Space of modular forms of level 8400, weight 2, and trivial character

• Label: 8400.2.a
• Level: $8400$
• Weight: $2$
• Character orbit: 8400.a
• Rep. character: $\chi_{8400}(1,\cdot)$
• Character field: $\Q$
• Dimension: $114$
• Newform subspaces: $90$
• Sturm bound: $3840$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8400.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 90 \)
Sturm bound:			\(3840\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(11\), \(13\), \(17\), \(19\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(8400))\).

	Total	New	Old
Modular forms	1992	114	1878
Cusp forms	1849	114	1735
Eisenstein series	143	0	143

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

\(2\)	\(3\)	\(5\)	\(7\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	$+$	\(6\)
\(+\)	\(+\)	\(+\)	\(-\)	$-$	\(7\)
\(+\)	\(+\)	\(-\)	\(+\)	$-$	\(9\)
\(+\)	\(+\)	\(-\)	\(-\)	$+$	\(7\)
\(+\)	\(-\)	\(+\)	\(+\)	$-$	\(7\)
\(+\)	\(-\)	\(+\)	\(-\)	$+$	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)	$+$	\(7\)
\(+\)	\(-\)	\(-\)	\(-\)	$-$	\(9\)
\(-\)	\(+\)	\(+\)	\(+\)	$-$	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)	$+$	\(8\)
\(-\)	\(+\)	\(-\)	\(+\)	$+$	\(7\)
\(-\)	\(+\)	\(-\)	\(-\)	$-$	\(7\)
\(-\)	\(-\)	\(+\)	\(+\)	$+$	\(5\)
\(-\)	\(-\)	\(+\)	\(-\)	$-$	\(9\)
\(-\)	\(-\)	\(-\)	\(+\)	$-$	\(9\)
\(-\)	\(-\)	\(-\)	\(-\)	$+$	\(5\)
Plus space				\(+\)	\(51\)
Minus space				\(-\)	\(63\)

