$$$T^{18} -$$$$15\!\cdots\!60$$$$T^{19} +$$$$58\!\cdots\!47$$$$T^{20} -$$$$43\!\cdots\!80$$$$T^{21} +$$$$17\!\cdots\!47$$$$T^{22} -$$$$92\!\cdots\!60$$$$T^{23} +$$$$43\!\cdots\!47$$$$T^{24} -$$$$13\!\cdots\!80$$$$T^{25} +$$$$85\!\cdots\!11$$$$T^{26} -$$$$98\!\cdots\!80$$$$T^{27} +$$$$11\!\cdots\!51$$$$T^{28} -$$$$14\!\cdots\!24$$$$T^{29} +$$$$83\!\cdots\!51$$$$T^{30}$$
$73$ $$1 + 715627902 T + 725896042339854143 T^{2} +$$$$39\!\cdots\!04$$$$T^{3} +$$$$23\!\cdots\!09$$$$T^{4} +$$$$10\!\cdots\!42$$$$T^{5} +$$$$48\!\cdots\!43$$$$T^{6} +$$$$18\!\cdots\!64$$$$T^{7} +$$$$69\!\cdots\!09$$$$T^{8} +$$$$23\!\cdots\!46$$$$T^{9} +$$$$77\!\cdots\!47$$$$T^{10} +$$$$23\!\cdots\!12$$$$T^{11} +$$$$68\!\cdots\!33$$$$T^{12} +$$$$18\!\cdots\!86$$$$T^{13} +$$$$49\!\cdots\!39$$$$T^{14} +$$$$11\!\cdots\!52$$$$T^{15} +$$$$28\!\cdots\!07$$$$T^{16} +$$$$63\!\cdots\!34$$$$T^{17} +$$$$13\!\cdots\!01$$$$T^{18} +$$$$27\!\cdots\!32$$$$T^{19} +$$$$54\!\cdots\!71$$$$T^{20} +$$$$96\!\cdots\!14$$$$T^{21} +$$$$17\!\cdots\!53$$$$T^{22} +$$$$26\!\cdots\!44$$$$T^{23} +$$$$40\!\cdots\!39$$$$T^{24} +$$$$52\!\cdots\!58$$$$T^{25} +$$$$69\!\cdots\!33$$$$T^{26} +$$$$68\!\cdots\!24$$$$T^{27} +$$$$74\!\cdots\!79$$$$T^{28} +$$$$42\!\cdots\!78$$$$T^{29} +$$$$35\!\cdots\!57$$$$T^{30}$$
$79$ $$1 - 560681783 T + 849133708768547829 T^{2} -$$$$32\!\cdots\!85$$$$T^{3} +$$$$33\!\cdots\!49$$$$T^{4} -$$$$10\!\cdots\!12$$$$T^{5} +$$$$90\!\cdots\!32$$$$T^{6} -$$$$23\!\cdots\!28$$$$T^{7} +$$$$19\!\cdots\!98$$$$T^{8} -$$$$43\!\cdots\!74$$$$T^{9} +$$$$34\!\cdots\!01$$$$T^{10} -$$$$67\!\cdots\!51$$$$T^{11} +$$$$53\!\cdots\!45$$$$T^{12} -$$$$93\!\cdots\!63$$$$T^{13} +$$$$72\!\cdots\!51$$$$T^{14} -$$$$11\!\cdots\!88$$$$T^{15} +$$$$86\!\cdots\!69$$$$T^{16} -$$$$13\!\cdots\!43$$$$T^{17} +$$$$91\!\cdots\!55$$$$T^{18} -$$$$13\!\cdots\!71$$$$T^{19} +$$$$85\!\cdots\!99$$$$T^{20} -$$$$12\!\cdots\!94$$$$T^{21} +$$$$69\!\cdots\!22$$$$T^{22} -$$$$10\!\cdots\!48$$$$T^{23} +$$$$46\!\cdots\!28$$$$T^{24} -$$$$62\!\cdots\!12$$$$T^{25} +$$$$24\!\cdots\!31$$$$T^{26} -$$$$28\!\cdots\!85$$$$T^{27} +$$$$89\!\cdots\!11$$$$T^{28} -$$$$70\!\cdots\!43$$$$T^{29} +$$$$15\!\cdots\!99$$$$T^{30}$$
