Properties

 Label 1152.3.m.f Level 1152 Weight 3 Character orbit 1152.m Analytic conductor 31.390 Analytic rank 0 Dimension 16 CM no Inner twists 2

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1152 = 2^{7} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1152.m (of order $$4$$, degree $$2$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$31.3897264543$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 6 x^{14} - 4 x^{13} + 10 x^{12} + 56 x^{11} + 88 x^{10} - 128 x^{9} - 496 x^{8} - 512 x^{7} + 1408 x^{6} + 3584 x^{5} + 2560 x^{4} - 4096 x^{3} - 24576 x^{2} + 65536$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$2^{28}$$ Twist minimal: no (minimal twist has level 48) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta_{11} q^{5} + \beta_{3} q^{7} +O(q^{10})$$ $$q + \beta_{11} q^{5} + \beta_{3} q^{7} + ( 2 + 2 \beta_{2} - \beta_{5} - \beta_{10} ) q^{11} + ( \beta_{3} - \beta_{5} + \beta_{6} - \beta_{8} + \beta_{10} - \beta_{14} - \beta_{15} ) q^{13} + ( \beta_{1} + \beta_{3} - \beta_{6} - \beta_{7} - \beta_{9} ) q^{17} + ( 2 - 2 \beta_{2} - \beta_{6} + \beta_{9} - \beta_{12} + 2 \beta_{13} + \beta_{15} ) q^{19} + ( 8 - \beta_{1} - 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{13} + 2 \beta_{14} ) q^{23} + ( -5 \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{25} + ( 2 + 2 \beta_{2} - \beta_{3} + 3 \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} + 2 \beta_{14} + \beta_{15} ) q^{29} + ( 8 \beta_{2} + 2 \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} - \beta_{12} + 2 \beta_{15} ) q^{31} + ( 6 - 6 \beta_{2} - 2 \beta_{3} - 2 \beta_{6} - 2 \beta_{8} - 2 \beta_{9} + \beta_{11} - \beta_{12} - 2 \beta_{15} ) q^{35} + ( 6 - 6 \beta_{2} - \beta_{3} - \beta_{8} + 2 \beta_{9} - \beta_{11} - 2 \beta_{12} - \beta_{13} - \beta_{15} ) q^{37} + ( -3 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} + 3 \beta_{8} - \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{41} + ( -10 - 10 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{10} - 2 \beta_{14} + \beta_{15} ) q^{43} + ( 24 \beta_{2} + 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{47} + ( 7 - 4 \beta_{1} + 4 \beta_{3} - 5 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} - 5 \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{49} + ( -10 + 2 \beta_{1} + 10 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + \beta_{6} + 3 \beta_{8} + 2 \beta_{9} + \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - \beta_{15} ) q^{53} + ( -16 - 2 \beta_{1} - 2 \beta_{3} + 6 \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} + 6 \beta_{11} - \beta_{12} ) q^{55} + ( -8 - 8 \beta_{2} - 2 \beta_{3} - 4 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{10} + 2 \beta_{15} ) q^{59} + ( 2 - 4 \beta_{1} + 2 \beta_{2} + \beta_{3} + 4 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{8} + 5 \beta_{10} + \beta_{14} - \beta_{15} ) q^{61} + ( 2 - 5 \beta_{1} - 3 \beta_{3} - 2 \beta_{5} - 3 \beta_{6} + \beta_{7} + \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{13} - 2 \beta_{14} ) q^{65} + ( -20 + \beta_{1} + 20 \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{6} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{11} - 4 \beta_{12} - 2 \beta_{15} ) q^{67} + ( -32 + 2 \beta_{3} ) q^{71} + ( 6 \beta_{2} + 4 \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} + 3 \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} - 4 \beta_{15} ) q^{73} + ( 14 - 6 \beta_{1} + 14 \beta_{2} + 5 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} - 5 \beta_{8} + 2 \beta_{10} - \beta_{15} ) q^{77} + ( -8 \beta_{2} + 6 \beta_{4} - 6 \beta_{5} - 5 \beta_{8} + 2 \beta_{10} + 6 \beta_{11} - 2 \beta_{12} ) q^{79} + ( -10 + 10 \beta_{2} - 2 \beta_{6} - 2 \beta_{9} + 3 \beta_{11} - 3 \beta_{12} + 4 \beta_{13} + 2 \beta_{15} ) q^{83} + ( -10 - 8 \beta_{1} + 10 \beta_{2} + 4 \beta_{3} - 8 \beta_{4} - 2 \beta_{6} + 4 \beta_{8} - 2 \beta_{9} + 8 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - 4 \beta_{15} ) q^{85} + ( -10 \beta_{2} + 6 \beta_{4} - 8 \beta_{8} + 2 \beta_{10} - 2 \beta_{12} - 4 \beta_{15} ) q^{89} + ( 30 - 3 \beta_{1} + 30 \beta_{2} + 3 \beta_{4} - 8 \beta_{5} - 5 \beta_{6} + 3 \beta_{7} - 5 \beta_{10} + 2 \beta_{14} + 5 \beta_{15} ) q^{91} + ( -40 \beta_{2} + 5 \beta_{4} + 6 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{9} - 6 \beta_{11} - 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} + 6 \beta_{15} ) q^{95} + ( 6 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - 2 \beta_{9} + 6 \beta_{10} + 2 \beta_{11} + 4 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q + 32q^{11} + 32q^{19} + 128q^{23} + 32q^{29} + 96q^{35} + 96q^{37} - 160q^{43} + 112q^{49} - 160q^{53} - 256q^{55} - 128q^{59} + 32q^{61} + 32q^{65} - 320q^{67} - 512q^{71} + 224q^{77} - 160q^{83} - 160q^{85} + 480q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 6 x^{14} - 4 x^{13} + 10 x^{12} + 56 x^{11} + 88 x^{10} - 128 x^{9} - 496 x^{8} - 512 x^{7} + 1408 x^{6} + 3584 x^{5} + 2560 x^{4} - 4096 x^{3} - 24576 x^{2} + 65536$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$5 \nu^{15} - 18 \nu^{14} - 134 \nu^{13} - 168 \nu^{12} + 170 \nu^{11} + 1156 \nu^{10} + 1848 \nu^{9} - 2928 \nu^{8} - 15600 \nu^{7} - 25632 \nu^{6} + 1792 \nu^{5} + 73472 \nu^{4} + 112128 \nu^{3} - 23552 \nu^{2} - 528384 \nu - 753664$$$$)/12288$$ $$\beta_{2}$$ $$=$$ $$($$$$-347 \nu^{15} - 626 \nu^{14} + 1234 \nu^{13} + 4536 \nu^{12} + 5530 \nu^{11} - 11868 \nu^{10} - 59096 \nu^{9} - 66544 \nu^{8} + 88528 \nu^{7} + 450272 \nu^{6} + 454272 \nu^{5} - 499456 \nu^{4} - 2271744 \nu^{3} - 3177472 \nu^{2} + 3719168 \nu + 10231808$$$$)/368640$$ $$\beta_{3}$$ $$=$$ $$($$$$47 \nu^{15} + 110 \nu^{14} - 90 \nu^{13} - 528 \nu^{12} - 610 \nu^{11} + 1684 \nu^{10} + 8376 \nu^{9} + 11728 \nu^{8} - 5136 \nu^{7} - 52256 \nu^{6} - 60800 \nu^{5} + 73472 \nu^{4} + 350720 \nu^{3} + 537600 \nu^{2} - 172032 \nu - 1228800$$$$)/40960$$ $$\beta_{4}$$ $$=$$ $$($$$$-53 \nu^{15} - 89 \nu^{14} + 226 \nu^{13} + 714 \nu^{12} + 730 \nu^{11} - 2082 \nu^{10} - 9224 \nu^{9} - 8056 \nu^{8} + 19792 \nu^{7} + 75248 \nu^{6} + 67008 \nu^{5} - 110464 \nu^{4} - 377856 \nu^{3} - 429568 \nu^{2} + 837632 \nu + 1974272$$$$)/30720$$ $$\beta_{5}$$ $$=$$ $$($$$$-751 \nu^{15} - 448 \nu^{14} + 6626 \nu^{13} + 14076 \nu^{12} + 5306 \nu^{11} - 54888 \nu^{10} - 156376 \nu^{9} - 19520 \nu^{8} + 641360 \nu^{7} + 1598464 \nu^{6} + 829440 \nu^{5} - 3026432 \nu^{4} - 7073280 \nu^{3} - 4517888 \nu^{2} + 22466560 \nu + 41107456$$$$)/368640$$ $$\beta_{6}$$ $$=$$ $$($$$$119 \nu^{15} + 1230 \nu^{14} + 3430 \nu^{13} + 2064 \nu^{12} - 9010 \nu^{11} - 25772 \nu^{10} - 2088 \nu^{9} + 157776 \nu^{8} + 431088 \nu^{7} + 442848 \nu^{6} - 394880 \nu^{5} - 1689856 \nu^{4} - 1328640 \nu^{3} + 4019200 \nu^{2} + 13664256 \nu + 14581760$$$$)/122880$$ $$\beta_{7}$$ $$=$$ $$($$$$197 \nu^{15} - 244 \nu^{14} - 2572 \nu^{13} - 3852 \nu^{12} + 998 \nu^{11} + 22056 \nu^{10} + 40172 \nu^{9} - 35600 \nu^{8} - 275680 \nu^{7} - 517568 \nu^{6} - 54240 \nu^{5} + 1233664 \nu^{4} + 2280960 \nu^{3} - 88064 \nu^{2} - 9620480 \nu - 12972032$$$$)/92160$$ $$\beta_{8}$$ $$=$$ $$($$$$1063 \nu^{15} + 2242 \nu^{14} - 2666 \nu^{13} - 14184 \nu^{12} - 21122 \nu^{11} + 26652 \nu^{10} + 189400 \nu^{9} + 278960 \nu^{8} - 106640 \nu^{7} - 1269472 \nu^{6} - 1748352 \nu^{5} + 775424 \nu^{4} + 6971904 \nu^{3} + 12336128 \nu^{2} - 4268032 \nu - 26214400$$$$)/368640$$ $$\beta_{9}$$ $$=$$ $$($$$$-995 \nu^{15} - 3884 \nu^{14} - 4022 \nu^{13} + 5076 \nu^{12} + 30658 \nu^{11} + 28992 \nu^{10} - 122744 \nu^{9} - 468448 \nu^{8} - 716144 \nu^{7} + 