Space of modular forms of level 1392, weight 2, and trivial character

• Label: 1392.2.a
• Level: $1392$
• Weight: $2$
• Character orbit: 1392.a
• Rep. character: $\chi_{1392}(1,\cdot)$
• Character field: $\Q$
• Dimension: $28$
• Newform subspaces: $21$
• Sturm bound: $480$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1392 = 2^{4} \cdot 3 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1392.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 21 \)
Sturm bound:			\(480\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1392))\).

	Total	New	Old
Modular forms	252	28	224
Cusp forms	229	28	201
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.

\(2\)	\(3\)	\(29\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(24\)	\(4\)	\(20\)		\(22\)	\(4\)	\(18\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)		\(39\)	\(4\)	\(35\)		\(36\)	\(4\)	\(32\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)		\(33\)	\(3\)	\(30\)		\(30\)	\(3\)	\(27\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)		\(30\)	\(3\)	\(27\)		\(27\)	\(3\)	\(24\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)		\(33\)	\(5\)	\(28\)		\(30\)	\(5\)	\(25\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)		\(30\)	\(2\)	\(28\)		\(27\)	\(2\)	\(25\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)		\(36\)	\(2\)	\(34\)		\(33\)	\(2\)	\(31\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)		\(27\)	\(5\)	\(22\)		\(24\)	\(5\)	\(19\)		\(3\)	\(0\)	\(3\)
Plus space			\(+\)		\(120\)	\(11\)	\(109\)		\(109\)	\(11\)	\(98\)		\(11\)	\(0\)	\(11\)
Minus space			\(-\)		\(132\)	\(17\)	\(115\)		\(120\)	\(17\)	\(103\)		\(12\)	\(0\)	\(12\)

