Space of modular forms of level 2259, weight 2, and trivial character

• Label: 2259.2.a
• Level: $2259$
• Weight: $2$
• Character orbit: 2259.a
• Rep. character: $\chi_{2259}(1,\cdot)$
• Character field: $\Q$
• Dimension: $104$
• Newform subspaces: $13$
• Sturm bound: $504$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 2259 = 3^{2} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2259.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(504\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2259))\).

	Total	New	Old
Modular forms	256	104	152
Cusp forms	249	104	145
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.

\(3\)	\(251\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(21\)
\(+\)	\(-\)	$-$	\(21\)
\(-\)	\(+\)	$-$	\(38\)
\(-\)	\(-\)	$+$	\(24\)
Plus space		\(+\)	\(45\)
Minus space		\(-\)	\(59\)

Trace form

\( 104 q + 3 q^{2} + 101 q^{4} + 2 q^{5} + 9 q^{8} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2259))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	251
2259.2.a.a	$2259$	$2$	2259.a	1.a	$1$	$1$	$1$	$18.038$	\(\Q\)	None	✓	$1$	$2$	\(-2\)	\(0\)	\(-3\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}-3q^{5}-q^{7}+6q^{10}+\cdots\)
2259.2.a.b	$2259$	$2$	2259.a	1.a	$1$	$1$	$1$	$18.038$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(-1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}-3q^{5}-q^{7}+2q^{13}+4q^{16}+\cdots\)
2259.2.a.c	$2259$	$2$	2259.a	1.a	$1$	$1$	$1$	$18.038$	\(\Q\)	None	✓	$1$	$2$	\(0\)	\(0\)	\(-1\)	\(-5\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}-q^{5}-5q^{7}-6q^{11}-6q^{13}+\cdots\)
2259.2.a.d	$2259$	$2$	2259.a	1.a	$1$	$2$	$2$	$18.038$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(6\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(3+\beta )q^{5}+(-1+2\beta )q^{7}+\cdots\)
2259.2.a.e	$2259$	$2$	2259.a	1.a	$1$	$4$	$4$	$18.038$	4.4.19664.1	None	✓	$1$	$1$	\(-2\)	\(0\)	\(-4\)	\(6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(2+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
2259.2.a.f	$2259$	$2$	2259.a	1.a	$1$	$4$	$4$	$18.038$	4.4.725.1	None	✓	$1$	$1$	\(2\)	\(0\)	\(3\)	\(-3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{3})q^{2}-\beta _{3}q^{4}+(1-\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
2259.2.a.g	$2259$	$2$	2259.a	1.a	$1$	$6$	$6$	$18.038$	6.6.4448597.1	None	✓	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(5\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{4}q^{2}+(\beta _{1}-\beta _{4})q^{4}+\beta _{5}q^{5}+(1+\cdots)q^{7}+\cdots\)
2259.2.a.h	$2259$	$2$	2259.a	1.a	$1$	$6$	$6$	$18.038$	6.6.2812877.1	None	✓	$1$	$0$	\(6\)	\(0\)	\(4\)	\(-7\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1-\beta _{1}+\beta _{2})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots\)
2259.2.a.i	$2259$	$2$	2259.a	1.a	$1$	$9$	$9$	$18.038$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-14\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1-\beta _{2}+\beta _{3}+\beta _{5})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots\)
2259.2.a.j	$2259$	$2$	2259.a	1.a	$1$	$11$	$11$	$18.038$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	$1$	$0$	\(3\)	\(0\)	\(4\)	\(19\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{7}q^{5}+(2+\beta _{8}+\cdots)q^{7}+\cdots\)
2259.2.a.k	$2259$	$2$	2259.a	1.a	$1$	$17$	$17$	$18.038$	\(\mathbb{Q}[x]/(x^{17} - \cdots)\)	None	✓	$1$	$0$	\(-2\)	\(0\)	\(-3\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(2+\beta _{2})q^{4}+\beta _{7}q^{5}+(-\beta _{6}+\cdots)q^{7}+\cdots\)
2259.2.a.l	$2259$	$2$	2259.a	1.a	$1$	$21$	$21$	$18.038$		None	✓	$1$	$1$	\(-7\)	\(0\)	\(-12\)	\(0\)		$+$	$+$	$+$		$\mathrm{SU}(2)$
2259.2.a.m	$2259$	$2$	2259.a	1.a	$1$	$21$	$21$	$18.038$		None	✓	$1$	$0$	\(7\)	\(0\)	\(12\)	\(0\)		$+$	$-$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2259))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2259)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(251))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(753))\)\(^{\oplus 2}\)