Space of modular forms of level 456, weight 6, and trivial character

• Label: 456.6.a
• Level: $456$
• Weight: $6$
• Character orbit: 456.a
• Rep. character: $\chi_{456}(1,\cdot)$
• Character field: $\Q$
• Dimension: $44$
• Newform subspaces: $8$
• Sturm bound: $480$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 456 = 2^{3} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	456.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(480\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(456))\).

	Total	New	Old
Modular forms	408	44	364
Cusp forms	392	44	348
Eisenstein series	16	0	16

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

\(2\)	\(3\)	\(19\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(4\)
\(+\)	\(+\)	\(-\)	$-$	\(7\)
\(+\)	\(-\)	\(+\)	$-$	\(5\)
\(+\)	\(-\)	\(-\)	$+$	\(5\)
\(-\)	\(+\)	\(+\)	$-$	\(5\)
\(-\)	\(+\)	\(-\)	$+$	\(6\)
\(-\)	\(-\)	\(+\)	$+$	\(5\)
\(-\)	\(-\)	\(-\)	$-$	\(7\)
Plus space			\(+\)	\(20\)
Minus space			\(-\)	\(24\)

Trace form

\( 44 q - 196 q^{5} + 76 q^{7} + 3564 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(456))\) 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	19
456.6.a.a	$456$	$6$	456.a	1.a	$1$	$4$	$4$	$73.135$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-36\)	\(-20\)	\(70\)		$+$	$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-9q^{3}+(-5+\beta _{2})q^{5}+(19-4\beta _{1}+\cdots)q^{7}+\cdots\)
456.6.a.b	$456$	$6$	456.a	1.a	$1$	$5$	$5$	$73.135$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-45\)	\(-90\)	\(-2\)		$-$	$+$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q-9q^{3}+(-18+\beta _{2}-\beta _{3})q^{5}+(\beta _{1}+\cdots)q^{7}+\cdots\)
456.6.a.c	$456$	$6$	456.a	1.a	$1$	$5$	$5$	$73.135$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(45\)	\(-59\)	\(-149\)		$+$	$-$	$-$	$+$	$2^{6}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+9q^{3}+(-12-\beta _{1})q^{5}+(-30+\beta _{1}+\cdots)q^{7}+\cdots\)
456.6.a.d	$456$	$6$	456.a	1.a	$1$	$5$	$5$	$73.135$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(45\)	\(-54\)	\(70\)		$-$	$-$	$+$	$+$	$2^{6}\cdot 5$	$\mathrm{SU}(2)$	\(q+9q^{3}+(-11+\beta _{2})q^{5}+(14+\beta _{4})q^{7}+\cdots\)
456.6.a.e	$456$	$6$	456.a	1.a	$1$	$5$	$5$	$73.135$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(45\)	\(66\)	\(-2\)		$+$	$-$	$+$	$-$	$2^{6}\cdot 3$	$\mathrm{SU}(2)$	\(q+9q^{3}+(13+\beta _{1})q^{5}+\beta _{2}q^{7}+3^{4}q^{9}+\cdots\)
456.6.a.f	$456$	$6$	456.a	1.a	$1$	$6$	$6$	$73.135$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-54\)	\(-65\)	\(-149\)		$-$	$+$	$-$	$+$	$2^{9}$	$\mathrm{SU}(2)$	\(q-9q^{3}+(-11-\beta _{2})q^{5}+(-5^{2}-\beta _{4}+\cdots)q^{7}+\cdots\)
456.6.a.g	$456$	$6$	456.a	1.a	$1$	$7$	$7$	$73.135$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-63\)	\(55\)	\(119\)		$+$	$+$	$-$	$-$	$2^{12}$	$\mathrm{SU}(2)$	\(q-9q^{3}+(8-\beta _{1})q^{5}+(17+\beta _{1}+\beta _{3}+\cdots)q^{7}+\cdots\)
456.6.a.h	$456$	$6$	456.a	1.a	$1$	$7$	$7$	$73.135$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(63\)	\(-29\)	\(119\)		$-$	$-$	$-$	$-$	$2^{10}$	$\mathrm{SU}(2)$	\(q+9q^{3}+(-4-\beta _{1})q^{5}+(17-\beta _{3})q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(456))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(456)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(114))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(152))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(228))\)\(^{\oplus 2}\)