Space of modular forms of level 105, weight 10, and trivial character

• Label: 105.10.a
• Level: $105$
• Weight: $10$
• Character orbit: 105.a
• Rep. character: $\chi_{105}(1,\cdot)$
• Character field: $\Q$
• Dimension: $36$
• Newform subspaces: $8$
• Sturm bound: $160$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	105.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(160\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(105))\).

	Total	New	Old
Modular forms	148	36	112
Cusp forms	140	36	104
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.

\(3\)	\(5\)	\(7\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(4\)
\(+\)	\(+\)	\(-\)	\(-\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)	\(6\)
Plus space			\(+\)	\(16\)
Minus space			\(-\)	\(20\)

Trace form

\( 36 q - 68 q^{2} + 8872 q^{4} + 6156 q^{6} + 9604 q^{7} - 67932 q^{8} + 236196 q^{9} + O(q^{10}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(105))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5	7
105.10.a.a	$105$	$10$	105.a	1.a	$1$	$4$	$4$	$54.079$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	105.10.a.a	$1$	$1$	\(-41\)	\(-324\)	\(2500\)	\(9604\)		$+$	$-$	$-$	$+$	$2^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(-10+\beta _{1})q^{2}-3^{4}q^{3}+(120-21\beta _{1}+\cdots)q^{4}+\cdots\)
105.10.a.b	$105$	$10$	105.a	1.a	$1$	$4$	$4$	$54.079$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	105.10.a.b	$1$	$0$	\(-20\)	\(-324\)	\(2500\)	\(-9604\)		$+$	$-$	$+$	$-$	$2^{5}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-5+\beta _{1})q^{2}-3^{4}q^{3}+(106-8\beta _{1}+\cdots)q^{4}+\cdots\)
105.10.a.c	$105$	$10$	105.a	1.a	$1$	$4$	$4$	$54.079$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	105.10.a.c	$1$	$1$	\(-17\)	\(324\)	\(2500\)	\(-9604\)		$-$	$-$	$+$	$+$	$2^{2}\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+(-4-\beta _{1})q^{2}+3^{4}q^{3}+(235+9\beta _{1}+\cdots)q^{4}+\cdots\)
105.10.a.d	$105$	$10$	105.a	1.a	$1$	$4$	$4$	$54.079$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	105.10.a.d	$1$	$1$	\(-13\)	\(324\)	\(-2500\)	\(9604\)		$-$	$+$	$-$	$+$	$2^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{1})q^{2}+3^{4}q^{3}+(123+8\beta _{1}+\cdots)q^{4}+\cdots\)
105.10.a.e	$105$	$10$	105.a	1.a	$1$	$4$	$4$	$54.079$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	105.10.a.e	$1$	$1$	\(5\)	\(-324\)	\(-2500\)	\(-9604\)		$+$	$+$	$+$	$+$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}-3^{4}q^{3}+(237+3\beta _{1}+\cdots)q^{4}+\cdots\)
105.10.a.f	$105$	$10$	105.a	1.a	$1$	$4$	$4$	$54.079$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	105.10.a.f	$1$	$0$	\(8\)	\(324\)	\(-2500\)	\(-9604\)		$-$	$+$	$+$	$-$	$2^{6}\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{2}+3^{4}q^{3}+(106-\beta _{1}+\beta _{3})q^{4}+\cdots\)
105.10.a.g	$105$	$10$	105.a	1.a	$1$	$6$	$6$	$54.079$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	105.10.a.g	$1$	$0$	\(-16\)	\(-486\)	\(-3750\)	\(14406\)		$+$	$+$	$-$	$-$	$2^{7}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1})q^{2}-3^{4}q^{3}+(428-2\beta _{1}+\cdots)q^{4}+\cdots\)
105.10.a.h	$105$	$10$	105.a	1.a	$1$	$6$	$6$	$54.079$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	105.10.a.h	$1$	$0$	\(26\)	\(486\)	\(3750\)	\(14406\)		$-$	$-$	$-$	$-$	$2^{7}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(4+\beta _{1})q^{2}+3^{4}q^{3}+(426+3\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(105))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(105)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 2}\)