Space of modular forms of level 50, weight 18, and trivial character

• Label: 50.18.a
• Level: $50$
• Weight: $18$
• Character orbit: 50.a
• Rep. character: $\chi_{50}(1,\cdot)$
• Character field: $\Q$
• Dimension: $26$
• Newform subspaces: $11$
• Sturm bound: $135$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	50.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(135\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{18}(\Gamma_0(50))\).

	Total	New	Old
Modular forms	133	26	107
Cusp forms	121	26	95
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
\(+\)	\(+\)	\(+\)		\(32\)	\(6\)	\(26\)		\(29\)	\(6\)	\(23\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)		\(34\)	\(7\)	\(27\)		\(31\)	\(7\)	\(24\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(34\)	\(7\)	\(27\)		\(31\)	\(7\)	\(24\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(33\)	\(6\)	\(27\)		\(30\)	\(6\)	\(24\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(65\)	\(12\)	\(53\)		\(59\)	\(12\)	\(47\)		\(6\)	\(0\)	\(6\)
Minus space		\(-\)		\(68\)	\(14\)	\(54\)		\(62\)	\(14\)	\(48\)		\(6\)	\(0\)	\(6\)

Trace form

26 * q - 1120 * q^3 + 1703936 * q^4 - 6570496 * q^6 - 27174640 * q^7 + 1162074668 * q^9 + 35972442 * q^11 - 73400320 * q^12 + 7439368160 * q^13 + 5060246528 * q^14 + 111669149696 * q^16 - 29023708560 * q^17 - 31548047360 * q^18 - 128943601110 * q^19 + 538660941092 * q^21 + 400908042240 * q^22 - 48118999920 * q^23 - 430604025856 * q^24 - 1372905158656 * q^26 - 2180290903840 * q^27 - 1780917207040 * q^28 + 7005944358720 * q^29 + 4832653702052 * q^31 + 23031986265120 * q^33 - 28008516516352 * q^34 + 76157725442048 * q^36 + 56745143659040 * q^37 - 6160221634560 * q^38 - 115037669632584 * q^39 + 137145968838642 * q^41 + 25712151265280 * q^42 - 75298514988160 * q^43 + 2357489958912 * q^44 + 34063703880704 * q^46 + 701474319162960 * q^47 - 4810363371520 * q^48 - 637948324862238 * q^49 - 869582138709478 * q^51 + 487546431733760 * q^52 - 608491077759840 * q^53 - 1907230222497280 * q^54 + 331628316459008 * q^56 + 5358176265396160 * q^57 - 1335683381329920 * q^58 - 2323341416847360 * q^59 - 98126676284348 * q^61 + 6151916185927680 * q^62 - 10024991637285200 * q^63 + 7318349394477056 * q^64 - 3665805747026432 * q^66 + 15434120580764480 * q^67 - 1902097764188160 * q^68 - 2203902619373044 * q^69 - 28187057394437928 * q^71 - 2067532831784960 * q^72 + 8139650426443760 * q^73 - 14142145057211392 * q^74 - 8450447842344960 * q^76 + 16227343955382240 * q^77 - 1204964864573440 * q^78 - 16887025242880500 * q^79 + 124060872066979946 * q^81 - 17233013565358080 * q^82 - 8525273284147200 * q^83 + 35301683435405312 * q^84 + 78412997830313984 * q^86 - 147073861315207680 * q^87 + 26273909456240640 * q^88 + 5188920361079190 * q^89 + 175214609953716912 * q^91 - 3153526778757120 * q^92 + 190002362881254880 * q^93 + 105038558087784448 * q^94 - 28220065438498816 * q^96 - 420716960590654000 * q^97 + 129166697657794560 * q^98 + 302181383415857356 * q^99

