Properties

Label 392.2.a
Level $392$
Weight $2$
Character orbit 392.a
Rep. character $\chi_{392}(1,\cdot)$
Character field $\Q$
Dimension $10$
Newform subspaces $8$
Sturm bound $112$
Trace bound $9$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 392.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(112\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(392))\).

Total New Old
Modular forms 72 10 62
Cusp forms 41 10 31
Eisenstein series 31 0 31

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.
\(+\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(3\)
\(-\)\(-\)\(+\)\(2\)
Plus space\(+\)\(3\)
Minus space\(-\)\(7\)

Trace form

\( 10 q - 2 q^{3} + 2 q^{5} + 14 q^{9} + O(q^{10}) \) \( 10 q - 2 q^{3} + 2 q^{5} + 14 q^{9} + 4 q^{11} - 2 q^{13} + 12 q^{15} + 8 q^{17} - 6 q^{19} - 4 q^{23} + 6 q^{25} + 4 q^{27} - 8 q^{29} - 12 q^{31} + 12 q^{37} - 8 q^{39} + 10 q^{45} + 12 q^{47} - 4 q^{51} - 16 q^{53} + 8 q^{55} - 4 q^{57} - 6 q^{59} + 2 q^{61} - 36 q^{65} + 20 q^{67} - 16 q^{69} + 4 q^{73} - 22 q^{75} - 4 q^{79} + 10 q^{81} - 14 q^{83} - 24 q^{85} - 4 q^{87} - 4 q^{89} - 12 q^{93} - 28 q^{95} + 8 q^{97} - 64 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(392))\) 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
392.2.a.a 392.a 1.a $1$ $3.130$ \(\Q\) None \(0\) \(-3\) \(1\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+q^{5}+6q^{9}-q^{11}-2q^{13}+\cdots\)
392.2.a.b 392.a 1.a $1$ $3.130$ \(\Q\) None \(0\) \(-2\) \(4\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+4q^{5}+q^{9}-8q^{15}+2q^{17}+\cdots\)
392.2.a.c 392.a 1.a $1$ $3.130$ \(\Q\) None \(0\) \(-1\) \(-1\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}-2q^{9}+3q^{11}-6q^{13}+\cdots\)
392.2.a.d 392.a 1.a $1$ $3.130$ \(\Q\) None \(0\) \(0\) \(-2\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{5}-3q^{9}-4q^{11}-2q^{13}+6q^{17}+\cdots\)
392.2.a.e 392.a 1.a $1$ $3.130$ \(\Q\) None \(0\) \(1\) \(1\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-2q^{9}+3q^{11}+6q^{13}+\cdots\)
392.2.a.f 392.a 1.a $1$ $3.130$ \(\Q\) None \(0\) \(3\) \(-1\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}-q^{5}+6q^{9}-q^{11}+2q^{13}+\cdots\)
392.2.a.g 392.a 1.a $2$ $3.130$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+2\beta q^{5}-q^{9}+6q^{11}-4\beta q^{13}+\cdots\)
392.2.a.h 392.a 1.a $2$ $3.130$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+\beta q^{5}+5q^{9}-4q^{11}-\beta q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(392))\) into lower level spaces

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