Space of modular forms of level 1232, weight 4, and trivial character

• Label: 1232.4.a
• Level: $1232$
• Weight: $4$
• Character orbit: 1232.a
• Rep. character: $\chi_{1232}(1,\cdot)$
• Character field: $\Q$
• Dimension: $90$
• Newform subspaces: $30$
• Sturm bound: $768$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1232.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 30 \)
Sturm bound:			\(768\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1232))\).

	Total	New	Old
Modular forms	588	90	498
Cusp forms	564	90	474
Eisenstein series	24	0	24

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		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(77\)	\(11\)	\(66\)		\(74\)	\(11\)	\(63\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)		\(73\)	\(12\)	\(61\)		\(70\)	\(12\)	\(58\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)		\(70\)	\(9\)	\(61\)		\(67\)	\(9\)	\(58\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)		\(74\)	\(14\)	\(60\)		\(71\)	\(14\)	\(57\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)		\(72\)	\(10\)	\(62\)		\(69\)	\(10\)	\(59\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)		\(76\)	\(12\)	\(64\)		\(73\)	\(12\)	\(61\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)		\(75\)	\(12\)	\(63\)		\(72\)	\(12\)	\(60\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)		\(71\)	\(10\)	\(61\)		\(68\)	\(10\)	\(58\)		\(3\)	\(0\)	\(3\)
Plus space			\(+\)		\(302\)	\(49\)	\(253\)		\(290\)	\(49\)	\(241\)		\(12\)	\(0\)	\(12\)
Minus space			\(-\)		\(286\)	\(41\)	\(245\)		\(274\)	\(41\)	\(233\)		\(12\)	\(0\)	\(12\)

