Space of modular forms of level 364, weight 2, and trivial character

• Label: 364.2.a
• Level: $364$
• Weight: $2$
• Character orbit: 364.a
• Rep. character: $\chi_{364}(1,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $4$
• Sturm bound: $112$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 364 = 2^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	364.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(112\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(364))\).

	Total	New	Old
Modular forms	62	6	56
Cusp forms	51	6	45
Eisenstein series	11	0	11

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

\(2\)	\(7\)	\(13\)	Fricke	Dim
\(-\)	\(+\)	\(+\)	\(-\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)	\(2\)
Plus space			\(+\)	\(2\)
Minus space			\(-\)	\(4\)

Trace form

\( 6 q - 4 q^{5} + 6 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(364))\) 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	7	13
364.2.a.a	$364$	$2$	364.a	1.a	$1$	$1$	$1$	$2.907$	\(\Q\)	None	✓	✓	✓	✓	364.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(1\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}-q^{7}+q^{9}-4q^{11}+q^{13}+\cdots\)
364.2.a.b	$364$	$2$	364.a	1.a	$1$	$1$	$1$	$2.907$	\(\Q\)	None	✓	✓	✓	✓	364.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}+q^{7}-3q^{9}-2q^{11}-q^{13}+\cdots\)
364.2.a.c	$364$	$2$	364.a	1.a	$1$	$2$	$2$	$2.907$	\(\Q(\sqrt{6}) \)	None	✓	✓	✓	✓	364.2.a.c	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(-1+\beta )q^{5}-q^{7}+3q^{9}+\cdots\)
364.2.a.d	$364$	$2$	364.a	1.a	$1$	$2$	$2$	$2.907$	\(\Q(\sqrt{3}) \)	None	✓	✓	✓	✓	364.2.a.d	$1$	$0$	\(0\)	\(2\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-\beta q^{5}+q^{7}+(1+2\beta )q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(364))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(364)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(91))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(182))\)\(^{\oplus 2}\)