Space of modular forms of level 6416, weight 2, and trivial character

• Label: 6416.2.a
• Level: $6416$
• Weight: $2$
• Character orbit: 6416.a
• Rep. character: $\chi_{6416}(1,\cdot)$
• Character field: $\Q$
• Dimension: $200$
• Newform subspaces: $17$
• Sturm bound: $1608$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 6416 = 2^{4} \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6416.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 17 \)
Sturm bound:			\(1608\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6416))\).

	Total	New	Old
Modular forms	810	200	610
Cusp forms	799	200	599
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.

\(2\)	\(401\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(45\)
\(+\)	\(-\)	\(-\)	\(55\)
\(-\)	\(+\)	\(-\)	\(55\)
\(-\)	\(-\)	\(+\)	\(45\)
Plus space		\(+\)	\(90\)
Minus space		\(-\)	\(110\)

Trace form

\( 200 q + 2 q^{3} + 4 q^{7} + 200 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6416))\) 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	401
6416.2.a.a	$6416$	$2$	6416.a	1.a	$1$	$1$	$1$	$51.232$	\(\Q\)	None	✓				802.2.a.b	$1$	$0$	\(0\)	\(0\)	\(4\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{5}+2q^{7}-3q^{9}-3q^{11}+q^{13}+\cdots\)
6416.2.a.b	$6416$	$2$	6416.a	1.a	$1$	$1$	$1$	$51.232$	\(\Q\)	None	✓				802.2.a.a	$1$	$0$	\(0\)	\(2\)	\(-2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2q^{5}+q^{9}-4q^{15}+6q^{17}+\cdots\)
6416.2.a.c	$6416$	$2$	6416.a	1.a	$1$	$2$	$2$	$51.232$	\(\Q(\sqrt{5}) \)	None	✓				3208.2.a.c	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(-2+3\beta )q^{5}+2\beta q^{7}+(-2+\cdots)q^{9}+\cdots\)
6416.2.a.d	$6416$	$2$	6416.a	1.a	$1$	$2$	$2$	$51.232$	\(\Q(\sqrt{5}) \)	None	✓				3208.2.a.a	$1$	$0$	\(0\)	\(2\)	\(-5\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-3+\beta )q^{5}+(1+2\beta )q^{7}-2q^{9}+\cdots\)
6416.2.a.e	$6416$	$2$	6416.a	1.a	$1$	$2$	$2$	$51.232$	\(\Q(\sqrt{5}) \)	None	✓				3208.2.a.b	$1$	$0$	\(0\)	\(2\)	\(-1\)	\(7\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{3}-\beta q^{5}+(3+\beta )q^{7}+(1+4\beta )q^{9}+\cdots\)
6416.2.a.f	$6416$	$2$	6416.a	1.a	$1$	$5$	$5$	$51.232$	5.5.38569.1	None	✓				802.2.a.c	$1$	$1$	\(0\)	\(6\)	\(-9\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{4})q^{3}+(-2+\beta _{1}+\beta _{3})q^{5}+\cdots\)
6416.2.a.g	$6416$	$2$	6416.a	1.a	$1$	$7$	$7$	$51.232$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓				802.2.a.d	$1$	$1$	\(0\)	\(-6\)	\(5\)	\(-1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{5})q^{3}+(1-\beta _{1}+\beta _{2})q^{5}+\cdots\)
6416.2.a.h	$6416$	$2$	6416.a	1.a	$1$	$9$	$9$	$51.232$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓				802.2.a.e	$1$	$0$	\(0\)	\(6\)	\(-5\)	\(7\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{7})q^{3}+(-1+\beta _{3})q^{5}+(1-\beta _{1}+\cdots)q^{7}+\cdots\)
6416.2.a.i	$6416$	$2$	6416.a	1.a	$1$	$10$	$10$	$51.232$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓				802.2.a.f	$1$	$0$	\(0\)	\(-10\)	\(9\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(1+\beta _{8})q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
6416.2.a.j	$6416$	$2$	6416.a	1.a	$1$	$12$	$12$	$51.232$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓				1604.2.a.a	$1$	$1$	\(0\)	\(3\)	\(-7\)	\(0\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1-\beta _{6})q^{5}-\beta _{3}q^{7}+(\beta _{1}+\cdots)q^{9}+\cdots\)
6416.2.a.k	$6416$	$2$	6416.a	1.a	$1$	$12$	$12$	$51.232$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓				401.2.a.a	$1$	$0$	\(0\)	\(5\)	\(-7\)	\(20\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{5}q^{3}+(-1-\beta _{1}+\beta _{2}-\beta _{3}+\beta _{5}+\cdots)q^{5}+\cdots\)
6416.2.a.l	$6416$	$2$	6416.a	1.a	$1$	$19$	$19$	$51.232$	\(\mathbb{Q}[x]/(x^{19} - \cdots)\)	None	✓				3208.2.a.d	$1$	$1$	\(0\)	\(4\)	\(-11\)	\(3\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1-\beta _{11})q^{5}+\beta _{14}q^{7}+\cdots\)
6416.2.a.m	$6416$	$2$	6416.a	1.a	$1$	$21$	$21$	$51.232$		None	✓		✓		401.2.a.b	$1$	$1$	\(0\)	\(-3\)	\(3\)	\(-24\)		$-$	$-$	$+$		$\mathrm{SU}(2)$
6416.2.a.n	$6416$	$2$	6416.a	1.a	$1$	$22$	$22$	$51.232$		None	✓		✓		1604.2.a.b	$1$	$0$	\(0\)	\(-3\)	\(9\)	\(0\)		$-$	$+$	$-$		$\mathrm{SU}(2)$
6416.2.a.o	$6416$	$2$	6416.a	1.a	$1$	$24$	$24$	$51.232$		None	✓		✓		3208.2.a.f	$1$	$1$	\(0\)	\(-11\)	\(4\)	\(-5\)		$+$	$+$	$+$		$\mathrm{SU}(2)$
6416.2.a.p	$6416$	$2$	6416.a	1.a	$1$	$24$	$24$	$51.232$		None	✓				3208.2.a.e	$1$	$0$	\(0\)	\(11\)	\(-5\)	\(-4\)		$+$	$-$	$-$		$\mathrm{SU}(2)$
6416.2.a.q	$6416$	$2$	6416.a	1.a	$1$	$27$	$27$	$51.232$		None	✓		✓		3208.2.a.g	$1$	$0$	\(0\)	\(-5\)	\(19\)	\(-7\)		$+$	$-$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6416))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6416)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(401))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(802))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1604))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3208))\)\(^{\oplus 2}\)