Space of modular forms of level 50, weight 22, and Character 50.b

• Label: 50.22.b
• Level: $50$
• Weight: $22$
• Character orbit: 50.b
• Rep. character: $\chi_{50}(49,\cdot)$
• Character field: $\Q$
• Dimension: $32$
• Newform subspaces: $8$
• Sturm bound: $165$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	50.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(165\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{22}(50, [\chi])\).

	Total	New	Old
Modular forms	164	32	132
Cusp forms	152	32	120
Eisenstein series	12	0	12

Trace form

32 * q - 33554432 * q^4 + 340563968 * q^6 - 155571751486 * q^9 + 251334301254 * q^11 + 1366826430464 * q^14 + 35184372088832 * q^16 - 84567857658310 * q^19 + 301477533664364 * q^21 - 357107203309568 * q^24 - 1053707964649472 * q^26 - 1849807144850700 * q^29 - 12439392818559236 * q^31 + 14858053903202304 * q^34 + 163128804886183936 * q^36 - 43595682889725272 * q^39 - 230570519944985046 * q^41 - 263543116271714304 * q^44 + 672821726896631808 * q^46 - 3151324373307233304 * q^49 - 2174412750698246946 * q^51 - 1111424065882818560 * q^54 - 1433221391150219264 * q^56 + 15542243642589216600 * q^59 + 25569497095826363464 * q^61 - 36893488147419103232 * q^64 + 35748668481964087296 * q^66 - 136572317555961743892 * q^69 + 12145824590195381784 * q^71 - 24028576300427739136 * q^74 + 88675825911920066560 * q^76 + 42927810483386371820 * q^79 + 1365673972759609635232 * q^81 - 316122106339644145664 * q^84 + 1024481732876408086528 * q^86 - 444367294081951966230 * q^89 + 3355865240204686962944 * q^91 - 1032571005570896265216 * q^94 + 374454042817533575168 * q^96 - 3472672388511836862492 * q^99

Decomposition of \(S_{22}^{\mathrm{new}}(50, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
50.22.b.a	$50$	$22$	50.b	5.b	$2$	$2$	$2$	$139.739$	\(\Q(\sqrt{-1}) \)	None					2.22.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+512\beta q^{2}+35802\beta q^{3}-1048576 q^{4}+\cdots\)
50.22.b.b	$50$	$22$	50.b	5.b	$2$	$2$	$2$	$139.739$	\(\Q(\sqrt{-1}) \)	None					10.22.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+512\beta q^{2}+10962\beta q^{3}-1048576 q^{4}+\cdots\)
50.22.b.c	$50$	$22$	50.b	5.b	$2$	$2$	$2$	$139.739$	\(\Q(\sqrt{-1}) \)	None					2.22.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-512\beta q^{2}+29658\beta q^{3}-1048576 q^{4}+\cdots\)
50.22.b.d	$50$	$22$	50.b	5.b	$2$	$4$	$4$	$139.739$	\(\mathbb{Q}[x]/(x^{4} + \cdots)\)	None					10.22.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3^{2}\cdot 5^{2}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+2^{9}\beta _{1}q^{2}+(25077\beta _{1}-\beta _{2})q^{3}-2^{20}q^{4}+\cdots\)
50.22.b.e	$50$	$22$	50.b	5.b	$2$	$4$	$4$	$139.739$	\(\mathbb{Q}[x]/(x^{4} + \cdots)\)	None					10.22.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-2^{9}\beta _{1}q^{2}+(7743\beta _{1}+\beta _{2})q^{3}-2^{20}q^{4}+\cdots\)
50.22.b.f	$50$	$22$	50.b	5.b	$2$	$4$	$4$	$139.739$	\(\mathbb{Q}[x]/(x^{4} + \cdots)\)	None					10.22.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}\cdot 5^{2}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+2^{9}\beta _{1}q^{2}+(-31673\beta _{1}+\beta _{2})q^{3}+\cdots\)
50.22.b.g	$50$	$22$	50.b	5.b	$2$	$6$	$6$	$139.739$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None		✓			50.22.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}\cdot 3^{4}\cdot 5^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q-2^{10}\beta _{1}q^{2}+(15461\beta _{1}+\beta _{2})q^{3}+\cdots\)
50.22.b.h	$50$	$22$	50.b	5.b	$2$	$8$	$8$	$139.739$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		✓	✓		50.22.a.i	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{18}\cdot 3^{4}\cdot 5^{14}\cdot 7^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-2^{10}\beta _{3}q^{2}+(-\beta _{1}+24191\beta _{3})q^{3}+\cdots\)

Decomposition of \(S_{22}^{\mathrm{old}}(50, [\chi])\) into lower level spaces

\( S_{22}^{\mathrm{old}}(50, [\chi]) \simeq \) \(S_{22}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)