Space of modular forms of level 294, weight 5, and Character 294.g

• Label: 294.5.g
• Level: $294$
• Weight: $5$
• Character orbit: 294.g
• Rep. character: $\chi_{294}(19,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $52$
• Newform subspaces: $8$
• Sturm bound: $280$
• Trace bound: $10$

Defining parameters

Level:	\( N \)	\(=\)	\( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	294.g (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(280\)
Trace bound:			\(10\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(294, [\chi])\).

	Total	New	Old
Modular forms	480	52	428
Cusp forms	416	52	364
Eisenstein series	64	0	64

Trace form

52 * q - 18 * q^3 - 208 * q^4 + 108 * q^5 + 702 * q^9 - 96 * q^10 + 204 * q^11 + 144 * q^12 - 144 * q^15 - 1664 * q^16 - 864 * q^17 - 42 * q^19 - 256 * q^22 + 1200 * q^23 + 938 * q^25 - 1152 * q^26 + 4752 * q^29 - 1152 * q^30 + 2466 * q^31 - 1080 * q^33 - 11232 * q^36 - 2306 * q^37 - 8640 * q^38 + 4266 * q^39 + 768 * q^40 - 13004 * q^43 + 1632 * q^44 + 2916 * q^45 + 5568 * q^46 + 13932 * q^47 + 4608 * q^50 - 5400 * q^51 - 624 * q^52 - 13704 * q^53 + 8172 * q^57 + 15008 * q^58 + 5112 * q^59 + 576 * q^60 - 15936 * q^61 + 26624 * q^64 + 5028 * q^65 - 13158 * q^67 + 6912 * q^68 + 19608 * q^71 - 4926 * q^73 + 15360 * q^74 - 32166 * q^75 + 13824 * q^78 - 29594 * q^79 - 6912 * q^80 - 18954 * q^81 + 768 * q^82 - 47424 * q^85 + 10176 * q^86 + 34020 * q^87 + 1024 * q^88 + 27576 * q^89 - 19200 * q^92 - 2538 * q^93 + 42240 * q^94 + 53700 * q^95 + 11016 * q^99

Decomposition of \(S_{5}^{\mathrm{new}}(294, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
294.5.g.a	$294$	$5$	294.g	7.d	$6$	$4$	$2$	$30.391$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None					42.5.c.a	$2$	$0$	\(0\)	\(-18\)	\(24\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{1}+\beta _{3})q^{2}+(-3+3\beta _{2})q^{3}+(-8+\cdots)q^{4}+\cdots\)
294.5.g.b	$294$	$5$	294.g	7.d	$6$	$4$	$2$	$30.391$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None					42.5.c.a	$2$	$0$	\(0\)	\(18\)	\(-24\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{1}-\beta _{3})q^{2}+(3-3\beta _{2})q^{3}+(-8+\cdots)q^{4}+\cdots\)
294.5.g.c	$294$	$5$	294.g	7.d	$6$	$4$	$2$	$30.391$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None					42.5.g.a	$2$	$0$	\(0\)	\(18\)	\(66\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{1}q^{2}+(6+3\beta _{2})q^{3}+8\beta _{2}q^{4}+(11+\cdots)q^{5}+\cdots\)
294.5.g.d	$294$	$5$	294.g	7.d	$6$	$8$	$4$	$30.391$	8.0.339738624.1	None		✓			294.5.c.c	$2$	$0$	\(0\)	\(-36\)	\(0\)	\(0\)	$7^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q-2\beta _{5}q^{2}+(-3+3\beta _{4})q^{3}+(-8-8\beta _{4}+\cdots)q^{4}+\cdots\)
294.5.g.e	$294$	$5$	294.g	7.d	$6$	$8$	$4$	$30.391$	8.0.339738624.1	None		✓			294.5.c.b	$2$	$0$	\(0\)	\(-36\)	\(0\)	\(0\)	$7^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-2\beta _{2}-2\beta _{5})q^{2}+(-6-3\beta _{4})q^{3}+\cdots\)
294.5.g.f	$294$	$5$	294.g	7.d	$6$	$8$	$4$	$30.391$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					42.5.g.b	$2$	$0$	\(0\)	\(-36\)	\(42\)	\(0\)	$2^{4}\cdot 3^{2}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{3}q^{2}+(-6-3\beta _{1})q^{3}+8\beta _{1}q^{4}+\cdots\)
294.5.g.g	$294$	$5$	294.g	7.d	$6$	$8$	$4$	$30.391$	8.0.339738624.1	None		✓			294.5.c.b	$2$	$0$	\(0\)	\(36\)	\(0\)	\(0\)	$7^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(2\beta _{2}+2\beta _{5})q^{2}+(6+3\beta _{4})q^{3}+8\beta _{4}q^{4}+\cdots\)
294.5.g.h	$294$	$5$	294.g	7.d	$6$	$8$	$4$	$30.391$	8.0.339738624.1	None		✓			294.5.c.c	$2$	$0$	\(0\)	\(36\)	\(0\)	\(0\)	$7^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q-2\beta _{5}q^{2}+(3-3\beta _{4})q^{3}+(-8-8\beta _{4}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(294, [\chi])\) into lower level spaces

\( S_{5}^{\mathrm{old}}(294, [\chi]) \simeq \) \(S_{5}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(98, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(147, [\chi])\)\(^{\oplus 2}\)