Space of modular forms of level 338, weight 2, and trivial character

• Label: 338.2.a
• Level: $338$
• Weight: $2$
• Character orbit: 338.a
• Rep. character: $\chi_{338}(1,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $8$
• Sturm bound: $91$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	338.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(91\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(338))\).

	Total	New	Old
Modular forms	59	12	47
Cusp forms	32	12	20
Eisenstein series	27	0	27

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

\(2\)	\(13\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(2\)
\(+\)	\(-\)	\(-\)	\(4\)
\(-\)	\(+\)	\(-\)	\(5\)
\(-\)	\(-\)	\(+\)	\(1\)
Plus space		\(+\)	\(3\)
Minus space		\(-\)	\(9\)

Trace form

12 * q + 2 * q^3 + 12 * q^4 + 4 * q^5 + 4 * q^6 + 10 * q^9 - 2 * q^10 - 4 * q^11 + 2 * q^12 - 4 * q^14 + 12 * q^16 + 4 * q^17 - 8 * q^18 - 8 * q^19 + 4 * q^20 + 4 * q^21 + 6 * q^22 + 4 * q^24 + 10 * q^25 + 8 * q^27 - 14 * q^29 - 12 * q^30 - 12 * q^33 - 12 * q^35 + 10 * q^36 + 4 * q^37 - 6 * q^38 - 2 * q^40 - 4 * q^42 + 2 * q^43 - 4 * q^44 + 4 * q^46 - 16 * q^47 + 2 * q^48 + 16 * q^49 + 8 * q^50 - 20 * q^51 - 18 * q^53 + 4 * q^54 + 8 * q^55 - 4 * q^56 + 16 * q^57 + 4 * q^58 + 16 * q^59 - 10 * q^61 - 16 * q^62 - 8 * q^63 + 12 * q^64 - 8 * q^66 - 12 * q^67 + 4 * q^68 - 28 * q^69 + 4 * q^70 + 8 * q^71 - 8 * q^72 + 8 * q^73 - 18 * q^74 - 38 * q^75 - 8 * q^76 - 12 * q^77 - 20 * q^79 + 4 * q^80 - 20 * q^81 - 4 * q^82 - 12 * q^83 + 4 * q^84 - 12 * q^85 + 4 * q^86 - 20 * q^87 + 6 * q^88 - 2 * q^90 + 16 * q^93 - 16 * q^94 - 12 * q^95 + 4 * q^96 - 4 * q^97 + 24 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(338))\) 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	13
338.2.a.a	$338$	$2$	338.a	1.a	$1$	$1$	$1$	$2.699$	\(\Q\)	None	✓				26.2.a.b	$1$	$1$	\(-1\)	\(-3\)	\(1\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-3q^{3}+q^{4}+q^{5}+3q^{6}-q^{7}+\cdots\)
338.2.a.b	$338$	$2$	338.a	1.a	$1$	$1$	$1$	$2.699$	\(\Q\)	None	✓				26.2.b.a	$1$	$0$	\(-1\)	\(-1\)	\(3\)	\(3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+3q^{5}+q^{6}+3q^{7}+\cdots\)
338.2.a.c	$338$	$2$	338.a	1.a	$1$	$1$	$1$	$2.699$	\(\Q\)	None	✓				26.2.c.a	$1$	$1$	\(-1\)	\(0\)	\(1\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}-4q^{7}-q^{8}-3q^{9}+\cdots\)
338.2.a.d	$338$	$2$	338.a	1.a	$1$	$1$	$1$	$2.699$	\(\Q\)	None	✓		✓	✓	26.2.b.a	$1$	$1$	\(1\)	\(-1\)	\(-3\)	\(-3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-3q^{5}-q^{6}-3q^{7}+\cdots\)
338.2.a.e	$338$	$2$	338.a	1.a	$1$	$1$	$1$	$2.699$	\(\Q\)	None	✓				26.2.c.a	$1$	$0$	\(1\)	\(0\)	\(-1\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}+4q^{7}+q^{8}-3q^{9}+\cdots\)
338.2.a.f	$338$	$2$	338.a	1.a	$1$	$1$	$1$	$2.699$	\(\Q\)	None	✓				26.2.a.a	$1$	$0$	\(1\)	\(1\)	\(3\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+3q^{5}+q^{6}+q^{7}+\cdots\)
338.2.a.g	$338$	$2$	338.a	1.a	$1$	$3$	$3$	$2.699$	\(\Q(\zeta_{14})^+\)	None	✓	✓	✓		338.2.a.g	$1$	$0$	\(-3\)	\(3\)	\(-2\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(1-\beta _{1}-\beta _{2})q^{3}+q^{4}-2\beta _{1}q^{5}+\cdots\)
338.2.a.h	$338$	$2$	338.a	1.a	$1$	$3$	$3$	$2.699$	\(\Q(\zeta_{14})^+\)	None	✓	✓	✓		338.2.a.g	$1$	$0$	\(3\)	\(3\)	\(2\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(1-\beta _{1}-\beta _{2})q^{3}+q^{4}+2\beta _{1}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(338))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(338)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 2}\)