Space of modular forms of level 325, weight 6, and trivial character

• Label: 325.6.a
• Level: $325$
• Weight: $6$
• Character orbit: 325.a
• Rep. character: $\chi_{325}(1,\cdot)$
• Character field: $\Q$
• Dimension: $95$
• Newform subspaces: $13$
• Sturm bound: $210$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	325.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(210\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(325))\).

	Total	New	Old
Modular forms	180	95	85
Cusp forms	168	95	73
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
\(+\)	\(+\)	\(+\)	\(21\)
\(+\)	\(-\)	\(-\)	\(24\)
\(-\)	\(+\)	\(-\)	\(26\)
\(-\)	\(-\)	\(+\)	\(24\)
Plus space		\(+\)	\(45\)
Minus space		\(-\)	\(50\)

Trace form

\( 95 q + 2 q^{2} - 24 q^{3} + 1478 q^{4} + 196 q^{6} + 292 q^{7} + 60 q^{8} + 7947 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(325))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	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
325.6.a.a	$325$	$6$	325.a	1.a	$1$	$1$	$1$	$52.125$	\(\Q\)	None	✓				65.6.a.a	$1$	$0$	\(-5\)	\(-6\)	\(0\)	\(244\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{2}-6q^{3}-7q^{4}+30q^{6}+244q^{7}+\cdots\)
325.6.a.b	$325$	$6$	325.a	1.a	$1$	$2$	$2$	$52.125$	\(\Q(\sqrt{17}) \)	None	✓				13.6.a.a	$1$	$0$	\(5\)	\(28\)	\(0\)	\(36\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(3-\beta )q^{2}+(11+6\beta )q^{3}+(-19-5\beta )q^{4}+\cdots\)
325.6.a.c	$325$	$6$	325.a	1.a	$1$	$3$	$3$	$52.125$	3.3.168897.1	None	✓				13.6.a.b	$1$	$1$	\(-7\)	\(-8\)	\(0\)	\(60\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{1})q^{2}+(-3+\beta _{1}+\beta _{2})q^{3}+\cdots\)
325.6.a.d	$325$	$6$	325.a	1.a	$1$	$3$	$3$	$52.125$	3.3.49857.1	None	✓				65.6.a.b	$1$	$1$	\(2\)	\(16\)	\(0\)	\(208\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(5+\beta _{2})q^{3}+(-4+\beta _{1}+\cdots)q^{4}+\cdots\)
325.6.a.e	$325$	$6$	325.a	1.a	$1$	$4$	$4$	$52.125$	4.4.1878612.1	None	✓				65.6.a.c	$1$	$0$	\(9\)	\(4\)	\(0\)	\(136\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1}-\beta _{2})q^{2}+(1+\beta _{2}+\beta _{3})q^{3}+\cdots\)
325.6.a.f	$325$	$6$	325.a	1.a	$1$	$6$	$6$	$52.125$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				65.6.a.e	$1$	$0$	\(-2\)	\(-20\)	\(0\)	\(-172\)		$+$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-3+\beta _{2})q^{3}+(23-\beta _{1}+\cdots)q^{4}+\cdots\)
325.6.a.g	$325$	$6$	325.a	1.a	$1$	$6$	$6$	$52.125$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				65.6.a.d	$1$	$1$	\(0\)	\(-38\)	\(0\)	\(-220\)		$+$	$+$	$+$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-6-\beta _{1}-\beta _{2})q^{3}+(22+\cdots)q^{4}+\cdots\)
325.6.a.h	$325$	$6$	325.a	1.a	$1$	$9$	$9$	$52.125$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓			325.6.a.h	$1$	$1$	\(-5\)	\(-11\)	\(0\)	\(12\)		$-$	$-$	$+$	$2^{3}\cdot 5$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-1-\beta _{4})q^{3}+(11+\cdots)q^{4}+\cdots\)
325.6.a.i	$325$	$6$	325.a	1.a	$1$	$9$	$9$	$52.125$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓		325.6.a.h	$1$	$1$	\(5\)	\(11\)	\(0\)	\(-12\)		$+$	$+$	$+$	$2^{3}\cdot 5$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1+\beta _{4})q^{3}+(11-2\beta _{1}+\cdots)q^{4}+\cdots\)
325.6.a.j	$325$	$6$	325.a	1.a	$1$	$11$	$11$	$52.125$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	✓			325.6.a.j	$1$	$0$	\(-5\)	\(-11\)	\(0\)	\(-208\)		$-$	$+$	$-$	$2^{6}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1-\beta _{1}+\beta _{3})q^{3}+(17+\cdots)q^{4}+\cdots\)
325.6.a.k	$325$	$6$	325.a	1.a	$1$	$11$	$11$	$52.125$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	✓	✓		325.6.a.j	$1$	$0$	\(5\)	\(11\)	\(0\)	\(208\)		$+$	$-$	$-$	$2^{6}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{1}-\beta _{3})q^{3}+(17+\beta _{2}+\cdots)q^{4}+\cdots\)
325.6.a.l	$325$	$6$	325.a	1.a	$1$	$15$	$15$	$52.125$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓		✓		65.6.b.a	$1$	$1$	\(-12\)	\(-36\)	\(0\)	\(-306\)		$-$	$-$	$+$	$2^{7}\cdot 5^{6}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-2-\beta _{2})q^{3}+(2^{4}+\cdots)q^{4}+\cdots\)
325.6.a.m	$325$	$6$	325.a	1.a	$1$	$15$	$15$	$52.125$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓		✓		65.6.b.a	$1$	$0$	\(12\)	\(36\)	\(0\)	\(306\)		$-$	$+$	$-$	$2^{7}\cdot 5^{6}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(2+\beta _{2})q^{3}+(2^{4}-\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(325))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(325)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 2}\)