Space of modular forms of level 6525, weight 2, and trivial character

• Label: 6525.2.a
• Level: $6525$
• Weight: $2$
• Character orbit: 6525.a
• Rep. character: $\chi_{6525}(1,\cdot)$
• Character field: $\Q$
• Dimension: $221$
• Newform subspaces: $58$
• Sturm bound: $1800$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6525.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 58 \)
Sturm bound:			\(1800\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(2\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6525))\).

	Total	New	Old
Modular forms	924	221	703
Cusp forms	877	221	656
Eisenstein series	47	0	47

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

\(3\)	\(5\)	\(29\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(21\)
\(+\)	\(+\)	\(-\)	$-$	\(21\)
\(+\)	\(-\)	\(+\)	$-$	\(23\)
\(+\)	\(-\)	\(-\)	$+$	\(23\)
\(-\)	\(+\)	\(+\)	$-$	\(33\)
\(-\)	\(+\)	\(-\)	$+$	\(30\)
\(-\)	\(-\)	\(+\)	$+$	\(32\)
\(-\)	\(-\)	\(-\)	$-$	\(38\)
Plus space			\(+\)	\(106\)
Minus space			\(-\)	\(115\)

Trace form

\( 221 q - q^{2} + 223 q^{4} - 4 q^{7} - 9 q^{8} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6525))\) 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	29
6525.2.a.a	$6525$	$2$	6525.a	1.a	$1$	$1$	$1$	$52.102$	\(\Q\)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}+2q^{7}-3q^{11}+4q^{13}+\cdots\)
6525.2.a.b	$6525$	$2$	6525.a	1.a	$1$	$1$	$1$	$52.102$	\(\Q\)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}+2q^{7}+3q^{11}-4q^{13}+\cdots\)
6525.2.a.c	$6525$	$2$	6525.a	1.a	$1$	$1$	$1$	$52.102$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}-2q^{7}+3q^{8}+4q^{13}+\cdots\)
6525.2.a.d	$6525$	$2$	6525.a	1.a	$1$	$1$	$1$	$52.102$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+2q^{7}+3q^{8}+6q^{11}+\cdots\)
6525.2.a.e	$6525$	$2$	6525.a	1.a	$1$	$1$	$1$	$52.102$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+4q^{7}+3q^{8}-6q^{13}+\cdots\)
6525.2.a.f	$6525$	$2$	6525.a	1.a	$1$	$1$	$1$	$52.102$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}-2q^{7}-3q^{11}-2q^{13}+4q^{16}+\cdots\)
6525.2.a.g	$6525$	$2$	6525.a	1.a	$1$	$1$	$1$	$52.102$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}-q^{7}+2q^{11}+3q^{13}+4q^{16}+\cdots\)
6525.2.a.h	$6525$	$2$	6525.a	1.a	$1$	$1$	$1$	$52.102$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+q^{7}+2q^{11}-3q^{13}+4q^{16}+\cdots\)
6525.2.a.i	$6525$	$2$	6525.a	1.a	$1$	$1$	$1$	$52.102$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+2q^{7}-q^{11}-6q^{13}+4q^{16}+\cdots\)
6525.2.a.j	$6525$	$2$	6525.a	1.a	$1$	$1$	$1$	$52.102$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-4q^{7}-3q^{8}+4q^{11}+\cdots\)
6525.2.a.k	$6525$	$2$	6525.a	1.a	$1$	$1$	$1$	$52.102$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+2q^{7}-3q^{8}-4q^{13}+\cdots\)
6525.2.a.l	$6525$	$2$	6525.a	1.a	$1$	$1$	$1$	$52.102$	\(\Q\)	None	✓	$1$	$1$	\(2\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}-2q^{7}+3q^{11}+4q^{13}+\cdots\)
6525.2.a.m	$6525$	$2$	6525.a	1.a	$1$	$1$	$1$	$52.102$	\(\Q\)	None	✓	$1$	$0$	\(2\)	\(0\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}+2q^{7}+3q^{11}+4q^{13}+\cdots\)
6525.2.a.n	$6525$	$2$	6525.a	1.a	$1$	$2$	$2$	$52.102$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+\beta q^{7}+3q^{8}+(2-\beta )q^{11}+\cdots\)
6525.2.a.o	$6525$	$2$	6525.a	1.a	$1$	$2$	$2$	$52.102$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+(1-2\beta )q^{4}-2\beta q^{7}+\cdots\)
6525.2.a.p	$6525$	$2$	6525.a	1.a	$1$	$2$	$2$	$52.102$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+(1-2\beta )q^{4}+(2+2\beta )q^{7}+\cdots\)
6525.2.a.q	$6525$	$2$	6525.a	1.a	$1$	$2$	$2$	$52.102$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1+\beta )q^{4}+(1-2\beta )q^{7}+\cdots\)
6525.2.a.r	$6525$	$2$	6525.a	1.a	$1$	$2$	$2$	$52.102$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1+\beta )q^{4}+(1+2\beta )q^{7}+\cdots\)
