Space of modular forms of level 1386, weight 2, and trivial character

• Label: 1386.2.a
• Level: $1386$
• Weight: $2$
• Character orbit: 1386.a
• Rep. character: $\chi_{1386}(1,\cdot)$
• Character field: $\Q$
• Dimension: $26$
• Newform subspaces: $18$
• Sturm bound: $576$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1386.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 18 \)
Sturm bound:			\(576\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1386))\).

	Total	New	Old
Modular forms	304	26	278
Cusp forms	273	26	247
Eisenstein series	31	0	31

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

\(2\)	\(3\)	\(7\)	\(11\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)	\(1\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)	\(2\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)	\(3\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)	\(2\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)	\(2\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)	\(1\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)	\(2\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)	\(3\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)	\(3\)
Plus space				\(+\)	\(7\)
Minus space				\(-\)	\(19\)

Trace form

26 * q + 2 * q^2 + 26 * q^4 + 2 * q^8 + 4 * q^10 - 2 * q^11 + 4 * q^13 - 4 * q^14 + 26 * q^16 + 12 * q^17 + 8 * q^19 + 2 * q^22 + 8 * q^23 + 54 * q^25 + 16 * q^26 + 28 * q^29 + 16 * q^31 + 2 * q^32 + 12 * q^34 + 8 * q^35 + 28 * q^37 + 12 * q^38 + 4 * q^40 - 4 * q^41 + 8 * q^43 - 2 * q^44 - 24 * q^47 + 26 * q^49 + 6 * q^50 + 4 * q^52 - 4 * q^53 + 12 * q^55 - 4 * q^56 + 4 * q^58 - 4 * q^59 + 36 * q^61 - 8 * q^62 + 26 * q^64 - 16 * q^65 + 12 * q^68 + 4 * q^70 + 8 * q^71 - 28 * q^73 + 28 * q^74 + 8 * q^76 - 4 * q^77 - 32 * q^79 + 12 * q^82 + 16 * q^83 + 24 * q^85 - 8 * q^86 + 2 * q^88 + 20 * q^89 - 4 * q^91 + 8 * q^92 - 24 * q^94 - 8 * q^95 + 44 * q^97 + 2 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1386))\) 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	3	7	11
1386.2.a.a	$1386$	$2$	1386.a	1.a	$1$	$1$	$1$	$11.067$	\(\Q\)	None	✓		✓	✓	462.2.a.g	$1$	$0$	\(-1\)	\(0\)	\(-2\)	\(-1\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2q^{5}-q^{7}-q^{8}+2q^{10}+\cdots\)
1386.2.a.b	$1386$	$2$	1386.a	1.a	$1$	$1$	$1$	$11.067$	\(\Q\)	None	✓		✓	✓	154.2.a.c	$1$	$1$	\(-1\)	\(0\)	\(-2\)	\(-1\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2q^{5}-q^{7}-q^{8}+2q^{10}+\cdots\)
1386.2.a.c	$1386$	$2$	1386.a	1.a	$1$	$1$	$1$	$11.067$	\(\Q\)	None	✓				462.2.a.f	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(1\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{7}-q^{8}+q^{11}+2q^{13}+\cdots\)
1386.2.a.d	$1386$	$2$	1386.a	1.a	$1$	$1$	$1$	$11.067$	\(\Q\)	None	✓	✓	✓	✓	1386.2.a.d	$1$	$1$	\(-1\)	\(0\)	\(2\)	\(-1\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+2q^{5}-q^{7}-q^{8}-2q^{10}+\cdots\)
1386.2.a.e	$1386$	$2$	1386.a	1.a	$1$	$1$	$1$	$11.067$	\(\Q\)	None	✓				462.2.a.e	$1$	$0$	\(-1\)	\(0\)	\(4\)	\(1\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+4q^{5}+q^{7}-q^{8}-4q^{10}+\cdots\)