Trace form

\( 114 q + 2 q^{7} + 114 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(8400))\) 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	7
8400.2.a.a	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}-6q^{11}-q^{13}+3q^{17}+\cdots\)
8400.2.a.b	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}-4q^{11}+2q^{13}+\cdots\)
8400.2.a.c	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-1\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}-2q^{11}-4q^{13}+\cdots\)
8400.2.a.d	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-1\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}-2q^{11}-q^{13}+3q^{17}+\cdots\)
8400.2.a.e	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-1\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}-2q^{11}+6q^{13}+\cdots\)
8400.2.a.f	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}-2q^{13}-2q^{17}+\cdots\)
8400.2.a.g	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}+q^{11}-4q^{13}-2q^{17}+\cdots\)
8400.2.a.h	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-1\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}+q^{11}+2q^{13}-6q^{19}+\cdots\)
8400.2.a.i	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}+3q^{11}-4q^{13}+\cdots\)
8400.2.a.j	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-1\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}+3q^{11}+2q^{13}+\cdots\)
8400.2.a.k	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-1\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}+4q^{11}-6q^{13}+\cdots\)
8400.2.a.l	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}+4q^{11}+2q^{13}+\cdots\)
8400.2.a.m	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-1\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}+4q^{11}+2q^{13}+\cdots\)
8400.2.a.n	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-5q^{11}-2q^{13}+\cdots\)
8400.2.a.o	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-4q^{11}-2q^{13}+\cdots\)
8400.2.a.p	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-4q^{11}+2q^{13}+\cdots\)
8400.2.a.q	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-4q^{11}+2q^{13}+\cdots\)
8400.2.a.r	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-3q^{11}-2q^{17}+\cdots\)
8400.2.a.s	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-2q^{11}-3q^{13}+\cdots\)
8400.2.a.t	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-2q^{11}+q^{13}-q^{17}+\cdots\)
8400.2.a.u	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-2q^{11}+2q^{13}+\cdots\)
8400.2.a.v	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-2q^{11}+2q^{13}+\cdots\)
8400.2.a.w	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-2q^{11}+6q^{13}+\cdots\)
8400.2.a.x	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-q^{11}-4q^{13}-2q^{17}+\cdots\)
8400.2.a.y	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-6q^{13}+2q^{17}+\cdots\)
8400.2.a.z	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}+2q^{13}-6q^{17}+\cdots\)
8400.2.a.ba	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}+q^{11}-2q^{13}+8q^{17}+\cdots\)
8400.2.a.bb	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}+2q^{11}-7q^{13}+\cdots\)
8400.2.a.bc	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}+2q^{11}-4q^{13}+\cdots\)
8400.2.a.bd	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}+2q^{11}-2q^{13}+\cdots\)
8400.2.a.be	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}+2q^{11}+5q^{13}+\cdots\)
8400.2.a.bf	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}+3q^{11}+2q^{13}+\cdots\)
8400.2.a.bg	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}+4q^{11}-2q^{13}+\cdots\)
8400.2.a.bh	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}+4q^{11}+6q^{13}+\cdots\)
8400.2.a.bi	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}+5q^{11}+4q^{13}+\cdots\)
8400.2.a.bj	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}+6q^{11}-2q^{13}+\cdots\)
8400.2.a.bk	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}-5q^{11}+2q^{13}+\cdots\)
8400.2.a.bl	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}-4q^{11}+2q^{13}+\cdots\)
8400.2.a.bm	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}-4q^{11}+2q^{13}+\cdots\)
8400.2.a.bn	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}-4q^{11}+2q^{13}+\cdots\)
8400.2.a.bo	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}-3q^{11}+2q^{17}+\cdots\)
8400.2.a.bp	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}-2q^{11}-6q^{13}+\cdots\)
8400.2.a.bq	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}-2q^{11}-4q^{13}+\cdots\)
8400.2.a.br	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}-2q^{11}-2q^{13}+\cdots\)
8400.2.a.bs	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}-2q^{11}-2q^{13}+\cdots\)
8400.2.a.bt	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}-2q^{11}-q^{13}+q^{17}+\cdots\)
8400.2.a.bu	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}-2q^{11}+3q^{13}+\cdots\)
8400.2.a.bv	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}-q^{11}+4q^{13}+2q^{17}+\cdots\)
8400.2.a.bw	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}-6q^{13}+2q^{17}+\cdots\)
8400.2.a.bx	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}+2q^{13}-6q^{17}+\cdots\)
8400.2.a.by	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}+q^{11}+2q^{13}-8q^{17}+\cdots\)
8400.2.a.bz	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}+2q^{11}-5q^{13}+\cdots\)
8400.2.a.ca	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}+2q^{11}+2q^{13}+\cdots\)
8400.2.a.cb	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}+2q^{11}+7q^{13}+\cdots\)
8400.2.a.cc	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}+3q^{11}-2q^{13}+\cdots\)
8400.2.a.cd	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}+4q^{11}-6q^{13}+\cdots\)
8400.2.a.ce	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}+4q^{11}+2q^{13}+\cdots\)
8400.2.a.cf	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}+4q^{11}+6q^{13}+\cdots\)
8400.2.a.cg	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}+5q^{11}-4q^{13}+\cdots\)
8400.2.a.ch	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}+6q^{11}+2q^{13}+\cdots\)
8400.2.a.ci	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}-6q^{11}+q^{13}-3q^{17}+\cdots\)
8400.2.a.cj	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}-6q^{11}+4q^{13}+\cdots\)
