Space of modular forms of level 294, weight 2, and trivial character

• Label: 294.2.a
• Level: $294$
• Weight: $2$
• Character orbit: 294.a
• Rep. character: $\chi_{294}(1,\cdot)$
• Character field: $\Q$
• Dimension: $7$
• Newform subspaces: $7$
• Sturm bound: $112$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	294.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(112\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(294))\).

	Total	New	Old
Modular forms	72	7	65
Cusp forms	41	7	34
Eisenstein series	31	0	31

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	Dim
\(+\)	\(+\)	\(-\)	\(-\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)	\(2\)
Plus space			\(+\)	\(1\)
Minus space			\(-\)	\(6\)

Trace form

7 * q - q^2 + q^3 + 7 * q^4 + 2 * q^5 + q^6 - q^8 + 7 * q^9 + 2 * q^10 + 4 * q^11 + q^12 - 6 * q^13 - 2 * q^15 + 7 * q^16 - 2 * q^17 - q^18 + 4 * q^19 + 2 * q^20 + 8 * q^22 + q^24 + 21 * q^25 - 6 * q^26 + q^27 + 10 * q^29 + 2 * q^30 - q^32 - 4 * q^33 - 2 * q^34 + 7 * q^36 - 14 * q^37 + 4 * q^38 - 22 * q^39 + 2 * q^40 + 6 * q^41 - 12 * q^43 + 4 * q^44 + 2 * q^45 + q^48 - 39 * q^50 + 6 * q^51 - 6 * q^52 - 38 * q^53 + q^54 - 8 * q^55 - 28 * q^57 - 34 * q^58 - 4 * q^59 - 2 * q^60 - 6 * q^61 + 7 * q^64 - 4 * q^65 - 4 * q^66 - 12 * q^67 - 2 * q^68 + 8 * q^69 + 16 * q^71 - q^72 - 10 * q^73 - 22 * q^74 - q^75 + 4 * q^76 + 10 * q^78 - 12 * q^79 + 2 * q^80 + 7 * q^81 + 6 * q^82 + 4 * q^83 - 12 * q^85 + 12 * q^86 - 2 * q^87 + 8 * q^88 + 6 * q^89 + 2 * q^90 - 24 * q^93 + 32 * q^95 + q^96 + 14 * q^97 + 4 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(294))\) 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
294.2.a.a	$294$	$2$	294.a	1.a	$1$	$1$	$1$	$2.348$	\(\Q\)	None	✓				42.2.e.b	$1$	$0$	\(-1\)	\(-1\)	\(-3\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}-3q^{5}+q^{6}-q^{8}+\cdots\)
294.2.a.b	$294$	$2$	294.a	1.a	$1$	$1$	$1$	$2.348$	\(\Q\)	None	✓	✓			294.2.a.b	$1$	$0$	\(-1\)	\(-1\)	\(4\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+4q^{5}+q^{6}-q^{8}+\cdots\)
294.2.a.c	$294$	$2$	294.a	1.a	$1$	$1$	$1$	$2.348$	\(\Q\)	None	✓	✓	✓	✓	294.2.a.b	$1$	$1$	\(-1\)	\(1\)	\(-4\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-4q^{5}-q^{6}-q^{8}+\cdots\)
294.2.a.d	$294$	$2$	294.a	1.a	$1$	$1$	$1$	$2.348$	\(\Q\)	None	✓		✓	✓	42.2.e.b	$1$	$0$	\(-1\)	\(1\)	\(3\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+3q^{5}-q^{6}-q^{8}+\cdots\)
294.2.a.e	$294$	$2$	294.a	1.a	$1$	$1$	$1$	$2.348$	\(\Q\)	None	✓		✓	✓	42.2.e.a	$1$	$0$	\(1\)	\(-1\)	\(1\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{8}+\cdots\)
294.2.a.f	$294$	$2$	294.a	1.a	$1$	$1$	$1$	$2.348$	\(\Q\)	None	✓				42.2.e.a	$1$	$0$	\(1\)	\(1\)	\(-1\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+q^{8}+\cdots\)
294.2.a.g	$294$	$2$	294.a	1.a	$1$	$1$	$1$	$2.348$	\(\Q\)	None	✓				42.2.a.a	$1$	$0$	\(1\)	\(1\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+2q^{5}+q^{6}+q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(294))\) into lower level spaces

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