Space of modular forms of level 4212, weight 2, and trivial character

• Label: 4212.2.a
• Level: $4212$
• Weight: $2$
• Character orbit: 4212.a
• Rep. character: $\chi_{4212}(1,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $13$
• Sturm bound: $1512$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 4212 = 2^{2} \cdot 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4212.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(1512\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4212))\).

	Total	New	Old
Modular forms	792	48	744
Cusp forms	721	48	673
Eisenstein series	71	0	71

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

\(2\)	\(3\)	\(13\)	Fricke	Dim
\(-\)	\(+\)	\(+\)	$-$	\(14\)
\(-\)	\(+\)	\(-\)	$+$	\(10\)
\(-\)	\(-\)	\(+\)	$+$	\(10\)
\(-\)	\(-\)	\(-\)	$-$	\(14\)
Plus space			\(+\)	\(20\)
Minus space			\(-\)	\(28\)

Trace form

\( 48 q + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4212))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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	13
4212.2.a.a	$4212$	$2$	4212.a	1.a	$1$	$1$	$1$	$33.633$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}-6q^{11}+q^{13}+3q^{17}+2q^{19}+\cdots\)
4212.2.a.b	$4212$	$2$	4212.a	1.a	$1$	$1$	$1$	$33.633$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}+6q^{11}+q^{13}-3q^{17}+2q^{19}+\cdots\)
4212.2.a.c	$4212$	$2$	4212.a	1.a	$1$	$2$	$2$	$33.633$	\(\Q(\sqrt{13}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{5}+(-1-\beta )q^{7}-\beta q^{11}+q^{13}+\cdots\)
4212.2.a.d	$4212$	$2$	4212.a	1.a	$1$	$2$	$2$	$33.633$	\(\Q(\sqrt{3}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{5}-q^{7}+3\beta q^{11}+q^{13}+4\beta q^{17}+\cdots\)
4212.2.a.e	$4212$	$2$	4212.a	1.a	$1$	$2$	$2$	$33.633$	\(\Q(\sqrt{13}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+(-1-\beta )q^{7}+\beta q^{11}+q^{13}+\cdots\)
4212.2.a.f	$4212$	$2$	4212.a	1.a	$1$	$4$	$4$	$33.633$	4.4.113688.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{5}+(-\beta _{1}-\beta _{2})q^{7}+(1+\cdots)q^{11}+\cdots\)
4212.2.a.g	$4212$	$2$	4212.a	1.a	$1$	$4$	$4$	$33.633$	\(\Q(\sqrt{3}, \sqrt{7})\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{5}+\beta _{2}q^{7}-\beta _{3}q^{11}-q^{13}+\cdots\)
4212.2.a.h	$4212$	$2$	4212.a	1.a	$1$	$4$	$4$	$33.633$	4.4.113688.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(3\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{5}+(-\beta _{1}-\beta _{2})q^{7}+(-1+\cdots)q^{11}+\cdots\)
4212.2.a.i	$4212$	$2$	4212.a	1.a	$1$	$5$	$5$	$33.633$	5.5.8655345.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(-7\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{5}-\beta _{2}q^{7}+(1+\beta _{1}+\beta _{3}+\cdots)q^{11}+\cdots\)
4212.2.a.j	$4212$	$2$	4212.a	1.a	$1$	$5$	$5$	$33.633$	5.5.8655345.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(7\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{5}-\beta _{2}q^{7}+(-1-\beta _{1}-\beta _{3}+\cdots)q^{11}+\cdots\)
4212.2.a.k	$4212$	$2$	4212.a	1.a	$1$	$6$	$6$	$33.633$	6.6.35342001.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(0\)		$-$	$-$	$+$	$+$	$3$	$\mathrm{SU}(2)$	\(q-\beta _{4}q^{5}+(-\beta _{1}+\beta _{3})q^{7}+(\beta _{1}-\beta _{3}+\cdots)q^{11}+\cdots\)
4212.2.a.l	$4212$	$2$	4212.a	1.a	$1$	$6$	$6$	$33.633$	6.6.232742592.1	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}+(2-\beta _{3}-\beta _{4})q^{7}+(-\beta _{1}+\cdots)q^{11}+\cdots\)
4212.2.a.m	$4212$	$2$	4212.a	1.a	$1$	$6$	$6$	$33.633$	6.6.35342001.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$+$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q+\beta _{4}q^{5}+(-\beta _{1}+\beta _{3})q^{7}+(-\beta _{1}+\beta _{3}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4212))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4212)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(117))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(156))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(162))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(234))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(324))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(351))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(468))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(702))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1053))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1404))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2106))\)\(^{\oplus 2}\)