Space of modular forms of level 50, weight 4, and Character 50.b

• Label: 50.4.b
• Level: $50$
• Weight: $4$
• Character orbit: 50.b
• Rep. character: $\chi_{50}(49,\cdot)$
• Character field: $\Q$
• Dimension: $4$
• Newform subspaces: $2$
• Sturm bound: $30$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	50.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(30\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(50, [\chi])\).

	Total	New	Old
Modular forms	28	4	24
Cusp forms	16	4	12
Eisenstein series	12	0	12

Trace form

4 * q - 16 * q^4 - 4 * q^6 - 118 * q^9 + 78 * q^11 + 152 * q^14 + 64 * q^16 + 130 * q^19 - 412 * q^21 + 16 * q^24 - 344 * q^26 + 420 * q^29 + 668 * q^31 - 348 * q^34 + 472 * q^36 - 536 * q^39 - 162 * q^41 - 312 * q^44 + 216 * q^46 - 972 * q^49 - 762 * q^51 + 460 * q^54 - 608 * q^56 + 840 * q^59 + 1328 * q^61 - 256 * q^64 + 372 * q^66 + 3204 * q^69 - 552 * q^71 - 448 * q^74 - 520 * q^76 - 980 * q^79 - 2396 * q^81 + 1648 * q^84 - 464 * q^86 - 3090 * q^89 + 2368 * q^91 + 1152 * q^94 - 64 * q^96 - 2076 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(50, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
50.4.b.a	$50$	$4$	50.b	5.b	$2$	$2$	$2$	$2.950$	\(\Q(\sqrt{-1}) \)	None					10.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{2}+4\beta q^{3}-4 q^{4}-16 q^{6}+\cdots\)
50.4.b.b	$50$	$4$	50.b	5.b	$2$	$2$	$2$	$2.950$	\(\Q(\sqrt{-1}) \)	None		✓			50.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{2}-7 i q^{3}-4 q^{4}+14 q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(50, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(50, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)