$83$ $$1 - 1442854698 T + 2517006525957293097 T^{2} -$$$$25\!\cdots\!08$$$$T^{3} +$$$$27\!\cdots\!99$$$$T^{4} -$$$$22\!\cdots\!86$$$$T^{5} +$$$$19\!\cdots\!63$$$$T^{6} -$$$$13\!\cdots\!56$$$$T^{7} +$$$$92\!\cdots\!28$$$$T^{8} -$$$$56\!\cdots\!76$$$$T^{9} +$$$$33\!\cdots\!68$$$$T^{10} -$$$$18\!\cdots\!28$$$$T^{11} +$$$$97\!\cdots\!54$$$$T^{12} -$$$$47\!\cdots\!68$$$$T^{13} +$$$$22\!\cdots\!94$$$$T^{14} -$$$$98\!\cdots\!20$$$$T^{15} +$$$$42\!\cdots\!82$$$$T^{16} -$$$$16\!\cdots\!12$$$$T^{17} +$$$$64\!\cdots\!58$$$$T^{18} -$$$$22\!\cdots\!68$$$$T^{19} +$$$$77\!\cdots\!24$$$$T^{20} -$$$$24\!\cdots\!04$$$$T^{21} +$$$$73\!\cdots\!36$$$$T^{22} -$$$$19\!\cdots\!16$$$$T^{23} +$$$$53\!\cdots\!29$$$$T^{24} -$$$$11\!\cdots\!14$$$$T^{25} +$$$$27\!\cdots\!53$$$$T^{26} -$$$$47\!\cdots\!28$$$$T^{27} +$$$$85\!\cdots\!31$$$$T^{28} -$$$$91\!\cdots\!62$$$$T^{29} +$$$$11\!\cdots\!07$$$$T^{30}$$
$89$ $$1 - 396710008 T + 2452763342866776295 T^{2} -$$$$86\!\cdots\!68$$$$T^{3} +$$$$29\!\cdots\!85$$$$T^{4} -$$$$87\!\cdots\!92$$$$T^{5} +$$$$22\!\cdots\!31$$$$T^{6} -$$$$52\!\cdots\!68$$$$T^{7} +$$$$12\!\cdots\!05$$$$T^{8} -$$$$17\!\cdots\!32$$$$T^{9} +$$$$51\!\cdots\!71$$$$T^{10} -$$$$14\!\cdots\!24$$$$T^{11} +$$$$17\!\cdots\!37$$$$T^{12} +$$$$18\!\cdots\!80$$$$T^{13} +$$$$54\!\cdots\!11$$$$T^{14} +$$$$10\!\cdots\!56$$$$T^{15} +$$$$19\!\cdots\!99$$$$T^{16} +$$$$23\!\cdots\!80$$$$T^{17} +$$$$74\!\cdots\!73$$$$T^{18} -$$$$21\!\cdots\!64$$$$T^{19} +$$$$27\!\cdots\!79$$$$T^{20} -$$$$32\!\cdots\!12$$$$T^{21} +$$$$79\!\cdots\!45$$$$T^{22} -$$$$11\!\cdots\!28$$$$T^{23} +$$$$17\!\cdots\!59$$$$T^{24} -$$$$24\!\cdots\!92$$$$T^{25} +$$$$28\!\cdots\!65$$$$T^{26} -$$$$29\!\cdots\!08$$$$T^{27} +$$$$29\!\cdots\!55$$$$T^{28} -$$$$16\!\cdots\!88$$$$T^{29} +$$$$14\!\cdots\!49$$$$T^{30}$$
$97$ $$1 + 3063837815 T + 9536233785462653481 T^{2} +$$$$19\!\cdots\!71$$$$T^{3} +$$$$38\!\cdots\!99$$$$T^{4} +$$$$61\!\cdots\!48$$$$T^{5} +$$$$97\!\cdots\!08$$$$T^{6} +$$$$13\!\cdots\!04$$$$T^{7} +$$$$18\!\cdots\!19$$$$T^{8} +$$$$21\!\cdots\!95$$$$T^{9} +$$$$25\!\cdots\!96$$$$T^{10} +$$$$27\!\cdots\!24$$$$T^{11} +$$$$29\!\cdots\!54$$$$T^{12} +$$$$28\!\cdots\!39$$$$T^{13} +$$$$26\!\cdots\!31$$$$T^{14} +$$$$23\!\cdots\!02$$$$T^{15} +$$$$20\!\cdots\!27$$$$T^{16} +$$$$16\!\cdots\!71$$$$T^{17} +$$$$12\!\cdots\!02$$$$T^{18} +$$$$92\!\cdots\!04$$$$T^{19} +$$$$65\!\cdots\!72$$$$T^{20} +$$$$41\!\cdots\!55$$$$T^{21} +$$$$26\!\cdots\!87$$$$T^{22} +$$$$14\!\cdots\!64$$$$T^{23} +$$$$82\!\cdots\!76$$$$T^{24} +$$$$39\!\cdots\!52$$$$T^{25} +$$$$18\!\cdots\!67$$$$T^{26} +$$$$71\!\cdots\!31$$$$T^{27} +$$$$27\!\cdots\!97$$$$T^{28} +$$$$65\!\cdots\!35$$$$T^{29} +$$$$16\!\cdots\!93$$$$T^{30}$$