Space of modular forms of level 414, weight 2, and trivial character

• Label: 414.2.a
• Level: $414$
• Weight: $2$
• Character orbit: 414.a
• Rep. character: $\chi_{414}(1,\cdot)$
• Character field: $\Q$
• Dimension: $10$
• Newform subspaces: $7$
• Sturm bound: $144$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	414.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(144\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(414))\).

	Total	New	Old
Modular forms	80	10	70
Cusp forms	65	10	55
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.

\(2\)	\(3\)	\(23\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(6\)	\(0\)	\(6\)		\(5\)	\(0\)	\(5\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(14\)	\(2\)	\(12\)		\(12\)	\(2\)	\(10\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(11\)	\(2\)	\(9\)		\(9\)	\(2\)	\(7\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(9\)	\(1\)	\(8\)		\(7\)	\(1\)	\(6\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(8\)	\(2\)	\(6\)		\(6\)	\(2\)	\(4\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(12\)	\(0\)	\(12\)		\(10\)	\(0\)	\(10\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(9\)	\(1\)	\(8\)		\(7\)	\(1\)	\(6\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(11\)	\(2\)	\(9\)		\(9\)	\(2\)	\(7\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(36\)	\(2\)	\(34\)		\(29\)	\(2\)	\(27\)		\(7\)	\(0\)	\(7\)
Minus space			\(-\)		\(44\)	\(8\)	\(36\)		\(36\)	\(8\)	\(28\)		\(8\)	\(0\)	\(8\)

Trace form

10 * q + 10 * q^4 - 2 * q^5 + 4 * q^7 + 2 * q^10 + 10 * q^11 - 4 * q^13 - 4 * q^14 + 10 * q^16 + 8 * q^17 + 2 * q^19 - 2 * q^20 - 6 * q^22 + 18 * q^25 + 4 * q^28 + 16 * q^31 - 12 * q^34 + 32 * q^35 + 2 * q^37 + 10 * q^38 + 2 * q^40 - 16 * q^41 + 6 * q^43 + 10 * q^44 - 2 * q^46 - 8 * q^47 + 10 * q^49 + 4 * q^50 - 4 * q^52 - 18 * q^53 - 16 * q^55 - 4 * q^56 - 20 * q^58 - 8 * q^59 + 6 * q^61 + 8 * q^62 + 10 * q^64 - 12 * q^65 - 22 * q^67 + 8 * q^68 - 16 * q^73 - 34 * q^74 + 2 * q^76 - 24 * q^77 - 28 * q^79 - 2 * q^80 - 28 * q^82 - 34 * q^83 - 60 * q^85 + 6 * q^86 - 6 * q^88 - 12 * q^89 - 24 * q^91 - 8 * q^95 + 8 * q^97 - 16 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(414))\) 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	23
414.2.a.a	$414$	$2$	414.a	1.a	$1$	$1$	$1$	$3.306$	\(\Q\)	None	✓		✓	✓	138.2.a.c	$1$	$1$	\(-1\)	\(0\)	\(-2\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2q^{5}-q^{8}+2q^{10}-2q^{13}+\cdots\)
414.2.a.b	$414$	$2$	414.a	1.a	$1$	$1$	$1$	$3.306$	\(\Q\)	None	✓		✓	✓	46.2.a.a	$1$	$1$	\(1\)	\(0\)	\(-4\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-4q^{5}-4q^{7}+q^{8}-4q^{10}+\cdots\)
414.2.a.c	$414$	$2$	414.a	1.a	$1$	$1$	$1$	$3.306$	\(\Q\)	None	✓				138.2.a.b	$1$	$0$	\(1\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+2q^{7}+q^{8}+2q^{13}+2q^{14}+\cdots\)
414.2.a.d	$414$	$2$	414.a	1.a	$1$	$1$	$1$	$3.306$	\(\Q\)	None	✓				138.2.a.a	$1$	$0$	\(1\)	\(0\)	\(2\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+2q^{5}-2q^{7}+q^{8}+2q^{10}+\cdots\)
414.2.a.e	$414$	$2$	414.a	1.a	$1$	$2$	$2$	$3.306$	\(\Q(\sqrt{7}) \)	None	✓	✓	✓	✓	414.2.a.e	$1$	$0$	\(-2\)	\(0\)	\(-2\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(-1+\beta )q^{5}+2q^{7}-q^{8}+\cdots\)
414.2.a.f	$414$	$2$	414.a	1.a	$1$	$2$	$2$	$3.306$	\(\Q(\sqrt{5}) \)	None	✓		✓	✓	138.2.a.d	$1$	$0$	\(-2\)	\(0\)	\(2\)	\(0\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(1+\beta )q^{5}+2\beta q^{7}-q^{8}+\cdots\)
414.2.a.g	$414$	$2$	414.a	1.a	$1$	$2$	$2$	$3.306$	\(\Q(\sqrt{7}) \)	None	✓	✓	✓	✓	414.2.a.e	$1$	$0$	\(2\)	\(0\)	\(2\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(1+\beta )q^{5}+2q^{7}+q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(414))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(414)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(138))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(207))\)\(^{\oplus 2}\)