Properties

Label 77.2.m
Level 77
Weight 2
Character orbit m
Rep. character \(\chi_{77}(4,\cdot)\)
Character field \(\Q(\zeta_{15})\)
Dimension 48
Newform subspaces 2
Sturm bound 16
Trace bound 1

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Defining parameters

Level: \( N \) = \( 77 = 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 77.m (of order \(15\) and degree \(8\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 77 \)
Character field: \(\Q(\zeta_{15})\)
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 80 80 0
Cusp forms 48 48 0
Eisenstein series 32 32 0

Trace form

\( 48q - 5q^{2} - 3q^{3} - q^{4} - q^{5} - 12q^{6} - 7q^{7} - 28q^{8} + 5q^{9} + O(q^{10}) \) \( 48q - 5q^{2} - 3q^{3} - q^{4} - q^{5} - 12q^{6} - 7q^{7} - 28q^{8} + 5q^{9} + 4q^{10} - 5q^{11} - 16q^{12} - 4q^{13} - 24q^{15} + q^{16} - 11q^{17} + 18q^{18} - 7q^{19} - 40q^{20} - 18q^{21} + 40q^{22} + 2q^{23} - 7q^{24} + 21q^{25} - 15q^{26} - 6q^{27} + 4q^{28} + 24q^{29} + 21q^{30} - 9q^{31} - 12q^{32} - 26q^{33} + 72q^{34} - 19q^{35} + 22q^{36} + 11q^{37} + 3q^{38} + 33q^{39} + 5q^{40} + 62q^{41} - 60q^{42} - 44q^{43} + 6q^{44} - 16q^{45} + 16q^{46} + 19q^{47} + 146q^{48} - 21q^{49} + 6q^{50} - q^{51} - 3q^{52} + 21q^{53} + 44q^{54} + 4q^{55} + 24q^{56} + 20q^{57} - 38q^{58} + 3q^{59} - 43q^{60} + 18q^{61} - 80q^{62} + 22q^{63} + 100q^{64} - 10q^{65} - 47q^{66} - 76q^{67} - 21q^{68} - 126q^{69} + 17q^{70} + 4q^{71} - 48q^{72} + 26q^{73} - 55q^{74} - 11q^{75} - 144q^{76} + 32q^{77} - 140q^{78} + 30q^{79} - 3q^{80} - 57q^{81} + 13q^{82} - 28q^{83} - 27q^{84} - 98q^{85} + 66q^{87} + 27q^{88} - 34q^{89} - 18q^{90} + 36q^{91} - 94q^{92} + 20q^{93} + 45q^{94} + 22q^{95} - 55q^{96} - 68q^{97} + 120q^{98} - 40q^{99} + O(q^{100}) \)