66368 \nu^{6} + 1824000 \nu^{5} + 2557952 \nu^{4} - 2188800 \nu^{3} - 16144384 \nu^{2} - 20942848 \nu - 8568832$$$$)/368640$$ $$\beta_{10}$$ $$=$$ $$($$$$134 \nu^{15} + 20 \nu^{14} - 1153 \nu^{13} - 2232 \nu^{12} - 622 \nu^{11} + 9756 \nu^{10} + 25118 \nu^{9} - 1448 \nu^{8} - 113704 \nu^{7} - 257408 \nu^{6} - 102288 \nu^{5} + 502912 \nu^{4} + 1184256 \nu^{3} + 538624 \nu^{2} - 3969536 \nu - 6232064$$$$)/46080$$ $$\beta_{11}$$ $$=$$ $$($$$$545 \nu^{15} + 1574 \nu^{14} + 302 \nu^{13} - 5256 \nu^{12} - 12838 \nu^{11} + 1188 \nu^{10} + 82544 \nu^{9} + 190768 \nu^{8} + 127664 \nu^{7} - 372128 \nu^{6} - 897600 \nu^{5} - 303872 \nu^{4} + 2511360 \nu^{3} + 7066624 \nu^{2} + 3770368 \nu - 6053888$$$$)/184320$$ $$\beta_{12}$$ $$=$$ $$($$$$-1417 \nu^{15} - 4300 \nu^{14} - 1186 \nu^{13} + 13356 \nu^{12} + 35366 \nu^{11} - 528 \nu^{10} - 219304 \nu^{9} - 508256 \nu^{8} - 401488 \nu^{7} + 933184 \nu^{6} + 2421504 \nu^{5} + 922624 \nu^{4} - 6455808 \nu^{3} - 18839552 \nu^{2} - 11743232 \nu + 13975552$$$$)/368640$$ $$\beta_{13}$$ $$=$$ $$($$$$265 \nu^{15} + 526 \nu^{14} - 578 \nu^{13} - 2856 \nu^{12} - 4406 \nu^{11} + 6228 \nu^{10} + 41008 \nu^{9} + 59984 \nu^{8} - 23888 \nu^{7} - 271264 \nu^{6} - 338496 \nu^{5} + 217088 \nu^{4} + 1423872 \nu^{3} + 2561024 \nu^{2} - 1472512 \nu - 5570560$$$$)/61440$$ $$\beta_{14}$$ $$=$$ $$($$$$1181 \nu^{15} + 2693 \nu^{14} - 2326 \nu^{13} - 13986 \nu^{12} - 22786 \nu^{11} + 22218 \nu^{10} + 190016 \nu^{9} + 302680 \nu^{8} - 52720 \nu^{7} - 1210544 \nu^{6} - 1754880 \nu^{5} + 602752 \nu^{4} + 6497280 \nu^{3} + 12133888 \nu^{2} - 3891200 \nu - 25296896$$$$)/184320$$ $$\beta_{15}$$ $$=$$ $$($$$$-2519 \nu^{15} - 4382 \nu^{14} + 9178 \nu^{13} + 33552 \nu^{12} + 37810 \nu^{11} - 85236 \nu^{10} - 427352 \nu^{9} - 462928 \nu^{8} + 698896 \nu^{7} + 3347744 \nu^{6} + 3329664 \nu^{5} - 3795712 \nu^{4} - 16501248 \nu^{3} - 21873664 \nu^{2} + 28983296 \nu + 80101376$$$$)/368640$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-2 \beta_{12} + 2 \beta_{11} - 2 \beta_{8} - 2 \beta_{6} + \beta_{4} + 4 \beta_{2} - \beta_{1}$$$$)/16$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{14} + \beta_{12} + 2 \beta_{11} + \beta_{10} - \beta_{9} - \beta_{7} - 2 \beta_{5} - \beta_{4} - 2 \beta_{2} - \beta_{1} + 6$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{15} - 4 \beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} - 2 \beta_{9} + \beta_{8} + 2 \beta_{7} + \beta_{6} + \beta_{5} - 3 \beta_{4} + \beta_{3} + 10 \beta_{2} + 6$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} + 3 \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - 7 \beta_{5} - 4 \beta_{4} - \beta_{3} + 22 \beta_{2} - 2 \beta_{1} + 8$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-8 \beta_{15} - 8 \beta_{14} - 16 \beta_{13} + 2 \beta_{12} + 18 \beta_{11} + 8 \beta_{10} - 4 \beta_{9} - 14 \beta_{8} + 4 \beta_{7} + 14 \beta_{6} - 12 \beta_{5} + 23 \beta_{4} - 12 \beta_{3} - 28 \beta_{2} + 17 \beta_{1} - 80$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$($$$$12 \beta_{15} + 4 \beta_{14} + 2 \beta_{13} + 9 \beta_{12} + 2 \beta_{11} + 9 \beta_{10} + 5 \beta_{9} + 8 \beta_{8} - 5 \beta_{7} + 6 \beta_{6} - 2 \beta_{5} - 21 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} - 3 \beta_{1} - 102$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$-15 \beta_{15} - 4 \beta_{14} - 8 \beta_{13} - 3 \beta_{12} - 47 \beta_{11} - \beta_{10} - 10 \beta_{9} + 13 \beta_{8} + 14 \beta_{7} + 29 \beta_{6} - 33 \beta_{5} + 16 \beta_{4} + 11 \beta_{3} + 90 \beta_{2} - 5 \beta_{1} - 182$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$13 \beta_{15} + 21 \beta_{14} + 9 \beta_{13} + 26 \beta_{12} + 23 \beta_{11} - 24 \beta_{10} + 12 \beta_{9} + 9 \beta_{8} + 17 \beta_{7} + 8 \beta_{6} + 7 \beta_{5} + 12 \beta_{4} - 9 \beta_{3} - 