Space of modular forms of level 50, weight 10, and trivial character

• Label: 50.10.a
• Level: $50$
• Weight: $10$
• Character orbit: 50.a
• Rep. character: $\chi_{50}(1,\cdot)$
• Character field: $\Q$
• Dimension: $14$
• Newform subspaces: $10$
• Sturm bound: $75$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	50.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(75\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(50))\).

	Total	New	Old
Modular forms	73	14	59
Cusp forms	61	14	47
Eisenstein series	12	0	12

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
\(+\)	\(+\)	$+$	\(3\)
\(+\)	\(-\)	$-$	\(4\)
\(-\)	\(+\)	$-$	\(4\)
\(-\)	\(-\)	$+$	\(3\)
Plus space		\(+\)	\(6\)
Minus space		\(-\)	\(8\)

Trace form

\( 14 q + 140 q^{3} + 3584 q^{4} - 544 q^{6} + 1180 q^{7} + 79892 q^{9} + O(q^{10}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(50))\) 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
50.10.a.a	$50$	$10$	50.a	1.a	$1$	$1$	$1$	$25.752$	\(\Q\)	None	✓	$1$	$1$	\(-16\)	\(-174\)	\(0\)	\(-4658\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}-174q^{3}+2^{8}q^{4}+2784q^{6}+\cdots\)
50.10.a.b	$50$	$10$	50.a	1.a	$1$	$1$	$1$	$25.752$	\(\Q\)	None	✓	$1$	$1$	\(-16\)	\(21\)	\(0\)	\(1882\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+21q^{3}+2^{8}q^{4}-336q^{6}+\cdots\)
50.10.a.c	$50$	$10$	50.a	1.a	$1$	$1$	$1$	$25.752$	\(\Q\)	None	✓	$1$	$1$	\(-16\)	\(156\)	\(0\)	\(952\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+156q^{3}+2^{8}q^{4}-2496q^{6}+\cdots\)
50.10.a.d	$50$	$10$	50.a	1.a	$1$	$1$	$1$	$25.752$	\(\Q\)	None	✓	$1$	$0$	\(16\)	\(-46\)	\(0\)	\(10318\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}-46q^{3}+2^{8}q^{4}-736q^{6}+\cdots\)
50.10.a.e	$50$	$10$	50.a	1.a	$1$	$1$	$1$	$25.752$	\(\Q\)	None	✓	$1$	$1$	\(16\)	\(-21\)	\(0\)	\(-1882\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}-21q^{3}+2^{8}q^{4}-336q^{6}+\cdots\)
50.10.a.f	$50$	$10$	50.a	1.a	$1$	$1$	$1$	$25.752$	\(\Q\)	None	✓	$1$	$0$	\(16\)	\(204\)	\(0\)	\(-5432\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+204q^{3}+2^{8}q^{4}+3264q^{6}+\cdots\)
50.10.a.g	$50$	$10$	50.a	1.a	$1$	$2$	$2$	$25.752$	\(\Q(\sqrt{1009}) \)	None	✓	$1$	$0$	\(-32\)	\(-68\)	\(0\)	\(-56\)		$+$	$-$	$-$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+(-34-\beta )q^{3}+2^{8}q^{4}+(544+\cdots)q^{6}+\cdots\)
50.10.a.h	$50$	$10$	50.a	1.a	$1$	$2$	$2$	$25.752$	\(\Q(\sqrt{319}) \)	None	✓	$1$	$0$	\(-32\)	\(152\)	\(0\)	\(5784\)		$+$	$-$	$-$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+(76+\beta )q^{3}+2^{8}q^{4}+(-1216+\cdots)q^{6}+\cdots\)
50.10.a.i	$50$	$10$	50.a	1.a	$1$	$2$	$2$	$25.752$	\(\Q(\sqrt{319}) \)	None	✓	$1$	$1$	\(32\)	\(-152\)	\(0\)	\(-5784\)		$-$	$-$	$+$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+(-76+\beta )q^{3}+2^{8}q^{4}+(-1216+\cdots)q^{6}+\cdots\)
50.10.a.j	$50$	$10$	50.a	1.a	$1$	$2$	$2$	$25.752$	\(\Q(\sqrt{1009}) \)	None	✓	$1$	$0$	\(32\)	\(68\)	\(0\)	\(56\)		$-$	$+$	$-$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+(34-\beta )q^{3}+2^{8}q^{4}+(544+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(50))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(50)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)