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	Dim
\(+\)	\(+\)	\(+\)	$+$	\(3\)
\(+\)	\(-\)	\(+\)	$-$	\(2\)
\(+\)	\(-\)	\(-\)	$+$	\(2\)
\(-\)	\(+\)	\(+\)	$-$	\(3\)
\(-\)	\(+\)	\(-\)	$+$	\(1\)
\(-\)	\(-\)	\(-\)	$-$	\(4\)
Plus space			\(+\)	\(6\)
Minus space			\(-\)	\(9\)

Trace form

\( 15 q + q^{2} + 15 q^{4} + 6 q^{5} + 4 q^{6} - 8 q^{7} + q^{8} + 19 q^{9} + O(q^{10}) \)

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

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

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