Trace form

28 * q - 2 * q^3 - 4 * q^7 + 28 * q^9 + 4 * q^15 - 8 * q^19 - 24 * q^23 + 28 * q^25 - 2 * q^27 + 16 * q^31 + 4 * q^39 + 16 * q^41 + 36 * q^49 + 8 * q^51 + 8 * q^55 - 8 * q^57 + 8 * q^59 - 16 * q^61 - 4 * q^63 + 16 * q^65 + 20 * q^67 - 16 * q^69 + 32 * q^71 - 8 * q^73 + 2 * q^75 - 24 * q^79 + 28 * q^81 + 16 * q^83 + 6 * q^87 + 16 * q^89 + 12 * q^91 + 16 * q^93 + 64 * q^95 - 8 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1392))\) 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	29
1392.2.a.a	$1392$	$2$	1392.a	1.a	$1$	$1$	$1$	$11.115$	\(\Q\)	None	✓				348.2.a.c	$1$	$0$	\(0\)	\(-1\)	\(-4\)	\(3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-4q^{5}+3q^{7}+q^{9}+q^{11}-3q^{13}+\cdots\)
1392.2.a.b	$1392$	$2$	1392.a	1.a	$1$	$1$	$1$	$11.115$	\(\Q\)	None	✓				174.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(-3\)	\(-5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-3q^{5}-5q^{7}+q^{9}-6q^{11}+\cdots\)
1392.2.a.c	$1392$	$2$	1392.a	1.a	$1$	$1$	$1$	$11.115$	\(\Q\)	None	✓				696.2.a.e	$1$	$1$	\(0\)	\(-1\)	\(-3\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-3q^{5}-q^{7}+q^{9}+2q^{11}+4q^{13}+\cdots\)
1392.2.a.d	$1392$	$2$	1392.a	1.a	$1$	$1$	$1$	$11.115$	\(\Q\)	None	✓				696.2.a.f	$1$	$1$	\(0\)	\(-1\)	\(-2\)	\(1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}+q^{7}+q^{9}+3q^{11}-7q^{13}+\cdots\)
1392.2.a.e	$1392$	$2$	1392.a	1.a	$1$	$1$	$1$	$11.115$	\(\Q\)	None	✓				174.2.a.e	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-q^{7}+q^{9}+2q^{11}+q^{15}+\cdots\)
1392.2.a.f	$1392$	$2$	1392.a	1.a	$1$	$1$	$1$	$11.115$	\(\Q\)	None	✓				696.2.a.g	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(5\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+5q^{7}+q^{9}+5q^{11}+q^{13}+\cdots\)
1392.2.a.g	$1392$	$2$	1392.a	1.a	$1$	$1$	$1$	$11.115$	\(\Q\)	None	✓				348.2.a.d	$1$	$1$	\(0\)	\(-1\)	\(2\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}-q^{7}+q^{9}-q^{11}-3q^{13}+\cdots\)
1392.2.a.h	$1392$	$2$	1392.a	1.a	$1$	$1$	$1$	$11.115$	\(\Q\)	None	✓				174.2.a.c	$1$	$0$	\(0\)	\(-1\)	\(2\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}+q^{9}+4q^{11}+6q^{13}+\cdots\)
1392.2.a.i	$1392$	$2$	1392.a	1.a	$1$	$1$	$1$	$11.115$	\(\Q\)	None	✓				696.2.a.a	$1$	$1$	\(0\)	\(1\)	\(-2\)	\(-3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}-3q^{7}+q^{9}+5q^{11}+\cdots\)
1392.2.a.j	$1392$	$2$	1392.a	1.a	$1$	$1$	$1$	$11.115$	\(\Q\)	None	✓				348.2.a.a	$1$	$1$	\(0\)	\(1\)	\(-2\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}-q^{7}+q^{9}-3q^{11}+5q^{13}+\cdots\)
1392.2.a.k	$1392$	$2$	1392.a	1.a	$1$	$1$	$1$	$11.115$	\(\Q\)	None	✓				696.2.a.b	$1$	$0$	\(0\)	\(1\)	\(0\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{7}+q^{9}+3q^{11}+q^{13}-q^{17}+\cdots\)
1392.2.a.l	$1392$	$2$	1392.a	1.a	$1$	$1$	$1$	$11.115$	\(\Q\)	None	✓				348.2.a.b	$1$	$0$	\(0\)	\(1\)	\(0\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+3q^{7}+q^{9}+3q^{11}-3q^{13}+\cdots\)
1392.2.a.m	$1392$	$2$	1392.a	1.a	$1$	$1$	$1$	$11.115$	\(\Q\)	None	✓				174.2.a.d	$1$	$1$	\(0\)	\(1\)	\(1\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-q^{7}+q^{9}-6q^{11}-4q^{13}+\cdots\)
1392.2.a.n	$1392$	$2$	1392.a	1.a	$1$	$1$	$1$	$11.115$	\(\Q\)	None	✓				696.2.a.c	$1$	$0$	\(0\)	\(1\)	\(1\)	\(3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+3q^{7}+q^{9}+2q^{11}+4q^{13}+\cdots\)
1392.2.a.o	$1392$	$2$	1392.a	1.a	$1$	$1$	$1$	$11.115$	\(\Q\)	None	✓				174.2.a.a	$1$	$0$	\(0\)	\(1\)	\(3\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+3q^{5}+3q^{7}+q^{9}-6q^{11}+\cdots\)
1392.2.a.p	$1392$	$2$	1392.a	1.a	$1$	$1$	$1$	$11.115$	\(\Q\)	None	✓				696.2.a.d	$1$	$0$	\(0\)	\(1\)	\(4\)	\(-3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{5}-3q^{7}+q^{9}-q^{11}+q^{13}+\cdots\)
1392.2.a.q	$1392$	$2$	1392.a	1.a	$1$	$2$	$2$	$11.115$	\(\Q(\sqrt{5}) \)	None	✓		✓		87.2.a.a	$1$	$0$	\(0\)	\(-2\)	\(2\)	\(4\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}+(1+\beta )q^{5}+(2-\beta )q^{7}+q^{9}+\cdots\)
1392.2.a.r	$1392$	$2$	1392.a	1.a	$1$	$2$	$2$	$11.115$	\(\Q(\sqrt{41}) \)	None	✓		✓		696.2.a.i	$1$	$1$	\(0\)	\(-2\)	\(4\)	\(-3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}+(-2+\beta )q^{7}+q^{9}+(-2+\cdots)q^{11}+\cdots\)
1392.2.a.s	$1392$	$2$	1392.a	1.a	$1$	$2$	$2$	$11.115$	\(\Q(\sqrt{17}) \)	None	✓		✓		696.2.a.h	$1$	$1$	\(0\)	\(2\)	\(-3\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-1-\beta )q^{5}+(-1+2\beta )q^{7}+\cdots\)
1392.2.a.t	$1392$	$2$	1392.a	1.a	$1$	$3$	$3$	$11.115$	3.3.961.1	None	✓		✓		696.2.a.j	$1$	$0$	\(0\)	\(-3\)	\(1\)	\(-4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta _{1}q^{5}+(-1+\beta _{2})q^{7}+q^{9}+\cdots\)
1392.2.a.u	$1392$	$2$	1392.a	1.a	$1$	$3$	$3$	$11.115$	3.3.229.1	None	✓		✓		87.2.a.b	$1$	$0$	\(0\)	\(3\)	\(0\)	\(-4\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+(-\beta _{1}+\beta _{2})q^{5}+(-1+\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1392))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1392)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(58))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(87))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(116))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(174))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(232))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(348))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(464))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(696))\)\(^{\oplus 2}\)