Properties

Label 77.2.e
Level 77
Weight 2
Character orbit e
Rep. character \(\chi_{77}(23,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 12
Newform subspaces 2
Sturm bound 16
Trace bound 3

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Defining parameters

Level: \( N \) = \( 77 = 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 77.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 2 \)
Sturm bound: \(16\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\)

Dimensions

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

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

Trace form

\( 12q - 2q^{3} - 4q^{4} - 4q^{5} - 8q^{6} + 2q^{7} - 12q^{8} + O(q^{10}) \) \( 12q - 2q^{3} - 4q^{4} - 4q^{5} - 8q^{6} + 2q^{7} - 12q^{8} + 6q^{10} + 6q^{12} - 16q^{13} + 4q^{15} + 4q^{16} + 6q^{17} + 2q^{18} + 2q^{19} + 40q^{20} - 2q^{21} - 12q^{23} - 8q^{24} - 6q^{25} - 10q^{26} + 16q^{27} + 16q^{28} - 24q^{29} + 4q^{30} - 6q^{31} + 12q^{32} - 4q^{33} - 32q^{34} - 6q^{35} - 12q^{36} + 4q^{37} - 8q^{38} + 2q^{39} + 8q^{41} - 20q^{42} + 4q^{43} + 4q^{44} + 6q^{45} + 34q^{46} + 6q^{47} - 16q^{48} - 24q^{49} + 24q^{50} + 16q^{51} - 2q^{52} - 26q^{53} + 26q^{54} + 16q^{55} + 6q^{56} + 40q^{57} - 2q^{58} - 8q^{59} - 12q^{60} + 12q^{61} + 20q^{62} + 18q^{63} - 20q^{64} - 30q^{65} + 2q^{66} + 16q^{67} + 16q^{68} + 36q^{69} + 18q^{70} - 4q^{71} - 22q^{72} + 14q^{73} - 40q^{74} - 34q^{75} - 96q^{76} - 2q^{77} + 18q^{80} + 2q^{81} - 38q^{82} - 52q^{83} + 12q^{84} - 32q^{85} + 30q^{86} - 6q^{87} - 12q^{88} + 14q^{89} - 52q^{90} - 6q^{91} + 44q^{92} + 20q^{93} + 10q^{94} + 8q^{95} - 30q^{96} + 108q^{97} - 30q^{98} + O(q^{100}) \)