16 \beta_{2} + 24 \beta_{1} - 146$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-14 \beta_{15} + 24 \beta_{14} + 32 \beta_{13} - 84 \beta_{12} - 120 \beta_{11} + 66 \beta_{10} + 44 \beta_{9} + 24 \beta_{8} - 60 \beta_{7} + 28 \beta_{6} + 106 \beta_{5} + 81 \beta_{4} - 70 \beta_{3} + 40 \beta_{2} + 189 \beta_{1} - 276$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$88 \beta_{15} + 38 \beta_{14} + 80 \beta_{13} + 5 \beta_{12} - 166 \beta_{11} - 55 \beta_{10} + 59 \beta_{9} + 56 \beta_{8} - 33 \beta_{7} - 48 \beta_{6} + 86 \beta_{5} + 51 \beta_{4} + 128 \beta_{3} - 1018 \beta_{2} - 109 \beta_{1} - 786$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-167 \beta_{15} + 48 \beta_{14} + 36 \beta_{13} + 143 \beta_{12} - 105 \beta_{11} + 27 \beta_{10} - 70 \beta_{9} + 59 \beta_{8} - 74 \beta_{7} + 3 \beta_{6} + 27 \beta_{5} + 161 \beta_{4} + 199 \beta_{3} + 1006 \beta_{2} + 62 \beta_{1} + 1210$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$127 \beta_{15} + 165 \beta_{14} + 227 \beta_{13} - 175 \beta_{12} - 131 \beta_{11} - 207 \beta_{10} - 21 \beta_{9} + 17 \beta_{8} + 30 \beta_{7} - 350 \beta_{6} + 511 \beta_{5} - 84 \beta_{4} - 71 \beta_{3} - 238 \beta_{2} + 174 \beta_{1} + 8$$ $$\nu^{13}$$ $$=$$ $$($$$$-520 \beta_{15} - 904 \beta_{14} + 32 \beta_{13} - 322 \beta_{12} + 174 \beta_{11} + 296 \beta_{10} + 532 \beta_{9} - 18 \beta_{8} - 452 \beta_{7} + 114 \beta_{6} + 348 \beta_{5} - 31 \beta_{4} - 20 \beta_{3} - 2788 \beta_{2} - 281 \beta_{1} + 6448$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$204 \beta_{15} + 484 \beta_{14} - 426 \beta_{13} - 25 \beta_{12} - 210 \beta_{11} - 113 \beta_{10} - 285 \beta_{9} - 888 \beta_{8} - 395 \beta_{7} - 670 \beta_{6} + 458 \beta_{5} + 97 \beta_{4} + 692 \beta_{3} - 4926 \beta_{2} - 153 \beta_{1} + 7174$$ $$\nu^{15}$$ $$=$$ $$-1033 \beta_{15} - 1244 \beta_{14} + 920 \beta_{13} - 565 \beta_{12} + 1631 \beta_{11} + 1233 \beta_{10} - 654 \beta_{9} - 1245 \beta_{8} - 86 \beta_{7} - 1069 \beta_{6} - 695 \beta_{5} - 1836 \beta_{4} + 605 \beta_{3} + 14390 \beta_{2} - 3255 \beta_{1} + 4502$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$641$$ $$901$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-\beta_{2}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
415.1
 −1.87459 + 0.697079i 1.84258 − 0.777752i −0.455024 − 1.94755i −1.25564 − 1.55672i 1.78012 − 0.911682i −1.96679 − 0.362960i 0.125358 + 1.99607i 1.80398 + 0.863518i −1.87459 − 0.697079i 1.84258 + 0.777752i −0.455024 + 1.94755i −1.25564 + 1.55672i 1.78012 + 0.911682i −1.96679 + 0.362960i 0.125358 − 1.99607i 1.80398 − 0.863518i
0 0 0 −5.24354 5.24354i 0 −5.32796 0 0 0
415.2 0 0 0 −4.78830 4.78830i 0 −10.3302 0 0 0
415.3 0 0 0 −3.40572 3.40572i 0 12.1303 0 0 0
415.4 0 0 0 0.909023 + 0.909023i 0 −0.654713 0 0 0
415.5 0 0 0 1.00772 + 1.00772i 0 10.0236 0 0 0
415.6 0 0 0 1.69930 + 1.69930i 0 −5.74280 0 0 0
415.7 0 0 0 3.32679 + 3.32679i 0 −4.04088 0 0 0
415.8 0 0 0 6.49473 + 6.49473i 0 3.94273 0 0 0
991.1 0 0 0 −5.24354 + 5.24354i 0 −5.32796 0 0 0
991.2 0 0 0 −4.78830 + 4.78830i 0 −10.3302 0 0 0
991.3 0 0 0 −3.40572 + 3.40572i 0 12.1303 0 0 0
991.4 0 0 0 0.909023 0.909023i 0 −0.654713 0 0 0
991.5 0 0 0 1.00772 1.00772i 0 10.0236 0 0 0
991.6 0 0 0 1.69930 1.69930i 0 −5.74280 0 0 0
991.7 0 0 0 3.32679 3.32679i 0 −4.04088 0 0 0
991.8 0 0 0 6.49473 6.49473i 0 3.94273 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 991.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.3.m.f 16
3.b odd 2 1 384.3.l.a 16
4.b odd 2 1 1152.3.m.c 16
8.b even 2 1 144.3.m.c 16
8.d odd 2 1 576.3.m.c 16
12.b even 2 1 384.3.l.b 16
16.e even 4 1 576.3.m.c 16
16.e even 4 1 1152.3.m.c 16
16.f odd 4 1 144.3.m.c 16
16.f odd 4 1 inner 1152.3.m.f 16
24.f even 2 1 192.3.l.a 16
24.h odd 2 1 48.3.l.a 16
48.i odd 4 1 192.3.l.a 16
