Properties

Label 336.2.q
Level $336$
Weight $2$
Character orbit 336.q
Rep. character $\chi_{336}(193,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $16$
Newform subspaces $7$
Sturm bound $128$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 336.q (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 7 \)
Sturm bound: \(128\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(336, [\chi])\).

Total New Old
Modular forms 152 16 136
Cusp forms 104 16 88
Eisenstein series 48 0 48

Trace form

\( 16 q - 2 q^{3} - 2 q^{7} - 8 q^{9} + O(q^{10}) \) \( 16 q - 2 q^{3} - 2 q^{7} - 8 q^{9} + 4 q^{11} - 10 q^{19} + 16 q^{23} - 12 q^{25} + 4 q^{27} + 16 q^{29} + 14 q^{31} - 4 q^{33} + 36 q^{35} - 8 q^{37} + 2 q^{39} + 16 q^{41} - 12 q^{43} - 12 q^{47} - 16 q^{49} - 16 q^{53} - 56 q^{55} - 8 q^{57} - 16 q^{59} - 8 q^{61} - 2 q^{63} - 8 q^{65} - 6 q^{67} + 24 q^{71} + 12 q^{73} - 14 q^{75} + 8 q^{77} + 18 q^{79} - 8 q^{81} - 24 q^{83} - 16 q^{85} + 12 q^{87} + 16 q^{89} + 54 q^{91} + 8 q^{93} - 36 q^{95} + 8 q^{97} - 8 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(336, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
336.2.q.a 336.q 7.c $2$ $2.683$ \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(-2\) \(-5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{3}-2\zeta_{6}q^{5}+(-2-\zeta_{6})q^{7}+\cdots\)
336.2.q.b 336.q 7.c $2$ $2.683$ \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(-1\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{5}+(1-3\zeta_{6})q^{7}+\cdots\)
336.2.q.c 336.q 7.c $2$ $2.683$ \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(2\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{3}+2\zeta_{6}q^{5}+(-2+3\zeta_{6})q^{7}+\cdots\)
336.2.q.d 336.q 7.c $2$ $2.683$ \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(-3\) \(-5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}-3\zeta_{6}q^{5}+(-3+\zeta_{6})q^{7}+\cdots\)
336.2.q.e 336.q 7.c $2$ $2.683$ \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(1\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}+\zeta_{6}q^{5}+(1-3\zeta_{6})q^{7}+\cdots\)
336.2.q.f 336.q 7.c $2$ $2.683$ \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(2\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}+2\zeta_{6}q^{5}+(2+\zeta_{6})q^{7}+\cdots\)
336.2.q.g 336.q 7.c $4$ $2.683$ \(\Q(\sqrt{-3}, \sqrt{-19})\) None \(0\) \(-2\) \(1\) \(6\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{2})q^{3}+(-1+2\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(336, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(336, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 2}\)