Space of modular forms of level 1275, weight 2, and trivial character

• Label: 1275.2.a
• Level: $1275$
• Weight: $2$
• Character orbit: 1275.a
• Rep. character: $\chi_{1275}(1,\cdot)$
• Character field: $\Q$
• Dimension: $50$
• Newform subspaces: $22$
• Sturm bound: $360$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1275 = 3 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1275.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 22 \)
Sturm bound:			\(360\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1275))\).

	Total	New	Old
Modular forms	192	50	142
Cusp forms	169	50	119
Eisenstein series	23	0	23

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

\(3\)	\(5\)	\(17\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(6\)
\(+\)	\(+\)	\(-\)	$-$	\(8\)
\(+\)	\(-\)	\(+\)	$-$	\(8\)
\(+\)	\(-\)	\(-\)	$+$	\(4\)
\(-\)	\(+\)	\(+\)	$-$	\(6\)
\(-\)	\(+\)	\(-\)	$+$	\(4\)
\(-\)	\(-\)	\(+\)	$+$	\(5\)
\(-\)	\(-\)	\(-\)	$-$	\(9\)
Plus space			\(+\)	\(19\)
Minus space			\(-\)	\(31\)

Trace form

\( 50 q - 4 q^{2} - 2 q^{3} + 44 q^{4} + 2 q^{6} - 4 q^{7} + 50 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1275))\) 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	17
1275.2.a.a	$1275$	$2$	1275.a	1.a	$1$	$1$	$1$	$10.181$	\(\Q\)	None	✓	$1$	$1$	\(-2\)	\(-1\)	\(0\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-q^{3}+2q^{4}+2q^{6}+q^{7}+q^{9}+\cdots\)
1275.2.a.b	$1275$	$2$	1275.a	1.a	$1$	$1$	$1$	$10.181$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(-1\)	\(0\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}-q^{4}+q^{6}+4q^{7}+3q^{8}+\cdots\)
1275.2.a.c	$1275$	$2$	1275.a	1.a	$1$	$1$	$1$	$10.181$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{4}-q^{7}+q^{9}+2q^{11}+2q^{12}+\cdots\)
1275.2.a.d	$1275$	$2$	1275.a	1.a	$1$	$1$	$1$	$10.181$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{4}+4q^{7}+q^{9}-3q^{11}+\cdots\)
1275.2.a.e	$1275$	$2$	1275.a	1.a	$1$	$1$	$1$	$10.181$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{4}+q^{7}+q^{9}+2q^{11}-2q^{12}+\cdots\)
1275.2.a.f	$1275$	$2$	1275.a	1.a	$1$	$1$	$1$	$10.181$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(1\)	\(0\)	\(-4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}-q^{4}+q^{6}-4q^{7}-3q^{8}+\cdots\)
1275.2.a.g	$1275$	$2$	1275.a	1.a	$1$	$1$	$1$	$10.181$	\(\Q\)	None	✓	$1$	$0$	\(2\)	\(1\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+q^{3}+2q^{4}+2q^{6}-q^{7}+q^{9}+\cdots\)
1275.2.a.h	$1275$	$2$	1275.a	1.a	$1$	$2$	$2$	$10.181$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(-3\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+q^{3}+3\beta q^{4}+(-1+\cdots)q^{6}+\cdots\)
1275.2.a.i	$1275$	$2$	1275.a	1.a	$1$	$2$	$2$	$10.181$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(-1\)	\(2\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+q^{3}+(-1+\beta )q^{4}-\beta q^{6}+\cdots\)
1275.2.a.j	$1275$	$2$	1275.a	1.a	$1$	$2$	$2$	$10.181$	\(\Q(\sqrt{13}) \)	None	✓	$1$	$0$	\(-1\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+q^{3}+(1+\beta )q^{4}-\beta q^{6}+(1+\cdots)q^{7}+\cdots\)
1275.2.a.k	$1275$	$2$	1275.a	1.a	$1$	$2$	$2$	$10.181$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(6\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-q^{3}-\beta q^{6}+(3+\beta )q^{7}-2\beta q^{8}+\cdots\)
1275.2.a.l	$1275$	$2$	1275.a	1.a	$1$	$2$	$2$	$10.181$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-6\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{3}+\beta q^{6}+(-3+\beta )q^{7}+\cdots\)
1275.2.a.m	$1275$	$2$	1275.a	1.a	$1$	$2$	$2$	$10.181$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(1\)	\(-2\)	\(0\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-q^{3}+(-1+\beta )q^{4}-\beta q^{6}+\cdots\)
1275.2.a.n	$1275$	$2$	1275.a	1.a	$1$	$2$	$2$	$10.181$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(1\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{3}+(2+\beta )q^{4}+\beta q^{6}+(4+\cdots)q^{8}+\cdots\)
1275.2.a.o	$1275$	$2$	1275.a	1.a	$1$	$3$	$3$	$10.181$	3.3.148.1	None	✓	$1$	$1$	\(-2\)	\(3\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+q^{3}+(1-\beta _{1}+\beta _{2})q^{4}+\cdots\)
1275.2.a.p	$1275$	$2$	1275.a	1.a	$1$	$3$	$3$	$10.181$	3.3.229.1	None	✓	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
1275.2.a.q	$1275$	$2$	1275.a	1.a	$1$	$3$	$3$	$10.181$	3.3.148.1	None	✓	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}-q^{3}+(1-\beta _{1}-\beta _{2})q^{4}-\beta _{2}q^{6}+\cdots\)
1275.2.a.r	$1275$	$2$	1275.a	1.a	$1$	$3$	$3$	$10.181$	3.3.148.1	None	✓	$1$	$0$	\(0\)	\(3\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+q^{3}+(1-\beta _{1}-\beta _{2})q^{4}-\beta _{2}q^{6}+\cdots\)
1275.2.a.s	$1275$	$2$	1275.a	1.a	$1$	$3$	$3$	$10.181$	3.3.148.1	None	✓	$1$	$0$	\(2\)	\(-3\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-q^{3}+(1-\beta _{1}+\beta _{2})q^{4}+\cdots\)
1275.2.a.t	$1275$	$2$	1275.a	1.a	$1$	$4$	$4$	$10.181$	4.4.13768.1	None	✓	$1$	$0$	\(-1\)	\(-4\)	\(0\)	\(-4\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{2}-q^{3}+(2-\beta _{1})q^{4}-\beta _{3}q^{6}+\cdots\)
1275.2.a.u	$1275$	$2$	1275.a	1.a	$1$	$5$	$5$	$10.181$	5.5.3717884.1	None	✓	$1$	$0$	\(-2\)	\(-5\)	\(0\)	\(-2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
1275.2.a.v	$1275$	$2$	1275.a	1.a	$1$	$5$	$5$	$10.181$	5.5.3717884.1	None	✓	$1$	$0$	\(2\)	\(5\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1275))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1275)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(85))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(255))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(425))\)\(^{\oplus 2}\)