Space of modular forms of level 7600, weight 2, and trivial character

• Label: 7600.2.a
• Level: $7600$
• Weight: $2$
• Character orbit: 7600.a
• Rep. character: $\chi_{7600}(1,\cdot)$
• Character field: $\Q$
• Dimension: $171$
• Newform subspaces: $66$
• Sturm bound: $2400$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7600.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 66 \)
Sturm bound:			\(2400\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(3\), \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(7600))\).

	Total	New	Old
Modular forms	1236	171	1065
Cusp forms	1165	171	994
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.

\(2\)	\(5\)	\(19\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(19\)
\(+\)	\(+\)	\(-\)	\(-\)	\(22\)
\(+\)	\(-\)	\(+\)	\(-\)	\(25\)
\(+\)	\(-\)	\(-\)	\(+\)	\(19\)
\(-\)	\(+\)	\(+\)	\(-\)	\(20\)
\(-\)	\(+\)	\(-\)	\(+\)	\(20\)
\(-\)	\(-\)	\(+\)	\(+\)	\(23\)
\(-\)	\(-\)	\(-\)	\(-\)	\(23\)
Plus space			\(+\)	\(81\)
Minus space			\(-\)	\(90\)

Trace form

\( 171 q - 2 q^{7} + 171 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(7600))\) 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	5	19
7600.2.a.a	$7600$	$2$	7600.a	1.a	$1$	$1$	$1$	$60.686$	\(\Q\)	None	✓				190.2.a.b	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(-5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-5q^{7}+6q^{9}+4q^{11}+q^{13}+\cdots\)
7600.2.a.b	$7600$	$2$	7600.a	1.a	$1$	$1$	$1$	$60.686$	\(\Q\)	None	✓				152.2.a.a	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-3\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-3q^{7}+q^{9}+3q^{11}+4q^{13}+\cdots\)
7600.2.a.c	$7600$	$2$	7600.a	1.a	$1$	$1$	$1$	$60.686$	\(\Q\)	None	✓				19.2.a.a	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-q^{7}+q^{9}-3q^{11}+4q^{13}+\cdots\)
7600.2.a.d	$7600$	$2$	7600.a	1.a	$1$	$1$	$1$	$60.686$	\(\Q\)	None	✓				760.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{9}+4q^{11}-4q^{13}+2q^{17}+\cdots\)
7600.2.a.e	$7600$	$2$	7600.a	1.a	$1$	$1$	$1$	$60.686$	\(\Q\)	None	✓				760.2.d.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+2q^{7}+q^{9}-4q^{11}-8q^{17}+\cdots\)
7600.2.a.f	$7600$	$2$	7600.a	1.a	$1$	$1$	$1$	$60.686$	\(\Q\)	None	✓				760.2.a.b	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+4q^{7}+q^{9}-4q^{11}+4q^{13}+\cdots\)
7600.2.a.g	$7600$	$2$	7600.a	1.a	$1$	$1$	$1$	$60.686$	\(\Q\)	None	✓				190.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{7}-2q^{9}+3q^{13}+7q^{17}+\cdots\)
7600.2.a.h	$7600$	$2$	7600.a	1.a	$1$	$1$	$1$	$60.686$	\(\Q\)	None	✓				38.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+3q^{7}-2q^{9}-2q^{11}+q^{13}+\cdots\)
7600.2.a.i	$7600$	$2$	7600.a	1.a	$1$	$1$	$1$	$60.686$	\(\Q\)	None	✓				95.2.b.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-3q^{9}+4q^{11}-2q^{13}+4q^{17}+\cdots\)
7600.2.a.j	$7600$	$2$	7600.a	1.a	$1$	$1$	$1$	$60.686$	\(\Q\)	None	✓				380.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-3q^{9}+4q^{11}+4q^{13}-6q^{17}+\cdots\)
7600.2.a.k	$7600$	$2$	7600.a	1.a	$1$	$1$	$1$	$60.686$	\(\Q\)	None	✓				760.2.a.c	$1$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{9}+4q^{11}+6q^{13}+6q^{17}+q^{19}+\cdots\)
