Space of modular forms of level 1725, weight 2, and trivial character

• Label: 1725.2.a
• Level: $1725$
• Weight: $2$
• Character orbit: 1725.a
• Rep. character: $\chi_{1725}(1,\cdot)$
• Character field: $\Q$
• Dimension: $70$
• Newform subspaces: $38$
• Sturm bound: $480$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 1725 = 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1725.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 38 \)
Sturm bound:			\(480\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(2\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1725))\).

	Total	New	Old
Modular forms	252	70	182
Cusp forms	229	70	159
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.

\(3\)	\(5\)	\(23\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(24\)	\(8\)	\(16\)		\(22\)	\(8\)	\(14\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)		\(39\)	\(10\)	\(29\)		\(36\)	\(10\)	\(26\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)		\(37\)	\(11\)	\(26\)		\(34\)	\(11\)	\(23\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)		\(26\)	\(7\)	\(19\)		\(23\)	\(7\)	\(16\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)		\(30\)	\(8\)	\(22\)		\(27\)	\(8\)	\(19\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)		\(33\)	\(6\)	\(27\)		\(30\)	\(6\)	\(24\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)		\(35\)	\(8\)	\(27\)		\(32\)	\(8\)	\(24\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)		\(28\)	\(12\)	\(16\)		\(25\)	\(12\)	\(13\)		\(3\)	\(0\)	\(3\)
Plus space			\(+\)		\(118\)	\(29\)	\(89\)		\(107\)	\(29\)	\(78\)		\(11\)	\(0\)	\(11\)
Minus space			\(-\)		\(134\)	\(41\)	\(93\)		\(122\)	\(41\)	\(81\)		\(12\)	\(0\)	\(12\)

Trace form

70 * q + 2 * q^2 - 2 * q^3 + 70 * q^4 + 2 * q^6 + 18 * q^8 + 70 * q^9 + 2 * q^12 - 4 * q^13 + 20 * q^14 + 94 * q^16 + 8 * q^17 + 2 * q^18 - 12 * q^19 - 16 * q^21 - 8 * q^22 - 6 * q^24 - 12 * q^26 - 2 * q^27 - 28 * q^29 + 4 * q^31 + 18 * q^32 + 8 * q^33 - 44 * q^34 + 70 * q^36 + 4 * q^37 + 12 * q^38 - 8 * q^39 - 4 * q^41 - 8 * q^42 + 24 * q^44 + 4 * q^46 + 32 * q^47 + 2 * q^48 + 50 * q^49 - 4 * q^51 - 36 * q^52 + 2 * q^54 + 20 * q^56 - 12 * q^57 + 44 * q^58 + 32 * q^59 - 64 * q^61 - 56 * q^62 + 166 * q^64 + 32 * q^66 + 24 * q^67 + 56 * q^68 + 4 * q^69 + 72 * q^71 + 18 * q^72 + 36 * q^73 + 4 * q^74 - 56 * q^76 - 40 * q^77 - 4 * q^78 - 40 * q^79 + 70 * q^81 - 20 * q^82 - 16 * q^83 - 28 * q^84 + 4 * q^86 - 20 * q^87 + 16 * q^88 - 8 * q^89 - 52 * q^91 - 24 * q^93 + 56 * q^94 - 30 * q^96 + 4 * q^97 + 74 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1725))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5	23
