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Properties

Label 336.4.q
Level $336$
Weight $4$
Character orbit 336.q
Rep. character $\chi_{336}(193,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $48$
Newform subspaces $13$
Sturm bound $256$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 336.q (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 13 \)
Sturm bound: \(256\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(336, [\chi])\).

Total New Old
Modular forms 408 48 360
Cusp forms 360 48 312
Eisenstein series 48 0 48

Trace form

\( 48 q + 6 q^{3} - 18 q^{7} - 216 q^{9} + O(q^{10}) \) \( 48 q + 6 q^{3} - 18 q^{7} - 216 q^{9} + 20 q^{11} + 318 q^{19} + 80 q^{23} - 516 q^{25} - 108 q^{27} - 400 q^{29} - 402 q^{31} - 12 q^{33} + 468 q^{35} + 8 q^{37} + 210 q^{39} - 592 q^{41} - 892 q^{43} - 204 q^{47} + 592 q^{49} + 16 q^{53} - 376 q^{55} + 168 q^{57} + 688 q^{59} + 200 q^{61} - 162 q^{63} - 280 q^{65} - 750 q^{67} - 1704 q^{71} + 1092 q^{73} + 1050 q^{75} + 184 q^{77} - 942 q^{79} - 1944 q^{81} - 4152 q^{83} + 1296 q^{85} - 1044 q^{87} + 1712 q^{89} - 1138 q^{91} - 456 q^{93} - 36 q^{95} + 2328 q^{97} - 360 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(336, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
336.4.q.a 336.q 7.c $2$ $19.825$ \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(-7\) \(35\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-3+3\zeta_{6})q^{3}-7\zeta_{6}q^{5}+(21-7\zeta_{6})q^{7}+\cdots\)
336.4.q.b 336.q 7.c $2$ $19.825$ \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(-2\) \(-35\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-3+3\zeta_{6})q^{3}-2\zeta_{6}q^{5}+(-14+\cdots)q^{7}+\cdots\)
336.4.q.c 336.q 7.c $2$ $19.825$ \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(11\) \(-7\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-3+3\zeta_{6})q^{3}+11\zeta_{6}q^{5}+(7-21\zeta_{6})q^{7}+\cdots\)
336.4.q.d 336.q 7.c $2$ $19.825$ \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(15\) \(-35\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-3+3\zeta_{6})q^{3}+15\zeta_{6}q^{5}+(-21+\cdots)q^{7}+\cdots\)
336.4.q.e 336.q 7.c $2$ $19.825$ \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(3\) \(7\) $\mathrm{SU}(2)[C_{3}]$ \(q+(3-3\zeta_{6})q^{3}+3\zeta_{6}q^{5}+(-7+21\zeta_{6})q^{7}+\cdots\)
336.4.q.f 336.q 7.c $2$ $19.825$ \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(6\) \(7\) $\mathrm{SU}(2)[C_{3}]$ \(q+(3-3\zeta_{6})q^{3}+6\zeta_{6}q^{5}+(14-21\zeta_{6})q^{7}+\cdots\)
336.4.q.g 336.q 7.c $4$ $19.825$ \(\Q(\sqrt{-3}, \sqrt{505})\) None \(0\) \(-6\) \(-9\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-3+3\beta _{2})q^{3}+(-\beta _{1}-4\beta _{2})q^{5}+\cdots\)
336.4.q.h 336.q 7.c $4$ $19.825$ \(\Q(\sqrt{-3}, \sqrt{-19})\) None \(0\) \(-6\) \(3\) \(20\) $\mathrm{SU}(2)[C_{3}]$ \(q+3\beta _{2}q^{3}+(1+2\beta _{1}+2\beta _{2}-\beta _{3})q^{5}+\cdots\)
336.4.q.i 336.q 7.c $4$ $19.825$ \(\Q(\sqrt{-3}, \sqrt{193})\) None \(0\) \(6\) \(-11\) \(-6\) $\mathrm{SU}(2)[C_{3}]$ \(q+(3-3\beta _{2})q^{3}+(\beta _{1}-6\beta _{2})q^{5}+(1-2\beta _{1}+\cdots)q^{7}+\cdots\)
336.4.q.j 336.q 7.c $4$ $19.825$ \(\Q(\sqrt{-3}, \sqrt{1345})\) None \(0\) \(6\) \(-5\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(3-3\beta _{2})q^{3}+(-\beta _{1}-2\beta _{2})q^{5}+(-1+\cdots)q^{7}+\cdots\)
336.4.q.k 336.q 7.c $6$ $19.825$ 6.0.9924270768.1 None \(0\) \(-9\) \(-11\) \(13\) $\mathrm{SU}(2)[C_{3}]$ \(q+3\beta _{3}q^{3}+(-4+\beta _{2}-4\beta _{3}+\beta _{4}+\cdots)q^{5}+\cdots\)
336.4.q.l 336.q 7.c $6$ $19.825$ 6.0.\(\cdots\).1 None \(0\) \(9\) \(11\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(3-3\beta _{3})q^{3}+(4\beta _{3}-\beta _{4}-\beta _{5})q^{5}+\cdots\)
336.4.q.m 336.q 7.c $8$ $19.825$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(12\) \(-4\) \(-18\) $\mathrm{SU}(2)[C_{3}]$ \(q-3\beta _{1}q^{3}+(-1-\beta _{1}+\beta _{6})q^{5}+(-2+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(336, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(336, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 2}\)