Space of modular forms of level 9800, weight 2, and trivial character

• Label: 9800.2.a
• Level: $9800$
• Weight: $2$
• Character orbit: 9800.a
• Rep. character: $\chi_{9800}(1,\cdot)$
• Character field: $\Q$
• Dimension: $195$
• Newform subspaces: $81$
• Sturm bound: $3360$
• Trace bound: $23$

Defining parameters

Level:	\( N \)	\(=\)	\( 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9800.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 81 \)
Sturm bound:			\(3360\)
Trace bound:			\(23\)
Distinguishing \(T_p\):			\(3\), \(11\), \(13\), \(19\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(9800))\).

	Total	New	Old
Modular forms	1776	195	1581
Cusp forms	1585	195	1390
Eisenstein series	191	0	191

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

\(2\)	\(5\)	\(7\)	Fricke	Dim.
\(+\)	\(+\)	\(+\)	\(+\)	\(19\)
\(+\)	\(+\)	\(-\)	\(-\)	\(26\)
\(+\)	\(-\)	\(+\)	\(-\)	\(28\)
\(+\)	\(-\)	\(-\)	\(+\)	\(24\)
\(-\)	\(+\)	\(+\)	\(-\)	\(25\)
\(-\)	\(+\)	\(-\)	\(+\)	\(22\)
\(-\)	\(-\)	\(+\)	\(+\)	\(24\)
\(-\)	\(-\)	\(-\)	\(-\)	\(27\)
Plus space			\(+\)	\(89\)
Minus space			\(-\)	\(106\)

