Space of modular forms of level 315, weight 6, and trivial character

• Label: 315.6.a
• Level: $315$
• Weight: $6$
• Character orbit: 315.a
• Rep. character: $\chi_{315}(1,\cdot)$
• Character field: $\Q$
• Dimension: $50$
• Newform subspaces: $16$
• Sturm bound: $288$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(288\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(315))\).

	Total	New	Old
Modular forms	248	50	198
Cusp forms	232	50	182
Eisenstein series	16	0	16

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

\(3\)	\(5\)	\(7\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(28\)	\(6\)	\(22\)		\(26\)	\(6\)	\(20\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)		\(32\)	\(4\)	\(28\)		\(30\)	\(4\)	\(26\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(34\)	\(6\)	\(28\)		\(32\)	\(6\)	\(26\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(30\)	\(4\)	\(26\)		\(28\)	\(4\)	\(24\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(31\)	\(8\)	\(23\)		\(29\)	\(8\)	\(21\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(31\)	\(7\)	\(24\)		\(29\)	\(7\)	\(22\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(31\)	\(6\)	\(25\)		\(29\)	\(6\)	\(23\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(31\)	\(9\)	\(22\)		\(29\)	\(9\)	\(20\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(120\)	\(23\)	\(97\)		\(112\)	\(23\)	\(89\)		\(8\)	\(0\)	\(8\)
Minus space			\(-\)		\(128\)	\(27\)	\(101\)		\(120\)	\(27\)	\(93\)		\(8\)	\(0\)	\(8\)

Trace form

50 * q + 10 * q^2 + 842 * q^4 - 98 * q^7 - 582 * q^8 - 100 * q^10 + 974 * q^11 + 404 * q^13 - 1078 * q^14 + 14946 * q^16 - 2524 * q^17 - 3864 * q^19 - 460 * q^22 + 7448 * q^23 + 31250 * q^25 + 14436 * q^26 - 1274 * q^28 + 2390 * q^29 + 13404 * q^31 - 43254 * q^32 + 12552 * q^34 + 4900 * q^35 + 3168 * q^37 + 44692 * q^38 + 3000 * q^40 - 47132 * q^41 - 27068 * q^43 + 15864 * q^44 + 101480 * q^46 - 16420 * q^47 + 120050 * q^49 + 6250 * q^50 - 16760 * q^52 + 88604 * q^53 - 27800 * q^55 - 29106 * q^56 - 102344 * q^58 - 55276 * q^59 + 44160 * q^61 + 135744 * q^62 + 365242 * q^64 + 55850 * q^65 - 122660 * q^67 + 64396 * q^68 - 9800 * q^70 - 172936 * q^71 - 238424 * q^73 + 129876 * q^74 - 84380 * q^76 + 54880 * q^77 + 262926 * q^79 + 32000 * q^80 + 321060 * q^82 + 314320 * q^83 - 170150 * q^85 - 408000 * q^86 - 382832 * q^88 - 20568 * q^89 - 38318 * q^91 + 293760 * q^92 + 819772 * q^94 + 146500 * q^95 + 319524 * q^97 + 24010 * q^98

