Space of modular forms of level 354, weight 6, and trivial character

• Label: 354.6.a
• Level: $354$
• Weight: $6$
• Character orbit: 354.a
• Rep. character: $\chi_{354}(1,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $9$
• Sturm bound: $360$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 354 = 2 \cdot 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	354.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(360\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(354))\).

	Total	New	Old
Modular forms	304	48	256
Cusp forms	296	48	248
Eisenstein series	8	0	8

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

\(2\)	\(3\)	\(59\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(6\)
\(+\)	\(+\)	\(-\)	$-$	\(6\)
\(+\)	\(-\)	\(+\)	$-$	\(8\)
\(+\)	\(-\)	\(-\)	$+$	\(5\)
\(-\)	\(+\)	\(+\)	$-$	\(6\)
\(-\)	\(+\)	\(-\)	$+$	\(5\)
\(-\)	\(-\)	\(+\)	$+$	\(4\)
\(-\)	\(-\)	\(-\)	$-$	\(8\)
Plus space			\(+\)	\(20\)
Minus space			\(-\)	\(28\)

Trace form

\( 48 q - 8 q^{2} + 18 q^{3} + 768 q^{4} + 132 q^{5} - 276 q^{7} - 128 q^{8} + 3888 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(354))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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	59
354.6.a.a	$354$	$6$	354.a	1.a	$1$	$1$	$1$	$56.776$	\(\Q\)	None	✓	$1$	$0$	\(4\)	\(-9\)	\(10\)	\(144\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}-9q^{3}+2^{4}q^{4}+10q^{5}-6^{2}q^{6}+\cdots\)
354.6.a.b	$354$	$6$	354.a	1.a	$1$	$4$	$4$	$56.776$	4.4.32832.1	None	✓	$1$	$1$	\(16\)	\(36\)	\(-104\)	\(-162\)		$-$	$-$	$+$	$+$	$2^{2}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+4q^{2}+9q^{3}+2^{4}q^{4}+(-5^{2}+2\beta _{1}+\cdots)q^{5}+\cdots\)
354.6.a.c	$354$	$6$	354.a	1.a	$1$	$5$	$5$	$56.776$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(-20\)	\(45\)	\(-10\)	\(-162\)		$+$	$-$	$-$	$+$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q-4q^{2}+9q^{3}+2^{4}q^{4}+(-2+\beta _{1}+\cdots)q^{5}+\cdots\)
354.6.a.d	$354$	$6$	354.a	1.a	$1$	$5$	$5$	$56.776$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(20\)	\(-45\)	\(-24\)	\(-103\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+4q^{2}-9q^{3}+2^{4}q^{4}+(-5+\beta _{3}+\cdots)q^{5}+\cdots\)
354.6.a.e	$354$	$6$	354.a	1.a	$1$	$5$	$5$	$56.776$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(20\)	\(-45\)	\(166\)	\(-198\)		$-$	$+$	$+$	$-$	$2^{2}\cdot 3^{4}\cdot 5$	$\mathrm{SU}(2)$	\(q+4q^{2}-9q^{3}+2^{4}q^{4}+(33-\beta _{2})q^{5}+\cdots\)
354.6.a.f	$354$	$6$	354.a	1.a	$1$	$6$	$6$	$56.776$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(-24\)	\(-54\)	\(-46\)	\(-103\)		$+$	$+$	$+$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q-4q^{2}-9q^{3}+2^{4}q^{4}+(-8+\beta _{5})q^{5}+\cdots\)
354.6.a.g	$354$	$6$	354.a	1.a	$1$	$6$	$6$	$56.776$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(-24\)	\(-54\)	\(4\)	\(-54\)		$+$	$+$	$-$	$-$	$2^{3}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q-4q^{2}-9q^{3}+2^{4}q^{4}+(1-\beta _{5})q^{5}+\cdots\)
354.6.a.h	$354$	$6$	354.a	1.a	$1$	$8$	$8$	$56.776$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(-32\)	\(72\)	\(40\)	\(181\)		$+$	$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-4q^{2}+9q^{3}+2^{4}q^{4}+(5+\beta _{1})q^{5}+\cdots\)
354.6.a.i	$354$	$6$	354.a	1.a	$1$	$8$	$8$	$56.776$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(32\)	\(72\)	\(96\)	\(181\)		$-$	$-$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+4q^{2}+9q^{3}+2^{4}q^{4}+(12-\beta _{1})q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(354))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(354)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(59))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(118))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(177))\)\(^{\oplus 2}\)