Properties

Label 384.10.d
Level $384$
Weight $10$
Character orbit 384.d
Rep. character $\chi_{384}(193,\cdot)$
Character field $\Q$
Dimension $72$
Newform subspaces $6$
Sturm bound $640$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(640\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(384, [\chi])\).

Total New Old
Modular forms 592 72 520
Cusp forms 560 72 488
Eisenstein series 32 0 32

Trace form

\( 72 q - 472392 q^{9} + O(q^{10}) \) \( 72 q - 472392 q^{9} + 815984 q^{17} - 21237944 q^{25} - 15123824 q^{41} + 466693992 q^{49} + 316449504 q^{57} + 198366656 q^{65} - 2337658224 q^{73} + 3099363912 q^{81} + 1100315568 q^{89} + 6481449392 q^{97} + O(q^{100}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(384, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
384.10.d.a 384.d 8.b $8$ $197.774$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(0\) \(0\) \(-13632\) $\mathrm{SU}(2)[C_{2}]$ \(q+3^{4}\beta _{4}q^{3}+(140\beta _{4}-\beta _{5})q^{5}+(-1704+\cdots)q^{7}+\cdots\)
384.10.d.b 384.d 8.b $8$ $197.774$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(0\) \(0\) \(13632\) $\mathrm{SU}(2)[C_{2}]$ \(q-3^{4}\beta _{4}q^{3}+(140\beta _{4}-\beta _{5})q^{5}+(1704+\cdots)q^{7}+\cdots\)
384.10.d.c 384.d 8.b $10$ $197.774$ \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None \(0\) \(0\) \(0\) \(-5576\) $\mathrm{SU}(2)[C_{2}]$ \(q-3^{4}\beta _{5}q^{3}+(112\beta _{5}+\beta _{6})q^{5}+(-558+\cdots)q^{7}+\cdots\)
384.10.d.d 384.d 8.b $10$ $197.774$ \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None \(0\) \(0\) \(0\) \(5576\) $\mathrm{SU}(2)[C_{2}]$ \(q+3^{4}\beta _{5}q^{3}+(112\beta _{5}+\beta _{6})q^{5}+(558+\cdots)q^{7}+\cdots\)
384.10.d.e 384.d 8.b $16$ $197.774$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{8}q^{3}+\beta _{9}q^{5}+\beta _{1}q^{7}-3^{8}q^{9}+\cdots\)
384.10.d.f 384.d 8.b $20$ $197.774$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{3}-\beta _{7}q^{5}+\beta _{10}q^{7}-3^{8}q^{9}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(384, [\chi])\) into lower level spaces

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