Space of modular forms of level 50, weight 26, and Character 50.b

• Label: 50.26.b
• Level: $50$
• Weight: $26$
• Character orbit: 50.b
• Rep. character: $\chi_{50}(49,\cdot)$
• Character field: $\Q$
• Dimension: $38$
• Newform subspaces: $8$
• Sturm bound: $195$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 26 \)
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:			\(195\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{26}(50, [\chi])\).

	Total	New	Old
Modular forms	194	38	156
Cusp forms	182	38	144
Eisenstein series	12	0	12

Trace form

38 * q - 637534208 * q^4 + 8776835072 * q^6 - 12991242549664 * q^9 - 30472316016774 * q^11 - 330406594985984 * q^14 + 10696049115004928 * q^16 + 35852589007060790 * q^19 + 41011644003619556 * q^21 - 147250857799319552 * q^24 + 728574066604408832 * q^26 - 1161745650351107640 * q^29 - 5751369799647037724 * q^31 - 52394895581776994304 * q^34 + 217956882364103655424 * q^36 + 160834675135271007112 * q^39 - 490898088920940924174 * q^41 + 511240627833677021184 * q^44 + 2181510926430735777792 * q^46 - 2523886281983822036766 * q^49 - 2527743061026638989014 * q^51 - 13760058513872723025920 * q^54 + 5543302811904370540544 * q^56 - 17761878240663664993680 * q^59 - 26807480196318081709124 * q^61 - 179449926349046518120448 * q^64 + 147719411259239791878144 * q^66 + 591755414261249531006772 * q^69 - 171301832723967058366824 * q^71 - 216778552989432627052544 * q^74 - 601506629930684398960640 * q^76 - 1883216138048409165346540 * q^79 + 2610126012992200516185478 * q^81 - 688061209963830072836096 * q^84 + 3297807242412311350771712 * q^86 - 10356123934125502533417270 * q^89 + 10877049061682639587177136 * q^91 + 2111004639366149810651136 * q^94 + 2470459447484468776927232 * q^96 + 23067610244180080135064772 * q^99

Decomposition of \(S_{26}^{\mathrm{new}}(50, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
50.26.b.a	$50$	$26$	50.b	5.b	$2$	$2$	$2$	$197.998$	\(\Q(\sqrt{-1}) \)	None					2.26.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2048\beta q^{2}+48978\beta q^{3}-16777216 q^{4}+\cdots\)
50.26.b.b	$50$	$26$	50.b	5.b	$2$	$2$	$2$	$197.998$	\(\Q(\sqrt{-1}) \)	None					10.26.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-2048\beta q^{2}+81432\beta q^{3}-16777216 q^{4}+\cdots\)
50.26.b.c	$50$	$26$	50.b	5.b	$2$	$4$	$4$	$197.998$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None					10.26.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+2^{11}\beta _{1}q^{2}+(136303\beta _{1}-\beta _{3})q^{3}+\cdots\)
50.26.b.d	$50$	$26$	50.b	5.b	$2$	$4$	$4$	$197.998$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None					10.26.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}\cdot 3^{6}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-2^{11}\beta _{1}q^{2}+(-24273\beta _{1}-\beta _{3})q^{3}+\cdots\)
50.26.b.e	$50$	$26$	50.b	5.b	$2$	$4$	$4$	$197.998$	\(\mathbb{Q}[x]/(x^{4} + \cdots)\)	None					2.26.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{15}\cdot 3^{2}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+2^{11}\beta _{1}q^{2}+(-94962\beta _{1}-\beta _{2})q^{3}+\cdots\)
50.26.b.f	$50$	$26$	50.b	5.b	$2$	$4$	$4$	$197.998$	\(\mathbb{Q}[x]/(x^{4} + \cdots)\)	None					10.26.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3^{6}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-2^{11}\beta _{1}q^{2}+(109647\beta _{1}-\beta _{2})q^{3}+\cdots\)
50.26.b.g	$50$	$26$	50.b	5.b	$2$	$8$	$8$	$197.998$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		✓			50.26.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{20}\cdot 3^{10}\cdot 5^{14}$	$\mathrm{SU}(2)[C_{2}]$	\(q+2^{12}\beta _{3}q^{2}+(\beta _{1}-26589\beta _{3})q^{3}+\cdots\)
50.26.b.h	$50$	$26$	50.b	5.b	$2$	$10$	$10$	$197.998$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None		✓	✓		50.26.a.i	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{38}\cdot 3^{10}\cdot 5^{24}$	$\mathrm{SU}(2)[C_{2}]$	\(q+2^{12}\beta _{5}q^{2}+(-144799\beta _{5}+\beta _{6}+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{26}^{\mathrm{old}}(50, [\chi])\) into lower level spaces

\( S_{26}^{\mathrm{old}}(50, [\chi]) \simeq \) \(S_{26}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{26}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{26}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)