Space of modular forms of level 294, weight 6, and trivial character

• Label: 294.6.a
• Level: $294$
• Weight: $6$
• Character orbit: 294.a
• Rep. character: $\chi_{294}(1,\cdot)$
• Character field: $\Q$
• Dimension: $33$
• Newform subspaces: $23$
• Sturm bound: $336$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	294.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 23 \)
Sturm bound:			\(336\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(294))\).

	Total	New	Old
Modular forms	296	33	263
Cusp forms	264	33	231
Eisenstein series	32	0	32

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(3\)	\(7\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(34\)	\(3\)	\(31\)		\(30\)	\(3\)	\(27\)		\(4\)	\(0\)	\(4\)
\(+\)	\(+\)	\(-\)	\(-\)		\(39\)	\(5\)	\(34\)		\(35\)	\(5\)	\(30\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)		\(38\)	\(4\)	\(34\)		\(34\)	\(4\)	\(30\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)		\(37\)	\(4\)	\(33\)		\(33\)	\(4\)	\(29\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)		\(38\)	\(4\)	\(34\)		\(34\)	\(4\)	\(30\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)		\(36\)	\(4\)	\(32\)		\(32\)	\(4\)	\(28\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)		\(38\)	\(3\)	\(35\)		\(34\)	\(3\)	\(31\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)		\(36\)	\(6\)	\(30\)		\(32\)	\(6\)	\(26\)		\(4\)	\(0\)	\(4\)
Plus space			\(+\)		\(145\)	\(14\)	\(131\)		\(129\)	\(14\)	\(115\)		\(16\)	\(0\)	\(16\)
Minus space			\(-\)		\(151\)	\(19\)	\(132\)		\(135\)	\(19\)	\(116\)		\(16\)	\(0\)	\(16\)

