Space of modular forms of level 10, weight 38, and trivial character

• Label: 10.38.a
• Level: $10$
• Weight: $38$
• Character orbit: 10.a
• Rep. character: $\chi_{10}(1,\cdot)$
• Character field: $\Q$
• Dimension: $11$
• Newform subspaces: $4$
• Sturm bound: $57$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 10 = 2 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 38 \)
Character orbit:	\([\chi]\)	\(=\)	10.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(57\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{38}(\Gamma_0(10))\).

	Total	New	Old
Modular forms	57	11	46
Cusp forms	53	11	42
Eisenstein series	4	0	4

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
\(+\)	\(+\)	\(+\)	\(3\)
\(+\)	\(-\)	\(-\)	\(3\)
\(-\)	\(+\)	\(-\)	\(3\)
\(-\)	\(-\)	\(+\)	\(2\)
Plus space		\(+\)	\(5\)
Minus space		\(-\)	\(6\)

Trace form

11 * q - 262144 * q^2 - 101265596 * q^3 + 755914244096 * q^4 - 3814697265625 * q^5 + 467891507953664 * q^6 - 8136610409482472 * q^7 - 18014398509481984 * q^8 + 243341190951053383 * q^9 - 1000000000000000000 * q^10 - 8892179155992564828 * q^11 - 6958918768479174656 * q^12 + 909184511023571927314 * q^13 + 3327232089650860392448 * q^14 - 5228137954711914062500 * q^15 + 51946031311566097350656 * q^16 - 156847220844825553473882 * q^17 + 143600426236374980231168 * q^18 + 1095588869521461004993900 * q^19 - 262144000000000000000000 * q^20 + 955663803398391184621832 * q^21 - 1373747344466365658431488 * q^22 - 39511609522513754583347496 * q^23 + 32153259595793772213960704 * q^24 + 160071067512035369873046875 * q^25 + 208092682306919206209716224 * q^26 + 10245242176817474957683960 * q^27 - 559143609744326168403771392 * q^28 + 292895527842939812454548490 * q^29 - 1055077496000000000000000000 * q^30 + 11363939067624181352404144192 * q^31 - 1237940039285380274899124224 * q^32 + 21753087921420048362119123008 * q^33 - 15327716933637936682080141312 * q^34 - 8531250865334594726562500000 * q^35 + 16722279310471446667762597888 * q^36 + 187071649912861757278571509978 * q^37 - 361112668985263133933935001600 * q^38 + 886331438473179991957182124616 * q^39 - 68719476736000000000000000000 * q^40 + 1305526631117393752357594393542 * q^41 - 3387625646236979152701177724928 * q^42 - 162888835175696745465335348276 * q^43 - 611065898642575173686717841408 * q^44 - 3395628003346663105010986328125 * q^45 + 878097393314793086238524964864 * q^46 + 17251738736487345234779042140848 * q^47 - 478213256418218412873472802816 * q^48 - 5848181094864236037051726022053 * q^49 - 3814697265625000000000000000000 * q^50 - 55107088209370766132422595222808 * q^51 + 62478683854015886606677105967104 * q^52 - 59277127589292161917743812257686 * q^53 + 430089660405491227865868885032960 * q^54 - 106531279383847678573608398437500 * q^55 + 228645648180034967101214166089728 * q^56 + 211651039157794433062940140199600 * q^57 - 81658069382197038522402023669760 * q^58 - 856733149315655512159787916735420 * q^59 - 359274904551424000000000000000000 * q^60 + 4399567231272275173564257651660562 * q^61 + 669747197889664786812960224837632 * q^62 - 7056364533093793760838850782630616 * q^63 + 3569704090242693994614716226338816 * q^64 - 8397784495699781419700622558593750 * q^65 - 2717924964568076844089638725353472 * q^66 - 14834478129506952863355414739088252 * q^67 - 10778458943952243887926758082609152 * q^68 + 28879001594363816042061963850465176 * q^69 - 4925677525534772000000000000000000 * q^70 + 22540768664980123645298858057164632 * q^71 + 9868146150030254490968209278107648 * q^72 + 17162246578778779758533141811531214 * q^73 + 32492999935656010694023289672040448 * q^74 - 1473608368542045354843139648437500 * q^75 + 75288293831300578985409490871910400 * q^76 - 370837541095491940327721215526630544 * q^77 + 281686233996506324613321821557620736 * q^78 + 101688090861235228023425958298386160 * q^79 - 18014398509481984000000000000000000 * q^80 + 22041381624420509047402666808660011 * q^81 + 147587241779774488519290812282437632 * q^82 + 320926720613963505679306276287863844 * q^83 + 65672716505073020751447465081700352 * q^84 + 693758605291932583160087585449218750 * q^85 - 1644955491417931002763716563006652416 * q^86 + 1388952154489028011605118115175443160 * q^87 - 94403198679198193199051961865863168 * q^88 + 273310186853836130412557636671191390 * q^89 - 582547698925752653000000000000000000 * q^90 + 1121708147716715222710754750148696512 * q^91 - 2715217131384300026330361624375853056 * q^92 + 6623305052922340495062522712944463088 * q^93 - 985000758006950238352383192918392832 * q^94 - 15362438518294179388809204101562500 * q^95 + 2209555174779718893110955812946182144 * q^96 + 19574454273022155120560716121395075078 * q^97 - 5220610730153960440690972725497561088 * q^98 - 5270368491881054888822529251353949484 * q^99

