Space of modular forms of level 50, weight 6, and trivial character

• Label: 50.6.a
• Level: $50$
• Weight: $6$
• Character orbit: 50.a
• Rep. character: $\chi_{50}(1,\cdot)$
• Character field: $\Q$
• Dimension: $7$
• Newform subspaces: $7$
• Sturm bound: $45$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	50.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(45\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(50))\).

	Total	New	Old
Modular forms	43	7	36
Cusp forms	31	7	24
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
\(+\)	\(+\)	\(+\)	\(1\)
\(+\)	\(-\)	\(-\)	\(2\)
\(-\)	\(+\)	\(-\)	\(3\)
\(-\)	\(-\)	\(+\)	\(1\)
Plus space		\(+\)	\(2\)
Minus space		\(-\)	\(5\)

Trace form

7 * q + 4 * q^2 - 4 * q^3 + 112 * q^4 + 8 * q^6 + 312 * q^7 + 64 * q^8 + 221 * q^9 + 814 * q^11 - 64 * q^12 - 114 * q^13 + 176 * q^14 + 1792 * q^16 - 918 * q^17 + 3892 * q^18 + 5590 * q^19 + 3284 * q^21 - 3312 * q^22 - 3144 * q^23 + 128 * q^24 - 4152 * q^26 + 5480 * q^27 + 4992 * q^28 - 7890 * q^29 - 9036 * q^31 + 1024 * q^32 - 24288 * q^33 - 13744 * q^34 + 3536 * q^36 + 16902 * q^37 + 17360 * q^38 - 15168 * q^39 - 32636 * q^41 - 11392 * q^42 - 13164 * q^43 + 13024 * q^44 - 4912 * q^46 - 44448 * q^47 - 1024 * q^48 + 16599 * q^49 + 47354 * q^51 - 1824 * q^52 + 44406 * q^53 + 680 * q^54 + 2816 * q^56 + 33040 * q^57 - 37320 * q^58 - 10180 * q^59 + 22514 * q^61 + 20768 * q^62 + 42376 * q^63 + 28672 * q^64 - 2984 * q^66 - 10788 * q^67 - 14688 * q^68 + 70892 * q^69 + 64464 * q^71 + 62272 * q^72 + 9666 * q^73 - 46984 * q^74 + 89440 * q^76 + 28464 * q^77 - 112576 * q^78 + 140860 * q^79 - 91633 * q^81 + 12648 * q^82 + 220476 * q^83 + 52544 * q^84 - 153232 * q^86 - 233880 * q^87 - 52992 * q^88 - 245720 * q^89 - 1696 * q^91 - 50304 * q^92 - 117968 * q^93 + 66496 * q^94 + 2048 * q^96 + 224682 * q^97 - 2652 * q^98 - 504008 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(50))\) 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
50.6.a.a	$50$	$6$	50.a	1.a	$1$	$1$	$1$	$8.019$	\(\Q\)	None	✓	✓			50.6.a.a	$1$	$0$	\(-4\)	\(-11\)	\(0\)	\(-142\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}-11q^{3}+2^{4}q^{4}+44q^{6}-142q^{7}+\cdots\)
50.6.a.b	$50$	$6$	50.a	1.a	$1$	$1$	$1$	$8.019$	\(\Q\)	None	✓		✓	✓	10.6.a.c	$1$	$1$	\(-4\)	\(-6\)	\(0\)	\(118\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}-6q^{3}+2^{4}q^{4}+24q^{6}+118q^{7}+\cdots\)
50.6.a.c	$50$	$6$	50.a	1.a	$1$	$1$	$1$	$8.019$	\(\Q\)	None	✓				10.6.b.a	$1$	$0$	\(-4\)	\(14\)	\(0\)	\(158\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+14q^{3}+2^{4}q^{4}-56q^{6}+158q^{7}+\cdots\)
50.6.a.d	$50$	$6$	50.a	1.a	$1$	$1$	$1$	$8.019$	\(\Q\)	None	✓				10.6.a.b	$1$	$0$	\(4\)	\(-24\)	\(0\)	\(172\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}-24q^{3}+2^{4}q^{4}-96q^{6}+172q^{7}+\cdots\)
50.6.a.e	$50$	$6$	50.a	1.a	$1$	$1$	$1$	$8.019$	\(\Q\)	None	✓		✓	✓	10.6.b.a	$1$	$1$	\(4\)	\(-14\)	\(0\)	\(-158\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}-14q^{3}+2^{4}q^{4}-56q^{6}-158q^{7}+\cdots\)
50.6.a.f	$50$	$6$	50.a	1.a	$1$	$1$	$1$	$8.019$	\(\Q\)	None	✓	✓			50.6.a.a	$1$	$0$	\(4\)	\(11\)	\(0\)	\(142\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+11q^{3}+2^{4}q^{4}+44q^{6}+142q^{7}+\cdots\)
50.6.a.g	$50$	$6$	50.a	1.a	$1$	$1$	$1$	$8.019$	\(\Q\)	None	✓				10.6.a.a	$1$	$0$	\(4\)	\(26\)	\(0\)	\(22\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+26q^{3}+2^{4}q^{4}+104q^{6}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(50))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(50)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)