Properties

Label 33.2.e
Level 33
Weight 2
Character orbit e
Rep. character \(\chi_{33}(4,\cdot)\)
Character field \(\Q(\zeta_{5})\)
Dimension 8
Newform subspaces 2
Sturm bound 8
Trace bound 2

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Defining parameters

Level: \( N \) = \( 33 = 3 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 33.e (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 11 \)
Character field: \(\Q(\zeta_{5})\)
Newform subspaces: \( 2 \)
Sturm bound: \(8\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(33, [\chi])\).

Total New Old
Modular forms 24 8 16
Cusp forms 8 8 0
Eisenstein series 16 0 16

Trace form

\( 8q - 4q^{2} - 6q^{4} - 4q^{5} - 2q^{6} - 2q^{7} + 8q^{8} - 2q^{9} + O(q^{10}) \) \( 8q - 4q^{2} - 6q^{4} - 4q^{5} - 2q^{6} - 2q^{7} + 8q^{8} - 2q^{9} + 4q^{10} - 2q^{11} - 8q^{12} - 2q^{13} + 2q^{14} + 2q^{15} + 10q^{16} + 14q^{17} + 6q^{18} - 20q^{19} + 6q^{20} + 16q^{21} - 4q^{22} - 12q^{23} + 12q^{24} + 10q^{26} - 12q^{28} - 4q^{29} - 4q^{31} - 48q^{32} - 10q^{33} + 8q^{34} - 6q^{36} + 6q^{37} - 10q^{38} - 16q^{39} + 4q^{40} + 20q^{41} - 10q^{42} + 16q^{43} + 34q^{44} - 4q^{45} + 10q^{46} + 14q^{47} + 8q^{48} + 4q^{49} + 18q^{50} + 10q^{52} + 10q^{53} + 8q^{54} + 16q^{55} + 12q^{56} - 24q^{58} - 26q^{59} - 18q^{61} - 28q^{62} - 2q^{63} + 6q^{64} - 28q^{65} - 12q^{66} - 4q^{67} - 28q^{68} - 6q^{69} - 6q^{70} - 12q^{71} - 2q^{72} + 20q^{73} - 30q^{74} + 8q^{75} + 40q^{76} - 2q^{77} + 16q^{78} - 6q^{79} + 22q^{80} - 2q^{81} - 14q^{82} + 34q^{83} - 2q^{85} + 8q^{86} + 24q^{87} - 62q^{88} - 4q^{89} + 4q^{90} + 6q^{91} + 10q^{92} + 20q^{93} + 18q^{94} + 30q^{95} - 8q^{96} - 30q^{97} + 40q^{98} + 8q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(33, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
33.2.e.a \(4\) \(0.264\) \(\Q(\zeta_{10})\) None \(-3\) \(-1\) \(-1\) \(-3\) \(q+(-1-\zeta_{10}^{2})q^{2}-\zeta_{10}^{3}q^{3}+(\zeta_{10}+\cdots)q^{4}+\cdots\)
33.2.e.b \(4\) \(0.264\) \(\Q(\zeta_{10})\) None \(-1\) \(1\) \(-3\) \(1\) \(q+(-1+2\zeta_{10}-\zeta_{10}^{2})q^{2}+\zeta_{10}^{3}q^{3}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 3 T + 2 T^{2} + T^{4} + 8 T^{6} + 24 T^{7} + 16 T^{8} \))(\( 1 + T + 4 T^{2} + 2 T^{3} + 9 T^{4} + 4 T^{5} + 16 T^{6} + 8 T^{7} + 16 T^{8} \))
$3$ (\( 1 + T + T^{2} + T^{3} + T^{4} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))