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(315))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5	7
315.6.a.a	$315$	$6$	315.a	1.a	$1$	$1$	$1$	$50.521$	\(\Q\)	None	✓				35.6.a.a	$1$	$1$	\(8\)	\(0\)	\(-25\)	\(49\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+2^{5}q^{4}-5^{2}q^{5}+7^{2}q^{7}-200q^{10}+\cdots\)
315.6.a.b	$315$	$6$	315.a	1.a	$1$	$2$	$2$	$50.521$	\(\Q(\sqrt{65}) \)	None	✓				105.6.a.f	$1$	$1$	\(-3\)	\(0\)	\(-50\)	\(98\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+(-15+3\beta )q^{4}-5^{2}q^{5}+\cdots\)
315.6.a.c	$315$	$6$	315.a	1.a	$1$	$2$	$2$	$50.521$	\(\Q(\sqrt{65}) \)	None	✓				35.6.a.b	$1$	$1$	\(-1\)	\(0\)	\(50\)	\(-98\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-2^{4}+\beta )q^{4}+5^{2}q^{5}-7^{2}q^{7}+\cdots\)
315.6.a.d	$315$	$6$	315.a	1.a	$1$	$2$	$2$	$50.521$	\(\Q(\sqrt{233}) \)	None	✓				105.6.a.e	$1$	$1$	\(-1\)	\(0\)	\(50\)	\(-98\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(26+\beta )q^{4}+5^{2}q^{5}-7^{2}q^{7}+\cdots\)
315.6.a.e	$315$	$6$	315.a	1.a	$1$	$2$	$2$	$50.521$	\(\Q(\sqrt{73}) \)	None	✓				105.6.a.d	$1$	$0$	\(1\)	\(0\)	\(50\)	\(98\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-14+\beta )q^{4}+5^{2}q^{5}+7^{2}q^{7}+\cdots\)
315.6.a.f	$315$	$6$	315.a	1.a	$1$	$2$	$2$	$50.521$	\(\Q(\sqrt{2}) \)	None	✓				105.6.a.c	$1$	$0$	\(4\)	\(0\)	\(-50\)	\(-98\)		$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{2}+(4+4\beta )q^{4}-5^{2}q^{5}-7^{2}q^{7}+\cdots\)
315.6.a.g	$315$	$6$	315.a	1.a	$1$	$2$	$2$	$50.521$	\(\Q(\sqrt{5}) \)	None	✓				105.6.a.b	$1$	$1$	\(8\)	\(0\)	\(50\)	\(-98\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(4+\beta )q^{2}+(4+8\beta )q^{4}+5^{2}q^{5}-7^{2}q^{7}+\cdots\)
315.6.a.h	$315$	$6$	315.a	1.a	$1$	$2$	$2$	$50.521$	\(\Q(\sqrt{65}) \)	None	✓				105.6.a.a	$1$	$0$	\(13\)	\(0\)	\(-50\)	\(-98\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(7-\beta )q^{2}+(33-13\beta )q^{4}-5^{2}q^{5}+\cdots\)
315.6.a.i	$315$	$6$	315.a	1.a	$1$	$3$	$3$	$50.521$	3.3.577880.1	None	✓				35.6.a.c	$1$	$0$	\(6\)	\(0\)	\(75\)	\(147\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{2}+(38+2\beta _{2})q^{4}+5^{2}q^{5}+\cdots\)
315.6.a.j	$315$	$6$	315.a	1.a	$1$	$4$	$4$	$50.521$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		105.6.a.h	$1$	$1$	\(-10\)	\(0\)	\(-100\)	\(196\)		$-$	$+$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1})q^{2}+(30-3\beta _{1}+\beta _{2})q^{4}+\cdots\)
315.6.a.k	$315$	$6$	315.a	1.a	$1$	$4$	$4$	$50.521$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		105.6.a.g	$1$	$0$	\(-8\)	\(0\)	\(100\)	\(196\)		$-$	$-$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{2}+(29-\beta _{1}+\beta _{2})q^{4}+\cdots\)
315.6.a.l	$315$	$6$	315.a	1.a	$1$	$4$	$4$	$50.521$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		35.6.a.d	$1$	$0$	\(-7\)	\(0\)	\(-100\)	\(-196\)		$-$	$+$	$+$	$-$	$3^{2}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{2}+(13-4\beta _{1}+\beta _{2})q^{4}+\cdots\)
315.6.a.m	$315$	$6$	315.a	1.a	$1$	$4$	$4$	$50.521$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	315.6.a.m	$1$	$1$	\(-5\)	\(0\)	\(100\)	\(196\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}+(11+2\beta _{1}+\beta _{3})q^{4}+\cdots\)
315.6.a.n	$315$	$6$	315.a	1.a	$1$	$4$	$4$	$50.521$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	315.6.a.m	$1$	$0$	\(5\)	\(0\)	\(-100\)	\(196\)		$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+(11+2\beta _{1}+\beta _{3})q^{4}+\cdots\)
315.6.a.o	$315$	$6$	315.a	1.a	$1$	$6$	$6$	$50.521$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	315.6.a.o	$1$	$1$	\(-3\)	\(0\)	\(-150\)	\(-294\)		$+$	$+$	$+$	$+$	$2^{3}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(5^{2}+\beta _{2})q^{4}-5^{2}q^{5}+\cdots\)
315.6.a.p	$315$	$6$	315.a	1.a	$1$	$6$	$6$	$50.521$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	315.6.a.o	$1$	$0$	\(3\)	\(0\)	\(150\)	\(-294\)		$+$	$-$	$+$	$-$	$2^{3}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(5^{2}+\beta _{2})q^{4}+5^{2}q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(315))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(315)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 2}\)