1386.2.a.f	$1386$	$2$	1386.a	1.a	$1$	$1$	$1$	$11.067$	\(\Q\)	None	✓				154.2.a.b	$1$	$1$	\(1\)	\(0\)	\(-2\)	\(-1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-2q^{5}-q^{7}+q^{8}-2q^{10}+\cdots\)
1386.2.a.g	$1386$	$2$	1386.a	1.a	$1$	$1$	$1$	$11.067$	\(\Q\)	None	✓				462.2.a.c	$1$	$1$	\(1\)	\(0\)	\(-2\)	\(-1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-2q^{5}-q^{7}+q^{8}-2q^{10}+\cdots\)
1386.2.a.h	$1386$	$2$	1386.a	1.a	$1$	$1$	$1$	$11.067$	\(\Q\)	None	✓	✓	✓	✓	1386.2.a.d	$1$	$1$	\(1\)	\(0\)	\(-2\)	\(-1\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-2q^{5}-q^{7}+q^{8}-2q^{10}+\cdots\)
1386.2.a.i	$1386$	$2$	1386.a	1.a	$1$	$1$	$1$	$11.067$	\(\Q\)	None	✓				462.2.a.b	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{7}+q^{8}+q^{11}-2q^{13}+\cdots\)
1386.2.a.j	$1386$	$2$	1386.a	1.a	$1$	$1$	$1$	$11.067$	\(\Q\)	None	✓				462.2.a.d	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{7}+q^{8}+q^{11}+6q^{13}+\cdots\)
1386.2.a.k	$1386$	$2$	1386.a	1.a	$1$	$1$	$1$	$11.067$	\(\Q\)	None	✓				462.2.a.a	$1$	$0$	\(1\)	\(0\)	\(2\)	\(1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+2q^{5}+q^{7}+q^{8}+2q^{10}+\cdots\)
1386.2.a.l	$1386$	$2$	1386.a	1.a	$1$	$1$	$1$	$11.067$	\(\Q\)	None	✓				154.2.a.a	$1$	$0$	\(1\)	\(0\)	\(4\)	\(-1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+4q^{5}-q^{7}+q^{8}+4q^{10}+\cdots\)
1386.2.a.m	$1386$	$2$	1386.a	1.a	$1$	$2$	$2$	$11.067$	\(\Q(\sqrt{5}) \)	None	✓		✓	✓	154.2.a.d	$1$	$1$	\(-2\)	\(0\)	\(-2\)	\(2\)		$+$	$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(-1-\beta )q^{5}+q^{7}-q^{8}+\cdots\)
1386.2.a.n	$1386$	$2$	1386.a	1.a	$1$	$2$	$2$	$11.067$	\(\Q(\sqrt{10}) \)	None	✓	✓	✓	✓	1386.2.a.n	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+\beta q^{5}-q^{7}-q^{8}-\beta q^{10}+\cdots\)
1386.2.a.o	$1386$	$2$	1386.a	1.a	$1$	$2$	$2$	$11.067$	\(\Q(\sqrt{10}) \)	None	✓	✓	✓	✓	1386.2.a.n	$1$	$0$	\(2\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+\beta q^{5}-q^{7}+q^{8}+\beta q^{10}+\cdots\)
1386.2.a.p	$1386$	$2$	1386.a	1.a	$1$	$2$	$2$	$11.067$	\(\Q(\sqrt{3}) \)	None	✓		✓		462.2.a.h	$1$	$0$	\(2\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+\beta q^{5}+q^{7}+q^{8}+\beta q^{10}+\cdots\)
1386.2.a.q	$1386$	$2$	1386.a	1.a	$1$	$3$	$3$	$11.067$	3.3.1304.1	None	✓	✓	✓	✓	1386.2.a.q	$1$	$0$	\(-3\)	\(0\)	\(-2\)	\(3\)		$+$	$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(-1-\beta _{2})q^{5}+q^{7}-q^{8}+\cdots\)
1386.2.a.r	$1386$	$2$	1386.a	1.a	$1$	$3$	$3$	$11.067$	3.3.1304.1	None	✓	✓	✓	✓	1386.2.a.q	$1$	$0$	\(3\)	\(0\)	\(2\)	\(3\)		$-$	$+$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(1+\beta _{2})q^{5}+q^{7}+q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1386))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1386)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(154))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(198))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(231))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(462))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(693))\)\(^{\oplus 2}\)