Space of modular forms of level 1338, weight 2, and trivial character

• Label: 1338.2.a
• Level: $1338$
• Weight: $2$
• Character orbit: 1338.a
• Rep. character: $\chi_{1338}(1,\cdot)$
• Character field: $\Q$
• Dimension: $37$
• Newform subspaces: $10$
• Sturm bound: $448$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 1338 = 2 \cdot 3 \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1338.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(448\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1338))\).

	Total	New	Old
Modular forms	228	37	191
Cusp forms	221	37	184
Eisenstein series	7	0	7

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

\(2\)	\(3\)	\(223\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(4\)
\(+\)	\(+\)	\(-\)	$-$	\(5\)
\(+\)	\(-\)	\(+\)	$-$	\(5\)
\(+\)	\(-\)	\(-\)	$+$	\(4\)
\(-\)	\(+\)	\(+\)	$-$	\(7\)
\(-\)	\(+\)	\(-\)	$+$	\(3\)
\(-\)	\(-\)	\(+\)	$+$	\(3\)
\(-\)	\(-\)	\(-\)	$-$	\(6\)
Plus space			\(+\)	\(14\)
Minus space			\(-\)	\(23\)

Trace form

\( 37 q + q^{2} - q^{3} + 37 q^{4} + 6 q^{5} - q^{6} + q^{8} + 37 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1338))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	223
1338.2.a.a	$1338$	$2$	1338.a	1.a	$1$	$1$	$1$	$10.684$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(1\)	\(-3\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-3q^{5}-q^{6}-q^{8}+\cdots\)
1338.2.a.b	$1338$	$2$	1338.a	1.a	$1$	$2$	$2$	$10.684$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(-2\)	\(-2\)	\(2\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+2\beta q^{5}+q^{6}-q^{8}+\cdots\)
1338.2.a.c	$1338$	$2$	1338.a	1.a	$1$	$3$	$3$	$10.684$	3.3.473.1	None	✓	$1$	$0$	\(-3\)	\(-3\)	\(-1\)	\(9\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}-\beta _{2}q^{5}+q^{6}+(3+\cdots)q^{7}+\cdots\)
1338.2.a.d	$1338$	$2$	1338.a	1.a	$1$	$3$	$3$	$10.684$	3.3.257.1	None	✓	$1$	$1$	\(-3\)	\(3\)	\(-1\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-\beta _{1}q^{5}-q^{6}+(\beta _{1}+\cdots)q^{7}+\cdots\)
1338.2.a.e	$1338$	$2$	1338.a	1.a	$1$	$3$	$3$	$10.684$	\(\Q(\zeta_{18})^+\)	None	✓	$1$	$1$	\(3\)	\(-3\)	\(-3\)	\(-3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+(-1-\beta _{1})q^{5}-q^{6}+\cdots\)
1338.2.a.f	$1338$	$2$	1338.a	1.a	$1$	$3$	$3$	$10.684$	\(\Q(\zeta_{14})^+\)	None	✓	$1$	$1$	\(3\)	\(3\)	\(-7\)	\(-9\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+(-2-\beta _{1})q^{5}+q^{6}+\cdots\)
1338.2.a.g	$1338$	$2$	1338.a	1.a	$1$	$4$	$4$	$10.684$	4.4.10273.1	None	✓	$1$	$1$	\(-4\)	\(-4\)	\(2\)	\(-5\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+(1-\beta _{1})q^{5}+q^{6}+\cdots\)
1338.2.a.h	$1338$	$2$	1338.a	1.a	$1$	$5$	$5$	$10.684$	5.5.356173.1	None	✓	$1$	$0$	\(-5\)	\(5\)	\(5\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+(1-\beta _{1}+\beta _{2})q^{5}+\cdots\)
1338.2.a.i	$1338$	$2$	1338.a	1.a	$1$	$6$	$6$	$10.684$	6.6.232773917.1	None	✓	$1$	$0$	\(6\)	\(6\)	\(6\)	\(5\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+(1-\beta _{1})q^{5}+q^{6}+\cdots\)
1338.2.a.j	$1338$	$2$	1338.a	1.a	$1$	$7$	$7$	$10.684$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(7\)	\(-7\)	\(6\)	\(3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+(1+\beta _{4})q^{5}-q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1338))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1338)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(223))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(446))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(669))\)\(^{\oplus 2}\)