Space of modular forms of level 675, weight 4, and trivial character

• Label: 675.4.a
• Level: $675$
• Weight: $4$
• Character orbit: 675.a
• Rep. character: $\chi_{675}(1,\cdot)$
• Character field: $\Q$
• Dimension: $76$
• Newform subspaces: $29$
• Sturm bound: $360$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	675.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 29 \)
Sturm bound:			\(360\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(675))\).

	Total	New	Old
Modular forms	288	76	212
Cusp forms	252	76	176
Eisenstein series	36	0	36

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)	\(5\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(20\)
\(+\)	\(-\)	\(-\)	\(19\)
\(-\)	\(+\)	\(-\)	\(16\)
\(-\)	\(-\)	\(+\)	\(21\)
Plus space		\(+\)	\(41\)
Minus space		\(-\)	\(35\)

Trace form

\( 76 q + 298 q^{4} - 40 q^{7} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(675))\) 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
675.4.a.a	$675$	$4$	675.a	1.a	$1$	$1$	$1$	$39.826$	\(\Q\)	None	✓				27.4.a.a	$1$	$1$	\(-3\)	\(0\)	\(0\)	\(25\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+q^{4}+5^{2}q^{7}+21q^{8}-15q^{11}+\cdots\)
675.4.a.b	$675$	$4$	675.a	1.a	$1$	$1$	$1$	$39.826$	\(\Q\)	None	✓				135.4.a.a	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-4q^{4}+24q^{8}-10q^{11}+80q^{13}+\cdots\)
675.4.a.c	$675$	$4$	675.a	1.a	$1$	$1$	$1$	$39.826$	\(\Q\)	None	✓				135.4.a.b	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-7q^{4}+6q^{7}+15q^{8}-47q^{11}+\cdots\)
675.4.a.d	$675$	$4$	675.a	1.a	$1$	$1$	$1$	$39.826$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			675.4.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-37\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-8q^{4}-37q^{7}-70q^{13}+2^{6}q^{16}+\cdots\)
675.4.a.e	$675$	$4$	675.a	1.a	$1$	$1$	$1$	$39.826$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			675.4.a.e	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-17\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-8q^{4}-17q^{7}+70q^{13}+2^{6}q^{16}+\cdots\)
675.4.a.f	$675$	$4$	675.a	1.a	$1$	$1$	$1$	$39.826$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			675.4.a.e	$2$	$1$	\(0\)	\(0\)	\(0\)	\(17\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-8q^{4}+17q^{7}-70q^{13}+2^{6}q^{16}+\cdots\)
675.4.a.g	$675$	$4$	675.a	1.a	$1$	$1$	$1$	$39.826$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			675.4.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(37\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-8q^{4}+37q^{7}+70q^{13}+2^{6}q^{16}+\cdots\)
675.4.a.h	$675$	$4$	675.a	1.a	$1$	$1$	$1$	$39.826$	\(\Q\)	None	✓				135.4.a.b	$1$	$1$	\(1\)	\(0\)	\(0\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-7q^{4}+6q^{7}-15q^{8}+47q^{11}+\cdots\)
675.4.a.i	$675$	$4$	675.a	1.a	$1$	$1$	$1$	$39.826$	\(\Q\)	None	✓				135.4.a.a	$1$	$1$	\(2\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-4q^{4}-24q^{8}+10q^{11}+80q^{13}+\cdots\)
675.4.a.j	$675$	$4$	675.a	1.a	$1$	$1$	$1$	$39.826$	\(\Q\)	None	✓				27.4.a.a	$1$	$0$	\(3\)	\(0\)	\(0\)	\(25\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+q^{4}+5^{2}q^{7}-21q^{8}+15q^{11}+\cdots\)
675.4.a.k	$675$	$4$	675.a	1.a	$1$	$2$	$2$	$39.826$	\(\Q(\sqrt{5}) \)	\(\Q(\sqrt{-15}) \)	✓				135.4.b.a	$2$	$1$	\(-9\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+(-4-\beta )q^{2}+(9+9\beta )q^{4}+(-13+\cdots)q^{8}+\cdots\)
675.4.a.l	$675$	$4$	675.a	1.a	$1$	$2$	$2$	$39.826$	\(\Q(\sqrt{13}) \)	None	✓	✓			675.4.a.l	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-18\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+5q^{4}-9q^{7}+3\beta q^{8}-4\beta q^{11}+\cdots\)
675.4.a.m	$675$	$4$	675.a	1.a	$1$	$2$	$2$	$39.826$	\(\Q(\sqrt{13}) \)	None	✓	✓			675.4.a.l	$2$	$1$	\(0\)	\(0\)	\(0\)	\(18\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+5q^{4}+9q^{7}+3\beta q^{8}+4\beta q^{11}+\cdots\)
675.4.a.n	$675$	$4$	675.a	1.a	$1$	$2$	$2$	$39.826$	\(\Q(\sqrt{2}) \)	None	✓				27.4.a.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-22\)		$-$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+10q^{4}-11q^{7}+2\beta q^{8}-4\beta q^{11}+\cdots\)
