Space of modular forms of level 387, weight 2, and trivial character

• Label: 387.2.a
• Level: $387$
• Weight: $2$
• Character orbit: 387.a
• Rep. character: $\chi_{387}(1,\cdot)$
• Character field: $\Q$
• Dimension: $18$
• Newform subspaces: $10$
• Sturm bound: $88$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 387 = 3^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	387.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(88\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(387))\).

	Total	New	Old
Modular forms	48	18	30
Cusp forms	41	18	23
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.

\(3\)	\(43\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(10\)	\(2\)	\(8\)		\(9\)	\(2\)	\(7\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(14\)	\(6\)	\(8\)		\(12\)	\(6\)	\(6\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)		\(14\)	\(6\)	\(8\)		\(12\)	\(6\)	\(6\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)		\(10\)	\(4\)	\(6\)		\(8\)	\(4\)	\(4\)		\(2\)	\(0\)	\(2\)
Plus space		\(+\)		\(20\)	\(6\)	\(14\)		\(17\)	\(6\)	\(11\)		\(3\)	\(0\)	\(3\)
Minus space		\(-\)		\(28\)	\(12\)	\(16\)		\(24\)	\(12\)	\(12\)		\(4\)	\(0\)	\(4\)

Trace form

18 * q + q^2 + 15 * q^4 + 2 * q^5 - 4 * q^7 + 3 * q^8 + 2 * q^10 - 3 * q^11 + 7 * q^13 + 9 * q^16 + 5 * q^17 - 2 * q^19 + 18 * q^20 - 10 * q^22 - 19 * q^23 + 14 * q^25 - 24 * q^28 + 4 * q^29 + q^31 + 27 * q^32 - 8 * q^34 + 8 * q^35 + 10 * q^37 + 4 * q^38 + 6 * q^40 + 17 * q^41 + 2 * q^43 - 38 * q^44 - 6 * q^46 - 8 * q^47 + 22 * q^49 + 7 * q^50 + 20 * q^52 - 13 * q^53 - 20 * q^56 - 26 * q^58 - 8 * q^59 - 20 * q^61 - 42 * q^62 - 27 * q^64 + 16 * q^65 + 11 * q^67 + 28 * q^68 - 56 * q^70 - 18 * q^71 + 18 * q^74 - 16 * q^76 - 16 * q^77 + 12 * q^79 - 22 * q^80 + 16 * q^82 - 27 * q^83 - 20 * q^85 - 5 * q^86 - 36 * q^88 + 30 * q^89 - 20 * q^91 - 86 * q^92 - 24 * q^94 - 20 * q^95 - 9 * q^97 + 33 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(387))\) 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	43
387.2.a.a	$387$	$2$	387.a	1.a	$1$	$1$	$1$	$3.090$	\(\Q\)	None	✓				129.2.a.b	$1$	$0$	\(-1\)	\(0\)	\(-2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}-2q^{5}+3q^{8}+2q^{10}+\cdots\)
387.2.a.b	$387$	$2$	387.a	1.a	$1$	$1$	$1$	$3.090$	\(\Q\)	None	✓	✓			387.2.a.b	$1$	$1$	\(-1\)	\(0\)	\(1\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+q^{5}-3q^{7}+3q^{8}-q^{10}+\cdots\)
387.2.a.c	$387$	$2$	387.a	1.a	$1$	$1$	$1$	$3.090$	\(\Q\)	None	✓				129.2.a.a	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+2q^{5}-2q^{7}+5q^{11}+3q^{13}+\cdots\)
387.2.a.d	$387$	$2$	387.a	1.a	$1$	$1$	$1$	$3.090$	\(\Q\)	None	✓	✓			387.2.a.b	$1$	$1$	\(1\)	\(0\)	\(-1\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-q^{5}-3q^{7}-3q^{8}-q^{10}+\cdots\)
387.2.a.e	$387$	$2$	387.a	1.a	$1$	$1$	$1$	$3.090$	\(\Q\)	None	✓				43.2.a.a	$1$	$0$	\(2\)	\(0\)	\(4\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}+4q^{5}+8q^{10}-3q^{11}+\cdots\)
387.2.a.f	$387$	$2$	387.a	1.a	$1$	$2$	$2$	$3.090$	\(\Q(\sqrt{2}) \)	None	✓				129.2.a.c	$1$	$1$	\(-2\)	\(0\)	\(-2\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+(1-2\beta )q^{4}+(-1-\beta )q^{5}+\cdots\)
387.2.a.g	$387$	$2$	387.a	1.a	$1$	$2$	$2$	$3.090$	\(\Q(\sqrt{3}) \)	None	✓	✓			387.2.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+2\beta q^{5}+2q^{7}-3\beta q^{11}+5q^{13}+\cdots\)
387.2.a.h	$387$	$2$	387.a	1.a	$1$	$2$	$2$	$3.090$	\(\Q(\sqrt{2}) \)	None	✓				43.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-2-\beta )q^{5}+(-2-\beta )q^{7}+\cdots\)
387.2.a.i	$387$	$2$	387.a	1.a	$1$	$3$	$3$	$3.090$	3.3.568.1	None	✓		✓		129.2.a.d	$1$	$0$	\(2\)	\(0\)	\(4\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(3+\beta _{2})q^{4}+(1+\beta _{1}+\cdots)q^{5}+\cdots\)
387.2.a.j	$387$	$2$	387.a	1.a	$1$	$4$	$4$	$3.090$	\(\Q(\sqrt{5}, \sqrt{13})\)	None	✓	✓	✓		387.2.a.j	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(\beta _{1}+\beta _{3})q^{2}+3q^{4}+\beta _{1}q^{5}+(-1+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(387))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(387)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(43))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(129))\)\(^{\oplus 2}\)