Space of modular forms of level 1250, weight 2, and trivial character

• Label: 1250.2.a
• Level: $1250$
• Weight: $2$
• Character orbit: 1250.a
• Rep. character: $\chi_{1250}(1,\cdot)$
• Character field: $\Q$
• Dimension: $40$
• Newform subspaces: $12$
• Sturm bound: $375$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 1250 = 2 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1250.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(375\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1250))\).

	Total	New	Old
Modular forms	217	40	177
Cusp forms	158	40	118
Eisenstein series	59	0	59

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

\(2\)	\(5\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(8\)
\(+\)	\(-\)	$-$	\(12\)
\(-\)	\(+\)	$-$	\(14\)
\(-\)	\(-\)	$+$	\(6\)
Plus space		\(+\)	\(14\)
Minus space		\(-\)	\(26\)

Trace form

\( 40 q + 40 q^{4} + 40 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1250))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	5
1250.2.a.a	$1250$	$2$	1250.a	1.a	$1$	$2$	$2$	$9.981$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(-2\)	\(-3\)	\(0\)	\(-6\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(-1-\beta )q^{3}+q^{4}+(1+\beta )q^{6}+\cdots\)
1250.2.a.b	$1250$	$2$	1250.a	1.a	$1$	$2$	$2$	$9.981$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(-2\)	\(2\)	\(0\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{6}-\beta q^{7}-q^{8}+\cdots\)
1250.2.a.c	$1250$	$2$	1250.a	1.a	$1$	$2$	$2$	$9.981$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(2\)	\(-2\)	\(0\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{6}+\beta q^{7}+q^{8}+\cdots\)
1250.2.a.d	$1250$	$2$	1250.a	1.a	$1$	$2$	$2$	$9.981$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(2\)	\(3\)	\(0\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(1+\beta )q^{3}+q^{4}+(1+\beta )q^{6}+\cdots\)
1250.2.a.e	$1250$	$2$	1250.a	1.a	$1$	$4$	$4$	$9.981$	\(\Q(\zeta_{15})^+\)	None	✓	$1$	$0$	\(-4\)	\(-1\)	\(0\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(\beta _{2}+\beta _{3})q^{3}+q^{4}+(-\beta _{2}-\beta _{3})q^{6}+\cdots\)
1250.2.a.f	$1250$	$2$	1250.a	1.a	$1$	$4$	$4$	$9.981$	4.4.7625.1	None	✓	$1$	$1$	\(-4\)	\(-1\)	\(0\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(-1+\cdots)q^{7}+\cdots\)
1250.2.a.g	$1250$	$2$	1250.a	1.a	$1$	$4$	$4$	$9.981$	4.4.18625.1	None	✓	$1$	$0$	\(-4\)	\(-1\)	\(0\)	\(3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(\beta _{1}-\beta _{2})q^{3}+q^{4}+(-\beta _{1}+\beta _{2}+\cdots)q^{6}+\cdots\)
1250.2.a.h	$1250$	$2$	1250.a	1.a	$1$	$4$	$4$	$9.981$	\(\Q(\zeta_{20})^+\)	None	✓	$1$	$0$	\(-4\)	\(4\)	\(0\)	\(8\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(1+\beta _{1})q^{3}+q^{4}+(-1-\beta _{1}+\cdots)q^{6}+\cdots\)
1250.2.a.i	$1250$	$2$	1250.a	1.a	$1$	$4$	$4$	$9.981$	\(\Q(\zeta_{20})^+\)	None	✓	$1$	$1$	\(4\)	\(-4\)	\(0\)	\(-8\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(-1+\beta _{1})q^{3}+q^{4}+(-1+\beta _{1}+\cdots)q^{6}+\cdots\)
1250.2.a.j	$1250$	$2$	1250.a	1.a	$1$	$4$	$4$	$9.981$	4.4.18625.1	None	✓	$1$	$0$	\(4\)	\(1\)	\(0\)	\(-3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(-\beta _{1}+\beta _{2})q^{3}+q^{4}+(-\beta _{1}+\cdots)q^{6}+\cdots\)
1250.2.a.k	$1250$	$2$	1250.a	1.a	$1$	$4$	$4$	$9.981$	\(\Q(\zeta_{15})^+\)	None	✓	$1$	$0$	\(4\)	\(1\)	\(0\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(-\beta _{2}-\beta _{3})q^{3}+q^{4}+(-\beta _{2}+\cdots)q^{6}+\cdots\)
1250.2.a.l	$1250$	$2$	1250.a	1.a	$1$	$4$	$4$	$9.981$	4.4.7625.1	None	✓	$1$	$0$	\(4\)	\(1\)	\(0\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(1+\beta _{3})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1250))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1250)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(125))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(250))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(625))\)\(^{\oplus 2}\)