Space of modular forms of level 12, weight 8, and trivial character

• Label: 12.8.a
• Level: $12$
• Weight: $8$
• Character orbit: 12.a
• Rep. character: $\chi_{12}(1,\cdot)$
• Character field: $\Q$
• Dimension: $2$
• Newform subspaces: $2$
• Sturm bound: $16$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 12 = 2^{2} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	12.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(16\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(12))\).

	Total	New	Old
Modular forms	17	2	15
Cusp forms	11	2	9
Eisenstein series	6	0	6

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(3\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(5\)	\(0\)	\(5\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(4\)	\(0\)	\(4\)		\(2\)	\(0\)	\(2\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)		\(4\)	\(1\)	\(3\)		\(3\)	\(1\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(4\)	\(1\)	\(3\)		\(3\)	\(1\)	\(2\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(9\)	\(1\)	\(8\)		\(6\)	\(1\)	\(5\)		\(3\)	\(0\)	\(3\)
Minus space		\(-\)		\(8\)	\(1\)	\(7\)		\(5\)	\(1\)	\(4\)		\(3\)	\(0\)	\(3\)

Trace form

2 * q - 108 * q^5 + 280 * q^7 + 1458 * q^9 - 8208 * q^11 + 10300 * q^13 + 17496 * q^15 - 58860 * q^17 + 35440 * q^19 + 52488 * q^21 - 92880 * q^23 + 59534 * q^25 - 108 * q^29 - 120392 * q^31 - 87480 * q^33 + 614736 * q^35 - 244100 * q^37 - 524880 * q^39 + 418500 * q^41 - 518960 * q^43 - 78732 * q^45 + 1328400 * q^47 + 281682 * q^49 - 384912 * q^51 - 2043900 * q^53 - 606528 * q^55 + 1837080 * q^57 - 433728 * q^59 + 3900844 * q^61 + 204120 * q^63 - 6854760 * q^65 - 311360 * q^67 + 3709152 * q^69 + 1643760 * q^71 + 465460 * q^73 - 1889568 * q^75 - 4298400 * q^77 + 1171864 * q^79 + 1062882 * q^81 + 10620720 * q^83 - 1440504 * q^85 - 8485560 * q^87 + 3245940 * q^89 - 17453680 * q^91 - 2361960 * q^93 + 20131200 * q^95 + 12096100 * q^97 - 5983632 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(12))\) 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	3
12.8.a.a	$12$	$8$	12.a	1.a	$1$	$1$	$1$	$3.749$	\(\Q\)	None	✓	✓	✓	✓	12.8.a.a	$1$	$1$	\(0\)	\(-27\)	\(-378\)	\(-832\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}-378q^{5}-832q^{7}+3^{6}q^{9}+\cdots\)
12.8.a.b	$12$	$8$	12.a	1.a	$1$	$1$	$1$	$3.749$	\(\Q\)	None	✓	✓	✓	✓	12.8.a.b	$1$	$0$	\(0\)	\(27\)	\(270\)	\(1112\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+270q^{5}+1112q^{7}+3^{6}q^{9}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(12))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(12)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 2}\)