7600.2.a.l	$7600$	$2$	7600.a	1.a	$1$	$1$	$1$	$60.686$	\(\Q\)	None	✓				95.2.b.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}-3q^{9}+4q^{11}+2q^{13}-4q^{17}+\cdots\)
7600.2.a.m	$7600$	$2$	7600.a	1.a	$1$	$1$	$1$	$60.686$	\(\Q\)	None	✓				190.2.a.c	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}-2q^{9}+q^{13}+3q^{17}+\cdots\)
7600.2.a.n	$7600$	$2$	7600.a	1.a	$1$	$1$	$1$	$60.686$	\(\Q\)	None	✓				38.2.a.a	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}-2q^{9}+6q^{11}-5q^{13}+\cdots\)
7600.2.a.o	$7600$	$2$	7600.a	1.a	$1$	$1$	$1$	$60.686$	\(\Q\)	None	✓				152.2.a.b	$1$	$1$	\(0\)	\(1\)	\(0\)	\(3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+3q^{7}-2q^{9}-2q^{11}-q^{13}+\cdots\)
7600.2.a.p	$7600$	$2$	7600.a	1.a	$1$	$1$	$1$	$60.686$	\(\Q\)	None	✓				76.2.a.a	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-3q^{7}+q^{9}-5q^{11}+4q^{13}+\cdots\)
7600.2.a.q	$7600$	$2$	7600.a	1.a	$1$	$1$	$1$	$60.686$	\(\Q\)	None	✓				760.2.d.a	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2q^{7}+q^{9}-4q^{11}+8q^{17}+\cdots\)
7600.2.a.r	$7600$	$2$	7600.a	1.a	$1$	$1$	$1$	$60.686$	\(\Q\)	None	✓				380.2.a.b	$1$	$1$	\(0\)	\(2\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+2q^{7}+q^{9}-6q^{13}-2q^{17}+\cdots\)
7600.2.a.s	$7600$	$2$	7600.a	1.a	$1$	$1$	$1$	$60.686$	\(\Q\)	None	✓				760.2.a.d	$1$	$0$	\(0\)	\(2\)	\(0\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+4q^{7}+q^{9}+4q^{11}-6q^{17}+\cdots\)
7600.2.a.t	$7600$	$2$	7600.a	1.a	$1$	$1$	$1$	$60.686$	\(\Q\)	None	✓				760.2.a.e	$1$	$0$	\(0\)	\(3\)	\(0\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-q^{7}+6q^{9}-4q^{11}-q^{13}+\cdots\)
7600.2.a.u	$7600$	$2$	7600.a	1.a	$1$	$2$	$2$	$60.686$	\(\Q(\sqrt{2}) \)	None	✓				380.2.a.c	$1$	$1$	\(0\)	\(-4\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{3}+(-2-2\beta )q^{7}+(3-4\beta )q^{9}+\cdots\)
7600.2.a.v	$7600$	$2$	7600.a	1.a	$1$	$2$	$2$	$60.686$	\(\Q(\sqrt{2}) \)	None	✓				190.2.b.a	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-6\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+(-3-\beta )q^{7}-2\beta q^{9}+\cdots\)
7600.2.a.w	$7600$	$2$	7600.a	1.a	$1$	$2$	$2$	$60.686$	\(\Q(\sqrt{5}) \)	None	✓				760.2.d.b	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+2\beta q^{7}+(3+2\beta )q^{9}+\cdots\)
7600.2.a.x	$7600$	$2$	7600.a	1.a	$1$	$2$	$2$	$60.686$	\(\Q(\sqrt{2}) \)	None	✓				760.2.d.c	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+(1-\beta )q^{7}-2\beta q^{9}+\cdots\)
7600.2.a.y	$7600$	$2$	7600.a	1.a	$1$	$2$	$2$	$60.686$	\(\Q(\sqrt{17}) \)	None	✓				190.2.a.d	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}-\beta q^{7}+(1+\beta )q^{9}-4q^{11}+\cdots\)
7600.2.a.z	$7600$	$2$	7600.a	1.a	$1$	$2$	$2$	$60.686$	\(\Q(\sqrt{2}) \)	None	✓				760.2.d.d	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(-2+2\beta )q^{7}-q^{9}+(-2+\cdots)q^{11}+\cdots\)
7600.2.a.ba	$7600$	$2$	7600.a	1.a	$1$	$2$	$2$	$60.686$	\(\Q(\sqrt{2}) \)	None	✓				760.2.a.f	$1$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+2\beta q^{7}-q^{9}+(-2-2\beta )q^{11}+\cdots\)
