Space of modular forms of level 735, weight 4, and trivial character

• Label: 735.4.a
• Level: $735$
• Weight: $4$
• Character orbit: 735.a
• Rep. character: $\chi_{735}(1,\cdot)$
• Character field: $\Q$
• Dimension: $82$
• Newform subspaces: $29$
• Sturm bound: $448$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	735.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 29 \)
Sturm bound:			\(448\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(2\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(735))\).

	Total	New	Old
Modular forms	352	82	270
Cusp forms	320	82	238
Eisenstein series	32	0	32

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
\(+\)	\(+\)	\(+\)	\(+\)		\(48\)	\(11\)	\(37\)		\(44\)	\(11\)	\(33\)		\(4\)	\(0\)	\(4\)
\(+\)	\(+\)	\(-\)	\(-\)		\(41\)	\(10\)	\(31\)		\(37\)	\(10\)	\(27\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)		\(40\)	\(7\)	\(33\)		\(36\)	\(7\)	\(29\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)		\(47\)	\(13\)	\(34\)		\(43\)	\(13\)	\(30\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)		\(44\)	\(9\)	\(35\)		\(40\)	\(9\)	\(31\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)		\(45\)	\(11\)	\(34\)		\(41\)	\(11\)	\(30\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)		\(44\)	\(13\)	\(31\)		\(40\)	\(13\)	\(27\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)		\(43\)	\(8\)	\(35\)		\(39\)	\(8\)	\(31\)		\(4\)	\(0\)	\(4\)
Plus space			\(+\)		\(184\)	\(48\)	\(136\)		\(168\)	\(48\)	\(120\)		\(16\)	\(0\)	\(16\)
Minus space			\(-\)		\(168\)	\(34\)	\(134\)		\(152\)	\(34\)	\(118\)		\(16\)	\(0\)	\(16\)

