Space of modular forms of level 9300, weight 2, and trivial character

• Label: 9300.2.a
• Level: $9300$
• Weight: $2$
• Character orbit: 9300.a
• Rep. character: $\chi_{9300}(1,\cdot)$
• Character field: $\Q$
• Dimension: $94$
• Newform subspaces: $32$
• Sturm bound: $3840$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9300.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 32 \)
Sturm bound:			\(3840\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(9300))\).

	Total	New	Old
Modular forms	1956	94	1862
Cusp forms	1885	94	1791
Eisenstein series	71	0	71

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

\(2\)	\(3\)	\(5\)	\(31\)	Fricke		Total				Cusp				Eisenstein
						All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)		\(120\)	\(0\)	\(120\)		\(115\)	\(0\)	\(115\)		\(5\)	\(0\)	\(5\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)		\(123\)	\(0\)	\(123\)		\(117\)	\(0\)	\(117\)		\(6\)	\(0\)	\(6\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)		\(126\)	\(0\)	\(126\)		\(120\)	\(0\)	\(120\)		\(6\)	\(0\)	\(6\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)		\(123\)	\(0\)	\(123\)		\(117\)	\(0\)	\(117\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)		\(126\)	\(0\)	\(126\)		\(120\)	\(0\)	\(120\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)		\(123\)	\(0\)	\(123\)		\(117\)	\(0\)	\(117\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)		\(120\)	\(0\)	\(120\)		\(114\)	\(0\)	\(114\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)		\(123\)	\(0\)	\(123\)		\(117\)	\(0\)	\(117\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)		\(123\)	\(13\)	\(110\)		\(120\)	\(13\)	\(107\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)		\(120\)	\(10\)	\(110\)		\(117\)	\(10\)	\(107\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)		\(120\)	\(11\)	\(109\)		\(117\)	\(11\)	\(106\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)		\(123\)	\(13\)	\(110\)		\(120\)	\(13\)	\(107\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)		\(120\)	\(10\)	\(110\)		\(117\)	\(10\)	\(107\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)		\(123\)	\(13\)	\(110\)		\(120\)	\(13\)	\(107\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)		\(123\)	\(13\)	\(110\)		\(120\)	\(13\)	\(107\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)		\(120\)	\(11\)	\(109\)		\(117\)	\(11\)	\(106\)		\(3\)	\(0\)	\(3\)
Plus space				\(+\)		\(966\)	\(42\)	\(924\)		\(931\)	\(42\)	\(889\)		\(35\)	\(0\)	\(35\)
Minus space				\(-\)		\(990\)	\(52\)	\(938\)		\(954\)	\(52\)	\(902\)		\(36\)	\(0\)	\(36\)

