Space of modular forms of level 336, weight 5, and Character 336.bh

• Label: 336.5.bh
• Level: $336$
• Weight: $5$
• Character orbit: 336.bh
• Rep. character: $\chi_{336}(145,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $64$
• Newform subspaces: $9$
• Sturm bound: $320$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	336.bh (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(320\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(336, [\chi])\).

	Total	New	Old
Modular forms	536	64	472
Cusp forms	488	64	424
Eisenstein series	48	0	48

Trace form

64 * q - 16 * q^7 + 864 * q^9 - 96 * q^11 - 1248 * q^19 + 2016 * q^23 + 3632 * q^25 - 1728 * q^29 - 1104 * q^31 + 2160 * q^33 - 3744 * q^35 - 1824 * q^37 + 1584 * q^39 + 4864 * q^43 - 8640 * q^47 - 5280 * q^49 - 2112 * q^53 + 3744 * q^57 + 19584 * q^59 + 12384 * q^61 + 432 * q^63 + 2400 * q^65 - 7232 * q^67 + 5376 * q^71 + 5520 * q^73 + 4320 * q^77 + 4944 * q^79 - 23328 * q^81 + 5312 * q^85 - 29472 * q^91 + 7200 * q^93 + 29376 * q^95 - 5184 * q^99

Decomposition of \(S_{5}^{\mathrm{new}}(336, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
336.5.bh.a	$336$	$5$	336.bh	7.d	$6$	$2$	$1$	$34.732$	\(\Q(\sqrt{-3}) \)	None					21.5.f.b	$2$	$0$	\(0\)	\(-9\)	\(-3\)	\(91\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-3-3\zeta_{6})q^{3}+(-2+\zeta_{6})q^{5}+(35+\cdots)q^{7}+\cdots\)
336.5.bh.b	$336$	$5$	336.bh	7.d	$6$	$2$	$1$	$34.732$	\(\Q(\sqrt{-3}) \)	None					21.5.f.a	$2$	$0$	\(0\)	\(-9\)	\(18\)	\(-77\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-3-3\zeta_{6})q^{3}+(12-6\zeta_{6})q^{5}+(-56+\cdots)q^{7}+\cdots\)
336.5.bh.c	$336$	$5$	336.bh	7.d	$6$	$4$	$2$	$34.732$	\(\Q(\sqrt{-3}, \sqrt{-19})\)	None					84.5.m.a	$2$	$0$	\(0\)	\(-18\)	\(39\)	\(70\)	$3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-6-3\beta _{2})q^{3}+(5-8\beta _{2}-3\beta _{3})q^{5}+\cdots\)
336.5.bh.d	$336$	$5$	336.bh	7.d	$6$	$4$	$2$	$34.732$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None					42.5.g.a	$2$	$0$	\(0\)	\(18\)	\(-66\)	\(70\)	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{6}]$	\(q+(6+3\beta _{1})q^{3}+(-11+11\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
336.5.bh.e	$336$	$5$	336.bh	7.d	$6$	$6$	$3$	$34.732$	6.0.\(\cdots\).2	None					84.5.m.b	$2$	$0$	\(0\)	\(27\)	\(15\)	\(-15\)	$2^{2}\cdot 3^{2}\cdot 7$	$\mathrm{SU}(2)[C_{6}]$	\(q+(6-3\beta _{1})q^{3}+(1+2\beta _{1}+\beta _{3})q^{5}+\cdots\)
336.5.bh.f	$336$	$5$	336.bh	7.d	$6$	$6$	$3$	$34.732$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None					21.5.f.c	$2$	$0$	\(0\)	\(27\)	\(39\)	\(-23\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(3+3\beta _{2})q^{3}+(8+\beta _{1}-4\beta _{2}+2\beta _{3}+\cdots)q^{5}+\cdots\)
336.5.bh.g	$336$	$5$	336.bh	7.d	$6$	$8$	$4$	$34.732$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					42.5.g.b	$2$	$0$	\(0\)	\(-36\)	\(-42\)	\(-76\)	$2^{8}\cdot 3^{4}\cdot 7$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-3-3\beta _{1})q^{3}+(-7+4\beta _{1}+\beta _{6}+\cdots)q^{5}+\cdots\)
336.5.bh.h	$336$	$5$	336.bh	7.d	$6$	$16$	$8$	$34.732$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None					168.5.z.b	$2$	$0$	\(0\)	\(-72\)	\(-12\)	\(-16\)	$2^{36}\cdot 7^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-6+3\beta _{2})q^{3}+(-1-\beta _{3})q^{5}+(-1+\cdots)q^{7}+\cdots\)
336.5.bh.i	$336$	$5$	336.bh	7.d	$6$	$16$	$8$	$34.732$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None					168.5.z.a	$2$	$0$	\(0\)	\(72\)	\(12\)	\(-40\)	$2^{36}\cdot 7^{5}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(6-3\beta _{1})q^{3}+(\beta _{1}+\beta _{5}+\beta _{7})q^{5}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(336, [\chi])\) into lower level spaces

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