# Properties

 Label 16.4 Level 16 Weight 4 Dimension 11 Nonzero newspaces 2 Newform subspaces 2 Sturm bound 64 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$16 = 2^{4}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$2$$ Newform subspaces: $$2$$ Sturm bound: $$64$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(16))$$.

Total New Old
Modular forms 31 16 15
Cusp forms 17 11 6
Eisenstein series 14 5 9

## Trace form

 $$11q - 2q^{2} + 2q^{3} + 8q^{4} - 4q^{5} - 32q^{6} - 24q^{7} - 44q^{8} - 11q^{9} + O(q^{10})$$ $$11q - 2q^{2} + 2q^{3} + 8q^{4} - 4q^{5} - 32q^{6} - 24q^{7} - 44q^{8} - 11q^{9} - 68q^{10} + 62q^{11} + 100q^{12} + 20q^{13} + 188q^{14} - 132q^{15} + 280q^{16} + 46q^{17} + 174q^{18} - 70q^{19} - 196q^{20} - 44q^{21} - 588q^{22} + 56q^{23} - 848q^{24} - 121q^{25} - 264q^{26} + 32q^{27} + 280q^{28} - 4q^{29} + 1236q^{30} + 528q^{31} + 968q^{32} + 172q^{33} + 436q^{34} + 524q^{35} - 596q^{36} - 172q^{37} - 1232q^{38} + 88q^{39} - 1336q^{40} - 198q^{41} - 680q^{42} - 890q^{43} + 868q^{44} + 216q^{45} + 1132q^{46} - 1472q^{47} + 1768q^{48} + 327q^{49} + 726q^{50} - 1300q^{51} - 236q^{52} - 620q^{53} - 1376q^{54} - 88q^{55} - 488q^{56} - 176q^{57} + 8q^{58} + 2374q^{59} - 192q^{60} + 1460q^{61} - 80q^{62} + 2892q^{63} + 512q^{64} - 536q^{65} - 428q^{66} + 1754q^{67} - 880q^{68} + 804q^{69} + 160q^{70} - 728q^{71} + 1092q^{72} + 154q^{73} - 452q^{74} - 3438q^{75} - 1228q^{76} - 1324q^{77} - 772q^{78} - 3760q^{79} - 2648q^{80} + 171q^{81} - 704q^{82} - 2798q^{83} + 1960q^{84} - 112q^{85} + 3764q^{86} + 792q^{87} + 1528q^{88} + 714q^{89} + 1896q^{90} + 2804q^{91} + 632q^{92} - 1552q^{93} - 3248q^{94} + 6988q^{95} - 4432q^{96} - 482q^{97} + 314q^{98} + 4474q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(16))$$

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space $$S_k^{\mathrm{new}}(N, \chi)$$ we list the newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
16.4.a $$\chi_{16}(1, \cdot)$$ 16.4.a.a 1 1
16.4.b $$\chi_{16}(9, \cdot)$$ None 0 1
16.4.e $$\chi_{16}(5, \cdot)$$ 16.4.e.a 10 2

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(16))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(16)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ()($$1 + 2 T - 2 T^{2} + 8 T^{3} - 40 T^{4} - 352 T^{5} - 320 T^{6} + 512 T^{7} - 1024 T^{8} + 8192 T^{9} + 32768 T^{10}$$)