Trace form

94 * q + 4 * q^7 + 94 * q^9 - 4 * q^11 - 4 * q^17 - 12 * q^19 - 12 * q^23 + 24 * q^29 + 8 * q^33 + 16 * q^41 + 4 * q^43 - 12 * q^47 + 90 * q^49 + 4 * q^51 - 20 * q^53 + 8 * q^57 + 52 * q^61 + 4 * q^63 - 24 * q^67 + 8 * q^69 - 16 * q^71 - 24 * q^73 - 36 * q^77 - 8 * q^79 + 94 * q^81 - 4 * q^87 + 20 * q^89 - 4 * q^91 + 2 * q^93 + 16 * q^97 - 4 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(9300))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	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	31
9300.2.a.a	$9300$	$2$	9300.a	1.a	$1$	$1$	$1$	$74.261$	\(\Q\)	None	✓				372.2.a.c	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-4\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-4q^{7}+q^{9}-2q^{13}+4q^{19}+\cdots\)
9300.2.a.b	$9300$	$2$	9300.a	1.a	$1$	$1$	$1$	$74.261$	\(\Q\)	None	✓				1860.2.a.c	$1$	$2$	\(0\)	\(-1\)	\(0\)	\(-2\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{7}+q^{9}-4q^{11}-4q^{13}+\cdots\)
9300.2.a.c	$9300$	$2$	9300.a	1.a	$1$	$1$	$1$	$74.261$	\(\Q\)	None	✓	✓			9300.2.a.c	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{9}-3q^{11}+4q^{13}-q^{17}+\cdots\)
9300.2.a.d	$9300$	$2$	9300.a	1.a	$1$	$1$	$1$	$74.261$	\(\Q\)	None	✓				372.2.a.d	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-2q^{13}-q^{19}-q^{21}+\cdots\)
9300.2.a.e	$9300$	$2$	9300.a	1.a	$1$	$1$	$1$	$74.261$	\(\Q\)	None	✓				1860.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{7}+q^{9}-4q^{11}+4q^{13}+\cdots\)
9300.2.a.f	$9300$	$2$	9300.a	1.a	$1$	$1$	$1$	$74.261$	\(\Q\)	None	✓	✓			9300.2.a.f	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+4q^{7}+q^{9}+4q^{13}-3q^{17}+\cdots\)
9300.2.a.g	$9300$	$2$	9300.a	1.a	$1$	$1$	$1$	$74.261$	\(\Q\)	None	✓	✓			9300.2.a.g	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(4\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+4q^{7}+q^{9}+3q^{11}-2q^{13}+\cdots\)
9300.2.a.h	$9300$	$2$	9300.a	1.a	$1$	$1$	$1$	$74.261$	\(\Q\)	None	✓				372.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(5\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+5q^{7}+q^{9}+2q^{11}+4q^{13}+\cdots\)
9300.2.a.i	$9300$	$2$	9300.a	1.a	$1$	$1$	$1$	$74.261$	\(\Q\)	None	✓	✓			9300.2.a.f	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-4\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-4q^{7}+q^{9}-4q^{13}+3q^{17}+\cdots\)
9300.2.a.j	$9300$	$2$	9300.a	1.a	$1$	$1$	$1$	$74.261$	\(\Q\)	None	✓	✓			9300.2.a.g	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-4\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-4q^{7}+q^{9}+3q^{11}+2q^{13}+\cdots\)
9300.2.a.k	$9300$	$2$	9300.a	1.a	$1$	$1$	$1$	$74.261$	\(\Q\)	None	✓	✓			9300.2.a.c	$1$	$1$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{9}-3q^{11}-4q^{13}+q^{17}+\cdots\)
9300.2.a.l	$9300$	$2$	9300.a	1.a	$1$	$1$	$1$	$74.261$	\(\Q\)	None	✓				1860.2.a.a	$1$	$1$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{9}+2q^{13}+4q^{17}-4q^{19}+\cdots\)
9300.2.a.m	$9300$	$2$	9300.a	1.a	$1$	$1$	$1$	$74.261$	\(\Q\)	None	✓				372.2.a.a	$1$	$0$	\(0\)	\(1\)	\(0\)	\(1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}+6q^{13}+8q^{17}+\cdots\)
9300.2.a.n	$9300$	$2$	9300.a	1.a	$1$	$2$	$2$	$74.261$	\(\Q(\sqrt{3}) \)	None	✓				1860.2.a.e	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-2\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-1+\beta )q^{7}+q^{9}-4q^{11}+\cdots\)
9300.2.a.o	$9300$	$2$	9300.a	1.a	$1$	$2$	$2$	$74.261$	\(\Q(\sqrt{33}) \)	None	✓	✓			9300.2.a.o	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{9}+\beta q^{11}+(1+2\beta )q^{17}+\cdots\)
9300.2.a.p	$9300$	$2$	9300.a	1.a	$1$	$2$	$2$	$74.261$	\(\Q(\sqrt{6}) \)	None	✓				1860.2.a.d	$1$	$0$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta q^{7}+q^{9}+(2-\beta )q^{13}-2q^{17}+\cdots\)
9300.2.a.q	$9300$	$2$	9300.a	1.a	$1$	$2$	$2$	$74.261$	\(\Q(\sqrt{33}) \)	None	✓	✓			9300.2.a.o	$1$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{9}+\beta q^{11}+(-1-2\beta )q^{17}+\cdots\)
9300.2.a.r	$9300$	$2$	9300.a	1.a	$1$	$2$	$2$	$74.261$	\(\Q(\sqrt{17}) \)	None	✓				372.2.a.e	$1$	$1$	\(0\)	\(2\)	\(0\)	\(1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta q^{7}+q^{9}+2\beta q^{11}+(-2+\cdots)q^{13}+\cdots\)
9300.2.a.s	$9300$	$2$	9300.a	1.a	$1$	$3$	$3$	$74.261$	3.3.404.1	None	✓	✓			9300.2.a.s	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-\beta _{1}+\beta _{2})q^{7}+q^{9}+(-2+\cdots)q^{11}+\cdots\)