7600.2.a.bb	$7600$	$2$	7600.a	1.a	$1$	$2$	$2$	$60.686$	\(\Q(\sqrt{2}) \)	None	✓				760.2.d.d	$1$	$1$	\(0\)	\(0\)	\(0\)	\(4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(2+2\beta )q^{7}-q^{9}+(-2-2\beta )q^{11}+\cdots\)
7600.2.a.bc	$7600$	$2$	7600.a	1.a	$1$	$2$	$2$	$60.686$	\(\Q(\sqrt{2}) \)	None	✓				760.2.d.c	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(-1-\beta )q^{7}+2\beta q^{9}+\cdots\)
7600.2.a.bd	$7600$	$2$	7600.a	1.a	$1$	$2$	$2$	$60.686$	\(\Q(\sqrt{3}) \)	None	✓				760.2.a.g	$1$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(1+2\beta )q^{9}-2q^{11}+(-1+\cdots)q^{13}+\cdots\)
7600.2.a.be	$7600$	$2$	7600.a	1.a	$1$	$2$	$2$	$60.686$	\(\Q(\sqrt{5}) \)	None	✓				760.2.d.b	$1$	$0$	\(0\)	\(2\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-2\beta q^{7}+(3+2\beta )q^{9}+(1+\cdots)q^{13}+\cdots\)
7600.2.a.bf	$7600$	$2$	7600.a	1.a	$1$	$2$	$2$	$60.686$	\(\Q(\sqrt{3}) \)	None	✓				380.2.a.d	$1$	$0$	\(0\)	\(2\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+2q^{7}+(1+2\beta )q^{9}+2\beta q^{11}+\cdots\)
7600.2.a.bg	$7600$	$2$	7600.a	1.a	$1$	$2$	$2$	$60.686$	\(\Q(\sqrt{2}) \)	None	✓				190.2.b.a	$1$	$0$	\(0\)	\(2\)	\(0\)	\(6\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(3-\beta )q^{7}+2\beta q^{9}-\beta q^{11}+\cdots\)
7600.2.a.bh	$7600$	$2$	7600.a	1.a	$1$	$3$	$3$	$60.686$	3.3.169.1	None	✓				475.2.a.e	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-1-\beta _{1})q^{7}+(1+\cdots)q^{9}+\cdots\)
7600.2.a.bi	$7600$	$2$	7600.a	1.a	$1$	$3$	$3$	$60.686$	3.3.568.1	None	✓				190.2.b.b	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-1-\beta _{1})q^{7}+(2+\cdots)q^{9}+\cdots\)
7600.2.a.bj	$7600$	$2$	7600.a	1.a	$1$	$3$	$3$	$60.686$	3.3.257.1	None	✓				1900.2.a.f	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-1+\beta _{1})q^{7}+(1+\cdots)q^{9}+\cdots\)
7600.2.a.bk	$7600$	$2$	7600.a	1.a	$1$	$3$	$3$	$60.686$	3.3.993.1	None	✓				950.2.a.j	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-1+\beta _{2})q^{7}+(2+\cdots)q^{9}+\cdots\)
7600.2.a.bl	$7600$	$2$	7600.a	1.a	$1$	$3$	$3$	$60.686$	3.3.321.1	None	✓				1900.2.a.g	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}+(-1+2\beta _{1}+\beta _{2})q^{7}+\cdots\)
7600.2.a.bm	$7600$	$2$	7600.a	1.a	$1$	$3$	$3$	$60.686$	3.3.257.1	None	✓				950.2.a.k	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1}+\beta _{2})q^{3}+(-1+\beta _{1}-2\beta _{2})q^{7}+\cdots\)
7600.2.a.bn	$7600$	$2$	7600.a	1.a	$1$	$3$	$3$	$60.686$	\(\Q(\zeta_{14})^+\)	None	✓				475.2.a.d	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}+(1-2\beta _{1}+\beta _{2})q^{7}+\cdots\)
7600.2.a.bo	$7600$	$2$	7600.a	1.a	$1$	$3$	$3$	$60.686$	3.3.568.1	None	✓				760.2.a.h	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-5\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-2-\beta _{2})q^{7}+(1+2\beta _{1}+\cdots)q^{9}+\cdots\)
7600.2.a.bp	$7600$	$2$	7600.a	1.a	$1$	$3$	$3$	$60.686$	3.3.316.1	None	✓				760.2.a.i	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1+2\beta _{1}-\beta _{2})q^{7}+\beta _{2}q^{9}+\cdots\)