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.m.a \(8\) \(0.615\) \(\Q(\zeta_{15})\) None \(-2\) \(1\) \(-5\) \(-5\) \(q+(-1+\zeta_{15}^{4}-\zeta_{15}^{5}+\zeta_{15}^{7})q^{2}+\cdots\)
77.2.m.b \(40\) \(0.615\) None \(-3\) \(-4\) \(4\) \(-2\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} - 6 T^{3} - 17 T^{4} - 24 T^{5} - T^{6} + 47 T^{7} + 103 T^{8} + 94 T^{9} - 4 T^{10} - 192 T^{11} - 272 T^{12} - 192 T^{13} + 128 T^{14} + 256 T^{15} + 256 T^{16} \))
$3$ (\( 1 - T + 3 T^{2} - 8 T^{3} + 8 T^{4} + 7 T^{5} + 6 T^{6} + 56 T^{7} - 137 T^{8} + 168 T^{9} + 54 T^{10} + 189 T^{11} + 648 T^{12} - 1944 T^{13} + 2187 T^{14} - 2187 T^{15} + 6561 T^{16} \))
$5$ (\( ( 1 + 5 T + 15 T^{2} + 25 T^{3} + 25 T^{4} )^{2}( 1 - 5 T + 10 T^{2} - 25 T^{3} + 75 T^{4} - 125 T^{5} + 250 T^{6} - 625 T^{7} + 625 T^{8} ) \))
$7$ (\( 1 + 5 T + 18 T^{2} + 55 T^{3} + 149 T^{4} + 385 T^{5} + 882 T^{6} + 1715 T^{7} + 2401 T^{8} \))
$11$ (\( 1 - 4 T + 10 T^{2} + 64 T^{3} - 261 T^{4} + 704 T^{5} + 1210 T^{6} - 5324 T^{7} + 14641 T^{8} \))
$13$ (\( ( 1 + 5 T + 27 T^{2} + 115 T^{3} + 584 T^{4} + 1495 T^{5} + 4563 T^{6} + 10985 T^{7} + 28561 T^{8} )^{2} \))
$17$ (\( 1 + 4 T + 17 T^{2} - 120 T^{3} - 740 T^{4} - 3144 T^{5} - 649 T^{6} + 49990 T^{7} + 275299 T^{8} + 849830 T^{9} - 187561 T^{10} - 15446472 T^{11} - 61805540 T^{12} - 170382840 T^{13} + 410338673 T^{14} + 1641354692 T^{15} + 6975757441 T^{16} \))
$19$ (\( 1 + 3 T - 26 T^{2} - 21 T^{3} + 252 T^{4} - 1302 T^{5} - 1792 T^{6} + 24084 T^{7} + 90161 T^{8} + 457596 T^{9} - 646912 T^{10} - 8930418 T^{11} + 32840892 T^{12} - 51998079 T^{13} - 1223192906 T^{14} + 2681615217 T^{15} + 16983563041 T^{16} \))
$23$ (\( ( 1 - 8 T + 7 T^{2} - 88 T^{3} + 1248 T^{4} - 2024 T^{5} + 3703 T^{6} - 97336 T^{7} + 279841 T^{8} )^{2} \))
$29$ (\( ( 1 - 12 T + 25 T^{2} + 288 T^{3} - 2471 T^{4} + 8352 T^{5} + 21025 T^{6} - 292668 T^{7} + 707281 T^{8} )^{2} \))
$31$ (\( ( 1 - 19 T + 210 T^{2} - 1691 T^{3} + 10649 T^{4} - 52421 T^{5} + 201810 T^{6} - 566029 T^{7} + 923521 T^{8} )( 1 + 11 T + 60 T^{2} - 11 T^{3} - 991 T^{4} - 341 T^{5} + 57660 T^{6} + 327701 T^{7} + 923521 T^{8} ) \))
$37$ (\( 1 + 13 T + 112 T^{2} + 71 T^{3} - 4102 T^{4} - 46406 T^{5} - 63616 T^{6} + 1263548 T^{7} + 15622363 T^{8} + 46751276 T^{9} - 87090304 T^{10} - 2350603118 T^{11} - 7687808422 T^{12} + 4923420947 T^{13} + 287361357808 T^{14} + 1234114402729 T^{15} + 3512479453921 T^{16} \))
$41$ (\( ( 1 - T - 25 T^{2} + 221 T^{3} + 1064 T^{4} + 9061 T^{5} - 42025 T^{6} - 68921 T^{7} + 2825761 T^{8} )^{2} \))
$43$ (\( ( 1 - 7 T + 87 T^{2} - 301 T^{3} + 1849 T^{4} )^{4} \))
$47$ (\( 1 - 6 T + 7 T^{2} + 690 T^{3} - 6420 T^{4} + 23196 T^{5} + 57521 T^{6} - 1846680 T^{7} + 13766039 T^{8} - 86793960 T^{9} + 127063889 T^{10} + 2408278308 T^{11} - 31327552020 T^{12} + 158248054830 T^{13} + 75454507303 T^{14} - 3039738722778 T^{15} + 23811286661761 T^{16} \))
$53$ (\( 1 + 12 T + 103 T^{2} - 420 T^{3} - 9840 T^{4} - 102312 T^{5} - 99751 T^{6} + 3945120 T^{7} + 55833959 T^{8} + 209091360 T^{9} - 280200559 T^{10} - 15231903624 T^{11} - 77642333040 T^{12} - 175642107060 T^{13} + 2282929196287 T^{14} + 14096533678044 T^{15} + 62259690411361 T^{16} \))
$59$ (\( 1 + 18 T + 239 T^{2} + 1374 T^{3} + 5292 T^{4} - 7812 T^{5} + 470053 T^{6} + 8281674 T^{7} + 95738891 T^{8} + 488618766 T^{9} + 1636254493 T^{10} - 1604420748 T^{11} + 64125074412 T^{12} + 982305986826 T^{13} + 10081147540199 T^{14} + 44795726726742 T^{15} + 146830437604321 T^{16} \))
$61$ (\( 1 - 18 T + 141 T^{2} + 538 T^{3} - 19524 T^{4} + 199396 T^{5} - 460901 T^{6} - 9059916 T^{7} + 127866487 T^{8} - 552654876 T^{9} - 1715012621 T^{10} + 45259103476 T^{11} - 270326199684 T^{12} + 454392809938 T^{13} + 7264372784901 T^{14} - 56569371048378 T^{15} + 191707312997281 T^{16} \))
$67$ (\( ( 1 + 19 T + 138 T^{2} + 1691 T^{3} + 21053 T^{4} + 113297 T^{5} + 619482 T^{6} + 5714497 T^{7} + 20151121 T^{8} )^{2} \))
$71$ (\( ( 1 + 8 T - 47 T^{2} - 434 T^{3} + 1365 T^{4} - 30814 T^{5} - 236927 T^{6} + 2863288 T^{7} + 25411681 T^{8} )^{2} \))
$73$ (\( 1 - 15 T + 208 T^{2} - 2445 T^{3} + 29070 T^{4} - 272130 T^{5} + 2789288 T^{6} - 23441760 T^{7} + 210261239 T^{8} - 1711248480 T^{9} + 14864115752 T^{10} - 105863196210 T^{11} + 825536865870 T^{12} - 5068660044885 T^{13} + 31477519068112 T^{14} - 165710977786455 T^{15} + 806460091894081 T^{16} \))
$79$ (\( 1 - 9 T - 11 T^{2} + 1446 T^{3} - 17334 T^{4} + 85743 T^{5} + 200726 T^{6} - 10774872 T^{7} + 128238887 T^{8} - 851214888 T^{9} + 1252730966 T^{10} + 42274642977 T^{11} - 675160704054 T^{12} + 4449423552954 T^{13} - 2673962010731 T^{14} - 172835180875431 T^{15} + 1517108809906561 T^{16} \))
$83$ (\( ( 1 - 9 T - 2 T^{2} + 765 T^{3} - 6719 T^{4} + 63495 T^{5} - 13778 T^{6} - 5146083 T^{7} + 47458321 T^{8} )^{2} \))
$89$ (\( ( 1 + 12 T - 50 T^{2} + 192 T^{3} + 16899 T^{4} + 17088 T^{5} - 396050 T^{6} + 8459628 T^{7} + 62742241 T^{8} )^{2} \))
$97$ (\( ( 1 + 7 T - 63 T^{2} + 185 T^{3} + 11276 T^{4} + 17945 T^{5} - 592767 T^{6} + 6388711 T^{7} + 88529281 T^{8} )^{2} \))
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