Space of modular forms of level 7800, weight 2, and trivial character

• Label: 7800.2.a
• Level: $7800$
• Weight: $2$
• Character orbit: 7800.a
• Rep. character: $\chi_{7800}(1,\cdot)$
• Character field: $\Q$
• Dimension: $114$
• Newform subspaces: $53$
• Sturm bound: $3360$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7800.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 53 \)
Sturm bound:			\(3360\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(7\), \(11\), \(17\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(7800))\).

	Total	New	Old
Modular forms	1728	114	1614
Cusp forms	1633	114	1519
Eisenstein series	95	0	95

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

\(2\)	\(3\)	\(5\)	\(13\)	Fricke	Dim.
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)	\(7\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)	\(7\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)	\(7\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)	\(7\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)	\(8\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)	\(5\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)	\(7\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)	\(9\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)	\(8\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)	\(5\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)	\(7\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)	\(9\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)	\(7\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)	\(7\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)	\(7\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)	\(7\)
Plus space				\(+\)	\(52\)
Minus space				\(-\)	\(62\)

Trace form

\( 114 q + 4 q^{7} + 114 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(7800))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	5	13
7800.2.a.a	$7800$	$2$	7800.a	1.a	$1$	$1$	$1$	$62.283$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-4\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-4q^{7}+q^{9}-q^{13}-2q^{17}+\cdots\)
7800.2.a.b	$7800$	$2$	7800.a	1.a	$1$	$1$	$1$	$62.283$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-1\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}+3q^{11}+q^{13}+7q^{17}+\cdots\)
7800.2.a.c	$7800$	$2$	7800.a	1.a	$1$	$1$	$1$	$62.283$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(0\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{9}-4q^{11}+q^{13}+8q^{17}+\cdots\)
7800.2.a.d	$7800$	$2$	7800.a	1.a	$1$	$1$	$1$	$62.283$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{9}-q^{13}-2q^{17}-4q^{19}+\cdots\)
7800.2.a.e	$7800$	$2$	7800.a	1.a	$1$	$1$	$1$	$62.283$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{9}+q^{13}-2q^{17}-4q^{23}+\cdots\)
7800.2.a.f	$7800$	$2$	7800.a	1.a	$1$	$1$	$1$	$62.283$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{9}+6q^{11}+q^{13}-2q^{17}+\cdots\)
7800.2.a.g	$7800$	$2$	7800.a	1.a	$1$	$1$	$1$	$62.283$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-5q^{11}-q^{13}-3q^{17}+\cdots\)
7800.2.a.h	$7800$	$2$	7800.a	1.a	$1$	$1$	$1$	$62.283$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(2\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{7}+q^{9}-q^{13}+4q^{17}+\cdots\)
7800.2.a.i	$7800$	$2$	7800.a	1.a	$1$	$1$	$1$	$62.283$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(3\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+3q^{7}+q^{9}-3q^{11}+q^{13}+\cdots\)
7800.2.a.j	$7800$	$2$	7800.a	1.a	$1$	$1$	$1$	$62.283$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(4\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+4q^{7}+q^{9}-2q^{11}+q^{13}+\cdots\)
7800.2.a.k	$7800$	$2$	7800.a	1.a	$1$	$1$	$1$	$62.283$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(4\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+4q^{7}+q^{9}+4q^{11}-q^{13}+\cdots\)
7800.2.a.l	$7800$	$2$	7800.a	1.a	$1$	$1$	$1$	$62.283$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(5\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+5q^{7}+q^{9}+5q^{11}-q^{13}+\cdots\)
7800.2.a.m	$7800$	$2$	7800.a	1.a	$1$	$1$	$1$	$62.283$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-5\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-5q^{7}+q^{9}+5q^{11}+q^{13}+\cdots\)
7800.2.a.n	$7800$	$2$	7800.a	1.a	$1$	$1$	$1$	$62.283$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-4\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-4q^{7}+q^{9}-q^{13}-2q^{17}+\cdots\)
7800.2.a.o	$7800$	$2$	7800.a	1.a	$1$	$1$	$1$	$62.283$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-4\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-4q^{7}+q^{9}+4q^{11}-q^{13}+\cdots\)
