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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(43\)	\(21\)	\(22\)		\(40\)	\(21\)	\(19\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)		\(46\)	\(24\)	\(22\)		\(43\)	\(24\)	\(19\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(47\)	\(26\)	\(21\)		\(44\)	\(26\)	\(18\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(44\)	\(24\)	\(20\)		\(41\)	\(24\)	\(17\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(87\)	\(45\)	\(42\)		\(81\)	\(45\)	\(36\)		\(6\)	\(0\)	\(6\)
Minus space		\(-\)		\(93\)	\(50\)	\(43\)		\(87\)	\(50\)	\(37\)		\(6\)	\(0\)	\(6\)

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 - 472 * q^11 + 302 * q^12 + 169 * q^13 + 1610 * q^14 + 25122 * q^16 - 1418 * q^17 - 5714 * q^18 + 1668 * q^19 - 916 * q^21 - 7016 * q^22 - 1744 * q^23 + 4184 * q^24 - 2028 * q^26 + 780 * q^27 - 9436 * q^28 + 11106 * q^29 + 1840 * q^31 + 4312 * q^32 + 21012 * q^33 - 24564 * q^34 + 147008 * q^36 + 8762 * q^37 + 6440 * q^38 - 6084 * q^39 - 52134 * q^41 + 43794 * q^42 + 1516 * q^43 + 85300 * q^44 + 11744 * q^46 + 12892 * q^47 + 16226 * q^48 + 246559 * q^49 + 81944 * q^51 - 11492 * q^52 + 15626 * q^53 - 60676 * q^54 - 61146 * q^56 + 66460 * q^57 + 170860 * q^58 - 40932 * q^59 - 39710 * q^61 - 108636 * q^62 - 18244 * q^63 + 211282 * q^64 - 98180 * q^66 + 133832 * q^67 - 10526 * q^68 + 133028 * q^69 + 27060 * q^71 - 157680 * q^72 + 60486 * q^73 - 64866 * q^74 - 9900 * q^76 - 107636 * q^77 - 36842 * q^78 - 7100 * q^79 + 417343 * q^81 + 72384 * q^82 - 140224 * q^83 + 602148 * q^84 - 697424 * q^86 - 198140 * q^87 - 11540 * q^88 - 245614 * q^89 + 4056 * q^91 - 355288 * q^92 - 186148 * q^93 + 678938 * q^94 + 1021752 * q^96 - 430118 * q^97 + 24874 * q^98 - 1077416 * q^99

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

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