Space of modular forms of level 1232, weight 2, and trivial character

• Label: 1232.2.a
• Level: $1232$
• Weight: $2$
• Character orbit: 1232.a
• Rep. character: $\chi_{1232}(1,\cdot)$
• Character field: $\Q$
• Dimension: $30$
• Newform subspaces: $19$
• Sturm bound: $384$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1232.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 19 \)
Sturm bound:			\(384\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1232))\).

	Total	New	Old
Modular forms	204	30	174
Cusp forms	181	30	151
Eisenstein series	23	0	23

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

\(2\)	\(7\)	\(11\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(4\)
\(+\)	\(+\)	\(-\)	\(-\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)	\(5\)
\(-\)	\(+\)	\(-\)	\(+\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)	\(5\)
Plus space			\(+\)	\(11\)
Minus space			\(-\)	\(19\)

Trace form

\( 30 q + 4 q^{5} + 30 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1232))\) 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	7	11
1232.2.a.a	$1232$	$2$	1232.a	1.a	$1$	$1$	$1$	$9.838$	\(\Q\)	None	✓				77.2.a.c	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-2q^{5}+q^{7}+q^{9}-q^{11}+4q^{13}+\cdots\)
1232.2.a.b	$1232$	$2$	1232.a	1.a	$1$	$1$	$1$	$9.838$	\(\Q\)	None	✓				616.2.a.e	$1$	$0$	\(0\)	\(-2\)	\(2\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+2q^{5}-q^{7}+q^{9}+q^{11}-4q^{15}+\cdots\)
1232.2.a.c	$1232$	$2$	1232.a	1.a	$1$	$1$	$1$	$9.838$	\(\Q\)	None	✓				154.2.a.b	$1$	$1$	\(0\)	\(-2\)	\(2\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+2q^{5}+q^{7}+q^{9}-q^{11}-4q^{13}+\cdots\)
1232.2.a.d	$1232$	$2$	1232.a	1.a	$1$	$1$	$1$	$9.838$	\(\Q\)	None	✓				77.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(3\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+3q^{5}-q^{7}-2q^{9}+q^{11}-4q^{13}+\cdots\)
1232.2.a.e	$1232$	$2$	1232.a	1.a	$1$	$1$	$1$	$9.838$	\(\Q\)	None	✓				154.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-4\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{5}+q^{7}-3q^{9}+q^{11}+2q^{13}+\cdots\)
1232.2.a.f	$1232$	$2$	1232.a	1.a	$1$	$1$	$1$	$9.838$	\(\Q\)	None	✓				616.2.a.c	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-q^{7}-3q^{9}+q^{11}+2q^{13}+\cdots\)
1232.2.a.g	$1232$	$2$	1232.a	1.a	$1$	$1$	$1$	$9.838$	\(\Q\)	None	✓		✓	✓	616.2.a.d	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}-3q^{9}+q^{11}-6q^{13}+2q^{19}+\cdots\)
1232.2.a.h	$1232$	$2$	1232.a	1.a	$1$	$1$	$1$	$9.838$	\(\Q\)	None	✓				154.2.a.c	$1$	$0$	\(0\)	\(0\)	\(2\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+q^{7}-3q^{9}+q^{11}+2q^{13}+\cdots\)
1232.2.a.i	$1232$	$2$	1232.a	1.a	$1$	$1$	$1$	$9.838$	\(\Q\)	None	✓				616.2.a.b	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-q^{7}-2q^{9}-q^{11}-q^{15}+\cdots\)
1232.2.a.j	$1232$	$2$	1232.a	1.a	$1$	$1$	$1$	$9.838$	\(\Q\)	None	✓				308.2.a.a	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+q^{7}-2q^{9}-q^{11}-4q^{13}+\cdots\)
1232.2.a.k	$1232$	$2$	1232.a	1.a	$1$	$1$	$1$	$9.838$	\(\Q\)	None	✓				616.2.a.a	$1$	$0$	\(0\)	\(2\)	\(2\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+2q^{5}-q^{7}+q^{9}+q^{11}+4q^{13}+\cdots\)
1232.2.a.l	$1232$	$2$	1232.a	1.a	$1$	$1$	$1$	$9.838$	\(\Q\)	None	✓				77.2.a.a	$1$	$0$	\(0\)	\(3\)	\(-1\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-q^{5}+q^{7}+6q^{9}+q^{11}-4q^{13}+\cdots\)
1232.2.a.m	$1232$	$2$	1232.a	1.a	$1$	$2$	$2$	$9.838$	\(\Q(\sqrt{5}) \)	None	✓		✓		77.2.a.d	$1$	$1$	\(0\)	\(-2\)	\(-4\)	\(-2\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}-2q^{5}-q^{7}+(3+2\beta )q^{9}+\cdots\)
1232.2.a.n	$1232$	$2$	1232.a	1.a	$1$	$2$	$2$	$9.838$	\(\Q(\sqrt{6}) \)	None	✓		✓		308.2.a.b	$1$	$0$	\(0\)	\(0\)	\(4\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+2q^{5}+q^{7}+3q^{9}+q^{11}+\cdots\)
1232.2.a.o	$1232$	$2$	1232.a	1.a	$1$	$2$	$2$	$9.838$	\(\Q(\sqrt{17}) \)	None	✓				616.2.a.f	$1$	$0$	\(0\)	\(1\)	\(-3\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(-2+\beta )q^{5}+q^{7}+(1+\beta )q^{9}+\cdots\)
1232.2.a.p	$1232$	$2$	1232.a	1.a	$1$	$2$	$2$	$9.838$	\(\Q(\sqrt{5}) \)	None	✓				154.2.a.d	$1$	$0$	\(0\)	\(2\)	\(2\)	\(-2\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(1+\beta )q^{5}-q^{7}+(3+2\beta )q^{9}+\cdots\)
1232.2.a.q	$1232$	$2$	1232.a	1.a	$1$	$3$	$3$	$9.838$	3.3.229.1	None	✓		✓		616.2.a.g	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(-3\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}-\beta _{2}q^{5}-q^{7}+(1-\beta _{1})q^{9}+\cdots\)
1232.2.a.r	$1232$	$2$	1232.a	1.a	$1$	$3$	$3$	$9.838$	3.3.1016.1	None	✓		✓		308.2.a.c	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(-3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{2})q^{5}-q^{7}+(1+\cdots)q^{9}+\cdots\)
1232.2.a.s	$1232$	$2$	1232.a	1.a	$1$	$4$	$4$	$9.838$	4.4.11348.1	None	✓		✓		616.2.a.h	$1$	$0$	\(0\)	\(-1\)	\(5\)	\(4\)		$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(1-\beta _{1})q^{5}+q^{7}+(2+\beta _{2}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1232))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1232)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(154))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(176))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(308))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(616))\)\(^{\oplus 2}\)