Decomposition of \(S_{38}^{\mathrm{new}}(\Gamma_0(10))\) 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
10.38.a.a	$10$	$38$	10.a	1.a	$1$	$2$	$2$	$86.714$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓	✓	✓	✓	10.38.a.a	$1$	$1$	\(524288\)	\(-185500908\)	\(76\!\cdots\!50\)	\(-65\!\cdots\!56\)		$-$	$-$	$+$	$2^{2}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+2^{18}q^{2}+(-92750454-9\beta )q^{3}+\cdots\)
10.38.a.b	$10$	$38$	10.a	1.a	$1$	$3$	$3$	$86.714$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓	✓	10.38.a.b	$1$	$0$	\(-786432\)	\(-550394388\)	\(11\!\cdots\!75\)	\(-45\!\cdots\!16\)		$+$	$-$	$-$	$2^{9}\cdot 3^{5}\cdot 5^{3}\cdot 7$	$\mathrm{SU}(2)$	\(q-2^{18}q^{2}+(-183464796+\beta _{1})q^{3}+\cdots\)
10.38.a.c	$10$	$38$	10.a	1.a	$1$	$3$	$3$	$86.714$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓	✓	10.38.a.c	$1$	$1$	\(-786432\)	\(-392670638\)	\(-11\!\cdots\!75\)	\(-58\!\cdots\!66\)		$+$	$+$	$+$	$2^{13}\cdot 3\cdot 5^{3}\cdot 7$	$\mathrm{SU}(2)$	\(q-2^{18}q^{2}+(-130890213-\beta _{1})q^{3}+\cdots\)
10.38.a.d	$10$	$38$	10.a	1.a	$1$	$3$	$3$	$86.714$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓	✓	10.38.a.d	$1$	$0$	\(786432\)	\(1027300338\)	\(-11\!\cdots\!75\)	\(29\!\cdots\!66\)		$-$	$+$	$-$	$2^{13}\cdot 3^{5}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+2^{18}q^{2}+(342433446+\beta _{1})q^{3}+\cdots\)

Decomposition of \(S_{38}^{\mathrm{old}}(\Gamma_0(10))\) into lower level spaces

\( S_{38}^{\mathrm{old}}(\Gamma_0(10)) \simeq \) \(S_{38}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{38}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)\(\oplus\)\(S_{38}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)