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.e.a \(6\) \(0.615\) \(\Q(\zeta_{18})\) None \(0\) \(-3\) \(-6\) \(0\) \(q+(-\zeta_{18}+\zeta_{18}^{2}+\zeta_{18}^{4}-\zeta_{18}^{5})q^{2}+\cdots\)
77.2.e.b \(6\) \(0.615\) 6.0.1783323.2 None \(0\) \(1\) \(2\) \(2\) \(q+(-\beta _{3}+\beta _{5})q^{2}+\beta _{1}q^{3}+(-1-\beta _{1}+\cdots)q^{4}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 3 T^{2} - 2 T^{3} + 3 T^{4} + 3 T^{5} - T^{6} + 6 T^{7} + 12 T^{8} - 16 T^{9} - 48 T^{10} + 64 T^{12} \))(\( 1 - T^{2} + 6 T^{3} - T^{4} - 3 T^{5} + 23 T^{6} - 6 T^{7} - 4 T^{8} + 48 T^{9} - 16 T^{10} + 64 T^{12} \))
$3$ (\( 1 + 3 T - 7 T^{3} + 3 T^{4} + 18 T^{5} + 19 T^{6} + 54 T^{7} + 27 T^{8} - 189 T^{9} + 729 T^{11} + 729 T^{12} \))(\( 1 - T - 4 T^{2} + T^{3} + 7 T^{4} + 6 T^{5} - 21 T^{6} + 18 T^{7} + 63 T^{8} + 27 T^{9} - 324 T^{10} - 243 T^{11} + 729 T^{12} \))
$5$ (\( 1 + 6 T + 12 T^{2} + 22 T^{3} + 90 T^{4} + 174 T^{5} + 191 T^{6} + 870 T^{7} + 2250 T^{8} + 2750 T^{9} + 7500 T^{10} + 18750 T^{11} + 15625 T^{12} \))(\( 1 - 2 T - 4 T^{2} + 14 T^{3} - 6 T^{4} - 10 T^{5} + 55 T^{6} - 50 T^{7} - 150 T^{8} + 1750 T^{9} - 2500 T^{10} - 6250 T^{11} + 15625 T^{12} \))
$7$ (\( 1 + 17 T^{3} + 343 T^{6} \))(\( 1 - 2 T + 14 T^{2} - 23 T^{3} + 98 T^{4} - 98 T^{5} + 343 T^{6} \))
$11$ (\( ( 1 + T + T^{2} )^{3} \))(\( ( 1 - T + T^{2} )^{3} \))
$13$ (\( ( 1 - 3 T + 33 T^{2} - 79 T^{3} + 429 T^{4} - 507 T^{5} + 2197 T^{6} )^{2} \))(\( ( 1 + 11 T + 75 T^{2} + 321 T^{3} + 975 T^{4} + 1859 T^{5} + 2197 T^{6} )^{2} \))
$17$ (\( 1 - 3 T - 6 T^{2} - 95 T^{3} + 45 T^{4} + 1038 T^{5} + 4025 T^{6} + 17646 T^{7} + 13005 T^{8} - 466735 T^{9} - 501126 T^{10} - 4259571 T^{11} + 24137569 T^{12} \))(\( 1 - 3 T - 40 T^{2} + 43 T^{3} + 1283 T^{4} - 524 T^{5} - 24567 T^{6} - 8908 T^{7} + 370787 T^{8} + 211259 T^{9} - 3340840 T^{10} - 4259571 T^{11} + 24137569 T^{12} \))
$19$ (\( 1 + 9 T + 6 T^{2} - 27 T^{3} + 1041 T^{4} + 3924 T^{5} + 803 T^{6} + 74556 T^{7} + 375801 T^{8} - 185193 T^{9} + 781926 T^{10} + 22284891 T^{11} + 47045881 T^{12} \))(\( 1 - 11 T + 44 T^{2} - 125 T^{3} + 495 T^{4} + 418 T^{5} - 12293 T^{6} + 7942 T^{7} + 178695 T^{8} - 857375 T^{9} + 5734124 T^{10} - 27237089 T^{11} + 47045881 T^{12} \))
$23$ (\( 1 - 12 T^{2} - 214 T^{3} - 132 T^{4} + 1284 T^{5} + 32471 T^{6} + 29532 T^{7} - 69828 T^{8} - 2603738 T^{9} - 3358092 T^{10} + 148035889 T^{12} \))(\( 1 + 12 T + 32 T^{2} + 146 T^{3} + 2780 T^{4} + 11612 T^{5} + 18447 T^{6} + 267076 T^{7} + 1470620 T^{8} + 1776382 T^{9} + 8954912 T^{10} + 77236116 T^{11} + 148035889 T^{12} \))
