Space of modular forms of level 212, weight 2, and trivial character

• Label: 212.2.a
• Level: $212$
• Weight: $2$
• Character orbit: 212.a
• Rep. character: $\chi_{212}(1,\cdot)$
• Character field: $\Q$
• Dimension: $5$
• Newform subspaces: $3$
• Sturm bound: $54$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 212 = 2^{2} \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	212.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(54\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(212))\).

	Total	New	Old
Modular forms	30	5	25
Cusp forms	25	5	20
Eisenstein series	5	0	5

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

\(2\)	\(53\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(6\)	\(0\)	\(6\)		\(5\)	\(0\)	\(5\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(9\)	\(0\)	\(9\)		\(7\)	\(0\)	\(7\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)		\(9\)	\(4\)	\(5\)		\(8\)	\(4\)	\(4\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(6\)	\(1\)	\(5\)		\(5\)	\(1\)	\(4\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(12\)	\(1\)	\(11\)		\(10\)	\(1\)	\(9\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(18\)	\(4\)	\(14\)		\(15\)	\(4\)	\(11\)		\(3\)	\(0\)	\(3\)

Trace form

5 * q - 2 * q^3 + 4 * q^7 + 5 * q^9 + 4 * q^11 + 6 * q^13 + 2 * q^17 + 4 * q^19 + 2 * q^21 - 8 * q^23 + 7 * q^25 - 14 * q^27 + 2 * q^29 - 12 * q^31 + 2 * q^33 - 20 * q^35 - 2 * q^37 - 12 * q^39 - 2 * q^41 - 6 * q^45 + 16 * q^47 + 5 * q^49 - 20 * q^51 - 3 * q^53 - 28 * q^57 - 20 * q^59 + 28 * q^63 + 10 * q^65 - 4 * q^67 - 16 * q^69 - 6 * q^71 + 28 * q^73 + 2 * q^75 - 4 * q^77 + 18 * q^79 + 5 * q^81 - 20 * q^83 + 22 * q^85 + 10 * q^87 - 26 * q^89 + 44 * q^91 + 48 * q^93 - 12 * q^95 + 2 * q^97 - 44 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(212))\) 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	53
212.2.a.a	$212$	$2$	212.a	1.a	$1$	$1$	$1$	$1.693$	\(\Q\)	None	✓	✓	✓	✓	212.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(-2\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}-2q^{7}-2q^{9}+2q^{11}+\cdots\)
212.2.a.b	$212$	$2$	212.a	1.a	$1$	$1$	$1$	$1.693$	\(\Q\)	None	✓	✓			212.2.a.b	$1$	$0$	\(0\)	\(2\)	\(2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+2q^{5}+q^{9}-4q^{11}-2q^{13}+\cdots\)
212.2.a.c	$212$	$2$	212.a	1.a	$1$	$3$	$3$	$1.693$	3.3.756.1	None	✓	✓	✓		212.2.a.c	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}-\beta _{2}q^{5}+(2+\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(212))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(212)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(53))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(106))\)\(^{\oplus 2}\)