Space of modular forms of level 50, weight 22, and trivial character

• Label: 50.22.a
• Level: $50$
• Weight: $22$
• Character orbit: 50.a
• Rep. character: $\chi_{50}(1,\cdot)$
• Character field: $\Q$
• Dimension: $33$
• Newform subspaces: $12$
• Sturm bound: $165$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	50.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(165\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{22}(\Gamma_0(50))\).

	Total	New	Old
Modular forms	163	33	130
Cusp forms	151	33	118
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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(40\)	\(7\)	\(33\)		\(37\)	\(7\)	\(30\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)		\(42\)	\(9\)	\(33\)		\(39\)	\(9\)	\(30\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(41\)	\(9\)	\(32\)		\(38\)	\(9\)	\(29\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(40\)	\(8\)	\(32\)		\(37\)	\(8\)	\(29\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(80\)	\(15\)	\(65\)		\(74\)	\(15\)	\(59\)		\(6\)	\(0\)	\(6\)
Minus space		\(-\)		\(83\)	\(18\)	\(65\)		\(77\)	\(18\)	\(59\)		\(6\)	\(0\)	\(6\)

Trace form

33 * q + 1024 * q^2 - 113584 * q^3 + 34603008 * q^4 + 86116352 * q^6 - 447807468 * q^7 + 1073741824 * q^8 + 139175397539 * q^9 - 63774758294 * q^11 - 119101456384 * q^12 - 713622573774 * q^13 - 1345500368896 * q^14 + 36283883716608 * q^16 + 9496469506542 * q^17 + 36747218793472 * q^18 - 21599508459830 * q^19 - 238791783940924 * q^21 + 19326219522048 * q^22 + 228622672329996 * q^23 + 90299539914752 * q^24 - 770370679093248 * q^26 - 1110380713249600 * q^27 - 469560163565568 * q^28 + 1951357002008610 * q^29 + 6285497958136676 * q^31 + 1125899906842624 * q^32 - 20199937642899168 * q^33 - 8534745318404096 * q^34 + 145935981649854464 * q^36 - 5158686632250678 * q^37 - 27094676893020160 * q^38 + 50719307227178088 * q^39 - 278594906750294684 * q^41 + 207117905012031488 * q^42 + 400770549268257816 * q^43 - 66872680952889344 * q^44 + 187027736864346112 * q^46 - 239274419923828308 * q^47 - 124886928729309184 * q^48 + 3947331976677296121 * q^49 - 1921571507443393474 * q^51 - 748287503917645824 * q^52 - 1358173479072543414 * q^53 + 4743684008162600960 * q^54 - 1410859394815492096 * q^56 + 13678977430787618560 * q^57 - 4067886245344757760 * q^58 - 12435193014219790780 * q^59 + 7909971833572427286 * q^61 - 2451403867105779712 * q^62 + 13747131336285796 * q^63 + 38046409652025950208 * q^64 - 15766300993213147136 * q^66 + 36477017414410978152 * q^67 + 9957770009291784192 * q^68 - 48500074657583885572 * q^69 - 54543310020449679384 * q^71 + 38532251693583695872 * q^72 - 74759842719966019434 * q^73 - 126951820323228583936 * q^74 - 22648726182774702080 * q^76 + 411003646824772766064 * q^77 - 225394292695998447616 * q^78 - 402630313979646461900 * q^79 + 300477134124291781073 * q^81 + 367925800652845836288 * q^82 - 212343231236446733184 * q^83 - 250391333637638324224 * q^84 - 47663231304417906688 * q^86 + 1815864716059146848160 * q^87 + 20265009961551003648 * q^88 - 2744784623698691339480 * q^89 - 179322572237394623224 * q^91 + 239728247261097885696 * q^92 - 1274444289334196295008 * q^93 - 317356071961450725376 * q^94 + 94685930365650993152 * q^96 + 2968256985110241511902 * q^97 - 58856295254402030592 * q^98 - 4405537641480644117552 * q^99

