Properties

Label 384.7.b
Level $384$
Weight $7$
Character orbit 384.b
Rep. character $\chi_{384}(319,\cdot)$
Character field $\Q$
Dimension $48$
Newform subspaces $5$
Sturm bound $448$
Trace bound $17$

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Defining parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 384.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 8 \)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(448\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(384, [\chi])\).

Total New Old
Modular forms 400 48 352
Cusp forms 368 48 320
Eisenstein series 32 0 32

Trace form

\( 48 q + 11664 q^{9} + O(q^{10}) \) \( 48 q + 11664 q^{9} - 19552 q^{17} - 108816 q^{25} - 217120 q^{41} - 394896 q^{49} + 544320 q^{57} - 1089152 q^{65} - 921888 q^{73} + 2834352 q^{81} - 881760 q^{89} + 3764256 q^{97} + O(q^{100}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(384, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
384.7.b.a 384.b 8.d $4$ $88.341$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+3\zeta_{12}q^{3}-7^{2}\zeta_{12}^{2}q^{5}-13\zeta_{12}^{3}q^{7}+\cdots\)
384.7.b.b 384.b 8.d $4$ $88.341$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+3\zeta_{12}q^{3}-5\zeta_{12}^{2}q^{5}-\zeta_{12}^{3}q^{7}+\cdots\)
384.7.b.c 384.b 8.d $8$ $88.341$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{4}q^{3}-\beta _{1}q^{5}+\beta _{5}q^{7}+3^{5}q^{9}+\cdots\)
384.7.b.d 384.b 8.d $8$ $88.341$ 8.0.1731891456.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{3}+5\beta _{3}q^{5}+(3\beta _{2}+\beta _{4})q^{7}+\cdots\)
384.7.b.e 384.b 8.d $24$ $88.341$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{7}^{\mathrm{old}}(384, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(384, [\chi]) \cong \) \(S_{7}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(128, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 2}\)