Space of modular forms of level 217, weight 2, and trivial character

• Label: 217.2.a
• Level: $217$
• Weight: $2$
• Character orbit: 217.a
• Rep. character: $\chi_{217}(1,\cdot)$
• Character field: $\Q$
• Dimension: $15$
• Newform subspaces: $4$
• Sturm bound: $42$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 217 = 7 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	217.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(42\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(217))\).

	Total	New	Old
Modular forms	22	15	7
Cusp forms	19	15	4
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.

\(7\)	\(31\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(3\)
\(+\)	\(-\)	$-$	\(5\)
\(-\)	\(+\)	$-$	\(4\)
\(-\)	\(-\)	$+$	\(3\)
Plus space		\(+\)	\(6\)
Minus space		\(-\)	\(9\)

Trace form

\( 15 q - 3 q^{2} + 17 q^{4} - 2 q^{5} + 8 q^{6} - q^{7} - 3 q^{8} + 7 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(217))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		7	31
217.2.a.a	$217$	$2$	217.a	1.a	$1$	$3$	$3$	$1.733$	\(\Q(\zeta_{18})^+\)	None	✓	$1$	$1$	\(-3\)	\(-3\)	\(-6\)	\(3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-1+\beta _{1}-\beta _{2})q^{3}+\cdots\)
217.2.a.b	$217$	$2$	217.a	1.a	$1$	$3$	$3$	$1.733$	\(\Q(\zeta_{18})^+\)	None	✓	$1$	$1$	\(-3\)	\(-3\)	\(0\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-1-\beta _{1}+\beta _{2})q^{3}+\cdots\)
217.2.a.c	$217$	$2$	217.a	1.a	$1$	$4$	$4$	$1.733$	4.4.6809.1	None	✓	$1$	$0$	\(0\)	\(3\)	\(4\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{3})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
217.2.a.d	$217$	$2$	217.a	1.a	$1$	$5$	$5$	$1.733$	5.5.138136.1	None	✓	$1$	$0$	\(3\)	\(3\)	\(0\)	\(-5\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{4})q^{2}+(1-\beta _{2})q^{3}+(2+\beta _{3}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(217))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(217)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(31))\)\(^{\oplus 2}\)