Trace form

33 * q + 4 * q^2 + 9 * q^3 + 528 * q^4 + 22 * q^5 + 36 * q^6 + 64 * q^8 + 2673 * q^9 - 360 * q^10 - 652 * q^11 + 144 * q^12 + 694 * q^13 + 198 * q^15 + 8448 * q^16 - 1054 * q^17 + 324 * q^18 + 3124 * q^19 + 352 * q^20 - 1696 * q^22 + 5424 * q^23 + 576 * q^24 + 24427 * q^25 - 4392 * q^26 + 729 * q^27 + 1262 * q^29 - 5976 * q^30 + 616 * q^31 + 1024 * q^32 + 180 * q^33 + 13704 * q^34 + 42768 * q^36 - 31658 * q^37 + 2192 * q^38 - 10134 * q^39 - 5760 * q^40 - 20742 * q^41 + 3684 * q^43 - 10432 * q^44 + 1782 * q^45 + 14080 * q^46 + 13776 * q^47 + 2304 * q^48 - 89124 * q^50 + 11718 * q^51 + 11104 * q^52 - 63538 * q^53 + 2916 * q^54 - 69672 * q^55 + 80100 * q^57 + 2440 * q^58 - 80900 * q^59 + 3168 * q^60 + 86566 * q^61 + 106848 * q^62 + 135168 * q^64 + 146500 * q^65 + 37008 * q^66 + 123956 * q^67 - 16864 * q^68 - 88920 * q^69 + 161888 * q^71 + 5184 * q^72 + 105274 * q^73 + 349528 * q^74 + 48807 * q^75 + 49984 * q^76 - 48024 * q^78 + 212748 * q^79 + 5632 * q^80 + 216513 * q^81 + 13800 * q^82 + 62612 * q^83 + 158524 * q^85 - 172368 * q^86 + 188982 * q^87 - 27136 * q^88 - 121398 * q^89 - 29160 * q^90 + 86784 * q^92 + 235656 * q^93 + 129024 * q^94 + 302848 * q^95 + 9216 * q^96 - 163646 * q^97 - 52812 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(294))\) 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					A-L signs			Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	7
294.6.a.a	$294$	$6$	294.a	1.a	$1$	$1$	$1$	$47.153$	\(\Q\)	None	✓	✓			294.6.a.a	$1$	$0$	\(-4\)	\(-9\)	\(-26\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}-9q^{3}+2^{4}q^{4}-26q^{5}+6^{2}q^{6}+\cdots\)
294.6.a.b	$294$	$6$	294.a	1.a	$1$	$1$	$1$	$47.153$	\(\Q\)	None	✓				42.6.a.d	$1$	$0$	\(-4\)	\(-9\)	\(-26\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}-9q^{3}+2^{4}q^{4}-26q^{5}+6^{2}q^{6}+\cdots\)
294.6.a.c	$294$	$6$	294.a	1.a	$1$	$1$	$1$	$47.153$	\(\Q\)	None	✓				42.6.a.c	$1$	$0$	\(-4\)	\(-9\)	\(72\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}-9q^{3}+2^{4}q^{4}+72q^{5}+6^{2}q^{6}+\cdots\)
294.6.a.d	$294$	$6$	294.a	1.a	$1$	$1$	$1$	$47.153$	\(\Q\)	None	✓				42.6.e.b	$1$	$1$	\(-4\)	\(-9\)	\(86\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}-9q^{3}+2^{4}q^{4}+86q^{5}+6^{2}q^{6}+\cdots\)
294.6.a.e	$294$	$6$	294.a	1.a	$1$	$1$	$1$	$47.153$	\(\Q\)	None	✓				42.6.e.b	$1$	$1$	\(-4\)	\(9\)	\(-86\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+9q^{3}+2^{4}q^{4}-86q^{5}-6^{2}q^{6}+\cdots\)
294.6.a.f	$294$	$6$	294.a	1.a	$1$	$1$	$1$	$47.153$	\(\Q\)	None	✓				42.6.a.b	$1$	$1$	\(-4\)	\(9\)	\(-44\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+9q^{3}+2^{4}q^{4}-44q^{5}-6^{2}q^{6}+\cdots\)
294.6.a.g	$294$	$6$	294.a	1.a	$1$	$1$	$1$	$47.153$	\(\Q\)	None	✓	✓			294.6.a.a	$1$	$1$	\(-4\)	\(9\)	\(26\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+9q^{3}+2^{4}q^{4}+26q^{5}-6^{2}q^{6}+\cdots\)
294.6.a.h	$294$	$6$	294.a	1.a	$1$	$1$	$1$	$47.153$	\(\Q\)	None	✓				42.6.a.a	$1$	$1$	\(-4\)	\(9\)	\(54\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+9q^{3}+2^{4}q^{4}+54q^{5}-6^{2}q^{6}+\cdots\)
294.6.a.i	$294$	$6$	294.a	1.a	$1$	$1$	$1$	$47.153$	\(\Q\)	None	✓				42.6.a.f	$1$	$1$	\(4\)	\(-9\)	\(-24\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}-9q^{3}+2^{4}q^{4}-24q^{5}-6^{2}q^{6}+\cdots\)