48.i odd 4 1 384.3.l.b 16
48.k even 4 1 48.3.l.a 16
48.k even 4 1 384.3.l.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.3.l.a 16 24.h odd 2 1
48.3.l.a 16 48.k even 4 1
144.3.m.c 16 8.b even 2 1
144.3.m.c 16 16.f odd 4 1
192.3.l.a 16 24.f even 2 1
192.3.l.a 16 48.i odd 4 1
384.3.l.a 16 3.b odd 2 1
384.3.l.a 16 48.k even 4 1
384.3.l.b 16 12.b even 2 1
384.3.l.b 16 48.i odd 4 1
576.3.m.c 16 8.d odd 2 1
576.3.m.c 16 16.e even 4 1
1152.3.m.c 16 4.b odd 2 1
1152.3.m.c 16 16.e even 4 1
1152.3.m.f 16 1.a even 1 1 trivial
1152.3.m.f 16 16.f odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1152, [\chi])$$:

 $$T_{5}^{16} + \cdots$$ $$T_{7}^{8} - 224 T_{7}^{6} - 448 T_{7}^{5} + 13704 T_{7}^{4} + 53248 T_{7}^{3} - 136576 T_{7}^{2} - 720640 T_{7} - 400880$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 + 32 T^{3} - 344 T^{4} + 5664 T^{5} + 512 T^{6} + 145600 T^{7} + 223452 T^{8} - 2255168 T^{9} + 20875776 T^{10} - 67282720 T^{11} + 753060504 T^{12} + 1881828576 T^{13} + 2740220928 T^{14} + 75471830656 T^{15} - 399298967994 T^{16} + 1886795766400 T^{17} + 1712638080000 T^{18} + 29403571500000 T^{19} + 294164259375000 T^{20} - 657057812500000 T^{21} + 5096625000000000 T^{22} - 13764453125000000 T^{23} + 34096069335937500 T^{24} + 555419921875000000 T^{25} + 48828125000000000 T^{26} + 13504028320312500000 T^{27} - 20503997802734375000 T^{28} + 47683715820312500000 T^{29} +$$$$23\!\cdots\!25$$$$T^{32}$$
$7$ $$( 1 + 168 T^{2} - 448 T^{3} + 15076 T^{4} - 56512 T^{5} + 1070392 T^{6} - 3649664 T^{7} + 60103046 T^{8} - 178833536 T^{9} + 2570011192 T^{10} - 6648580288 T^{11} + 86910139876 T^{12} - 126548911552 T^{13} + 2325336249768 T^{14} + 33232930569601 T^{16} )^{2}$$
$11$ $$1 - 32 T + 512 T^{2} - 8480 T^{3} + 137032 T^{4} - 1636576 T^{5} + 18165248 T^{6} - 219655136 T^{7} + 2263228700 T^{8} - 21867108000 T^{9} + 253620152832 T^{10} - 2916953293728 T^{11} + 33797606438392 T^{12} - 431768458252384 T^{13} + 5293166227138048 T^{14} - 61910274521995104 T^{15} + 703855220885889990 T^{16} - 7491143217161407584 T^{17} + 77497246731528160768 T^{18} -$$$$76\!\cdots\!24$$$$T^{19} +$$$$72\!\cdots\!52$$$$T^{20} -$$$$75\!\cdots\!28$$$$T^{21} +$$$$79\!\cdots\!72$$$$T^{22} -$$$$83\!\cdots\!00$$$$T^{23} +$$$$10\!\cdots\!00$$$$T^{24} -$$$$12\!\cdots\!16$$$$T^{25} +$$$$12\!\cdots\!48$$$$T^{26} -$$$$13\!\cdots\!96$$$$T^{27} +$$$$13\!\cdots\!12$$$$T^{28} -$$$$10\!\cdots\!80$$$$T^{29} +$$$$73\!\cdots\!72$$$$T^{30} -$$$$55\!\cdots\!32$$$$T^{31} +$$$$21\!\cdots\!21$$$$T^{32}$$
$13$ $$1 - 3200 T^{3} + 7608 T^{4} - 95360 T^{5} + 5120000 T^{6} - 68335872 T^{7} + 2004669468 T^{8} - 7270355200 T^{9} + 184268595200 T^{10} - 4889456013184 T^{11} + 5354592144136 T^{12} - 669839496880000 T^{13} + 7008632866619392 T^{14} - 70586941744778752 T^{15} + 2398056097119178950 T^{16} - 11929193154867609088 T^{17} +$$$$20\!\cdots\!12$$$$T^{18} -$$$$32\!\cdots\!00$$$$T^{19} +$$$$43\!\cdots\!56$$$$T^{20} -$$$$67\!\cdots\!16$$$$T^{21} +$$$$42\!\cdots\!00$$$$T^{22} -$$$$28\!\cdots\!00$$$$T^{23} +$$$$13\!\cdots\!88$$$$T^{24} -$$$$76\!\cdots\!88$$$$T^{25} +$$$$97\!\cdots\!00$$$$T^{26} -$$$$30\!\cdots\!40$$$$T^{27} +$$$$41\!\cdots\!88$$$$T^{28} -$$$$29\!\cdots\!00$$$$T^{29} +$$$$44\!\cdots\!81$$$$T^{32}$$
$17$ $$( 1 + 968 T^{2} + 2944 T^{3} + 516540 T^{4} + 3209600 T^{5} + 201700088 T^{6} + 1543904000 T^{7} + 63894476806 T^{8} + 446188256000 T^{9} + 16846193049848 T^{10} + 77471941462400 T^{11} + 3603257748574140 T^{12} + 5935086042921856 T^{13} + 563978325638408648 T^{14} + 48661191875666868481 T^{16} )^{2}$$
