Space of modular forms of level 294, weight 6, and Character 294.e

• Label: 294.6.e
• Level: $294$
• Weight: $6$
• Character orbit: 294.e
• Rep. character: $\chi_{294}(67,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $68$
• Newform subspaces: $26$
• Sturm bound: $336$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	294.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 26 \)
Sturm bound:			\(336\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(294, [\chi])\).

	Total	New	Old
Modular forms	592	68	524
Cusp forms	528	68	460
Eisenstein series	64	0	64

Trace form

68 * q - 544 * q^4 + 44 * q^5 + 144 * q^6 - 2754 * q^9 - 648 * q^10 - 296 * q^11 + 3312 * q^13 + 396 * q^15 - 8704 * q^16 - 2612 * q^17 - 1020 * q^19 - 1408 * q^20 + 7120 * q^22 - 8652 * q^23 - 1152 * q^24 - 16568 * q^25 - 880 * q^26 - 29720 * q^29 + 8352 * q^30 + 18834 * q^31 - 6714 * q^33 + 12384 * q^34 + 88128 * q^36 + 43320 * q^37 - 816 * q^38 + 7344 * q^39 - 10368 * q^40 - 53424 * q^41 - 13808 * q^43 - 4736 * q^44 + 3564 * q^45 + 24080 * q^46 + 99492 * q^47 - 43904 * q^50 + 23328 * q^51 - 26496 * q^52 + 45556 * q^53 - 5832 * q^54 + 119028 * q^55 + 122256 * q^57 - 23560 * q^58 - 73096 * q^59 - 3168 * q^60 + 47592 * q^61 + 18080 * q^62 + 278528 * q^64 - 287620 * q^65 - 34848 * q^66 - 140164 * q^67 - 41792 * q^68 + 197784 * q^69 + 78808 * q^71 + 2544 * q^73 - 102704 * q^74 - 9432 * q^75 + 32640 * q^76 - 93312 * q^78 - 138962 * q^79 + 11264 * q^80 - 223074 * q^81 - 75264 * q^82 - 446624 * q^83 + 168872 * q^85 + 7664 * q^86 + 119574 * q^87 - 56960 * q^88 + 127008 * q^89 + 104976 * q^90 + 276864 * q^92 + 4824 * q^93 - 108192 * q^94 + 572036 * q^95 - 18432 * q^96 + 342372 * q^97 + 47952 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(294, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
294.6.e.a	$294$	$6$	294.e	7.c	$3$	$2$	$1$	$47.153$	\(\Q(\sqrt{-3}) \)	None					6.6.a.a	$2$	$0$	\(-4\)	\(-9\)	\(-66\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-4\zeta_{6}q^{2}+(-9+9\zeta_{6})q^{3}+(-2^{4}+\cdots)q^{4}+\cdots\)
294.6.e.b	$294$	$6$	294.e	7.c	$3$	$2$	$1$	$47.153$	\(\Q(\sqrt{-3}) \)	None					42.6.a.f	$2$	$0$	\(-4\)	\(-9\)	\(-24\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-4\zeta_{6}q^{2}+(-9+9\zeta_{6})q^{3}+(-2^{4}+\cdots)q^{4}+\cdots\)
294.6.e.c	$294$	$6$	294.e	7.c	$3$	$2$	$1$	$47.153$	\(\Q(\sqrt{-3}) \)	None					42.6.a.e	$2$	$0$	\(-4\)	\(-9\)	\(76\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-4\zeta_{6}q^{2}+(-9+9\zeta_{6})q^{3}+(-2^{4}+\cdots)q^{4}+\cdots\)
294.6.e.d	$294$	$6$	294.e	7.c	$3$	$2$	$1$	$47.153$	\(\Q(\sqrt{-3}) \)	None					42.6.a.e	$2$	$0$	\(-4\)	\(9\)	\(-76\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-4\zeta_{6}q^{2}+(9-9\zeta_{6})q^{3}+(-2^{4}+2^{4}\zeta_{6})q^{4}+\cdots\)