9300.2.a.t	$9300$	$2$	9300.a	1.a	$1$	$3$	$3$	$74.261$	3.3.7636.1	None	✓				1860.2.a.h	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(0\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta _{1}q^{7}+q^{9}+2q^{11}-\beta _{1}q^{13}+\cdots\)
9300.2.a.u	$9300$	$2$	9300.a	1.a	$1$	$3$	$3$	$74.261$	3.3.404.1	None	✓				1860.2.a.g	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(1-\beta _{2})q^{7}+q^{9}+2\beta _{2}q^{11}+\cdots\)
9300.2.a.v	$9300$	$2$	9300.a	1.a	$1$	$3$	$3$	$74.261$	3.3.564.1	None	✓		✓		1860.2.a.f	$1$	$1$	\(0\)	\(3\)	\(0\)	\(-4\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-2+\beta _{1}-\beta _{2})q^{7}+q^{9}-2\beta _{1}q^{11}+\cdots\)
9300.2.a.w	$9300$	$2$	9300.a	1.a	$1$	$3$	$3$	$74.261$	3.3.404.1	None	✓	✓			9300.2.a.s	$1$	$1$	\(0\)	\(3\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(\beta _{1}-\beta _{2})q^{7}+q^{9}+(-2+\beta _{1}+\cdots)q^{11}+\cdots\)
9300.2.a.x	$9300$	$2$	9300.a	1.a	$1$	$4$	$4$	$74.261$	4.4.224148.1	None	✓				1860.2.a.i	$1$	$0$	\(0\)	\(4\)	\(0\)	\(4\)		$-$	$-$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+(1-\beta _{1})q^{7}+q^{9}+(\beta _{1}+\beta _{2}+\cdots)q^{11}+\cdots\)
9300.2.a.y	$9300$	$2$	9300.a	1.a	$1$	$6$	$6$	$74.261$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓		9300.2.a.y	$1$	$0$	\(0\)	\(-6\)	\(0\)	\(-4\)		$-$	$+$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}+(-1-\beta _{5})q^{7}+q^{9}+(-1-\beta _{4}+\cdots)q^{11}+\cdots\)
9300.2.a.z	$9300$	$2$	9300.a	1.a	$1$	$6$	$6$	$74.261$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓			9300.2.a.z	$1$	$0$	\(0\)	\(-6\)	\(0\)	\(-4\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-1+\beta _{1})q^{7}+q^{9}-\beta _{4}q^{11}+\cdots\)
9300.2.a.ba	$9300$	$2$	9300.a	1.a	$1$	$6$	$6$	$74.261$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓			9300.2.a.y	$1$	$0$	\(0\)	\(6\)	\(0\)	\(4\)		$-$	$-$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+(1+\beta _{5})q^{7}+q^{9}+(-1-\beta _{4}+\cdots)q^{11}+\cdots\)
9300.2.a.bb	$9300$	$2$	9300.a	1.a	$1$	$6$	$6$	$74.261$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓		9300.2.a.z	$1$	$0$	\(0\)	\(6\)	\(0\)	\(4\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(1-\beta _{1})q^{7}+q^{9}-\beta _{4}q^{11}+\cdots\)
9300.2.a.bc	$9300$	$2$	9300.a	1.a	$1$	$7$	$7$	$74.261$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		1860.2.g.a	$1$	$1$	\(0\)	\(-7\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-1-\beta _{2})q^{7}+q^{9}+(1-\beta _{4}+\cdots)q^{11}+\cdots\)
9300.2.a.bd	$9300$	$2$	9300.a	1.a	$1$	$7$	$7$	$74.261$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		1860.2.g.b	$1$	$0$	\(0\)	\(-7\)	\(0\)	\(4\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(1-\beta _{4})q^{7}+q^{9}+(\beta _{3}-\beta _{5}+\cdots)q^{11}+\cdots\)
9300.2.a.be	$9300$	$2$	9300.a	1.a	$1$	$7$	$7$	$74.261$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		1860.2.g.b	$1$	$1$	\(0\)	\(7\)	\(0\)	\(-4\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-1+\beta _{4})q^{7}+q^{9}+(\beta _{3}-\beta _{5}+\cdots)q^{11}+\cdots\)
9300.2.a.bf	$9300$	$2$	9300.a	1.a	$1$	$7$	$7$	$74.261$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		1860.2.g.a	$1$	$0$	\(0\)	\(7\)	\(0\)	\(4\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(1+\beta _{2})q^{7}+q^{9}+(1-\beta _{4})q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(9300))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(9300)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(31))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(62))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(93))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(124))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(155))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(186))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(310))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(372))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(465))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(620))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(775))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(930))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1550))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1860))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2325))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3100))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(4650))\)\(^{\oplus 2}\)