$19$ $$1 - 32 T + 512 T^{2} + 2656 T^{3} - 523448 T^{4} + 8424608 T^{5} + 1945088 T^{6} - 4454446304 T^{7} + 107916937244 T^{8} + 703649376 T^{9} - 30601835632128 T^{10} + 698985761087712 T^{11} + 998616856187896 T^{12} - 253693358084547040 T^{13} + 3161998119961945600 T^{14} + 30474951661580761248 T^{15} -$$$$19\!\cdots\!42$$$$T^{16} +$$$$11\!\cdots\!28$$$$T^{17} +$$$$41\!\cdots\!00$$$$T^{18} -$$$$11\!\cdots\!40$$$$T^{19} +$$$$16\!\cdots\!36$$$$T^{20} +$$$$42\!\cdots\!12$$$$T^{21} -$$$$67\!\cdots\!08$$$$T^{22} +$$$$56\!\cdots\!96$$$$T^{23} +$$$$31\!\cdots\!64$$$$T^{24} -$$$$46\!\cdots\!64$$$$T^{25} +$$$$73\!\cdots\!88$$$$T^{26} +$$$$11\!\cdots\!88$$$$T^{27} -$$$$25\!\cdots\!08$$$$T^{28} +$$$$46\!\cdots\!36$$$$T^{29} +$$$$32\!\cdots\!92$$$$T^{30} -$$$$73\!\cdots\!32$$$$T^{31} +$$$$83\!\cdots\!61$$$$T^{32}$$
$23$ $$( 1 - 64 T + 3496 T^{2} - 127936 T^{3} + 4410332 T^{4} - 130001728 T^{5} + 3673719192 T^{6} - 94049622208 T^{7} + 2261818535238 T^{8} - 49752250148032 T^{9} + 1028057252408472 T^{10} - 19244921376016192 T^{11} + 345377444336323292 T^{12} - 5299942138629398464 T^{13} + 76613527014343042216 T^{14} -$$$$74\!\cdots\!76$$$$T^{15} +$$$$61\!\cdots\!61$$$$T^{16} )^{2}$$
$29$ $$1 - 32 T + 512 T^{2} + 18368 T^{3} - 1552984 T^{4} + 20596992 T^{5} + 304715776 T^{6} - 25469097376 T^{7} + 491466517980 T^{8} + 9791032230816 T^{9} - 347423504794624 T^{10} + 2649303176415616 T^{11} + 694517140133881240 T^{12} - 20658732330776531008 T^{13} +$$$$19\!\cdots\!28$$$$T^{14} +$$$$11\!\cdots\!40$$$$T^{15} -$$$$82\!\cdots\!10$$$$T^{16} +$$$$96\!\cdots\!40$$$$T^{17} +$$$$13\!\cdots\!68$$$$T^{18} -$$$$12\!\cdots\!68$$$$T^{19} +$$$$34\!\cdots\!40$$$$T^{20} +$$$$11\!\cdots\!16$$$$T^{21} -$$$$12\!\cdots\!84$$$$T^{22} +$$$$29\!\cdots\!96$$$$T^{23} +$$$$12\!\cdots\!80$$$$T^{24} -$$$$53\!\cdots\!36$$$$T^{25} +$$$$53\!\cdots\!76$$$$T^{26} +$$$$30\!\cdots\!72$$$$T^{27} -$$$$19\!\cdots\!04$$$$T^{28} +$$$$19\!\cdots\!28$$$$T^{29} +$$$$45\!\cdots\!32$$$$T^{30} -$$$$23\!\cdots\!32$$$$T^{31} +$$$$62\!\cdots\!41$$$$T^{32}$$
$31$ $$1 - 7312 T^{2} + 29025544 T^{4} - 80335806576 T^{6} + 171125889681052 T^{8} - 295006946315669072 T^{10} +$$$$42\!\cdots\!64$$$$T^{12} -$$$$51\!\cdots\!00$$$$T^{14} +$$$$53\!\cdots\!38$$$$T^{16} -$$$$47\!\cdots\!00$$$$T^{18} +$$$$36\!\cdots\!24$$$$T^{20} -$$$$23\!\cdots\!92$$$$T^{22} +$$$$12\!\cdots\!12$$$$T^{24} -$$$$53\!\cdots\!76$$$$T^{26} +$$$$18\!\cdots\!24$$$$T^{28} -$$$$41\!\cdots\!92$$$$T^{30} +$$$$52\!\cdots\!61$$$$T^{32}$$
$37$ $$1 - 96 T + 4608 T^{2} - 145952 T^{3} + 4040888 T^{4} - 217733344 T^{5} + 12932982272 T^{6} - 602883756192 T^{7} + 21839639792924 T^{8} - 655265530977504 T^{9} + 21703692469355008 T^{10} - 815191556064282016 T^{11} + 35433653736114978312 T^{12} -$$$$14\!\cdots\!40$$$$T^{13} +$$$$50\!\cdots\!52$$$$T^{14} -$$$$14\!\cdots\!84$$$$T^{15} +$$$$43\!\cdots\!90$$$$T^{16} -$$$$20\!\cdots\!96$$$$T^{17} +$$$$95\!\cdots\!72$$$$T^{18} -$$$$37\!\cdots\!60$$$$T^{19} +$$$$12\!\cdots\!52$$$$T^{20} -$$$$39\!\cdots\!84$$$$T^{21} +$$$$14\!\cdots\!48$$$$T^{22} -$$$$59\!\cdots\!56$$$$T^{23} +$$$$26\!\cdots\!84$$$$T^{24} -$$$$10\!\cdots\!68$$$$T^{25} +$$$$29\!\cdots\!72$$$$T^{26} -$$$$68\!\cdots\!36$$$$T^{27} +$$$$17\!\cdots\!68$$$$T^{28} -$$$$86\!\cdots\!68$$$$T^{29} +$$$$37\!\cdots\!68$$$$T^{30} -$$$$10\!\cdots\!04$$$$T^{31} +$$$$15\!\cdots\!81$$$$T^{32}$$
$41$ $$1 - 13840 T^{2} + 102706104 T^{4} - 524939980080 T^{6} + 2044068651261084 T^{8} - 6376104819902485008 T^{10} +$$$$16\!\cdots\!68$$$$T^{12} -$$$$35\!\cdots\!72$$$$T^{14} +$$$$64\!\cdots\!06$$$$T^{16} -$$$$99\!\cdots\!92$$$$T^{18} +$$$$13\!\cdots\!28$$$$T^{20} -$$$$14\!\cdots\!48$$$$T^{22} +$$$$13\!\cdots\!44$$$$T^{24} -$$$$94\!\cdots\!80$$$$T^{26} +$$$$52\!\cdots\!44$$$$T^{28} -$$$$19\!\cdots\!40$$$$T^{30} +$$$$40\!\cdots\!81$$$$T^{32}$$