$3$ ($$1 - 4 T + 27 T^{2}$$)($$1 + 2 T + 2 T^{2} - 42 T^{3} - 571 T^{4} + 760 T^{5} + 3544 T^{6} + 66792 T^{7} + 57874 T^{8} - 3046228 T^{9} - 6701044 T^{10} - 82248156 T^{11} + 42190146 T^{12} + 1314666936 T^{13} + 1883426904 T^{14} + 10905169320 T^{15} - 221217099219 T^{16} - 439334834526 T^{17} + 564859072962 T^{18} + 15251194969974 T^{19} + 205891132094649 T^{20}$$)
$5$ ($$1 + 2 T + 125 T^{2}$$)($$1 + 2 T + 2 T^{2} - 966 T^{3} - 13723 T^{4} + 18040 T^{5} + 530104 T^{6} - 11981288 T^{7} - 28535006 T^{8} + 2301854348 T^{9} + 23672040908 T^{10} + 287731793500 T^{11} - 445859468750 T^{12} - 23400953125000 T^{13} + 129419921875000 T^{14} + 550537109375000 T^{15} - 52349090576171875 T^{16} - 460624694824218750 T^{17} + 119209289550781250 T^{18} + 14901161193847656250 T^{19} +$$$$93\!\cdots\!25$$$$T^{20}$$)
$7$ ($$1 + 24 T + 343 T^{2}$$)($$1 - 1762 T^{2} + 1539965 T^{4} - 932087576 T^{6} + 440869947922 T^{8} - 168121217547916 T^{10} + 51867908503075378 T^{12} - 12901291835899914776 T^{14} +$$$$25\!\cdots\!85$$$$T^{16} -$$$$33\!\cdots\!62$$$$T^{18} +$$$$22\!\cdots\!49$$$$T^{20}$$)
$11$ ($$1 - 44 T + 1331 T^{2}$$)($$1 - 18 T + 162 T^{2} - 122934 T^{3} + 4077397 T^{4} + 79597000 T^{5} + 5463099864 T^{6} - 313798751208 T^{7} - 6887886337838 T^{8} + 101615185776500 T^{9} + 18755914132083020 T^{10} + 135249812268521500 T^{11} - 12202310808546625118 T^{12} -$$$$73\!\cdots\!28$$$$T^{13} +$$$$17\!\cdots\!44$$$$T^{14} +$$$$33\!\cdots\!00$$$$T^{15} +$$$$22\!\cdots\!57$$$$T^{16} -$$$$90\!\cdots\!74$$$$T^{17} +$$$$15\!\cdots\!42$$$$T^{18} -$$$$23\!\cdots\!78$$$$T^{19} +$$$$17\!\cdots\!01$$$$T^{20}$$)
$13$ ($$1 - 22 T + 2197 T^{2}$$)($$1 + 2 T + 2 T^{2} - 45206 T^{3} - 2401451 T^{4} + 29395960 T^{5} + 1085386040 T^{6} - 384228102440 T^{7} - 9400034983966 T^{8} + 1531455561616908 T^{9} + 23489689415409228 T^{10} + 3364607868872346876 T^{11} - 45372173460921944494 T^{12} -$$$$40\!\cdots\!20$$$$T^{13} +$$$$25\!\cdots\!40$$$$T^{14} +$$$$15\!\cdots\!20$$$$T^{15} -$$$$27\!\cdots\!79$$$$T^{16} -$$$$11\!\cdots\!78$$$$T^{17} +$$$$10\!\cdots\!22$$$$T^{18} +$$$$23\!\cdots\!34$$$$T^{19} +$$$$26\!\cdots\!49$$$$T^{20}$$)
$17$ ($$1 - 50 T + 4913 T^{2}$$)($$( 1 + 2 T + 12653 T^{2} + 102520 T^{3} + 98460610 T^{4} + 354493580 T^{5} + 483736976930 T^{6} + 2474583573880 T^{7} + 1500492401316541 T^{8} + 1165244474459522 T^{9} + 2862423051509815793 T^{10} )^{2}$$)
$19$ ($$1 + 44 T + 6859 T^{2}$$)($$1 + 26 T + 338 T^{2} - 339906 T^{3} - 64153371 T^{4} + 2461461784 T^{5} + 143449890200 T^{6} + 36783398837960 T^{7} + 1011857007777554 T^{8} - 259766590630759364 T^{9} - 7366645907488092948 T^{10} -$$$$17\!\cdots\!76$$$$T^{11} +$$$$47\!\cdots\!74$$$$T^{12} +$$$$11\!\cdots\!40$$$$T^{13} +$$$$31\!\cdots\!00$$$$T^{14} +$$$$37\!\cdots\!16$$$$T^{15} -$$$$66\!\cdots\!11$$$$T^{16} -$$$$24\!\cdots\!14$$$$T^{17} +$$$$16\!\cdots\!98$$$$T^{18} +$$$$87\!\cdots\!14$$$$T^{19} +$$$$23\!\cdots\!01$$$$T^{20}$$)
