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Properties

Label 336.3.bh
Level $336$
Weight $3$
Character orbit 336.bh
Rep. character $\chi_{336}(145,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $32$
Newform subspaces $8$
Sturm bound $192$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 336.bh (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 8 \)
Sturm bound: \(192\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(336, [\chi])\).

Total New Old
Modular forms 280 32 248
Cusp forms 232 32 200
Eisenstein series 48 0 48

Trace form

\( 32 q - 8 q^{7} + 48 q^{9} + O(q^{10}) \) \( 32 q - 8 q^{7} + 48 q^{9} + 16 q^{11} - 48 q^{19} - 16 q^{23} + 88 q^{25} + 32 q^{29} + 24 q^{31} - 72 q^{33} + 240 q^{35} + 48 q^{37} + 24 q^{39} + 128 q^{43} + 288 q^{47} + 48 q^{49} + 96 q^{53} - 48 q^{57} - 192 q^{59} - 144 q^{61} + 24 q^{63} - 80 q^{65} - 160 q^{67} - 384 q^{71} - 120 q^{73} - 80 q^{77} - 24 q^{79} - 144 q^{81} - 32 q^{85} + 48 q^{91} - 48 q^{93} + 96 q^{95} + 96 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(336, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
336.3.bh.a 336.bh 7.d $2$ $9.155$ \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(-9\) \(7\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-1-\zeta_{6})q^{3}+(-6+3\zeta_{6})q^{5}+(7+\cdots)q^{7}+\cdots\)
336.3.bh.b 336.bh 7.d $2$ $9.155$ \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(3\) \(-13\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-1-\zeta_{6})q^{3}+(2-\zeta_{6})q^{5}+(-5+\cdots)q^{7}+\cdots\)
336.3.bh.c 336.bh 7.d $2$ $9.155$ \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(-6\) \(7\) $\mathrm{SU}(2)[C_{6}]$ \(q+(1+\zeta_{6})q^{3}+(-4+2\zeta_{6})q^{5}+7\zeta_{6}q^{7}+\cdots\)
336.3.bh.d 336.bh 7.d $2$ $9.155$ \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(9\) \(-13\) $\mathrm{SU}(2)[C_{6}]$ \(q+(1+\zeta_{6})q^{3}+(6-3\zeta_{6})q^{5}+(-5-3\zeta_{6})q^{7}+\cdots\)
336.3.bh.e 336.bh 7.d $4$ $9.155$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(-6\) \(12\) \(10\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-1+\beta _{1})q^{3}+(4+2\beta _{1}+\beta _{3})q^{5}+\cdots\)
336.3.bh.f 336.bh 7.d $4$ $9.155$ \(\Q(\sqrt{-3}, \sqrt{65})\) None \(0\) \(6\) \(-9\) \(-2\) $\mathrm{SU}(2)[C_{6}]$ \(q+(2-\beta _{1})q^{3}+(-1-2\beta _{1}-\beta _{3})q^{5}+\cdots\)
336.3.bh.g 336.bh 7.d $8$ $9.155$ 8.0.\(\cdots\).9 None \(0\) \(-12\) \(-6\) \(-8\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-1-\beta _{1})q^{3}+(-1+\beta _{1}-\beta _{6})q^{5}+\cdots\)
336.3.bh.h 336.bh 7.d $8$ $9.155$ 8.0.\(\cdots\).2 None \(0\) \(12\) \(6\) \(4\) $\mathrm{SU}(2)[C_{6}]$ \(q+(1-\beta _{3})q^{3}+(1+\beta _{4})q^{5}+(-\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(336, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(336, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 2}\)