# Properties

 Label 77.2.i Level 77 Weight 2 Character orbit i Rep. character $$\chi_{77}(10,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 12 Newform subspaces 1 Sturm bound 16 Trace bound 0

# Learn more about

## Defining parameters

 Level: $$N$$ = $$77 = 7 \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 77.i (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$77$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$1$$ Sturm bound: $$16$$ Trace bound: $$0$$

## Dimensions

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

Total New Old
Modular forms 20 20 0
Cusp forms 12 12 0
Eisenstein series 8 8 0

## Trace form

 $$12q - 6q^{3} + 4q^{4} - 4q^{9} + O(q^{10})$$ $$12q - 6q^{3} + 4q^{4} - 4q^{9} - 4q^{11} - 18q^{12} + 8q^{14} - 20q^{15} + 12q^{16} - 4q^{22} - 20q^{23} + 14q^{25} + 18q^{26} + 6q^{31} + 18q^{33} - 12q^{36} + 16q^{37} - 48q^{38} + 16q^{42} + 20q^{44} + 54q^{45} - 18q^{47} + 16q^{49} - 2q^{53} + 18q^{56} - 6q^{58} - 12q^{59} + 28q^{64} - 42q^{66} - 24q^{67} - 58q^{70} + 20q^{71} - 78q^{75} - 50q^{77} + 8q^{78} + 30q^{80} + 14q^{81} + 54q^{82} - 38q^{86} - 4q^{88} - 66q^{89} + 22q^{91} - 20q^{92} + 12q^{93} + 12q^{99} + O(q^{100})$$

