Space of modular forms of level 1323, weight 2, and trivial character

• Label: 1323.2.a
• Level: $1323$
• Weight: $2$
• Character orbit: 1323.a
• Rep. character: $\chi_{1323}(1,\cdot)$
• Character field: $\Q$
• Dimension: $55$
• Newform subspaces: $31$
• Sturm bound: $336$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1323.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 31 \)
Sturm bound:			\(336\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(2\), \(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1323))\).

	Total	New	Old
Modular forms	192	55	137
Cusp forms	145	55	90
Eisenstein series	47	0	47

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

\(3\)	\(7\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(44\)	\(12\)	\(32\)		\(33\)	\(12\)	\(21\)		\(11\)	\(0\)	\(11\)
\(+\)	\(-\)	\(-\)		\(50\)	\(16\)	\(34\)		\(38\)	\(16\)	\(22\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(-\)		\(52\)	\(15\)	\(37\)		\(40\)	\(15\)	\(25\)		\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(+\)		\(46\)	\(12\)	\(34\)		\(34\)	\(12\)	\(22\)		\(12\)	\(0\)	\(12\)
Plus space		\(+\)		\(90\)	\(24\)	\(66\)		\(67\)	\(24\)	\(43\)		\(23\)	\(0\)	\(23\)
Minus space		\(-\)		\(102\)	\(31\)	\(71\)		\(78\)	\(31\)	\(47\)		\(24\)	\(0\)	\(24\)

