Space of modular forms of level 50, weight 4, and trivial character

• Label: 50.4.a
• Level: $50$
• Weight: $4$
• Character orbit: 50.a
• Rep. character: $\chi_{50}(1,\cdot)$
• Character field: $\Q$
• Dimension: $5$
• Newform subspaces: $5$
• Sturm bound: $30$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	50.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(30\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(50))\).

	Total	New	Old
Modular forms	29	5	24
Cusp forms	17	5	12
Eisenstein series	12	0	12

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

\(2\)	\(5\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(2\)
\(+\)	\(-\)	\(-\)	\(1\)
\(-\)	\(-\)	\(+\)	\(2\)
Plus space		\(+\)	\(4\)
Minus space		\(-\)	\(1\)

Trace form

5 * q - 2 * q^2 + 8 * q^3 + 20 * q^4 + 20 * q^6 + 4 * q^7 - 8 * q^8 + 35 * q^9 + 10 * q^11 + 32 * q^12 + 58 * q^13 - 40 * q^14 + 80 * q^16 - 66 * q^17 - 74 * q^18 - 150 * q^19 - 340 * q^21 - 24 * q^22 - 132 * q^23 + 80 * q^24 - 180 * q^26 + 80 * q^27 + 16 * q^28 - 150 * q^29 + 260 * q^31 - 32 * q^32 + 96 * q^33 - 40 * q^34 + 140 * q^36 + 34 * q^37 + 200 * q^38 + 120 * q^39 + 760 * q^41 - 64 * q^42 - 32 * q^43 + 40 * q^44 - 280 * q^46 + 204 * q^47 + 128 * q^48 + 1965 * q^49 - 490 * q^51 + 232 * q^52 - 222 * q^53 - 700 * q^54 - 160 * q^56 - 800 * q^57 + 180 * q^58 - 1300 * q^59 + 1510 * q^61 - 304 * q^62 + 148 * q^63 + 320 * q^64 + 340 * q^66 + 1024 * q^67 - 264 * q^68 - 2380 * q^69 - 3240 * q^71 - 296 * q^72 - 362 * q^73 + 1460 * q^74 - 600 * q^76 + 48 * q^77 - 928 * q^78 - 300 * q^79 - 1195 * q^81 + 876 * q^82 - 72 * q^83 - 1360 * q^84 + 920 * q^86 - 720 * q^87 - 96 * q^88 + 1300 * q^89 + 2760 * q^91 - 528 * q^92 + 1216 * q^93 + 160 * q^94 + 320 * q^96 - 1106 * q^97 + 654 * q^98 + 2920 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(50))\) 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	5
50.4.a.a	$50$	$4$	50.a	1.a	$1$	$1$	$1$	$2.950$	\(\Q\)	None	✓	✓			50.4.a.a	$1$	$0$	\(-2\)	\(-7\)	\(0\)	\(34\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-7q^{3}+4q^{4}+14q^{6}+34q^{7}+\cdots\)
50.4.a.b	$50$	$4$	50.a	1.a	$1$	$1$	$1$	$2.950$	\(\Q\)	None	✓		✓	✓	10.4.b.a	$1$	$1$	\(-2\)	\(-2\)	\(0\)	\(-26\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-2q^{3}+4q^{4}+4q^{6}-26q^{7}+\cdots\)
50.4.a.c	$50$	$4$	50.a	1.a	$1$	$1$	$1$	$2.950$	\(\Q\)	None	✓				10.4.a.a	$1$	$0$	\(-2\)	\(8\)	\(0\)	\(4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+8q^{3}+4q^{4}-2^{4}q^{6}+4q^{7}+\cdots\)
50.4.a.d	$50$	$4$	50.a	1.a	$1$	$1$	$1$	$2.950$	\(\Q\)	None	✓				10.4.b.a	$1$	$0$	\(2\)	\(2\)	\(0\)	\(26\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{3}+4q^{4}+4q^{6}+26q^{7}+\cdots\)
50.4.a.e	$50$	$4$	50.a	1.a	$1$	$1$	$1$	$2.950$	\(\Q\)	None	✓	✓			50.4.a.a	$1$	$0$	\(2\)	\(7\)	\(0\)	\(-34\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+7q^{3}+4q^{4}+14q^{6}-34q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(50))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(50)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)