Decomposition of \(S_{22}^{\mathrm{new}}(\Gamma_0(50))\) 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
50.22.a.a	$50$	$22$	50.a	1.a	$1$	$1$	$1$	$139.739$	\(\Q\)	None	✓				2.22.a.b	$1$	$1$	\(-1024\)	\(-59316\)	\(0\)	\(-1427425832\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{10}q^{2}-59316q^{3}+2^{20}q^{4}+60739584q^{6}+\cdots\)
50.22.a.b	$50$	$22$	50.a	1.a	$1$	$1$	$1$	$139.739$	\(\Q\)	None	✓				10.22.a.a	$1$	$1$	\(-1024\)	\(21924\)	\(0\)	\(722753248\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{10}q^{2}+21924q^{3}+2^{20}q^{4}-22450176q^{6}+\cdots\)
50.22.a.c	$50$	$22$	50.a	1.a	$1$	$1$	$1$	$139.739$	\(\Q\)	None	✓				2.22.a.a	$1$	$0$	\(1024\)	\(-71604\)	\(0\)	\(853202392\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{10}q^{2}-71604q^{3}+2^{20}q^{4}-73322496q^{6}+\cdots\)
50.22.a.d	$50$	$22$	50.a	1.a	$1$	$2$	$2$	$139.739$	\(\Q(\sqrt{1179649}) \)	None	✓				10.22.a.d	$1$	$1$	\(-2048\)	\(-30972\)	\(0\)	\(439959356\)		$+$	$+$	$+$	$2^{4}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q-2^{10}q^{2}+(-15486-\beta )q^{3}+2^{20}q^{4}+\cdots\)
50.22.a.e	$50$	$22$	50.a	1.a	$1$	$2$	$2$	$139.739$	\(\Q(\sqrt{474529}) \)	None	✓				10.22.a.c	$1$	$0$	\(2048\)	\(-100308\)	\(0\)	\(-1328895316\)		$-$	$+$	$-$	$2^{2}\cdot 3\cdot 5\cdot 7$	$\mathrm{SU}(2)$	\(q+2^{10}q^{2}+(-50154-\beta )q^{3}+2^{20}q^{4}+\cdots\)
50.22.a.f	$50$	$22$	50.a	1.a	$1$	$2$	$2$	$139.739$	\(\Q(\sqrt{157921}) \)	None	✓				10.22.a.b	$1$	$0$	\(2048\)	\(126692\)	\(0\)	\(292598684\)		$-$	$+$	$-$	$2^{4}\cdot 5\cdot 7$	$\mathrm{SU}(2)$	\(q+2^{10}q^{2}+(63346-\beta )q^{3}+2^{20}q^{4}+\cdots\)
50.22.a.g	$50$	$22$	50.a	1.a	$1$	$3$	$3$	$139.739$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓		50.22.a.g	$1$	$1$	\(-3072\)	\(-46383\)	\(0\)	\(911775234\)		$+$	$+$	$+$	$2^{5}\cdot 3^{2}\cdot 5^{4}$	$\mathrm{SU}(2)$	\(q-2^{10}q^{2}+(-15461+\beta _{1})q^{3}+2^{20}q^{4}+\cdots\)
50.22.a.h	$50$	$22$	50.a	1.a	$1$	$3$	$3$	$139.739$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓			50.22.a.g	$1$	$1$	\(3072\)	\(46383\)	\(0\)	\(-911775234\)		$-$	$-$	$+$	$2^{5}\cdot 3^{2}\cdot 5^{4}$	$\mathrm{SU}(2)$	\(q+2^{10}q^{2}+(15461-\beta _{1})q^{3}+2^{20}q^{4}+\cdots\)
50.22.a.i	$50$	$22$	50.a	1.a	$1$	$4$	$4$	$139.739$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			50.22.a.i	$1$	$0$	\(-4096\)	\(-96764\)	\(0\)	\(-162760528\)		$+$	$-$	$-$	$2^{9}\cdot 3^{2}\cdot 5^{7}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q-2^{10}q^{2}+(-24191+\beta _{1})q^{3}+2^{20}q^{4}+\cdots\)
50.22.a.j	$50$	$22$	50.a	1.a	$1$	$4$	$4$	$139.739$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓		50.22.a.i	$1$	$0$	\(4096\)	\(96764\)	\(0\)	\(162760528\)		$-$	$+$	$-$	$2^{9}\cdot 3^{2}\cdot 5^{7}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+2^{10}q^{2}+(24191-\beta _{1})q^{3}+2^{20}q^{4}+\cdots\)
50.22.a.k	$50$	$22$	50.a	1.a	$1$	$5$	$5$	$139.739$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		10.22.b.a	$1$	$0$	\(-5120\)	\(112670\)	\(0\)	\(-51222610\)		$+$	$-$	$-$	$2^{13}\cdot 3^{2}\cdot 5^{12}\cdot 7$	$\mathrm{SU}(2)$	\(q-2^{10}q^{2}+(22534-\beta _{1})q^{3}+2^{20}q^{4}+\cdots\)
50.22.a.l	$50$	$22$	50.a	1.a	$1$	$5$	$5$	$139.739$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		10.22.b.a	$1$	$1$	\(5120\)	\(-112670\)	\(0\)	\(51222610\)		$-$	$-$	$+$	$2^{13}\cdot 3^{2}\cdot 5^{12}\cdot 7$	$\mathrm{SU}(2)$	\(q+2^{10}q^{2}+(-22534+\beta _{1})q^{3}+2^{20}q^{4}+\cdots\)

Decomposition of \(S_{22}^{\mathrm{old}}(\Gamma_0(50))\) into lower level spaces

\( S_{22}^{\mathrm{old}}(\Gamma_0(50)) \simeq \) \(S_{22}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 6}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 3}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 2}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)