1725.2.a.a	$1725$	$2$	1725.a	1.a	$1$	$1$	$1$	$13.774$	\(\Q\)	None	✓				345.2.a.f	$1$	$1$	\(-2\)	\(1\)	\(0\)	\(-3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+q^{3}+2q^{4}-2q^{6}-3q^{7}+\cdots\)
1725.2.a.b	$1725$	$2$	1725.a	1.a	$1$	$1$	$1$	$13.774$	\(\Q\)	None	✓				345.2.a.e	$1$	$0$	\(-1\)	\(-1\)	\(0\)	\(-4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}-q^{4}+q^{6}-4q^{7}+3q^{8}+\cdots\)
1725.2.a.c	$1725$	$2$	1725.a	1.a	$1$	$1$	$1$	$13.774$	\(\Q\)	None	✓	✓			1725.2.a.c	$1$	$1$	\(-1\)	\(-1\)	\(0\)	\(-3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}-q^{4}+q^{6}-3q^{7}+3q^{8}+\cdots\)
1725.2.a.d	$1725$	$2$	1725.a	1.a	$1$	$1$	$1$	$13.774$	\(\Q\)	None	✓				345.2.b.b	$1$	$0$	\(-1\)	\(-1\)	\(0\)	\(-2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}-q^{4}+q^{6}-2q^{7}+3q^{8}+\cdots\)
1725.2.a.e	$1725$	$2$	1725.a	1.a	$1$	$1$	$1$	$13.774$	\(\Q\)	None	✓				69.2.a.a	$1$	$0$	\(-1\)	\(-1\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}-q^{4}+q^{6}+2q^{7}+3q^{8}+\cdots\)
1725.2.a.f	$1725$	$2$	1725.a	1.a	$1$	$1$	$1$	$13.774$	\(\Q\)	None	✓	✓			1725.2.a.f	$1$	$0$	\(-1\)	\(-1\)	\(0\)	\(3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}-q^{4}+q^{6}+3q^{7}+3q^{8}+\cdots\)
1725.2.a.g	$1725$	$2$	1725.a	1.a	$1$	$1$	$1$	$13.774$	\(\Q\)	None	✓	✓			1725.2.a.g	$1$	$1$	\(-1\)	\(1\)	\(0\)	\(-5\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}-q^{4}-q^{6}-5q^{7}+3q^{8}+\cdots\)
1725.2.a.h	$1725$	$2$	1725.a	1.a	$1$	$1$	$1$	$13.774$	\(\Q\)	None	✓				345.2.b.a	$1$	$1$	\(-1\)	\(1\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}-q^{4}-q^{6}+3q^{8}+q^{9}+\cdots\)
1725.2.a.i	$1725$	$2$	1725.a	1.a	$1$	$1$	$1$	$13.774$	\(\Q\)	None	✓	✓			1725.2.a.i	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{4}-3q^{7}+q^{9}+2q^{11}+\cdots\)
1725.2.a.j	$1725$	$2$	1725.a	1.a	$1$	$1$	$1$	$13.774$	\(\Q\)	None	✓	✓			1725.2.a.j	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{4}+q^{7}+q^{9}-6q^{11}+2q^{12}+\cdots\)
1725.2.a.k	$1725$	$2$	1725.a	1.a	$1$	$1$	$1$	$13.774$	\(\Q\)	None	✓				345.2.a.d	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{4}+3q^{7}+q^{9}-4q^{11}+\cdots\)
1725.2.a.l	$1725$	$2$	1725.a	1.a	$1$	$1$	$1$	$13.774$	\(\Q\)	None	✓	✓			1725.2.a.j	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{4}-q^{7}+q^{9}-6q^{11}-2q^{12}+\cdots\)
1725.2.a.m	$1725$	$2$	1725.a	1.a	$1$	$1$	$1$	$13.774$	\(\Q\)	None	✓				345.2.a.c	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{4}-q^{7}+q^{9}+4q^{11}-2q^{12}+\cdots\)
1725.2.a.n	$1725$	$2$	1725.a	1.a	$1$	$1$	$1$	$13.774$	\(\Q\)	None	✓	✓			1725.2.a.i	$1$	$0$	\(0\)	\(1\)	\(0\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{4}+3q^{7}+q^{9}+2q^{11}+\cdots\)
