Properties

Label 384.8.d
Level $384$
Weight $8$
Character orbit 384.d
Rep. character $\chi_{384}(193,\cdot)$
Character field $\Q$
Dimension $56$
Newform subspaces $6$
Sturm bound $512$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 384.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 8 \)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(512\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(384, [\chi])\).

Total New Old
Modular forms 464 56 408
Cusp forms 432 56 376
Eisenstein series 32 0 32

Trace form

\( 56 q - 40824 q^{9} + O(q^{10}) \) \( 56 q - 40824 q^{9} - 11632 q^{17} - 810504 q^{25} - 1765136 q^{41} + 4544152 q^{49} - 6204384 q^{57} + 12748352 q^{65} + 11043568 q^{73} + 29760696 q^{81} + 28686288 q^{89} - 60259504 q^{97} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(384, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
384.8.d.a $6$ $119.956$ \(\mathbb{Q}[x]/(x^{6} + \cdots)\) None \(0\) \(0\) \(0\) \(-136\) \(q+3^{3}\beta _{1}q^{3}+(-37\beta _{1}+\beta _{3})q^{5}+(-23+\cdots)q^{7}+\cdots\)
384.8.d.b $6$ $119.956$ \(\mathbb{Q}[x]/(x^{6} + \cdots)\) None \(0\) \(0\) \(0\) \(136\) \(q-3^{3}\beta _{1}q^{3}+(-37\beta _{1}+\beta _{3})q^{5}+(23+\cdots)q^{7}+\cdots\)
384.8.d.c $8$ $119.956$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(0\) \(0\) \(-2880\) \(q-3^{3}\beta _{1}q^{3}+(28\beta _{1}+\beta _{5})q^{5}+(-360+\cdots)q^{7}+\cdots\)
384.8.d.d $8$ $119.956$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(0\) \(0\) \(2880\) \(q+3^{3}\beta _{1}q^{3}+(28\beta _{1}+\beta _{5})q^{5}+(360+\cdots)q^{7}+\cdots\)
384.8.d.e $12$ $119.956$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+3\beta _{1}q^{3}-\beta _{8}q^{5}+(\beta _{6}-\beta _{7})q^{7}-3^{6}q^{9}+\cdots\)
384.8.d.f $16$ $119.956$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{8}q^{3}-\beta _{1}q^{5}+\beta _{10}q^{7}-3^{6}q^{9}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(384, [\chi])\) into lower level spaces

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