$43$ $$1 + 160 T + 12800 T^{2} + 978464 T^{3} + 71106632 T^{4} + 3813053664 T^{5} + 178619596288 T^{6} + 8719368905312 T^{7} + 336417491247900 T^{8} + 9339737479444512 T^{9} + 288453906337733120 T^{10} + 7137460469658328480 T^{11} -$$$$12\!\cdots\!76$$$$T^{12} -$$$$13\!\cdots\!84$$$$T^{13} -$$$$44\!\cdots\!20$$$$T^{14} -$$$$28\!\cdots\!76$$$$T^{15} -$$$$17\!\cdots\!30$$$$T^{16} -$$$$53\!\cdots\!24$$$$T^{17} -$$$$15\!\cdots\!20$$$$T^{18} -$$$$83\!\cdots\!16$$$$T^{19} -$$$$15\!\cdots\!76$$$$T^{20} +$$$$15\!\cdots\!20$$$$T^{21} +$$$$11\!\cdots\!20$$$$T^{22} +$$$$69\!\cdots\!88$$$$T^{23} +$$$$45\!\cdots\!00$$$$T^{24} +$$$$22\!\cdots\!88$$$$T^{25} +$$$$83\!\cdots\!88$$$$T^{26} +$$$$32\!\cdots\!36$$$$T^{27} +$$$$11\!\cdots\!32$$$$T^{28} +$$$$28\!\cdots\!36$$$$T^{29} +$$$$69\!\cdots\!00$$$$T^{30} +$$$$16\!\cdots\!40$$$$T^{31} +$$$$18\!\cdots\!01$$$$T^{32}$$
$47$ $$1 - 24144 T^{2} + 280869112 T^{4} - 2097883923184 T^{6} + 11327375509374492 T^{8} - 47271044690493269328 T^{10} +$$$$15\!\cdots\!16$$$$T^{12} -$$$$44\!\cdots\!04$$$$T^{14} +$$$$10\!\cdots\!58$$$$T^{16} -$$$$21\!\cdots\!24$$$$T^{18} +$$$$37\!\cdots\!76$$$$T^{20} -$$$$54\!\cdots\!48$$$$T^{22} +$$$$64\!\cdots\!32$$$$T^{24} -$$$$58\!\cdots\!84$$$$T^{26} +$$$$37\!\cdots\!72$$$$T^{28} -$$$$15\!\cdots\!84$$$$T^{30} +$$$$32\!\cdots\!41$$$$T^{32}$$
$53$ $$1 + 160 T + 12800 T^{2} + 602944 T^{3} + 3948712 T^{4} - 1481707264 T^{5} - 105845942272 T^{6} - 3791430241760 T^{7} + 34861972067036 T^{8} + 14471440004155872 T^{9} + 1098860393015073792 T^{10} + 54880211634179791488 T^{11} +$$$$13\!\cdots\!12$$$$T^{12} -$$$$17\!\cdots\!00$$$$T^{13} -$$$$18\!\cdots\!48$$$$T^{14} +$$$$16\!\cdots\!08$$$$T^{15} +$$$$51\!\cdots\!54$$$$T^{16} +$$$$46\!\cdots\!72$$$$T^{17} -$$$$14\!\cdots\!88$$$$T^{18} -$$$$38\!\cdots\!00$$$$T^{19} +$$$$84\!\cdots\!32$$$$T^{20} +$$$$95\!\cdots\!12$$$$T^{21} +$$$$53\!\cdots\!72$$$$T^{22} +$$$$19\!\cdots\!68$$$$T^{23} +$$$$13\!\cdots\!56$$$$T^{24} -$$$$41\!\cdots\!40$$$$T^{25} -$$$$32\!\cdots\!72$$$$T^{26} -$$$$12\!\cdots\!76$$$$T^{27} +$$$$95\!\cdots\!72$$$$T^{28} +$$$$40\!\cdots\!76$$$$T^{29} +$$$$24\!\cdots\!00$$$$T^{30} +$$$$85\!\cdots\!40$$$$T^{31} +$$$$15\!\cdots\!41$$$$T^{32}$$
$59$ $$1 + 128 T + 8192 T^{2} + 1121408 T^{3} + 136226184 T^{4} + 9279937408 T^{5} + 700645040128 T^{6} + 71627082366848 T^{7} + 5234572115355804 T^{8} + 316007889653226112 T^{9} + 25502997282495045632 T^{10} +$$$$19\!\cdots\!80$$$$T^{11} +$$$$10\!\cdots\!40$$$$T^{12} +$$$$69\!\cdots\!16$$$$T^{13} +$$$$51\!\cdots\!56$$$$T^{14} +$$$$29\!\cdots\!24$$$$T^{15} +$$$$15\!\cdots\!38$$$$T^{16} +$$$$10\!\cdots\!44$$$$T^{17} +$$$$62\!\cdots\!16$$$$T^{18} +$$$$29\!\cdots\!56$$$$T^{19} +$$$$16\!\cdots\!40$$$$T^{20} +$$$$98\!\cdots\!80$$$$T^{21} +$$$$45\!\cdots\!92$$$$T^{22} +$$$$19\!\cdots\!32$$$$T^{23} +$$$$11\!\cdots\!64$$$$T^{24} +$$$$53\!\cdots\!08$$$$T^{25} +$$$$18\!\cdots\!28$$$$T^{26} +$$$$84\!\cdots\!48$$$$T^{27} +$$$$43\!\cdots\!24$$$$T^{28} +$$$$12\!\cdots\!28$$$$T^{29} +$$$$31\!\cdots\!32$$$$T^{30} +$$$$17\!\cdots\!28$$$$T^{31} +$$$$46\!\cdots\!81$$$$T^{32}$$
$61$ $$1 - 32 T + 512 T^{2} + 38048 T^{3} - 60439624 T^{4} + 1520787552 T^{5} - 16996289024 T^{6} - 2981900088544 T^{7} + 2018049968078364 T^{8} - 40394929489472928 T^{9} + 275088896591278592 T^{10} +$$$$10\!\cdots\!56$$$$T^{11} -$$$$45\!\cdots\!20$$$$T^{12} +$$$$71\!\cdots\!16$$$$T^{13} -$$$$18\!\cdots\!28$$$$T^{14} -$$$$23\!\cdots\!64$$$$T^{15} +$$$$72\!\cdots\!42$$$$T^{16} -$$$$88\!\cdots\!44$$$$T^{17} -$$$$26\!\cdots\!48$$$$T^{18} +$$$$36\!\cdots\!76$$$$T^{19} -$$$$86\!\cdots\!20$$$$T^{20} +$$$$75\!\cdots\!56$$$$T^{21} +$$$$73\!\cdots\!32$$$$T^{22} -$$$$39\!\cdots\!48$$$$T^{23} +$$$$74\!\cdots\!04$$$$T^{24} -$$$$40\!\cdots\!64$$$$T^{25} -$$$$86\!\cdots\!24$$$$T^{26} +$$$$28\!\cdots\!92$$$$T^{27} -$$$$42\!\cdots\!84$$$$T^{28} +$$$$99\!\cdots\!28$$$$T^{29} +$$$$49\!\cdots\!72$$$$T^{30} -$$$$11\!\cdots\!32$$$$T^{31} +$$$$13\!\cdots\!21$$$$T^{32}$$
