Properties

Label 392.6.a
Level $392$
Weight $6$
Character orbit 392.a
Rep. character $\chi_{392}(1,\cdot)$
Character field $\Q$
Dimension $51$
Newform subspaces $14$
Sturm bound $336$
Trace bound $9$

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Defining parameters

Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 392.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 14 \)
Sturm bound: \(336\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(392))\).

Total New Old
Modular forms 296 51 245
Cusp forms 264 51 213
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.

\(2\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(11\)
\(+\)\(-\)\(-\)\(14\)
\(-\)\(+\)\(-\)\(13\)
\(-\)\(-\)\(+\)\(13\)
Plus space\(+\)\(24\)
Minus space\(-\)\(27\)

Trace form

\( 51 q + 2 q^{3} - 24 q^{5} + 3827 q^{9} + O(q^{10}) \) \( 51 q + 2 q^{3} - 24 q^{5} + 3827 q^{9} + 144 q^{11} + 700 q^{13} - 1388 q^{15} - 1126 q^{17} - 3938 q^{19} - 1260 q^{23} + 26957 q^{25} - 2500 q^{27} + 6134 q^{29} + 5252 q^{31} + 11776 q^{33} - 5070 q^{37} + 1600 q^{39} + 8922 q^{41} + 26588 q^{43} + 20168 q^{45} + 4812 q^{47} + 5044 q^{51} - 15642 q^{53} + 65312 q^{55} - 57316 q^{57} - 50714 q^{59} - 49120 q^{61} + 130144 q^{65} - 75480 q^{67} - 49536 q^{69} + 97864 q^{71} + 66158 q^{73} + 117142 q^{75} - 158484 q^{79} + 407611 q^{81} - 141530 q^{83} + 32228 q^{85} - 283628 q^{87} + 203158 q^{89} + 41732 q^{93} + 291388 q^{95} - 113302 q^{97} + 21148 q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(392))\) 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 7
392.6.a.a 392.a 1.a $1$ $62.870$ \(\Q\) None \(0\) \(-30\) \(-32\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-30q^{3}-2^{5}q^{5}+657q^{9}-624q^{11}+\cdots\)
392.6.a.b 392.a 1.a $1$ $62.870$ \(\Q\) None \(0\) \(-20\) \(74\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-20q^{3}+74q^{5}+157q^{9}+124q^{11}+\cdots\)
392.6.a.c 392.a 1.a $1$ $62.870$ \(\Q\) None \(0\) \(6\) \(-4\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+6q^{3}-4q^{5}-207q^{9}-240q^{11}+\cdots\)
392.6.a.d 392.a 1.a $2$ $62.870$ \(\Q(\sqrt{345}) \) None \(0\) \(6\) \(-82\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(3+\beta )q^{3}+(-41+3\beta )q^{5}+(111+\cdots)q^{9}+\cdots\)
392.6.a.e 392.a 1.a $2$ $62.870$ \(\Q(\sqrt{193}) \) None \(0\) \(14\) \(-42\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(7-\beta )q^{3}+(-21+5\beta )q^{5}+(-1+\cdots)q^{9}+\cdots\)
392.6.a.f 392.a 1.a $2$ $62.870$ \(\Q(\sqrt{177}) \) None \(0\) \(26\) \(62\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(13-\beta )q^{3}+(31-5\beta )q^{5}+(103-26\beta )q^{9}+\cdots\)
392.6.a.g 392.a 1.a $4$ $62.870$ \(\Q(\sqrt{86}, \sqrt{134})\) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+\beta _{2}q^{5}+(-23+\beta _{3})q^{9}+\cdots\)
392.6.a.h 392.a 1.a $4$ $62.870$ 4.4.2732674592.1 None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{3}+(-\beta _{1}-2\beta _{2})q^{5}+(209-\beta _{3})q^{9}+\cdots\)
392.6.a.i 392.a 1.a $5$ $62.870$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(-13\) \(-31\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-3+\beta _{1})q^{3}+(-6-\beta _{3})q^{5}+(47+\cdots)q^{9}+\cdots\)
392.6.a.j 392.a 1.a $5$ $62.870$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(-5\) \(81\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{3}+(2^{4}+\beta _{1}+\beta _{2})q^{5}+\cdots\)
392.6.a.k 392.a 1.a $5$ $62.870$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(5\) \(-81\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{3}+(-2^{4}-\beta _{1}-\beta _{2})q^{5}+\cdots\)
392.6.a.l 392.a 1.a $5$ $62.870$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(13\) \(31\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(3-\beta _{1})q^{3}+(6+\beta _{3})q^{5}+(47-4\beta _{1}+\cdots)q^{9}+\cdots\)
392.6.a.m 392.a 1.a $6$ $62.870$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{3}+(\beta _{1}-\beta _{3})q^{5}+(-20+\beta _{4}+\cdots)q^{9}+\cdots\)
392.6.a.n 392.a 1.a $8$ $62.870$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{3}+(-\beta _{1}-\beta _{4})q^{5}+(116+\beta _{3}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(392))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(392)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 2}\)