7800.2.a.p	$7800$	$2$	7800.a	1.a	$1$	$1$	$1$	$62.283$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{7}+q^{9}+4q^{11}+q^{13}+\cdots\)
7800.2.a.q	$7800$	$2$	7800.a	1.a	$1$	$1$	$1$	$62.283$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(0\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{9}-4q^{11}-q^{13}-8q^{17}+\cdots\)
7800.2.a.r	$7800$	$2$	7800.a	1.a	$1$	$1$	$1$	$62.283$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(0\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{9}-4q^{11}-q^{13}-2q^{17}+\cdots\)
7800.2.a.s	$7800$	$2$	7800.a	1.a	$1$	$1$	$1$	$62.283$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{9}-2q^{11}+q^{13}-2q^{17}+\cdots\)
7800.2.a.t	$7800$	$2$	7800.a	1.a	$1$	$1$	$1$	$62.283$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(0\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{9}-q^{13}+6q^{17}+4q^{19}+\cdots\)
7800.2.a.u	$7800$	$2$	7800.a	1.a	$1$	$1$	$1$	$62.283$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{9}+4q^{11}-q^{13}+2q^{17}+\cdots\)
7800.2.a.v	$7800$	$2$	7800.a	1.a	$1$	$1$	$1$	$62.283$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(1\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}+3q^{11}-q^{13}-7q^{17}+\cdots\)
7800.2.a.w	$7800$	$2$	7800.a	1.a	$1$	$1$	$1$	$62.283$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(4\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{7}+q^{9}-2q^{11}+q^{13}+\cdots\)
7800.2.a.x	$7800$	$2$	7800.a	1.a	$1$	$1$	$1$	$62.283$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(5\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+5q^{7}+q^{9}+q^{11}-q^{13}+3q^{17}+\cdots\)
7800.2.a.y	$7800$	$2$	7800.a	1.a	$1$	$2$	$2$	$62.283$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta q^{7}+q^{9}+\beta q^{11}+q^{13}+\cdots\)
7800.2.a.z	$7800$	$2$	7800.a	1.a	$1$	$2$	$2$	$62.283$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(1+\beta )q^{7}+q^{9}+(-1+\beta )q^{11}+\cdots\)
7800.2.a.ba	$7800$	$2$	7800.a	1.a	$1$	$2$	$2$	$62.283$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-2\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-1+\beta )q^{7}+q^{9}+(-1-\beta )q^{11}+\cdots\)
7800.2.a.bb	$7800$	$2$	7800.a	1.a	$1$	$2$	$2$	$62.283$	\(\Q(\sqrt{33}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-\beta q^{7}+q^{9}-\beta q^{11}+q^{13}+\cdots\)
7800.2.a.bc	$7800$	$2$	7800.a	1.a	$1$	$2$	$2$	$62.283$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta q^{7}+q^{9}-\beta q^{11}-q^{13}+\cdots\)
7800.2.a.bd	$7800$	$2$	7800.a	1.a	$1$	$2$	$2$	$62.283$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$1$	\(0\)	\(2\)	\(0\)	\(1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta q^{7}+q^{9}+(-4+\beta )q^{11}+\cdots\)
7800.2.a.be	$7800$	$2$	7800.a	1.a	$1$	$2$	$2$	$62.283$	\(\Q(\sqrt{41}) \)	None	✓	$1$	$1$	\(0\)	\(2\)	\(0\)	\(1\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta q^{7}+q^{9}+(-2-\beta )q^{11}+\cdots\)
7800.2.a.bf	$7800$	$2$	7800.a	1.a	$1$	$3$	$3$	$62.283$	3.3.1849.1	None	✓	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-5\)		$+$	$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}+(-2+\beta _{1})q^{7}+q^{9}+(1+\beta _{2})q^{11}+\cdots\)
7800.2.a.bg	$7800$	$2$	7800.a	1.a	$1$	$3$	$3$	$62.283$	3.3.788.1	None	✓	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-1\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-\beta _{1}q^{7}+q^{9}+(-1-\beta _{1}+\beta _{2})q^{11}+\cdots\)
7800.2.a.bh	$7800$	$2$	7800.a	1.a	$1$	$3$	$3$	$62.283$	3.3.3732.1	None	✓	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(-1\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-\beta _{2}q^{7}+q^{9}+(2-\beta _{1})q^{11}+\cdots\)
7800.2.a.bi	$7800$	$2$	7800.a	1.a	$1$	$3$	$3$	$62.283$	3.3.940.1	None	✓	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(-1\)		$+$	$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}-\beta _{2}q^{7}+q^{9}+(2+\beta _{1})q^{11}+\cdots\)
7800.2.a.bj	$7800$	$2$	7800.a	1.a	$1$	$3$	$3$	$62.283$	3.3.568.1	None	✓	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(0\)		$+$	$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{9}+(-2-\beta _{1})q^{11}-q^{13}+\cdots\)
7800.2.a.bk	$7800$	$2$	7800.a	1.a	$1$	$3$	$3$	$62.283$	3.3.2700.1	None	✓	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(0\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta _{1}q^{7}+q^{9}-\beta _{1}q^{11}-q^{13}+\cdots\)
7800.2.a.bl	$7800$	$2$	7800.a	1.a	$1$	$3$	$3$	$62.283$	3.3.148.1	None	✓	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(1\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(\beta _{1}+2\beta _{2})q^{7}+q^{9}+(-1+\beta _{2})q^{11}+\cdots\)