$23$ ($$1 - 56 T + 12167 T^{2}$$)($$1 - 76386 T^{2} + 2913757597 T^{4} - 73253961622040 T^{6} + 1342371312768300946 T^{8} -$$$$18\!\cdots\!84$$$$T^{10} +$$$$19\!\cdots\!94$$$$T^{12} -$$$$16\!\cdots\!40$$$$T^{14} +$$$$94\!\cdots\!93$$$$T^{16} -$$$$36\!\cdots\!26$$$$T^{18} +$$$$71\!\cdots\!49$$$$T^{20}$$)
$29$ ($$1 - 198 T + 24389 T^{2}$$)($$1 + 202 T + 20402 T^{2} - 1177934 T^{3} + 398569397 T^{4} + 239164019416 T^{5} + 40873283338616 T^{6} + 2529271278095288 T^{7} + 194871598558001506 T^{8} +$$$$12\!\cdots\!32$$$$T^{9} +$$$$30\!\cdots\!36$$$$T^{10} +$$$$30\!\cdots\!48$$$$T^{11} +$$$$11\!\cdots\!26$$$$T^{12} +$$$$36\!\cdots\!72$$$$T^{13} +$$$$14\!\cdots\!56$$$$T^{14} +$$$$20\!\cdots\!84$$$$T^{15} +$$$$83\!\cdots\!17$$$$T^{16} -$$$$60\!\cdots\!86$$$$T^{17} +$$$$25\!\cdots\!62$$$$T^{18} +$$$$61\!\cdots\!18$$$$T^{19} +$$$$74\!\cdots\!01$$$$T^{20}$$)
$31$ ($$1 - 160 T + 29791 T^{2}$$)($$( 1 - 184 T + 134043 T^{2} - 19809056 T^{3} + 7638677322 T^{4} - 852982867024 T^{5} + 227563836099702 T^{6} - 17580610117135136 T^{7} + 3544046273282822853 T^{8} -$$$$14\!\cdots\!24$$$$T^{9} +$$$$23\!\cdots\!51$$$$T^{10} )^{2}$$)
$37$ ($$1 + 162 T + 50653 T^{2}$$)($$1 + 10 T + 50 T^{2} + 1972962 T^{3} + 1630465317 T^{4} - 153991562664 T^{5} + 324850634232 T^{6} - 5252842710654600 T^{7} + 3474549392106364962 T^{8} + 27985624577691139772 T^{9} +$$$$10\!\cdots\!24$$$$T^{10} +$$$$14\!\cdots\!16$$$$T^{11} +$$$$89\!\cdots\!58$$$$T^{12} -$$$$68\!\cdots\!00$$$$T^{13} +$$$$21\!\cdots\!92$$$$T^{14} -$$$$51\!\cdots\!52$$$$T^{15} +$$$$27\!\cdots\!93$$$$T^{16} +$$$$16\!\cdots\!94$$$$T^{17} +$$$$21\!\cdots\!50$$$$T^{18} +$$$$21\!\cdots\!30$$$$T^{19} +$$$$11\!\cdots\!49$$$$T^{20}$$)
$41$ ($$1 + 198 T + 68921 T^{2}$$)($$1 - 441018 T^{2} + 97166156061 T^{4} - 13934678680622904 T^{6} +$$$$14\!\cdots\!14$$$$T^{8} -$$$$11\!\cdots\!88$$$$T^{10} +$$$$68\!\cdots\!74$$$$T^{12} -$$$$31\!\cdots\!24$$$$T^{14} +$$$$10\!\cdots\!81$$$$T^{16} -$$$$22\!\cdots\!98$$$$T^{18} +$$$$24\!\cdots\!01$$$$T^{20}$$)