$29$ (\( ( 1 + 3 T + 51 T^{2} + 225 T^{3} + 1479 T^{4} + 2523 T^{5} + 24389 T^{6} )^{2} \))(\( ( 1 + 9 T + 67 T^{2} + 469 T^{3} + 1943 T^{4} + 7569 T^{5} + 24389 T^{6} )^{2} \))
$31$ (\( 1 + 9 T - 36 T^{2} - 101 T^{3} + 4869 T^{4} + 10872 T^{5} - 109689 T^{6} + 337032 T^{7} + 4679109 T^{8} - 3008891 T^{9} - 33246756 T^{10} + 257662359 T^{11} + 887503681 T^{12} \))(\( 1 - 3 T - 40 T^{2} + 11 T^{3} + 645 T^{4} + 2360 T^{5} - 17257 T^{6} + 73160 T^{7} + 619845 T^{8} + 327701 T^{9} - 36940840 T^{10} - 85887453 T^{11} + 887503681 T^{12} \))
$37$ (\( 1 - 75 T^{2} + 144 T^{3} + 2850 T^{4} - 5400 T^{5} - 101635 T^{6} - 199800 T^{7} + 3901650 T^{8} + 7294032 T^{9} - 140562075 T^{10} + 2565726409 T^{12} \))(\( 1 - 4 T - 59 T^{2} - 12 T^{3} + 2274 T^{4} + 5924 T^{5} - 107987 T^{6} + 219188 T^{7} + 3113106 T^{8} - 607836 T^{9} - 110575499 T^{10} - 277375828 T^{11} + 2565726409 T^{12} \))
$41$ (\( ( 1 - 9 T + 129 T^{2} - 739 T^{3} + 5289 T^{4} - 15129 T^{5} + 68921 T^{6} )^{2} \))(\( ( 1 + 5 T + 43 T^{2} + 301 T^{3} + 1763 T^{4} + 8405 T^{5} + 68921 T^{6} )^{2} \))
$43$ (\( ( 1 + 120 T^{2} + 9 T^{3} + 5160 T^{4} + 79507 T^{6} )^{2} \))(\( ( 1 - 2 T + 104 T^{2} - 131 T^{3} + 4472 T^{4} - 3698 T^{5} + 79507 T^{6} )^{2} \))
$47$ (\( 1 - 3 T - 54 T^{2} - 271 T^{3} + 1131 T^{4} + 13722 T^{5} + 2903 T^{6} + 644934 T^{7} + 2498379 T^{8} - 28136033 T^{9} - 263502774 T^{10} - 688035021 T^{11} + 10779215329 T^{12} \))(\( 1 - 3 T - 130 T^{2} + 133 T^{3} + 11963 T^{4} - 5654 T^{5} - 644697 T^{6} - 265738 T^{7} + 26426267 T^{8} + 13808459 T^{9} - 634358530 T^{10} - 688035021 T^{11} + 10779215329 T^{12} \))
$53$ (\( 1 + 9 T - 24 T^{2} - 45 T^{3} + 1005 T^{4} - 22914 T^{5} - 269075 T^{6} - 1214442 T^{7} + 2823045 T^{8} - 6699465 T^{9} - 189371544 T^{10} + 3763759437 T^{11} + 22164361129 T^{12} \))(\( 1 + 17 T + 56 T^{2} + 315 T^{3} + 10949 T^{4} + 52646 T^{5} - 68035 T^{6} + 2790238 T^{7} + 30755741 T^{8} + 46896255 T^{9} + 441866936 T^{10} + 7109323381 T^{11} + 22164361129 T^{12} \))
$59$ (\( 1 - 84 T^{2} + 38 T^{3} + 2100 T^{4} - 1596 T^{5} - 5185 T^{6} - 94164 T^{7} + 7310100 T^{8} + 7804402 T^{9} - 1017858324 T^{10} + 42180533641 T^{12} \))(\( 1 + 8 T + 44 T^{2} + 918 T^{3} + 2252 T^{4} + 1388 T^{5} + 308015 T^{6} + 81892 T^{7} + 7839212 T^{8} + 188537922 T^{9} + 533163884 T^{10} + 5719394392 T^{11} + 42180533641 T^{12} \))