Trace form

90 * q - 4 * q^5 + 810 * q^9 + 66 * q^11 + 92 * q^13 - 132 * q^15 + 52 * q^17 - 48 * q^19 - 276 * q^23 + 2206 * q^25 - 708 * q^27 + 284 * q^29 + 528 * q^31 - 660 * q^37 + 600 * q^39 - 236 * q^41 - 180 * q^45 - 1164 * q^47 + 4410 * q^49 + 1488 * q^51 + 572 * q^53 - 848 * q^57 + 2040 * q^59 + 2604 * q^61 + 232 * q^65 + 804 * q^67 + 1056 * q^69 - 668 * q^71 + 740 * q^73 - 5476 * q^75 + 2832 * q^79 + 7594 * q^81 + 456 * q^83 + 2696 * q^85 + 2768 * q^87 - 900 * q^89 + 1092 * q^91 + 4272 * q^93 + 1896 * q^95 - 1396 * q^97 + 1782 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1232))\) 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
1232.4.a.a	$1232$	$4$	1232.a	1.a	$1$	$1$	$1$	$72.690$	\(\Q\)	None	✓				616.4.a.a	$1$	$1$	\(0\)	\(-9\)	\(15\)	\(7\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-9q^{3}+15q^{5}+7q^{7}+54q^{9}-11q^{11}+\cdots\)
1232.4.a.b	$1232$	$4$	1232.a	1.a	$1$	$1$	$1$	$72.690$	\(\Q\)	None	✓				154.4.a.e	$1$	$0$	\(0\)	\(-7\)	\(3\)	\(-7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-7q^{3}+3q^{5}-7q^{7}+22q^{9}+11q^{11}+\cdots\)
1232.4.a.c	$1232$	$4$	1232.a	1.a	$1$	$1$	$1$	$72.690$	\(\Q\)	None	✓				308.4.a.b	$1$	$1$	\(0\)	\(-4\)	\(-12\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}-12q^{5}-7q^{7}-11q^{9}-11q^{11}+\cdots\)
1232.4.a.d	$1232$	$4$	1232.a	1.a	$1$	$1$	$1$	$72.690$	\(\Q\)	None	✓				77.4.a.a	$1$	$1$	\(0\)	\(-4\)	\(12\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}+12q^{5}-7q^{7}-11q^{9}-11q^{11}+\cdots\)
1232.4.a.e	$1232$	$4$	1232.a	1.a	$1$	$1$	$1$	$72.690$	\(\Q\)	None	✓				154.4.a.b	$1$	$0$	\(0\)	\(0\)	\(2\)	\(7\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+7q^{7}-3^{3}q^{9}-11q^{11}+26q^{13}+\cdots\)
1232.4.a.f	$1232$	$4$	1232.a	1.a	$1$	$1$	$1$	$72.690$	\(\Q\)	None	✓				154.4.a.d	$1$	$0$	\(0\)	\(2\)	\(18\)	\(-7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+18q^{5}-7q^{7}-23q^{9}+11q^{11}+\cdots\)
1232.4.a.g	$1232$	$4$	1232.a	1.a	$1$	$1$	$1$	$72.690$	\(\Q\)	None	✓				154.4.a.a	$1$	$1$	\(0\)	\(5\)	\(-1\)	\(7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{3}-q^{5}+7q^{7}-2q^{9}+11q^{11}+\cdots\)
1232.4.a.h	$1232$	$4$	1232.a	1.a	$1$	$1$	$1$	$72.690$	\(\Q\)	None	✓				308.4.a.a	$1$	$1$	\(0\)	\(7\)	\(-1\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+7q^{3}-q^{5}-7q^{7}+22q^{9}-11q^{11}+\cdots\)
1232.4.a.i	$1232$	$4$	1232.a	1.a	$1$	$1$	$1$	$72.690$	\(\Q\)	None	✓				154.4.a.c	$1$	$0$	\(0\)	\(10\)	\(-14\)	\(-7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+10q^{3}-14q^{5}-7q^{7}+73q^{9}+11q^{11}+\cdots\)
1232.4.a.j	$1232$	$4$	1232.a	1.a	$1$	$2$	$2$	$72.690$	\(\Q(\sqrt{37}) \)	None	✓				154.4.a.g	$1$	$1$	\(0\)	\(-6\)	\(26\)	\(-14\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-3-\beta )q^{3}+(13-\beta )q^{5}-7q^{7}+\cdots\)
1232.4.a.k	$1232$	$4$	1232.a	1.a	$1$	$2$	$2$	$72.690$	\(\Q(\sqrt{5}) \)	None	✓				616.4.a.c	$1$	$1$	\(0\)	\(2\)	\(-8\)	\(14\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+3\beta )q^{3}+(-4+2\beta )q^{5}+7q^{7}+\cdots\)
1232.4.a.l	$1232$	$4$	1232.a	1.a	$1$	$2$	$2$	$72.690$	\(\Q(\sqrt{5}) \)	None	✓				616.4.a.d	$1$	$0$	\(0\)	\(2\)	\(6\)	\(14\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1+3\beta )q^{3}+(3+\beta )q^{5}+7q^{7}+(19+\cdots)q^{9}+\cdots\)
1232.4.a.m	$1232$	$4$	1232.a	1.a	$1$	$2$	$2$	$72.690$	\(\Q(\sqrt{2}) \)	None	✓				77.4.a.b	$1$	$0$	\(0\)	\(4\)	\(-4\)	\(-14\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+2q^{3}+(-2-3\beta )q^{5}-7q^{7}-23q^{9}+\cdots\)
1232.4.a.n	$1232$	$4$	1232.a	1.a	$1$	$2$	$2$	$72.690$	\(\Q(\sqrt{37}) \)	None	✓				616.4.a.b	$1$	$1$	\(0\)	\(4\)	\(0\)	\(14\)		$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+2q^{3}-\beta q^{5}+7q^{7}-23q^{9}-11q^{11}+\cdots\)
1232.4.a.o	$1232$	$4$	1232.a	1.a	$1$	$2$	$2$	$72.690$	\(\Q(\sqrt{57}) \)	None	✓				154.4.a.h	$1$	$1$	\(0\)	\(5\)	\(-17\)	\(14\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(3-\beta )q^{3}+(-7-3\beta )q^{5}+7q^{7}+\cdots\)
1232.4.a.p	$1232$	$4$	1232.a	1.a	$1$	$2$	$2$	$72.690$	\(\Q(\sqrt{137}) \)	None	✓				154.4.a.f	$1$	$0$	\(0\)	\(5\)	\(-7\)	\(-14\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(3-\beta )q^{3}+(-3-\beta )q^{5}-7q^{7}+(2^{4}+\cdots)q^{9}+\cdots\)
