Space of modular forms of level 399, weight 2, and trivial character

• Label: 399.2.a
• Level: $399$
• Weight: $2$
• Character orbit: 399.a
• Rep. character: $\chi_{399}(1,\cdot)$
• Character field: $\Q$
• Dimension: $19$
• Newform subspaces: $7$
• Sturm bound: $106$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 399 = 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	399.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(106\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(399))\).

	Total	New	Old
Modular forms	56	19	37
Cusp forms	49	19	30
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.

\(3\)	\(7\)	\(19\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)	\(5\)
\(+\)	\(-\)	\(-\)	\(+\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)	\(5\)
Plus space			\(+\)	\(2\)
Minus space			\(-\)	\(17\)

Trace form

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(399))\) 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	7	19
399.2.a.a	$399$	$2$	399.a	1.a	$1$	$1$	$1$	$3.186$	\(\Q\)	None	✓	✓	✓	✓	399.2.a.a	$1$	$1$	\(-1\)	\(-1\)	\(0\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}-q^{4}+q^{6}+q^{7}+3q^{8}+\cdots\)
399.2.a.b	$399$	$2$	399.a	1.a	$1$	$1$	$1$	$3.186$	\(\Q\)	None	✓	✓			399.2.a.b	$1$	$0$	\(-1\)	\(1\)	\(4\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}-q^{4}+4q^{5}-q^{6}-q^{7}+\cdots\)
399.2.a.c	$399$	$2$	399.a	1.a	$1$	$1$	$1$	$3.186$	\(\Q\)	None	✓	✓	✓	✓	399.2.a.c	$1$	$1$	\(1\)	\(-1\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}-q^{4}-q^{6}-q^{7}-3q^{8}+\cdots\)
399.2.a.d	$399$	$2$	399.a	1.a	$1$	$3$	$3$	$3.186$	3.3.148.1	None	✓	✓	✓	✓	399.2.a.d	$1$	$0$	\(1\)	\(-3\)	\(4\)	\(-3\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-q^{3}+(\beta _{1}+\beta _{2})q^{4}+(1+\beta _{1}+\cdots)q^{5}+\cdots\)
399.2.a.e	$399$	$2$	399.a	1.a	$1$	$3$	$3$	$3.186$	3.3.404.1	None	✓	✓	✓		399.2.a.e	$1$	$0$	\(1\)	\(3\)	\(0\)	\(-3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}+q^{3}+(3+\beta _{1}-\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
399.2.a.f	$399$	$2$	399.a	1.a	$1$	$5$	$5$	$3.186$	5.5.1240016.1	None	✓	✓	✓	✓	399.2.a.f	$1$	$0$	\(1\)	\(5\)	\(-2\)	\(5\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{2}+q^{3}+(1+\beta _{4})q^{4}+\beta _{2}q^{5}+\cdots\)
399.2.a.g	$399$	$2$	399.a	1.a	$1$	$5$	$5$	$3.186$	5.5.368464.1	None	✓	✓	✓	✓	399.2.a.g	$1$	$0$	\(3\)	\(-5\)	\(4\)	\(5\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(399))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(399)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(133))\)\(^{\oplus 2}\)