Space of modular forms of level 10, weight 20, and trivial character

• Label: 10.20.a
• Level: $10$
• Weight: $20$
• Character orbit: 10.a
• Rep. character: $\chi_{10}(1,\cdot)$
• Character field: $\Q$
• Dimension: $5$
• Newform subspaces: $4$
• Sturm bound: $30$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 10 = 2 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 20 \)
Character orbit:	\([\chi]\)	\(=\)	10.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(30\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{20}(\Gamma_0(10))\).

	Total	New	Old
Modular forms	31	5	26
Cusp forms	27	5	22
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
\(+\)	\(+\)	\(+\)	\(1\)
\(+\)	\(-\)	\(-\)	\(1\)
\(-\)	\(+\)	\(-\)	\(1\)
\(-\)	\(-\)	\(+\)	\(2\)
Plus space		\(+\)	\(3\)
Minus space		\(-\)	\(2\)

Trace form

5 * q + 512 * q^2 + 70372 * q^3 + 1310720 * q^4 + 1953125 * q^5 + 23736320 * q^6 - 272909624 * q^7 + 134217728 * q^8 - 93890015 * q^9 + 1000000000 * q^10 - 23642432940 * q^11 + 18447597568 * q^12 + 42687059302 * q^13 + 48757749760 * q^14 + 145179687500 * q^15 + 343597383680 * q^16 - 130413305574 * q^17 + 78566050304 * q^18 + 291111663100 * q^19 + 512000000000 * q^20 - 8319287671240 * q^21 + 3580788086784 * q^22 + 2244987950952 * q^23 + 6222333870080 * q^24 + 19073486328125 * q^25 + 45955751480320 * q^26 + 18146233127800 * q^27 - 71541620473856 * q^28 + 175893113518350 * q^29 - 56168000000000 * q^30 + 115494568728160 * q^31 + 35184372088832 * q^32 + 326638787488704 * q^33 - 359833343523840 * q^34 + 294259093750000 * q^35 - 24612704092160 * q^36 - 755742421616594 * q^37 + 2989779870791680 * q^38 - 1449156339672280 * q^39 + 262144000000000 * q^40 - 3038800820912790 * q^41 + 356527902244864 * q^42 - 5890433108424788 * q^43 - 6197721940623360 * q^44 + 3756323744140625 * q^45 + 999717406033920 * q^46 + 4064006442579936 * q^47 + 4835927016865792 * q^48 - 1207601469859035 * q^49 + 1953125000000000 * q^50 - 41519778873297240 * q^51 + 11190156473663488 * q^52 + 9595387533001662 * q^53 - 1831455688499200 * q^54 + 39899202914062500 * q^55 + 12781551553085440 * q^56 + 76710950311905680 * q^57 - 28210833504875520 * q^58 - 206648874333696300 * q^59 + 38057984000000000 * q^60 - 121654830052901690 * q^61 + 207032923877453824 * q^62 + 57309083325363992 * q^63 + 90071992547409920 * q^64 + 48211764496093750 * q^65 + 94190736319856640 * q^66 + 1131032827584999556 * q^67 - 34187065576390656 * q^68 - 1336220511562261080 * q^69 + 359264156000000000 * q^70 + 523198357960986360 * q^71 + 20595618690891776 * q^72 - 2364737054355928958 * q^73 - 1018464761013847040 * q^74 + 268447875976562500 * q^75 + 76313175811686400 * q^76 + 3133800838977636432 * q^77 - 1613105060298678272 * q^78 - 220529221087134800 * q^79 + 134217728000000000 * q^80 - 3189494015937417995 * q^81 + 295060066566902784 * q^82 - 235062914463794748 * q^83 - 2180851347289538560 * q^84 + 2973477664347656250 * q^85 + 4534157417538027520 * q^86 + 4440506494908073080 * q^87 + 938682112221904896 * q^88 - 2647956784372483950 * q^89 + 356454757000000000 * q^90 - 2805575271928755040 * q^91 + 588510121414361088 * q^92 - 1754715828918044656 * q^93 - 7676930749559101440 * q^94 - 5339289156289062500 * q^95 + 1631147490038251520 * q^96 + 17503635495991682266 * q^97 - 190754708351061504 * q^98 + 26994792784658592420 * q^99

Decomposition of \(S_{20}^{\mathrm{new}}(\Gamma_0(10))\) 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
10.20.a.a	$10$	$20$	10.a	1.a	$1$	$1$	$1$	$22.882$	\(\Q\)	None	✓	✓	✓	✓	10.20.a.a	$1$	$0$	\(-512\)	\(-26622\)	\(-1953125\)	\(-39884026\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{9}q^{2}-26622q^{3}+2^{18}q^{4}-5^{9}q^{5}+\cdots\)
10.20.a.b	$10$	$20$	10.a	1.a	$1$	$1$	$1$	$22.882$	\(\Q\)	None	✓	✓	✓	✓	10.20.a.b	$1$	$1$	\(-512\)	\(38628\)	\(1953125\)	\(-144185776\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2^{9}q^{2}+38628q^{3}+2^{18}q^{4}+5^{9}q^{5}+\cdots\)
10.20.a.c	$10$	$20$	10.a	1.a	$1$	$1$	$1$	$22.882$	\(\Q\)	None	✓	✓	✓	✓	10.20.a.c	$1$	$1$	\(512\)	\(24642\)	\(-1953125\)	\(-171901114\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{9}q^{2}+24642q^{3}+2^{18}q^{4}-5^{9}q^{5}+\cdots\)
10.20.a.d	$10$	$20$	10.a	1.a	$1$	$2$	$2$	$22.882$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓	✓	✓	✓	10.20.a.d	$1$	$0$	\(1024\)	\(33724\)	\(3906250\)	\(83061292\)		$-$	$-$	$+$	$2^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+2^{9}q^{2}+(16862-\beta )q^{3}+2^{18}q^{4}+\cdots\)

Decomposition of \(S_{20}^{\mathrm{old}}(\Gamma_0(10))\) into lower level spaces

\( S_{20}^{\mathrm{old}}(\Gamma_0(10)) \simeq \) \(S_{20}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{20}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)\(\oplus\)\(S_{20}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)