Space of modular forms of level 735, weight 2, and trivial character

• Label: 735.2.a
• Level: $735$
• Weight: $2$
• Character orbit: 735.a
• Rep. character: $\chi_{735}(1,\cdot)$
• Character field: $\Q$
• Dimension: $28$
• Newform subspaces: $15$
• Sturm bound: $224$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	735.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(224\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(735))\).

	Total	New	Old
Modular forms	128	28	100
Cusp forms	97	28	69
Eisenstein series	31	0	31

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
\(+\)	\(+\)	\(+\)	\(+\)		\(12\)	\(2\)	\(10\)		\(9\)	\(2\)	\(7\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)		\(19\)	\(4\)	\(15\)		\(15\)	\(4\)	\(11\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)		\(20\)	\(6\)	\(14\)		\(16\)	\(6\)	\(10\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)		\(13\)	\(1\)	\(12\)		\(9\)	\(1\)	\(8\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)		\(16\)	\(5\)	\(11\)		\(12\)	\(5\)	\(7\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)		\(15\)	\(3\)	\(12\)		\(11\)	\(3\)	\(8\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)		\(16\)	\(1\)	\(15\)		\(12\)	\(1\)	\(11\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)		\(17\)	\(6\)	\(11\)		\(13\)	\(6\)	\(7\)		\(4\)	\(0\)	\(4\)
Plus space			\(+\)		\(56\)	\(7\)	\(49\)		\(41\)	\(7\)	\(34\)		\(15\)	\(0\)	\(15\)
Minus space			\(-\)		\(72\)	\(21\)	\(51\)		\(56\)	\(21\)	\(35\)		\(16\)	\(0\)	\(16\)

