Space of modular forms of level 1323, weight 4, and trivial character

• Label: 1323.4.a
• Level: $1323$
• Weight: $4$
• Character orbit: 1323.a
• Rep. character: $\chi_{1323}(1,\cdot)$
• Character field: $\Q$
• Dimension: $164$
• Newform subspaces: $43$
• Sturm bound: $672$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1323.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 43 \)
Sturm bound:			\(672\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(2\), \(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1323))\).

	Total	New	Old
Modular forms	528	164	364
Cusp forms	480	164	316
Eisenstein series	48	0	48

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	Dim
\(+\)	\(+\)	$+$	\(41\)
\(+\)	\(-\)	$-$	\(40\)
\(-\)	\(+\)	$-$	\(39\)
\(-\)	\(-\)	$+$	\(44\)
Plus space		\(+\)	\(85\)
Minus space		\(-\)	\(79\)

Trace form

\( 164 q + 662 q^{4} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1323))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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
1323.4.a.a	$1323$	$4$	1323.a	1.a	$1$	$1$	$1$	$78.060$	\(\Q\)	None	✓	$1$	$1$	\(-3\)	\(0\)	\(-12\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+q^{4}-12q^{5}+21q^{8}+6^{2}q^{10}+\cdots\)
1323.4.a.b	$1323$	$4$	1323.a	1.a	$1$	$1$	$1$	$78.060$	\(\Q\)	None	✓	$1$	$1$	\(-3\)	\(0\)	\(-6\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+q^{4}-6q^{5}+21q^{8}+18q^{10}+\cdots\)
1323.4.a.c	$1323$	$4$	1323.a	1.a	$1$	$1$	$1$	$78.060$	\(\Q\)	None	✓	$1$	$1$	\(-3\)	\(0\)	\(6\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+q^{4}+6q^{5}+21q^{8}-18q^{10}+\cdots\)
1323.4.a.d	$1323$	$4$	1323.a	1.a	$1$	$1$	$1$	$78.060$	\(\Q\)	None	✓	$1$	$1$	\(-3\)	\(0\)	\(15\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+q^{4}+15q^{5}+21q^{8}-45q^{10}+\cdots\)
1323.4.a.e	$1323$	$4$	1323.a	1.a	$1$	$1$	$1$	$78.060$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-21\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{4}-21q^{5}-21q^{11}-2q^{13}+\cdots\)
1323.4.a.f	$1323$	$4$	1323.a	1.a	$1$	$1$	$1$	$78.060$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-8q^{4}-89q^{13}+2^{6}q^{16}-56q^{19}+\cdots\)
1323.4.a.g	$1323$	$4$	1323.a	1.a	$1$	$1$	$1$	$78.060$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-8q^{4}-19q^{13}+2^{6}q^{16}+56q^{19}+\cdots\)
1323.4.a.h	$1323$	$4$	1323.a	1.a	$1$	$1$	$1$	$78.060$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-8q^{4}+19q^{13}+2^{6}q^{16}-56q^{19}+\cdots\)
1323.4.a.i	$1323$	$4$	1323.a	1.a	$1$	$1$	$1$	$78.060$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-8q^{4}+89q^{13}+2^{6}q^{16}+56q^{19}+\cdots\)
1323.4.a.j	$1323$	$4$	1323.a	1.a	$1$	$1$	$1$	$78.060$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(21\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{4}+21q^{5}+21q^{11}-2q^{13}+\cdots\)
1323.4.a.k	$1323$	$4$	1323.a	1.a	$1$	$1$	$1$	$78.060$	\(\Q\)	None	✓	$1$	$0$	\(3\)	\(0\)	\(-15\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+q^{4}-15q^{5}-21q^{8}-45q^{10}+\cdots\)
1323.4.a.l	$1323$	$4$	1323.a	1.a	$1$	$1$	$1$	$78.060$	\(\Q\)	None	✓	$1$	$0$	\(3\)	\(0\)	\(-6\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+q^{4}-6q^{5}-21q^{8}-18q^{10}+\cdots\)
1323.4.a.m	$1323$	$4$	1323.a	1.a	$1$	$1$	$1$	$78.060$	\(\Q\)	None	✓	$1$	$0$	\(3\)	\(0\)	\(6\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+q^{4}+6q^{5}-21q^{8}+18q^{10}+\cdots\)
