# Properties

 Label 177.2 Level 177 Weight 2 Dimension 811 Nonzero newspaces 4 Newform subspaces 10 Sturm bound 4640 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$177 = 3 \cdot 59$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$4$$ Newform subspaces: $$10$$ Sturm bound: $$4640$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(177))$$.

Total New Old
Modular forms 1276 927 349
Cusp forms 1045 811 234
Eisenstein series 231 116 115

## Trace form

 $$811q - 3q^{2} - 30q^{3} - 65q^{4} - 6q^{5} - 32q^{6} - 66q^{7} - 15q^{8} - 30q^{9} + O(q^{10})$$ $$811q - 3q^{2} - 30q^{3} - 65q^{4} - 6q^{5} - 32q^{6} - 66q^{7} - 15q^{8} - 30q^{9} - 76q^{10} - 12q^{11} - 36q^{12} - 72q^{13} - 24q^{14} - 35q^{15} - 89q^{16} - 18q^{17} - 32q^{18} - 78q^{19} - 42q^{20} - 37q^{21} - 94q^{22} - 24q^{23} - 44q^{24} - 89q^{25} - 42q^{26} - 30q^{27} - 114q^{28} - 30q^{29} - 47q^{30} - 90q^{31} - 63q^{32} - 41q^{33} - 112q^{34} - 48q^{35} - 36q^{36} - 96q^{37} - 60q^{38} - 43q^{39} - 148q^{40} - 42q^{41} - 53q^{42} - 102q^{43} - 84q^{44} - 6q^{45} - 14q^{46} + 10q^{47} + 114q^{48} + q^{49} + 139q^{50} + 69q^{51} + 76q^{52} + 62q^{53} + 171q^{54} + 44q^{55} + 344q^{56} + 96q^{57} + 26q^{58} + 57q^{59} + 248q^{60} - 4q^{61} + 20q^{62} + 108q^{63} + 279q^{64} + 90q^{65} + 138q^{66} - 10q^{67} + 106q^{68} + 63q^{69} + 30q^{70} + 44q^{71} + 130q^{72} - 74q^{73} + 2q^{74} - 31q^{75} - 198q^{76} - 96q^{77} - 71q^{78} - 138q^{79} - 186q^{80} - 30q^{81} - 184q^{82} - 84q^{83} - 85q^{84} - 166q^{85} - 132q^{86} - 59q^{87} - 238q^{88} - 90q^{89} - 47q^{90} - 170q^{91} - 168q^{92} - 61q^{93} - 202q^{94} - 120q^{95} - 92q^{96} - 156q^{97} + 3q^{98} - 41q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(177))$$

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
177.2.a $$\chi_{177}(1, \cdot)$$ 177.2.a.a 2 1
177.2.a.b 2
177.2.a.c 2
177.2.a.d 3
177.2.d $$\chi_{177}(176, \cdot)$$ 177.2.d.a 6 1
177.2.d.b 6
177.2.d.c 6
177.2.e $$\chi_{177}(4, \cdot)$$ 177.2.e.a 140 28
177.2.e.b 140
177.2.f $$\chi_{177}(2, \cdot)$$ 177.2.f.a 504 28

