Space of modular forms of level 10, weight 6, and trivial character

• Label: 10.6.a
• Level: $10$
• Weight: $6$
• Character orbit: 10.a
• Rep. character: $\chi_{10}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $3$
• Sturm bound: $9$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 10 = 2 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	10.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(9\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(10))\).

	Total	New	Old
Modular forms	9	3	6
Cusp forms	5	3	2
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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(2\)	\(1\)	\(1\)		\(1\)	\(1\)	\(0\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(3\)	\(1\)	\(2\)		\(2\)	\(1\)	\(1\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(2\)	\(1\)	\(1\)		\(1\)	\(1\)	\(0\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(4\)	\(1\)	\(3\)		\(2\)	\(1\)	\(1\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(5\)	\(2\)	\(3\)		\(3\)	\(2\)	\(1\)		\(2\)	\(0\)	\(2\)

Trace form

3 * q - 4 * q^2 + 4 * q^3 + 48 * q^4 - 25 * q^5 + 32 * q^6 - 312 * q^7 - 64 * q^8 + 559 * q^9 - 100 * q^10 - 444 * q^11 + 64 * q^12 + 114 * q^13 + 304 * q^14 + 1100 * q^15 + 768 * q^16 + 918 * q^17 - 3892 * q^18 - 1140 * q^19 - 400 * q^20 - 4264 * q^21 + 3312 * q^22 + 3144 * q^23 + 512 * q^24 + 1875 * q^25 + 8392 * q^26 - 5480 * q^27 - 4992 * q^28 + 2490 * q^29 - 5600 * q^30 - 8544 * q^31 - 1024 * q^32 + 24288 * q^33 + 2424 * q^34 - 800 * q^35 + 8944 * q^36 - 16902 * q^37 - 17360 * q^38 - 14872 * q^39 - 1600 * q^40 + 2406 * q^41 + 11392 * q^42 + 13164 * q^43 - 7104 * q^44 + 2675 * q^45 - 48 * q^46 + 44448 * q^47 + 1024 * q^48 - 6429 * q^49 - 2500 * q^50 - 10584 * q^51 + 1824 * q^52 - 44406 * q^53 + 320 * q^54 + 17700 * q^55 + 4864 * q^56 - 33040 * q^57 + 37320 * q^58 + 1380 * q^59 + 17600 * q^60 + 12306 * q^61 - 20768 * q^62 - 42376 * q^63 + 12288 * q^64 - 50150 * q^65 - 87936 * q^66 + 10788 * q^67 + 14688 * q^68 + 146568 * q^69 + 26800 * q^70 + 88056 * q^71 - 62272 * q^72 - 9666 * q^73 + 23464 * q^74 + 2500 * q^75 - 18240 * q^76 - 28464 * q^77 + 112576 * q^78 - 81360 * q^79 - 6400 * q^80 + 28243 * q^81 - 12648 * q^82 - 220476 * q^83 - 68224 * q^84 - 34050 * q^85 - 72128 * q^86 + 233880 * q^87 + 52992 * q^88 - 14130 * q^89 + 30700 * q^90 + 33216 * q^91 + 50304 * q^92 + 117968 * q^93 - 72816 * q^94 + 53500 * q^95 + 8192 * q^96 - 224682 * q^97 + 2652 * q^98 - 328332 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(10))\) 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
10.6.a.a	$10$	$6$	10.a	1.a	$1$	$1$	$1$	$1.604$	\(\Q\)	None	✓	✓	✓	✓	10.6.a.a	$1$	$1$	\(-4\)	\(-26\)	\(-25\)	\(-22\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}-26q^{3}+2^{4}q^{4}-5^{2}q^{5}+104q^{6}+\cdots\)
10.6.a.b	$10$	$6$	10.a	1.a	$1$	$1$	$1$	$1.604$	\(\Q\)	None	✓	✓	✓	✓	10.6.a.b	$1$	$0$	\(-4\)	\(24\)	\(25\)	\(-172\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+24q^{3}+2^{4}q^{4}+5^{2}q^{5}-96q^{6}+\cdots\)
10.6.a.c	$10$	$6$	10.a	1.a	$1$	$1$	$1$	$1.604$	\(\Q\)	None	✓	✓	✓	✓	10.6.a.c	$1$	$0$	\(4\)	\(6\)	\(-25\)	\(-118\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+6q^{3}+2^{4}q^{4}-5^{2}q^{5}+24q^{6}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(10))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(10)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)