1323.4.a.n	$1323$	$4$	1323.a	1.a	$1$	$1$	$1$	$78.060$	\(\Q\)	None	✓	$1$	$0$	\(3\)	\(0\)	\(12\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+q^{4}+12q^{5}-21q^{8}+6^{2}q^{10}+\cdots\)
1323.4.a.o	$1323$	$4$	1323.a	1.a	$1$	$2$	$2$	$78.060$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(-6\)	\(0\)	\(6\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-3+\beta )q^{2}+(4-6\beta )q^{4}+(3-4\beta )q^{5}+\cdots\)
1323.4.a.p	$1323$	$4$	1323.a	1.a	$1$	$2$	$2$	$78.060$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(-22\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+2\beta )q^{2}+(1-4\beta )q^{4}+(-11+\cdots)q^{5}+\cdots\)
1323.4.a.q	$1323$	$4$	1323.a	1.a	$1$	$2$	$2$	$78.060$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(-2\)	\(0\)	\(22\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+2\beta )q^{2}+(1-4\beta )q^{4}+(11-\beta )q^{5}+\cdots\)
1323.4.a.r	$1323$	$4$	1323.a	1.a	$1$	$2$	$2$	$78.060$	\(\Q(\sqrt{3}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-5q^{4}-\beta q^{5}-13\beta q^{8}-3q^{10}+\cdots\)
1323.4.a.s	$1323$	$4$	1323.a	1.a	$1$	$2$	$2$	$78.060$	\(\Q(\sqrt{7}) \)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-q^{4}+\beta q^{5}-9\beta q^{8}+7q^{10}+\cdots\)
1323.4.a.t	$1323$	$4$	1323.a	1.a	$1$	$2$	$2$	$78.060$	\(\Q(\sqrt{2}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+10q^{4}+4\beta q^{5}+2\beta q^{8}+72q^{10}+\cdots\)
1323.4.a.u	$1323$	$4$	1323.a	1.a	$1$	$2$	$2$	$78.060$	\(\Q(\sqrt{21}) \)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+13q^{4}+\beta q^{5}-5\beta q^{8}-21q^{10}+\cdots\)
1323.4.a.v	$1323$	$4$	1323.a	1.a	$1$	$2$	$2$	$78.060$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(2\)	\(0\)	\(-22\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+2\beta )q^{2}+(1+4\beta )q^{4}+(-11-\beta )q^{5}+\cdots\)
1323.4.a.w	$1323$	$4$	1323.a	1.a	$1$	$2$	$2$	$78.060$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(2\)	\(0\)	\(22\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+2\beta )q^{2}+(1+4\beta )q^{4}+(11+\beta )q^{5}+\cdots\)
1323.4.a.x	$1323$	$4$	1323.a	1.a	$1$	$2$	$2$	$78.060$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$1$	\(6\)	\(0\)	\(-6\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(3+\beta )q^{2}+(4+6\beta )q^{4}+(-3-4\beta )q^{5}+\cdots\)
1323.4.a.y	$1323$	$4$	1323.a	1.a	$1$	$3$	$3$	$78.060$	3.3.3576.1	None	✓	$1$	$1$	\(-1\)	\(0\)	\(2\)	\(0\)		$+$	$-$	$-$	$3$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(6-2\beta _{1}+\beta _{2})q^{4}+(1+2\beta _{1}+\cdots)q^{5}+\cdots\)
1323.4.a.z	$1323$	$4$	1323.a	1.a	$1$	$3$	$3$	$78.060$	3.3.3576.1	None	✓	$1$	$0$	\(1\)	\(0\)	\(-2\)	\(0\)		$-$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(6-2\beta _{1}+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
1323.4.a.ba	$1323$	$4$	1323.a	1.a	$1$	$4$	$4$	$78.060$	\(\Q(\sqrt{5}, \sqrt{13})\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-\beta _{1}-\beta _{2})q^{2}+(6-\beta _{3})q^{4}+(4\beta _{1}+\cdots)q^{5}+\cdots\)
1323.4.a.bb	$1323$	$4$	1323.a	1.a	$1$	$6$	$6$	$78.060$	6.6.346909504.1	None	✓	$2$	$1$	\(0\)	\(0\)	\(-24\)	\(0\)		$+$	$-$	$-$	$2^{4}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(3+\beta _{2})q^{4}+(-4+\beta _{4})q^{5}+\cdots\)
1323.4.a.bc	$1323$	$4$	1323.a	1.a	$1$	$6$	$6$	$78.060$	6.6.346909504.1	None	✓	$2$	$0$	\(0\)	\(0\)	\(24\)	\(0\)		$-$	$-$	$+$	$2^{4}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(3+\beta _{2})q^{4}+(4-\beta _{4})q^{5}+\cdots\)