Decomposition of \(S_{18}^{\mathrm{new}}(\Gamma_0(50))\) 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
50.18.a.a	$50$	$18$	50.a	1.a	$1$	$1$	$1$	$91.611$	\(\Q\)	None	✓				2.18.a.a	$1$	$1$	\(-256\)	\(-6084\)	\(0\)	\(22465912\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{8}q^{2}-78^{2}q^{3}+2^{16}q^{4}+1248^{2}q^{6}+\cdots\)
50.18.a.b	$50$	$18$	50.a	1.a	$1$	$1$	$1$	$91.611$	\(\Q\)	None	✓				10.18.a.a	$1$	$1$	\(-256\)	\(14976\)	\(0\)	\(-14808668\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{8}q^{2}+14976q^{3}+2^{16}q^{4}-3833856q^{6}+\cdots\)
50.18.a.c	$50$	$18$	50.a	1.a	$1$	$2$	$2$	$91.611$	\(\Q(\sqrt{2941}) \)	None	✓				10.18.a.d	$1$	$1$	\(-512\)	\(-17628\)	\(0\)	\(-27684196\)		$+$	$+$	$+$	$2^{5}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q-2^{8}q^{2}+(-8814-\beta )q^{3}+2^{16}q^{4}+\cdots\)
50.18.a.d	$50$	$18$	50.a	1.a	$1$	$2$	$2$	$91.611$	\(\Q(\sqrt{37401}) \)	None	✓	✓			50.18.a.d	$1$	$1$	\(-512\)	\(11592\)	\(0\)	\(-13386856\)		$+$	$+$	$+$	$2\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q-2^{8}q^{2}+(5796-\beta )q^{3}+2^{16}q^{4}+\cdots\)
50.18.a.e	$50$	$18$	50.a	1.a	$1$	$2$	$2$	$91.611$	\(\Q(\sqrt{37401}) \)	None	✓	✓			50.18.a.d	$1$	$1$	\(512\)	\(-11592\)	\(0\)	\(13386856\)		$-$	$-$	$+$	$2\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+2^{8}q^{2}+(-5796-\beta )q^{3}+2^{16}q^{4}+\cdots\)
50.18.a.f	$50$	$18$	50.a	1.a	$1$	$2$	$2$	$91.611$	\(\Q(\sqrt{83281}) \)	None	✓				10.18.a.c	$1$	$0$	\(512\)	\(1308\)	\(0\)	\(-603844\)		$-$	$+$	$-$	$2^{2}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+2^{8}q^{2}+(654-\beta )q^{3}+2^{16}q^{4}+(167424+\cdots)q^{6}+\cdots\)
50.18.a.g	$50$	$18$	50.a	1.a	$1$	$2$	$2$	$91.611$	\(\Q(\sqrt{36061}) \)	None	✓				10.18.a.b	$1$	$0$	\(512\)	\(6308\)	\(0\)	\(-6543844\)		$-$	$+$	$-$	$2^{5}\cdot 5$	$\mathrm{SU}(2)$	\(q+2^{8}q^{2}+(3154-\beta )q^{3}+2^{16}q^{4}+\cdots\)
50.18.a.h	$50$	$18$	50.a	1.a	$1$	$3$	$3$	$91.611$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓			50.18.a.h	$1$	$0$	\(-768\)	\(7513\)	\(0\)	\(-17909334\)		$+$	$-$	$-$	$2^{4}\cdot 3^{2}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q-2^{8}q^{2}+(2504+\beta _{1})q^{3}+2^{16}q^{4}+\cdots\)
50.18.a.i	$50$	$18$	50.a	1.a	$1$	$3$	$3$	$91.611$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓		50.18.a.h	$1$	$0$	\(768\)	\(-7513\)	\(0\)	\(17909334\)		$-$	$+$	$-$	$2^{4}\cdot 3^{2}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+2^{8}q^{2}+(-2504-\beta _{1})q^{3}+2^{16}q^{4}+\cdots\)
50.18.a.j	$50$	$18$	50.a	1.a	$1$	$4$	$4$	$91.611$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		10.18.b.a	$1$	$0$	\(-1024\)	\(1904\)	\(0\)	\(27852528\)		$+$	$-$	$-$	$2^{8}\cdot 3^{2}\cdot 5^{7}$	$\mathrm{SU}(2)$	\(q-2^{8}q^{2}+(476+\beta _{1})q^{3}+2^{16}q^{4}+\cdots\)
50.18.a.k	$50$	$18$	50.a	1.a	$1$	$4$	$4$	$91.611$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		10.18.b.a	$1$	$1$	\(1024\)	\(-1904\)	\(0\)	\(-27852528\)		$-$	$-$	$+$	$2^{8}\cdot 3^{2}\cdot 5^{7}$	$\mathrm{SU}(2)$	\(q+2^{8}q^{2}+(-476-\beta _{1})q^{3}+2^{16}q^{4}+\cdots\)

Decomposition of \(S_{18}^{\mathrm{old}}(\Gamma_0(50))\) into lower level spaces

\( S_{18}^{\mathrm{old}}(\Gamma_0(50)) \simeq \) \(S_{18}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 6}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 3}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 2}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)