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(177))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(177)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(59))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}$$)($$1 + T + 3 T^{2} + 2 T^{3} + 4 T^{4}$$)($$1 - T + 3 T^{2} - 2 T^{3} + 4 T^{4}$$)($$1 + 2 T^{2} - T^{3} + 4 T^{4} + 8 T^{6}$$)($$( 1 + 2 T + 2 T^{2} + T^{3} + 4 T^{4} + 8 T^{5} + 8 T^{6} )^{2}$$)($$( 1 + 2 T^{2} )^{6}$$)($$( 1 - 2 T + 2 T^{2} - T^{3} + 4 T^{4} - 8 T^{5} + 8 T^{6} )^{2}$$)
$3$ ($$( 1 - T )^{2}$$)($$( 1 + T )^{2}$$)($$( 1 - T )^{2}$$)($$( 1 + T )^{3}$$)($$1 + T + 3 T^{2} + 2 T^{3} + 9 T^{4} + 9 T^{5} + 27 T^{6}$$)($$1 - 7 T^{3} + 27 T^{6}$$)($$1 + T + 3 T^{2} + 2 T^{3} + 9 T^{4} + 9 T^{5} + 27 T^{6}$$)
$5$ ($$( 1 + 3 T + 5 T^{2} )^{2}$$)($$1 + 5 T^{2} + 25 T^{4}$$)($$( 1 - T + 5 T^{2} )^{2}$$)($$1 + 2 T + 10 T^{2} + 18 T^{3} + 50 T^{4} + 50 T^{5} + 125 T^{6}$$)($$1 - 13 T^{2} + 106 T^{4} - 637 T^{6} + 2650 T^{8} - 8125 T^{10} + 15625 T^{12}$$)($$( 1 - 21 T^{3} + 125 T^{6} )( 1 + 21 T^{3} + 125 T^{6} )$$)($$1 - 13 T^{2} + 106 T^{4} - 637 T^{6} + 2650 T^{8} - 8125 T^{10} + 15625 T^{12}$$)
$7$ ($$1 + 7 T + 25 T^{2} + 49 T^{3} + 49 T^{4}$$)($$1 + 7 T + 25 T^{2} + 49 T^{3} + 49 T^{4}$$)($$1 - T + 13 T^{2} - 7 T^{3} + 49 T^{4}$$)($$1 - 9 T + 44 T^{2} - 142 T^{3} + 308 T^{4} - 441 T^{5} + 343 T^{6}$$)($$( 1 + 17 T^{2} + T^{3} + 119 T^{4} + 343 T^{6} )^{2}$$)($$( 1 - 29 T^{3} + 343 T^{6} )^{2}$$)($$( 1 + 17 T^{2} + T^{3} + 119 T^{4} + 343 T^{6} )^{2}$$)
$11$ ($$1 + 2 T + 3 T^{2} + 22 T^{3} + 121 T^{4}$$)($$1 + 17 T^{2} + 121 T^{4}$$)($$1 - 4 T + 21 T^{2} - 44 T^{3} + 121 T^{4}$$)($$1 + 2 T + 22 T^{2} + 48 T^{3} + 242 T^{4} + 242 T^{5} + 1331 T^{6}$$)($$( 1 - 10 T + 60 T^{2} - 234 T^{3} + 660 T^{4} - 1210 T^{5} + 1331 T^{6} )^{2}$$)($$( 1 + 11 T^{2} )^{6}$$)($$( 1 + 10 T + 60 T^{2} + 234 T^{3} + 660 T^{4} + 1210 T^{5} + 1331 T^{6} )^{2}$$)
$13$ ($$1 - 19 T^{2} + 169 T^{4}$$)($$1 + 8 T + 37 T^{2} + 104 T^{3} + 169 T^{4}$$)($$( 1 + T + 13 T^{2} )^{2}$$)($$1 - 4 T + 32 T^{2} - 78 T^{3} + 416 T^{4} - 676 T^{5} + 2197 T^{6}$$)($$1 - 20 T^{2} + 516 T^{4} - 6542 T^{6} + 87204 T^{8} - 571220 T^{10} + 4826809 T^{12}$$)($$( 1 - 13 T^{2} )^{6}$$)($$1 - 20 T^{2} + 516 T^{4} - 6542 T^{6} + 87204 T^{8} - 571220 T^{10} + 4826809 T^{12}$$)
