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

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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:

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