# Properties

 Label 60.2 Level 60 Weight 2 Dimension 34 Nonzero newspaces 5 Newform subspaces 6 Sturm bound 384 Trace bound 4

## Defining parameters

 Level: $$N$$ = $$60\( 60 = 2^{2} \cdot 3 \cdot 5$$ \) Weight: $$k$$ = $$2$$ Nonzero newspaces: $$5$$ Newform subspaces: $$6$$ Sturm bound: $$384$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(60))$$.

Total New Old
Modular forms 136 42 94
Cusp forms 57 34 23
Eisenstein series 79 8 71

## Trace form

 $$34q + 2q^{3} - 8q^{4} + 2q^{5} - 8q^{6} - 4q^{7} - 12q^{8} - 10q^{9} + O(q^{10})$$ $$34q + 2q^{3} - 8q^{4} + 2q^{5} - 8q^{6} - 4q^{7} - 12q^{8} - 10q^{9} - 20q^{10} - 8q^{11} - 4q^{12} - 24q^{13} - 14q^{15} - 20q^{17} + 16q^{18} + 20q^{20} - 12q^{21} + 8q^{22} + 24q^{24} - 14q^{25} + 16q^{26} + 14q^{27} + 24q^{28} + 12q^{29} + 36q^{30} + 24q^{31} + 20q^{32} + 12q^{33} + 24q^{34} + 16q^{35} + 16q^{38} + 16q^{40} - 4q^{41} + 8q^{42} - 12q^{43} - 2q^{45} - 24q^{46} - 20q^{48} - 18q^{49} - 16q^{50} - 12q^{51} - 40q^{52} + 4q^{53} - 56q^{54} - 8q^{55} - 64q^{56} + 20q^{57} - 56q^{58} - 8q^{59} - 64q^{60} - 36q^{61} - 56q^{62} + 8q^{63} - 56q^{64} + 20q^{65} - 64q^{66} - 4q^{67} - 16q^{68} + 32q^{69} - 8q^{70} + 12q^{72} + 48q^{73} + 2q^{75} + 32q^{76} + 48q^{77} + 16q^{78} + 24q^{79} + 4q^{80} + 26q^{81} + 56q^{82} + 56q^{84} + 56q^{85} + 64q^{86} + 20q^{87} + 104q^{88} + 20q^{89} + 60q^{90} + 24q^{91} + 56q^{92} + 24q^{93} + 104q^{94} + 64q^{96} - 32q^{97} + 24q^{98} + 8q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(60))$$

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
60.2.a $$\chi_{60}(1, \cdot)$$ None 0 1
60.2.d $$\chi_{60}(49, \cdot)$$ 60.2.d.a 2 1
60.2.e $$\chi_{60}(11, \cdot)$$ 60.2.e.a 8 1
60.2.h $$\chi_{60}(59, \cdot)$$ 60.2.h.a 4 1
60.2.h.b 4
60.2.i $$\chi_{60}(17, \cdot)$$ 60.2.i.a 4 2
60.2.j $$\chi_{60}(7, \cdot)$$ 60.2.j.a 12 2

