Space of modular forms of level 725, weight 2, and trivial character

• Label: 725.2.a
• Level: $725$
• Weight: $2$
• Character orbit: 725.a
• Rep. character: $\chi_{725}(1,\cdot)$
• Character field: $\Q$
• Dimension: $45$
• Newform subspaces: $12$
• Sturm bound: $150$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 725 = 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	725.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(150\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(725))\).

	Total	New	Old
Modular forms	80	45	35
Cusp forms	69	45	24
Eisenstein series	11	0	11

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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(15\)	\(9\)	\(6\)		\(13\)	\(9\)	\(4\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(24\)	\(12\)	\(12\)		\(21\)	\(12\)	\(9\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(22\)	\(15\)	\(7\)		\(19\)	\(15\)	\(4\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(19\)	\(9\)	\(10\)		\(16\)	\(9\)	\(7\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(34\)	\(18\)	\(16\)		\(29\)	\(18\)	\(11\)		\(5\)	\(0\)	\(5\)
Minus space		\(-\)		\(46\)	\(27\)	\(19\)		\(40\)	\(27\)	\(13\)		\(6\)	\(0\)	\(6\)

Trace form

45 * q + q^2 + 2 * q^3 + 43 * q^4 - 2 * q^6 + 4 * q^7 - 3 * q^8 + 51 * q^9 - 6 * q^11 + 6 * q^12 + 12 * q^13 - 12 * q^14 + 23 * q^16 + 10 * q^17 + 15 * q^18 + 4 * q^19 + 20 * q^21 - 14 * q^22 - 16 * q^23 - 6 * q^24 - 36 * q^26 - 10 * q^27 + 12 * q^28 - 3 * q^29 + 2 * q^31 - 15 * q^32 + 10 * q^33 - 6 * q^34 + 29 * q^36 + 2 * q^37 + 24 * q^38 - 18 * q^39 - 6 * q^41 - 24 * q^42 + 2 * q^43 - 30 * q^44 - 52 * q^46 - 10 * q^47 + 6 * q^48 + 45 * q^49 + 16 * q^51 + 24 * q^52 - 16 * q^53 - 50 * q^54 - 48 * q^56 - 4 * q^57 + q^58 - 4 * q^59 - 10 * q^61 - 14 * q^62 + 4 * q^63 - q^64 + 6 * q^66 - 16 * q^67 + 26 * q^68 - 20 * q^69 - 8 * q^71 + 39 * q^72 + 18 * q^73 - 2 * q^74 - 28 * q^76 - 12 * q^77 + 2 * q^78 + 18 * q^79 + 105 * q^81 - 42 * q^82 - 20 * q^83 + 60 * q^84 + 6 * q^86 + 6 * q^87 - 50 * q^88 - 2 * q^89 + 20 * q^91 - 24 * q^92 - 26 * q^93 + 26 * q^94 - 30 * q^96 + 42 * q^97 - 67 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(725))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	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
725.2.a.a	$725$	$2$	725.a	1.a	$1$	$1$	$1$	$5.789$	\(\Q\)	None	✓				145.2.a.a	$1$	$1$	\(1\)	\(0\)	\(0\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+2q^{7}-3q^{8}-3q^{9}-6q^{11}+\cdots\)
725.2.a.b	$725$	$2$	725.a	1.a	$1$	$2$	$2$	$5.789$	\(\Q(\sqrt{2}) \)	None	✓				29.2.a.a	$1$	$0$	\(2\)	\(-2\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(-1-\beta )q^{3}+(1+2\beta )q^{4}+\cdots\)
725.2.a.c	$725$	$2$	725.a	1.a	$1$	$2$	$2$	$5.789$	\(\Q(\sqrt{2}) \)	None	✓				145.2.a.b	$1$	$0$	\(2\)	\(4\)	\(0\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+2q^{3}+(1+2\beta )q^{4}+(2+\cdots)q^{6}+\cdots\)
725.2.a.d	$725$	$2$	725.a	1.a	$1$	$3$	$3$	$5.789$	3.3.148.1	None	✓				145.2.a.d	$1$	$0$	\(-3\)	\(2\)	\(0\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{2}+(1-\beta _{1}-\beta _{2})q^{3}+\cdots\)
725.2.a.e	$725$	$2$	725.a	1.a	$1$	$3$	$3$	$5.789$	3.3.148.1	None	✓				145.2.a.c	$1$	$1$	\(-1\)	\(-2\)	\(0\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1+\beta _{1}+\beta _{2})q^{3}+(\beta _{1}+\cdots)q^{4}+\cdots\)
725.2.a.f	$725$	$2$	725.a	1.a	$1$	$4$	$4$	$5.789$	\(\Q(\zeta_{24})^+\)	None	✓				145.2.b.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-\beta _{1}-\beta _{3})q^{3}+\beta _{2}q^{4}+\cdots\)
725.2.a.g	$725$	$2$	725.a	1.a	$1$	$4$	$4$	$5.789$	\(\Q(\sqrt{3}, \sqrt{11})\)	None	✓				145.2.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+\beta _{1}q^{3}+q^{4}+(-2-\beta _{3})q^{6}+\cdots\)
725.2.a.h	$725$	$2$	725.a	1.a	$1$	$5$	$5$	$5.789$	5.5.240881.1	None	✓	✓	✓		725.2.a.h	$1$	$1$	\(-2\)	\(-6\)	\(0\)	\(-6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1+\beta _{4})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
725.2.a.i	$725$	$2$	725.a	1.a	$1$	$5$	$5$	$5.789$	5.5.294577.1	None	✓	✓	✓		725.2.a.i	$1$	$1$	\(0\)	\(-6\)	\(0\)	\(-10\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-1+\beta _{3})q^{3}+(1+\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots\)
725.2.a.j	$725$	$2$	725.a	1.a	$1$	$5$	$5$	$5.789$	5.5.294577.1	None	✓	✓			725.2.a.i	$1$	$0$	\(0\)	\(6\)	\(0\)	\(10\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1-\beta _{3})q^{3}+(1+\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots\)
725.2.a.k	$725$	$2$	725.a	1.a	$1$	$5$	$5$	$5.789$	5.5.240881.1	None	✓	✓	✓		725.2.a.h	$1$	$0$	\(2\)	\(6\)	\(0\)	\(6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1-\beta _{4})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
725.2.a.l	$725$	$2$	725.a	1.a	$1$	$6$	$6$	$5.789$	6.6.337383424.1	None	✓		✓		145.2.b.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(\beta _{1}-\beta _{4})q^{3}+(2+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(725))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(725)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(145))\)\(^{\oplus 2}\)