Space of modular forms of level 612, weight 2, and trivial character

• Label: 612.2.a
• Level: $612$
• Weight: $2$
• Character orbit: 612.a
• Rep. character: $\chi_{612}(1,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $5$
• Sturm bound: $216$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 612 = 2^{2} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	612.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(216\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(612))\).

	Total	New	Old
Modular forms	120	6	114
Cusp forms	97	6	91
Eisenstein series	23	0	23

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

\(2\)	\(3\)	\(17\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(13\)	\(0\)	\(13\)		\(10\)	\(0\)	\(10\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)		\(17\)	\(0\)	\(17\)		\(13\)	\(0\)	\(13\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)		\(16\)	\(0\)	\(16\)		\(12\)	\(0\)	\(12\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)		\(16\)	\(0\)	\(16\)		\(12\)	\(0\)	\(12\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)		\(17\)	\(1\)	\(16\)		\(15\)	\(1\)	\(14\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(13\)	\(1\)	\(12\)		\(11\)	\(1\)	\(10\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(14\)	\(1\)	\(13\)		\(12\)	\(1\)	\(11\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(14\)	\(3\)	\(11\)		\(12\)	\(3\)	\(9\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(56\)	\(2\)	\(54\)		\(45\)	\(2\)	\(43\)		\(11\)	\(0\)	\(11\)
Minus space			\(-\)		\(64\)	\(4\)	\(60\)		\(52\)	\(4\)	\(48\)		\(12\)	\(0\)	\(12\)

Trace form

6 * q + 6 * q^7 - 2 * q^11 + 2 * q^17 - 8 * q^19 + 6 * q^23 + 14 * q^25 + 8 * q^29 - 2 * q^31 - 8 * q^35 + 8 * q^37 + 12 * q^41 - 8 * q^43 - 4 * q^47 - 10 * q^49 + 8 * q^53 + 8 * q^55 - 12 * q^59 + 32 * q^65 + 8 * q^67 + 6 * q^71 + 20 * q^73 - 12 * q^77 + 10 * q^79 + 8 * q^83 - 4 * q^85 - 40 * q^89 - 8 * q^91 - 24 * q^95 - 12 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(612))\) 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	17
612.2.a.a	$612$	$2$	612.a	1.a	$1$	$1$	$1$	$4.887$	\(\Q\)	None	✓	✓	✓	✓	612.2.a.a	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}+2q^{7}-3q^{11}-q^{13}+q^{17}+\cdots\)
612.2.a.b	$612$	$2$	612.a	1.a	$1$	$1$	$1$	$4.887$	\(\Q\)	None	✓		✓	✓	204.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-5q^{11}-5q^{13}-q^{17}+q^{19}+\cdots\)
612.2.a.c	$612$	$2$	612.a	1.a	$1$	$1$	$1$	$4.887$	\(\Q\)	None	✓				204.2.a.a	$1$	$0$	\(0\)	\(0\)	\(1\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+4q^{7}-3q^{11}+3q^{13}+q^{17}+\cdots\)
612.2.a.d	$612$	$2$	612.a	1.a	$1$	$1$	$1$	$4.887$	\(\Q\)	None	✓	✓	✓	✓	612.2.a.a	$1$	$0$	\(0\)	\(0\)	\(3\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}+2q^{7}+3q^{11}-q^{13}-q^{17}+\cdots\)
612.2.a.e	$612$	$2$	612.a	1.a	$1$	$2$	$2$	$4.887$	\(\Q(\sqrt{3}) \)	None	✓		✓		68.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{5}+(-1-\beta )q^{7}+(3-\beta )q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(612))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(612)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(68))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(102))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(153))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(204))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(306))\)\(^{\oplus 2}\)