Space of modular forms of level 10, weight 8, and trivial character

• Label: 10.8.a
• Level: $10$
• Weight: $8$
• Character orbit: 10.a
• Rep. character: $\chi_{10}(1,\cdot)$
• Character field: $\Q$
• Dimension: $1$
• Newform subspaces: $1$
• Sturm bound: $12$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 10 = 2 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	10.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(12\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(10))\).

	Total	New	Old
Modular forms	13	1	12
Cusp forms	9	1	8
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
\(+\)	\(+\)	\(+\)		\(4\)	\(0\)	\(4\)		\(3\)	\(0\)	\(3\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(3\)	\(1\)	\(2\)		\(2\)	\(1\)	\(1\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(7\)	\(1\)	\(6\)		\(5\)	\(1\)	\(4\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(6\)	\(0\)	\(6\)		\(4\)	\(0\)	\(4\)		\(2\)	\(0\)	\(2\)

Trace form

q + 8 * q^2 + 28 * q^3 + 64 * q^4 + 125 * q^5 + 224 * q^6 + 104 * q^7 + 512 * q^8 - 1403 * q^9 + 1000 * q^10 - 5148 * q^11 + 1792 * q^12 - 8602 * q^13 + 832 * q^14 + 3500 * q^15 + 4096 * q^16 + 20274 * q^17 - 11224 * q^18 + 45500 * q^19 + 8000 * q^20 + 2912 * q^21 - 41184 * q^22 - 72072 * q^23 + 14336 * q^24 + 15625 * q^25 - 68816 * q^26 - 100520 * q^27 + 6656 * q^28 + 231510 * q^29 + 28000 * q^30 - 80128 * q^31 + 32768 * q^32 - 144144 * q^33 + 162192 * q^34 + 13000 * q^35 - 89792 * q^36 + 104654 * q^37 + 364000 * q^38 - 240856 * q^39 + 64000 * q^40 + 584922 * q^41 + 23296 * q^42 - 795532 * q^43 - 329472 * q^44 - 175375 * q^45 - 576576 * q^46 + 425664 * q^47 + 114688 * q^48 - 812727 * q^49 + 125000 * q^50 + 567672 * q^51 - 550528 * q^52 + 1500798 * q^53 - 804160 * q^54 - 643500 * q^55 + 53248 * q^56 + 1274000 * q^57 + 1852080 * q^58 + 246420 * q^59 + 224000 * q^60 + 893942 * q^61 - 641024 * q^62 - 145912 * q^63 + 262144 * q^64 - 1075250 * q^65 - 1153152 * q^66 - 2336836 * q^67 + 1297536 * q^68 - 2018016 * q^69 + 104000 * q^70 - 203688 * q^71 - 718336 * q^72 - 3805702 * q^73 + 837232 * q^74 + 437500 * q^75 + 2912000 * q^76 - 535392 * q^77 - 1926848 * q^78 + 5053040 * q^79 + 512000 * q^80 + 253801 * q^81 + 4679376 * q^82 - 45492 * q^83 + 186368 * q^84 + 2534250 * q^85 - 6364256 * q^86 + 6482280 * q^87 - 2635776 * q^88 + 980010 * q^89 - 1403000 * q^90 - 894608 * q^91 - 4612608 * q^92 - 2243584 * q^93 + 3405312 * q^94 + 5687500 * q^95 + 917504 * q^96 - 5247646 * q^97 - 6501816 * q^98 + 7222644 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(10))\) 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
10.8.a.a	$10$	$8$	10.a	1.a	$1$	$1$	$1$	$3.124$	\(\Q\)	None	✓	✓	✓	✓	10.8.a.a	$1$	$0$	\(8\)	\(28\)	\(125\)	\(104\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+28q^{3}+2^{6}q^{4}+5^{3}q^{5}+224q^{6}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(10))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(10)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)