$61$ (\( 1 + 12 T - 3 T^{2} - 1212 T^{3} - 5214 T^{4} + 48180 T^{5} + 814133 T^{6} + 2938980 T^{7} - 19401294 T^{8} - 275100972 T^{9} - 41537523 T^{10} + 10135155612 T^{11} + 51520374361 T^{12} \))(\( 1 - 24 T + 221 T^{2} - 1912 T^{3} + 25074 T^{4} - 222784 T^{5} + 1539557 T^{6} - 13589824 T^{7} + 93300354 T^{8} - 433987672 T^{9} + 3059930861 T^{10} - 20270311224 T^{11} + 51520374361 T^{12} \))
$67$ (\( 1 - 189 T^{2} - 16 T^{3} + 23058 T^{4} + 1512 T^{5} - 1791717 T^{6} + 101304 T^{7} + 103507362 T^{8} - 4812208 T^{9} - 3808561869 T^{10} + 90458382169 T^{12} \))(\( 1 - 16 T - 13 T^{2} + 128 T^{3} + 18882 T^{4} - 91192 T^{5} - 485189 T^{6} - 6109864 T^{7} + 84761298 T^{8} + 38497664 T^{9} - 261964573 T^{10} - 21602001712 T^{11} + 90458382169 T^{12} \))
$71$ (\( ( 1 + 9 T + 123 T^{2} + 477 T^{3} + 8733 T^{4} + 45369 T^{5} + 357911 T^{6} )^{2} \))(\( ( 1 - 7 T + 127 T^{2} - 575 T^{3} + 9017 T^{4} - 35287 T^{5} + 357911 T^{6} )^{2} \))
$73$ (\( 1 + 6 T - 192 T^{2} - 386 T^{3} + 30078 T^{4} + 32202 T^{5} - 2439513 T^{6} + 2350746 T^{7} + 160285662 T^{8} - 150160562 T^{9} - 5452462272 T^{10} + 12438429558 T^{11} + 151334226289 T^{12} \))(\( 1 - 20 T + 156 T^{2} - 290 T^{3} - 4176 T^{4} + 45920 T^{5} - 432669 T^{6} + 3352160 T^{7} - 22253904 T^{8} - 112814930 T^{9} + 4430125596 T^{10} - 41461431860 T^{11} + 151334226289 T^{12} \))
$79$ (\( 1 - 3 T - 114 T^{2} + 653 T^{3} + 3879 T^{4} - 23274 T^{5} - 4161 T^{6} - 1838646 T^{7} + 24208839 T^{8} + 321954467 T^{9} - 4440309234 T^{10} - 9231169197 T^{11} + 243087455521 T^{12} \))(\( 1 + 3 T - 190 T^{2} - 69 T^{3} + 22881 T^{4} - 9336 T^{5} - 2087681 T^{6} - 737544 T^{7} + 142800321 T^{8} - 34019691 T^{9} - 7400515390 T^{10} + 9231169197 T^{11} + 243087455521 T^{12} \))
$83$ (\( ( 1 + 15 T + 267 T^{2} + 2223 T^{3} + 22161 T^{4} + 103335 T^{5} + 571787 T^{6} )^{2} \))(\( ( 1 + 11 T + 265 T^{2} + 1823 T^{3} + 21995 T^{4} + 75779 T^{5} + 571787 T^{6} )^{2} \))
$89$ (\( 1 - 15 T - 114 T^{2} + 477 T^{3} + 43035 T^{4} - 156444 T^{5} - 2866295 T^{6} - 13923516 T^{7} + 340880235 T^{8} + 336270213 T^{9} - 7152615474 T^{10} - 83760891735 T^{11} + 496981290961 T^{12} \))(\( 1 + T - 258 T^{2} - 91 T^{3} + 43855 T^{4} + 7144 T^{5} - 4537567 T^{6} + 635816 T^{7} + 347375455 T^{8} - 64152179 T^{9} - 16187498178 T^{10} + 5584059449 T^{11} + 496981290961 T^{12} \))
$97$ (\( ( 1 - 45 T + 963 T^{2} - 12059 T^{3} + 93411 T^{4} - 423405 T^{5} + 912673 T^{6} )^{2} \))(\( ( 1 - 9 T + 279 T^{2} - 1699 T^{3} + 27063 T^{4} - 84681 T^{5} + 912673 T^{6} )^{2} \))
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