Space of modular forms of level 50, weight 12, and Character 50.b

• Label: 50.12.b
• Level: $50$
• Weight: $12$
• Character orbit: 50.b
• Rep. character: $\chi_{50}(49,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $6$
• Sturm bound: $90$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	50.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(90\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(50, [\chi])\).

	Total	New	Old
Modular forms	88	16	72
Cusp forms	76	16	60
Eisenstein series	12	0	12

Trace form

16 * q - 16384 * q^4 - 18496 * q^6 - 711622 * q^9 + 627762 * q^11 + 2100608 * q^14 + 16777216 * q^16 + 36373390 * q^19 - 101516548 * q^21 + 18939904 * q^24 - 54490496 * q^26 + 71084820 * q^29 - 785611948 * q^31 + 871852608 * q^34 + 728700928 * q^36 - 3616815224 * q^39 - 290144718 * q^41 - 642828288 * q^44 + 4411300224 * q^46 - 10488744408 * q^49 + 18427748202 * q^51 - 2972290880 * q^54 - 2151022592 * q^56 - 29952162360 * q^59 + 27837074072 * q^61 - 17179869184 * q^64 - 40563271872 * q^66 + 26185672956 * q^69 - 45239755368 * q^71 + 55603862528 * q^74 - 37246351360 * q^76 - 63853443500 * q^79 + 115622884096 * q^81 + 103952945152 * q^84 - 130978846976 * q^86 - 77857879710 * q^89 + 772575952 * q^91 + 486817102848 * q^94 - 19394461696 * q^96 - 299335762404 * q^99

Decomposition of \(S_{12}^{\mathrm{new}}(50, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
50.12.b.a	$50$	$12$	50.b	5.b	$2$	$2$	$2$	$38.417$	\(\Q(\sqrt{-1}) \)	None					10.12.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+16\beta q^{2}+369\beta q^{3}-1024 q^{4}+\cdots\)
50.12.b.b	$50$	$12$	50.b	5.b	$2$	$2$	$2$	$38.417$	\(\Q(\sqrt{-1}) \)	None					10.12.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+16\beta q^{2}+159\beta q^{3}-1024 q^{4}+\cdots\)
50.12.b.c	$50$	$12$	50.b	5.b	$2$	$2$	$2$	$38.417$	\(\Q(\sqrt{-1}) \)	None					10.12.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-16\beta q^{2}+6\beta q^{3}-1024 q^{4}+384 q^{6}+\cdots\)
50.12.b.d	$50$	$12$	50.b	5.b	$2$	$2$	$2$	$38.417$	\(\Q(\sqrt{-1}) \)	None		✓			50.12.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-32 i q^{2}+207 i q^{3}-1024 q^{4}+\cdots\)
50.12.b.e	$50$	$12$	50.b	5.b	$2$	$4$	$4$	$38.417$	\(\mathbb{Q}[x]/(x^{4} + \cdots)\)	None		✓			50.12.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-2^{5}\beta _{1}q^{2}+(-28\beta _{1}+\beta _{2})q^{3}-2^{10}q^{4}+\cdots\)
50.12.b.f	$50$	$12$	50.b	5.b	$2$	$4$	$4$	$38.417$	\(\Q(i, \sqrt{1969})\)	None					10.12.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-2^{4}\beta _{1}q^{2}+(151\beta _{1}-\beta _{2})q^{3}-2^{10}q^{4}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(50, [\chi])\) into lower level spaces

\( S_{12}^{\mathrm{old}}(50, [\chi]) \simeq \) \(S_{12}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)