# Properties

 Label 77.2.l Level 77 Weight 2 Character orbit l Rep. character $$\chi_{77}(6,\cdot)$$ Character field $$\Q(\zeta_{10})$$ Dimension 24 Newform subspaces 2 Sturm bound 16 Trace bound 1

# Related objects

## Defining parameters

 Level: $$N$$ = $$77 = 7 \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 77.l (of order $$10$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$77$$ Character field: $$\Q(\zeta_{10})$$ Newform subspaces: $$2$$ Sturm bound: $$16$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(77, [\chi])$$.

Total New Old
Modular forms 40 40 0
Cusp forms 24 24 0
Eisenstein series 16 16 0

## Trace form

 $$24q - 10q^{2} - 10q^{7} + 2q^{9} + O(q^{10})$$ $$24q - 10q^{2} - 10q^{7} + 2q^{9} - 2q^{11} - 13q^{14} - 6q^{16} - 35q^{18} + 28q^{22} + 8q^{23} - 40q^{25} + 25q^{28} + 10q^{29} - 40q^{30} + 40q^{35} + 50q^{36} - 14q^{37} + 30q^{39} + 50q^{42} + 15q^{44} + 5q^{46} + 22q^{49} - 60q^{50} - 10q^{51} + 30q^{53} - 46q^{56} - 90q^{57} + 17q^{58} + 120q^{60} - 20q^{63} - 72q^{64} + 4q^{67} - 104q^{71} - 45q^{72} + 90q^{74} - 12q^{77} - 20q^{78} + 10q^{79} - 34q^{81} - 70q^{84} + 80q^{85} + 29q^{86} - 94q^{88} - 30q^{91} + 45q^{92} - 40q^{93} - 60q^{95} + 84q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(77, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
77.2.l.a $$8$$ $$0.615$$ 8.0.37515625.1 $$\Q(\sqrt{-7})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+(1-\beta _{1}+\beta _{3}+\beta _{4}+\beta _{5}-\beta _{6})q^{2}+\cdots$$
77.2.l.b $$16$$ $$0.615$$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$-10$$ $$0$$ $$0$$ $$-10$$ $$q+(-1-\beta _{9}-\beta _{11})q^{2}+(-\beta _{1}-\beta _{2}+\cdots)q^{3}+\cdots$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$( 1 - T - T^{2} + 3 T^{3} - T^{4} + 6 T^{5} - 4 T^{6} - 8 T^{7} + 16 T^{8} )( 1 + T - T^{2} - 3 T^{3} - T^{4} - 6 T^{5} - 4 T^{6} + 8 T^{7} + 16 T^{8} )$$)($$( 1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4} )^{4}( 1 - T + 4 T^{2} - 2 T^{3} + 9 T^{4} - 4 T^{5} + 16 T^{6} - 8 T^{7} + 16 T^{8} )^{2}$$)
$3$ ($$( 1 + 3 T^{2} + 9 T^{4} + 27 T^{6} + 81 T^{8} )^{2}$$)($$1 + 2 T^{2} + 10 T^{6} + 65 T^{8} + 466 T^{10} + 912 T^{12} + 1220 T^{14} + 6865 T^{16} + 10980 T^{18} + 73872 T^{20} + 339714 T^{22} + 426465 T^{24} + 590490 T^{26} + 9565938 T^{30} + 43046721 T^{32}$$)
$5$ ($$( 1 - 5 T + 15 T^{2} - 25 T^{3} + 25 T^{4} )^{2}( 1 + 5 T + 15 T^{2} + 25 T^{3} + 25 T^{4} )^{2}$$)($$1 + 25 T^{2} + 235 T^{4} + 830 T^{6} - 985 T^{8} - 8750 T^{10} + 76100 T^{12} + 846725 T^{14} + 4674775 T^{16} + 21168125 T^{18} + 47562500 T^{20} - 136718750 T^{22} - 384765625 T^{24} + 8105468750 T^{26} + 57373046875 T^{28} + 152587890625 T^{30} + 152587890625 T^{32}$$)
