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

• Label: 8.10.a
• Level: $8$
• Weight: $10$
• Character orbit: 8.a
• Rep. character: $\chi_{8}(1,\cdot)$
• Character field: $\Q$
• Dimension: $2$
• Newform subspaces: $2$
• Sturm bound: $10$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 8 = 2^{3} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	8.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(10\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

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

	Total	New	Old
Modular forms	11	2	9
Cusp forms	7	2	5
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.

\(2\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(5\)	\(1\)	\(4\)		\(3\)	\(1\)	\(2\)		\(2\)	\(0\)	\(2\)
\(-\)		\(6\)	\(1\)	\(5\)		\(4\)	\(1\)	\(3\)		\(2\)	\(0\)	\(2\)

Trace form

2 * q + 8 * q^3 - 564 * q^5 + 5904 * q^7 - 31142 * q^9 + 97560 * q^11 - 188836 * q^13 + 227120 * q^15 - 171228 * q^17 - 53720 * q^19 + 957504 * q^21 - 2602704 * q^23 + 2675326 * q^25 - 216496 * q^27 + 1154940 * q^29 - 5041088 * q^31 - 5352352 * q^33 + 24483936 * q^35 - 13132788 * q^37 - 11273872 * q^39 - 15066924 * q^41 + 45379928 * q^43 + 10617052 * q^45 - 79274400 * q^47 + 43184626 * q^49 + 29265040 * q^51 + 39751980 * q^53 - 188304496 * q^55 + 67197088 * q^57 + 173485944 * q^59 - 12522052 * q^61 - 84460080 * q^63 - 241267032 * q^65 + 195391624 * q^67 - 85728064 * q^69 + 335441808 * q^71 - 7997164 * q^73 - 118666760 * q^75 - 366658368 * q^77 - 34286944 * q^79 + 323563378 * q^81 - 291672408 * q^83 + 886884952 * q^85 - 187171344 * q^87 - 6641100 * q^89 - 1756556064 * q^91 + 479023360 * q^93 + 1902684144 * q^95 + 204662980 * q^97 - 1565047496 * q^99

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(8))\) 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
8.10.a.a	$8$	$10$	8.a	1.a	$1$	$1$	$1$	$4.120$	\(\Q\)	None	✓	✓	✓	✓	8.10.a.a	$1$	$1$	\(0\)	\(-60\)	\(-2074\)	\(-4344\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-60q^{3}-2074q^{5}-4344q^{7}-16083q^{9}+\cdots\)
8.10.a.b	$8$	$10$	8.a	1.a	$1$	$1$	$1$	$4.120$	\(\Q\)	None	✓	✓	✓	✓	8.10.a.b	$1$	$0$	\(0\)	\(68\)	\(1510\)	\(10248\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+68q^{3}+1510q^{5}+10248q^{7}-15059q^{9}+\cdots\)

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

\( S_{10}^{\mathrm{old}}(\Gamma_0(8)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 2}\)