294.6.e.e	$294$	$6$	294.e	7.c	$3$	$2$	$1$	$47.153$	\(\Q(\sqrt{-3}) \)	None					42.6.e.a	$2$	$0$	\(-4\)	\(9\)	\(-6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-4\zeta_{6}q^{2}+(9-9\zeta_{6})q^{3}+(-2^{4}+2^{4}\zeta_{6})q^{4}+\cdots\)
294.6.e.f	$294$	$6$	294.e	7.c	$3$	$2$	$1$	$47.153$	\(\Q(\sqrt{-3}) \)	None					42.6.a.f	$2$	$0$	\(-4\)	\(9\)	\(24\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-4\zeta_{6}q^{2}+(9-9\zeta_{6})q^{3}+(-2^{4}+2^{4}\zeta_{6})q^{4}+\cdots\)
294.6.e.g	$294$	$6$	294.e	7.c	$3$	$2$	$1$	$47.153$	\(\Q(\sqrt{-3}) \)	None					6.6.a.a	$2$	$0$	\(-4\)	\(9\)	\(66\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-4\zeta_{6}q^{2}+(9-9\zeta_{6})q^{3}+(-2^{4}+2^{4}\zeta_{6})q^{4}+\cdots\)
294.6.e.h	$294$	$6$	294.e	7.c	$3$	$2$	$1$	$47.153$	\(\Q(\sqrt{-3}) \)	None					42.6.a.a	$2$	$0$	\(4\)	\(-9\)	\(-54\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+4\zeta_{6}q^{2}+(-9+9\zeta_{6})q^{3}+(-2^{4}+\cdots)q^{4}+\cdots\)
294.6.e.i	$294$	$6$	294.e	7.c	$3$	$2$	$1$	$47.153$	\(\Q(\sqrt{-3}) \)	None					42.6.a.d	$2$	$0$	\(4\)	\(-9\)	\(-26\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+4\zeta_{6}q^{2}+(-9+9\zeta_{6})q^{3}+(-2^{4}+\cdots)q^{4}+\cdots\)
294.6.e.j	$294$	$6$	294.e	7.c	$3$	$2$	$1$	$47.153$	\(\Q(\sqrt{-3}) \)	None		✓			294.6.a.a	$2$	$0$	\(4\)	\(-9\)	\(-26\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+4\zeta_{6}q^{2}+(-9+9\zeta_{6})q^{3}+(-2^{4}+\cdots)q^{4}+\cdots\)
294.6.e.k	$294$	$6$	294.e	7.c	$3$	$2$	$1$	$47.153$	\(\Q(\sqrt{-3}) \)	None					42.6.a.b	$2$	$0$	\(4\)	\(-9\)	\(44\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+4\zeta_{6}q^{2}+(-9+9\zeta_{6})q^{3}+(-2^{4}+\cdots)q^{4}+\cdots\)
294.6.e.l	$294$	$6$	294.e	7.c	$3$	$2$	$1$	$47.153$	\(\Q(\sqrt{-3}) \)	None					42.6.a.c	$2$	$0$	\(4\)	\(-9\)	\(72\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+4\zeta_{6}q^{2}+(-9+9\zeta_{6})q^{3}+(-2^{4}+\cdots)q^{4}+\cdots\)
294.6.e.m	$294$	$6$	294.e	7.c	$3$	$2$	$1$	$47.153$	\(\Q(\sqrt{-3}) \)	None					42.6.e.b	$2$	$0$	\(4\)	\(-9\)	\(86\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+4\zeta_{6}q^{2}+(-9+9\zeta_{6})q^{3}+(-2^{4}+\cdots)q^{4}+\cdots\)
294.6.e.n	$294$	$6$	294.e	7.c	$3$	$2$	$1$	$47.153$	\(\Q(\sqrt{-3}) \)	None					42.6.a.c	$2$	$0$	\(4\)	\(9\)	\(-72\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+4\zeta_{6}q^{2}+(9-9\zeta_{6})q^{3}+(-2^{4}+2^{4}\zeta_{6})q^{4}+\cdots\)
294.6.e.o	$294$	$6$	294.e	7.c	$3$	$2$	$1$	$47.153$	\(\Q(\sqrt{-3}) \)	None					42.6.a.b	$2$	$0$	\(4\)	\(9\)	\(-44\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+4\zeta_{6}q^{2}+(9-9\zeta_{6})q^{3}+(-2^{4}+2^{4}\zeta_{6})q^{4}+\cdots\)
294.6.e.p	$294$	$6$	294.e	7.c	$3$	$2$	$1$	$47.153$	\(\Q(\sqrt{-3}) \)	None					42.6.a.d	$2$	$1$	\(4\)	\(9\)	\(26\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+4\zeta_{6}q^{2}+(9-9\zeta_{6})q^{3}+(-2^{4}+2^{4}\zeta_{6})q^{4}+\cdots\)