$17$ ($$1 + 3 T + 35 T^{2} + 51 T^{3} + 289 T^{4}$$)($$1 + 3 T + 25 T^{2} + 51 T^{3} + 289 T^{4}$$)($$1 - T + 23 T^{2} - 17 T^{3} + 289 T^{4}$$)($$1 - 3 T + 8 T^{2} - 4 T^{3} + 136 T^{4} - 867 T^{5} + 4913 T^{6}$$)($$1 - 75 T^{2} + 2696 T^{4} - 57772 T^{6} + 779144 T^{8} - 6264075 T^{10} + 24137569 T^{12}$$)($$( 1 - 3 T + 17 T^{2} )^{3}( 1 + 3 T + 17 T^{2} )^{3}$$)($$1 - 75 T^{2} + 2696 T^{4} - 57772 T^{6} + 779144 T^{8} - 6264075 T^{10} + 24137569 T^{12}$$)
$19$ ($$1 + 5 T + 33 T^{2} + 95 T^{3} + 361 T^{4}$$)($$1 + 5 T + 13 T^{2} + 95 T^{3} + 361 T^{4}$$)($$1 + 5 T + 33 T^{2} + 95 T^{3} + 361 T^{4}$$)($$1 - 7 T + 68 T^{2} - 270 T^{3} + 1292 T^{4} - 2527 T^{5} + 6859 T^{6}$$)($$( 1 - 2 T + 31 T^{2} - 113 T^{3} + 589 T^{4} - 722 T^{5} + 6859 T^{6} )^{2}$$)($$( 1 - 119 T^{3} + 6859 T^{6} )^{2}$$)($$( 1 - 2 T + 31 T^{2} - 113 T^{3} + 589 T^{4} - 722 T^{5} + 6859 T^{6} )^{2}$$)
$23$ ($$1 + 5 T + 51 T^{2} + 115 T^{3} + 529 T^{4}$$)($$1 + 7 T + 57 T^{2} + 161 T^{3} + 529 T^{4}$$)($$1 - 3 T + 37 T^{2} - 69 T^{3} + 529 T^{4}$$)($$1 - T + 42 T^{2} + 18 T^{3} + 966 T^{4} - 529 T^{5} + 12167 T^{6}$$)($$( 1 - 9 T + 92 T^{2} - 428 T^{3} + 2116 T^{4} - 4761 T^{5} + 12167 T^{6} )^{2}$$)($$( 1 + 23 T^{2} )^{6}$$)($$( 1 + 9 T + 92 T^{2} + 428 T^{3} + 2116 T^{4} + 4761 T^{5} + 12167 T^{6} )^{2}$$)
$29$ ($$1 + 11 T + 87 T^{2} + 319 T^{3} + 841 T^{4}$$)($$1 - 15 T + 113 T^{2} - 435 T^{3} + 841 T^{4}$$)($$1 - 5 T + 63 T^{2} - 145 T^{3} + 841 T^{4}$$)($$1 + 11 T + 96 T^{2} + 564 T^{3} + 2784 T^{4} + 9251 T^{5} + 24389 T^{6}$$)($$1 - 120 T^{2} + 6965 T^{4} - 249781 T^{6} + 5857565 T^{8} - 84873720 T^{10} + 594823321 T^{12}$$)($$( 1 - 159 T^{3} + 24389 T^{6} )( 1 + 159 T^{3} + 24389 T^{6} )$$)($$1 - 120 T^{2} + 6965 T^{4} - 249781 T^{6} + 5857565 T^{8} - 84873720 T^{10} + 594823321 T^{12}$$)
$31$ ($$1 + 7 T + 73 T^{2} + 217 T^{3} + 961 T^{4}$$)($$1 + T - 39 T^{2} + 31 T^{3} + 961 T^{4}$$)($$1 + T + 51 T^{2} + 31 T^{3} + 961 T^{4}$$)($$1 - 13 T + 130 T^{2} - 778 T^{3} + 4030 T^{4} - 12493 T^{5} + 29791 T^{6}$$)($$1 - 71 T^{2} + 3744 T^{4} - 137216 T^{6} + 3597984 T^{8} - 65569991 T^{10} + 887503681 T^{12}$$)($$( 1 - 31 T^{2} )^{6}$$)($$1 - 71 T^{2} + 3744 T^{4} - 137216 T^{6} + 3597984 T^{8} - 65569991 T^{10} + 887503681 T^{12}$$)
