Space of modular forms of level 381, weight 2, and trivial character

• Label: 381.2.a
• Level: $381$
• Weight: $2$
• Character orbit: 381.a
• Rep. character: $\chi_{381}(1,\cdot)$
• Character field: $\Q$
• Dimension: $21$
• Newform subspaces: $5$
• Sturm bound: $85$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 381 = 3 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	381.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(85\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(381))\).

	Total	New	Old
Modular forms	44	21	23
Cusp forms	41	21	20
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.

\(3\)	\(127\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(8\)	\(5\)	\(3\)		\(8\)	\(5\)	\(3\)		\(0\)	\(0\)	\(0\)
\(+\)	\(-\)	\(-\)		\(13\)	\(5\)	\(8\)		\(12\)	\(5\)	\(7\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(14\)	\(10\)	\(4\)		\(13\)	\(10\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(9\)	\(1\)	\(8\)		\(8\)	\(1\)	\(7\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(17\)	\(6\)	\(11\)		\(16\)	\(6\)	\(10\)		\(1\)	\(0\)	\(1\)
Minus space		\(-\)		\(27\)	\(15\)	\(12\)		\(25\)	\(15\)	\(10\)		\(2\)	\(0\)	\(2\)

Trace form

21 * q + q^2 + q^3 + 21 * q^4 - 6 * q^5 - q^6 + 4 * q^7 - 3 * q^8 + 21 * q^9 - 2 * q^10 + 8 * q^11 + 7 * q^12 + 2 * q^13 + 4 * q^14 + 2 * q^15 + 33 * q^16 - 10 * q^17 + q^18 + 4 * q^19 - 26 * q^20 + 4 * q^21 - 16 * q^22 - 9 * q^24 + 19 * q^25 - 18 * q^26 + q^27 - 4 * q^28 - 2 * q^29 + 6 * q^30 - 12 * q^31 - 25 * q^32 + 12 * q^33 - 4 * q^34 - 20 * q^35 + 21 * q^36 + 18 * q^37 - 22 * q^38 + 6 * q^39 + 2 * q^40 - 2 * q^41 - 12 * q^42 + 12 * q^43 + 58 * q^44 - 6 * q^45 - 8 * q^46 + 31 * q^48 + 33 * q^49 - 21 * q^50 - 6 * q^51 - 12 * q^52 - 26 * q^53 - q^54 + 24 * q^55 + 16 * q^56 + 12 * q^57 + 30 * q^58 + 16 * q^59 - 14 * q^60 - 22 * q^61 - 34 * q^62 + 4 * q^63 + 9 * q^64 - 16 * q^65 + 4 * q^66 + 24 * q^67 - 38 * q^68 - 12 * q^69 - 88 * q^70 + 36 * q^71 - 3 * q^72 - 18 * q^73 - 56 * q^74 + 23 * q^75 - 12 * q^76 - 28 * q^77 - 38 * q^78 + 8 * q^79 - 86 * q^80 + 21 * q^81 - 22 * q^82 - 20 * q^83 - 4 * q^84 - 48 * q^85 + 28 * q^86 + 14 * q^87 - 66 * q^88 + 2 * q^89 - 2 * q^90 + 56 * q^91 - 12 * q^92 - 60 * q^94 - 8 * q^95 - 49 * q^96 + 26 * q^97 - 39 * q^98 + 8 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(381))\) 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	127
381.2.a.a	$381$	$2$	381.a	1.a	$1$	$1$	$1$	$3.042$	\(\Q\)	None	✓	✓	✓	✓	381.2.a.a	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{4}-q^{5}-2q^{7}+q^{9}-4q^{11}+\cdots\)
381.2.a.b	$381$	$2$	381.a	1.a	$1$	$1$	$1$	$3.042$	\(\Q\)	None	✓	✓			381.2.a.b	$1$	$0$	\(2\)	\(1\)	\(3\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+q^{3}+2q^{4}+3q^{5}+2q^{6}+\cdots\)
381.2.a.c	$381$	$2$	381.a	1.a	$1$	$5$	$5$	$3.042$	5.5.81509.1	None	✓	✓	✓	✓	381.2.a.c	$1$	$1$	\(-1\)	\(-5\)	\(-5\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-q^{3}+\beta _{2}q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots\)
381.2.a.d	$381$	$2$	381.a	1.a	$1$	$5$	$5$	$3.042$	5.5.246832.1	None	✓	✓	✓	✓	381.2.a.d	$1$	$0$	\(2\)	\(-5\)	\(1\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}-q^{3}+(\beta _{1}+\beta _{2}+\beta _{3}+\beta _{4})q^{4}+\cdots\)
381.2.a.e	$381$	$2$	381.a	1.a	$1$	$9$	$9$	$3.042$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓		381.2.a.e	$1$	$0$	\(-2\)	\(9\)	\(-4\)	\(10\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}-\beta _{5}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(381))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(381)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(127))\)\(^{\oplus 2}\)