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Properties

Label 384.10.a
Level $384$
Weight $10$
Character orbit 384.a
Rep. character $\chi_{384}(1,\cdot)$
Character field $\Q$
Dimension $72$
Newform subspaces $16$
Sturm bound $640$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 384.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 16 \)
Sturm bound: \(640\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(384))\).

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

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)FrickeDim.
\(+\)\(+\)\(+\)\(17\)
\(+\)\(-\)\(-\)\(19\)
\(-\)\(+\)\(-\)\(19\)
\(-\)\(-\)\(+\)\(17\)
Plus space\(+\)\(34\)
Minus space\(-\)\(38\)

Trace form

\( 72 q + 472392 q^{9} + O(q^{10}) \) \( 72 q + 472392 q^{9} - 815984 q^{17} + 35012056 q^{25} - 15123824 q^{41} + 363437352 q^{49} + 316449504 q^{57} - 952039712 q^{65} + 562858800 q^{73} + 3099363912 q^{81} - 1100315568 q^{89} - 3010260848 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(384))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3
384.10.a.a 384.a 1.a $4$ $197.774$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(-324\) \(-1728\) \(4840\) $+$ $+$ $\mathrm{SU}(2)$ \(q-3^{4}q^{3}+(-432-\beta _{1})q^{5}+(1210+\cdots)q^{7}+\cdots\)
384.10.a.b 384.a 1.a $4$ $197.774$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(-324\) \(-240\) \(-4840\) $+$ $+$ $\mathrm{SU}(2)$ \(q-3^{4}q^{3}+(-60-\beta _{2})q^{5}+(-1210+\cdots)q^{7}+\cdots\)
384.10.a.c 384.a 1.a $4$ $197.774$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(-324\) \(240\) \(4840\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3^{4}q^{3}+(60+\beta _{2})q^{5}+(1210+\beta _{1}+\cdots)q^{7}+\cdots\)
384.10.a.d 384.a 1.a $4$ $197.774$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(-324\) \(1728\) \(-4840\) $+$ $+$ $\mathrm{SU}(2)$ \(q-3^{4}q^{3}+(432+\beta _{1})q^{5}+(-1210+\cdots)q^{7}+\cdots\)
384.10.a.e 384.a 1.a $4$ $197.774$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(324\) \(-1728\) \(-4840\) $-$ $-$ $\mathrm{SU}(2)$ \(q+3^{4}q^{3}+(-432-\beta _{1})q^{5}+(-1210+\cdots)q^{7}+\cdots\)
384.10.a.f 384.a 1.a $4$ $197.774$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(324\) \(-240\) \(4840\) $-$ $-$ $\mathrm{SU}(2)$ \(q+3^{4}q^{3}+(-60-\beta _{2})q^{5}+(1210+\beta _{1}+\cdots)q^{7}+\cdots\)
384.10.a.g 384.a 1.a $4$ $197.774$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(324\) \(240\) \(-4840\) $-$ $-$ $\mathrm{SU}(2)$ \(q+3^{4}q^{3}+(60+\beta _{2})q^{5}+(-1210-\beta _{1}+\cdots)q^{7}+\cdots\)
384.10.a.h 384.a 1.a $4$ $197.774$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(324\) \(1728\) \(4840\) $+$ $-$ $\mathrm{SU}(2)$ \(q+3^{4}q^{3}+(432+\beta _{1})q^{5}+(1210+\beta _{1}+\cdots)q^{7}+\cdots\)
384.10.a.i 384.a 1.a $5$ $197.774$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(-405\) \(-772\) \(-38\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3^{4}q^{3}+(-154+\beta _{1})q^{5}+(-8+\beta _{2}+\cdots)q^{7}+\cdots\)
384.10.a.j 384.a 1.a $5$ $197.774$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(-405\) \(-240\) \(-38\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3^{4}q^{3}+(-48-\beta _{1})q^{5}+(-8+\beta _{1}+\cdots)q^{7}+\cdots\)
384.10.a.k 384.a 1.a $5$ $197.774$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(-405\) \(240\) \(38\) $+$ $+$ $\mathrm{SU}(2)$ \(q-3^{4}q^{3}+(48+\beta _{1})q^{5}+(8-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
384.10.a.l 384.a 1.a $5$ $197.774$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(-405\) \(772\) \(38\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3^{4}q^{3}+(154-\beta _{1})q^{5}+(8-\beta _{2})q^{7}+\cdots\)
384.10.a.m 384.a 1.a $5$ $197.774$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(405\) \(-772\) \(38\) $-$ $-$ $\mathrm{SU}(2)$ \(q+3^{4}q^{3}+(-154+\beta _{1})q^{5}+(8-\beta _{2}+\cdots)q^{7}+\cdots\)
384.10.a.n 384.a 1.a $5$ $197.774$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(405\) \(-240\) \(38\) $+$ $-$ $\mathrm{SU}(2)$ \(q+3^{4}q^{3}+(-48-\beta _{1})q^{5}+(8-\beta _{1}+\cdots)q^{7}+\cdots\)
384.10.a.o 384.a 1.a $5$ $197.774$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(405\) \(240\) \(-38\) $+$ $-$ $\mathrm{SU}(2)$ \(q+3^{4}q^{3}+(48+\beta _{1})q^{5}+(-8+\beta _{1}+\cdots)q^{7}+\cdots\)
384.10.a.p 384.a 1.a $5$ $197.774$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(405\) \(772\) \(-38\) $+$ $-$ $\mathrm{SU}(2)$ \(q+3^{4}q^{3}+(154-\beta _{1})q^{5}+(-8+\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(384))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(384)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 14}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 12}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 7}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 10}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 8}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 5}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 2}\)