Trace form

82 * q - 4 * q^2 + 354 * q^4 - 6 * q^6 + 36 * q^8 + 738 * q^9 - 50 * q^10 + 52 * q^11 + 24 * q^12 - 64 * q^13 + 30 * q^15 + 1338 * q^16 + 24 * q^17 - 36 * q^18 + 248 * q^19 + 40 * q^20 + 196 * q^22 + 368 * q^23 - 342 * q^24 + 2050 * q^25 - 84 * q^26 - 312 * q^29 + 60 * q^30 - 200 * q^31 + 1124 * q^32 - 36 * q^33 + 780 * q^34 + 3186 * q^36 + 1152 * q^37 + 1040 * q^38 + 1020 * q^39 - 450 * q^40 - 372 * q^41 + 1512 * q^43 + 1468 * q^44 + 1480 * q^46 - 504 * q^47 + 912 * q^48 - 100 * q^50 - 780 * q^51 + 296 * q^52 - 1184 * q^53 - 54 * q^54 - 740 * q^55 - 528 * q^57 + 244 * q^58 + 180 * q^59 - 90 * q^60 + 988 * q^61 + 384 * q^62 + 7234 * q^64 + 580 * q^65 - 996 * q^66 - 400 * q^67 + 2936 * q^68 + 1152 * q^69 + 152 * q^71 + 324 * q^72 - 2124 * q^73 + 140 * q^74 + 3936 * q^76 + 3984 * q^78 - 4768 * q^79 + 1200 * q^80 + 6642 * q^81 + 1984 * q^82 - 3368 * q^83 + 780 * q^85 + 12208 * q^86 - 1164 * q^87 + 9148 * q^88 + 2036 * q^89 - 450 * q^90 + 8872 * q^92 - 2304 * q^93 + 288 * q^94 - 520 * q^95 - 2262 * q^96 + 3196 * q^97 + 468 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(735))\) 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
735.4.a.a	$735$	$4$	735.a	1.a	$1$	$1$	$1$	$43.366$	\(\Q\)	None	✓	✓			735.4.a.a	$1$	$1$	\(-4\)	\(-3\)	\(-5\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}-3q^{3}+8q^{4}-5q^{5}+12q^{6}+\cdots\)
735.4.a.b	$735$	$4$	735.a	1.a	$1$	$1$	$1$	$43.366$	\(\Q\)	None	✓	✓			735.4.a.a	$1$	$1$	\(-4\)	\(3\)	\(5\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+3q^{3}+8q^{4}+5q^{5}-12q^{6}+\cdots\)
735.4.a.c	$735$	$4$	735.a	1.a	$1$	$1$	$1$	$43.366$	\(\Q\)	None	✓				105.4.a.a	$1$	$0$	\(0\)	\(3\)	\(-5\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-8q^{4}-5q^{5}+9q^{9}+42q^{11}+\cdots\)
735.4.a.d	$735$	$4$	735.a	1.a	$1$	$1$	$1$	$43.366$	\(\Q\)	None	✓	✓			735.4.a.d	$1$	$1$	\(1\)	\(-3\)	\(-5\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-3q^{3}-7q^{4}-5q^{5}-3q^{6}+\cdots\)
735.4.a.e	$735$	$4$	735.a	1.a	$1$	$1$	$1$	$43.366$	\(\Q\)	None	✓				15.4.a.a	$1$	$1$	\(1\)	\(-3\)	\(-5\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-3q^{3}-7q^{4}-5q^{5}-3q^{6}+\cdots\)
735.4.a.f	$735$	$4$	735.a	1.a	$1$	$1$	$1$	$43.366$	\(\Q\)	None	✓	✓			735.4.a.d	$1$	$1$	\(1\)	\(3\)	\(5\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+3q^{3}-7q^{4}+5q^{5}+3q^{6}+\cdots\)
735.4.a.g	$735$	$4$	735.a	1.a	$1$	$1$	$1$	$43.366$	\(\Q\)	None	✓				105.4.i.a	$1$	$0$	\(3\)	\(-3\)	\(-5\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}-3q^{3}+q^{4}-5q^{5}-9q^{6}+\cdots\)
735.4.a.h	$735$	$4$	735.a	1.a	$1$	$1$	$1$	$43.366$	\(\Q\)	None	✓				105.4.i.a	$1$	$1$	\(3\)	\(3\)	\(5\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+3q^{3}+q^{4}+5q^{5}+9q^{6}+\cdots\)
735.4.a.i	$735$	$4$	735.a	1.a	$1$	$1$	$1$	$43.366$	\(\Q\)	None	✓				15.4.a.b	$1$	$1$	\(3\)	\(3\)	\(5\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+3q^{3}+q^{4}+5q^{5}+9q^{6}+\cdots\)
735.4.a.j	$735$	$4$	735.a	1.a	$1$	$1$	$1$	$43.366$	\(\Q\)	None	✓				105.4.a.b	$1$	$0$	\(5\)	\(3\)	\(-5\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{2}+3q^{3}+17q^{4}-5q^{5}+15q^{6}+\cdots\)
735.4.a.k	$735$	$4$	735.a	1.a	$1$	$2$	$2$	$43.366$	\(\Q(\sqrt{17}) \)	None	✓				105.4.a.c	$1$	$1$	\(-7\)	\(6\)	\(10\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-3-\beta )q^{2}+3q^{3}+(5+7\beta )q^{4}+\cdots\)
735.4.a.l	$735$	$4$	735.a	1.a	$1$	$2$	$2$	$43.366$	\(\Q(\sqrt{2}) \)	None	✓				105.4.i.b	$1$	$0$	\(-4\)	\(-6\)	\(-10\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{2}-3q^{3}+(-2-4\beta )q^{4}+\cdots\)
735.4.a.m	$735$	$4$	735.a	1.a	$1$	$2$	$2$	$43.366$	\(\Q(\sqrt{5}) \)	None	✓				105.4.a.d	$1$	$0$	\(-4\)	\(-6\)	\(10\)	\(0\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-2-\beta )q^{2}-3q^{3}+(1+4\beta )q^{4}+\cdots\)
735.4.a.n	$735$	$4$	735.a	1.a	$1$	$2$	$2$	$43.366$	\(\Q(\sqrt{2}) \)	None	✓				105.4.i.b	$1$	$1$	\(-4\)	\(6\)	\(10\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{2}+3q^{3}+(-2-4\beta )q^{4}+\cdots\)
