Properties

Label 384.5.b
Level $384$
Weight $5$
Character orbit 384.b
Rep. character $\chi_{384}(319,\cdot)$
Character field $\Q$
Dimension $32$
Newform subspaces $4$
Sturm bound $320$
Trace bound $17$

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Defining parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 5 \)
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: \(320\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(384, [\chi])\).

Total New Old
Modular forms 272 32 240
Cusp forms 240 32 208
Eisenstein series 32 0 32

Trace form

\( 32 q + 864 q^{9} + O(q^{10}) \) \( 32 q + 864 q^{9} + 960 q^{17} - 2656 q^{25} + 2880 q^{41} - 16992 q^{49} - 14976 q^{57} + 26880 q^{65} + 46400 q^{73} + 23328 q^{81} - 37440 q^{89} - 50496 q^{97} + O(q^{100}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(384, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
384.5.b.a 384.b 8.d $4$ $39.694$ \(\Q(\sqrt{3}, \sqrt{-19})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+3\beta _{1}q^{3}+\beta _{2}q^{5}+\beta _{3}q^{7}+3^{3}q^{9}+\cdots\)
384.5.b.b 384.b 8.d $4$ $39.694$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{12}q^{3}+\zeta_{12}^{2}q^{5}-\zeta_{12}^{3}q^{7}+3^{3}q^{9}+\cdots\)
384.5.b.c 384.b 8.d $8$ $39.694$ 8.0.49787136.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{3}+(\beta _{1}-\beta _{3})q^{5}-\beta _{5}q^{7}+3^{3}q^{9}+\cdots\)
384.5.b.d 384.b 8.d $16$ $39.694$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{3}+\beta _{11}q^{5}-\beta _{5}q^{7}+3^{3}q^{9}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(384, [\chi])\) into lower level spaces

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