Space of modular forms of level 49, weight 6, and trivial character

• Label: 49.6.a
• Level: $49$
• Weight: $6$
• Character orbit: 49.a
• Rep. character: $\chi_{49}(1,\cdot)$
• Character field: $\Q$
• Dimension: $14$
• Newform subspaces: $7$
• Sturm bound: $28$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	49.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(28\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(49))\).

	Total	New	Old
Modular forms	28	19	9
Cusp forms	20	14	6
Eisenstein series	8	5	3

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

\(7\)	Dim
\(+\)	\(6\)
\(-\)	\(8\)

Trace form

\( 14 q + 20 q^{3} + 140 q^{4} + 74 q^{5} + 58 q^{6} + 420 q^{8} + 490 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(49))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		7
49.6.a.a	$49$	$6$	49.a	1.a	$1$	$1$	$1$	$7.859$	\(\Q\)	None	✓	$1$	$0$	\(-10\)	\(14\)	\(56\)	\(0\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-10q^{2}+14q^{3}+68q^{4}+56q^{5}+\cdots\)
49.6.a.b	$49$	$6$	49.a	1.a	$1$	$1$	$1$	$7.859$	\(\Q\)	\(\Q(\sqrt{-7}) \)	✓	$2$	$0$	\(11\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+11q^{2}+89q^{4}+627q^{8}-3^{5}q^{9}+\cdots\)
49.6.a.c	$49$	$6$	49.a	1.a	$1$	$2$	$2$	$7.859$	\(\Q(\sqrt{39}) \)	None	✓	$2$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-2q^{2}+\beta q^{3}-28q^{4}+3\beta q^{5}-2\beta q^{6}+\cdots\)
49.6.a.d	$49$	$6$	49.a	1.a	$1$	$2$	$2$	$7.859$	\(\Q(\sqrt{37}) \)	None	✓	$1$	$1$	\(2\)	\(-8\)	\(-38\)	\(0\)		$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(-4-\beta )q^{3}+(6+2\beta )q^{4}+\cdots\)
49.6.a.e	$49$	$6$	49.a	1.a	$1$	$2$	$2$	$7.859$	\(\Q(\sqrt{37}) \)	None	✓	$1$	$0$	\(2\)	\(8\)	\(38\)	\(0\)		$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(4+\beta )q^{3}+(6+2\beta )q^{4}+\cdots\)
49.6.a.f	$49$	$6$	49.a	1.a	$1$	$2$	$2$	$7.859$	\(\Q(\sqrt{57}) \)	None	✓	$1$	$0$	\(9\)	\(6\)	\(18\)	\(0\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(5-\beta )q^{2}+(6-6\beta )q^{3}+(7-9\beta )q^{4}+\cdots\)
49.6.a.g	$49$	$6$	49.a	1.a	$1$	$4$	$4$	$7.859$	\(\Q(\sqrt{2}, \sqrt{113})\)	None	✓	$2$	$1$	\(-10\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$7$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{2}+(2\beta _{2}-\beta _{3})q^{3}-5\beta _{1}q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(49))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(49)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)