Space of modular forms of level 259, weight 2, and trivial character

• Label: 259.2.a
• Level: $259$
• Weight: $2$
• Character orbit: 259.a
• Rep. character: $\chi_{259}(1,\cdot)$
• Character field: $\Q$
• Dimension: $19$
• Newform subspaces: $7$
• Sturm bound: $50$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 259 = 7 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	259.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(50\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(259))\).

	Total	New	Old
Modular forms	26	19	7
Cusp forms	23	19	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.

\(7\)	\(37\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(3\)
\(+\)	\(-\)	$-$	\(6\)
\(-\)	\(+\)	$-$	\(7\)
\(-\)	\(-\)	$+$	\(3\)
Plus space		\(+\)	\(6\)
Minus space		\(-\)	\(13\)

Trace form

\( 19 q + q^{2} + 17 q^{4} + 6 q^{5} - 12 q^{6} + q^{7} - 3 q^{8} + 15 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(259))\) 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	37
259.2.a.a	$259$	$2$	259.a	1.a	$1$	$1$	$1$	$2.068$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(4\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+4q^{5}+q^{7}-3q^{8}-3q^{9}+\cdots\)
259.2.a.b	$259$	$2$	259.a	1.a	$1$	$2$	$2$	$2.068$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(6\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{3}-2q^{4}+(3+\beta )q^{5}-q^{7}+5q^{9}+\cdots\)
259.2.a.c	$259$	$2$	259.a	1.a	$1$	$2$	$2$	$2.068$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(1\)	\(0\)	\(1\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(2+\beta )q^{4}+(1-\beta )q^{5}+q^{7}+\cdots\)
259.2.a.d	$259$	$2$	259.a	1.a	$1$	$3$	$3$	$2.068$	\(\Q(\zeta_{18})^+\)	None	✓	$1$	$1$	\(-3\)	\(0\)	\(-6\)	\(3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}-\beta _{2}q^{3}+(1-2\beta _{1}+\cdots)q^{4}+\cdots\)
259.2.a.e	$259$	$2$	259.a	1.a	$1$	$3$	$3$	$2.068$	\(\Q(\zeta_{14})^+\)	None	✓	$1$	$1$	\(1\)	\(-2\)	\(-6\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-1-\beta _{2})q^{3}+\beta _{2}q^{4}+(-1+\cdots)q^{5}+\cdots\)
259.2.a.f	$259$	$2$	259.a	1.a	$1$	$4$	$4$	$2.068$	4.4.26825.1	None	✓	$1$	$0$	\(0\)	\(2\)	\(6\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+(1-\beta _{1})q^{3}+(2+\beta _{1})q^{4}+\cdots\)
259.2.a.g	$259$	$2$	259.a	1.a	$1$	$4$	$4$	$2.068$	4.4.22545.1	None	✓	$1$	$0$	\(1\)	\(0\)	\(1\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-\beta _{1}-\beta _{3})q^{3}+(1+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(259))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(259)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 2}\)