Space of modular forms of level 4016, weight 2, and trivial character

• Label: 4016.2.a
• Level: $4016$
• Weight: $2$
• Character orbit: 4016.a
• Rep. character: $\chi_{4016}(1,\cdot)$
• Character field: $\Q$
• Dimension: $125$
• Newform subspaces: $13$
• Sturm bound: $1008$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(1008\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4016))\).

	Total	New	Old
Modular forms	510	125	385
Cusp forms	499	125	374
Eisenstein series	11	0	11

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

\(2\)	\(251\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(21\)
\(+\)	\(-\)	$-$	\(42\)
\(-\)	\(+\)	$-$	\(31\)
\(-\)	\(-\)	$+$	\(31\)
Plus space		\(+\)	\(52\)
Minus space		\(-\)	\(73\)

Trace form

\( 125 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 125 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4016))\) 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	251
4016.2.a.a	$4016$	$2$	4016.a	1.a	$1$	$2$	$2$	$32.068$	\(\Q(\sqrt{13}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(5\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(2+\beta )q^{5}+q^{7}-2q^{9}-3q^{11}+\cdots\)
4016.2.a.b	$4016$	$2$	4016.a	1.a	$1$	$2$	$2$	$32.068$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(-3\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-2+\beta )q^{5}+(1+2\beta )q^{7}-2q^{9}+\cdots\)
4016.2.a.c	$4016$	$2$	4016.a	1.a	$1$	$4$	$4$	$32.068$	4.4.725.1	None	✓	$1$	$1$	\(0\)	\(2\)	\(-3\)	\(3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(-1+\beta _{1}+\beta _{2}-\beta _{3})q^{5}+\cdots\)
4016.2.a.d	$4016$	$2$	4016.a	1.a	$1$	$5$	$5$	$32.068$	5.5.138917.1	None	✓	$1$	$0$	\(0\)	\(-2\)	\(7\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{4}q^{3}+(2+\beta _{1}+\beta _{2}-\beta _{3})q^{5}+(\beta _{2}+\cdots)q^{7}+\cdots\)
4016.2.a.e	$4016$	$2$	4016.a	1.a	$1$	$5$	$5$	$32.068$	5.5.242773.1	None	✓	$1$	$1$	\(0\)	\(1\)	\(-6\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1+\beta _{2})q^{5}+(1-2\beta _{1}+\cdots)q^{7}+\cdots\)
4016.2.a.f	$4016$	$2$	4016.a	1.a	$1$	$6$	$6$	$32.068$	6.6.60853001.1	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(-6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{4}q^{3}+(-1+\beta _{1}-\beta _{3})q^{5}+(-1+\cdots)q^{7}+\cdots\)
4016.2.a.g	$4016$	$2$	4016.a	1.a	$1$	$7$	$7$	$32.068$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(3\)	\(-2\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{5}q^{5}+(1+\beta _{6})q^{7}+(-1+\cdots)q^{9}+\cdots\)
4016.2.a.h	$4016$	$2$	4016.a	1.a	$1$	$9$	$9$	$32.068$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(-5\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1-\beta _{5})q^{5}+(-\beta _{6}+\beta _{7}+\cdots)q^{7}+\cdots\)
4016.2.a.i	$4016$	$2$	4016.a	1.a	$1$	$12$	$12$	$32.068$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-3\)	\(5\)	\(-5\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+\beta _{8}q^{5}-\beta _{5}q^{7}+(\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots\)
4016.2.a.j	$4016$	$2$	4016.a	1.a	$1$	$14$	$14$	$32.068$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-3\)	\(-2\)	\(-8\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+\beta _{4}q^{5}+(-1+\beta _{9})q^{7}+(1+\cdots)q^{9}+\cdots\)
4016.2.a.k	$4016$	$2$	4016.a	1.a	$1$	$17$	$17$	$32.068$	\(\mathbb{Q}[x]/(x^{17} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(3\)	\(-3\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{4}q^{3}+\beta _{6}q^{5}+\beta _{13}q^{7}+(1+\beta _{3}+\cdots)q^{9}+\cdots\)
4016.2.a.l	$4016$	$2$	4016.a	1.a	$1$	$19$	$19$	$32.068$	\(\mathbb{Q}[x]/(x^{19} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(6\)	\(-8\)	\(11\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{4}q^{5}+(1+\beta _{13})q^{7}+(1+\cdots)q^{9}+\cdots\)
4016.2.a.m	$4016$	$2$	4016.a	1.a	$1$	$23$	$23$	$32.068$		None	✓	$1$	$0$	\(0\)	\(-2\)	\(8\)	\(-2\)		$+$	$-$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4016))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4016)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(251))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(502))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1004))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2008))\)\(^{\oplus 2}\)