7800.2.a.bm	$7800$	$2$	7800.a	1.a	$1$	$3$	$3$	$62.283$	3.3.568.1	None	✓	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(2\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(1-\beta _{1})q^{7}+q^{9}+(1+\beta _{1}+2\beta _{2})q^{11}+\cdots\)
7800.2.a.bn	$7800$	$2$	7800.a	1.a	$1$	$3$	$3$	$62.283$	3.3.568.1	None	✓	$1$	$0$	\(0\)	\(3\)	\(0\)	\(-2\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-1+\beta _{1})q^{7}+q^{9}+(1+\beta _{1}+\cdots)q^{11}+\cdots\)
7800.2.a.bo	$7800$	$2$	7800.a	1.a	$1$	$3$	$3$	$62.283$	3.3.148.1	None	✓	$1$	$1$	\(0\)	\(3\)	\(0\)	\(-1\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-\beta _{1}-2\beta _{2})q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
7800.2.a.bp	$7800$	$2$	7800.a	1.a	$1$	$3$	$3$	$62.283$	3.3.568.1	None	✓	$1$	$1$	\(0\)	\(3\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{9}+(-2-\beta _{1})q^{11}+q^{13}+\cdots\)
7800.2.a.bq	$7800$	$2$	7800.a	1.a	$1$	$3$	$3$	$62.283$	3.3.2700.1	None	✓	$1$	$0$	\(0\)	\(3\)	\(0\)	\(0\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-\beta _{1}q^{7}+q^{9}-\beta _{1}q^{11}+q^{13}+\cdots\)
7800.2.a.br	$7800$	$2$	7800.a	1.a	$1$	$3$	$3$	$62.283$	3.3.788.1	None	✓	$1$	$1$	\(0\)	\(3\)	\(0\)	\(1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta _{1}q^{7}+q^{9}+(-1-\beta _{1}+\beta _{2})q^{11}+\cdots\)
7800.2.a.bs	$7800$	$2$	7800.a	1.a	$1$	$3$	$3$	$62.283$	3.3.3732.1	None	✓	$1$	$0$	\(0\)	\(3\)	\(0\)	\(1\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta _{2}q^{7}+q^{9}+(2-\beta _{1})q^{11}+\cdots\)
7800.2.a.bt	$7800$	$2$	7800.a	1.a	$1$	$4$	$4$	$62.283$	4.4.30056.1	None	✓	$1$	$1$	\(0\)	\(-4\)	\(0\)	\(-7\)		$+$	$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}+(-2-\beta _{1})q^{7}+q^{9}+(\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)
7800.2.a.bu	$7800$	$2$	7800.a	1.a	$1$	$4$	$4$	$62.283$	4.4.461844.1	None	✓	$1$	$0$	\(0\)	\(-4\)	\(0\)	\(-3\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-1+\beta _{1})q^{7}+q^{9}+(1+\beta _{1}+\cdots)q^{11}+\cdots\)
7800.2.a.bv	$7800$	$2$	7800.a	1.a	$1$	$4$	$4$	$62.283$	4.4.10304.1	None	✓	$1$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)		$-$	$+$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{3}-\beta _{3}q^{7}+q^{9}+(-2-\beta _{1})q^{11}+\cdots\)
7800.2.a.bw	$7800$	$2$	7800.a	1.a	$1$	$4$	$4$	$62.283$	4.4.10304.1	None	✓	$1$	$1$	\(0\)	\(4\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta _{3}q^{7}+q^{9}+(-2-\beta _{1})q^{11}+\cdots\)
7800.2.a.bx	$7800$	$2$	7800.a	1.a	$1$	$4$	$4$	$62.283$	4.4.461844.1	None	✓	$1$	$0$	\(0\)	\(4\)	\(0\)	\(3\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(1-\beta _{1})q^{7}+q^{9}+(1+\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)
7800.2.a.by	$7800$	$2$	7800.a	1.a	$1$	$4$	$4$	$62.283$	4.4.30056.1	None	✓	$1$	$0$	\(0\)	\(4\)	\(0\)	\(7\)		$-$	$-$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+(2+\beta _{1})q^{7}+q^{9}+(\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)
7800.2.a.bz	$7800$	$2$	7800.a	1.a	$1$	$5$	$5$	$62.283$	5.5.504568.1	None	✓	$1$	$1$	\(0\)	\(-5\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{3}-\beta _{1}q^{7}+q^{9}+(1-\beta _{3})q^{11}+\cdots\)
7800.2.a.ca	$7800$	$2$	7800.a	1.a	$1$	$5$	$5$	$62.283$	5.5.504568.1	None	✓	$1$	$0$	\(0\)	\(5\)	\(0\)	\(1\)		$+$	$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta _{1}q^{7}+q^{9}+(1-\beta _{3})q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(7800))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(7800)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(130))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(156))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(195))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(260))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(312))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(325))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(390))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(520))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(600))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(650))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(780))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(975))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1300))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1560))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1950))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2600))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3900))\)\(^{\oplus 2}\)