$37$ ($$1 + T + 43 T^{2} + 37 T^{3} + 1369 T^{4}$$)($$1 + 7 T + 75 T^{2} + 259 T^{3} + 1369 T^{4}$$)($$1 + 9 T + 93 T^{2} + 333 T^{3} + 1369 T^{4}$$)($$1 + 5 T + 92 T^{2} + 384 T^{3} + 3404 T^{4} + 6845 T^{5} + 50653 T^{6}$$)($$1 + T^{2} + 1068 T^{4} + 37816 T^{6} + 1462092 T^{8} + 1874161 T^{10} + 2565726409 T^{12}$$)($$( 1 - 37 T^{2} )^{6}$$)($$1 + T^{2} + 1068 T^{4} + 37816 T^{6} + 1462092 T^{8} + 1874161 T^{10} + 2565726409 T^{12}$$)
$41$ ($$1 + T + 51 T^{2} + 41 T^{3} + 1681 T^{4}$$)($$1 - 5 T + 57 T^{2} - 205 T^{3} + 1681 T^{4}$$)($$1 + T + 51 T^{2} + 41 T^{3} + 1681 T^{4}$$)($$1 + T + 84 T^{2} + 156 T^{3} + 3444 T^{4} + 1681 T^{5} + 68921 T^{6}$$)($$1 - 188 T^{2} + 15985 T^{4} - 816533 T^{6} + 26870785 T^{8} - 531243068 T^{10} + 4750104241 T^{12}$$)($$( 1 - 525 T^{3} + 68921 T^{6} )( 1 + 525 T^{3} + 68921 T^{6} )$$)($$1 - 188 T^{2} + 15985 T^{4} - 816533 T^{6} + 26870785 T^{8} - 531243068 T^{10} + 4750104241 T^{12}$$)
$43$ ($$1 - 12 T + 77 T^{2} - 516 T^{3} + 1849 T^{4}$$)($$1 + 4 T + 45 T^{2} + 172 T^{3} + 1849 T^{4}$$)($$1 + 2 T + 7 T^{2} + 86 T^{3} + 1849 T^{4}$$)($$1 - 6 T + 38 T^{2} + 76 T^{3} + 1634 T^{4} - 11094 T^{5} + 79507 T^{6}$$)($$1 - 52 T^{2} + 5868 T^{4} - 182854 T^{6} + 10849932 T^{8} - 177777652 T^{10} + 6321363049 T^{12}$$)($$( 1 - 43 T^{2} )^{6}$$)($$1 - 52 T^{2} + 5868 T^{4} - 182854 T^{6} + 10849932 T^{8} - 177777652 T^{10} + 6321363049 T^{12}$$)
$47$ ($$1 + 3 T + 85 T^{2} + 141 T^{3} + 2209 T^{4}$$)($$1 + 15 T + 139 T^{2} + 705 T^{3} + 2209 T^{4}$$)($$1 - 11 T + 123 T^{2} - 517 T^{3} + 2209 T^{4}$$)($$1 - 11 T + 104 T^{2} - 538 T^{3} + 4888 T^{4} - 24299 T^{5} + 103823 T^{6}$$)($$( 1 + 23 T + 312 T^{2} + 2568 T^{3} + 14664 T^{4} + 50807 T^{5} + 103823 T^{6} )^{2}$$)($$( 1 + 47 T^{2} )^{6}$$)($$( 1 - 23 T + 312 T^{2} - 2568 T^{3} + 14664 T^{4} - 50807 T^{5} + 103823 T^{6} )^{2}$$)