Trace form

55 * q + 54 * q^4 + 4 * q^10 + 7 * q^13 + 32 * q^16 - 5 * q^19 - 8 * q^22 + 73 * q^25 - 16 * q^31 + 36 * q^34 + 19 * q^37 + 48 * q^40 - 46 * q^43 + 4 * q^46 + 18 * q^52 - 52 * q^55 - 8 * q^58 + 13 * q^61 + 56 * q^64 - 47 * q^67 - q^73 - 54 * q^76 - 67 * q^79 - 28 * q^82 + 28 * q^85 + 12 * q^88 + 12 * q^94 + 47 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1323))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	7
1323.2.a.a	$1323$	$2$	1323.a	1.a	$1$	$1$	$1$	$10.564$	\(\Q\)	None	✓	✓			1323.2.a.a	$1$	$1$	\(-2\)	\(0\)	\(-3\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}-3q^{5}+6q^{10}-2q^{11}+\cdots\)
1323.2.a.b	$1323$	$2$	1323.a	1.a	$1$	$1$	$1$	$10.564$	\(\Q\)	None	✓				189.2.a.a	$1$	$1$	\(-2\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}+q^{5}-2q^{10}-4q^{11}+\cdots\)
1323.2.a.c	$1323$	$2$	1323.a	1.a	$1$	$1$	$1$	$10.564$	\(\Q\)	None	✓	✓			1323.2.a.a	$1$	$0$	\(-2\)	\(0\)	\(3\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}+3q^{5}-6q^{10}-2q^{11}+\cdots\)
1323.2.a.d	$1323$	$2$	1323.a	1.a	$1$	$1$	$1$	$10.564$	\(\Q\)	None	✓				189.2.e.a	$1$	$0$	\(-1\)	\(0\)	\(-4\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}-4q^{5}+3q^{8}+4q^{10}+\cdots\)
1323.2.a.e	$1323$	$2$	1323.a	1.a	$1$	$1$	$1$	$10.564$	\(\Q\)	None	✓	✓			1323.2.a.e	$1$	$1$	\(-1\)	\(0\)	\(-3\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}-3q^{5}+3q^{8}+3q^{10}+\cdots\)
1323.2.a.f	$1323$	$2$	1323.a	1.a	$1$	$1$	$1$	$10.564$	\(\Q\)	None	✓	✓			1323.2.a.e	$1$	$0$	\(-1\)	\(0\)	\(3\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+3q^{5}+3q^{8}-3q^{10}+\cdots\)
1323.2.a.g	$1323$	$2$	1323.a	1.a	$1$	$1$	$1$	$10.564$	\(\Q\)	None	✓				189.2.e.a	$1$	$0$	\(-1\)	\(0\)	\(4\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+4q^{5}+3q^{8}-4q^{10}+\cdots\)
1323.2.a.h	$1323$	$2$	1323.a	1.a	$1$	$1$	$1$	$10.564$	\(\Q\)	None	✓				189.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}-3q^{5}+6q^{11}+4q^{13}+4q^{16}+\cdots\)
1323.2.a.i	$1323$	$2$	1323.a	1.a	$1$	$1$	$1$	$10.564$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				27.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{4}-5q^{13}+4q^{16}+7q^{19}-5q^{25}+\cdots\)
1323.2.a.j	$1323$	$2$	1323.a	1.a	$1$	$1$	$1$	$10.564$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				189.2.e.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{4}-2q^{13}+4q^{16}+7q^{19}-5q^{25}+\cdots\)
1323.2.a.k	$1323$	$2$	1323.a	1.a	$1$	$1$	$1$	$10.564$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				189.2.e.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{4}+2q^{13}+4q^{16}-7q^{19}-5q^{25}+\cdots\)
1323.2.a.l	$1323$	$2$	1323.a	1.a	$1$	$1$	$1$	$10.564$	\(\Q\)	None	✓				189.2.a.b	$1$	$0$	\(0\)	\(0\)	\(3\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+3q^{5}-6q^{11}+4q^{13}+4q^{16}+\cdots\)
1323.2.a.m	$1323$	$2$	1323.a	1.a	$1$	$1$	$1$	$10.564$	\(\Q\)	None	✓				189.2.e.a	$1$	$0$	\(1\)	\(0\)	\(-4\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-4q^{5}-3q^{8}-4q^{10}+\cdots\)
1323.2.a.n	$1323$	$2$	1323.a	1.a	$1$	$1$	$1$	$10.564$	\(\Q\)	None	✓	✓			1323.2.a.e	$1$	$0$	\(1\)	\(0\)	\(-3\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-3q^{5}-3q^{8}-3q^{10}+\cdots\)
1323.2.a.o	$1323$	$2$	1323.a	1.a	$1$	$1$	$1$	$10.564$	\(\Q\)	None	✓	✓			1323.2.a.e	$1$	$1$	\(1\)	\(0\)	\(3\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+3q^{5}-3q^{8}+3q^{10}+\cdots\)
1323.2.a.p	$1323$	$2$	1323.a	1.a	$1$	$1$	$1$	$10.564$	\(\Q\)	None	✓				189.2.e.a	$1$	$0$	\(1\)	\(0\)	\(4\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+4q^{5}-3q^{8}+4q^{10}+\cdots\)
1323.2.a.q	$1323$	$2$	1323.a	1.a	$1$	$1$	$1$	$10.564$	\(\Q\)	None	✓	✓			1323.2.a.a	$1$	$1$	\(2\)	\(0\)	\(-3\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}-3q^{5}-6q^{10}+2q^{11}+\cdots\)
1323.2.a.r	$1323$	$2$	1323.a	1.a	$1$	$1$	$1$	$10.564$	\(\Q\)	None	✓				189.2.a.a	$1$	$0$	\(2\)	\(0\)	\(-1\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}-q^{5}-2q^{10}+4q^{11}+\cdots\)
1323.2.a.s	$1323$	$2$	1323.a	1.a	$1$	$1$	$1$	$10.564$	\(\Q\)	None	✓	✓			1323.2.a.a	$1$	$0$	\(2\)	\(0\)	\(3\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}+3q^{5}+6q^{10}+2q^{11}+\cdots\)
1323.2.a.t	$1323$	$2$	1323.a	1.a	$1$	$2$	$2$	$10.564$	\(\Q(\sqrt{3}) \)	None	✓				189.2.a.e	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{4}-\beta q^{5}-\beta q^{8}-3q^{10}+\cdots\)
1323.2.a.u	$1323$	$2$	1323.a	1.a	$1$	$2$	$2$	$10.564$	\(\Q(\sqrt{6}) \)	None	✓				189.2.e.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+4q^{4}-\beta q^{5}+2\beta q^{8}-6q^{10}+\cdots\)
1323.2.a.v	$1323$	$2$	1323.a	1.a	$1$	$2$	$2$	$10.564$	\(\Q(\sqrt{6}) \)	None	✓				189.2.e.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+4q^{4}+\beta q^{5}+2\beta q^{8}+6q^{10}+\cdots\)
1323.2.a.w	$1323$	$2$	1323.a	1.a	$1$	$2$	$2$	$10.564$	\(\Q(\sqrt{7}) \)	None	✓				189.2.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+5q^{4}+\beta q^{5}+3\beta q^{8}+7q^{10}+\cdots\)
1323.2.a.x	$1323$	$2$	1323.a	1.a	$1$	$3$	$3$	$10.564$	3.3.321.1	None	✓				189.2.e.e	$1$	$1$	\(-2\)	\(0\)	\(-1\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
1323.2.a.y	$1323$	$2$	1323.a	1.a	$1$	$3$	$3$	$10.564$	3.3.321.1	None	✓		✓		189.2.e.e	$1$	$1$	\(-2\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
1323.2.a.z	$1323$	$2$	1323.a	1.a	$1$	$3$	$3$	$10.564$	3.3.321.1	None	✓		✓		189.2.e.e	$1$	$0$	\(2\)	\(0\)	\(-1\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+\beta _{2}q^{5}+\cdots\)
1323.2.a.ba	$1323$	$2$	1323.a	1.a	$1$	$3$	$3$	$10.564$	3.3.321.1	None	✓				189.2.e.e	$1$	$0$	\(2\)	\(0\)	\(1\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}-\beta _{2}q^{5}+\cdots\)
1323.2.a.bb	$1323$	$2$	1323.a	1.a	$1$	$4$	$4$	$10.564$	\(\Q(\sqrt{2}, \sqrt{5})\)	None	✓	✓			1323.2.a.bb	$2$	$1$	\(0\)	\(0\)	\(-8\)	\(0\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{3})q^{4}+(-2-\beta _{3})q^{5}+\cdots\)
1323.2.a.bc	$1323$	$2$	1323.a	1.a	$1$	$4$	$4$	$10.564$	4.4.7168.1	None	✓	✓			1323.2.a.bc	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{1}q^{5}+\beta _{3}q^{8}+\cdots\)
1323.2.a.bd	$1323$	$2$	1323.a	1.a	$1$	$4$	$4$	$10.564$	4.4.7168.1	None	✓	✓			1323.2.a.bc	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{1}q^{5}+\beta _{3}q^{8}+\cdots\)
1323.2.a.be	$1323$	$2$	1323.a	1.a	$1$	$4$	$4$	$10.564$	\(\Q(\sqrt{2}, \sqrt{5})\)	None	✓	✓			1323.2.a.bb	$2$	$0$	\(0\)	\(0\)	\(8\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{3})q^{4}+(2+\beta _{3})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1323))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1323)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(189))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(441))\)\(^{\oplus 2}\)