Properties

Label 392.2.i
Level $392$
Weight $2$
Character orbit 392.i
Rep. character $\chi_{392}(177,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $20$
Newform subspaces $8$
Sturm bound $112$
Trace bound $9$

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Defining parameters

Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 392.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 8 \)
Sturm bound: \(112\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(392, [\chi])\).

Total New Old
Modular forms 144 20 124
Cusp forms 80 20 60
Eisenstein series 64 0 64

Trace form

\( 20 q + 2 q^{3} - 2 q^{5} - 8 q^{9} + O(q^{10}) \) \( 20 q + 2 q^{3} - 2 q^{5} - 8 q^{9} + 2 q^{11} + 8 q^{13} + 12 q^{15} - 2 q^{17} + 6 q^{19} - 14 q^{23} - 24 q^{25} - 28 q^{27} + 8 q^{29} - 6 q^{31} - 6 q^{33} - 6 q^{37} + 20 q^{39} + 24 q^{41} + 24 q^{43} - 4 q^{45} + 6 q^{47} + 22 q^{51} + 10 q^{53} + 4 q^{55} - 44 q^{57} + 18 q^{59} - 2 q^{61} + 36 q^{65} - 2 q^{67} + 4 q^{69} - 48 q^{71} + 14 q^{73} - 8 q^{75} - 14 q^{79} + 2 q^{81} - 16 q^{83} - 60 q^{85} - 20 q^{87} - 2 q^{89} + 6 q^{93} - 2 q^{95} - 8 q^{97} - 80 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(392, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
392.2.i.a $2$ $3.130$ \(\Q(\sqrt{-3}) \) None \(0\) \(-2\) \(4\) \(0\) \(q+(-2+2\zeta_{6})q^{3}+4\zeta_{6}q^{5}-\zeta_{6}q^{9}+\cdots\)
392.2.i.b $2$ $3.130$ \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(-1\) \(0\) \(q+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{5}+2\zeta_{6}q^{9}+\cdots\)
392.2.i.c $2$ $3.130$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-2\) \(0\) \(q-2\zeta_{6}q^{5}+3\zeta_{6}q^{9}+(4-4\zeta_{6})q^{11}+\cdots\)
392.2.i.d $2$ $3.130$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(2\) \(0\) \(q+2\zeta_{6}q^{5}+3\zeta_{6}q^{9}+(4-4\zeta_{6})q^{11}+\cdots\)
392.2.i.e $2$ $3.130$ \(\Q(\sqrt{-3}) \) None \(0\) \(2\) \(-4\) \(0\) \(q+(2-2\zeta_{6})q^{3}-4\zeta_{6}q^{5}-\zeta_{6}q^{9}-8q^{15}+\cdots\)
392.2.i.f $2$ $3.130$ \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(-1\) \(0\) \(q+(3-3\zeta_{6})q^{3}-\zeta_{6}q^{5}-6\zeta_{6}q^{9}+(1+\cdots)q^{11}+\cdots\)
392.2.i.g $4$ $3.130$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{3})q^{5}+5\beta _{2}q^{9}+\cdots\)
392.2.i.h $4$ $3.130$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{3}+(-2\beta _{1}-2\beta _{3})q^{5}-\beta _{2}q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(392, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(392, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(98, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(196, [\chi])\)\(^{\oplus 2}\)