Space of modular forms of level 415, weight 2, and trivial character

• Label: 415.2.a
• Level: $415$
• Weight: $2$
• Character orbit: 415.a
• Rep. character: $\chi_{415}(1,\cdot)$
• Character field: $\Q$
• Dimension: $27$
• Newform subspaces: $5$
• Sturm bound: $84$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 415 = 5 \cdot 83 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	415.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(84\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(415))\).

	Total	New	Old
Modular forms	44	27	17
Cusp forms	41	27	14
Eisenstein series	3	0	3

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

\(5\)	\(83\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(7\)
\(+\)	\(-\)	$-$	\(6\)
\(-\)	\(+\)	$-$	\(12\)
\(-\)	\(-\)	$+$	\(2\)
Plus space		\(+\)	\(9\)
Minus space		\(-\)	\(18\)

Trace form

\( 27 q - q^{2} + 25 q^{4} + q^{5} - 8 q^{7} - 9 q^{8} + 35 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(415))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		5	83
415.2.a.a	$415$	$2$	415.a	1.a	$1$	$1$	$1$	$3.314$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(3\)	\(1\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+3q^{3}-q^{4}+q^{5}+3q^{6}+q^{7}+\cdots\)
415.2.a.b	$415$	$2$	415.a	1.a	$1$	$2$	$2$	$3.314$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(-1\)	\(-1\)	\(2\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1+\beta )q^{3}+(-1+\beta )q^{4}+\cdots\)
415.2.a.c	$415$	$2$	415.a	1.a	$1$	$6$	$6$	$3.314$	6.6.7783241.1	None	✓	$1$	$0$	\(2\)	\(3\)	\(-6\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(\beta _{2}-\beta _{4})q^{3}+\beta _{2}q^{4}-q^{5}+\cdots\)
415.2.a.d	$415$	$2$	415.a	1.a	$1$	$7$	$7$	$3.314$	7.7.179711353.1	None	✓	$1$	$1$	\(-3\)	\(-5\)	\(-7\)	\(-6\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{6}q^{2}+(-1+\beta _{1})q^{3}+(1-\beta _{5}+\beta _{6})q^{4}+\cdots\)
415.2.a.e	$415$	$2$	415.a	1.a	$1$	$11$	$11$	$3.314$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(11\)	\(-3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{7}q^{3}+(2+\beta _{2})q^{4}+q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(415))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(415)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(83))\)\(^{\oplus 2}\)