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(60))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(60)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + T^{2} + 4 T^{4} + 4 T^{6} + 16 T^{8}$$)($$( 1 + 2 T^{2} )^{2}$$)($$1 - T^{2} + 4 T^{4}$$)($$1 + 4 T^{3} - 3 T^{4} - 4 T^{5} + 8 T^{6} - 8 T^{7} - 12 T^{8} + 32 T^{9} + 64 T^{12}$$)
$3$ ($$1 + T^{2}$$)($$1 + 2 T^{2} + 2 T^{4} + 18 T^{6} + 81 T^{8}$$)($$1 - 4 T^{2} + 9 T^{4}$$)($$( 1 + 3 T^{2} )^{2}$$)($$1 - 2 T + 2 T^{2} - 6 T^{3} + 9 T^{4}$$)($$( 1 + T^{4} )^{3}$$)
$5$ ($$1 - 2 T + 5 T^{2}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + 5 T^{2} )^{2}$$)($$( 1 - 5 T^{2} )^{2}$$)($$( 1 - 5 T^{2} )^{2}$$)($$( 1 + 5 T^{2} + 8 T^{3} + 25 T^{4} + 125 T^{6} )^{2}$$)
$7$ ($$1 + 2 T^{2} + 49 T^{4}$$)($$( 1 - 18 T^{2} + 162 T^{4} - 882 T^{6} + 2401 T^{8} )^{2}$$)($$( 1 + 4 T^{2} + 49 T^{4} )^{2}$$)($$( 1 + 7 T^{2} )^{4}$$)($$( 1 + 2 T + 2 T^{2} + 14 T^{3} + 49 T^{4} )^{2}$$)($$1 - 66 T^{4} + 671 T^{8} + 41476 T^{12} + 1611071 T^{16} - 380476866 T^{20} + 13841287201 T^{24}$$)
$11$ ($$( 1 + 4 T + 11 T^{2} )^{2}$$)($$( 1 + 24 T^{2} + 318 T^{4} + 2904 T^{6} + 14641 T^{8} )^{2}$$)($$( 1 + 11 T^{2} )^{4}$$)($$( 1 + 11 T^{2} )^{4}$$)($$( 1 - 2 T^{2} + 121 T^{4} )^{2}$$)($$( 1 - 30 T^{2} + 491 T^{4} - 6076 T^{6} + 59411 T^{8} - 439230 T^{10} + 1771561 T^{12} )^{2}$$)
$13$ ($$( 1 - 13 T^{2} )^{2}$$)($$( 1 + 2 T + 10 T^{2} + 26 T^{3} + 169 T^{4} )^{4}$$)($$( 1 - 13 T^{2} )^{4}$$)($$( 1 - 13 T^{2} )^{4}$$)($$( 1 + 6 T + 18 T^{2} + 78 T^{3} + 169 T^{4} )^{2}$$)($$( 1 + 2 T + 2 T^{2} - 6 T^{3} + 27 T^{4} + 580 T^{5} + 1124 T^{6} + 7540 T^{7} + 4563 T^{8} - 13182 T^{9} + 57122 T^{10} + 742586 T^{11} + 4826809 T^{12} )^{2}$$)
$17$ ($$1 - 18 T^{2} + 289 T^{4}$$)($$( 1 - 8 T + 17 T^{2} )^{4}( 1 + 8 T + 17 T^{2} )^{4}$$)($$( 1 + 17 T^{2} )^{4}$$)($$( 1 + 14 T^{2} + 289 T^{4} )^{2}$$)($$1 - 2 T^{4} + 83521 T^{8}$$)($$( 1 + 10 T + 50 T^{2} + 250 T^{3} + 1155 T^{4} + 4260 T^{5} + 16100 T^{6} + 72420 T^{7} + 333795 T^{8} + 1228250 T^{9} + 4176050 T^{10} + 14198570 T^{11} + 24137569 T^{12} )^{2}$$)
$19$ ($$( 1 + 19 T^{2} )^{2}$$)($$( 1 - 56 T^{2} + 1438 T^{4} - 20216 T^{6} + 130321 T^{8} )^{2}$$)($$( 1 - 19 T^{2} )^{4}$$)($$( 1 - 4 T + 19 T^{2} )^{2}( 1 + 4 T + 19 T^{2} )^{2}$$)($$( 1 - 34 T^{2} + 361 T^{4} )^{2}$$)($$( 1 + 74 T^{2} + 2775 T^{4} + 65228 T^{6} + 1001775 T^{8} + 9643754 T^{10} + 47045881 T^{12} )^{2}$$)
$23$ ($$1 - 30 T^{2} + 529 T^{4}$$)($$( 1 + 34 T^{2} + 514 T^{4} + 17986 T^{6} + 279841 T^{8} )^{2}$$)($$( 1 - 44 T^{2} + 529 T^{4} )^{2}$$)($$( 1 - 34 T^{2} + 529 T^{4} )^{2}$$)($$1 + 238 T^{4} + 279841 T^{8}$$)($$1 + 1190 T^{4} + 607727 T^{8} + 223349076 T^{12} + 170066931407 T^{16} + 93190072484390 T^{20} + 21914624432020321 T^{24}$$)
