Space of modular forms of level 325, weight 4, and trivial character

• Label: 325.4.a
• Level: $325$
• Weight: $4$
• Character orbit: 325.a
• Rep. character: $\chi_{325}(1,\cdot)$
• Character field: $\Q$
• Dimension: $57$
• Newform subspaces: $15$
• Sturm bound: $140$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	325.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(140\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(325))\).

	Total	New	Old
Modular forms	112	57	55
Cusp forms	100	57	43
Eisenstein series	12	0	12

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
\(+\)	\(+\)	$+$	\(15\)
\(+\)	\(-\)	$-$	\(12\)
\(-\)	\(+\)	$-$	\(14\)
\(-\)	\(-\)	$+$	\(16\)
Plus space		\(+\)	\(31\)
Minus space		\(-\)	\(26\)

Trace form

\( 57 q - 4 q^{2} + 6 q^{3} + 250 q^{4} + 28 q^{6} - 6 q^{7} + 36 q^{8} + 423 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(325))\) 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
325.4.a.a	$325$	$4$	325.a	1.a	$1$	$1$	$1$	$19.176$	\(\Q\)	None	✓	$1$	$1$	\(-5\)	\(-2\)	\(0\)	\(12\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{2}-2q^{3}+17q^{4}+10q^{6}+12q^{7}+\cdots\)
325.4.a.b	$325$	$4$	325.a	1.a	$1$	$1$	$1$	$19.176$	\(\Q\)	None	✓	$1$	$0$	\(-3\)	\(-4\)	\(0\)	\(28\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}-4q^{3}+q^{4}+12q^{6}+28q^{7}+\cdots\)
325.4.a.c	$325$	$4$	325.a	1.a	$1$	$1$	$1$	$19.176$	\(\Q\)	None	✓	$1$	$1$	\(3\)	\(4\)	\(0\)	\(-28\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+4q^{3}+q^{4}+12q^{6}-28q^{7}+\cdots\)
325.4.a.d	$325$	$4$	325.a	1.a	$1$	$1$	$1$	$19.176$	\(\Q\)	None	✓	$1$	$0$	\(5\)	\(7\)	\(0\)	\(13\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{2}+7q^{3}+17q^{4}+35q^{6}+13q^{7}+\cdots\)
325.4.a.e	$325$	$4$	325.a	1.a	$1$	$2$	$2$	$19.176$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(-4\)	\(10\)	\(0\)	\(36\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{2}+(5-3\beta )q^{3}+(-1-4\beta )q^{4}+\cdots\)
325.4.a.f	$325$	$4$	325.a	1.a	$1$	$2$	$2$	$19.176$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$1$	\(-1\)	\(-5\)	\(0\)	\(9\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-4+3\beta )q^{3}+(-4+\beta )q^{4}+\cdots\)
325.4.a.g	$325$	$4$	325.a	1.a	$1$	$2$	$2$	$19.176$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-58\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(2-4\beta )q^{3}+(-4+\beta )q^{4}+\cdots\)
325.4.a.h	$325$	$4$	325.a	1.a	$1$	$2$	$2$	$19.176$	\(\Q(\sqrt{6}) \)	None	✓	$1$	$1$	\(2\)	\(4\)	\(0\)	\(20\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(2-\beta )q^{3}+(-1+2\beta )q^{4}+\cdots\)
325.4.a.i	$325$	$4$	325.a	1.a	$1$	$5$	$5$	$19.176$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(-1\)	\(-1\)	\(0\)	\(-26\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{3}+(5+\beta _{2}+\beta _{3}+\cdots)q^{4}+\cdots\)
325.4.a.j	$325$	$4$	325.a	1.a	$1$	$5$	$5$	$19.176$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-8\)	\(0\)	\(-38\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-2-\beta _{1}-\beta _{3})q^{3}+(7-\beta _{2}+\cdots)q^{4}+\cdots\)
325.4.a.k	$325$	$4$	325.a	1.a	$1$	$5$	$5$	$19.176$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(1\)	\(1\)	\(0\)	\(26\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-\beta _{1}-\beta _{2})q^{3}+(5+\beta _{2}+\cdots)q^{4}+\cdots\)
325.4.a.l	$325$	$4$	325.a	1.a	$1$	$7$	$7$	$19.176$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(-1\)	\(-1\)	\(0\)	\(-2\)		$+$	$+$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-\beta _{1}+\beta _{3})q^{3}+(7+\beta _{2}+\cdots)q^{4}+\cdots\)
325.4.a.m	$325$	$4$	325.a	1.a	$1$	$7$	$7$	$19.176$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(1\)	\(1\)	\(0\)	\(2\)		$-$	$-$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(\beta _{1}-\beta _{3})q^{3}+(7+\beta _{2})q^{4}+\cdots\)
325.4.a.n	$325$	$4$	325.a	1.a	$1$	$8$	$8$	$19.176$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(-9\)	\(-16\)	\(0\)	\(-12\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}+(-2-\beta _{3})q^{3}+(4+\cdots)q^{4}+\cdots\)
325.4.a.o	$325$	$4$	325.a	1.a	$1$	$8$	$8$	$19.176$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(9\)	\(16\)	\(0\)	\(12\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+(2+\beta _{3})q^{3}+(4+\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(325))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(325)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 2}\)