1323.4.a.bd	$1323$	$4$	1323.a	1.a	$1$	$6$	$6$	$78.060$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(5+\beta _{3})q^{4}+\beta _{4}q^{5}+(\beta _{1}+\cdots)q^{8}+\cdots\)
1323.4.a.be	$1323$	$4$	1323.a	1.a	$1$	$6$	$6$	$78.060$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(5+\beta _{3})q^{4}-\beta _{4}q^{5}+(\beta _{1}+\cdots)q^{8}+\cdots\)
1323.4.a.bf	$1323$	$4$	1323.a	1.a	$1$	$6$	$6$	$78.060$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{3}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(6+\beta _{2})q^{4}+(-2\beta _{1}-\beta _{3}+\cdots)q^{5}+\cdots\)
1323.4.a.bg	$1323$	$4$	1323.a	1.a	$1$	$6$	$6$	$78.060$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{3}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(6+\beta _{2})q^{4}+(2\beta _{1}+\beta _{3})q^{5}+\cdots\)
1323.4.a.bh	$1323$	$4$	1323.a	1.a	$1$	$7$	$7$	$78.060$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(-1\)	\(0\)		$+$	$-$	$-$	$3^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(4+\beta _{1}+\beta _{2})q^{4}-\beta _{4}q^{5}+\cdots\)
1323.4.a.bi	$1323$	$4$	1323.a	1.a	$1$	$7$	$7$	$78.060$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(1\)	\(0\)		$-$	$+$	$-$	$3^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(4+\beta _{1}+\beta _{2})q^{4}+\beta _{4}q^{5}+\cdots\)
1323.4.a.bj	$1323$	$4$	1323.a	1.a	$1$	$7$	$7$	$78.060$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(-1\)	\(0\)		$+$	$+$	$+$	$3^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(4+\beta _{1}+\beta _{2})q^{4}-\beta _{4}q^{5}+\cdots\)
1323.4.a.bk	$1323$	$4$	1323.a	1.a	$1$	$7$	$7$	$78.060$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$+$	$3^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(4+\beta _{1}+\beta _{2})q^{4}+\beta _{4}q^{5}+\cdots\)
1323.4.a.bl	$1323$	$4$	1323.a	1.a	$1$	$8$	$8$	$78.060$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$7$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(6-\beta _{2}+\beta _{4})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
1323.4.a.bm	$1323$	$4$	1323.a	1.a	$1$	$8$	$8$	$78.060$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$7$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(6-\beta _{2}+\beta _{4})q^{4}+(\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots\)
1323.4.a.bn	$1323$	$4$	1323.a	1.a	$1$	$8$	$8$	$78.060$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{5}\cdot 3^{4}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+(6+\beta _{1})q^{4}-\beta _{3}q^{5}+(-5\beta _{2}+\cdots)q^{8}+\cdots\)
1323.4.a.bo	$1323$	$4$	1323.a	1.a	$1$	$8$	$8$	$78.060$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{5}\cdot 3^{4}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+(6+\beta _{1})q^{4}+\beta _{3}q^{5}+(-5\beta _{2}+\cdots)q^{8}+\cdots\)
1323.4.a.bp	$1323$	$4$	1323.a	1.a	$1$	$12$	$12$	$78.060$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(-40\)	\(0\)		$-$	$+$	$-$	$2^{6}\cdot 3^{4}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(4+\beta _{2})q^{4}+(-3-\beta _{5})q^{5}+\cdots\)
1323.4.a.bq	$1323$	$4$	1323.a	1.a	$1$	$12$	$12$	$78.060$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(40\)	\(0\)		$+$	$+$	$+$	$2^{6}\cdot 3^{4}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(4+\beta _{2})q^{4}+(3+\beta _{5})q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1323))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1323)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(189))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(441))\)\(^{\oplus 2}\)