Space of modular forms of level 314, weight 2, and trivial character

• Label: 314.2.a
• Level: $314$
• Weight: $2$
• Character orbit: 314.a
• Rep. character: $\chi_{314}(1,\cdot)$
• Character field: $\Q$
• Dimension: $14$
• Newform subspaces: $3$
• Sturm bound: $79$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 314 = 2 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	314.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(79\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(314))\).

	Total	New	Old
Modular forms	41	14	27
Cusp forms	38	14	24
Eisenstein series	3	0	3

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

\(2\)	\(157\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(6\)	\(1\)	\(5\)		\(6\)	\(1\)	\(5\)		\(0\)	\(0\)	\(0\)
\(+\)	\(-\)	\(-\)		\(14\)	\(6\)	\(8\)		\(13\)	\(6\)	\(7\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(13\)	\(7\)	\(6\)		\(12\)	\(7\)	\(5\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(8\)	\(0\)	\(8\)		\(7\)	\(0\)	\(7\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(14\)	\(1\)	\(13\)		\(13\)	\(1\)	\(12\)		\(1\)	\(0\)	\(1\)
Minus space		\(-\)		\(27\)	\(13\)	\(14\)		\(25\)	\(13\)	\(12\)		\(2\)	\(0\)	\(2\)

Trace form

14 * q + 2 * q^3 + 14 * q^4 + 4 * q^5 - 4 * q^6 + 4 * q^7 + 20 * q^9 + 2 * q^10 + 6 * q^11 + 2 * q^12 + 10 * q^13 + 4 * q^14 + 8 * q^15 + 14 * q^16 + 8 * q^18 + 14 * q^19 + 4 * q^20 - 4 * q^21 - 8 * q^22 - 4 * q^23 - 4 * q^24 + 24 * q^25 + 4 * q^26 - 16 * q^27 + 4 * q^28 - 12 * q^30 + 12 * q^31 - 24 * q^33 + 8 * q^34 - 24 * q^35 + 20 * q^36 + 10 * q^37 - 12 * q^38 - 4 * q^39 + 2 * q^40 - 20 * q^41 - 12 * q^42 + 8 * q^43 + 6 * q^44 - 28 * q^45 + 4 * q^46 - 20 * q^47 + 2 * q^48 + 38 * q^49 - 28 * q^51 + 10 * q^52 - 40 * q^54 - 4 * q^55 + 4 * q^56 - 40 * q^57 + 10 * q^58 - 28 * q^59 + 8 * q^60 - 8 * q^61 - 12 * q^62 - 32 * q^63 + 14 * q^64 - 52 * q^65 - 24 * q^66 + 10 * q^67 - 40 * q^69 - 48 * q^70 + 4 * q^71 + 8 * q^72 + 8 * q^73 - 16 * q^74 - 82 * q^75 + 14 * q^76 - 52 * q^77 - 40 * q^78 - 16 * q^79 + 4 * q^80 + 30 * q^81 + 4 * q^82 - 24 * q^83 - 4 * q^84 + 24 * q^85 - 22 * q^86 - 24 * q^87 - 8 * q^88 - 4 * q^89 - 2 * q^90 + 48 * q^91 - 4 * q^92 + 36 * q^93 - 4 * q^94 - 36 * q^95 - 4 * q^96 + 4 * q^97 - 8 * q^98 + 34 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(314))\) 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	157
314.2.a.a	$314$	$2$	314.a	1.a	$1$	$1$	$1$	$2.507$	\(\Q\)	None	✓	✓	✓	✓	314.2.a.a	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-3q^{7}-q^{8}-3q^{9}-2q^{11}+\cdots\)
314.2.a.b	$314$	$2$	314.a	1.a	$1$	$6$	$6$	$2.507$	6.6.48599781.1	None	✓	✓	✓	✓	314.2.a.b	$1$	$0$	\(-6\)	\(3\)	\(1\)	\(3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(1+\beta _{3})q^{3}+q^{4}+(1-\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots\)
314.2.a.c	$314$	$2$	314.a	1.a	$1$	$7$	$7$	$2.507$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓	✓	314.2.a.c	$1$	$0$	\(7\)	\(-1\)	\(3\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-\beta _{2}q^{3}+q^{4}-\beta _{6}q^{5}-\beta _{2}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(314))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(314)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(157))\)\(^{\oplus 2}\)