$43$ ($$1 + 52 T + 79507 T^{2}$$)($$1 + 838 T + 351122 T^{2} + 132133650 T^{3} + 56398378005 T^{4} + 20936462157416 T^{5} + 6471694737204248 T^{6} + 1952595983380873720 T^{7} +$$$$59\!\cdots\!10$$$$T^{8} +$$$$17\!\cdots\!68$$$$T^{9} +$$$$49\!\cdots\!52$$$$T^{10} +$$$$13\!\cdots\!76$$$$T^{11} +$$$$37\!\cdots\!90$$$$T^{12} +$$$$98\!\cdots\!60$$$$T^{13} +$$$$25\!\cdots\!48$$$$T^{14} +$$$$66\!\cdots\!12$$$$T^{15} +$$$$14\!\cdots\!45$$$$T^{16} +$$$$26\!\cdots\!50$$$$T^{17} +$$$$56\!\cdots\!22$$$$T^{18} +$$$$10\!\cdots\!66$$$$T^{19} +$$$$10\!\cdots\!49$$$$T^{20}$$)
$47$ ($$1 + 528 T + 103823 T^{2}$$)($$( 1 + 472 T + 462219 T^{2} + 171516064 T^{3} + 90105579914 T^{4} + 25593405310224 T^{5} + 9355031623411222 T^{6} + 1848808586238545056 T^{7} +$$$$51\!\cdots\!73$$$$T^{8} +$$$$54\!\cdots\!52$$$$T^{9} +$$$$12\!\cdots\!43$$$$T^{10} )^{2}$$)
$53$ ($$1 + 242 T + 148877 T^{2}$$)($$1 + 378 T + 71442 T^{2} - 52753550 T^{3} + 3016286341 T^{4} - 526686651752 T^{5} + 976891435665272 T^{6} - 1674349213754452168 T^{7} - 1581505854923305054 T^{8} -$$$$14\!\cdots\!20$$$$T^{9} +$$$$29\!\cdots\!08$$$$T^{10} -$$$$22\!\cdots\!40$$$$T^{11} -$$$$35\!\cdots\!66$$$$T^{12} -$$$$55\!\cdots\!44$$$$T^{13} +$$$$47\!\cdots\!52$$$$T^{14} -$$$$38\!\cdots\!64$$$$T^{15} +$$$$32\!\cdots\!49$$$$T^{16} -$$$$85\!\cdots\!50$$$$T^{17} +$$$$17\!\cdots\!02$$$$T^{18} +$$$$13\!\cdots\!86$$$$T^{19} +$$$$53\!\cdots\!49$$$$T^{20}$$)
$59$ ($$1 - 668 T + 205379 T^{2}$$)($$1 - 1706 T + 1455218 T^{2} - 989315358 T^{3} + 555128806581 T^{4} - 232658117164632 T^{5} + 78453755015006616 T^{6} - 20092577710244830152 T^{7} +$$$$57\!\cdots\!98$$$$T^{8} +$$$$22\!\cdots\!08$$$$T^{9} -$$$$12\!\cdots\!80$$$$T^{10} +$$$$45\!\cdots\!32$$$$T^{11} +$$$$24\!\cdots\!18$$$$T^{12} -$$$$17\!\cdots\!28$$$$T^{13} +$$$$13\!\cdots\!96$$$$T^{14} -$$$$85\!\cdots\!68$$$$T^{15} +$$$$41\!\cdots\!01$$$$T^{16} -$$$$15\!\cdots\!22$$$$T^{17} +$$$$46\!\cdots\!98$$$$T^{18} -$$$$11\!\cdots\!14$$$$T^{19} +$$$$13\!\cdots\!01$$$$T^{20}$$)
$61$ ($$1 - 550 T + 226981 T^{2}$$)($$1 - 910 T + 414050 T^{2} + 45940410 T^{3} + 20471098485 T^{4} - 72823214590920 T^{5} + 58848327585507000 T^{6} - 6001380717052735080 T^{7} +$$$$59\!\cdots\!50$$$$T^{8} -$$$$32\!\cdots\!00$$$$T^{9} +$$$$38\!\cdots\!00$$$$T^{10} -$$$$73\!\cdots\!00$$$$T^{11} +$$$$30\!\cdots\!50$$$$T^{12} -$$$$70\!\cdots\!80$$$$T^{13} +$$$$15\!\cdots\!00$$$$T^{14} -$$$$43\!\cdots\!20$$$$T^{15} +$$$$27\!\cdots\!85$$$$T^{16} +$$$$14\!\cdots\!10$$$$T^{17} +$$$$29\!\cdots\!50$$$$T^{18} -$$$$14\!\cdots\!10$$$$T^{19} +$$$$36\!\cdots\!01$$$$T^{20}$$)
