Space of modular forms of level 525, weight 6, and trivial character

• Label: 525.6.a
• Level: $525$
• Weight: $6$
• Character orbit: 525.a
• Rep. character: $\chi_{525}(1,\cdot)$
• Character field: $\Q$
• Dimension: $96$
• Newform subspaces: $24$
• Sturm bound: $480$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	525.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 24 \)
Sturm bound:			\(480\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(525))\).

	Total	New	Old
Modular forms	412	96	316
Cusp forms	388	96	292
Eisenstein series	24	0	24

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

\(3\)	\(5\)	\(7\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(12\)
\(+\)	\(+\)	\(-\)	$-$	\(10\)
\(+\)	\(-\)	\(+\)	$-$	\(13\)
\(+\)	\(-\)	\(-\)	$+$	\(13\)
\(-\)	\(+\)	\(+\)	$-$	\(13\)
\(-\)	\(+\)	\(-\)	$+$	\(9\)
\(-\)	\(-\)	\(+\)	$+$	\(11\)
\(-\)	\(-\)	\(-\)	$-$	\(15\)
Plus space			\(+\)	\(45\)
Minus space			\(-\)	\(51\)

Trace form

\( 96 q - 6 q^{2} + 1610 q^{4} - 180 q^{6} - 98 q^{7} - 498 q^{8} + 7776 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(525))\) 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	7
525.6.a.a	$525$	$6$	525.a	1.a	$1$	$1$	$1$	$84.202$	\(\Q\)	None	✓	$1$	$0$	\(-10\)	\(-9\)	\(0\)	\(49\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-10q^{2}-9q^{3}+68q^{4}+90q^{6}+7^{2}q^{7}+\cdots\)
525.6.a.b	$525$	$6$	525.a	1.a	$1$	$1$	$1$	$84.202$	\(\Q\)	None	✓	$1$	$0$	\(-5\)	\(-9\)	\(0\)	\(49\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{2}-9q^{3}-7q^{4}+45q^{6}+7^{2}q^{7}+\cdots\)
525.6.a.c	$525$	$6$	525.a	1.a	$1$	$1$	$1$	$84.202$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(9\)	\(0\)	\(49\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+9q^{3}-31q^{4}-9q^{6}+7^{2}q^{7}+\cdots\)
525.6.a.d	$525$	$6$	525.a	1.a	$1$	$1$	$1$	$84.202$	\(\Q\)	None	✓	$1$	$0$	\(6\)	\(9\)	\(0\)	\(-49\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+6q^{2}+9q^{3}+4q^{4}+54q^{6}-7^{2}q^{7}+\cdots\)
525.6.a.e	$525$	$6$	525.a	1.a	$1$	$2$	$2$	$84.202$	\(\Q(\sqrt{65}) \)	None	✓	$1$	$0$	\(-3\)	\(18\)	\(0\)	\(-98\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+9q^{3}+(-15+3\beta )q^{4}+\cdots\)
525.6.a.f	$525$	$6$	525.a	1.a	$1$	$2$	$2$	$84.202$	\(\Q(\sqrt{233}) \)	None	✓	$1$	$1$	\(-1\)	\(18\)	\(0\)	\(98\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+9q^{3}+(26+\beta )q^{4}-9\beta q^{6}+\cdots\)
525.6.a.g	$525$	$6$	525.a	1.a	$1$	$2$	$2$	$84.202$	\(\Q(\sqrt{73}) \)	None	✓	$1$	$1$	\(1\)	\(-18\)	\(0\)	\(-98\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-9q^{3}+(-14+\beta )q^{4}-9\beta q^{6}+\cdots\)
525.6.a.h	$525$	$6$	525.a	1.a	$1$	$2$	$2$	$84.202$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(4\)	\(18\)	\(0\)	\(98\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{2}+9q^{3}+(4+4\beta )q^{4}+(18+\cdots)q^{6}+\cdots\)
525.6.a.i	$525$	$6$	525.a	1.a	$1$	$2$	$2$	$84.202$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(8\)	\(-18\)	\(0\)	\(98\)		$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(4+\beta )q^{2}-9q^{3}+(4+8\beta )q^{4}+(-6^{2}+\cdots)q^{6}+\cdots\)
525.6.a.j	$525$	$6$	525.a	1.a	$1$	$2$	$2$	$84.202$	\(\Q(\sqrt{65}) \)	None	✓	$1$	$0$	\(13\)	\(-18\)	\(0\)	\(98\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(7-\beta )q^{2}-9q^{3}+(33-13\beta )q^{4}+\cdots\)
525.6.a.k	$525$	$6$	525.a	1.a	$1$	$4$	$4$	$84.202$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(-15\)	\(36\)	\(0\)	\(196\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-4+\beta _{1})q^{2}+9q^{3}+(14-7\beta _{1}+\cdots)q^{4}+\cdots\)
