Space of modular forms of level 105, weight 2, and trivial character

• Label: 105.2.a
• Level: $105$
• Weight: $2$
• Character orbit: 105.a
• Rep. character: $\chi_{105}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $2$
• Sturm bound: $32$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	105.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(32\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(105))\).

	Total	New	Old
Modular forms	20	3	17
Cusp forms	13	3	10
Eisenstein series	7	0	7

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		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(-\)	\(-\)		\(4\)	\(2\)	\(2\)		\(3\)	\(2\)	\(1\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)		\(4\)	\(0\)	\(4\)		\(3\)	\(0\)	\(3\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(4\)	\(0\)	\(4\)		\(3\)	\(0\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(2\)	\(1\)	\(1\)		\(1\)	\(1\)	\(0\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(8\)	\(0\)	\(8\)		\(5\)	\(0\)	\(5\)		\(3\)	\(0\)	\(3\)
Minus space			\(-\)		\(12\)	\(3\)	\(9\)		\(8\)	\(3\)	\(5\)		\(4\)	\(0\)	\(4\)

Trace form

3 * q + q^2 - q^3 + 5 * q^4 - q^5 + q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9 + q^10 + 4 * q^11 - 7 * q^12 - 6 * q^13 + q^14 + 3 * q^15 - 3 * q^16 - 2 * q^17 + q^18 - 4 * q^19 - 7 * q^20 - q^21 - 20 * q^22 + 16 * q^23 - 3 * q^24 + 3 * q^25 - 26 * q^26 - q^27 + 5 * q^28 - 6 * q^29 + q^30 + 16 * q^31 + 5 * q^32 - 4 * q^33 + 2 * q^34 - q^35 + 5 * q^36 + 2 * q^37 + 12 * q^38 - 6 * q^39 - 3 * q^40 - 10 * q^41 + q^42 + 4 * q^43 + 12 * q^44 - q^45 + 8 * q^46 + 16 * q^47 + q^48 + 3 * q^49 + q^50 + 6 * q^51 + 6 * q^52 - 6 * q^53 + q^54 - 4 * q^55 - 3 * q^56 - 12 * q^57 - 2 * q^58 + 4 * q^59 + 5 * q^60 - 6 * q^61 + 24 * q^62 + 3 * q^63 - 19 * q^64 - 6 * q^65 + 20 * q^66 - 4 * q^67 - 14 * q^68 + q^70 + 8 * q^71 - 3 * q^72 - 18 * q^73 + 38 * q^74 - q^75 + 20 * q^76 + 4 * q^77 + 14 * q^78 + 16 * q^79 + q^80 + 3 * q^81 - 6 * q^82 - 20 * q^83 - 7 * q^84 + 6 * q^85 - 36 * q^86 + 2 * q^87 - 20 * q^88 - 10 * q^89 + q^90 - 6 * q^91 + 16 * q^92 - 8 * q^93 - 32 * q^94 - 12 * q^95 + 5 * q^96 - 10 * q^97 + q^98 + 4 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(105))\) 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
105.2.a.a	$105$	$2$	105.a	1.a	$1$	$1$	$1$	$0.838$	\(\Q\)	None	✓	✓	✓	✓	105.2.a.a	$1$	$0$	\(1\)	\(1\)	\(1\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}-q^{4}+q^{5}+q^{6}+q^{7}+\cdots\)
105.2.a.b	$105$	$2$	105.a	1.a	$1$	$2$	$2$	$0.838$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓	✓	105.2.a.b	$1$	$0$	\(0\)	\(-2\)	\(-2\)	\(2\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{2}-q^{3}+3q^{4}-q^{5}+\beta q^{6}+q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(105))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(105)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 2}\)