Properties

Label 72.2.d
Level 72
Weight 2
Character orbit d
Rep. character \(\chi_{72}(37,\cdot)\)
Character field \(\Q\)
Dimension 4
Newform subspaces 2
Sturm bound 24
Trace bound 2

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Defining parameters

Level: \( N \) = \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 72.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 8 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(24\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(72, [\chi])\).

Total New Old
Modular forms 16 6 10
Cusp forms 8 4 4
Eisenstein series 8 2 6

Trace form

\( 4q + 2q^{2} - 4q^{4} - 4q^{8} + O(q^{10}) \) \( 4q + 2q^{2} - 4q^{4} - 4q^{8} - 4q^{10} - 4q^{14} + 4q^{17} + 8q^{20} + 16q^{22} - 8q^{23} - 4q^{25} + 8q^{26} - 8q^{28} - 16q^{31} - 8q^{32} + 4q^{34} - 8q^{38} + 24q^{40} - 4q^{41} - 8q^{46} + 24q^{47} - 12q^{49} + 2q^{50} + 16q^{52} + 32q^{55} + 8q^{56} - 20q^{58} + 4q^{62} - 16q^{64} - 16q^{65} - 24q^{70} - 24q^{71} + 16q^{73} - 16q^{74} - 16q^{76} - 4q^{82} + 8q^{86} - 32q^{88} + 20q^{89} + 24q^{94} + 16q^{95} - 6q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(72, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
72.2.d.a \(2\) \(0.575\) \(\Q(\sqrt{-2}) \) \(\Q(\sqrt{-6}) \) \(0\) \(0\) \(0\) \(4\) \(q+\beta q^{2}-2q^{4}+2\beta q^{5}+2q^{7}-2\beta q^{8}+\cdots\)
72.2.d.b \(2\) \(0.575\) \(\Q(\sqrt{-1}) \) None \(2\) \(0\) \(0\) \(-4\) \(q+(1+i)q^{2}+2iq^{4}-2iq^{5}-2q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(72, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(72, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T^{2} \))(\( 1 - 2 T + 2 T^{2} \))
$3$ 1
$5$ (\( 1 - 2 T^{2} + 25 T^{4} \))(\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))
$7$ (\( ( 1 - 2 T + 7 T^{2} )^{2} \))(\( ( 1 + 2 T + 7 T^{2} )^{2} \))
$11$ (\( 1 + 10 T^{2} + 121 T^{4} \))(\( ( 1 - 11 T^{2} )^{2} \))
$13$ (\( ( 1 - 13 T^{2} )^{2} \))(\( ( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \))
$17$ (\( ( 1 + 17 T^{2} )^{2} \))(\( ( 1 - 2 T + 17 T^{2} )^{2} \))
$19$ (\( ( 1 - 19 T^{2} )^{2} \))(\( 1 - 22 T^{2} + 361 T^{4} \))
$23$ (\( ( 1 + 23 T^{2} )^{2} \))(\( ( 1 + 4 T + 23 T^{2} )^{2} \))
$29$ (\( 1 - 50 T^{2} + 841 T^{4} \))(\( 1 - 22 T^{2} + 841 T^{4} \))
$31$ (\( ( 1 + 10 T + 31 T^{2} )^{2} \))(\( ( 1 - 2 T + 31 T^{2} )^{2} \))
$37$ (\( ( 1 - 37 T^{2} )^{2} \))(\( 1 - 10 T^{2} + 1369 T^{4} \))
$41$ (\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 2 T + 41 T^{2} )^{2} \))
$43$ (\( ( 1 - 43 T^{2} )^{2} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))
$47$ (\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 - 12 T + 47 T^{2} )^{2} \))
$53$ (\( 1 + 94 T^{2} + 2809 T^{4} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))
$59$ (\( 1 + 10 T^{2} + 3481 T^{4} \))(\( 1 - 102 T^{2} + 3481 T^{4} \))
$61$ (\( ( 1 - 61 T^{2} )^{2} \))(\( ( 1 - 61 T^{2} )^{2} \))
$67$ (\( ( 1 - 67 T^{2} )^{2} \))(\( 1 + 10 T^{2} + 4489 T^{4} \))
$71$ (\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))
$73$ (\( ( 1 - 14 T + 73 T^{2} )^{2} \))(\( ( 1 + 6 T + 73 T^{2} )^{2} \))
$79$ (\( ( 1 + 10 T + 79 T^{2} )^{2} \))(\( ( 1 - 10 T + 79 T^{2} )^{2} \))
$83$ (\( 1 - 134 T^{2} + 6889 T^{4} \))(\( 1 + 90 T^{2} + 6889 T^{4} \))
$89$ (\( ( 1 + 89 T^{2} )^{2} \))(\( ( 1 - 10 T + 89 T^{2} )^{2} \))
$97$ (\( ( 1 - 2 T + 97 T^{2} )^{2} \))(\( ( 1 + 2 T + 97 T^{2} )^{2} \))
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