525.6.a.l	$525$	$6$	525.a	1.a	$1$	$4$	$4$	$84.202$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(-10\)	\(-36\)	\(0\)	\(-196\)		$+$	$+$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1})q^{2}-9q^{3}+(30-3\beta _{1}+\cdots)q^{4}+\cdots\)
525.6.a.m	$525$	$6$	525.a	1.a	$1$	$4$	$4$	$84.202$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(-8\)	\(36\)	\(0\)	\(-196\)		$-$	$+$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{2}+9q^{3}+(29-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
525.6.a.n	$525$	$6$	525.a	1.a	$1$	$4$	$4$	$84.202$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(-3\)	\(36\)	\(0\)	\(-196\)		$-$	$-$	$+$	$+$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+9q^{3}+(12+\beta _{3})q^{4}+\cdots\)
525.6.a.o	$525$	$6$	525.a	1.a	$1$	$4$	$4$	$84.202$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(3\)	\(-36\)	\(0\)	\(196\)		$+$	$+$	$-$	$-$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-9q^{3}+(12+\beta _{3})q^{4}+\cdots\)
525.6.a.p	$525$	$6$	525.a	1.a	$1$	$4$	$4$	$84.202$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(15\)	\(-36\)	\(0\)	\(-196\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(4-\beta _{1})q^{2}-9q^{3}+(14-7\beta _{1}+\beta _{3})q^{4}+\cdots\)
525.6.a.q	$525$	$6$	525.a	1.a	$1$	$6$	$6$	$84.202$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(-6\)	\(54\)	\(0\)	\(294\)		$-$	$-$	$-$	$-$	$2\cdot 3\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+9q^{3}+(22+\beta _{2})q^{4}+\cdots\)
525.6.a.r	$525$	$6$	525.a	1.a	$1$	$6$	$6$	$84.202$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(-4\)	\(-54\)	\(0\)	\(294\)		$+$	$-$	$-$	$+$	$2\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}-9q^{3}+(22-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
525.6.a.s	$525$	$6$	525.a	1.a	$1$	$6$	$6$	$84.202$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(4\)	\(54\)	\(0\)	\(-294\)		$-$	$+$	$+$	$-$	$2\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+9q^{3}+(22-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
525.6.a.t	$525$	$6$	525.a	1.a	$1$	$6$	$6$	$84.202$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(6\)	\(-54\)	\(0\)	\(-294\)		$+$	$+$	$+$	$+$	$2\cdot 3\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-9q^{3}+(22+\beta _{2})q^{4}+\cdots\)
525.6.a.u	$525$	$6$	525.a	1.a	$1$	$7$	$7$	$84.202$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(-5\)	\(-63\)	\(0\)	\(343\)		$+$	$-$	$-$	$+$	$2^{9}\cdot 3\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}-9q^{3}+(13-\beta _{1}-\beta _{4}+\cdots)q^{4}+\cdots\)
525.6.a.v	$525$	$6$	525.a	1.a	$1$	$7$	$7$	$84.202$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(5\)	\(63\)	\(0\)	\(-343\)		$-$	$-$	$+$	$+$	$2^{9}\cdot 3\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+9q^{3}+(13-\beta _{1}-\beta _{4}+\cdots)q^{4}+\cdots\)
525.6.a.w	$525$	$6$	525.a	1.a	$1$	$9$	$9$	$84.202$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$0$	\(-5\)	\(-81\)	\(0\)	\(-441\)		$+$	$-$	$+$	$-$	$2^{10}\cdot 5^{5}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}-9q^{3}+(21-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
525.6.a.x	$525$	$6$	525.a	1.a	$1$	$9$	$9$	$84.202$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$0$	\(5\)	\(81\)	\(0\)	\(441\)		$-$	$-$	$-$	$-$	$2^{10}\cdot 5^{5}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+9q^{3}+(21-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(525))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(525)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 2}\)