6525.2.a.s	$6525$	$2$	6525.a	1.a	$1$	$2$	$2$	$52.102$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(6\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1+\beta )q^{4}+3q^{7}+(-1+\cdots)q^{8}+\cdots\)
6525.2.a.t	$6525$	$2$	6525.a	1.a	$1$	$2$	$2$	$52.102$	\(\Q(\sqrt{21}) \)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(3+\beta )q^{4}-q^{7}+(-5-2\beta )q^{8}+\cdots\)
6525.2.a.u	$6525$	$2$	6525.a	1.a	$1$	$2$	$2$	$52.102$	\(\Q(\sqrt{5}) \)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-2q^{4}-2\beta q^{7}-5q^{11}+2\beta q^{13}+\cdots\)
6525.2.a.v	$6525$	$2$	6525.a	1.a	$1$	$2$	$2$	$52.102$	\(\Q(\sqrt{5}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-2q^{4}+2\beta q^{7}+5q^{11}-2\beta q^{13}+\cdots\)
6525.2.a.w	$6525$	$2$	6525.a	1.a	$1$	$2$	$2$	$52.102$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-q^{7}-2\beta q^{8}+2q^{11}-q^{13}+\cdots\)
6525.2.a.x	$6525$	$2$	6525.a	1.a	$1$	$2$	$2$	$52.102$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{7}-2\beta q^{8}+2q^{11}+q^{13}+\cdots\)
6525.2.a.y	$6525$	$2$	6525.a	1.a	$1$	$2$	$2$	$52.102$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+3q^{4}-2q^{7}-\beta q^{8}+2q^{11}+\cdots\)
6525.2.a.z	$6525$	$2$	6525.a	1.a	$1$	$2$	$2$	$52.102$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(1\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+\beta )q^{4}+(1-2\beta )q^{7}+\cdots\)
6525.2.a.ba	$6525$	$2$	6525.a	1.a	$1$	$2$	$2$	$52.102$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(1\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+\beta )q^{4}+(1+2\beta )q^{7}+\cdots\)
6525.2.a.bb	$6525$	$2$	6525.a	1.a	$1$	$2$	$2$	$52.102$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(1\)	\(0\)	\(0\)	\(4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+\beta )q^{4}+(1+2\beta )q^{7}+\cdots\)
6525.2.a.bc	$6525$	$2$	6525.a	1.a	$1$	$2$	$2$	$52.102$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(2+\beta )q^{4}+(-2+2\beta )q^{7}+\cdots\)
6525.2.a.bd	$6525$	$2$	6525.a	1.a	$1$	$2$	$2$	$52.102$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(2\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-\beta q^{7}-3q^{8}+(2-\beta )q^{11}+\cdots\)
6525.2.a.be	$6525$	$2$	6525.a	1.a	$1$	$3$	$3$	$52.102$	3.3.148.1	None	✓	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{4}+(-1-\beta _{1}+\cdots)q^{7}+\cdots\)
6525.2.a.bf	$6525$	$2$	6525.a	1.a	$1$	$3$	$3$	$52.102$	3.3.469.1	None	✓	$1$	$1$	\(1\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(2-\beta _{1}+\beta _{2})q^{7}+\cdots\)
6525.2.a.bg	$6525$	$2$	6525.a	1.a	$1$	$3$	$3$	$52.102$	3.3.229.1	None	✓	$1$	$0$	\(2\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{2}+(2+\beta _{1})q^{4}+(-1-\beta _{1}+\cdots)q^{7}+\cdots\)
6525.2.a.bh	$6525$	$2$	6525.a	1.a	$1$	$3$	$3$	$52.102$	3.3.148.1	None	✓	$1$	$0$	\(3\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+(1+\cdots)q^{7}+\cdots\)
6525.2.a.bi	$6525$	$2$	6525.a	1.a	$1$	$4$	$4$	$52.102$	4.4.2225.1	None	✓	$1$	$1$	\(-3\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
6525.2.a.bj	$6525$	$2$	6525.a	1.a	$1$	$4$	$4$	$52.102$	\(\Q(\zeta_{24})^+\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(\beta _{1}-\beta _{3})q^{7}+\beta _{3}q^{8}+\cdots\)
6525.2.a.bk	$6525$	$2$	6525.a	1.a	$1$	$4$	$4$	$52.102$	\(\Q(\sqrt{3}, \sqrt{11})\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}+q^{4}-2\beta _{1}q^{7}-\beta _{2}q^{8}+(-3+\cdots)q^{11}+\cdots\)
6525.2.a.bl	$6525$	$2$	6525.a	1.a	$1$	$5$	$5$	$52.102$	5.5.246832.1	None	✓	$1$	$1$	\(-3\)	\(0\)	\(0\)	\(8\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(1-\beta _{1}+\beta _{2})q^{4}+\cdots\)
6525.2.a.bm	$6525$	$2$	6525.a	1.a	$1$	$5$	$5$	$52.102$	5.5.240881.1	None	✓	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(6\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1+\beta _{1}-\beta _{3}+\cdots)q^{7}+\cdots\)
6525.2.a.bn	$6525$	$2$	6525.a	1.a	$1$	$5$	$5$	$52.102$	5.5.331312.1	None	✓	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{2}+(1+\beta _{1}-\beta _{3}-\beta _{4})q^{4}+(1+\cdots)q^{7}+\cdots\)
