Space of modular forms of level 5625, weight 2, and trivial character

• Label: 5625.2.a
• Level: $5625$
• Weight: $2$
• Character orbit: 5625.a
• Rep. character: $\chi_{5625}(1,\cdot)$
• Character field: $\Q$
• Dimension: $192$
• Newform subspaces: $32$
• Sturm bound: $1500$
• Trace bound: $16$

Defining parameters

Level:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 32 \)
Sturm bound:			\(1500\)
Trace bound:			\(16\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(5625))\).

	Total	New	Old
Modular forms	810	208	602
Cusp forms	691	192	499
Eisenstein series	119	16	103

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

\(3\)	\(5\)	Fricke	Dim.
\(+\)	\(+\)	\(+\)	\(36\)
\(+\)	\(-\)	\(-\)	\(44\)
\(-\)	\(+\)	\(-\)	\(58\)
\(-\)	\(-\)	\(+\)	\(54\)
Plus space		\(+\)	\(90\)
Minus space		\(-\)	\(102\)

Trace form

\( 192 q + 184 q^{4} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(5625))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5
5625.2.a.a	$5625$	$2$	5625.a	1.a	$1$	$2$	$2$	$44.916$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+(-2+4\beta )q^{7}+3q^{8}+\cdots\)
5625.2.a.b	$5625$	$2$	5625.a	1.a	$1$	$2$	$2$	$44.916$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1+\beta )q^{4}-2q^{7}+(-1+\cdots)q^{8}+\cdots\)
5625.2.a.c	$5625$	$2$	5625.a	1.a	$1$	$2$	$2$	$44.916$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1+\beta )q^{4}+(-1+3\beta )q^{7}+\cdots\)
5625.2.a.d	$5625$	$2$	5625.a	1.a	$1$	$2$	$2$	$44.916$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1+\beta )q^{4}+(1-\beta )q^{7}+(-1+\cdots)q^{8}+\cdots\)
5625.2.a.e	$5625$	$2$	5625.a	1.a	$1$	$2$	$2$	$44.916$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+\beta )q^{4}+(1-3\beta )q^{7}+\cdots\)
5625.2.a.f	$5625$	$2$	5625.a	1.a	$1$	$2$	$2$	$44.916$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+\beta )q^{4}+(-1+\beta )q^{7}+\cdots\)
5625.2.a.g	$5625$	$2$	5625.a	1.a	$1$	$2$	$2$	$44.916$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+\beta )q^{4}+2q^{7}+(1-2\beta )q^{8}+\cdots\)
5625.2.a.h	$5625$	$2$	5625.a	1.a	$1$	$2$	$2$	$44.916$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(2\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+(2-4\beta )q^{7}-3q^{8}-2\beta q^{11}+\cdots\)
5625.2.a.i	$5625$	$2$	5625.a	1.a	$1$	$4$	$4$	$44.916$	4.4.5125.1	None	✓	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(2+\beta _{1}+\beta _{2})q^{4}+(-1+\beta _{1}+\cdots)q^{7}+\cdots\)
5625.2.a.j	$5625$	$2$	5625.a	1.a	$1$	$4$	$4$	$44.916$	\(\Q(\zeta_{15})^+\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(5\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(\beta _{2}+\beta _{3})q^{2}+\beta _{1}q^{4}+(2+\beta _{2}+2\beta _{3})q^{7}+\cdots\)
5625.2.a.k	$5625$	$2$	5625.a	1.a	$1$	$4$	$4$	$44.916$	\(\Q(\zeta_{15})^+\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-5\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{4}+(-\beta _{2}+2\beta _{3})q^{7}+(-1-3\beta _{1}+\cdots)q^{13}+\cdots\)
5625.2.a.l	$5625$	$2$	5625.a	1.a	$1$	$4$	$4$	$44.916$	\(\Q(\zeta_{15})^+\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(5\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{4}+(\beta _{2}-2\beta _{3})q^{7}+(1+3\beta _{1}+\beta _{3})q^{13}+\cdots\)
5625.2.a.m	$5625$	$2$	5625.a	1.a	$1$	$4$	$4$	$44.916$	\(\Q(\zeta_{15})^+\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-5\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-\beta _{2}-\beta _{3})q^{2}+\beta _{1}q^{4}+(-2-\beta _{2}+\cdots)q^{7}+\cdots\)
5625.2.a.n	$5625$	$2$	5625.a	1.a	$1$	$4$	$4$	$44.916$	4.4.5125.1	None	✓	$1$	$0$	\(2\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{1}+\beta _{2})q^{4}+(1-\beta _{1}+\cdots)q^{7}+\cdots\)
5625.2.a.o	$5625$	$2$	5625.a	1.a	$1$	$6$	$6$	$44.916$	6.6.46840000.1	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(-\beta _{2}+\beta _{3}+\cdots)q^{7}+\cdots\)