Trace form

28 * q + 4 * q^2 + 2 * q^3 + 32 * q^4 - 2 * q^6 + 12 * q^8 + 28 * q^9 + 8 * q^11 + 6 * q^12 + 8 * q^13 - 2 * q^15 + 48 * q^16 + 4 * q^18 + 8 * q^20 + 24 * q^22 - 8 * q^23 + 6 * q^24 + 28 * q^25 + 24 * q^26 + 2 * q^27 + 24 * q^29 - 2 * q^30 - 16 * q^31 + 28 * q^32 + 32 * q^36 - 4 * q^37 - 8 * q^38 - 24 * q^39 - 20 * q^43 + 8 * q^44 + 8 * q^46 - 24 * q^47 - 2 * q^48 + 4 * q^50 + 4 * q^51 - 8 * q^52 + 32 * q^53 - 2 * q^54 + 8 * q^55 - 12 * q^57 + 16 * q^58 - 6 * q^60 + 8 * q^61 - 24 * q^62 + 72 * q^64 + 8 * q^65 - 16 * q^66 - 20 * q^67 + 16 * q^68 + 16 * q^71 + 12 * q^72 + 8 * q^73 - 32 * q^74 + 2 * q^75 - 16 * q^76 - 4 * q^78 - 28 * q^79 + 28 * q^81 + 16 * q^82 + 8 * q^83 - 8 * q^85 - 112 * q^86 - 4 * q^87 - 112 * q^88 + 16 * q^89 - 160 * q^92 - 20 * q^93 + 40 * q^94 + 16 * q^95 - 10 * q^96 + 8 * q^97 + 8 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(735))\) 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
735.2.a.a	$735$	$2$	735.a	1.a	$1$	$1$	$1$	$5.869$	\(\Q\)	None	✓				105.2.i.b	$1$	$0$	\(-2\)	\(-1\)	\(-1\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-q^{3}+2q^{4}-q^{5}+2q^{6}+q^{9}+\cdots\)
735.2.a.b	$735$	$2$	735.a	1.a	$1$	$1$	$1$	$5.869$	\(\Q\)	None	✓		✓	✓	105.2.i.b	$1$	$1$	\(-2\)	\(1\)	\(1\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+q^{3}+2q^{4}+q^{5}-2q^{6}+q^{9}+\cdots\)
735.2.a.c	$735$	$2$	735.a	1.a	$1$	$1$	$1$	$5.869$	\(\Q\)	None	✓				15.2.a.a	$1$	$1$	\(-1\)	\(1\)	\(-1\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}-q^{4}-q^{5}-q^{6}+3q^{8}+\cdots\)
735.2.a.d	$735$	$2$	735.a	1.a	$1$	$1$	$1$	$5.869$	\(\Q\)	None	✓		✓	✓	105.2.i.a	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{4}+q^{5}+q^{9}+2q^{12}+q^{13}+\cdots\)
735.2.a.e	$735$	$2$	735.a	1.a	$1$	$1$	$1$	$5.869$	\(\Q\)	None	✓				105.2.i.a	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{4}-q^{5}+q^{9}-2q^{12}-q^{13}+\cdots\)
735.2.a.f	$735$	$2$	735.a	1.a	$1$	$1$	$1$	$5.869$	\(\Q\)	None	✓				105.2.a.a	$1$	$0$	\(1\)	\(-1\)	\(-1\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}-q^{4}-q^{5}-q^{6}-3q^{8}+\cdots\)
735.2.a.g	$735$	$2$	735.a	1.a	$1$	$2$	$2$	$5.869$	\(\Q(\sqrt{3}) \)	None	✓				105.2.i.d	$1$	$0$	\(-2\)	\(-2\)	\(2\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}-q^{3}+(2-2\beta )q^{4}+q^{5}+\cdots\)
735.2.a.h	$735$	$2$	735.a	1.a	$1$	$2$	$2$	$5.869$	\(\Q(\sqrt{3}) \)	None	✓		✓		105.2.i.d	$1$	$1$	\(-2\)	\(2\)	\(-2\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+q^{3}+(2-2\beta )q^{4}-q^{5}+\cdots\)
735.2.a.i	$735$	$2$	735.a	1.a	$1$	$2$	$2$	$5.869$	\(\Q(\sqrt{2}) \)	None	✓		✓	✓	105.2.i.c	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-q^{3}-q^{5}-\beta q^{6}-2\beta q^{8}+\cdots\)
735.2.a.j	$735$	$2$	735.a	1.a	$1$	$2$	$2$	$5.869$	\(\Q(\sqrt{2}) \)	None	✓				105.2.i.c	$1$	$0$	\(0\)	\(2\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{3}+q^{5}+\beta q^{6}-2\beta q^{8}+\cdots\)
735.2.a.k	$735$	$2$	735.a	1.a	$1$	$2$	$2$	$5.869$	\(\Q(\sqrt{5}) \)	None	✓				105.2.a.b	$1$	$0$	\(0\)	\(2\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+q^{3}+3q^{4}+q^{5}-\beta q^{6}-\beta q^{8}+\cdots\)
735.2.a.l	$735$	$2$	735.a	1.a	$1$	$2$	$2$	$5.869$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓		735.2.a.l	$1$	$0$	\(2\)	\(-2\)	\(-2\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}-q^{3}+(1+2\beta )q^{4}-q^{5}+\cdots\)
735.2.a.m	$735$	$2$	735.a	1.a	$1$	$2$	$2$	$5.869$	\(\Q(\sqrt{2}) \)	None	✓	✓			735.2.a.l	$1$	$0$	\(2\)	\(2\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+q^{3}+(1+2\beta )q^{4}+q^{5}+\cdots\)
735.2.a.n	$735$	$2$	735.a	1.a	$1$	$4$	$4$	$5.869$	\(\Q(\sqrt{10 +4 \sqrt{2}})\)	None	✓	✓	✓		735.2.a.n	$1$	$0$	\(4\)	\(-4\)	\(4\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1}-\beta _{3})q^{2}-q^{3}+(2+\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
735.2.a.o	$735$	$2$	735.a	1.a	$1$	$4$	$4$	$5.869$	\(\Q(\sqrt{10 +4 \sqrt{2}})\)	None	✓	✓	✓		735.2.a.n	$1$	$0$	\(4\)	\(4\)	\(-4\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1}-\beta _{3})q^{2}+q^{3}+(2+\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(735))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(735)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(245))\)\(^{\oplus 2}\)