Space of modular forms of level 245, weight 2, and trivial character

• Label: 245.2.a
• Level: $245$
• Weight: $2$
• Character orbit: 245.a
• Rep. character: $\chi_{245}(1,\cdot)$
• Character field: $\Q$
• Dimension: $13$
• Newform subspaces: $8$
• Sturm bound: $56$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	245.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(56\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(245))\).

	Total	New	Old
Modular forms	36	13	23
Cusp forms	21	13	8
Eisenstein series	15	0	15

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

\(5\)	\(7\)	Fricke	Dim.
\(+\)	\(+\)	\(+\)	\(2\)
\(+\)	\(-\)	\(-\)	\(5\)
\(-\)	\(+\)	\(-\)	\(4\)
\(-\)	\(-\)	\(+\)	\(2\)
Plus space		\(+\)	\(4\)
Minus space		\(-\)	\(9\)

Trace form

\( 13 q - q^{2} + 11 q^{4} - q^{5} + 8 q^{6} + 3 q^{8} + 13 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(245))\) 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	7
245.2.a.a	$245$	$2$	245.a	1.a	$1$	$1$	$1$	$1.956$	\(\Q\)	None	✓	$1$	$1$	\(-2\)	\(-3\)	\(1\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-3q^{3}+2q^{4}+q^{5}+6q^{6}+\cdots\)
245.2.a.b	$245$	$2$	245.a	1.a	$1$	$1$	$1$	$1.956$	\(\Q\)	None	✓	$1$	$0$	\(-2\)	\(3\)	\(-1\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+2q^{4}-q^{5}-6q^{6}+\cdots\)
245.2.a.c	$245$	$2$	245.a	1.a	$1$	$1$	$1$	$1.956$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{4}+q^{5}-2q^{9}-3q^{11}+\cdots\)
245.2.a.d	$245$	$2$	245.a	1.a	$1$	$2$	$2$	$1.956$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(-1\)	\(1\)	\(-2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(1-\beta )q^{3}+(2+\beta )q^{4}-q^{5}+\cdots\)
245.2.a.e	$245$	$2$	245.a	1.a	$1$	$2$	$2$	$1.956$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1-\beta )q^{3}-q^{5}+(-2-\beta )q^{6}+\cdots\)
245.2.a.f	$245$	$2$	245.a	1.a	$1$	$2$	$2$	$1.956$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(1+\beta )q^{3}+q^{5}+(2+\beta )q^{6}+\cdots\)
245.2.a.g	$245$	$2$	245.a	1.a	$1$	$2$	$2$	$1.956$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(2\)	\(-2\)	\(-2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(-1+\beta )q^{3}+(1+2\beta )q^{4}+\cdots\)
245.2.a.h	$245$	$2$	245.a	1.a	$1$	$2$	$2$	$1.956$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(2\)	\(2\)	\(2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(1-\beta )q^{3}+(1+2\beta )q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(245))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(245)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 2}\)