294.6.a.j	$294$	$6$	294.a	1.a	$1$	$1$	$1$	$47.153$	\(\Q\)	None	✓				42.6.e.a	$1$	$1$	\(4\)	\(-9\)	\(6\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}-9q^{3}+2^{4}q^{4}+6q^{5}-6^{2}q^{6}+\cdots\)
294.6.a.k	$294$	$6$	294.a	1.a	$1$	$1$	$1$	$47.153$	\(\Q\)	None	✓				42.6.a.e	$1$	$0$	\(4\)	\(9\)	\(-76\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+9q^{3}+2^{4}q^{4}-76q^{5}+6^{2}q^{6}+\cdots\)
294.6.a.l	$294$	$6$	294.a	1.a	$1$	$1$	$1$	$47.153$	\(\Q\)	None	✓				42.6.e.a	$1$	$1$	\(4\)	\(9\)	\(-6\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+9q^{3}+2^{4}q^{4}-6q^{5}+6^{2}q^{6}+\cdots\)
294.6.a.m	$294$	$6$	294.a	1.a	$1$	$1$	$1$	$47.153$	\(\Q\)	None	✓				6.6.a.a	$1$	$0$	\(4\)	\(9\)	\(66\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+9q^{3}+2^{4}q^{4}+66q^{5}+6^{2}q^{6}+\cdots\)
294.6.a.n	$294$	$6$	294.a	1.a	$1$	$2$	$2$	$47.153$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓		294.6.a.n	$1$	$1$	\(-8\)	\(-18\)	\(-108\)	\(0\)		$+$	$+$	$+$	$+$	$7$	$\mathrm{SU}(2)$	\(q-4q^{2}-9q^{3}+2^{4}q^{4}+(-54+5\beta )q^{5}+\cdots\)
294.6.a.o	$294$	$6$	294.a	1.a	$1$	$2$	$2$	$47.153$	\(\Q(\sqrt{505}) \)	None	✓		✓		42.6.e.d	$1$	$0$	\(-8\)	\(-18\)	\(-17\)	\(0\)		$+$	$+$	$-$	$-$	$7$	$\mathrm{SU}(2)$	\(q-4q^{2}-9q^{3}+2^{4}q^{4}+(-9-\beta )q^{5}+\cdots\)
294.6.a.p	$294$	$6$	294.a	1.a	$1$	$2$	$2$	$47.153$	\(\Q(\sqrt{505}) \)	None	✓				42.6.e.d	$1$	$0$	\(-8\)	\(18\)	\(17\)	\(0\)		$+$	$-$	$+$	$-$	$7$	$\mathrm{SU}(2)$	\(q-4q^{2}+9q^{3}+2^{4}q^{4}+(9+\beta )q^{5}+\cdots\)
294.6.a.q	$294$	$6$	294.a	1.a	$1$	$2$	$2$	$47.153$	\(\Q(\sqrt{2}) \)	None	✓	✓			294.6.a.n	$1$	$0$	\(-8\)	\(18\)	\(108\)	\(0\)		$+$	$-$	$+$	$-$	$7$	$\mathrm{SU}(2)$	\(q-4q^{2}+9q^{3}+2^{4}q^{4}+(54+5\beta )q^{5}+\cdots\)
294.6.a.r	$294$	$6$	294.a	1.a	$1$	$2$	$2$	$47.153$	\(\Q(\sqrt{9601}) \)	None	✓				42.6.e.c	$1$	$0$	\(8\)	\(-18\)	\(-53\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}-9q^{3}+2^{4}q^{4}+(-26-\beta )q^{5}+\cdots\)
294.6.a.s	$294$	$6$	294.a	1.a	$1$	$2$	$2$	$47.153$	\(\Q(\sqrt{4705}) \)	None	✓	✓	✓		294.6.a.s	$1$	$1$	\(8\)	\(-18\)	\(-18\)	\(0\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+4q^{2}-9q^{3}+2^{4}q^{4}+(-9-\beta )q^{5}+\cdots\)
294.6.a.t	$294$	$6$	294.a	1.a	$1$	$2$	$2$	$47.153$	\(\Q(\sqrt{2}) \)	None	✓	✓			294.6.a.t	$1$	$0$	\(8\)	\(-18\)	\(108\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}-9q^{3}+2^{4}q^{4}+(54+5\beta )q^{5}+\cdots\)
294.6.a.u	$294$	$6$	294.a	1.a	$1$	$2$	$2$	$47.153$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓		294.6.a.t	$1$	$1$	\(8\)	\(18\)	\(-108\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+9q^{3}+2^{4}q^{4}+(-54+5\beta )q^{5}+\cdots\)
294.6.a.v	$294$	$6$	294.a	1.a	$1$	$2$	$2$	$47.153$	\(\Q(\sqrt{4705}) \)	None	✓	✓			294.6.a.s	$1$	$0$	\(8\)	\(18\)	\(18\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+4q^{2}+9q^{3}+2^{4}q^{4}+(9-\beta )q^{5}+\cdots\)
294.6.a.w	$294$	$6$	294.a	1.a	$1$	$2$	$2$	$47.153$	\(\Q(\sqrt{9601}) \)	None	✓				42.6.e.c	$1$	$0$	\(8\)	\(18\)	\(53\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+9q^{3}+2^{4}q^{4}+(3^{3}-\beta )q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(294))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(294)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 2}\)