Space of modular forms of level 12, weight 10, and trivial character

• Label: 12.10.a
• Level: $12$
• Weight: $10$
• Character orbit: 12.a
• Rep. character: $\chi_{12}(1,\cdot)$
• Character field: $\Q$
• Dimension: $1$
• Newform subspaces: $1$
• Sturm bound: $20$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 12 = 2^{2} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	12.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(20\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(12))\).

	Total	New	Old
Modular forms	21	1	20
Cusp forms	15	1	14
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
\(+\)	\(+\)	\(+\)		\(5\)	\(0\)	\(5\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(6\)	\(0\)	\(6\)		\(4\)	\(0\)	\(4\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)		\(5\)	\(1\)	\(4\)		\(4\)	\(1\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(5\)	\(0\)	\(5\)		\(4\)	\(0\)	\(4\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(10\)	\(0\)	\(10\)		\(7\)	\(0\)	\(7\)		\(3\)	\(0\)	\(3\)
Minus space		\(-\)		\(11\)	\(1\)	\(10\)		\(8\)	\(1\)	\(7\)		\(3\)	\(0\)	\(3\)

Trace form

q - 81 * q^3 + 990 * q^5 + 8576 * q^7 + 6561 * q^9 + 70596 * q^11 - 2530 * q^13 - 80190 * q^15 - 200574 * q^17 - 695620 * q^19 - 694656 * q^21 + 2472696 * q^23 - 973025 * q^25 - 531441 * q^27 + 5474214 * q^29 + 3732104 * q^31 - 5718276 * q^33 + 8490240 * q^35 - 21898522 * q^37 + 204930 * q^39 - 23818950 * q^41 + 10612676 * q^43 + 6495390 * q^45 + 2398464 * q^47 + 33194169 * q^49 + 16246494 * q^51 - 8994978 * q^53 + 69890040 * q^55 + 56345220 * q^57 - 143417916 * q^59 - 19804258 * q^61 + 56267136 * q^63 - 2504700 * q^65 - 165625156 * q^67 - 200288376 * q^69 - 194801400 * q^71 + 148729418 * q^73 + 78815025 * q^75 + 605431296 * q^77 - 30134152 * q^79 + 43046721 * q^81 + 302054076 * q^83 - 198568260 * q^85 - 443411334 * q^87 + 909502650 * q^89 - 21697280 * q^91 - 302300424 * q^93 - 688663800 * q^95 - 872463358 * q^97 + 463180356 * q^99

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(12))\) 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
12.10.a.a	$12$	$10$	12.a	1.a	$1$	$1$	$1$	$6.180$	\(\Q\)	None	✓	✓	✓	✓	12.10.a.a	$1$	$0$	\(0\)	\(-81\)	\(990\)	\(8576\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3^{4}q^{3}+990q^{5}+8576q^{7}+3^{8}q^{9}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(12))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(12)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 2}\)