Space of modular forms of level 50321, weight 2, and trivial character

• Label: 50321.2.a
• Level: $50321$
• Weight: $2$
• Character orbit: 50321.a
• Rep. character: $\chi_{50321}(1,\cdot)$
• Character field: $\Q$
• Dimension: $4193$
• Newform subspaces: $3$
• Sturm bound: $8387$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 50321 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	50321.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(8387\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(50321))\).

	Total	New	Old
Modular forms	4194	4194	0
Cusp forms	4193	4193	0
Eisenstein series	1	1	0

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

\(50321\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(2007\)	\(2007\)	\(0\)		\(2007\)	\(2007\)	\(0\)		\(0\)	\(0\)	\(0\)
\(-\)		\(2187\)	\(2187\)	\(0\)		\(2186\)	\(2186\)	\(0\)		\(1\)	\(1\)	\(0\)

Trace form

4193 * q - q^2 + 4195 * q^4 - 4 * q^5 + 4 * q^7 - 3 * q^8 + 4189 * q^9 - 10 * q^10 + 2 * q^12 + 2 * q^13 + 4 * q^14 + 10 * q^15 + 4203 * q^16 - 14 * q^17 + 9 * q^18 + 2 * q^19 - 2 * q^20 - 16 * q^21 + 8 * q^22 + 6 * q^24 + 4189 * q^25 + 2 * q^26 + 12 * q^27 + 28 * q^28 - 4 * q^29 + 2 * q^30 + 18 * q^31 + 23 * q^32 - 14 * q^33 + 20 * q^34 - 8 * q^35 + 4189 * q^36 - 4 * q^37 - 18 * q^38 + 20 * q^39 - 8 * q^40 - 22 * q^41 - 2 * q^42 - 18 * q^43 - 16 * q^44 - 24 * q^45 + 30 * q^46 + 20 * q^48 + 4187 * q^49 + 27 * q^50 + 4 * q^51 + 46 * q^52 + 10 * q^53 - 42 * q^54 + 12 * q^55 - 6 * q^56 + 16 * q^57 + 24 * q^58 - 8 * q^59 + 74 * q^60 + 8 * q^61 + 14 * q^62 + 4269 * q^64 - 30 * q^65 - 26 * q^66 + 14 * q^67 - 34 * q^68 - 10 * q^69 + 12 * q^70 + 34 * q^71 - 13 * q^72 - 18 * q^73 - 16 * q^74 - 24 * q^75 + 18 * q^76 - 28 * q^77 - 72 * q^78 + 44 * q^79 - 12 * q^80 + 4161 * q^81 - 6 * q^82 - 28 * q^83 - 122 * q^84 - 36 * q^85 - 34 * q^86 - 12 * q^87 + 36 * q^88 - 32 * q^89 - 50 * q^90 + 26 * q^91 + 30 * q^92 - 8 * q^93 - 16 * q^94 - 22 * q^95 + 24 * q^96 + 14 * q^97 - 19 * q^98 - 24 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(50321))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		50321
50321.2.a.a	$50321$	$2$	50321.a	1.a	$1$	$2$	$2$	$401.815$	\(\Q(\sqrt{13}) \)	None	✓	✓			50321.2.a.a	$1$		\(0\)	\(2\)	\(-2\)	\(-3\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{4}-q^{5}+(-2+\beta )q^{7}-2q^{9}+\cdots\)
50321.2.a.b	$50321$	$2$	50321.a	1.a	$1$	$2007$	$2007$	$401.815$		None	✓	✓	✓	✓	50321.2.a.b	$1$		\(-43\)	\(-68\)	\(-49\)	\(-197\)		$+$	$+$		$\mathrm{SU}(2)$
50321.2.a.c	$50321$	$2$	50321.a	1.a	$1$	$2184$	$2184$	$401.815$		None	✓	✓	✓		50321.2.a.c	$1$		\(42\)	\(66\)	\(47\)	\(204\)		$-$	$-$		$\mathrm{SU}(2)$