Space of modular forms of level 456, weight 2, and trivial character

• Label: 456.2.a
• Level: $456$
• Weight: $2$
• Character orbit: 456.a
• Rep. character: $\chi_{456}(1,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $6$
• Sturm bound: $160$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 456 = 2^{3} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	456.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(160\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(456))\).

	Total	New	Old
Modular forms	88	8	80
Cusp forms	73	8	65
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.

\(2\)	\(3\)	\(19\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(8\)	\(0\)	\(8\)		\(7\)	\(0\)	\(7\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(13\)	\(3\)	\(10\)		\(11\)	\(3\)	\(8\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(10\)	\(1\)	\(9\)		\(8\)	\(1\)	\(7\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(11\)	\(1\)	\(10\)		\(9\)	\(1\)	\(8\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(14\)	\(0\)	\(14\)		\(12\)	\(0\)	\(12\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(9\)	\(1\)	\(8\)		\(7\)	\(1\)	\(6\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(12\)	\(0\)	\(12\)		\(10\)	\(0\)	\(10\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(11\)	\(2\)	\(9\)		\(9\)	\(2\)	\(7\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(40\)	\(2\)	\(38\)		\(33\)	\(2\)	\(31\)		\(7\)	\(0\)	\(7\)
Minus space			\(-\)		\(48\)	\(6\)	\(42\)		\(40\)	\(6\)	\(34\)		\(8\)	\(0\)	\(8\)

Trace form

8 * q + 4 * q^5 - 4 * q^7 + 8 * q^9 + 4 * q^13 - 4 * q^15 + 4 * q^17 + 6 * q^19 + 12 * q^23 + 20 * q^25 - 4 * q^29 - 12 * q^31 + 12 * q^33 + 12 * q^35 + 4 * q^37 - 8 * q^39 - 4 * q^41 + 4 * q^43 + 4 * q^45 + 24 * q^47 + 12 * q^49 - 8 * q^51 - 12 * q^53 - 4 * q^55 - 2 * q^57 + 16 * q^59 - 20 * q^61 - 4 * q^63 - 32 * q^65 - 32 * q^67 + 4 * q^73 - 16 * q^75 - 28 * q^77 + 12 * q^79 + 8 * q^81 - 12 * q^83 - 4 * q^85 - 4 * q^87 - 36 * q^89 - 40 * q^91 - 8 * q^93 - 16 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(456))\) 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	3	19
456.2.a.a	$456$	$2$	456.a	1.a	$1$	$1$	$1$	$3.641$	\(\Q\)	None	✓	✓	✓	✓	456.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(-3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-3q^{7}+q^{9}-5q^{11}-2q^{13}+\cdots\)
456.2.a.b	$456$	$2$	456.a	1.a	$1$	$1$	$1$	$3.641$	\(\Q\)	None	✓	✓			456.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(4\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+4q^{5}+4q^{7}+q^{9}-4q^{11}+\cdots\)
456.2.a.c	$456$	$2$	456.a	1.a	$1$	$1$	$1$	$3.641$	\(\Q\)	None	✓	✓	✓	✓	456.2.a.c	$1$	$1$	\(0\)	\(1\)	\(-3\)	\(-3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-3q^{5}-3q^{7}+q^{9}-q^{11}-2q^{13}+\cdots\)
456.2.a.d	$456$	$2$	456.a	1.a	$1$	$1$	$1$	$3.641$	\(\Q\)	None	✓	✓	✓	✓	456.2.a.d	$1$	$0$	\(0\)	\(1\)	\(2\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}+q^{9}+2q^{13}+2q^{15}+\cdots\)
456.2.a.e	$456$	$2$	456.a	1.a	$1$	$2$	$2$	$3.641$	\(\Q(\sqrt{41}) \)	None	✓	✓	✓		456.2.a.e	$1$	$0$	\(0\)	\(-2\)	\(-1\)	\(-3\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-\beta q^{5}+(-2+\beta )q^{7}+q^{9}+(2+\cdots)q^{11}+\cdots\)
456.2.a.f	$456$	$2$	456.a	1.a	$1$	$2$	$2$	$3.641$	\(\Q(\sqrt{17}) \)	None	✓	✓	✓	✓	456.2.a.f	$1$	$0$	\(0\)	\(2\)	\(1\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta q^{5}+\beta q^{7}+q^{9}+(4-\beta )q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(456))\) into lower level spaces

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