Trace form

\( 195 q - 2 q^{3} + 197 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(9800))\) 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	5	7
9800.2.a.a	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+6q^{9}-5q^{11}-5q^{13}-7q^{17}+\cdots\)
9800.2.a.b	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+6q^{9}-q^{11}-2q^{13}-3q^{17}+\cdots\)
9800.2.a.c	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+6q^{9}+q^{11}+4q^{13}+5q^{17}+\cdots\)
9800.2.a.d	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{9}-4q^{11}-4q^{13}+4q^{19}+\cdots\)
9800.2.a.e	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{9}-3q^{11}+2q^{13}-4q^{17}+\cdots\)
9800.2.a.f	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{9}-3q^{11}+2q^{13}-4q^{17}+\cdots\)
9800.2.a.g	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{9}-q^{11}-3q^{13}-2q^{17}+\cdots\)
9800.2.a.h	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{9}+q^{11}-4q^{13}-6q^{19}+\cdots\)
9800.2.a.i	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{9}+4q^{11}-2q^{13}+2q^{19}+\cdots\)
9800.2.a.j	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{9}+4q^{11}+2q^{13}+3q^{17}+\cdots\)
9800.2.a.k	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{9}+4q^{11}+2q^{13}+3q^{17}+\cdots\)
9800.2.a.l	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{9}+5q^{11}-8q^{17}+2q^{19}+\cdots\)
9800.2.a.m	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$2$	\(0\)	\(-1\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{9}-5q^{11}-7q^{13}+3q^{17}+\cdots\)
9800.2.a.n	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{9}-5q^{11}+q^{13}+3q^{17}+\cdots\)
9800.2.a.o	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{9}-2q^{11}+4q^{13}-6q^{19}+\cdots\)
9800.2.a.p	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{9}-q^{11}+q^{13}+3q^{17}+\cdots\)
9800.2.a.q	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{9}-q^{11}+6q^{13}-7q^{17}+\cdots\)
9800.2.a.r	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{9}+2q^{11}-4q^{17}-2q^{19}+\cdots\)
9800.2.a.s	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{9}+3q^{11}-6q^{13}-5q^{17}+\cdots\)
9800.2.a.t	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{9}+3q^{11}+q^{13}-5q^{17}+\cdots\)
9800.2.a.u	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{9}-4q^{11}+2q^{13}-6q^{17}-8q^{19}+\cdots\)
9800.2.a.v	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{9}+q^{11}-2q^{13}-4q^{17}+2q^{19}+\cdots\)
9800.2.a.w	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{9}+q^{11}+2q^{13}+4q^{17}+2q^{19}+\cdots\)
9800.2.a.x	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{9}+4q^{11}-2q^{13}+2q^{17}-4q^{19}+\cdots\)
9800.2.a.y	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{9}-5q^{11}+7q^{13}-3q^{17}+\cdots\)
9800.2.a.z	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{9}-2q^{11}-4q^{13}+6q^{19}+\cdots\)
9800.2.a.ba	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{9}-q^{11}-6q^{13}+7q^{17}+\cdots\)
9800.2.a.bb	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{9}-q^{11}-q^{13}-3q^{17}+\cdots\)
9800.2.a.bc	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{9}+2q^{11}+4q^{17}+2q^{19}+\cdots\)
9800.2.a.bd	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{9}+3q^{11}-q^{13}+5q^{17}+\cdots\)
9800.2.a.be	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{9}+3q^{11}+6q^{13}+5q^{17}+\cdots\)
9800.2.a.bf	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{9}-4q^{11}+4q^{13}+4q^{19}+\cdots\)
9800.2.a.bg	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{9}-3q^{11}-2q^{13}+4q^{17}+\cdots\)
9800.2.a.bh	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{9}-3q^{11}-2q^{13}+4q^{17}+\cdots\)
9800.2.a.bi	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{9}-q^{11}+3q^{13}+2q^{17}+\cdots\)
9800.2.a.bj	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{9}-2q^{17}+2q^{19}-8q^{23}+\cdots\)
9800.2.a.bk	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{9}+q^{11}+4q^{13}-6q^{19}+\cdots\)
9800.2.a.bl	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{9}+4q^{11}-2q^{13}-3q^{17}+\cdots\)
9800.2.a.bm	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(2\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{9}+4q^{11}-2q^{13}-3q^{17}+\cdots\)