$5$ (\( 1 + T + T^{2} + 11 T^{3} + 36 T^{4} + 55 T^{5} + 25 T^{6} + 125 T^{7} + 625 T^{8} \))(\( 1 + 3 T - T^{2} - 3 T^{3} + 16 T^{4} - 15 T^{5} - 25 T^{6} + 375 T^{7} + 625 T^{8} \))
$7$ (\( 1 + 3 T + 2 T^{2} - 15 T^{3} - 59 T^{4} - 105 T^{5} + 98 T^{6} + 1029 T^{7} + 2401 T^{8} \))(\( 1 - T - 6 T^{2} + 13 T^{3} + 29 T^{4} + 91 T^{5} - 294 T^{6} - 343 T^{7} + 2401 T^{8} \))
$11$ (\( 1 - 9 T + 41 T^{2} - 99 T^{3} + 121 T^{4} \))(\( 1 + 11 T + 51 T^{2} + 121 T^{3} + 121 T^{4} \))
$13$ (\( 1 + 9 T + 18 T^{2} - 115 T^{3} - 789 T^{4} - 1495 T^{5} + 3042 T^{6} + 19773 T^{7} + 28561 T^{8} \))(\( 1 - 7 T + 6 T^{2} + 49 T^{3} - 181 T^{4} + 637 T^{5} + 1014 T^{6} - 15379 T^{7} + 28561 T^{8} \))
$17$ (\( 1 - 2 T - 13 T^{2} - 20 T^{3} + 341 T^{4} - 340 T^{5} - 3757 T^{6} - 9826 T^{7} + 83521 T^{8} \))(\( 1 - 12 T + 37 T^{2} + 180 T^{3} - 1619 T^{4} + 3060 T^{5} + 10693 T^{6} - 58956 T^{7} + 83521 T^{8} \))
$19$ (\( 1 + 10 T + 21 T^{2} - 70 T^{3} - 469 T^{4} - 1330 T^{5} + 7581 T^{6} + 68590 T^{7} + 130321 T^{8} \))(\( 1 + 10 T + 21 T^{2} - 70 T^{3} - 469 T^{4} - 1330 T^{5} + 7581 T^{6} + 68590 T^{7} + 130321 T^{8} \))
$23$ (\( ( 1 + 2 T + 27 T^{2} + 46 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 + 4 T + 45 T^{2} + 92 T^{3} + 529 T^{4} )^{2} \))
$29$ (\( 1 + 10 T + 31 T^{2} + 200 T^{3} + 1821 T^{4} + 5800 T^{5} + 26071 T^{6} + 243890 T^{7} + 707281 T^{8} \))(\( 1 - 6 T + 7 T^{2} + 132 T^{3} - 995 T^{4} + 3828 T^{5} + 5887 T^{6} - 146334 T^{7} + 707281 T^{8} \))
$31$ (\( 1 - 8 T + 3 T^{2} - 46 T^{3} + 1175 T^{4} - 1426 T^{5} + 2883 T^{6} - 238328 T^{7} + 923521 T^{8} \))(\( 1 + 12 T + 63 T^{2} + 434 T^{3} + 3255 T^{4} + 13454 T^{5} + 60543 T^{6} + 357492 T^{7} + 923521 T^{8} \))
$37$ (\( 1 + 3 T - 18 T^{2} + 155 T^{3} + 1851 T^{4} + 5735 T^{5} - 24642 T^{6} + 151959 T^{7} + 1874161 T^{8} \))(\( 1 - 9 T - 6 T^{2} + 307 T^{3} - 1581 T^{4} + 11359 T^{5} - 8214 T^{6} - 455877 T^{7} + 1874161 T^{8} \))
$41$ (\( 1 - 23 T + 208 T^{2} - 961 T^{3} + 3975 T^{4} - 39401 T^{5} + 349648 T^{6} - 1585183 T^{7} + 2825761 T^{8} \))(\( 1 + 3 T - 22 T^{2} + 171 T^{3} + 2215 T^{4} + 7011 T^{5} - 36982 T^{6} + 206763 T^{7} + 2825761 T^{8} \))
$43$ (\( ( 1 - 8 T + 97 T^{2} - 344 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 41 T^{2} + 1849 T^{4} )^{2} \))
