Space of modular forms of level 343, weight 4, and trivial character

• Label: 343.4.a
• Level: $343$
• Weight: $4$
• Character orbit: 343.a
• Rep. character: $\chi_{343}(1,\cdot)$
• Character field: $\Q$
• Dimension: $72$
• Newform subspaces: $6$
• Sturm bound: $130$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 343 = 7^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	343.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(130\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(343))\).

	Total	New	Old
Modular forms	105	72	33
Cusp forms	91	72	19
Eisenstein series	14	0	14

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

\(7\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(56\)	\(39\)	\(17\)		\(49\)	\(39\)	\(10\)		\(7\)	\(0\)	\(7\)
\(-\)		\(49\)	\(33\)	\(16\)		\(42\)	\(33\)	\(9\)		\(7\)	\(0\)	\(7\)

Trace form

72 * q + q^2 + 283 * q^4 - 9 * q^8 + 664 * q^9 - 6 * q^11 + 24 * q^15 + 1187 * q^16 + 53 * q^18 - 2 * q^22 - 90 * q^23 + 1868 * q^25 + 4 * q^29 + 130 * q^30 + 69 * q^32 + 2733 * q^36 - 104 * q^37 - 224 * q^39 - 122 * q^43 + 160 * q^44 - 212 * q^46 - 2719 * q^50 - 80 * q^51 - 1924 * q^53 + 416 * q^57 - 1118 * q^58 + 210 * q^60 + 5237 * q^64 + 28 * q^65 + 610 * q^67 - 542 * q^71 - 285 * q^72 + 1846 * q^74 + 322 * q^78 + 498 * q^79 + 5980 * q^81 + 3496 * q^85 - 5563 * q^86 - 4921 * q^88 - 3907 * q^92 + 1348 * q^93 - 108 * q^95 - 10650 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(343))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		7
343.4.a.a	$343$	$4$	343.a	1.a	$1$	$3$	$3$	$20.238$	\(\Q(\zeta_{14})^+\)	\(\Q(\sqrt{-7}) \)	✓	✓			343.4.a.a	$2$	$0$	\(-1\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+(-1+3\beta _{1}+\beta _{2})q^{2}+(18-5\beta _{1}+\cdots)q^{4}+\cdots\)
343.4.a.b	$343$	$4$	343.a	1.a	$1$	$3$	$3$	$20.238$	\(\Q(\zeta_{14})^+\)	\(\Q(\sqrt{-7}) \)	✓	✓			343.4.a.b	$2$	$1$	\(6\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+(2-\beta _{1}-\beta _{2})q^{2}+(1-3\beta _{1}-2\beta _{2})q^{4}+\cdots\)
343.4.a.c	$343$	$4$	343.a	1.a	$1$	$12$	$12$	$20.238$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓			343.4.a.c	$2$	$1$	\(-22\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$7^{3}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{3}+\beta _{6})q^{2}+\beta _{4}q^{3}+(1+2\beta _{1}+\cdots)q^{4}+\cdots\)
343.4.a.d	$343$	$4$	343.a	1.a	$1$	$18$	$18$	$20.238$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓	✓	✓		343.4.a.d	$1$	$1$	\(2\)	\(-22\)	\(-62\)	\(0\)		$-$	$-$	$2^{3}\cdot 7^{10}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-1+\beta _{5})q^{3}+(3+\beta _{2})q^{4}+\cdots\)
343.4.a.e	$343$	$4$	343.a	1.a	$1$	$18$	$18$	$20.238$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓	✓			343.4.a.d	$1$	$0$	\(2\)	\(22\)	\(62\)	\(0\)		$+$	$+$	$2^{3}\cdot 7^{10}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1-\beta _{5})q^{3}+(3+\beta _{2})q^{4}+\cdots\)
343.4.a.f	$343$	$4$	343.a	1.a	$1$	$18$	$18$	$20.238$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓	✓			343.4.a.f	$2$	$0$	\(14\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$7^{11}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{2}+\beta _{12}q^{3}+(5+\beta _{2}-\beta _{5}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(343))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(343)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 2}\)