8400.2.a.ck	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}-2q^{11}+q^{13}-3q^{17}+\cdots\)
8400.2.a.cl	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(1\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}-2q^{13}-2q^{17}+\cdots\)
8400.2.a.cm	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}-2q^{13}+6q^{17}+\cdots\)
8400.2.a.cn	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}-2q^{13}+6q^{17}+\cdots\)
8400.2.a.co	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}+6q^{13}-2q^{17}+\cdots\)
8400.2.a.cp	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}+q^{11}-2q^{13}-6q^{19}+\cdots\)
8400.2.a.cq	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(1\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}+q^{11}+4q^{13}+2q^{17}+\cdots\)
8400.2.a.cr	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(1\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}+3q^{11}-2q^{13}+\cdots\)
8400.2.a.cs	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}+3q^{11}+4q^{13}+\cdots\)
8400.2.a.ct	$8400$	$2$	8400.a	1.a	$1$	$1$	$1$	$67.074$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}+6q^{11}-2q^{13}+\cdots\)
8400.2.a.cu	$8400$	$2$	8400.a	1.a	$1$	$2$	$2$	$67.074$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-2\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}+(-1-\beta )q^{11}+(2+\cdots)q^{13}+\cdots\)
8400.2.a.cv	$8400$	$2$	8400.a	1.a	$1$	$2$	$2$	$67.074$	\(\Q(\sqrt{73}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}-\beta q^{11}+(1+\beta )q^{13}+\cdots\)
8400.2.a.cw	$8400$	$2$	8400.a	1.a	$1$	$2$	$2$	$67.074$	\(\Q(\sqrt{13}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}+3q^{11}+(1+\beta )q^{13}+\cdots\)
8400.2.a.cx	$8400$	$2$	8400.a	1.a	$1$	$2$	$2$	$67.074$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}+(-2-\beta )q^{11}-\beta q^{13}+\cdots\)
8400.2.a.cy	$8400$	$2$	8400.a	1.a	$1$	$2$	$2$	$67.074$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}+(1+2\beta )q^{11}+(3+\cdots)q^{13}+\cdots\)
8400.2.a.cz	$8400$	$2$	8400.a	1.a	$1$	$2$	$2$	$67.074$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}+2q^{11}-\beta q^{13}+\cdots\)
8400.2.a.da	$8400$	$2$	8400.a	1.a	$1$	$2$	$2$	$67.074$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-2\)		$-$	$-$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}+(1+2\beta )q^{11}+(-3+\cdots)q^{13}+\cdots\)
8400.2.a.db	$8400$	$2$	8400.a	1.a	$1$	$2$	$2$	$67.074$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-2\)		$+$	$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}+2q^{11}-\beta q^{13}+\cdots\)
8400.2.a.dc	$8400$	$2$	8400.a	1.a	$1$	$2$	$2$	$67.074$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$1$	\(0\)	\(2\)	\(0\)	\(2\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}+(-1-\beta )q^{11}+(-2+\cdots)q^{13}+\cdots\)
8400.2.a.dd	$8400$	$2$	8400.a	1.a	$1$	$2$	$2$	$67.074$	\(\Q(\sqrt{73}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(0\)	\(2\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}-\beta q^{11}+(-1-\beta )q^{13}+\cdots\)
8400.2.a.de	$8400$	$2$	8400.a	1.a	$1$	$2$	$2$	$67.074$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(0\)	\(2\)	\(0\)	\(2\)		$+$	$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}+\beta q^{11}-2q^{13}+\cdots\)
8400.2.a.df	$8400$	$2$	8400.a	1.a	$1$	$2$	$2$	$67.074$	\(\Q(\sqrt{13}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(0\)	\(2\)		$-$	$-$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}+3q^{11}+(-1+\beta )q^{13}+\cdots\)
8400.2.a.dg	$8400$	$2$	8400.a	1.a	$1$	$3$	$3$	$67.074$	3.3.148.1	None	✓	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-3\)		$-$	$+$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}-2q^{11}+(2-\beta _{2})q^{13}+\cdots\)
8400.2.a.dh	$8400$	$2$	8400.a	1.a	$1$	$3$	$3$	$67.074$	3.3.148.1	None	✓	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(-3\)		$+$	$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}+(-1-\beta _{1}-\beta _{2})q^{11}+\cdots\)
8400.2.a.di	$8400$	$2$	8400.a	1.a	$1$	$3$	$3$	$67.074$	3.3.148.1	None	✓	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(-3\)		$+$	$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}+q^{9}+(1+\beta _{1}-\beta _{2})q^{11}+\cdots\)
8400.2.a.dj	$8400$	$2$	8400.a	1.a	$1$	$3$	$3$	$67.074$	3.3.148.1	None	✓	$1$	$1$	\(0\)	\(3\)	\(0\)	\(3\)		$-$	$-$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}-2q^{11}+(-2+\beta _{2})q^{13}+\cdots\)
8400.2.a.dk	$8400$	$2$	8400.a	1.a	$1$	$3$	$3$	$67.074$	3.3.148.1	None	✓	$1$	$0$	\(0\)	\(3\)	\(0\)	\(3\)		$+$	$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}+(-1-\beta _{1}-\beta _{2})q^{11}+\cdots\)
8400.2.a.dl	$8400$	$2$	8400.a	1.a	$1$	$3$	$3$	$67.074$	3.3.148.1	None	✓	$1$	$0$	\(0\)	\(3\)	\(0\)	\(3\)		$+$	$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}+(1+\beta _{1}-\beta _{2})q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(8400))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(8400)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 24}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 20}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 24}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 15}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 20}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(210))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(240))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(280))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(336))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(350))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(400))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(420))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(525))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(560))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(600))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(700))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(840))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1050))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1200))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1400))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1680))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2100))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2800))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(4200))\)\(^{\oplus 2}\)