$67$ $$1 + 320 T + 51200 T^{2} + 6047552 T^{3} + 641735304 T^{4} + 64228593856 T^{5} + 5982745065472 T^{6} + 525110406070976 T^{7} + 43992629224199580 T^{8} + 3502096836597496384 T^{9} +$$$$26\!\cdots\!12$$$$T^{10} +$$$$19\!\cdots\!80$$$$T^{11} +$$$$14\!\cdots\!20$$$$T^{12} +$$$$10\!\cdots\!88$$$$T^{13} +$$$$71\!\cdots\!04$$$$T^{14} +$$$$48\!\cdots\!52$$$$T^{15} +$$$$32\!\cdots\!74$$$$T^{16} +$$$$21\!\cdots\!28$$$$T^{17} +$$$$14\!\cdots\!84$$$$T^{18} +$$$$93\!\cdots\!72$$$$T^{19} +$$$$58\!\cdots\!20$$$$T^{20} +$$$$36\!\cdots\!20$$$$T^{21} +$$$$21\!\cdots\!32$$$$T^{22} +$$$$12\!\cdots\!36$$$$T^{23} +$$$$72\!\cdots\!80$$$$T^{24} +$$$$38\!\cdots\!84$$$$T^{25} +$$$$19\!\cdots\!72$$$$T^{26} +$$$$95\!\cdots\!84$$$$T^{27} +$$$$42\!\cdots\!84$$$$T^{28} +$$$$18\!\cdots\!88$$$$T^{29} +$$$$69\!\cdots\!00$$$$T^{30} +$$$$19\!\cdots\!80$$$$T^{31} +$$$$27\!\cdots\!61$$$$T^{32}$$
$71$ $$( 1 + 256 T + 68104 T^{2} + 10692864 T^{3} + 1610923548 T^{4} + 179723087616 T^{5} + 18972832358712 T^{6} + 1588998739085056 T^{7} + 125568612540426694 T^{8} + 8010142643727767296 T^{9} +$$$$48\!\cdots\!72$$$$T^{10} +$$$$23\!\cdots\!36$$$$T^{11} +$$$$10\!\cdots\!28$$$$T^{12} +$$$$34\!\cdots\!64$$$$T^{13} +$$$$11\!\cdots\!64$$$$T^{14} +$$$$21\!\cdots\!36$$$$T^{15} +$$$$41\!\cdots\!21$$$$T^{16} )^{2}$$
$73$ $$1 - 42768 T^{2} + 946714744 T^{4} - 14391245893936 T^{6} + 167549428359087132 T^{8} -$$$$15\!\cdots\!24$$$$T^{10} +$$$$12\!\cdots\!76$$$$T^{12} -$$$$83\!\cdots\!96$$$$T^{14} +$$$$47\!\cdots\!22$$$$T^{16} -$$$$23\!\cdots\!36$$$$T^{18} +$$$$10\!\cdots\!56$$$$T^{20} -$$$$36\!\cdots\!04$$$$T^{22} +$$$$10\!\cdots\!52$$$$T^{24} -$$$$26\!\cdots\!36$$$$T^{26} +$$$$49\!\cdots\!04$$$$T^{28} -$$$$63\!\cdots\!08$$$$T^{30} +$$$$42\!\cdots\!21$$$$T^{32}$$
$79$ $$1 - 62928 T^{2} + 1905826568 T^{4} - 37296559235888 T^{6} + 534425714020543644 T^{8} -$$$$60\!\cdots\!08$$$$T^{10} +$$$$55\!\cdots\!96$$$$T^{12} -$$$$43\!\cdots\!36$$$$T^{14} +$$$$29\!\cdots\!62$$$$T^{16} -$$$$16\!\cdots\!16$$$$T^{18} +$$$$84\!\cdots\!56$$$$T^{20} -$$$$35\!\cdots\!28$$$$T^{22} +$$$$12\!\cdots\!24$$$$T^{24} -$$$$33\!\cdots\!88$$$$T^{26} +$$$$66\!\cdots\!08$$$$T^{28} -$$$$85\!\cdots\!08$$$$T^{30} +$$$$52\!\cdots\!41$$$$T^{32}$$
$83$ $$1 + 160 T + 12800 T^{2} + 895904 T^{3} + 107479624 T^{4} + 16432771168 T^{5} + 1654826188288 T^{6} + 174484645067104 T^{7} + 18280323695716892 T^{8} + 1483531366054758688 T^{9} +$$$$11\!\cdots\!96$$$$T^{10} +$$$$11\!\cdots\!36$$$$T^{11} +$$$$13\!\cdots\!00$$$$T^{12} +$$$$12\!\cdots\!44$$$$T^{13} +$$$$89\!\cdots\!76$$$$T^{14} +$$$$74\!\cdots\!76$$$$T^{15} +$$$$61\!\cdots\!66$$$$T^{16} +$$$$51\!\cdots\!64$$$$T^{17} +$$$$42\!\cdots\!96$$$$T^{18} +$$$$39\!\cdots\!36$$$$T^{19} +$$$$30\!\cdots\!00$$$$T^{20} +$$$$17\!\cdots\!64$$$$T^{21} +$$$$12\!\cdots\!56$$$$T^{22} +$$$$10\!\cdots\!52$$$$T^{23} +$$$$92\!\cdots\!52$$$$T^{24} +$$$$60\!\cdots\!36$$$$T^{25} +$$$$39\!\cdots\!88$$$$T^{26} +$$$$27\!\cdots\!52$$$$T^{27} +$$$$12\!\cdots\!04$$$$T^{28} +$$$$70\!\cdots\!76$$$$T^{29} +$$$$69\!\cdots\!00$$$$T^{30} +$$$$59\!\cdots\!40$$$$T^{31} +$$$$25\!\cdots\!61$$$$T^{32}$$
$89$ $$1 - 81008 T^{2} + 3201135736 T^{4} - 82544801381712 T^{6} + 1567286911309649436 T^{8} -$$$$23\!\cdots\!04$$$$T^{10} +$$$$28\!\cdots\!72$$$$T^{12} -$$$$29\!\cdots\!36$$$$T^{14} +$$$$25\!\cdots\!10$$$$T^{16} -$$$$18\!\cdots\!76$$$$T^{18} +$$$$11\!\cdots\!32$$$$T^{20} -$$$$57\!\cdots\!84$$$$T^{22} +$$$$24\!\cdots\!96$$$$T^{24} -$$$$80\!\cdots\!12$$$$T^{26} +$$$$19\!\cdots\!76$$$$T^{28} -$$$$31\!\cdots\!48$$$$T^{30} +$$$$24\!\cdots\!21$$$$T^{32}$$
$97$ $$( 1 + 38216 T^{2} + 116224 T^{3} + 770481564 T^{4} + 3485408768 T^{5} + 10857255215864 T^{6} + 49274039499776 T^{7} + 116292098553803590 T^{8} + 463619437653392384 T^{9} +$$$$96\!\cdots\!84$$$$T^{10} +$$$$29\!\cdots\!72$$$$T^{11} +$$$$60\!\cdots\!04$$$$T^{12} +$$$$85\!\cdots\!76$$$$T^{13} +$$$$26\!\cdots\!56$$$$T^{14} +$$$$61\!\cdots\!21$$$$T^{16} )^{2}$$