5625.2.a.p	$5625$	$2$	5625.a	1.a	$1$	$6$	$6$	$44.916$	6.6.44400625.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1-\beta _{3}+\beta _{4})q^{4}+(-1+\beta _{1}+\cdots)q^{7}+\cdots\)
5625.2.a.q	$5625$	$2$	5625.a	1.a	$1$	$6$	$6$	$44.916$	6.6.44400625.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1-\beta _{3}+\beta _{4})q^{4}+(1-\beta _{1}+\cdots)q^{7}+\cdots\)
5625.2.a.r	$5625$	$2$	5625.a	1.a	$1$	$6$	$6$	$44.916$	6.6.46840000.1	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(\beta _{2}-\beta _{3})q^{7}+\cdots\)
5625.2.a.s	$5625$	$2$	5625.a	1.a	$1$	$8$	$8$	$44.916$	8.8.6152203125.1	None	✓	$1$	$1$	\(-5\)	\(0\)	\(0\)	\(10\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
5625.2.a.t	$5625$	$2$	5625.a	1.a	$1$	$8$	$8$	$44.916$	8.8.5444000000.1	None	✓	$1$	$1$	\(-4\)	\(0\)	\(0\)	\(8\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(1-\beta _{1}+\beta _{2})q^{4}+\cdots\)
5625.2.a.u	$5625$	$2$	5625.a	1.a	$1$	$8$	$8$	$44.916$	8.8.\(\cdots\).1	None	✓	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(12\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{4}+\beta _{5})q^{4}+(1+\beta _{2}+\cdots)q^{7}+\cdots\)
5625.2.a.v	$5625$	$2$	5625.a	1.a	$1$	$8$	$8$	$44.916$	8.8.\(\cdots\).1	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1+\beta _{2}+\beta _{4}+\cdots)q^{7}+\cdots\)
5625.2.a.w	$5625$	$2$	5625.a	1.a	$1$	$8$	$8$	$44.916$	8.8.\(\cdots\).1	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1-\beta _{2}-\beta _{4}+\cdots)q^{7}+\cdots\)
5625.2.a.x	$5625$	$2$	5625.a	1.a	$1$	$8$	$8$	$44.916$	8.8.\(\cdots\).2	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-\beta _{1}-\beta _{6}+\cdots)q^{7}+\cdots\)
5625.2.a.y	$5625$	$2$	5625.a	1.a	$1$	$8$	$8$	$44.916$	8.8.\(\cdots\).2	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-10\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(-\beta _{3}+2\beta _{5}+\cdots)q^{7}+\cdots\)
5625.2.a.z	$5625$	$2$	5625.a	1.a	$1$	$8$	$8$	$44.916$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-10\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(-1-\beta _{2}+\beta _{4}+\cdots)q^{7}+\cdots\)
5625.2.a.ba	$5625$	$2$	5625.a	1.a	$1$	$8$	$8$	$44.916$	8.8.\(\cdots\).2	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(10\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(\beta _{3}-2\beta _{5})q^{7}+\cdots\)
5625.2.a.bb	$5625$	$2$	5625.a	1.a	$1$	$8$	$8$	$44.916$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(10\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(1+\beta _{2}-\beta _{4}+\cdots)q^{7}+\cdots\)
5625.2.a.bc	$5625$	$2$	5625.a	1.a	$1$	$8$	$8$	$44.916$	8.8.\(\cdots\).1	None	✓	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-12\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{4}+\beta _{5})q^{4}+(-1-\beta _{2}+\cdots)q^{7}+\cdots\)
5625.2.a.bd	$5625$	$2$	5625.a	1.a	$1$	$8$	$8$	$44.916$	8.8.5444000000.1	None	✓	$1$	$1$	\(4\)	\(0\)	\(0\)	\(-8\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{4}+(-1-\beta _{3}+\cdots)q^{7}+\cdots\)
5625.2.a.be	$5625$	$2$	5625.a	1.a	$1$	$8$	$8$	$44.916$	8.8.6152203125.1	None	✓	$1$	$0$	\(5\)	\(0\)	\(0\)	\(-10\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+(-1+\cdots)q^{7}+\cdots\)
5625.2.a.bf	$5625$	$2$	5625.a	1.a	$1$	$24$	$24$	$44.916$		None	✓	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(5625))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(5625)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(125))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(375))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(625))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1125))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1875))\)\(^{\oplus 2}\)