Space of modular forms of level 1575, weight 2, and trivial character

• Label: 1575.2.a
• Level: $1575$
• Weight: $2$
• Character orbit: 1575.a
• Rep. character: $\chi_{1575}(1,\cdot)$
• Character field: $\Q$
• Dimension: $47$
• Newform subspaces: $26$
• Sturm bound: $480$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 26 \)
Sturm bound:			\(480\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(2\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1575))\).

	Total	New	Old
Modular forms	264	47	217
Cusp forms	217	47	170
Eisenstein series	47	0	47

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	Dim
\(+\)	\(+\)	\(+\)	$+$	\(2\)
\(+\)	\(+\)	\(-\)	$-$	\(8\)
\(+\)	\(-\)	\(+\)	$-$	\(6\)
\(+\)	\(-\)	\(-\)	$+$	\(2\)
\(-\)	\(+\)	\(+\)	$-$	\(8\)
\(-\)	\(+\)	\(-\)	$+$	\(5\)
\(-\)	\(-\)	\(+\)	$+$	\(7\)
\(-\)	\(-\)	\(-\)	$-$	\(9\)
Plus space			\(+\)	\(16\)
Minus space			\(-\)	\(31\)

Trace form

\( 47 q - q^{2} + 51 q^{4} + q^{7} - 9 q^{8} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1575))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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
1575.2.a.a	$1575$	$2$	1575.a	1.a	$1$	$1$	$1$	$12.576$	\(\Q\)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}-q^{7}+3q^{11}+q^{13}+\cdots\)
1575.2.a.b	$1575$	$2$	1575.a	1.a	$1$	$1$	$1$	$12.576$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}-q^{7}+3q^{8}+4q^{13}+q^{14}+\cdots\)
1575.2.a.c	$1575$	$2$	1575.a	1.a	$1$	$1$	$1$	$12.576$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+q^{7}+3q^{8}-4q^{11}+2q^{13}+\cdots\)
1575.2.a.d	$1575$	$2$	1575.a	1.a	$1$	$1$	$1$	$12.576$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+q^{7}+3q^{8}-4q^{13}-q^{14}+\cdots\)
1575.2.a.e	$1575$	$2$	1575.a	1.a	$1$	$1$	$1$	$12.576$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+q^{7}+3q^{8}+6q^{11}+2q^{13}+\cdots\)
1575.2.a.f	$1575$	$2$	1575.a	1.a	$1$	$1$	$1$	$12.576$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}-q^{7}+3q^{11}-5q^{13}+4q^{16}+\cdots\)
1575.2.a.g	$1575$	$2$	1575.a	1.a	$1$	$1$	$1$	$12.576$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-q^{7}-3q^{8}+4q^{13}-q^{14}+\cdots\)
1575.2.a.h	$1575$	$2$	1575.a	1.a	$1$	$1$	$1$	$12.576$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-q^{7}-3q^{8}+6q^{13}-q^{14}+\cdots\)
1575.2.a.i	$1575$	$2$	1575.a	1.a	$1$	$1$	$1$	$12.576$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-q^{7}-3q^{8}+6q^{11}-2q^{13}+\cdots\)
1575.2.a.j	$1575$	$2$	1575.a	1.a	$1$	$1$	$1$	$12.576$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(0\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+q^{7}-3q^{8}-4q^{13}+q^{14}+\cdots\)
1575.2.a.k	$1575$	$2$	1575.a	1.a	$1$	$1$	$1$	$12.576$	\(\Q\)	None	✓	$1$	$0$	\(2\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}+q^{7}+3q^{11}-q^{13}+\cdots\)
1575.2.a.l	$1575$	$2$	1575.a	1.a	$1$	$2$	$2$	$12.576$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(-3\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+3\beta q^{4}-q^{7}+(-1+\cdots)q^{8}+\cdots\)
1575.2.a.m	$1575$	$2$	1575.a	1.a	$1$	$2$	$2$	$12.576$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+(1-2\beta )q^{4}+q^{7}+(-3+\cdots)q^{8}+\cdots\)
1575.2.a.n	$1575$	$2$	1575.a	1.a	$1$	$2$	$2$	$12.576$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1+\beta )q^{4}+q^{7}+(-1+2\beta )q^{8}+\cdots\)
1575.2.a.o	$1575$	$2$	1575.a	1.a	$1$	$2$	$2$	$12.576$	\(\Q(\sqrt{13}) \)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(1+\beta )q^{4}+q^{7}-3q^{8}+3q^{11}+\cdots\)
1575.2.a.p	$1575$	$2$	1575.a	1.a	$1$	$2$	$2$	$12.576$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(2+\beta )q^{4}+q^{7}+(-4-\beta )q^{8}+\cdots\)
1575.2.a.q	$1575$	$2$	1575.a	1.a	$1$	$2$	$2$	$12.576$	\(\Q(\sqrt{3}) \)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{4}-q^{7}-\beta q^{8}-2\beta q^{11}+\cdots\)
1575.2.a.r	$1575$	$2$	1575.a	1.a	$1$	$2$	$2$	$12.576$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+3q^{4}-q^{7}-\beta q^{8}+(-2-2\beta )q^{11}+\cdots\)
1575.2.a.s	$1575$	$2$	1575.a	1.a	$1$	$2$	$2$	$12.576$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+\beta )q^{4}-q^{7}+(1-2\beta )q^{8}+\cdots\)
1575.2.a.t	$1575$	$2$	1575.a	1.a	$1$	$2$	$2$	$12.576$	\(\Q(\sqrt{13}) \)	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(1+\beta )q^{4}-q^{7}+3q^{8}+3q^{11}+\cdots\)
1575.2.a.u	$1575$	$2$	1575.a	1.a	$1$	$2$	$2$	$12.576$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(2\)	\(0\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(1+2\beta )q^{4}+q^{7}+(3+\beta )q^{8}+\cdots\)
1575.2.a.v	$1575$	$2$	1575.a	1.a	$1$	$2$	$2$	$12.576$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(3\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+3\beta q^{4}+q^{7}+(1+4\beta )q^{8}+\cdots\)
1575.2.a.w	$1575$	$2$	1575.a	1.a	$1$	$3$	$3$	$12.576$	3.3.148.1	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-3\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}-q^{7}+(-2+\cdots)q^{8}+\cdots\)
1575.2.a.x	$1575$	$2$	1575.a	1.a	$1$	$3$	$3$	$12.576$	3.3.148.1	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(3\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}+q^{7}+(2+\cdots)q^{8}+\cdots\)
1575.2.a.y	$1575$	$2$	1575.a	1.a	$1$	$4$	$4$	$12.576$	4.4.174928.1	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(3+\beta _{2})q^{4}-q^{7}+(2\beta _{1}+\beta _{3})q^{8}+\cdots\)
1575.2.a.z	$1575$	$2$	1575.a	1.a	$1$	$4$	$4$	$12.576$	4.4.174928.1	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(3+\beta _{2})q^{4}+q^{7}+(2\beta _{1}+\beta _{3})q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1575))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1575)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(315))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(525))\)\(^{\oplus 2}\)