Space of modular forms of level 39, weight 10, and trivial character

• Label: 39.10.a
• Level: $39$
• Weight: $10$
• Character orbit: 39.a
• Rep. character: $\chi_{39}(1,\cdot)$
• Character field: $\Q$
• Dimension: $18$
• Newform subspaces: $4$
• Sturm bound: $46$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 39 = 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	39.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(46\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(39))\).

	Total	New	Old
Modular forms	44	18	26
Cusp forms	40	18	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.

\(3\)	\(13\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(10\)	\(4\)	\(6\)		\(9\)	\(4\)	\(5\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(12\)	\(5\)	\(7\)		\(11\)	\(5\)	\(6\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(12\)	\(6\)	\(6\)		\(11\)	\(6\)	\(5\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(10\)	\(3\)	\(7\)		\(9\)	\(3\)	\(6\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(20\)	\(7\)	\(13\)		\(18\)	\(7\)	\(11\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(24\)	\(11\)	\(13\)		\(22\)	\(11\)	\(11\)		\(2\)	\(0\)	\(2\)

Trace form

18 * q + 68 * q^2 + 4952 * q^4 + 1136 * q^5 - 8748 * q^6 - 7472 * q^7 + 69360 * q^8 + 118098 * q^9 + 1140 * q^10 - 146784 * q^11 + 57996 * q^12 - 57122 * q^13 - 422192 * q^14 - 163944 * q^15 + 1448116 * q^16 + 508092 * q^17 + 446148 * q^18 - 6600 * q^19 + 1883972 * q^20 - 777924 * q^21 - 1335524 * q^22 + 6183352 * q^23 - 3486564 * q^24 + 15416062 * q^25 - 1439136 * q^28 - 6426996 * q^29 + 11195172 * q^30 - 7182800 * q^31 + 20255228 * q^32 - 5045328 * q^33 - 583976 * q^34 + 14447496 * q^35 + 32490072 * q^36 - 6385380 * q^37 + 46890376 * q^38 - 9253764 * q^39 + 32200748 * q^40 + 1444888 * q^41 + 21632832 * q^42 + 65294560 * q^43 - 130900556 * q^44 + 7453296 * q^45 + 3550600 * q^46 - 140350704 * q^47 - 81659664 * q^48 + 8923834 * q^49 + 298072388 * q^50 + 64611432 * q^51 + 8568300 * q^52 - 151297060 * q^53 - 57395628 * q^54 + 208227280 * q^55 - 311032696 * q^56 + 33192180 * q^57 - 742537264 * q^58 - 244845008 * q^59 - 232221492 * q^60 + 81916348 * q^61 + 299496776 * q^62 - 49023792 * q^63 - 272343632 * q^64 + 66718496 * q^65 + 282779100 * q^66 - 246892584 * q^67 - 360526128 * q^68 + 196425000 * q^69 + 109480096 * q^70 - 671315792 * q^71 + 455070960 * q^72 + 17117980 * q^73 - 213267560 * q^74 - 350058024 * q^75 + 460786136 * q^76 - 1338237784 * q^77 - 74030112 * q^78 + 1678620952 * q^79 + 112669252 * q^80 + 774840978 * q^81 - 790671972 * q^82 + 1651848096 * q^83 + 387239616 * q^84 + 1626588736 * q^85 - 3001123200 * q^86 - 107658072 * q^87 + 1326272628 * q^88 + 2353799584 * q^89 + 7479540 * q^90 - 883791584 * q^91 + 5149123720 * q^92 + 993518460 * q^93 - 1508905124 * q^94 + 1044594712 * q^95 - 2339530776 * q^96 - 2175343956 * q^97 - 4653889676 * q^98 - 963049824 * q^99

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(39))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	13
39.10.a.a	$39$	$10$	39.a	1.a	$1$	$3$	$3$	$20.086$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓	✓	39.10.a.a	$1$	$1$	\(-18\)	\(243\)	\(-888\)	\(-14406\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-6+\beta _{1})q^{2}+3^{4}q^{3}+(328-6\beta _{1}+\cdots)q^{4}+\cdots\)
39.10.a.b	$39$	$10$	39.a	1.a	$1$	$4$	$4$	$20.086$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	39.10.a.b	$1$	$1$	\(36\)	\(-324\)	\(-1044\)	\(5868\)		$+$	$+$	$+$	$2^{5}$	$\mathrm{SU}(2)$	\(q+(9+\beta _{1})q^{2}-3^{4}q^{3}+(118+23\beta _{1}+\cdots)q^{4}+\cdots\)
39.10.a.c	$39$	$10$	39.a	1.a	$1$	$5$	$5$	$20.086$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	39.10.a.c	$1$	$0$	\(52\)	\(-405\)	\(2624\)	\(-4802\)		$+$	$-$	$-$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+(10+\beta _{1})q^{2}-3^{4}q^{3}+(326+8\beta _{1}+\cdots)q^{4}+\cdots\)
39.10.a.d	$39$	$10$	39.a	1.a	$1$	$6$	$6$	$20.086$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	39.10.a.d	$1$	$0$	\(-2\)	\(486\)	\(444\)	\(5868\)		$-$	$+$	$-$	$2^{5}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+3^{4}q^{3}+(311-6\beta _{1}+\beta _{3}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(39))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(39)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 2}\)