Space of modular forms of level 294, weight 4, and Character 294.e

• Label: 294.4.e
• Level: $294$
• Weight: $4$
• Character orbit: 294.e
• Rep. character: $\chi_{294}(67,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $40$
• Newform subspaces: $15$
• Sturm bound: $224$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	294.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(224\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(294, [\chi])\).

	Total	New	Old
Modular forms	368	40	328
Cusp forms	304	40	264
Eisenstein series	64	0	64

Trace form

40 * q + 6 * q^3 - 80 * q^4 - 16 * q^5 - 24 * q^6 - 180 * q^9 + 52 * q^10 + 28 * q^11 + 24 * q^12 - 12 * q^13 - 84 * q^15 - 320 * q^16 - 260 * q^17 + 50 * q^19 + 128 * q^20 - 568 * q^22 + 324 * q^23 + 48 * q^24 + 10 * q^25 + 152 * q^26 - 108 * q^27 + 40 * q^29 - 48 * q^30 - 156 * q^31 + 138 * q^33 - 304 * q^34 + 1440 * q^36 - 1210 * q^37 + 72 * q^38 - 294 * q^39 + 208 * q^40 + 936 * q^41 + 1516 * q^43 + 112 * q^44 - 144 * q^45 + 200 * q^46 + 72 * q^47 - 192 * q^48 + 1984 * q^50 - 216 * q^51 + 24 * q^52 - 2396 * q^53 + 108 * q^54 + 380 * q^55 + 300 * q^57 - 492 * q^58 - 112 * q^59 + 168 * q^60 + 1764 * q^61 + 3536 * q^62 + 2560 * q^64 + 584 * q^65 + 528 * q^66 + 2306 * q^67 - 1040 * q^68 - 1896 * q^69 - 4976 * q^71 - 746 * q^73 + 1720 * q^74 + 738 * q^75 - 400 * q^76 + 1728 * q^78 - 584 * q^79 - 256 * q^80 - 1620 * q^81 - 768 * q^82 + 3256 * q^83 - 1816 * q^85 - 40 * q^86 - 462 * q^87 + 1136 * q^88 - 864 * q^89 - 936 * q^90 - 2592 * q^92 + 2346 * q^93 + 2160 * q^94 - 4264 * q^95 + 192 * q^96 - 4340 * q^97 - 504 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(294, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
294.4.e.a	$294$	$4$	294.e	7.c	$3$	$2$	$1$	$17.347$	\(\Q(\sqrt{-3}) \)	None					42.4.a.b	$2$	$0$	\(-2\)	\(-3\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-2\zeta_{6}q^{2}+(-3+3\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
294.4.e.b	$294$	$4$	294.e	7.c	$3$	$2$	$1$	$17.347$	\(\Q(\sqrt{-3}) \)	None					42.4.a.a	$2$	$0$	\(-2\)	\(-3\)	\(18\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-2\zeta_{6}q^{2}+(-3+3\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
294.4.e.c	$294$	$4$	294.e	7.c	$3$	$2$	$1$	$17.347$	\(\Q(\sqrt{-3}) \)	None					42.4.a.a	$2$	$0$	\(-2\)	\(3\)	\(-18\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-2\zeta_{6}q^{2}+(3-3\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
294.4.e.d	$294$	$4$	294.e	7.c	$3$	$2$	$1$	$17.347$	\(\Q(\sqrt{-3}) \)	None					42.4.a.b	$2$	$0$	\(-2\)	\(3\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-2\zeta_{6}q^{2}+(3-3\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
294.4.e.e	$294$	$4$	294.e	7.c	$3$	$2$	$1$	$17.347$	\(\Q(\sqrt{-3}) \)	None					42.4.e.b	$2$	$0$	\(2\)	\(-3\)	\(-15\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\zeta_{6}q^{2}+(-3+3\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
294.4.e.f	$294$	$4$	294.e	7.c	$3$	$2$	$1$	$17.347$	\(\Q(\sqrt{-3}) \)	None		✓			294.4.a.b	$2$	$0$	\(2\)	\(-3\)	\(-8\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\zeta_{6}q^{2}+(-3+3\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
294.4.e.g	$294$	$4$	294.e	7.c	$3$	$2$	$1$	$17.347$	\(\Q(\sqrt{-3}) \)	None					6.4.a.a	$2$	$0$	\(2\)	\(-3\)	\(6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\zeta_{6}q^{2}+(-3+3\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
294.4.e.h	$294$	$4$	294.e	7.c	$3$	$2$	$1$	$17.347$	\(\Q(\sqrt{-3}) \)	None					6.4.a.a	$2$	$0$	\(2\)	\(3\)	\(-6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\zeta_{6}q^{2}+(3-3\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
294.4.e.i	$294$	$4$	294.e	7.c	$3$	$2$	$1$	$17.347$	\(\Q(\sqrt{-3}) \)	None					42.4.e.a	$2$	$0$	\(2\)	\(3\)	\(-6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\zeta_{6}q^{2}+(3-3\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
294.4.e.j	$294$	$4$	294.e	7.c	$3$	$2$	$1$	$17.347$	\(\Q(\sqrt{-3}) \)	None		✓			294.4.a.b	$2$	$0$	\(2\)	\(3\)	\(8\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\zeta_{6}q^{2}+(3-3\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
294.4.e.k	$294$	$4$	294.e	7.c	$3$	$4$	$2$	$17.347$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		✓			294.4.a.l	$2$	$0$	\(-4\)	\(-6\)	\(-12\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\beta _{2}q^{2}+(-3-3\beta _{2})q^{3}+(-4-4\beta _{2}+\cdots)q^{4}+\cdots\)
294.4.e.l	$294$	$4$	294.e	7.c	$3$	$4$	$2$	$17.347$	\(\Q(\sqrt{-3}, \sqrt{1345})\)	None					42.4.e.c	$2$	$0$	\(-4\)	\(6\)	\(5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-2\beta _{2}q^{2}+(3-3\beta _{2})q^{3}+(-4+4\beta _{2}+\cdots)q^{4}+\cdots\)
294.4.e.m	$294$	$4$	294.e	7.c	$3$	$4$	$2$	$17.347$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		✓			294.4.a.l	$2$	$0$	\(-4\)	\(6\)	\(12\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\beta _{2}q^{2}+(3+3\beta _{2})q^{3}+(-4-4\beta _{2}+\cdots)q^{4}+\cdots\)
294.4.e.n	$294$	$4$	294.e	7.c	$3$	$4$	$2$	$17.347$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		✓			294.4.a.j	$2$	$0$	\(4\)	\(-6\)	\(12\)	\(0\)	$7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2+2\beta _{2})q^{2}+3\beta _{2}q^{3}+4\beta _{2}q^{4}+\cdots\)
294.4.e.o	$294$	$4$	294.e	7.c	$3$	$4$	$2$	$17.347$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		✓			294.4.a.j	$2$	$0$	\(4\)	\(6\)	\(-12\)	\(0\)	$7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2+2\beta _{2})q^{2}-3\beta _{2}q^{3}+4\beta _{2}q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(294, [\chi])\) into lower level spaces

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