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Properties

Label 48.9.g
Level $48$
Weight $9$
Character orbit 48.g
Rep. character $\chi_{48}(31,\cdot)$
Character field $\Q$
Dimension $8$
Newform subspaces $3$
Sturm bound $72$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 48.g (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 4 \)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(72\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(48, [\chi])\).

Total New Old
Modular forms 70 8 62
Cusp forms 58 8 50
Eisenstein series 12 0 12

Trace form

\( 8 q + 1008 q^{5} - 17496 q^{9} + O(q^{10}) \) \( 8 q + 1008 q^{5} - 17496 q^{9} + 59984 q^{13} + 115920 q^{17} - 121824 q^{21} + 2112280 q^{25} + 206640 q^{29} + 2760480 q^{33} - 3960560 q^{37} + 13472208 q^{41} - 2204496 q^{45} + 6149768 q^{49} - 42196176 q^{53} + 775008 q^{57} + 16762000 q^{61} - 65829024 q^{65} - 89954800 q^{73} + 39265920 q^{77} + 38263752 q^{81} + 147417696 q^{85} - 18967536 q^{89} + 129711456 q^{93} - 36206320 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{9}^{\mathrm{new}}(48, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
48.9.g.a 48.g 4.b $2$ $19.554$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-180\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+3^{3}\zeta_{6}q^{3}-90q^{5}+532\zeta_{6}q^{7}-3^{7}q^{9}+\cdots\)
48.9.g.b 48.g 4.b $2$ $19.554$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(1452\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-3\zeta_{6}q^{3}+726q^{5}-14^{2}\zeta_{6}q^{7}-3^{7}q^{9}+\cdots\)
48.9.g.c 48.g 4.b $4$ $19.554$ \(\Q(\sqrt{-3}, \sqrt{1801})\) None \(0\) \(0\) \(-264\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-3^{3}\beta _{1}q^{3}+(-66+\beta _{2})q^{5}+(772\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{9}^{\mathrm{old}}(48, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(48, [\chi]) \cong \) \(S_{9}^{\mathrm{new}}(4, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 2}\)