$67$ ($$1 + 188 T + 300763 T^{2}$$)($$1 - 1942 T + 1885682 T^{2} - 1530965298 T^{3} + 1338066168261 T^{4} - 1068751594464168 T^{5} + 724275679990789656 T^{6} -$$$$46\!\cdots\!12$$$$T^{7} +$$$$29\!\cdots\!06$$$$T^{8} -$$$$17\!\cdots\!64$$$$T^{9} +$$$$97\!\cdots\!68$$$$T^{10} -$$$$52\!\cdots\!32$$$$T^{11} +$$$$26\!\cdots\!14$$$$T^{12} -$$$$12\!\cdots\!64$$$$T^{13} +$$$$59\!\cdots\!16$$$$T^{14} -$$$$26\!\cdots\!24$$$$T^{15} +$$$$99\!\cdots\!49$$$$T^{16} -$$$$34\!\cdots\!66$$$$T^{17} +$$$$12\!\cdots\!22$$$$T^{18} -$$$$39\!\cdots\!66$$$$T^{19} +$$$$60\!\cdots\!49$$$$T^{20}$$)
$71$ ($$1 + 728 T + 357911 T^{2}$$)($$1 - 2500418 T^{2} + 3072516920573 T^{4} - 2420243241413642648 T^{6} +$$$$13\!\cdots\!66$$$$T^{8} -$$$$55\!\cdots\!32$$$$T^{10} +$$$$17\!\cdots\!86$$$$T^{12} -$$$$39\!\cdots\!68$$$$T^{14} +$$$$64\!\cdots\!53$$$$T^{16} -$$$$67\!\cdots\!58$$$$T^{18} +$$$$34\!\cdots\!01$$$$T^{20}$$)
$73$ ($$1 - 154 T + 389017 T^{2}$$)($$1 - 3134282 T^{2} + 4627160201821 T^{4} - 4242124717558516472 T^{6} +$$$$26\!\cdots\!74$$$$T^{8} -$$$$12\!\cdots\!92$$$$T^{10} +$$$$40\!\cdots\!86$$$$T^{12} -$$$$97\!\cdots\!12$$$$T^{14} +$$$$16\!\cdots\!49$$$$T^{16} -$$$$16\!\cdots\!62$$$$T^{18} +$$$$79\!\cdots\!49$$$$T^{20}$$)
$79$ ($$1 - 656 T + 493039 T^{2}$$)($$( 1 + 2208 T + 3816107 T^{2} + 4320867712 T^{3} + 4245684014154 T^{4} + 3176789940661184 T^{5} + 2093287800654474006 T^{6} +$$$$10\!\cdots\!52$$$$T^{7} +$$$$45\!\cdots\!33$$$$T^{8} +$$$$13\!\cdots\!28$$$$T^{9} +$$$$29\!\cdots\!99$$$$T^{10} )^{2}$$)
$83$ ($$1 + 236 T + 571787 T^{2}$$)($$1 + 2562 T + 3281922 T^{2} + 2891460918 T^{3} + 1934799974629 T^{4} + 1191348439341176 T^{5} + 882645219418437336 T^{6} +$$$$79\!\cdots\!28$$$$T^{7} +$$$$89\!\cdots\!74$$$$T^{8} +$$$$97\!\cdots\!88$$$$T^{9} +$$$$82\!\cdots\!76$$$$T^{10} +$$$$55\!\cdots\!56$$$$T^{11} +$$$$29\!\cdots\!06$$$$T^{12} +$$$$14\!\cdots\!84$$$$T^{13} +$$$$94\!\cdots\!96$$$$T^{14} +$$$$72\!\cdots\!32$$$$T^{15} +$$$$67\!\cdots\!61$$$$T^{16} +$$$$57\!\cdots\!94$$$$T^{17} +$$$$37\!\cdots\!62$$$$T^{18} +$$$$16\!\cdots\!74$$$$T^{19} +$$$$37\!\cdots\!49$$$$T^{20}$$)
$89$ ($$1 - 714 T + 704969 T^{2}$$)($$1 - 3643178 T^{2} + 6505439011133 T^{4} - 7720859292932177528 T^{6} +$$$$69\!\cdots\!98$$$$T^{8} -$$$$52\!\cdots\!28$$$$T^{10} +$$$$34\!\cdots\!78$$$$T^{12} -$$$$19\!\cdots\!88$$$$T^{14} +$$$$79\!\cdots\!73$$$$T^{16} -$$$$22\!\cdots\!98$$$$T^{18} +$$$$30\!\cdots\!01$$$$T^{20}$$)
$97$ ($$1 + 478 T + 912673 T^{2}$$)($$( 1 + 2 T + 2565789 T^{2} - 723000584 T^{3} + 3231145430658 T^{4} - 1257303020547316 T^{5} + 2948979193634928834 T^{6} -$$$$60\!\cdots\!36$$$$T^{7} +$$$$19\!\cdots\!13$$$$T^{8} +$$$$13\!\cdots\!82$$$$T^{9} +$$$$63\!\cdots\!93$$$$T^{10} )^{2}$$)