$47$ (\( 1 + 3 T - 43 T^{2} - 45 T^{3} + 2116 T^{4} - 2115 T^{5} - 94987 T^{6} + 311469 T^{7} + 4879681 T^{8} \))(\( 1 - 17 T + 67 T^{2} + 335 T^{3} - 4344 T^{4} + 15745 T^{5} + 148003 T^{6} - 1764991 T^{7} + 4879681 T^{8} \))
$53$ (\( 1 - 6 T + 23 T^{2} + 120 T^{3} - 1319 T^{4} + 6360 T^{5} + 64607 T^{6} - 893262 T^{7} + 7890481 T^{8} \))(\( 1 - 4 T - 47 T^{2} + 160 T^{3} + 2121 T^{4} + 8480 T^{5} - 132023 T^{6} - 595508 T^{7} + 7890481 T^{8} \))
$59$ (\( 1 + 20 T + 131 T^{2} + 530 T^{3} + 3851 T^{4} + 31270 T^{5} + 456011 T^{6} + 4107580 T^{7} + 12117361 T^{8} \))(\( 1 + 6 T + 17 T^{2} - 222 T^{3} - 1685 T^{4} - 13098 T^{5} + 59177 T^{6} + 1232274 T^{7} + 12117361 T^{8} \))
$61$ (\( 1 - 3 T - 7 T^{2} - 441 T^{3} + 4900 T^{4} - 26901 T^{5} - 26047 T^{6} - 680943 T^{7} + 13845841 T^{8} \))(\( 1 + 21 T + 245 T^{2} + 2559 T^{3} + 23404 T^{4} + 156099 T^{5} + 911645 T^{6} + 4766601 T^{7} + 13845841 T^{8} \))
$67$ (\( ( 1 - T + 33 T^{2} - 67 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + 3 T + 125 T^{2} + 201 T^{3} + 4489 T^{4} )^{2} \))
$71$ (\( 1 + 27 T + 253 T^{2} + 819 T^{3} + 100 T^{4} + 58149 T^{5} + 1275373 T^{6} + 9663597 T^{7} + 25411681 T^{8} \))(\( 1 - 15 T + 119 T^{2} - 1455 T^{3} + 17296 T^{4} - 103305 T^{5} + 599879 T^{6} - 5368665 T^{7} + 25411681 T^{8} \))
$73$ (\( 1 - 6 T - 57 T^{2} + 130 T^{3} + 4761 T^{4} + 9490 T^{5} - 303753 T^{6} - 2334102 T^{7} + 28398241 T^{8} \))(\( 1 - 14 T + 63 T^{2} - 850 T^{3} + 12521 T^{4} - 62050 T^{5} + 335727 T^{6} - 5446238 T^{7} + 28398241 T^{8} \))
$79$ (\( 1 - 5 T + 6 T^{2} - 715 T^{3} + 9821 T^{4} - 56485 T^{5} + 37446 T^{6} - 2465195 T^{7} + 38950081 T^{8} \))(\( 1 + 11 T + 42 T^{2} - 407 T^{3} - 7795 T^{4} - 32153 T^{5} + 262122 T^{6} + 5423429 T^{7} + 38950081 T^{8} \))
$83$ (\( 1 - 21 T + 88 T^{2} + 915 T^{3} - 13199 T^{4} + 75945 T^{5} + 606232 T^{6} - 12007527 T^{7} + 47458321 T^{8} \))(\( 1 - 13 T - 14 T^{2} + 421 T^{3} + 1449 T^{4} + 34943 T^{5} - 96446 T^{6} - 7433231 T^{7} + 47458321 T^{8} \))
$89$ (\( ( 1 - 10 T + 183 T^{2} - 890 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 12 T + 209 T^{2} + 1068 T^{3} + 7921 T^{4} )^{2} \))
$97$ (\( 1 + 33 T + 537 T^{2} + 6655 T^{3} + 71196 T^{4} + 645535 T^{5} + 5052633 T^{6} + 30118209 T^{7} + 88529281 T^{8} \))(\( 1 - 3 T - 43 T^{2} - 765 T^{3} + 11236 T^{4} - 74205 T^{5} - 404587 T^{6} - 2738019 T^{7} + 88529281 T^{8} \))
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