$29$ ($$( 1 - 6 T + 29 T^{2} )^{2}$$)($$( 1 - 80 T^{2} + 3214 T^{4} - 67280 T^{6} + 707281 T^{8} )^{2}$$)($$( 1 - 6 T + 29 T^{2} )^{2}( 1 + 6 T + 29 T^{2} )^{2}$$)($$( 1 - 29 T^{2} )^{4}$$)($$( 1 + 38 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 154 T^{2} + 10395 T^{4} - 392596 T^{6} + 8742195 T^{8} - 108921274 T^{10} + 594823321 T^{12} )^{2}$$)
$31$ ($$( 1 - 4 T + 31 T^{2} )^{2}$$)($$( 1 - 96 T^{2} + 4158 T^{4} - 92256 T^{6} + 923521 T^{8} )^{2}$$)($$( 1 - 31 T^{2} )^{4}$$)($$( 1 - 8 T + 31 T^{2} )^{2}( 1 + 8 T + 31 T^{2} )^{2}$$)($$( 1 - 4 T + 31 T^{2} )^{4}$$)($$( 1 - 74 T^{2} + 4175 T^{4} - 143436 T^{6} + 4012175 T^{8} - 68340554 T^{10} + 887503681 T^{12} )^{2}$$)
$37$ ($$1 - 10 T^{2} + 1369 T^{4}$$)($$( 1 - 2 T + 58 T^{2} - 74 T^{3} + 1369 T^{4} )^{4}$$)($$( 1 - 37 T^{2} )^{4}$$)($$( 1 - 37 T^{2} )^{4}$$)($$( 1 + 6 T + 18 T^{2} + 222 T^{3} + 1369 T^{4} )^{2}$$)($$( 1 - 10 T + 93 T^{2} - 472 T^{3} + 3441 T^{4} - 13690 T^{5} + 50653 T^{6} )^{2}( 1 + 8 T - 11 T^{2} - 424 T^{3} - 407 T^{4} + 10952 T^{5} + 50653 T^{6} )^{2}$$)
$41$ ($$( 1 + 10 T + 41 T^{2} )^{2}$$)($$( 1 - 112 T^{2} + 5886 T^{4} - 188272 T^{6} + 2825761 T^{8} )^{2}$$)($$( 1 - 12 T + 41 T^{2} )^{2}( 1 + 12 T + 41 T^{2} )^{2}$$)($$( 1 - 41 T^{2} )^{4}$$)($$( 1 - 2 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 - 4 T + 103 T^{2} - 264 T^{3} + 4223 T^{4} - 6724 T^{5} + 68921 T^{6} )^{4}$$)
$43$ ($$1 - 70 T^{2} + 1849 T^{4}$$)($$( 1 - 110 T^{2} + 5890 T^{4} - 203390 T^{6} + 3418801 T^{8} )^{2}$$)($$( 1 + 76 T^{2} + 1849 T^{4} )^{2}$$)($$( 1 + 43 T^{2} )^{4}$$)($$( 1 + 6 T + 18 T^{2} + 258 T^{3} + 1849 T^{4} )^{2}$$)($$1 - 6826 T^{4} + 23542879 T^{8} - 52748850124 T^{12} + 80488418268079 T^{16} - 79783655094904426 T^{20} + 39959630797262576401 T^{24}$$)
$47$ ($$1 - 78 T^{2} + 2209 T^{4}$$)($$( 1 + 178 T^{2} + 12322 T^{4} + 393202 T^{6} + 4879681 T^{8} )^{2}$$)($$( 1 + 4 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 + 14 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 - 14 T + 98 T^{2} - 658 T^{3} + 2209 T^{4} )( 1 + 14 T + 98 T^{2} + 658 T^{3} + 2209 T^{4} )$$)($$1 + 1606 T^{4} + 1555663 T^{8} - 11534742380 T^{12} + 7591139183503 T^{16} + 38240926378788166 T^{20} +$$$$11\!\cdots\!41$$$$T^{24}$$)
$53$ ($$1 + 38 T^{2} + 2809 T^{4}$$)($$( 1 - 44 T^{2} + 1750 T^{4} - 123596 T^{6} + 7890481 T^{8} )^{2}$$)($$( 1 + 53 T^{2} )^{4}$$)($$( 1 + 86 T^{2} + 2809 T^{4} )^{2}$$)($$1 + 3598 T^{4} + 7890481 T^{8}$$)($$( 1 - 2 T + 2 T^{2} - 90 T^{3} + 5291 T^{4} - 7252 T^{5} + 7972 T^{6} - 384356 T^{7} + 14862419 T^{8} - 13398930 T^{9} + 15780962 T^{10} - 836390986 T^{11} + 22164361129 T^{12} )^{2}$$)
$59$ ($$( 1 + 4 T + 59 T^{2} )^{2}$$)($$( 1 - 16 T^{2} + 1518 T^{4} - 55696 T^{6} + 12117361 T^{8} )^{2}$$)($$( 1 + 59 T^{2} )^{4}$$)($$( 1 + 59 T^{2} )^{4}$$)($$( 1 + 38 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 + 254 T^{2} + 30475 T^{4} + 2237948 T^{6} + 106083475 T^{8} + 3077809694 T^{10} + 42180533641 T^{12} )^{2}$$)
