Space of modular forms of level 418, weight 2, and trivial character

• Label: 418.2.a
• Level: $418$
• Weight: $2$
• Character orbit: 418.a
• Rep. character: $\chi_{418}(1,\cdot)$
• Character field: $\Q$
• Dimension: $15$
• Newform subspaces: $8$
• Sturm bound: $120$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 418 = 2 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	418.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(120\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(418))\).

	Total	New	Old
Modular forms	64	15	49
Cusp forms	57	15	42
Eisenstein series	7	0	7

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

\(2\)	\(11\)	\(19\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(5\)	\(3\)	\(2\)		\(5\)	\(3\)	\(2\)		\(0\)	\(0\)	\(0\)
\(+\)	\(+\)	\(-\)	\(-\)		\(10\)	\(0\)	\(10\)		\(9\)	\(0\)	\(9\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)		\(9\)	\(2\)	\(7\)		\(8\)	\(2\)	\(6\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)		\(7\)	\(2\)	\(5\)		\(6\)	\(2\)	\(4\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)		\(7\)	\(3\)	\(4\)		\(6\)	\(3\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(10\)	\(1\)	\(9\)		\(9\)	\(1\)	\(8\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(8\)	\(0\)	\(8\)		\(7\)	\(0\)	\(7\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(8\)	\(4\)	\(4\)		\(7\)	\(4\)	\(3\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(30\)	\(6\)	\(24\)		\(27\)	\(6\)	\(21\)		\(3\)	\(0\)	\(3\)
Minus space			\(-\)		\(34\)	\(9\)	\(25\)		\(30\)	\(9\)	\(21\)		\(4\)	\(0\)	\(4\)

Trace form

15 * q + q^2 + 15 * q^4 + 6 * q^5 + 4 * q^6 - 8 * q^7 + q^8 + 19 * q^9 + 6 * q^10 + q^11 - 2 * q^13 - 16 * q^15 + 15 * q^16 - 14 * q^17 - 3 * q^18 - q^19 + 6 * q^20 - 16 * q^21 - q^22 - 20 * q^23 + 4 * q^24 + 13 * q^25 + 10 * q^26 - 8 * q^28 + 6 * q^29 + 8 * q^30 - 4 * q^31 + q^32 + 2 * q^34 - 8 * q^35 + 19 * q^36 - 14 * q^37 + 5 * q^38 - 44 * q^39 + 6 * q^40 + 2 * q^41 + 12 * q^42 - 20 * q^43 + q^44 - 18 * q^45 + 16 * q^46 - 16 * q^47 + 15 * q^49 - q^50 - 32 * q^51 - 2 * q^52 - 14 * q^53 + 4 * q^54 + 6 * q^55 - 4 * q^57 + 18 * q^58 - 8 * q^59 - 16 * q^60 + 6 * q^61 - 36 * q^63 + 15 * q^64 + 12 * q^65 + 4 * q^66 + 8 * q^67 - 14 * q^68 + 16 * q^69 - 8 * q^70 - 12 * q^71 - 3 * q^72 - 14 * q^73 + 10 * q^74 + 32 * q^75 - q^76 + 8 * q^77 - 24 * q^78 - 16 * q^79 + 6 * q^80 + 7 * q^81 + 18 * q^82 + 20 * q^83 - 16 * q^84 + 52 * q^85 - 4 * q^86 - 28 * q^87 - q^88 - 26 * q^89 - 10 * q^90 - 16 * q^91 - 20 * q^92 + 8 * q^93 - 6 * q^95 + 4 * q^96 + 38 * q^97 - 23 * q^98 + 13 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(418))\) 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	11	19
418.2.a.a	$418$	$2$	418.a	1.a	$1$	$1$	$1$	$3.338$	\(\Q\)	None	✓	✓	✓	✓	418.2.a.a	$1$	$1$	\(1\)	\(-1\)	\(-2\)	\(-3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-2q^{5}-q^{6}-3q^{7}+\cdots\)
418.2.a.b	$418$	$2$	418.a	1.a	$1$	$1$	$1$	$3.338$	\(\Q\)	None	✓	✓			418.2.a.b	$1$	$0$	\(1\)	\(0\)	\(2\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+2q^{5}+2q^{7}+q^{8}-3q^{9}+\cdots\)
418.2.a.c	$418$	$2$	418.a	1.a	$1$	$1$	$1$	$3.338$	\(\Q\)	None	✓	✓			418.2.a.c	$1$	$0$	\(1\)	\(3\)	\(-2\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+3q^{3}+q^{4}-2q^{5}+3q^{6}+q^{7}+\cdots\)
418.2.a.d	$418$	$2$	418.a	1.a	$1$	$2$	$2$	$3.338$	\(\Q(\sqrt{13}) \)	None	✓	✓	✓	✓	418.2.a.d	$1$	$1$	\(-2\)	\(-3\)	\(-1\)	\(-1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(-1-\beta )q^{3}+q^{4}+(-1+\beta )q^{5}+\cdots\)
418.2.a.e	$418$	$2$	418.a	1.a	$1$	$2$	$2$	$3.338$	\(\Q(\sqrt{17}) \)	None	✓	✓	✓	✓	418.2.a.e	$1$	$0$	\(-2\)	\(1\)	\(4\)	\(3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(1-\beta )q^{3}+q^{4}+2q^{5}+(-1+\cdots)q^{6}+\cdots\)
418.2.a.f	$418$	$2$	418.a	1.a	$1$	$2$	$2$	$3.338$	\(\Q(\sqrt{21}) \)	None	✓	✓	✓		418.2.a.f	$1$	$0$	\(2\)	\(-1\)	\(3\)	\(-5\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-\beta q^{3}+q^{4}+(2-\beta )q^{5}-\beta q^{6}+\cdots\)
418.2.a.g	$418$	$2$	418.a	1.a	$1$	$3$	$3$	$3.338$	3.3.621.1	None	✓	✓	✓	✓	418.2.a.g	$1$	$1$	\(-3\)	\(0\)	\(-3\)	\(-6\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+\beta _{1}q^{3}+q^{4}+(-1-\beta _{1}-\beta _{2})q^{5}+\cdots\)
418.2.a.h	$418$	$2$	418.a	1.a	$1$	$3$	$3$	$3.338$	3.3.469.1	None	✓	✓	✓	✓	418.2.a.h	$1$	$0$	\(3\)	\(1\)	\(5\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta _{1}q^{3}+q^{4}+(2-\beta _{1})q^{5}+\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(418))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(418)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(209))\)\(^{\oplus 2}\)