Properties

Label 60.2.h
Level $60$
Weight $2$
Character orbit 60.h
Rep. character $\chi_{60}(59,\cdot)$
Character field $\Q$
Dimension $8$
Newform subspaces $2$
Sturm bound $24$
Trace bound $4$

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Defining parameters

Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 60.h (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 60 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(24\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(60, [\chi])\).

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

Trace form

\( 8q - 6q^{4} + 2q^{6} - 4q^{9} + O(q^{10}) \) \( 8q - 6q^{4} + 2q^{6} - 4q^{9} - 10q^{10} + 2q^{16} - 20q^{21} + 26q^{24} + 20q^{30} + 20q^{34} - 22q^{36} + 10q^{40} + 20q^{45} - 4q^{46} - 16q^{49} - 46q^{54} - 30q^{60} + 24q^{61} - 54q^{64} + 28q^{69} - 40q^{70} + 60q^{76} + 32q^{81} + 40q^{84} - 40q^{85} + 30q^{90} + 92q^{94} - 22q^{96} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(60, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
60.2.h.a \(4\) \(0.479\) \(\Q(\sqrt{-2}, \sqrt{-5})\) \(\Q(\sqrt{-5}) \) \(0\) \(0\) \(0\) \(0\) \(q-\beta _{2}q^{2}-\beta _{1}q^{3}-2q^{4}-\beta _{3}q^{5}+(-1+\cdots)q^{6}+\cdots\)
60.2.h.b \(4\) \(0.479\) \(\Q(\sqrt{-3}, \sqrt{5})\) \(\Q(\sqrt{-15}) \) \(0\) \(0\) \(0\) \(0\) \(q-\beta _{2}q^{2}+(-\beta _{1}+\beta _{2})q^{3}+\beta _{3}q^{4}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + 2 T^{2} )^{2} \))(\( 1 - T^{2} + 4 T^{4} \))
$3$ (\( 1 - 4 T^{2} + 9 T^{4} \))(\( ( 1 + 3 T^{2} )^{2} \))
$5$ (\( ( 1 + 5 T^{2} )^{2} \))(\( ( 1 - 5 T^{2} )^{2} \))
$7$ (\( ( 1 + 4 T^{2} + 49 T^{4} )^{2} \))(\( ( 1 + 7 T^{2} )^{4} \))
$11$ (\( ( 1 + 11 T^{2} )^{4} \))(\( ( 1 + 11 T^{2} )^{4} \))
$13$ (\( ( 1 - 13 T^{2} )^{4} \))(\( ( 1 - 13 T^{2} )^{4} \))
$17$ (\( ( 1 + 17 T^{2} )^{4} \))(\( ( 1 + 14 T^{2} + 289 T^{4} )^{2} \))
$19$ (\( ( 1 - 19 T^{2} )^{4} \))(\( ( 1 - 4 T + 19 T^{2} )^{2}( 1 + 4 T + 19 T^{2} )^{2} \))
$23$ (\( ( 1 - 44 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 34 T^{2} + 529 T^{4} )^{2} \))
$29$ (\( ( 1 - 6 T + 29 T^{2} )^{2}( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 29 T^{2} )^{4} \))
$31$ (\( ( 1 - 31 T^{2} )^{4} \))(\( ( 1 - 8 T + 31 T^{2} )^{2}( 1 + 8 T + 31 T^{2} )^{2} \))
$37$ (\( ( 1 - 37 T^{2} )^{4} \))(\( ( 1 - 37 T^{2} )^{4} \))
$41$ (\( ( 1 - 12 T + 41 T^{2} )^{2}( 1 + 12 T + 41 T^{2} )^{2} \))(\( ( 1 - 41 T^{2} )^{4} \))
$43$ (\( ( 1 + 76 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 + 43 T^{2} )^{4} \))
$47$ (\( ( 1 + 4 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 14 T^{2} + 2209 T^{4} )^{2} \))
$53$ (\( ( 1 + 53 T^{2} )^{4} \))(\( ( 1 + 86 T^{2} + 2809 T^{4} )^{2} \))
$59$ (\( ( 1 + 59 T^{2} )^{4} \))(\( ( 1 + 59 T^{2} )^{4} \))
$61$ (\( ( 1 - 8 T + 61 T^{2} )^{4} \))(\( ( 1 + 2 T + 61 T^{2} )^{4} \))
$67$ (\( ( 1 - 116 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 + 67 T^{2} )^{4} \))
$71$ (\( ( 1 + 71 T^{2} )^{4} \))(\( ( 1 + 71 T^{2} )^{4} \))
$73$ (\( ( 1 - 73 T^{2} )^{4} \))(\( ( 1 - 73 T^{2} )^{4} \))
$79$ (\( ( 1 - 79 T^{2} )^{4} \))(\( ( 1 - 16 T + 79 T^{2} )^{2}( 1 + 16 T + 79 T^{2} )^{2} \))
$83$ (\( ( 1 + 76 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 154 T^{2} + 6889 T^{4} )^{2} \))
$89$ (\( ( 1 - 6 T + 89 T^{2} )^{2}( 1 + 6 T + 89 T^{2} )^{2} \))(\( ( 1 - 89 T^{2} )^{4} \))
$97$ (\( ( 1 - 97 T^{2} )^{4} \))(\( ( 1 - 97 T^{2} )^{4} \))
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