$7$ ($$1 - 7 T^{2} + 49 T^{4} - 343 T^{6} + 2401 T^{8}$$)($$1 + 10 T + 46 T^{2} + 110 T^{3} + 67 T^{4} - 430 T^{5} - 1612 T^{6} - 3140 T^{7} - 6135 T^{8} - 21980 T^{9} - 78988 T^{10} - 147490 T^{11} + 160867 T^{12} + 1848770 T^{13} + 5411854 T^{14} + 8235430 T^{15} + 5764801 T^{16}$$)
$11$ ($$1 + 4 T + 5 T^{2} - 24 T^{3} - 151 T^{4} - 264 T^{5} + 605 T^{6} + 5324 T^{7} + 14641 T^{8}$$)($$( 1 - T - 20 T^{2} + T^{3} + 309 T^{4} + 11 T^{5} - 2420 T^{6} - 1331 T^{7} + 14641 T^{8} )^{2}$$)
$13$ ($$( 1 - 13 T^{2} + 169 T^{4} - 2197 T^{6} + 28561 T^{8} )^{2}$$)($$1 - 37 T^{2} + 905 T^{4} - 17590 T^{6} + 252535 T^{8} - 3649636 T^{10} + 49961932 T^{12} - 672618165 T^{14} + 9550335135 T^{16} - 113672469885 T^{18} + 1426962739852 T^{20} - 17616095891524 T^{22} + 206000557627735 T^{24} - 2424930871623910 T^{26} + 21084767035845305 T^{28} - 145682926270873693 T^{30} + 665416609183179841 T^{32}$$)
$17$ ($$( 1 - 17 T^{2} + 289 T^{4} - 4913 T^{6} + 83521 T^{8} )^{2}$$)($$1 - 53 T^{2} + 935 T^{4} + 10075 T^{6} - 695885 T^{8} + 10961626 T^{10} - 1176848 T^{12} - 2883558960 T^{14} + 66310473675 T^{16} - 833348539440 T^{18} - 98291521808 T^{20} + 264587003927194 T^{22} - 4854324966830285 T^{24} + 20311138547023675 T^{26} + 544751791809826535 T^{28} - 8924024807648249237 T^{30} + 48661191875666868481 T^{32}$$)
$19$ ($$( 1 - 19 T^{2} + 361 T^{4} - 6859 T^{6} + 130321 T^{8} )^{2}$$)($$1 - 131 T^{2} + 7835 T^{4} - 269555 T^{6} + 5344225 T^{8} - 36384548 T^{10} - 1284840662 T^{12} + 54300385290 T^{14} - 1231352034765 T^{16} + 19602439089690 T^{18} - 167441719912502 T^{20} - 1711743115446788 T^{22} + 90763982192788225 T^{24} - 1652659565121548555 T^{26} + 17341322390883371435 T^{28} -$$$$10\!\cdots\!51$$$$T^{30} +$$$$28\!\cdots\!81$$$$T^{32}$$)
$23$ ($$( 1 - 8 T + 41 T^{2} - 144 T^{3} + 209 T^{4} - 3312 T^{5} + 21689 T^{6} - 97336 T^{7} + 279841 T^{8} )^{2}$$)($$( 1 + T + 35 T^{2} + 23 T^{3} + 529 T^{4} )^{8}$$)
$29$ ($$( 1 - 2 T - 25 T^{2} + 108 T^{3} + 509 T^{4} + 3132 T^{5} - 21025 T^{6} - 48778 T^{7} + 707281 T^{8} )( 1 + 2 T - 25 T^{2} - 108 T^{3} + 509 T^{4} - 3132 T^{5} - 21025 T^{6} + 48778 T^{7} + 707281 T^{8} )$$)($$( 1 - 5 T + 66 T^{2} - 555 T^{3} + 3920 T^{4} - 28200 T^{5} + 189134 T^{6} - 1026380 T^{7} + 6343899 T^{8} - 29765020 T^{9} + 159061694 T^{10} - 687769800 T^{11} + 2772541520 T^{12} - 11383687695 T^{13} + 39258339186 T^{14} - 86249381545 T^{15} + 500246412961 T^{16} )^{2}$$)
$31$ ($$( 1 + 31 T^{2} + 961 T^{4} + 29791 T^{6} + 923521 T^{8} )^{2}$$)($$1 + 39 T^{2} + 3430 T^{4} + 163065 T^{6} + 6773220 T^{8} + 296986872 T^{10} + 10144743998 T^{12} + 345739802400 T^{14} + 11501584169495 T^{16} + 332255950106400 T^{18} + 9368884121776958 T^{20} + 263576942108675832 T^{22} + 5776818632616130020 T^{24} +$$$$13\!\cdots\!65$$$$T^{26} +$$$$27\!\cdots\!30$$$$T^{28} +$$$$29\!\cdots\!19$$$$T^{30} +$$$$72\!\cdots\!81$$$$T^{32}$$)