$53$ ($$1 - 2 T + 87 T^{2} - 106 T^{3} + 2809 T^{4}$$)($$1 - 8 T + 117 T^{2} - 424 T^{3} + 2809 T^{4}$$)($$1 + 2 T + 27 T^{2} + 106 T^{3} + 2809 T^{4}$$)($$1 - 2 T + 70 T^{2} - 270 T^{3} + 3710 T^{4} - 5618 T^{5} + 148877 T^{6}$$)($$1 - 197 T^{2} + 19498 T^{4} - 1253909 T^{6} + 54769882 T^{8} - 1554424757 T^{10} + 22164361129 T^{12}$$)($$( 1 - 327 T^{3} + 148877 T^{6} )( 1 + 327 T^{3} + 148877 T^{6} )$$)($$1 - 197 T^{2} + 19498 T^{4} - 1253909 T^{6} + 54769882 T^{8} - 1554424757 T^{10} + 22164361129 T^{12}$$)
$59$ ($$( 1 - T )^{2}$$)($$( 1 + T )^{2}$$)($$( 1 + T )^{2}$$)($$( 1 - T )^{3}$$)($$1 - 10 T + 85 T^{2} - 732 T^{3} + 5015 T^{4} - 34810 T^{5} + 205379 T^{6}$$)($$( 1 + 59 T^{2} )^{3}$$)($$1 + 10 T + 85 T^{2} + 732 T^{3} + 5015 T^{4} + 34810 T^{5} + 205379 T^{6}$$)
$61$ ($$1 - 13 T + 133 T^{2} - 793 T^{3} + 3721 T^{4}$$)($$1 + 13 T + 153 T^{2} + 793 T^{3} + 3721 T^{4}$$)($$1 + T + 111 T^{2} + 61 T^{3} + 3721 T^{4}$$)($$1 + T + 82 T^{2} + 220 T^{3} + 5002 T^{4} + 3721 T^{5} + 226981 T^{6}$$)($$1 - 167 T^{2} + 17204 T^{4} - 1305528 T^{6} + 64016084 T^{8} - 2312255447 T^{10} + 51520374361 T^{12}$$)($$( 1 - 61 T^{2} )^{6}$$)($$1 - 167 T^{2} + 17204 T^{4} - 1305528 T^{6} + 64016084 T^{8} - 2312255447 T^{10} + 51520374361 T^{12}$$)
$67$ ($$1 + 4 T + 133 T^{2} + 268 T^{3} + 4489 T^{4}$$)($$1 - 8 T + 105 T^{2} - 536 T^{3} + 4489 T^{4}$$)($$1 - 6 T + 63 T^{2} - 402 T^{3} + 4489 T^{4}$$)($$1 - 10 T + 82 T^{2} - 556 T^{3} + 5494 T^{4} - 44890 T^{5} + 300763 T^{6}$$)($$1 - 344 T^{2} + 52788 T^{4} - 4582178 T^{6} + 236965332 T^{8} - 6931985624 T^{10} + 90458382169 T^{12}$$)($$( 1 - 67 T^{2} )^{6}$$)($$1 - 344 T^{2} + 52788 T^{4} - 4582178 T^{6} + 236965332 T^{8} - 6931985624 T^{10} + 90458382169 T^{12}$$)
$71$ ($$1 - 4 T + 141 T^{2} - 284 T^{3} + 5041 T^{4}$$)($$1 - 2 T + 63 T^{2} - 142 T^{3} + 5041 T^{4}$$)($$1 - 4 T + 101 T^{2} - 284 T^{3} + 5041 T^{4}$$)($$1 - 26 T + 406 T^{2} - 4116 T^{3} + 28826 T^{4} - 131066 T^{5} + 357911 T^{6}$$)($$1 - 120 T^{2} + 132 T^{4} + 542342 T^{6} + 665412 T^{8} - 3049401720 T^{10} + 128100283921 T^{12}$$)($$( 1 - 15 T + 71 T^{2} )^{3}( 1 + 15 T + 71 T^{2} )^{3}$$)($$1 - 120 T^{2} + 132 T^{4} + 542342 T^{6} + 665412 T^{8} - 3049401720 T^{10} + 128100283921 T^{12}$$)
