Space of modular forms of level 325, weight 2, and trivial character

• Label: 325.2.a
• Level: $325$
• Weight: $2$
• Character orbit: 325.a
• Rep. character: $\chi_{325}(1,\cdot)$
• Character field: $\Q$
• Dimension: $19$
• Newform subspaces: $11$
• Sturm bound: $70$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	325.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(70\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(325))\).

	Total	New	Old
Modular forms	40	19	21
Cusp forms	29	19	10
Eisenstein series	11	0	11

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

\(5\)	\(13\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(3\)
\(+\)	\(-\)	$-$	\(6\)
\(-\)	\(+\)	$-$	\(6\)
\(-\)	\(-\)	$+$	\(4\)
Plus space		\(+\)	\(7\)
Minus space		\(-\)	\(12\)

Trace form

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(325))\) 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	13
325.2.a.a	$325$	$2$	325.a	1.a	$1$	$1$	$1$	$2.595$	\(\Q\)	None	✓	$1$	$0$	\(-2\)	\(-1\)	\(0\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-q^{3}+2q^{4}+2q^{6}-2q^{7}+\cdots\)
325.2.a.b	$325$	$2$	325.a	1.a	$1$	$1$	$1$	$2.595$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{4}+4q^{7}-2q^{9}-6q^{11}+\cdots\)
325.2.a.c	$325$	$2$	325.a	1.a	$1$	$1$	$1$	$2.595$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{4}-4q^{7}-2q^{9}-6q^{11}+\cdots\)
325.2.a.d	$325$	$2$	325.a	1.a	$1$	$1$	$1$	$2.595$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(2\)	\(0\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+2q^{3}-q^{4}+2q^{6}+4q^{7}-3q^{8}+\cdots\)
325.2.a.e	$325$	$2$	325.a	1.a	$1$	$1$	$1$	$2.595$	\(\Q\)	None	✓	$1$	$0$	\(2\)	\(1\)	\(0\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+q^{3}+2q^{4}+2q^{6}+2q^{7}+\cdots\)
325.2.a.f	$325$	$2$	325.a	1.a	$1$	$2$	$2$	$2.595$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}-2\beta q^{3}+(1-2\beta )q^{4}+\cdots\)
325.2.a.g	$325$	$2$	325.a	1.a	$1$	$2$	$2$	$2.595$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1-\beta )q^{3}+q^{4}+(-3-\beta )q^{6}+\cdots\)
325.2.a.h	$325$	$2$	325.a	1.a	$1$	$2$	$2$	$2.595$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(2\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}-2\beta q^{3}+(1+2\beta )q^{4}+(-4+\cdots)q^{6}+\cdots\)
325.2.a.i	$325$	$2$	325.a	1.a	$1$	$2$	$2$	$2.595$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(2\)	\(0\)	\(0\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+\beta q^{3}+(1+2\beta )q^{4}+(2+\cdots)q^{6}+\cdots\)
325.2.a.j	$325$	$2$	325.a	1.a	$1$	$3$	$3$	$2.595$	3.3.148.1	None	✓	$1$	$1$	\(-3\)	\(-4\)	\(0\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{2}+(-1-\beta _{1})q^{3}+(2+\cdots)q^{4}+\cdots\)
325.2.a.k	$325$	$2$	325.a	1.a	$1$	$3$	$3$	$2.595$	3.3.148.1	None	✓	$1$	$0$	\(3\)	\(4\)	\(0\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{2}+(1+\beta _{1})q^{3}+(2-\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(325))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(325)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 2}\)