9800.2.a.bn	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(2\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{9}+4q^{11}+2q^{13}-2q^{19}+\cdots\)
9800.2.a.bo	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{9}+5q^{11}+8q^{17}+2q^{19}+\cdots\)
9800.2.a.bp	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(3\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+6q^{9}-q^{11}+2q^{13}+3q^{17}+\cdots\)
9800.2.a.bq	$9800$	$2$	9800.a	1.a	$1$	$1$	$1$	$78.253$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(3\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+6q^{9}+q^{11}-4q^{13}-5q^{17}+\cdots\)
9800.2.a.br	$9800$	$2$	9800.a	1.a	$1$	$2$	$2$	$78.253$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}-2\beta q^{9}+(-2+2\beta )q^{11}+\cdots\)
9800.2.a.bs	$9800$	$2$	9800.a	1.a	$1$	$2$	$2$	$78.253$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}-2\beta q^{9}-q^{11}+(-1+\cdots)q^{13}+\cdots\)
9800.2.a.bt	$9800$	$2$	9800.a	1.a	$1$	$2$	$2$	$78.253$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(1+\beta )q^{9}+(-3+2\beta )q^{11}+\cdots\)
9800.2.a.bu	$9800$	$2$	9800.a	1.a	$1$	$2$	$2$	$78.253$	\(\Q(\sqrt{33}) \)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(5+\beta )q^{9}+(4-\beta )q^{11}+(2+\cdots)q^{13}+\cdots\)
9800.2.a.bv	$9800$	$2$	9800.a	1.a	$1$	$2$	$2$	$78.253$	\(\Q(\sqrt{2}) \)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-q^{9}+6q^{11}-4\beta q^{13}+\beta q^{17}+\cdots\)
9800.2.a.bw	$9800$	$2$	9800.a	1.a	$1$	$2$	$2$	$78.253$	\(\Q(\sqrt{2}) \)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+5q^{9}-4q^{11}-\beta q^{13}-2\beta q^{17}+\cdots\)
9800.2.a.bx	$9800$	$2$	9800.a	1.a	$1$	$2$	$2$	$78.253$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(1+\beta )q^{9}+(-3+2\beta )q^{11}+\cdots\)
9800.2.a.by	$9800$	$2$	9800.a	1.a	$1$	$2$	$2$	$78.253$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(1+\beta )q^{9}-\beta q^{11}+(2-3\beta )q^{13}+\cdots\)
9800.2.a.bz	$9800$	$2$	9800.a	1.a	$1$	$2$	$2$	$78.253$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+2\beta q^{9}+(-2-2\beta )q^{11}+\cdots\)
9800.2.a.ca	$9800$	$2$	9800.a	1.a	$1$	$2$	$2$	$78.253$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+2\beta q^{9}-q^{11}+(1-\beta )q^{13}+\cdots\)
9800.2.a.cb	$9800$	$2$	9800.a	1.a	$1$	$3$	$3$	$78.253$	\(\Q(\zeta_{18})^+\)	None	✓	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-2\beta _{1}+\beta _{2})q^{9}+\cdots\)
9800.2.a.cc	$9800$	$2$	9800.a	1.a	$1$	$3$	$3$	$78.253$	\(\Q(\zeta_{18})^+\)	None	✓	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-2\beta _{1}+\beta _{2})q^{9}+\cdots\)
9800.2.a.cd	$9800$	$2$	9800.a	1.a	$1$	$3$	$3$	$78.253$	3.3.568.1	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1+2\beta _{1}+\beta _{2})q^{9}+(2-\beta _{2})q^{11}+\cdots\)
9800.2.a.ce	$9800$	$2$	9800.a	1.a	$1$	$3$	$3$	$78.253$	3.3.1944.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(3+\beta _{1}+\beta _{2})q^{9}+(1+\beta _{1}+\cdots)q^{11}+\cdots\)
9800.2.a.cf	$9800$	$2$	9800.a	1.a	$1$	$3$	$3$	$78.253$	3.3.1944.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(3+\beta _{1}+\beta _{2})q^{9}+(1+\beta _{1}+\cdots)q^{11}+\cdots\)
9800.2.a.cg	$9800$	$2$	9800.a	1.a	$1$	$3$	$3$	$78.253$	3.3.568.1	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1+2\beta _{1}+\beta _{2})q^{9}+(2-\beta _{2})q^{11}+\cdots\)
9800.2.a.ch	$9800$	$2$	9800.a	1.a	$1$	$3$	$3$	$78.253$	\(\Q(\zeta_{18})^+\)	None	✓	$1$	$1$	\(0\)	\(3\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-2\beta _{1}+\beta _{2})q^{9}+(-2+\cdots)q^{11}+\cdots\)
9800.2.a.ci	$9800$	$2$	9800.a	1.a	$1$	$3$	$3$	$78.253$	\(\Q(\zeta_{18})^+\)	None	✓	$1$	$0$	\(0\)	\(3\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-2\beta _{1}+\beta _{2})q^{9}+(-2+\cdots)q^{11}+\cdots\)
9800.2.a.cj	$9800$	$2$	9800.a	1.a	$1$	$4$	$4$	$78.253$	4.4.43449.1	None	✓	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(2-\beta _{1}+\beta _{2})q^{9}+\cdots\)
9800.2.a.ck	$9800$	$2$	9800.a	1.a	$1$	$4$	$4$	$78.253$	4.4.43449.1	None	✓	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(2-\beta _{1}+\beta _{2})q^{9}+\cdots\)
9800.2.a.cl	$9800$	$2$	9800.a	1.a	$1$	$4$	$4$	$78.253$	4.4.16448.2	None	✓	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1+\beta _{1}+\beta _{2})q^{9}+(\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)
