Properties

Label 36.2.h
Level 36
Weight 2
Character orbit h
Rep. character \(\chi_{36}(11,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 8
Newform subspaces 1
Sturm bound 12
Trace bound 0

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 36.h (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 36 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 1 \)
Sturm bound: \(12\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(36, [\chi])\).

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

Trace form

\( 8q - 3q^{2} - q^{4} - 6q^{5} - 3q^{6} - 6q^{9} + O(q^{10}) \) \( 8q - 3q^{2} - q^{4} - 6q^{5} - 3q^{6} - 6q^{9} - 8q^{10} + 6q^{12} - 2q^{13} + 12q^{14} - q^{16} + 18q^{18} + 18q^{20} - 6q^{21} + 3q^{22} + 3q^{24} - 6q^{25} - 12q^{28} + 6q^{29} - 18q^{30} - 33q^{32} + 24q^{33} + 7q^{34} - 33q^{36} - 8q^{37} - 27q^{38} + 10q^{40} + 24q^{41} - 18q^{42} + 6q^{45} + 12q^{46} + 21q^{48} - 10q^{49} + 21q^{50} + 16q^{52} + 39q^{54} + 18q^{56} + 6q^{57} + 4q^{58} + 6q^{60} - 2q^{61} + 26q^{64} - 30q^{65} - 24q^{66} - 15q^{68} - 30q^{69} - 6q^{70} - 21q^{72} + 4q^{73} - 30q^{74} - 3q^{76} - 30q^{77} - 12q^{78} - 30q^{81} + 10q^{82} + 30q^{84} + 8q^{85} + 21q^{86} - 21q^{88} + 6q^{90} + 24q^{92} + 30q^{93} - 18q^{94} + 12q^{96} + 4q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(36, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
36.2.h.a \(8\) \(0.287\) 8.0.170772624.1 None \(-3\) \(0\) \(-6\) \(0\) \(q+(1-\beta _{1}+\beta _{4}+\beta _{7})q^{2}+(-\beta _{2}-\beta _{3}+\cdots)q^{3}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T + 5 T^{2} + 6 T^{3} + 6 T^{4} + 12 T^{5} + 20 T^{6} + 24 T^{7} + 16 T^{8} \)
$3$ \( 1 + 3 T^{2} + 12 T^{4} + 27 T^{6} + 81 T^{8} \)
$5$ \( ( 1 + 3 T + 11 T^{2} + 24 T^{3} + 54 T^{4} + 120 T^{5} + 275 T^{6} + 375 T^{7} + 625 T^{8} )^{2} \)
$7$ \( 1 + 19 T^{2} + 181 T^{4} + 1558 T^{6} + 12310 T^{8} + 76342 T^{10} + 434581 T^{12} + 2235331 T^{14} + 5764801 T^{16} \)
$11$ \( 1 - 32 T^{2} + 559 T^{4} - 7136 T^{6} + 77680 T^{8} - 863456 T^{10} + 8184319 T^{12} - 56689952 T^{14} + 214358881 T^{16} \)
$13$ \( ( 1 + T - 17 T^{2} - 8 T^{3} + 142 T^{4} - 104 T^{5} - 2873 T^{6} + 2197 T^{7} + 28561 T^{8} )^{2} \)
$17$ \( ( 1 - 61 T^{2} + 1500 T^{4} - 17629 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 - 49 T^{2} + 1248 T^{4} - 17689 T^{6} + 130321 T^{8} )^{2} \)
$23$ \( 1 - 77 T^{2} + 3397 T^{4} - 113498 T^{6} + 2994742 T^{8} - 60040442 T^{10} + 950619877 T^{12} - 11398763453 T^{14} + 78310985281 T^{16} \)
$29$ \( ( 1 - 3 T + 59 T^{2} - 168 T^{3} + 2382 T^{4} - 4872 T^{5} + 49619 T^{6} - 73167 T^{7} + 707281 T^{8} )^{2} \)
$31$ \( 1 + 55 T^{2} + 1345 T^{4} - 13310 T^{6} - 1008146 T^{8} - 12790910 T^{10} + 1242135745 T^{12} + 48812702455 T^{14} + 852891037441 T^{16} \)
$37$ \( ( 1 + 2 T + 42 T^{2} + 74 T^{3} + 1369 T^{4} )^{4} \)
$41$ \( ( 1 - 12 T + 131 T^{2} - 996 T^{3} + 7176 T^{4} - 40836 T^{5} + 220211 T^{6} - 827052 T^{7} + 2825761 T^{8} )^{2} \)
$43$ \( 1 + 64 T^{2} - 593 T^{4} + 63424 T^{6} + 10994416 T^{8} + 117270976 T^{10} - 2027348993 T^{12} + 404567235136 T^{14} + 11688200277601 T^{16} \)
$47$ \( 1 - 53 T^{2} + 2053 T^{4} + 194086 T^{6} - 10513226 T^{8} + 428735974 T^{10} + 10017985093 T^{12} - 571298412437 T^{14} + 23811286661761 T^{16} \)
$53$ \( ( 1 - 136 T^{2} + 9054 T^{4} - 382024 T^{6} + 7890481 T^{8} )^{2} \)
$59$ \( 1 - 56 T^{2} - 617 T^{4} + 179704 T^{6} - 8948768 T^{8} + 625549624 T^{10} - 7476411737 T^{12} - 2362109883896 T^{14} + 146830437604321 T^{16} \)
$61$ \( ( 1 + T - 113 T^{2} - 8 T^{3} + 9214 T^{4} - 488 T^{5} - 420473 T^{6} + 226981 T^{7} + 13845841 T^{8} )^{2} \)
$67$ \( 1 + 160 T^{2} + 10255 T^{4} + 1018720 T^{6} + 100399504 T^{8} + 4573034080 T^{10} + 206649745855 T^{12} + 14473341147040 T^{14} + 406067677556641 T^{16} \)
$71$ \( ( 1 + 140 T^{2} + 10230 T^{4} + 705740 T^{6} + 25411681 T^{8} )^{2} \)
$73$ \( ( 1 - T + 138 T^{2} - 73 T^{3} + 5329 T^{4} )^{4} \)
$79$ \( 1 + 115 T^{2} - 2555 T^{4} + 379270 T^{6} + 127521094 T^{8} + 2367024070 T^{10} - 99517456955 T^{12} + 27955057384915 T^{14} + 1517108809906561 T^{16} \)
$83$ \( 1 - 221 T^{2} + 22861 T^{4} - 2696642 T^{6} + 291036430 T^{8} - 18577166738 T^{10} + 1084944676381 T^{12} - 72253822514549 T^{14} + 2252292232139041 T^{16} \)
$89$ \( ( 1 - 184 T^{2} + 21006 T^{4} - 1457464 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 - 2 T - 59 T^{2} + 262 T^{3} - 5828 T^{4} + 25414 T^{5} - 555131 T^{6} - 1825346 T^{7} + 88529281 T^{8} )^{2} \)
show more
show less