$37$ ($$( 1 + 6 T + 37 T^{2} )^{4}( 1 - 6 T - T^{2} + 228 T^{3} - 1331 T^{4} + 8436 T^{5} - 1369 T^{6} - 303918 T^{7} + 1874161 T^{8} )$$)($$( 1 - 2 T - 31 T^{2} + 88 T^{3} - 122 T^{4} - 18302 T^{5} + 81031 T^{6} + 289434 T^{7} - 1619457 T^{8} + 10709058 T^{9} + 110931439 T^{10} - 927051206 T^{11} - 228647642 T^{12} + 6102268216 T^{13} - 79537518679 T^{14} - 189863754266 T^{15} + 3512479453921 T^{16} )^{2}$$)
$41$ ($$( 1 - 41 T^{2} + 1681 T^{4} - 68921 T^{6} + 2825761 T^{8} )^{2}$$)($$1 - 54 T^{2} + 5160 T^{4} - 368270 T^{6} + 20358825 T^{8} - 1151359622 T^{10} + 57899888128 T^{12} - 2567003976540 T^{14} + 113908617753985 T^{16} - 4315133684563740 T^{18} + 163611245776465408 T^{20} - 5469078223378356902 T^{22} +$$$$16\!\cdots\!25$$$$T^{24} -$$$$49\!\cdots\!70$$$$T^{26} +$$$$11\!\cdots\!60$$$$T^{28} -$$$$20\!\cdots\!94$$$$T^{30} +$$$$63\!\cdots\!41$$$$T^{32}$$)
$43$ ($$( 1 - 12 T + 101 T^{2} - 696 T^{3} + 4009 T^{4} - 29928 T^{5} + 186749 T^{6} - 954084 T^{7} + 3418801 T^{8} )( 1 + 12 T + 101 T^{2} + 696 T^{3} + 4009 T^{4} + 29928 T^{5} + 186749 T^{6} + 954084 T^{7} + 3418801 T^{8} )$$)($$( 1 - 13 T + 43 T^{2} )^{8}( 1 + 13 T + 43 T^{2} )^{8}$$)
$47$ ($$( 1 + 47 T^{2} + 2209 T^{4} + 103823 T^{6} + 4879681 T^{8} )^{2}$$)($$1 + 173 T^{2} + 13870 T^{4} + 773020 T^{6} + 37466570 T^{8} + 1409514809 T^{10} + 35630259662 T^{12} + 1022803179550 T^{14} + 50909696806195 T^{16} + 2259372223625950 T^{18} + 173864301097727822 T^{20} + 15193463635625307161 T^{22} +$$$$89\!\cdots\!70$$$$T^{24} +$$$$40\!\cdots\!80$$$$T^{26} +$$$$16\!\cdots\!70$$$$T^{28} +$$$$44\!\cdots\!37$$$$T^{30} +$$$$56\!\cdots\!21$$$$T^{32}$$)
$53$ ($$( 1 - 10 T + 53 T^{2} )^{4}( 1 + 10 T + 47 T^{2} - 60 T^{3} - 3091 T^{4} - 3180 T^{5} + 132023 T^{6} + 1488770 T^{7} + 7890481 T^{8} )$$)($$( 1 - 71 T^{2} - 150 T^{3} + 6432 T^{4} + 1950 T^{5} - 488983 T^{6} + 192300 T^{7} + 26002355 T^{8} + 10191900 T^{9} - 1373553247 T^{10} + 290310150 T^{11} + 50751573792 T^{12} - 62729323950 T^{13} - 1573669640159 T^{14} + 62259690411361 T^{16} )^{2}$$)
$59$ ($$( 1 + 59 T^{2} + 3481 T^{4} + 205379 T^{6} + 12117361 T^{8} )^{2}$$)($$1 + 181 T^{2} + 12520 T^{4} + 229955 T^{6} - 15999940 T^{8} - 623745812 T^{10} + 48156426998 T^{12} + 5123867326280 T^{14} + 307569246051235 T^{16} + 17836182162780680 T^{18} + 583528810404912278 T^{20} - 26309931206498861492 T^{22} -$$$$23\!\cdots\!40$$$$T^{24} +$$$$11\!\cdots\!55$$$$T^{26} +$$$$22\!\cdots\!20$$$$T^{28} +$$$$11\!\cdots\!41$$$$T^{30} +$$$$21\!\cdots\!41$$$$T^{32}$$)
$61$ ($$( 1 - 61 T^{2} + 3721 T^{4} - 226981 T^{6} + 13845841 T^{8} )^{2}$$)($$1 - 189 T^{2} + 16680 T^{4} - 1152085 T^{6} + 74808120 T^{8} - 4782904102 T^{10} + 347814490078 T^{12} - 24590631974850 T^{14} + 1546838296855255 T^{16} - 91501741578416850 T^{18} + 4815784127116065598 T^{20} -$$$$24\!\cdots\!22$$$$T^{22} +$$$$14\!\cdots\!20$$$$T^{24} -$$$$82\!\cdots\!85$$$$T^{26} +$$$$44\!\cdots\!80$$$$T^{28} -$$$$18\!\cdots\!49$$$$T^{30} +$$$$36\!\cdots\!61$$$$T^{32}$$)