7600.2.a.bq	$7600$	$2$	7600.a	1.a	$1$	$3$	$3$	$60.686$	3.3.229.1	None	✓				760.2.a.j	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+\beta _{1}q^{7}+(1-\beta _{1})q^{9}+(-2+\cdots)q^{13}+\cdots\)
7600.2.a.br	$7600$	$2$	7600.a	1.a	$1$	$3$	$3$	$60.686$	\(\Q(\zeta_{18})^+\)	None	✓				3800.2.a.s	$1$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-\beta _{1}+2\beta _{2})q^{7}+(-1+\beta _{2})q^{9}+\cdots\)
7600.2.a.bs	$7600$	$2$	7600.a	1.a	$1$	$3$	$3$	$60.686$	\(\Q(\zeta_{18})^+\)	None	✓				3800.2.a.s	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(\beta _{1}-2\beta _{2})q^{7}+(-1+\beta _{2})q^{9}+\cdots\)
7600.2.a.bt	$7600$	$2$	7600.a	1.a	$1$	$3$	$3$	$60.686$	3.3.257.1	None	✓				3800.2.a.u	$1$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+\beta _{2}q^{7}+(\beta _{1}-\beta _{2})q^{9}+(1+\cdots)q^{11}+\cdots\)
7600.2.a.bu	$7600$	$2$	7600.a	1.a	$1$	$3$	$3$	$60.686$	3.3.257.1	None	✓				3800.2.a.u	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}-\beta _{2}q^{7}+(\beta _{1}-\beta _{2})q^{9}+(1+\cdots)q^{11}+\cdots\)
7600.2.a.bv	$7600$	$2$	7600.a	1.a	$1$	$3$	$3$	$60.686$	3.3.961.1	None	✓				152.2.a.c	$1$	$0$	\(0\)	\(1\)	\(0\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(2-\beta _{1}+\beta _{2})q^{7}+(5-\beta _{1}+\cdots)q^{9}+\cdots\)
7600.2.a.bw	$7600$	$2$	7600.a	1.a	$1$	$3$	$3$	$60.686$	\(\Q(\zeta_{14})^+\)	None	✓				475.2.a.d	$1$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(1-\beta _{1}+2\beta _{2})q^{7}+(-2\beta _{1}+\cdots)q^{9}+\cdots\)
7600.2.a.bx	$7600$	$2$	7600.a	1.a	$1$	$3$	$3$	$60.686$	3.3.148.1	None	✓				95.2.a.a	$1$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(\beta _{1}-\beta _{2})q^{7}+(1-\beta _{1}+\cdots)q^{9}+\cdots\)
7600.2.a.by	$7600$	$2$	7600.a	1.a	$1$	$3$	$3$	$60.686$	3.3.257.1	None	✓				1900.2.a.f	$1$	$0$	\(0\)	\(2\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(1-\beta _{1})q^{7}+(1-2\beta _{1}+\cdots)q^{9}+\cdots\)
7600.2.a.bz	$7600$	$2$	7600.a	1.a	$1$	$3$	$3$	$60.686$	3.3.993.1	None	✓				950.2.a.j	$1$	$0$	\(0\)	\(2\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(1-\beta _{2})q^{7}+(2-2\beta _{1}+\cdots)q^{9}+\cdots\)
7600.2.a.ca	$7600$	$2$	7600.a	1.a	$1$	$3$	$3$	$60.686$	3.3.321.1	None	✓				1900.2.a.g	$1$	$0$	\(0\)	\(2\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}+(1-2\beta _{1}-\beta _{2})q^{7}+(2+\cdots)q^{9}+\cdots\)
7600.2.a.cb	$7600$	$2$	7600.a	1.a	$1$	$3$	$3$	$60.686$	3.3.257.1	None	✓				950.2.a.k	$1$	$0$	\(0\)	\(2\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1}-\beta _{2})q^{3}+(1-\beta _{1}+2\beta _{2})q^{7}+\cdots\)
7600.2.a.cc	$7600$	$2$	7600.a	1.a	$1$	$3$	$3$	$60.686$	3.3.169.1	None	✓				475.2.a.e	$1$	$1$	\(0\)	\(2\)	\(0\)	\(4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(1+\beta _{1})q^{7}+(1-\beta _{1}+\cdots)q^{9}+\cdots\)
7600.2.a.cd	$7600$	$2$	7600.a	1.a	$1$	$3$	$3$	$60.686$	3.3.568.1	None	✓				190.2.b.b	$1$	$1$	\(0\)	\(2\)	\(0\)	\(4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(1+\beta _{1})q^{7}+(2+\beta _{2})q^{9}+\cdots\)
7600.2.a.ce	$7600$	$2$	7600.a	1.a	$1$	$4$	$4$	$60.686$	\(\Q(\sqrt{2}, \sqrt{5})\)	None	✓				380.2.c.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{3})q^{7}+\beta _{2}q^{9}+\cdots\)
