↑

Properties

Label 336.3.bn
Level $336$
Weight $3$
Character orbit 336.bn
Rep. character $\chi_{336}(65,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $60$
Newform subspaces $8$
Sturm bound $192$
Trace bound $7$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 336.bn (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 21 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 8 \)
Sturm bound: \(192\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(13\)

Dimensions

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

Total New Old
Modular forms 280 68 212
Cusp forms 232 60 172
Eisenstein series 48 8 40

Trace form

\( 60 q + q^{3} - 4 q^{7} - q^{9} + O(q^{10}) \) \( 60 q + q^{3} - 4 q^{7} - q^{9} - 8 q^{13} + 22 q^{15} - 14 q^{19} + 15 q^{21} + 116 q^{25} + 4 q^{27} + 58 q^{31} - 19 q^{33} - 2 q^{37} + 58 q^{39} + 232 q^{43} - 11 q^{45} + 44 q^{49} - 37 q^{51} + 108 q^{55} - 18 q^{57} - 2 q^{61} - 135 q^{63} + 98 q^{67} - 214 q^{69} - 74 q^{73} + 102 q^{75} - 70 q^{79} - 73 q^{81} - 76 q^{85} - 80 q^{87} - 200 q^{91} + 97 q^{93} - 216 q^{97} - 506 q^{99} + O(q^{100}) \)

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.bn.a 336.bn 21.h $2$ $9.155$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(-3\) \(0\) \(-11\) $\mathrm{U}(1)[D_{6}]$ \(q-3\zeta_{6}q^{3}+(-3-5\zeta_{6})q^{7}+(-9+9\zeta_{6})q^{9}+\cdots\)
336.3.bn.b 336.bn 21.h $2$ $9.155$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(-3\) \(0\) \(13\) $\mathrm{U}(1)[D_{6}]$ \(q-3\zeta_{6}q^{3}+(5+3\zeta_{6})q^{7}+(-9+9\zeta_{6})q^{9}+\cdots\)
336.3.bn.c 336.bn 21.h $4$ $9.155$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(-2\) \(0\) \(14\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\beta _{1}-\beta _{2}-\beta _{3})q^{3}+3\beta _{1}q^{5}+(7-7\beta _{2}+\cdots)q^{7}+\cdots\)
336.3.bn.d 336.bn 21.h $4$ $9.155$ \(\Q(\sqrt{-3}, \sqrt{-35})\) None \(0\) \(-1\) \(0\) \(28\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{2}-\beta _{3})q^{3}+(-1+2\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
336.3.bn.e 336.bn 21.h $4$ $9.155$ \(\Q(\sqrt{-3}, \sqrt{-5})\) None \(0\) \(4\) \(0\) \(-22\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\beta _{1}+2\beta _{2}-\beta _{3})q^{3}+3\beta _{1}q^{5}+(-8+\cdots)q^{7}+\cdots\)
336.3.bn.f 336.bn 21.h $4$ $9.155$ \(\Q(\sqrt{-3}, \sqrt{-5})\) None \(0\) \(4\) \(0\) \(-14\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{1}+2\beta _{2}+\beta _{3})q^{3}+\beta _{1}q^{5}-7\beta _{2}q^{7}+\cdots\)
336.3.bn.g 336.bn 21.h $8$ $9.155$ 8.0.4857532416.2 None \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{7}q^{3}+(2\beta _{1}+\beta _{4})q^{5}+(3+\beta _{3}+6\beta _{4}+\cdots)q^{7}+\cdots\)
336.3.bn.h 336.bn 21.h $32$ $9.155$ None \(0\) \(0\) \(0\) \(-12\) $\mathrm{SU}(2)[C_{6}]$

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

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