Space of modular forms of level 50, weight 8, and trivial character

• Label: 50.8.a
• Level: $50$
• Weight: $8$
• Character orbit: 50.a
• Rep. character: $\chi_{50}(1,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $10$
• Sturm bound: $60$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	50.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(60\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(50))\).

	Total	New	Old
Modular forms	59	12	47
Cusp forms	47	12	35
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
\(+\)	\(+\)	\(+\)	\(3\)
\(+\)	\(-\)	\(-\)	\(3\)
\(-\)	\(+\)	\(-\)	\(2\)
\(-\)	\(-\)	\(+\)	\(4\)
Plus space		\(+\)	\(7\)
Minus space		\(-\)	\(5\)

Trace form

12 * q - 40 * q^3 + 768 * q^4 - 592 * q^6 - 1120 * q^7 + 15554 * q^9 - 3366 * q^11 - 2560 * q^12 + 7220 * q^13 + 11744 * q^14 + 49152 * q^16 - 34980 * q^17 - 5120 * q^18 - 52310 * q^19 + 138764 * q^21 + 49920 * q^22 + 3360 * q^23 - 37888 * q^24 + 102848 * q^26 + 151280 * q^27 - 71680 * q^28 + 292500 * q^29 - 173116 * q^31 + 131040 * q^33 + 193744 * q^34 + 995456 * q^36 - 265180 * q^37 - 683520 * q^38 - 451512 * q^39 - 1029606 * q^41 + 74240 * q^42 + 1426280 * q^43 - 215424 * q^44 + 1391008 * q^46 - 898320 * q^47 - 163840 * q^48 + 320436 * q^49 - 3612286 * q^51 + 462080 * q^52 - 6780 * q^53 - 440560 * q^54 + 751616 * q^56 - 794720 * q^57 - 2672640 * q^58 + 1721400 * q^59 - 12392256 * q^61 + 2461440 * q^62 + 2221600 * q^63 + 3145728 * q^64 + 9608656 * q^66 + 2462840 * q^67 - 2238720 * q^68 - 13381852 * q^69 + 227784 * q^71 - 327680 * q^72 + 2825420 * q^73 - 9998176 * q^74 - 3347840 * q^76 - 574080 * q^77 + 2059520 * q^78 + 3599020 * q^79 + 25551692 * q^81 - 4592640 * q^82 - 5627400 * q^83 + 8880896 * q^84 - 8043712 * q^86 - 5251440 * q^87 + 3194880 * q^88 + 37214070 * q^89 + 7183184 * q^91 + 215040 * q^92 - 487040 * q^93 + 10224704 * q^94 - 2424832 * q^96 - 3434500 * q^97 + 8171520 * q^98 - 3678572 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(50))\) 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
50.8.a.a	$50$	$8$	50.a	1.a	$1$	$1$	$1$	$15.619$	\(\Q\)	None	✓	✓			50.8.a.a	$1$	$0$	\(-8\)	\(-43\)	\(0\)	\(-974\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}-43q^{3}+2^{6}q^{4}+344q^{6}+\cdots\)
50.8.a.b	$50$	$8$	50.a	1.a	$1$	$1$	$1$	$15.619$	\(\Q\)	None	✓				10.8.a.a	$1$	$0$	\(-8\)	\(-28\)	\(0\)	\(-104\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}-28q^{3}+2^{6}q^{4}+224q^{6}+\cdots\)
50.8.a.c	$50$	$8$	50.a	1.a	$1$	$1$	$1$	$15.619$	\(\Q\)	None	✓	✓			50.8.a.c	$1$	$1$	\(-8\)	\(57\)	\(0\)	\(-1174\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+57q^{3}+2^{6}q^{4}-456q^{6}+\cdots\)
50.8.a.d	$50$	$8$	50.a	1.a	$1$	$1$	$1$	$15.619$	\(\Q\)	None	✓	✓			50.8.a.d	$1$	$0$	\(-8\)	\(87\)	\(0\)	\(1366\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+87q^{3}+2^{6}q^{4}-696q^{6}+\cdots\)
50.8.a.e	$50$	$8$	50.a	1.a	$1$	$1$	$1$	$15.619$	\(\Q\)	None	✓	✓			50.8.a.d	$1$	$0$	\(8\)	\(-87\)	\(0\)	\(-1366\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}-87q^{3}+2^{6}q^{4}-696q^{6}+\cdots\)
50.8.a.f	$50$	$8$	50.a	1.a	$1$	$1$	$1$	$15.619$	\(\Q\)	None	✓	✓			50.8.a.c	$1$	$1$	\(8\)	\(-57\)	\(0\)	\(1174\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}-57q^{3}+2^{6}q^{4}-456q^{6}+\cdots\)
50.8.a.g	$50$	$8$	50.a	1.a	$1$	$1$	$1$	$15.619$	\(\Q\)	None	✓				2.8.a.a	$1$	$1$	\(8\)	\(-12\)	\(0\)	\(-1016\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}-12q^{3}+2^{6}q^{4}-96q^{6}-1016q^{7}+\cdots\)
50.8.a.h	$50$	$8$	50.a	1.a	$1$	$1$	$1$	$15.619$	\(\Q\)	None	✓	✓			50.8.a.a	$1$	$0$	\(8\)	\(43\)	\(0\)	\(974\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+43q^{3}+2^{6}q^{4}+344q^{6}+\cdots\)
50.8.a.i	$50$	$8$	50.a	1.a	$1$	$2$	$2$	$15.619$	\(\Q(\sqrt{31}) \)	None	✓		✓		10.8.b.a	$1$	$1$	\(-16\)	\(-56\)	\(0\)	\(-408\)		$+$	$-$	$-$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q-8q^{2}+(-28+\beta )q^{3}+2^{6}q^{4}+(224+\cdots)q^{6}+\cdots\)
50.8.a.j	$50$	$8$	50.a	1.a	$1$	$2$	$2$	$15.619$	\(\Q(\sqrt{31}) \)	None	✓		✓		10.8.b.a	$1$	$0$	\(16\)	\(56\)	\(0\)	\(408\)		$-$	$-$	$+$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q+8q^{2}+(28+\beta )q^{3}+2^{6}q^{4}+(224+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(50))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(50)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)