7600.2.a.cf	$7600$	$2$	7600.a	1.a	$1$	$4$	$4$	$60.686$	4.4.11344.1	None	✓		✓		95.2.a.b	$1$	$0$	\(0\)	\(2\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(1-\beta _{1})q^{7}+(2-\beta _{3})q^{9}+\cdots\)
7600.2.a.cg	$7600$	$2$	7600.a	1.a	$1$	$6$	$6$	$60.686$	6.6.253565184.1	None	✓				760.2.d.e	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-6\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1-\beta _{1}-\beta _{3})q^{7}+(1+\beta _{1}+\cdots)q^{9}+\cdots\)
7600.2.a.ch	$7600$	$2$	7600.a	1.a	$1$	$6$	$6$	$60.686$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		3800.2.a.ba	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-\beta _{1}-\beta _{4})q^{7}+(1+\beta _{1}+\cdots)q^{9}+\cdots\)
7600.2.a.ci	$7600$	$2$	7600.a	1.a	$1$	$6$	$6$	$60.686$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		3800.2.a.bb	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+\beta _{3}q^{7}+(1+\beta _{1}+\beta _{2})q^{9}+\cdots\)
7600.2.a.cj	$7600$	$2$	7600.a	1.a	$1$	$6$	$6$	$60.686$	6.6.56310016.1	None	✓		✓		380.2.c.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{3}-\beta _{4}q^{7}+(2-\beta _{1}+\beta _{5})q^{9}+\cdots\)
7600.2.a.ck	$7600$	$2$	7600.a	1.a	$1$	$6$	$6$	$60.686$	6.6.66064384.1	None	✓		✓		95.2.b.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{5}q^{3}+(-\beta _{2}-\beta _{5})q^{7}+(3+\beta _{1}+\cdots)q^{9}+\cdots\)
7600.2.a.cl	$7600$	$2$	7600.a	1.a	$1$	$6$	$6$	$60.686$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		3800.2.a.ba	$1$	$0$	\(0\)	\(2\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(\beta _{1}+\beta _{4})q^{7}+(1+\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots\)
7600.2.a.cm	$7600$	$2$	7600.a	1.a	$1$	$6$	$6$	$60.686$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				3800.2.a.bb	$1$	$0$	\(0\)	\(2\)	\(0\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{3}q^{7}+(1+\beta _{1}+\beta _{2})q^{9}+\cdots\)
7600.2.a.cn	$7600$	$2$	7600.a	1.a	$1$	$6$	$6$	$60.686$	6.6.253565184.1	None	✓				760.2.d.e	$1$	$0$	\(0\)	\(2\)	\(0\)	\(6\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1+\beta _{1}+\beta _{3})q^{7}+(1+\beta _{1}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(7600))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(7600)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 15}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(95))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(152))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(190))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(304))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(380))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(400))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(475))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(760))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(950))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1520))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1900))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3800))\)\(^{\oplus 2}\)