$67$ ($$( 1 - 4 T - 51 T^{2} + 472 T^{3} + 1529 T^{4} + 31624 T^{5} - 228939 T^{6} - 1203052 T^{7} + 20151121 T^{8} )^{2}$$)($$( 1 + T + 159 T^{2} - 238 T^{3} + 12569 T^{4} - 15946 T^{5} + 713751 T^{6} + 300763 T^{7} + 20151121 T^{8} )^{4}$$)
$71$ ($$( 1 + 16 T + 71 T^{2} )^{4}( 1 - 16 T + 185 T^{2} - 1824 T^{3} + 16049 T^{4} - 129504 T^{5} + 932585 T^{6} - 5726576 T^{7} + 25411681 T^{8} )$$)($$( 1 + 28 T + 311 T^{2} + 1907 T^{3} + 10031 T^{4} - 8251 T^{5} - 1509411 T^{6} - 19087514 T^{7} - 158934393 T^{8} - 1355213494 T^{9} - 7608940851 T^{10} - 2953123661 T^{11} + 254904572111 T^{12} + 3440665372357 T^{13} + 39839188299431 T^{14} + 254663364434948 T^{15} + 645753531245761 T^{16} )^{2}$$)
$73$ ($$( 1 - 73 T^{2} + 5329 T^{4} - 389017 T^{6} + 28398241 T^{8} )^{2}$$)($$1 + 23 T^{2} + 14710 T^{4} + 27475 T^{6} + 102964580 T^{8} - 4851528886 T^{10} + 427407600722 T^{12} - 53265322210310 T^{14} + 1562285407061275 T^{16} - 283850902058741990 T^{18} + 12137624050535130002 T^{20} -$$$$73\!\cdots\!54$$$$T^{22} +$$$$83\!\cdots\!80$$$$T^{24} +$$$$11\!\cdots\!75$$$$T^{26} +$$$$33\!\cdots\!10$$$$T^{28} +$$$$28\!\cdots\!07$$$$T^{30} +$$$$65\!\cdots\!61$$$$T^{32}$$)
$79$ ($$( 1 + 8 T + 79 T^{2} )^{4}( 1 + 8 T - 15 T^{2} - 752 T^{3} - 4831 T^{4} - 59408 T^{5} - 93615 T^{6} + 3944312 T^{7} + 38950081 T^{8} )$$)($$( 1 - 25 T + 501 T^{2} - 8100 T^{3} + 108715 T^{4} - 1313450 T^{5} + 14241744 T^{6} - 141629175 T^{7} + 1311396489 T^{8} - 11188704825 T^{9} + 88882724304 T^{10} - 647582074550 T^{11} + 4234458055915 T^{12} - 24924156831900 T^{13} + 121786815216021 T^{14} - 480097724653975 T^{15} + 1517108809906561 T^{16} )^{2}$$)
$83$ ($$( 1 - 83 T^{2} + 6889 T^{4} - 571787 T^{6} + 47458321 T^{8} )^{2}$$)($$1 - 97 T^{2} + 16850 T^{4} - 1726555 T^{6} + 145607140 T^{8} - 9460226116 T^{10} + 616698856582 T^{12} - 21019950912420 T^{14} + 1372062609692295 T^{16} - 144806441835661380 T^{18} + 29267492296001518822 T^{20} -$$$$30\!\cdots\!04$$$$T^{22} +$$$$32\!\cdots\!40$$$$T^{24} -$$$$26\!\cdots\!95$$$$T^{26} +$$$$18\!\cdots\!50$$$$T^{28} -$$$$71\!\cdots\!13$$$$T^{30} +$$$$50\!\cdots\!81$$$$T^{32}$$)
$89$ ($$( 1 - 89 T^{2} )^{8}$$)($$( 1 - 547 T^{2} + 136828 T^{4} - 20978449 T^{6} + 2212310245 T^{8} - 166170294529 T^{10} + 8584895351548 T^{12} - 271848766155667 T^{14} + 3936588805702081 T^{16} )^{2}$$)
$97$ ($$( 1 + 97 T^{2} + 9409 T^{4} + 912673 T^{6} + 88529281 T^{8} )^{2}$$)($$1 + 378 T^{2} + 43195 T^{4} - 1829100 T^{6} - 951404880 T^{8} - 97555724586 T^{10} - 1788063513133 T^{12} + 794596480873200 T^{14} + 120216740714426075 T^{16} + 7476358288535938800 T^{18} -$$$$15\!\cdots\!73$$$$T^{20} -$$$$81\!\cdots\!94$$$$T^{22} -$$$$74\!\cdots\!80$$$$T^{24} -$$$$13\!\cdots\!00$$$$T^{26} +$$$$29\!\cdots\!95$$$$T^{28} +$$$$24\!\cdots\!82$$$$T^{30} +$$$$61\!\cdots\!21$$$$T^{32}$$)