1725.2.a.o	$1725$	$2$	1725.a	1.a	$1$	$1$	$1$	$13.774$	\(\Q\)	None	✓				345.2.a.b	$1$	$0$	\(1\)	\(-1\)	\(0\)	\(-4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}-q^{4}-q^{6}-4q^{7}-3q^{8}+\cdots\)
1725.2.a.p	$1725$	$2$	1725.a	1.a	$1$	$1$	$1$	$13.774$	\(\Q\)	None	✓				345.2.b.a	$1$	$1$	\(1\)	\(-1\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}-q^{4}-q^{6}-3q^{8}+q^{9}+\cdots\)
1725.2.a.q	$1725$	$2$	1725.a	1.a	$1$	$1$	$1$	$13.774$	\(\Q\)	None	✓	✓			1725.2.a.g	$1$	$0$	\(1\)	\(-1\)	\(0\)	\(5\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}-q^{4}-q^{6}+5q^{7}-3q^{8}+\cdots\)
1725.2.a.r	$1725$	$2$	1725.a	1.a	$1$	$1$	$1$	$13.774$	\(\Q\)	None	✓	✓			1725.2.a.f	$1$	$1$	\(1\)	\(1\)	\(0\)	\(-3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}-q^{4}+q^{6}-3q^{7}-3q^{8}+\cdots\)
1725.2.a.s	$1725$	$2$	1725.a	1.a	$1$	$1$	$1$	$13.774$	\(\Q\)	None	✓				345.2.b.b	$1$	$0$	\(1\)	\(1\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}-q^{4}+q^{6}+2q^{7}-3q^{8}+\cdots\)
1725.2.a.t	$1725$	$2$	1725.a	1.a	$1$	$1$	$1$	$13.774$	\(\Q\)	None	✓	✓			1725.2.a.c	$1$	$0$	\(1\)	\(1\)	\(0\)	\(3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}-q^{4}+q^{6}+3q^{7}-3q^{8}+\cdots\)
1725.2.a.u	$1725$	$2$	1725.a	1.a	$1$	$1$	$1$	$13.774$	\(\Q\)	None	✓				345.2.a.a	$1$	$0$	\(2\)	\(-1\)	\(0\)	\(5\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-q^{3}+2q^{4}-2q^{6}+5q^{7}+\cdots\)
1725.2.a.v	$1725$	$2$	1725.a	1.a	$1$	$2$	$2$	$13.774$	\(\Q(\sqrt{5}) \)	None	✓	✓			1725.2.a.v	$1$	$1$	\(-3\)	\(2\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+q^{3}+3\beta q^{4}+(-1+\cdots)q^{6}+\cdots\)
1725.2.a.w	$1725$	$2$	1725.a	1.a	$1$	$2$	$2$	$13.774$	\(\Q(\sqrt{13}) \)	None	✓	✓			1725.2.a.w	$1$	$1$	\(-1\)	\(-2\)	\(0\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}-q^{3}+(1+\beta )q^{4}+\beta q^{6}+(1+\cdots)q^{7}+\cdots\)
1725.2.a.x	$1725$	$2$	1725.a	1.a	$1$	$2$	$2$	$13.774$	\(\Q(\sqrt{21}) \)	None	✓	✓			1725.2.a.x	$1$	$1$	\(-1\)	\(2\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+q^{3}+(3+\beta )q^{4}-\beta q^{6}+(-1+\cdots)q^{7}+\cdots\)
1725.2.a.y	$1725$	$2$	1725.a	1.a	$1$	$2$	$2$	$13.774$	\(\Q(\sqrt{6}) \)	None	✓				345.2.a.i	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-q^{3}+4q^{4}-\beta q^{6}+q^{7}+2\beta q^{8}+\cdots\)
1725.2.a.z	$1725$	$2$	1725.a	1.a	$1$	$2$	$2$	$13.774$	\(\Q(\sqrt{2}) \)	None	✓				345.2.a.h	$1$	$1$	\(0\)	\(2\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{3}+\beta q^{6}+(1-2\beta )q^{7}-2\beta q^{8}+\cdots\)
1725.2.a.ba	$1725$	$2$	1725.a	1.a	$1$	$2$	$2$	$13.774$	\(\Q(\sqrt{5}) \)	None	✓				69.2.a.b	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+q^{3}+3q^{4}-\beta q^{6}+(-1+\beta )q^{7}+\cdots\)