$73$ ($$1 - 5 T + 121 T^{2} - 365 T^{3} + 5329 T^{4}$$)($$1 + 5 T + 141 T^{2} + 365 T^{3} + 5329 T^{4}$$)($$1 - 3 T + 147 T^{2} - 219 T^{3} + 5329 T^{4}$$)($$1 - 7 T + 78 T^{2} - 304 T^{3} + 5694 T^{4} - 37303 T^{5} + 389017 T^{6}$$)($$1 - 191 T^{2} + 21072 T^{4} - 1746740 T^{6} + 112292688 T^{8} - 5424064031 T^{10} + 151334226289 T^{12}$$)($$( 1 - 73 T^{2} )^{6}$$)($$1 - 191 T^{2} + 21072 T^{4} - 1746740 T^{6} + 112292688 T^{8} - 5424064031 T^{10} + 151334226289 T^{12}$$)
$79$ ($$( 1 + 3 T + 79 T^{2} )^{2}$$)($$( 1 + 3 T + 79 T^{2} )^{2}$$)($$1 - 10 T + 103 T^{2} - 790 T^{3} + 6241 T^{4}$$)($$1 - 2 T + 206 T^{2} - 348 T^{3} + 16274 T^{4} - 12482 T^{5} + 493039 T^{6}$$)($$( 1 + 7 T + 152 T^{2} + 759 T^{3} + 12008 T^{4} + 43687 T^{5} + 493039 T^{6} )^{2}$$)($$( 1 + 835 T^{3} + 493039 T^{6} )^{2}$$)($$( 1 + 7 T + 152 T^{2} + 759 T^{3} + 12008 T^{4} + 43687 T^{5} + 493039 T^{6} )^{2}$$)
$83$ ($$1 + 9 T + 175 T^{2} + 747 T^{3} + 6889 T^{4}$$)($$1 + T + 165 T^{2} + 83 T^{3} + 6889 T^{4}$$)($$1 - 13 T + 197 T^{2} - 1079 T^{3} + 6889 T^{4}$$)($$1 + 3 T + 50 T^{2} + 350 T^{3} + 4150 T^{4} + 20667 T^{5} + 571787 T^{6}$$)($$( 1 - 25 T + 400 T^{2} - 4164 T^{3} + 33200 T^{4} - 172225 T^{5} + 571787 T^{6} )^{2}$$)($$( 1 + 83 T^{2} )^{6}$$)($$( 1 + 25 T + 400 T^{2} + 4164 T^{3} + 33200 T^{4} + 172225 T^{5} + 571787 T^{6} )^{2}$$)
$89$ ($$1 - T + 147 T^{2} - 89 T^{3} + 7921 T^{4}$$)($$1 + 3 T + 29 T^{2} + 267 T^{3} + 7921 T^{4}$$)($$1 + 5 T + 153 T^{2} + 445 T^{3} + 7921 T^{4}$$)($$1 + 23 T + 358 T^{2} + 3816 T^{3} + 31862 T^{4} + 182183 T^{5} + 704969 T^{6}$$)($$( 1 + 13 T + 242 T^{2} + 1796 T^{3} + 21538 T^{4} + 102973 T^{5} + 704969 T^{6} )^{2}$$)($$( 1 + 89 T^{2} )^{6}$$)($$( 1 - 13 T + 242 T^{2} - 1796 T^{3} + 21538 T^{4} - 102973 T^{5} + 704969 T^{6} )^{2}$$)
$97$ ($$1 + 10 T + 199 T^{2} + 970 T^{3} + 9409 T^{4}$$)($$( 1 - 3 T + 97 T^{2} )^{2}$$)($$1 + 4 T + 153 T^{2} + 388 T^{3} + 9409 T^{4}$$)($$1 - 14 T + 266 T^{2} - 2514 T^{3} + 25802 T^{4} - 131726 T^{5} + 912673 T^{6}$$)($$1 - 240 T^{2} + 15788 T^{4} - 353866 T^{6} + 148549292 T^{8} - 21247027440 T^{10} + 832972004929 T^{12}$$)($$( 1 - 97 T^{2} )^{6}$$)($$1 - 240 T^{2} + 15788 T^{4} - 353866 T^{6} + 148549292 T^{8} - 21247027440 T^{10} + 832972004929 T^{12}$$)