Space of modular forms of level 725, weight 4, and trivial character

• Label: 725.4.a
• Level: $725$
• Weight: $4$
• Character orbit: 725.a
• Rep. character: $\chi_{725}(1,\cdot)$
• Character field: $\Q$
• Dimension: $133$
• Newform subspaces: $13$
• Sturm bound: $300$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 725 = 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	725.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(300\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(725))\).

	Total	New	Old
Modular forms	232	133	99
Cusp forms	220	133	87
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.

\(5\)	\(29\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(33\)
\(+\)	\(-\)	$-$	\(30\)
\(-\)	\(+\)	$-$	\(32\)
\(-\)	\(-\)	$+$	\(38\)
Plus space		\(+\)	\(71\)
Minus space		\(-\)	\(62\)

Trace form

\( 133 q - 2 q^{2} + 6 q^{3} + 544 q^{4} + 8 q^{6} + 16 q^{7} + 18 q^{8} + 1135 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(725))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		5	29
725.4.a.a	$725$	$4$	725.a	1.a	$1$	$1$	$1$	$42.776$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(8\)	\(0\)	\(14\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+8q^{3}-7q^{4}-8q^{6}+14q^{7}+\cdots\)
725.4.a.b	$725$	$4$	725.a	1.a	$1$	$2$	$2$	$42.776$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(2\)	\(10\)	\(0\)	\(16\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(5-3\beta )q^{3}+(-5+2\beta )q^{4}+\cdots\)
725.4.a.c	$725$	$4$	725.a	1.a	$1$	$5$	$5$	$42.776$	5.5.13458092.1	None	✓	$1$	$0$	\(0\)	\(-8\)	\(0\)	\(-40\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-1+\beta _{1}+\beta _{3}+\beta _{4})q^{3}+\cdots\)
725.4.a.d	$725$	$4$	725.a	1.a	$1$	$6$	$6$	$42.776$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(1\)	\(1\)	\(0\)	\(-3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(\beta _{3}+\beta _{4})q^{3}+(4+2\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
725.4.a.e	$725$	$4$	725.a	1.a	$1$	$6$	$6$	$42.776$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(7\)	\(13\)	\(0\)	\(79\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+(2-\beta _{1}-\beta _{4})q^{3}+(3+\cdots)q^{4}+\cdots\)
725.4.a.f	$725$	$4$	725.a	1.a	$1$	$7$	$7$	$42.776$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(-6\)	\(-1\)	\(0\)	\(-17\)		$+$	$+$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{4})q^{2}+\beta _{3}q^{3}+(8-\beta _{4}-\beta _{5}+\cdots)q^{4}+\cdots\)
725.4.a.g	$725$	$4$	725.a	1.a	$1$	$8$	$8$	$42.776$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(-5\)	\(-17\)	\(0\)	\(-33\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-2+\beta _{3})q^{3}+(5+\cdots)q^{4}+\cdots\)
725.4.a.h	$725$	$4$	725.a	1.a	$1$	$14$	$14$	$42.776$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓	$1$	$1$	\(-4\)	\(-25\)	\(0\)	\(-48\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-2-\beta _{8})q^{3}+(4+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
725.4.a.i	$725$	$4$	725.a	1.a	$1$	$14$	$14$	$42.776$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-11\)	\(0\)	\(-64\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1+\beta _{6})q^{3}+(4+\beta _{2})q^{4}+\cdots\)
725.4.a.j	$725$	$4$	725.a	1.a	$1$	$14$	$14$	$42.776$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(11\)	\(0\)	\(64\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1-\beta _{6})q^{3}+(4+\beta _{2})q^{4}+\cdots\)
725.4.a.k	$725$	$4$	725.a	1.a	$1$	$14$	$14$	$42.776$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓	$1$	$0$	\(4\)	\(25\)	\(0\)	\(48\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{8})q^{3}+(4+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
725.4.a.l	$725$	$4$	725.a	1.a	$1$	$18$	$18$	$42.776$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{8}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{12}q^{3}+(2+\beta _{2})q^{4}+(-1+\cdots)q^{6}+\cdots\)
725.4.a.m	$725$	$4$	725.a	1.a	$1$	$24$	$24$	$42.776$		None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$		$\mathrm{SU}(2)$

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(725))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(725)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(145))\)\(^{\oplus 2}\)