1725.2.a.bb	$1725$	$2$	1725.a	1.a	$1$	$2$	$2$	$13.774$	\(\Q(\sqrt{21}) \)	None	✓	✓			1725.2.a.x	$1$	$0$	\(1\)	\(-2\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-q^{3}+(3+\beta )q^{4}-\beta q^{6}+(1+\cdots)q^{7}+\cdots\)
1725.2.a.bc	$1725$	$2$	1725.a	1.a	$1$	$2$	$2$	$13.774$	\(\Q(\sqrt{13}) \)	None	✓	✓			1725.2.a.w	$1$	$0$	\(1\)	\(2\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{3}+(1+\beta )q^{4}+\beta q^{6}+(-1+\cdots)q^{7}+\cdots\)
1725.2.a.bd	$1725$	$2$	1725.a	1.a	$1$	$2$	$2$	$13.774$	\(\Q(\sqrt{3}) \)	None	✓				345.2.a.g	$1$	$0$	\(2\)	\(2\)	\(0\)	\(6\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+q^{3}+(2+2\beta )q^{4}+(1+\beta )q^{6}+\cdots\)
1725.2.a.be	$1725$	$2$	1725.a	1.a	$1$	$2$	$2$	$13.774$	\(\Q(\sqrt{5}) \)	None	✓	✓			1725.2.a.v	$1$	$0$	\(3\)	\(-2\)	\(0\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}-q^{3}+3\beta q^{4}+(-1-\beta )q^{6}+\cdots\)
1725.2.a.bf	$1725$	$2$	1725.a	1.a	$1$	$3$	$3$	$13.774$	3.3.229.1	None	✓	✓			1725.2.a.bf	$1$	$1$	\(-2\)	\(-3\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{2}-q^{3}+(2+\beta _{1})q^{4}+\cdots\)
1725.2.a.bg	$1725$	$2$	1725.a	1.a	$1$	$3$	$3$	$13.774$	3.3.148.1	None	✓		✓		345.2.b.c	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}-q^{3}+(1-\beta _{1}-\beta _{2})q^{4}+\beta _{2}q^{6}+\cdots\)
1725.2.a.bh	$1725$	$2$	1725.a	1.a	$1$	$3$	$3$	$13.774$	3.3.148.1	None	✓		✓		345.2.b.c	$1$	$1$	\(0\)	\(3\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}+q^{3}+(1-\beta _{1}-\beta _{2})q^{4}+\beta _{2}q^{6}+\cdots\)
1725.2.a.bi	$1725$	$2$	1725.a	1.a	$1$	$3$	$3$	$13.774$	3.3.316.1	None	✓				345.2.a.j	$1$	$1$	\(1\)	\(-3\)	\(0\)	\(-6\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
1725.2.a.bj	$1725$	$2$	1725.a	1.a	$1$	$3$	$3$	$13.774$	3.3.229.1	None	✓	✓			1725.2.a.bf	$1$	$0$	\(2\)	\(3\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{2}+q^{3}+(2+\beta _{1})q^{4}+(1+\cdots)q^{6}+\cdots\)
1725.2.a.bk	$1725$	$2$	1725.a	1.a	$1$	$7$	$7$	$13.774$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		345.2.b.d	$1$	$0$	\(-2\)	\(-7\)	\(0\)	\(5\)		$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}-q^{3}+(2-\beta _{3})q^{4}+\beta _{2}q^{6}+\cdots\)
1725.2.a.bl	$1725$	$2$	1725.a	1.a	$1$	$7$	$7$	$13.774$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		345.2.b.d	$1$	$0$	\(2\)	\(7\)	\(0\)	\(-5\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}+q^{3}+(2-\beta _{3})q^{4}+\beta _{2}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1725))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1725)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(115))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(345))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(575))\)\(^{\oplus 2}\)