294.6.e.q	$294$	$6$	294.e	7.c	$3$	$2$	$1$	$47.153$	\(\Q(\sqrt{-3}) \)	None		✓			294.6.a.a	$2$	$0$	\(4\)	\(9\)	\(26\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+4\zeta_{6}q^{2}+(9-9\zeta_{6})q^{3}+(-2^{4}+2^{4}\zeta_{6})q^{4}+\cdots\)
294.6.e.r	$294$	$6$	294.e	7.c	$3$	$2$	$1$	$47.153$	\(\Q(\sqrt{-3}) \)	None					42.6.a.a	$2$	$0$	\(4\)	\(9\)	\(54\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+4\zeta_{6}q^{2}+(9-9\zeta_{6})q^{3}+(-2^{4}+2^{4}\zeta_{6})q^{4}+\cdots\)
294.6.e.s	$294$	$6$	294.e	7.c	$3$	$4$	$2$	$47.153$	\(\Q(\sqrt{-3}, \sqrt{9601})\)	None					42.6.e.c	$2$	$0$	\(-8\)	\(-18\)	\(-53\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-4\beta _{2}q^{2}+(-9+9\beta _{2})q^{3}+(-2^{4}+\cdots)q^{4}+\cdots\)
294.6.e.t	$294$	$6$	294.e	7.c	$3$	$4$	$2$	$47.153$	\(\Q(\sqrt{-3}, \sqrt{4705})\)	None		✓			294.6.a.s	$2$	$0$	\(-8\)	\(-18\)	\(-18\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q-4\beta _{1}q^{2}+(-9+9\beta _{1})q^{3}+(-2^{4}+\cdots)q^{4}+\cdots\)
294.6.e.u	$294$	$6$	294.e	7.c	$3$	$4$	$2$	$47.153$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		✓			294.6.a.t	$2$	$0$	\(-8\)	\(-18\)	\(108\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+4\beta _{2}q^{2}+(-9-9\beta _{2})q^{3}+(-2^{4}+\cdots)q^{4}+\cdots\)
294.6.e.v	$294$	$6$	294.e	7.c	$3$	$4$	$2$	$47.153$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		✓			294.6.a.t	$2$	$0$	\(-8\)	\(18\)	\(-108\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+4\beta _{2}q^{2}+(9+9\beta _{2})q^{3}+(-2^{4}-2^{4}\beta _{2}+\cdots)q^{4}+\cdots\)
294.6.e.w	$294$	$6$	294.e	7.c	$3$	$4$	$2$	$47.153$	\(\Q(\sqrt{-3}, \sqrt{4705})\)	None		✓			294.6.a.s	$2$	$0$	\(-8\)	\(18\)	\(18\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q-4\beta _{1}q^{2}+(9-9\beta _{1})q^{3}+(-2^{4}+2^{4}\beta _{1}+\cdots)q^{4}+\cdots\)
294.6.e.x	$294$	$6$	294.e	7.c	$3$	$4$	$2$	$47.153$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		✓			294.6.a.n	$2$	$0$	\(8\)	\(-18\)	\(-108\)	\(0\)	$7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(4+4\beta _{2})q^{2}+9\beta _{2}q^{3}+2^{4}\beta _{2}q^{4}+\cdots\)
294.6.e.y	$294$	$6$	294.e	7.c	$3$	$4$	$2$	$47.153$	\(\Q(\sqrt{-3}, \sqrt{505})\)	None					42.6.e.d	$2$	$0$	\(8\)	\(18\)	\(17\)	\(0\)	$7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+4\beta _{1}q^{2}+(9-9\beta _{1})q^{3}+(-2^{4}+2^{4}\beta _{1}+\cdots)q^{4}+\cdots\)
294.6.e.z	$294$	$6$	294.e	7.c	$3$	$4$	$2$	$47.153$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		✓			294.6.a.n	$2$	$0$	\(8\)	\(18\)	\(108\)	\(0\)	$7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q-4\beta _{2}q^{2}+(9+9\beta _{2})q^{3}+(-2^{4}-2^{4}\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(294, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(294, [\chi]) \simeq \) \(S_{6}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(98, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(147, [\chi])\)\(^{\oplus 2}\)