Space of modular forms of level 416, weight 2, and trivial character

• Label: 416.2.a
• Level: $416$
• Weight: $2$
• Character orbit: 416.a
• Rep. character: $\chi_{416}(1,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $6$
• Sturm bound: $112$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(112\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(416))\).

	Total	New	Old
Modular forms	64	12	52
Cusp forms	49	12	37
Eisenstein series	15	0	15

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

\(2\)	\(13\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(13\)	\(2\)	\(11\)		\(10\)	\(2\)	\(8\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)		\(19\)	\(5\)	\(14\)		\(15\)	\(5\)	\(10\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)		\(19\)	\(4\)	\(15\)		\(15\)	\(4\)	\(11\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)		\(13\)	\(1\)	\(12\)		\(9\)	\(1\)	\(8\)		\(4\)	\(0\)	\(4\)
Plus space		\(+\)		\(26\)	\(3\)	\(23\)		\(19\)	\(3\)	\(16\)		\(7\)	\(0\)	\(7\)
Minus space		\(-\)		\(38\)	\(9\)	\(29\)		\(30\)	\(9\)	\(21\)		\(8\)	\(0\)	\(8\)

Trace form

12 * q + 20 * q^9 + 8 * q^17 + 16 * q^21 + 28 * q^25 + 16 * q^29 - 24 * q^41 - 32 * q^45 + 28 * q^49 - 40 * q^53 - 16 * q^57 + 8 * q^61 - 32 * q^69 + 40 * q^73 - 40 * q^77 - 36 * q^81 - 16 * q^85 - 40 * q^89 - 16 * q^93 - 24 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(416))\) 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	13
416.2.a.a	$416$	$2$	416.a	1.a	$1$	$1$	$1$	$3.322$	\(\Q\)	None	✓	✓	✓	✓	416.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(-3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-3q^{7}-2q^{9}-2q^{11}+\cdots\)
416.2.a.b	$416$	$2$	416.a	1.a	$1$	$1$	$1$	$3.322$	\(\Q\)	None	✓	✓			416.2.a.a	$1$	$0$	\(0\)	\(1\)	\(1\)	\(3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+3q^{7}-2q^{9}+2q^{11}+\cdots\)
416.2.a.c	$416$	$2$	416.a	1.a	$1$	$2$	$2$	$3.322$	\(\Q(\sqrt{17}) \)	None	✓	✓	✓	✓	416.2.a.c	$1$	$1$	\(0\)	\(-1\)	\(-3\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(-2+\beta )q^{5}+(-2+\beta )q^{7}+\cdots\)
416.2.a.d	$416$	$2$	416.a	1.a	$1$	$2$	$2$	$3.322$	\(\Q(\sqrt{5}) \)	None	✓	✓			416.2.a.d	$2$	$0$	\(0\)	\(0\)	\(6\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+3q^{5}-\beta q^{7}+2q^{9}+2\beta q^{11}+\cdots\)
416.2.a.e	$416$	$2$	416.a	1.a	$1$	$2$	$2$	$3.322$	\(\Q(\sqrt{17}) \)	None	✓	✓			416.2.a.c	$1$	$0$	\(0\)	\(1\)	\(-3\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(-2+\beta )q^{5}+(2-\beta )q^{7}+(1+\cdots)q^{9}+\cdots\)
416.2.a.f	$416$	$2$	416.a	1.a	$1$	$4$	$4$	$3.322$	4.4.13448.1	None	✓	✓	✓		416.2.a.f	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+\beta _{3}q^{5}+(\beta _{1}-\beta _{2})q^{7}+(3+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(416))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(416)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(208))\)\(^{\oplus 2}\)