Space of modular forms of level 385, weight 2, and trivial character

• Label: 385.2.a
• Level: $385$
• Weight: $2$
• Character orbit: 385.a
• Rep. character: $\chi_{385}(1,\cdot)$
• Character field: $\Q$
• Dimension: $19$
• Newform subspaces: $8$
• Sturm bound: $96$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 385 = 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	385.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(96\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(385))\).

	Total	New	Old
Modular forms	52	19	33
Cusp forms	45	19	26
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.

\(5\)	\(7\)	\(11\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(3\)
\(+\)	\(+\)	\(-\)	$-$	\(2\)
\(+\)	\(-\)	\(+\)	$-$	\(2\)
\(+\)	\(-\)	\(-\)	$+$	\(3\)
\(-\)	\(+\)	\(+\)	$-$	\(4\)
\(-\)	\(+\)	\(-\)	$+$	\(1\)
\(-\)	\(-\)	\(+\)	$+$	\(1\)
\(-\)	\(-\)	\(-\)	$-$	\(3\)
Plus space			\(+\)	\(8\)
Minus space			\(-\)	\(11\)

Trace form

\( 19 q - 3 q^{2} - 4 q^{3} + 21 q^{4} - q^{5} + 20 q^{6} - q^{7} + 9 q^{8} + 7 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(385))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces					A-L signs			Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		5	7	11
385.2.a.a	$385$	$2$	385.a	1.a	$1$	$1$	$1$	$3.074$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(-2\)	\(1\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-2q^{3}-q^{4}+q^{5}+2q^{6}+q^{7}+\cdots\)
385.2.a.b	$385$	$2$	385.a	1.a	$1$	$1$	$1$	$3.074$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(1\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+q^{5}-q^{7}+3q^{8}-3q^{9}+\cdots\)
385.2.a.c	$385$	$2$	385.a	1.a	$1$	$2$	$2$	$3.074$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(-2\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(1+\beta )q^{3}+q^{4}-q^{5}+(3+\beta )q^{6}+\cdots\)
385.2.a.d	$385$	$2$	385.a	1.a	$1$	$2$	$2$	$3.074$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(2\)	\(0\)	\(-2\)	\(-2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+\beta q^{3}+(1+2\beta )q^{4}-q^{5}+\cdots\)
385.2.a.e	$385$	$2$	385.a	1.a	$1$	$3$	$3$	$3.074$	3.3.148.1	None	✓	$1$	$1$	\(-3\)	\(-4\)	\(-3\)	\(3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{2}+(-1-\beta _{1})q^{3}+(2+\cdots)q^{4}+\cdots\)
385.2.a.f	$385$	$2$	385.a	1.a	$1$	$3$	$3$	$3.074$	3.3.148.1	None	✓	$1$	$1$	\(-3\)	\(-2\)	\(-3\)	\(-3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{2}+(-1+\beta _{1})q^{3}+(2+\cdots)q^{4}+\cdots\)
385.2.a.g	$385$	$2$	385.a	1.a	$1$	$3$	$3$	$3.074$	3.3.148.1	None	✓	$1$	$0$	\(1\)	\(0\)	\(3\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{2}q^{3}+(\beta _{1}+\beta _{2})q^{4}+q^{5}+\cdots\)
385.2.a.h	$385$	$2$	385.a	1.a	$1$	$4$	$4$	$3.074$	4.4.11348.1	None	✓	$1$	$0$	\(2\)	\(2\)	\(4\)	\(-4\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{2}+(1+\beta _{3})q^{3}+(2+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(385))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(385)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 2}\)