1232.4.a.q	$1232$	$4$	1232.a	1.a	$1$	$3$	$3$	$72.690$	3.3.7636.1	None	✓				154.4.a.i	$1$	$0$	\(0\)	\(-6\)	\(26\)	\(21\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{3}+(9-\beta _{2})q^{5}+7q^{7}+\cdots\)
1232.4.a.r	$1232$	$4$	1232.a	1.a	$1$	$3$	$3$	$72.690$	3.3.5925.1	None	✓				308.4.a.c	$1$	$1$	\(0\)	\(-5\)	\(9\)	\(21\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{3}+(2+3\beta _{1})q^{5}+7q^{7}+\cdots\)
1232.4.a.s	$1232$	$4$	1232.a	1.a	$1$	$4$	$4$	$72.690$	4.4.522072.1	None	✓		✓		77.4.a.d	$1$	$1$	\(0\)	\(-14\)	\(10\)	\(28\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-4-\beta _{2})q^{3}+(4-\beta _{1}+2\beta _{2})q^{5}+\cdots\)
1232.4.a.t	$1232$	$4$	1232.a	1.a	$1$	$4$	$4$	$72.690$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		616.4.a.e	$1$	$1$	\(0\)	\(3\)	\(-19\)	\(28\)		$+$	$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-5-\beta _{2})q^{5}+7q^{7}+\cdots\)
1232.4.a.u	$1232$	$4$	1232.a	1.a	$1$	$4$	$4$	$72.690$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				616.4.a.f	$1$	$0$	\(0\)	\(3\)	\(-13\)	\(-28\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-4+\beta _{1}-\beta _{2})q^{5}+\cdots\)
1232.4.a.v	$1232$	$4$	1232.a	1.a	$1$	$4$	$4$	$72.690$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				308.4.a.d	$1$	$0$	\(0\)	\(3\)	\(1\)	\(28\)		$-$	$-$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+\beta _{2}q^{5}+7q^{7}+(-5+\cdots)q^{9}+\cdots\)
1232.4.a.w	$1232$	$4$	1232.a	1.a	$1$	$4$	$4$	$72.690$	4.4.509800.1	None	✓				77.4.a.c	$1$	$0$	\(0\)	\(12\)	\(-18\)	\(28\)		$-$	$-$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(3-\beta _{2})q^{3}+(-5+\beta _{1}-\beta _{2}-\beta _{3})q^{5}+\cdots\)
1232.4.a.x	$1232$	$4$	1232.a	1.a	$1$	$5$	$5$	$72.690$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		308.4.a.e	$1$	$0$	\(0\)	\(-5\)	\(15\)	\(-35\)		$-$	$+$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(3+\beta _{2})q^{5}-7q^{7}+\cdots\)
1232.4.a.y	$1232$	$4$	1232.a	1.a	$1$	$5$	$5$	$72.690$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		77.4.a.e	$1$	$1$	\(0\)	\(-2\)	\(-24\)	\(-35\)		$-$	$+$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(-4+\beta _{2}+\beta _{3})q^{5}-7q^{7}+\cdots\)
1232.4.a.z	$1232$	$4$	1232.a	1.a	$1$	$5$	$5$	$72.690$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				616.4.a.g	$1$	$0$	\(0\)	\(-2\)	\(-14\)	\(35\)		$+$	$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-2-\beta _{1}-\beta _{2})q^{5}+7q^{7}+\cdots\)
1232.4.a.ba	$1232$	$4$	1232.a	1.a	$1$	$6$	$6$	$72.690$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				616.4.a.h	$1$	$1$	\(0\)	\(0\)	\(-14\)	\(-42\)		$+$	$+$	$-$	$-$	$2^{6}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(-2-\beta _{3})q^{5}-7q^{7}+(4+\cdots)q^{9}+\cdots\)
1232.4.a.bb	$1232$	$4$	1232.a	1.a	$1$	$6$	$6$	$72.690$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				616.4.a.i	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-42\)		$+$	$+$	$-$	$-$	$2^{6}\cdot 5$	$\mathrm{SU}(2)$	\(q-\beta _{4}q^{3}+(-\beta _{2}+\beta _{4})q^{5}-7q^{7}+(11+\cdots)q^{9}+\cdots\)
1232.4.a.bc	$1232$	$4$	1232.a	1.a	$1$	$7$	$7$	$72.690$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		616.4.a.k	$1$	$0$	\(0\)	\(-3\)	\(11\)	\(-49\)		$+$	$+$	$+$	$+$	$2^{6}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(2-\beta _{2})q^{5}-7q^{7}+(15+\beta _{3}+\cdots)q^{9}+\cdots\)
1232.4.a.bd	$1232$	$4$	1232.a	1.a	$1$	$7$	$7$	$72.690$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		616.4.a.j	$1$	$0$	\(0\)	\(0\)	\(6\)	\(49\)		$+$	$-$	$-$	$+$	$2^{6}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1-\beta _{2})q^{5}+7q^{7}+(15-\beta _{2}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1232))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1232)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(154))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(176))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(308))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(616))\)\(^{\oplus 2}\)