735.4.a.o	$735$	$4$	735.a	1.a	$1$	$2$	$2$	$43.366$	\(\Q(\sqrt{2}) \)	None	✓				105.4.a.e	$1$	$0$	\(-2\)	\(6\)	\(-10\)	\(0\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+3q^{3}+(1-2\beta )q^{4}+\cdots\)
735.4.a.p	$735$	$4$	735.a	1.a	$1$	$2$	$2$	$43.366$	\(\Q(\sqrt{65}) \)	None	✓				105.4.a.f	$1$	$1$	\(1\)	\(-6\)	\(-10\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-3q^{3}+(8+\beta )q^{4}-5q^{5}-3\beta q^{6}+\cdots\)
735.4.a.q	$735$	$4$	735.a	1.a	$1$	$2$	$2$	$43.366$	\(\Q(\sqrt{41}) \)	None	✓				105.4.a.g	$1$	$0$	\(3\)	\(-6\)	\(10\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}-3q^{3}+(3+3\beta )q^{4}+5q^{5}+\cdots\)
735.4.a.r	$735$	$4$	735.a	1.a	$1$	$3$	$3$	$43.366$	3.3.4892.1	None	✓				105.4.i.c	$1$	$1$	\(3\)	\(-9\)	\(15\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-3q^{3}+(-\beta _{1}+\beta _{2})q^{4}+\cdots\)
735.4.a.s	$735$	$4$	735.a	1.a	$1$	$3$	$3$	$43.366$	3.3.4892.1	None	✓				105.4.i.c	$1$	$0$	\(3\)	\(9\)	\(-15\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+3q^{3}+(-\beta _{1}+\beta _{2})q^{4}+\cdots\)
735.4.a.t	$735$	$4$	735.a	1.a	$1$	$4$	$4$	$43.366$	4.4.51264.1	None	✓	✓	✓		735.4.a.t	$1$	$1$	\(-6\)	\(-12\)	\(20\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1}+\beta _{2})q^{2}-3q^{3}+(6-\beta _{1}+\cdots)q^{4}+\cdots\)
735.4.a.u	$735$	$4$	735.a	1.a	$1$	$4$	$4$	$43.366$	4.4.51264.1	None	✓	✓			735.4.a.t	$1$	$1$	\(-6\)	\(12\)	\(-20\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1}+\beta _{2})q^{2}+3q^{3}+(6-\beta _{1}+\cdots)q^{4}+\cdots\)
735.4.a.v	$735$	$4$	735.a	1.a	$1$	$4$	$4$	$43.366$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			735.4.a.v	$1$	$0$	\(1\)	\(-12\)	\(20\)	\(0\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-3q^{3}+(10+\beta _{2})q^{4}+5q^{5}+\cdots\)
735.4.a.w	$735$	$4$	735.a	1.a	$1$	$4$	$4$	$43.366$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓		735.4.a.v	$1$	$0$	\(1\)	\(12\)	\(-20\)	\(0\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+3q^{3}+(10+\beta _{2})q^{4}-5q^{5}+\cdots\)
735.4.a.x	$735$	$4$	735.a	1.a	$1$	$5$	$5$	$43.366$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		105.4.i.e	$1$	$0$	\(-1\)	\(-15\)	\(25\)	\(0\)		$+$	$-$	$-$	$+$	$2\cdot 7$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-3q^{3}+(4+\beta _{1}+\beta _{2})q^{4}+\cdots\)
735.4.a.y	$735$	$4$	735.a	1.a	$1$	$5$	$5$	$43.366$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		105.4.i.e	$1$	$1$	\(-1\)	\(15\)	\(-25\)	\(0\)		$-$	$+$	$+$	$-$	$2\cdot 7$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+3q^{3}+(4+\beta _{1}+\beta _{2})q^{4}+\cdots\)
735.4.a.z	$735$	$4$	735.a	1.a	$1$	$5$	$5$	$43.366$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		105.4.i.d	$1$	$1$	\(3\)	\(-15\)	\(-25\)	\(0\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-3q^{3}+(5+\beta _{2})q^{4}-5q^{5}+\cdots\)
735.4.a.ba	$735$	$4$	735.a	1.a	$1$	$5$	$5$	$43.366$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				105.4.i.d	$1$	$0$	\(3\)	\(15\)	\(25\)	\(0\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+3q^{3}+(5+\beta _{2})q^{4}+5q^{5}+\cdots\)
735.4.a.bb	$735$	$4$	735.a	1.a	$1$	$8$	$8$	$43.366$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		735.4.a.bb	$1$	$0$	\(2\)	\(-24\)	\(-40\)	\(0\)		$+$	$+$	$+$	$+$	$2\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-3q^{3}+(6+\beta _{1}+\beta _{2})q^{4}+\cdots\)
735.4.a.bc	$735$	$4$	735.a	1.a	$1$	$8$	$8$	$43.366$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		735.4.a.bb	$1$	$0$	\(2\)	\(24\)	\(40\)	\(0\)		$-$	$-$	$+$	$+$	$2\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+3q^{3}+(6+\beta _{1}+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(735))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(735)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(245))\)\(^{\oplus 2}\)