9800.2.a.cm	$9800$	$2$	9800.a	1.a	$1$	$4$	$4$	$78.253$	\(\Q(\sqrt{2}, \sqrt{5})\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{3}q^{9}-q^{11}+(\beta _{1}-\beta _{2}+\cdots)q^{13}+\cdots\)
9800.2.a.cn	$9800$	$2$	9800.a	1.a	$1$	$4$	$4$	$78.253$	\(\Q(\sqrt{2}, \sqrt{5})\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{3}q^{9}-q^{11}+(\beta _{1}-\beta _{2}+\cdots)q^{13}+\cdots\)
9800.2.a.co	$9800$	$2$	9800.a	1.a	$1$	$4$	$4$	$78.253$	4.4.13448.1	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1+\beta _{2})q^{9}+(2+\beta _{2})q^{11}+\cdots\)
9800.2.a.cp	$9800$	$2$	9800.a	1.a	$1$	$4$	$4$	$78.253$	4.4.13448.1	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1+\beta _{2})q^{9}+(2+\beta _{2})q^{11}+\cdots\)
9800.2.a.cq	$9800$	$2$	9800.a	1.a	$1$	$4$	$4$	$78.253$	\(\Q(\sqrt{7}, \sqrt{15})\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(2-\beta _{3})q^{9}+q^{11}+(\beta _{1}+\beta _{2}+\cdots)q^{13}+\cdots\)
9800.2.a.cr	$9800$	$2$	9800.a	1.a	$1$	$4$	$4$	$78.253$	\(\Q(\sqrt{7}, \sqrt{15})\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(2-\beta _{3})q^{9}+q^{11}+(\beta _{1}+\beta _{2}+\cdots)q^{13}+\cdots\)
9800.2.a.cs	$9800$	$2$	9800.a	1.a	$1$	$4$	$4$	$78.253$	4.4.16448.2	None	✓	$1$	$0$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1+\beta _{1}+\beta _{2})q^{9}+(\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)
9800.2.a.ct	$9800$	$2$	9800.a	1.a	$1$	$4$	$4$	$78.253$	4.4.43449.1	None	✓	$1$	$1$	\(0\)	\(3\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(2-\beta _{1}+\beta _{2})q^{9}+(-\beta _{2}+\cdots)q^{11}+\cdots\)
9800.2.a.cu	$9800$	$2$	9800.a	1.a	$1$	$4$	$4$	$78.253$	4.4.43449.1	None	✓	$1$	$0$	\(0\)	\(3\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(2-\beta _{1}+\beta _{2})q^{9}+(-\beta _{2}+\cdots)q^{11}+\cdots\)
9800.2.a.cv	$9800$	$2$	9800.a	1.a	$1$	$6$	$6$	$78.253$	6.6.239575536.1	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1+\beta _{2})q^{9}+(\beta _{1}+\beta _{5})q^{11}+\cdots\)
9800.2.a.cw	$9800$	$2$	9800.a	1.a	$1$	$6$	$6$	$78.253$	6.6.239575536.1	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1+\beta _{2})q^{9}+(\beta _{1}+\beta _{5})q^{11}+\cdots\)
9800.2.a.cx	$9800$	$2$	9800.a	1.a	$1$	$6$	$6$	$78.253$	6.6.239575536.1	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1+\beta _{2})q^{9}+(\beta _{1}+\beta _{5})q^{11}+\cdots\)
9800.2.a.cy	$9800$	$2$	9800.a	1.a	$1$	$6$	$6$	$78.253$	6.6.239575536.1	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1+\beta _{2})q^{9}+(\beta _{1}+\beta _{5})q^{11}+\cdots\)
9800.2.a.cz	$9800$	$2$	9800.a	1.a	$1$	$8$	$8$	$78.253$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(2+\beta _{2})q^{9}+(1+\beta _{3}+\beta _{6}+\cdots)q^{11}+\cdots\)
9800.2.a.da	$9800$	$2$	9800.a	1.a	$1$	$8$	$8$	$78.253$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(2+\beta _{2})q^{9}+(1+\beta _{3}+\beta _{6}+\cdots)q^{11}+\cdots\)
9800.2.a.db	$9800$	$2$	9800.a	1.a	$1$	$10$	$10$	$78.253$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$2^{6}$	$\mathrm{SU}(2)$	\(q+\beta _{7}q^{3}+(1-\beta _{4}-\beta _{8})q^{9}+(-1+\beta _{1}+\cdots)q^{11}+\cdots\)
9800.2.a.dc	$9800$	$2$	9800.a	1.a	$1$	$10$	$10$	$78.253$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$2^{6}$	$\mathrm{SU}(2)$	\(q+\beta _{7}q^{3}+(1-\beta _{4}-\beta _{8})q^{9}+(-1+\beta _{1}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(9800))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(9800)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(245))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(280))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(350))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(392))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(490))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(700))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(980))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1225))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1400))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1960))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2450))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(4900))\)\(^{\oplus 2}\)