Space of modular forms of level 402, weight 2, and trivial character

• Label: 402.2.a
• Level: $402$
• Weight: $2$
• Character orbit: 402.a
• Rep. character: $\chi_{402}(1,\cdot)$
• Character field: $\Q$
• Dimension: $11$
• Newform subspaces: $7$
• Sturm bound: $136$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 402 = 2 \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	402.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(136\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(402))\).

	Total	New	Old
Modular forms	72	11	61
Cusp forms	65	11	54
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.

\(2\)	\(3\)	\(67\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(1\)
\(+\)	\(+\)	\(-\)	$-$	\(2\)
\(+\)	\(-\)	\(+\)	$-$	\(1\)
\(+\)	\(-\)	\(-\)	$+$	\(1\)
\(-\)	\(+\)	\(+\)	$-$	\(3\)
\(-\)	\(-\)	\(-\)	$-$	\(3\)
Plus space			\(+\)	\(2\)
Minus space			\(-\)	\(9\)

Trace form

\( 11 q + q^{2} - q^{3} + 11 q^{4} + 6 q^{5} + q^{6} + 4 q^{7} + q^{8} + 11 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(402))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	67
402.2.a.a	$402$	$2$	402.a	1.a	$1$	$1$	$1$	$3.210$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(-1\)	\(1\)	\(-3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-3q^{7}+\cdots\)
402.2.a.b	$402$	$2$	402.a	1.a	$1$	$1$	$1$	$3.210$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(1\)	\(-3\)	\(-1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-3q^{5}-q^{6}-q^{7}+\cdots\)
402.2.a.c	$402$	$2$	402.a	1.a	$1$	$1$	$1$	$3.210$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(1\)	\(2\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+2q^{5}-q^{6}-q^{8}+\cdots\)
402.2.a.d	$402$	$2$	402.a	1.a	$1$	$1$	$1$	$3.210$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(-1\)	\(2\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+2q^{5}-q^{6}+2q^{7}+\cdots\)
402.2.a.e	$402$	$2$	402.a	1.a	$1$	$2$	$2$	$3.210$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(-2\)	\(-2\)	\(0\)	\(6\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+2\beta q^{5}+q^{6}+(3+\cdots)q^{7}+\cdots\)
402.2.a.f	$402$	$2$	402.a	1.a	$1$	$2$	$2$	$3.210$	\(\Q(\sqrt{41}) \)	None	✓	$1$	$0$	\(2\)	\(-2\)	\(1\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+\beta q^{5}-q^{6}-\beta q^{7}+\cdots\)
402.2.a.g	$402$	$2$	402.a	1.a	$1$	$3$	$3$	$3.210$	3.3.316.1	None	✓	$1$	$0$	\(3\)	\(3\)	\(3\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+(1-\beta _{2})q^{5}+q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(402))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(402)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(67))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(134))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(201))\)\(^{\oplus 2}\)