Space of modular forms of level 49, weight 10, and trivial character

• Label: 49.10.a
• Level: $49$
• Weight: $10$
• Character orbit: 49.a
• Rep. character: $\chi_{49}(1,\cdot)$
• Character field: $\Q$
• Dimension: $28$
• Newform subspaces: $7$
• Sturm bound: $46$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	49.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(46\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(49))\).

	Total	New	Old
Modular forms	46	33	13
Cusp forms	38	28	10
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
\(+\)	\(13\)
\(-\)	\(15\)

Trace form

\( 28 q - 32 q^{2} + 2 q^{3} + 6828 q^{4} + 684 q^{5} - 926 q^{6} - 33660 q^{8} + 192958 q^{9} + O(q^{10}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(49))\) 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
49.10.a.a	$49$	$10$	49.a	1.a	$1$	$1$	$1$	$25.237$	\(\Q\)	\(\Q(\sqrt{-7}) \)	✓	$2$	$0$	\(-5\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-5q^{2}-487q^{4}+4995q^{8}-3^{9}q^{9}+\cdots\)
49.10.a.b	$49$	$10$	49.a	1.a	$1$	$2$	$2$	$25.237$	\(\Q(\sqrt{193}) \)	None	✓	$1$	$0$	\(-6\)	\(86\)	\(2238\)	\(0\)		$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-3-\beta )q^{2}+(43-11\beta )q^{3}+(-310+\cdots)q^{4}+\cdots\)
49.10.a.c	$49$	$10$	49.a	1.a	$1$	$3$	$3$	$25.237$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$0$	\(21\)	\(-84\)	\(-1554\)	\(0\)		$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(7-\beta _{2})q^{2}+(-28+\beta _{1}+\beta _{2})q^{3}+\cdots\)
49.10.a.d	$49$	$10$	49.a	1.a	$1$	$4$	$4$	$25.237$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$2$	$0$	\(-12\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$2^{5}\cdot 3^{2}\cdot 7$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{3})q^{2}-\beta _{1}q^{3}+(698-6\beta _{3})q^{4}+\cdots\)
49.10.a.e	$49$	$10$	49.a	1.a	$1$	$5$	$5$	$25.237$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(18\)	\(-161\)	\(-1533\)	\(0\)		$+$	$+$	$2^{6}\cdot 3\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+(4-\beta _{1})q^{2}+(-33+\beta _{1}+\beta _{2})q^{3}+\cdots\)
49.10.a.f	$49$	$10$	49.a	1.a	$1$	$5$	$5$	$25.237$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(18\)	\(161\)	\(1533\)	\(0\)		$-$	$-$	$2^{6}\cdot 3\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+(4-\beta _{1})q^{2}+(33-\beta _{1}-\beta _{2})q^{3}+(191+\cdots)q^{4}+\cdots\)
49.10.a.g	$49$	$10$	49.a	1.a	$1$	$8$	$8$	$25.237$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$2$	$1$	\(-66\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$2^{3}\cdot 7^{6}$	$\mathrm{SU}(2)$	\(q+(-8+\beta _{1})q^{2}+(-\beta _{3}-\beta _{5})q^{3}+(209+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(49))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(49)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)