6525.2.a.bo	$6525$	$2$	6525.a	1.a	$1$	$5$	$5$	$52.102$	5.5.294577.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-10\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2}-\beta _{3})q^{4}+(-2-\beta _{1}+\cdots)q^{7}+\cdots\)
6525.2.a.bp	$6525$	$2$	6525.a	1.a	$1$	$5$	$5$	$52.102$	5.5.294577.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(10\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2}-\beta _{3})q^{4}+(2+\beta _{1}+\cdots)q^{7}+\cdots\)
6525.2.a.bq	$6525$	$2$	6525.a	1.a	$1$	$5$	$5$	$52.102$	5.5.240881.1	None	✓	$1$	$1$	\(2\)	\(0\)	\(0\)	\(-6\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1-\beta _{1}+\beta _{3}+\cdots)q^{7}+\cdots\)
6525.2.a.br	$6525$	$2$	6525.a	1.a	$1$	$5$	$5$	$52.102$	5.5.331312.1	None	✓	$1$	$0$	\(2\)	\(0\)	\(0\)	\(-4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{2}+(1+\beta _{1}-\beta _{3}-\beta _{4})q^{4}+(-1+\cdots)q^{7}+\cdots\)
6525.2.a.bs	$6525$	$2$	6525.a	1.a	$1$	$5$	$5$	$52.102$	5.5.246832.1	None	✓	$1$	$1$	\(3\)	\(0\)	\(0\)	\(-8\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1-\beta _{1}+\beta _{2})q^{4}+(-2+\cdots)q^{7}+\cdots\)
6525.2.a.bt	$6525$	$2$	6525.a	1.a	$1$	$6$	$6$	$52.102$	6.6.337383424.1	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(\beta _{1}-\beta _{4}-\beta _{5})q^{7}+\cdots\)
6525.2.a.bu	$6525$	$2$	6525.a	1.a	$1$	$7$	$7$	$52.102$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(\beta _{1}-\beta _{2}+\beta _{6})q^{7}+\cdots\)
6525.2.a.bv	$6525$	$2$	6525.a	1.a	$1$	$7$	$7$	$52.102$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-10\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(-1+\beta _{5})q^{7}+\cdots\)
6525.2.a.bw	$6525$	$2$	6525.a	1.a	$1$	$7$	$7$	$52.102$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-10\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(-1+\beta _{5})q^{7}+\cdots\)
6525.2.a.bx	$6525$	$2$	6525.a	1.a	$1$	$7$	$7$	$52.102$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(2\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
6525.2.a.by	$6525$	$2$	6525.a	1.a	$1$	$8$	$8$	$52.102$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(\beta _{1}-\beta _{4}-\beta _{5}+\cdots)q^{7}+\cdots\)
6525.2.a.bz	$6525$	$2$	6525.a	1.a	$1$	$8$	$8$	$52.102$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(2\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(-\beta _{1}+\beta _{4}+\cdots)q^{7}+\cdots\)
6525.2.a.ca	$6525$	$2$	6525.a	1.a	$1$	$9$	$9$	$52.102$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(-1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{4}q^{7}+(-1+\cdots)q^{8}+\cdots\)
6525.2.a.cb	$6525$	$2$	6525.a	1.a	$1$	$9$	$9$	$52.102$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{4}q^{7}+(-1+\cdots)q^{8}+\cdots\)
6525.2.a.cc	$6525$	$2$	6525.a	1.a	$1$	$9$	$9$	$52.102$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$0$	\(2\)	\(0\)	\(0\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{4}q^{7}+(1+\beta _{2}+\cdots)q^{8}+\cdots\)
6525.2.a.cd	$6525$	$2$	6525.a	1.a	$1$	$9$	$9$	$52.102$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$0$	\(2\)	\(0\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{4}q^{7}+(1+\beta _{2}+\cdots)q^{8}+\cdots\)
6525.2.a.ce	$6525$	$2$	6525.a	1.a	$1$	$12$	$12$	$52.102$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{4}q^{7}+(\beta _{1}+\cdots)q^{8}+\cdots\)
6525.2.a.cf	$6525$	$2$	6525.a	1.a	$1$	$12$	$12$	$52.102$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{4}q^{7}+(\beta _{1}+\cdots)q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6525))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6525)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(87))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(145))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(261))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(435))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(725))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1305))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2175))\)\(^{\oplus 2}\)