Properties

Label 36.2.e
Level 36
Weight 2
Character orbit e
Rep. character \(\chi_{36}(13,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 2
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.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 9 \)
Character field: \(\Q(\zeta_{3})\)
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 18 2 16
Cusp forms 6 2 4
Eisenstein series 12 0 12

Trace form

\( 2q - 3q^{5} + q^{7} - 6q^{9} + O(q^{10}) \) \( 2q - 3q^{5} + q^{7} - 6q^{9} - 3q^{11} + q^{13} + 9q^{15} + 12q^{17} - 8q^{19} + 3q^{21} + 3q^{23} - 4q^{25} - 3q^{29} - 5q^{31} - 9q^{33} - 6q^{35} + 4q^{37} - 3q^{39} - 3q^{41} + q^{43} + 9q^{45} + 9q^{47} + 6q^{49} - 12q^{53} + 18q^{55} + 3q^{59} + 13q^{61} - 3q^{63} + 3q^{65} + 7q^{67} - 9q^{69} - 24q^{71} - 20q^{73} - 12q^{75} + 3q^{77} - 11q^{79} + 18q^{81} + 9q^{83} - 18q^{85} - 9q^{87} + 12q^{89} + 2q^{91} + 15q^{93} + 12q^{95} - 11q^{97} + 9q^{99} + O(q^{100}) \)

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.e.a \(2\) \(0.287\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-3\) \(1\) \(q+(1-2\zeta_{6})q^{3}+(-3+3\zeta_{6})q^{5}+\zeta_{6}q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(36, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(36, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 3 T^{2} \)
$5$ \( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \)
$7$ \( ( 1 - 5 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} ) \)
$11$ \( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \)
$13$ \( 1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4} \)
$17$ \( ( 1 - 6 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 + 4 T + 19 T^{2} )^{2} \)
$23$ \( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \)
$29$ \( 1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4} \)
$31$ \( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \)
$37$ \( ( 1 - 2 T + 37 T^{2} )^{2} \)
$41$ \( 1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4} \)
$43$ \( 1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 9 T + 34 T^{2} - 423 T^{3} + 2209 T^{4} \)
$53$ \( ( 1 + 6 T + 53 T^{2} )^{2} \)
$59$ \( 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} \)
$61$ \( ( 1 - 14 T + 61 T^{2} )( 1 + T + 61 T^{2} ) \)
$67$ \( 1 - 7 T - 18 T^{2} - 469 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 12 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 + 10 T + 73 T^{2} )^{2} \)
$79$ \( 1 + 11 T + 42 T^{2} + 869 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4} \)
$89$ \( ( 1 - 6 T + 89 T^{2} )^{2} \)
$97$ \( 1 + 11 T + 24 T^{2} + 1067 T^{3} + 9409 T^{4} \)
show more
show less