Properties

Label 384.9.b
Level $384$
Weight $9$
Character orbit 384.b
Rep. character $\chi_{384}(319,\cdot)$
Character field $\Q$
Dimension $64$
Newform subspaces $4$
Sturm bound $576$
Trace bound $17$

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Defining parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 384.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 8 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(576\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(384, [\chi])\).

Total New Old
Modular forms 528 64 464
Cusp forms 496 64 432
Eisenstein series 32 0 32

Trace form

\( 64 q + 139968 q^{9} + O(q^{10}) \) \( 64 q + 139968 q^{9} + 309120 q^{17} - 6416576 q^{25} - 8749440 q^{41} - 76975296 q^{49} + 6200064 q^{57} + 29299200 q^{65} - 80055680 q^{73} + 306110016 q^{81} + 365339520 q^{89} - 121481856 q^{97} + O(q^{100}) \)

Decomposition of \(S_{9}^{\mathrm{new}}(384, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
384.9.b.a 384.b 8.d $8$ $156.433$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{3}-\beta _{5}q^{5}-7\beta _{4}q^{7}+3^{7}q^{9}+\cdots\)
384.9.b.b 384.b 8.d $8$ $156.433$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{3}+\beta _{2}q^{5}-\beta _{4}q^{7}+3^{7}q^{9}+\cdots\)
384.9.b.c 384.b 8.d $16$ $156.433$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{3}-\beta _{2}q^{5}+\beta _{5}q^{7}+3^{7}q^{9}+\cdots\)
384.9.b.d 384.b 8.d $32$ $156.433$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{9}^{\mathrm{old}}(384, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(384, [\chi]) \cong \) \(S_{9}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(128, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 2}\)