675.4.a.o	$675$	$4$	675.a	1.a	$1$	$2$	$2$	$39.826$	\(\Q(\sqrt{5}) \)	\(\Q(\sqrt{-15}) \)	✓				135.4.b.a	$2$	$0$	\(9\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+(5-\beta )q^{2}+(18-9\beta )q^{4}+(59-46\beta )q^{8}+\cdots\)
675.4.a.p	$675$	$4$	675.a	1.a	$1$	$3$	$3$	$39.826$	3.3.1772.1	None	✓				135.4.a.e	$1$	$0$	\(-5\)	\(0\)	\(0\)	\(4\)		$+$	$+$	$+$	$3$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{2}+(7-3\beta _{1}+\beta _{2})q^{4}+\cdots\)
675.4.a.q	$675$	$4$	675.a	1.a	$1$	$3$	$3$	$39.826$	3.3.5637.1	None	✓				135.4.a.f	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-44\)		$-$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(7+\beta _{1}+\beta _{2})q^{4}+(-14+\cdots)q^{7}+\cdots\)
675.4.a.r	$675$	$4$	675.a	1.a	$1$	$3$	$3$	$39.826$	3.3.5637.1	None	✓				135.4.a.f	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-44\)		$+$	$+$	$+$	$3$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(7+\beta _{1}+\beta _{2})q^{4}+(-14+\cdots)q^{7}+\cdots\)
675.4.a.s	$675$	$4$	675.a	1.a	$1$	$3$	$3$	$39.826$	3.3.1772.1	None	✓				135.4.a.e	$1$	$0$	\(5\)	\(0\)	\(0\)	\(4\)		$+$	$+$	$+$	$3$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{2}+(7-3\beta _{1}+\beta _{2})q^{4}+(2+\cdots)q^{7}+\cdots\)
675.4.a.t	$675$	$4$	675.a	1.a	$1$	$4$	$4$	$39.826$	4.4.467024.1	None	✓				135.4.b.b	$2$	$0$	\(-6\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{3})q^{2}+3\beta _{3}q^{4}-\beta _{2}q^{7}+\cdots\)
675.4.a.u	$675$	$4$	675.a	1.a	$1$	$4$	$4$	$39.826$	4.4.183945.1	None	✓	✓			675.4.a.u	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-4\)		$+$	$-$	$-$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}+(-1-3\beta _{1}+\cdots)q^{7}+\cdots\)
675.4.a.v	$675$	$4$	675.a	1.a	$1$	$4$	$4$	$39.826$	4.4.183945.1	None	✓	✓			675.4.a.u	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(4\)		$+$	$+$	$+$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}+(1+3\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
675.4.a.w	$675$	$4$	675.a	1.a	$1$	$4$	$4$	$39.826$	4.4.3173728.1	None	✓	✓			675.4.a.w	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-30\)		$+$	$+$	$+$	$3^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(7+\beta _{2})q^{4}+(-7+\beta _{2})q^{7}+\cdots\)
675.4.a.x	$675$	$4$	675.a	1.a	$1$	$4$	$4$	$39.826$	4.4.3173728.1	None	✓	✓			675.4.a.w	$2$	$0$	\(0\)	\(0\)	\(0\)	\(30\)		$-$	$-$	$+$	$3^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(7+\beta _{2})q^{4}+(7-\beta _{2})q^{7}+\cdots\)
675.4.a.y	$675$	$4$	675.a	1.a	$1$	$4$	$4$	$39.826$	4.4.183945.1	None	✓	✓			675.4.a.u	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-4\)		$-$	$-$	$+$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(5+\beta _{2})q^{4}+(-1-3\beta _{1}+\cdots)q^{7}+\cdots\)
675.4.a.z	$675$	$4$	675.a	1.a	$1$	$4$	$4$	$39.826$	4.4.183945.1	None	✓	✓	✓		675.4.a.u	$1$	$1$	\(1\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$-$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(5+\beta _{2})q^{4}+(1+3\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
675.4.a.ba	$675$	$4$	675.a	1.a	$1$	$4$	$4$	$39.826$	4.4.467024.1	None	✓				135.4.b.b	$2$	$1$	\(6\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(2+\beta _{3})q^{2}+3\beta _{3}q^{4}+\beta _{2}q^{7}+(-4+\cdots)q^{8}+\cdots\)
675.4.a.bb	$675$	$4$	675.a	1.a	$1$	$6$	$6$	$39.826$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		135.4.b.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-36\)		$+$	$-$	$-$	$2^{2}\cdot 3^{4}\cdot 5$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(5-\beta _{4})q^{4}+(-6+\beta _{2}+\beta _{4}+\cdots)q^{7}+\cdots\)
675.4.a.bc	$675$	$4$	675.a	1.a	$1$	$6$	$6$	$39.826$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		135.4.b.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(36\)		$-$	$-$	$+$	$2^{2}\cdot 3^{4}\cdot 5$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(5-\beta _{4})q^{4}+(6-\beta _{2}-\beta _{4}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(675))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(675)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(135))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 2}\)