Space of modular forms of level 675, weight 2, and trivial character

• Label: 675.2.a
• Level: $675$
• Weight: $2$
• Character orbit: 675.a
• Rep. character: $\chi_{675}(1,\cdot)$
• Character field: $\Q$
• Dimension: $25$
• Newform subspaces: $17$
• Sturm bound: $180$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	675.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 17 \)
Sturm bound:			\(180\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(2\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(675))\).

	Total	New	Old
Modular forms	108	25	83
Cusp forms	73	25	48
Eisenstein series	35	0	35

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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(24\)	\(4\)	\(20\)		\(16\)	\(4\)	\(12\)		\(8\)	\(0\)	\(8\)
\(+\)	\(-\)	\(-\)		\(30\)	\(8\)	\(22\)		\(21\)	\(8\)	\(13\)		\(9\)	\(0\)	\(9\)
\(-\)	\(+\)	\(-\)		\(30\)	\(8\)	\(22\)		\(21\)	\(8\)	\(13\)		\(9\)	\(0\)	\(9\)
\(-\)	\(-\)	\(+\)		\(24\)	\(5\)	\(19\)		\(15\)	\(5\)	\(10\)		\(9\)	\(0\)	\(9\)
Plus space		\(+\)		\(48\)	\(9\)	\(39\)		\(31\)	\(9\)	\(22\)		\(17\)	\(0\)	\(17\)
Minus space		\(-\)		\(60\)	\(16\)	\(44\)		\(42\)	\(16\)	\(26\)		\(18\)	\(0\)	\(18\)

Trace form

25 * q + 26 * q^4 + 3 * q^7 - 7 * q^13 + 48 * q^16 + 3 * q^19 + 20 * q^22 + 30 * q^28 + 14 * q^31 - 20 * q^34 - 29 * q^37 - 4 * q^43 + 8 * q^46 + 4 * q^49 - 14 * q^52 - 8 * q^58 - 19 * q^61 + 72 * q^64 + 45 * q^67 - 19 * q^73 - 62 * q^76 + 9 * q^79 - 68 * q^82 + 12 * q^88 - 43 * q^91 - 92 * q^94 + 13 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(675))\) 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
675.2.a.a	$675$	$2$	675.a	1.a	$1$	$1$	$1$	$5.390$	\(\Q\)	None	✓				135.2.a.a	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}+3q^{7}+2q^{11}+5q^{13}+\cdots\)
675.2.a.b	$675$	$2$	675.a	1.a	$1$	$1$	$1$	$5.390$	\(\Q\)	None	✓	✓			675.2.a.b	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+3q^{8}-5q^{11}+5q^{13}+\cdots\)
675.2.a.c	$675$	$2$	675.a	1.a	$1$	$1$	$1$	$5.390$	\(\Q\)	None	✓	✓			675.2.a.b	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+3q^{8}+5q^{11}-5q^{13}+\cdots\)
675.2.a.d	$675$	$2$	675.a	1.a	$1$	$1$	$1$	$5.390$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			675.2.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{4}-4q^{7}+5q^{13}+4q^{16}+8q^{19}+\cdots\)
675.2.a.e	$675$	$2$	675.a	1.a	$1$	$1$	$1$	$5.390$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				27.2.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{4}+q^{7}-5q^{13}+4q^{16}-7q^{19}+\cdots\)
675.2.a.f	$675$	$2$	675.a	1.a	$1$	$1$	$1$	$5.390$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			675.2.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{4}+4q^{7}-5q^{13}+4q^{16}+8q^{19}+\cdots\)
675.2.a.g	$675$	$2$	675.a	1.a	$1$	$1$	$1$	$5.390$	\(\Q\)	None	✓	✓			675.2.a.b	$1$	$1$	\(1\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-3q^{8}-5q^{11}-5q^{13}+\cdots\)
675.2.a.h	$675$	$2$	675.a	1.a	$1$	$1$	$1$	$5.390$	\(\Q\)	None	✓	✓			675.2.a.b	$1$	$0$	\(1\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-3q^{8}+5q^{11}+5q^{13}+\cdots\)
675.2.a.i	$675$	$2$	675.a	1.a	$1$	$1$	$1$	$5.390$	\(\Q\)	None	✓				135.2.a.a	$1$	$0$	\(2\)	\(0\)	\(0\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}+3q^{7}-2q^{11}+5q^{13}+\cdots\)
675.2.a.j	$675$	$2$	675.a	1.a	$1$	$2$	$2$	$5.390$	\(\Q(\sqrt{5}) \)	\(\Q(\sqrt{-15}) \)	✓				135.2.b.a	$2$	$1$	\(-3\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+(-1-\beta )q^{2}+3\beta q^{4}+(-1-4\beta )q^{8}+\cdots\)
675.2.a.k	$675$	$2$	675.a	1.a	$1$	$2$	$2$	$5.390$	\(\Q(\sqrt{13}) \)	None	✓		✓		135.2.a.c	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(1+\beta )q^{4}+(-2+2\beta )q^{7}+\cdots\)
675.2.a.l	$675$	$2$	675.a	1.a	$1$	$2$	$2$	$5.390$	\(\Q(\sqrt{2}) \)	None	✓				135.2.b.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-3q^{7}-2\beta q^{8}-3\beta q^{11}-3q^{13}+\cdots\)
675.2.a.m	$675$	$2$	675.a	1.a	$1$	$2$	$2$	$5.390$	\(\Q(\sqrt{2}) \)	None	✓				135.2.b.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+3q^{7}-2\beta q^{8}+3\beta q^{11}+3q^{13}+\cdots\)
675.2.a.n	$675$	$2$	675.a	1.a	$1$	$2$	$2$	$5.390$	\(\Q(\sqrt{7}) \)	None	✓	✓			675.2.a.n	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+5q^{4}-3q^{7}+3\beta q^{8}+2\beta q^{11}+\cdots\)
675.2.a.o	$675$	$2$	675.a	1.a	$1$	$2$	$2$	$5.390$	\(\Q(\sqrt{7}) \)	None	✓	✓			675.2.a.n	$2$	$0$	\(0\)	\(0\)	\(0\)	\(6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+5q^{4}+3q^{7}+3\beta q^{8}-2\beta q^{11}+\cdots\)
675.2.a.p	$675$	$2$	675.a	1.a	$1$	$2$	$2$	$5.390$	\(\Q(\sqrt{13}) \)	None	✓				135.2.a.c	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(1+\beta )q^{4}+(-2+2\beta )q^{7}+\cdots\)
675.2.a.q	$675$	$2$	675.a	1.a	$1$	$2$	$2$	$5.390$	\(\Q(\sqrt{5}) \)	\(\Q(\sqrt{-15}) \)	✓				135.2.b.a	$2$	$0$	\(3\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+(1+\beta )q^{2}+3\beta q^{4}+(1+4\beta )q^{8}+(5+\cdots)q^{16}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(675))\) into lower level spaces

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