## 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.i.a $$12$$ $$0.615$$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$-6$$ $$0$$ $$0$$ $$q+(\beta _{1}+\beta _{7})q^{2}+(-1+\beta _{2}-\beta _{8}-\beta _{9}+\cdots)q^{3}+\cdots$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T^{2} + 3 T^{4} - 6 T^{6} + T^{8} + 25 T^{10} + 37 T^{12} + 100 T^{14} + 16 T^{16} - 384 T^{18} + 768 T^{20} + 4096 T^{22} + 4096 T^{24}$$
$3$ $$( 1 + 3 T + 10 T^{2} + 21 T^{3} + 43 T^{4} + 78 T^{5} + 141 T^{6} + 234 T^{7} + 387 T^{8} + 567 T^{9} + 810 T^{10} + 729 T^{11} + 729 T^{12} )^{2}$$
$5$ $$( 1 + 4 T^{2} - 4 T^{4} + 12 T^{5} - 167 T^{6} + 60 T^{7} - 100 T^{8} + 2500 T^{10} + 15625 T^{12} )^{2}$$
$7$ $$1 - 8 T^{2} + 62 T^{4} - 203 T^{6} + 3038 T^{8} - 19208 T^{10} + 117649 T^{12}$$
$11$ $$1 + 4 T - 5 T^{2} - 44 T^{3} - 46 T^{4} - 20 T^{5} - 29 T^{6} - 220 T^{7} - 5566 T^{8} - 58564 T^{9} - 73205 T^{10} + 644204 T^{11} + 1771561 T^{12}$$
$13$ $$( 1 + 67 T^{2} + 1997 T^{4} + 33649 T^{6} + 337493 T^{8} + 1913587 T^{10} + 4826809 T^{12} )^{2}$$
$17$ $$1 - 63 T^{2} + 1818 T^{4} - 45823 T^{6} + 1179927 T^{8} - 23396904 T^{10} + 390708825 T^{12} - 6761705256 T^{14} + 98548682967 T^{16} - 1106055824287 T^{18} + 12681927027738 T^{20} - 127007615728287 T^{22} + 582622237229761 T^{24}$$
$19$ $$1 - 31 T^{2} - 322 T^{4} + 5085 T^{6} + 487627 T^{8} - 4100252 T^{10} - 122324807 T^{12} - 1480190972 T^{14} + 63548038267 T^{16} + 239228304885 T^{18} - 5468707299202 T^{20} - 190063053991831 T^{22} + 2213314919066161 T^{24}$$
$23$ $$( 1 + 10 T + 4 T^{2} - 2 T^{3} + 2198 T^{4} + 7306 T^{5} - 9173 T^{6} + 168038 T^{7} + 1162742 T^{8} - 24334 T^{9} + 1119364 T^{10} + 64363430 T^{11} + 148035889 T^{12} )^{2}$$
$29$ $$( 1 - 91 T^{2} + 4663 T^{4} - 160283 T^{6} + 3921583 T^{8} - 64362571 T^{10} + 594823321 T^{12} )^{2}$$
$31$ $$( 1 - 3 T + 84 T^{2} - 243 T^{3} + 4227 T^{4} - 11454 T^{5} + 151369 T^{6} - 355074 T^{7} + 4062147 T^{8} - 7239213 T^{9} + 77575764 T^{10} - 85887453 T^{11} + 887503681 T^{12} )^{2}$$
$37$ $$( 1 - 8 T - 51 T^{2} + 216 T^{3} + 4426 T^{4} - 8192 T^{5} - 153011 T^{6} - 303104 T^{7} + 6059194 T^{8} + 10941048 T^{9} - 95582211 T^{10} - 554751656 T^{11} + 2565726409 T^{12} )^{2}$$
$41$ $$( 1 + 147 T^{2} + 11733 T^{4} + 590425 T^{6} + 19723173 T^{8} + 415386867 T^{10} + 4750104241 T^{12} )^{2}$$
$43$ $$( 1 - 182 T^{2} + 16094 T^{4} - 864959 T^{6} + 29757806 T^{8} - 622221782 T^{10} + 6321363049 T^{12} )^{2}$$
$47$ $$( 1 + 9 T + 154 T^{2} + 1143 T^{3} + 12329 T^{4} + 71940 T^{5} + 665017 T^{6} + 3381180 T^{7} + 27234761 T^{8} + 118669689 T^{9} + 751470874 T^{10} + 2064105063 T^{11} + 10779215329 T^{12} )^{2}$$
$53$ $$( 1 + T - 92 T^{2} - 161 T^{3} + 3593 T^{4} + 4762 T^{5} - 147323 T^{6} + 252386 T^{7} + 10092737 T^{8} - 23969197 T^{9} - 725924252 T^{10} + 418195493 T^{11} + 22164361129 T^{12} )^{2}$$
$59$ $$( 1 + 6 T + 46 T^{2} + 204 T^{3} + 212 T^{4} - 2832 T^{5} - 290339 T^{6} - 167088 T^{7} + 737972 T^{8} + 41897316 T^{9} + 557398606 T^{10} + 4289545794 T^{11} + 42180533641 T^{12} )^{2}$$
$61$ $$1 - 154 T^{2} + 9581 T^{4} - 329790 T^{6} + 4982890 T^{8} + 1013739166 T^{10} - 115924161659 T^{12} + 3772123436686 T^{14} + 68992302660490 T^{16} - 16990904260514190 T^{18} + 1836747765826949261 T^{20} -$$$$10\!\cdots\!54$$$$T^{22} +$$$$26\!\cdots\!21$$$$T^{24}$$
$67$ $$( 1 + 12 T + 3 T^{2} - 1636 T^{3} - 9534 T^{4} + 61764 T^{5} + 1391979 T^{6} + 4138188 T^{7} - 42798126 T^{8} - 492048268 T^{9} + 60453363 T^{10} + 16201501284 T^{11} + 90458382169 T^{12} )^{2}$$
$71$ $$( 1 - 5 T + 201 T^{2} - 647 T^{3} + 14271 T^{4} - 25205 T^{5} + 357911 T^{6} )^{4}$$
$73$ $$1 - 370 T^{2} + 76046 T^{4} - 11079522 T^{6} + 1263173620 T^{8} - 118509141896 T^{10} + 9378689711155 T^{12} - 631535217163784 T^{14} + 35871908885602420 T^{16} - 1676710889521953858 T^{18} + 61328064148177283726 T^{20} -$$$$15\!\cdots\!30$$$$T^{22} +$$$$22\!\cdots\!21$$$$T^{24}$$
$79$ $$1 + 423 T^{2} + 101070 T^{4} + 16919663 T^{6} + 2188998081 T^{8} + 228833270262 T^{10} + 19801390804881 T^{12} + 1428148439705142 T^{14} + 85261652563794561 T^{16} + 4112957826942809423 T^{18} +$$$$15\!\cdots\!70$$$$T^{20} +$$$$40\!\cdots\!23$$$$T^{22} +$$$$59\!\cdots\!41$$$$T^{24}$$
$83$ $$( 1 + 27 T^{2} + 16191 T^{4} + 314125 T^{6} + 111539799 T^{8} + 1281374667 T^{10} + 326940373369 T^{12} )^{2}$$
$89$ $$( 1 + 33 T + 712 T^{2} + 11517 T^{3} + 157301 T^{4} + 1850406 T^{5} + 18759895 T^{6} + 164686134 T^{7} + 1245981221 T^{8} + 8119127973 T^{9} + 44672475592 T^{10} + 184273961817 T^{11} + 496981290961 T^{12} )^{2}$$
$97$ $$( 1 - 519 T^{2} + 117501 T^{4} - 14851811 T^{6} + 1105566909 T^{8} - 45946696839 T^{10} + 832972004929 T^{12} )^{2}$$
show more
show less