$61$ ($$( 1 - 2 T + 61 T^{2} )^{2}$$)($$( 1 + 2 T + 106 T^{2} + 122 T^{3} + 3721 T^{4} )^{4}$$)($$( 1 - 8 T + 61 T^{2} )^{4}$$)($$( 1 + 2 T + 61 T^{2} )^{4}$$)($$( 1 + 6 T + 61 T^{2} )^{4}$$)($$( 1 + 8 T + 83 T^{2} + 1152 T^{3} + 5063 T^{4} + 29768 T^{5} + 226981 T^{6} )^{4}$$)
$67$ ($$1 - 118 T^{2} + 4489 T^{4}$$)($$( 1 - 222 T^{2} + 21282 T^{4} - 996558 T^{6} + 20151121 T^{8} )^{2}$$)($$( 1 - 116 T^{2} + 4489 T^{4} )^{2}$$)($$( 1 + 67 T^{2} )^{4}$$)($$( 1 + 2 T + 2 T^{2} + 134 T^{3} + 4489 T^{4} )^{2}$$)($$1 - 6346 T^{4} + 60212671 T^{8} - 234156159244 T^{12} + 1213352819054191 T^{16} - 2576905481774443786 T^{20} +$$$$81\!\cdots\!61$$$$T^{24}$$)
$71$ ($$( 1 + 71 T^{2} )^{2}$$)($$( 1 + 228 T^{2} + 22806 T^{4} + 1149348 T^{6} + 25411681 T^{8} )^{2}$$)($$( 1 + 71 T^{2} )^{4}$$)($$( 1 + 71 T^{2} )^{4}$$)($$( 1 - 122 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 - 170 T^{2} + 18527 T^{4} - 1427916 T^{6} + 93394607 T^{8} - 4319985770 T^{10} + 128100283921 T^{12} )^{2}$$)
$73$ ($$1 - 82 T^{2} + 5329 T^{4}$$)($$( 1 + 78 T^{2} + 5329 T^{4} )^{4}$$)($$( 1 - 73 T^{2} )^{4}$$)($$( 1 - 73 T^{2} )^{4}$$)($$( 1 - 2 T + 2 T^{2} - 146 T^{3} + 5329 T^{4} )^{2}$$)($$( 1 - 22 T + 242 T^{2} - 2246 T^{3} + 20271 T^{4} - 179604 T^{5} + 1567964 T^{6} - 13111092 T^{7} + 108024159 T^{8} - 873732182 T^{9} + 6872374322 T^{10} - 45607575046 T^{11} + 151334226289 T^{12} )^{2}$$)
$79$ ($$( 1 - 12 T + 79 T^{2} )^{2}$$)($$( 1 - 168 T^{2} + 19470 T^{4} - 1048488 T^{6} + 38950081 T^{8} )^{2}$$)($$( 1 - 79 T^{2} )^{4}$$)($$( 1 - 16 T + 79 T^{2} )^{2}( 1 + 16 T + 79 T^{2} )^{2}$$)($$( 1 - 122 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 + 170 T^{2} + 10415 T^{4} + 507660 T^{6} + 65000015 T^{8} + 6621513770 T^{10} + 243087455521 T^{12} )^{2}$$)
$83$ ($$1 - 150 T^{2} + 6889 T^{4}$$)($$( 1 + 82 T^{2} + 4834 T^{4} + 564898 T^{6} + 47458321 T^{8} )^{2}$$)($$( 1 + 76 T^{2} + 6889 T^{4} )^{2}$$)($$( 1 - 154 T^{2} + 6889 T^{4} )^{2}$$)($$( 1 - 22 T + 242 T^{2} - 1826 T^{3} + 6889 T^{4} )( 1 + 22 T + 242 T^{2} + 1826 T^{3} + 6889 T^{4} )$$)($$1 + 7542 T^{4} - 16589569 T^{8} - 295904039564 T^{12} - 787313090853649 T^{16} + 16986788014792647222 T^{20} +$$$$10\!\cdots\!61$$$$T^{24}$$)
$89$ ($$( 1 - 10 T + 89 T^{2} )^{2}$$)($$( 1 - 212 T^{2} + 25990 T^{4} - 1679252 T^{6} + 62742241 T^{8} )^{2}$$)($$( 1 - 6 T + 89 T^{2} )^{2}( 1 + 6 T + 89 T^{2} )^{2}$$)($$( 1 - 89 T^{2} )^{4}$$)($$( 1 + 158 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 - 462 T^{2} + 94223 T^{4} - 10861604 T^{6} + 746340383 T^{8} - 28986915342 T^{10} + 496981290961 T^{12} )^{2}$$)
$97$ ($$( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} )$$)($$( 1 + 6 T + 97 T^{2} )^{8}$$)($$( 1 - 97 T^{2} )^{4}$$)($$( 1 - 97 T^{2} )^{4}$$)($$( 1 - 18 T + 162 T^{2} - 1746 T^{3} + 9409 T^{4} )^{2}$$)($$( 1 + 10 T + 50 T^{2} - 1078 T^{3} - 9153 T^{4} + 45804 T^{5} + 1496732 T^{6} + 4442988 T^{7} - 86120577 T^{8} - 983861494 T^{9} + 4426464050 T^{10} + 85873402570 T^{11} + 832972004929 T^{12} )^{2}$$)