Space of modular forms of level 315, weight 8, and trivial character

• Label: 315.8.a
• Level: $315$
• Weight: $8$
• Character orbit: 315.a
• Rep. character: $\chi_{315}(1,\cdot)$
• Character field: $\Q$
• Dimension: $70$
• Newform subspaces: $17$
• Sturm bound: $384$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 17 \)
Sturm bound:			\(384\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(315))\).

	Total	New	Old
Modular forms	344	70	274
Cusp forms	328	70	258
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	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(6\)
\(+\)	\(+\)	\(-\)	\(-\)	\(8\)
\(+\)	\(-\)	\(+\)	\(-\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)	\(8\)
\(-\)	\(+\)	\(+\)	\(-\)	\(10\)
\(-\)	\(+\)	\(-\)	\(+\)	\(11\)
\(-\)	\(-\)	\(+\)	\(+\)	\(12\)
\(-\)	\(-\)	\(-\)	\(-\)	\(9\)
Plus space			\(+\)	\(37\)
Minus space			\(-\)	\(33\)

Trace form

70 * q - 30 * q^2 + 4714 * q^4 + 686 * q^7 + 906 * q^8 - 6500 * q^10 + 5110 * q^11 - 9572 * q^13 - 8918 * q^14 + 308898 * q^16 - 62612 * q^17 + 33720 * q^19 - 184700 * q^22 - 47624 * q^23 + 1093750 * q^25 - 125404 * q^26 + 107702 * q^28 + 131186 * q^29 - 530052 * q^31 + 306778 * q^32 + 1001784 * q^34 - 171500 * q^35 + 495264 * q^37 + 100804 * q^38 - 1365000 * q^40 + 2349068 * q^41 + 2095844 * q^43 - 1647960 * q^44 - 2073176 * q^46 - 3620 * q^47 + 8235430 * q^49 - 468750 * q^50 + 5834120 * q^52 - 2337772 * q^53 - 811000 * q^55 - 1524978 * q^56 + 2851352 * q^58 - 6506252 * q^59 - 3382656 * q^61 + 22740672 * q^62 + 13580954 * q^64 - 2150250 * q^65 + 1684700 * q^67 - 17493108 * q^68 + 686000 * q^70 - 672024 * q^71 + 17383592 * q^73 + 872308 * q^74 - 959548 * q^76 + 2304960 * q^77 - 6066738 * q^79 - 15120000 * q^80 + 1520820 * q^82 - 15755040 * q^83 + 8672750 * q^85 + 47422128 * q^86 + 662336 * q^88 + 26929096 * q^89 - 11956294 * q^91 + 1016960 * q^92 + 5191388 * q^94 - 27587500 * q^95 - 33125748 * q^97 - 3529470 * q^98

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(315))\) 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
315.8.a.a	$315$	$8$	315.a	1.a	$1$	$1$	$1$	$98.401$	\(\Q\)	None	✓				105.8.a.b	$1$	$1$	\(-18\)	\(0\)	\(125\)	\(343\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-18q^{2}+14^{2}q^{4}+5^{3}q^{5}+7^{3}q^{7}+\cdots\)
315.8.a.b	$315$	$8$	315.a	1.a	$1$	$1$	$1$	$98.401$	\(\Q\)	None	✓				105.8.a.a	$1$	$1$	\(2\)	\(0\)	\(125\)	\(343\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-124q^{4}+5^{3}q^{5}+7^{3}q^{7}+\cdots\)
315.8.a.c	$315$	$8$	315.a	1.a	$1$	$2$	$2$	$98.401$	\(\Q(\sqrt{11}) \)	None	✓				35.8.a.a	$1$	$1$	\(-16\)	\(0\)	\(-250\)	\(-686\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-8+\beta )q^{2}+(-20-2^{4}\beta )q^{4}-5^{3}q^{5}+\cdots\)
315.8.a.d	$315$	$8$	315.a	1.a	$1$	$2$	$2$	$98.401$	\(\Q(\sqrt{2}) \)	None	✓				105.8.a.c	$1$	$0$	\(12\)	\(0\)	\(-250\)	\(686\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(6+2\beta )q^{2}+(6^{2}+24\beta )q^{4}-5^{3}q^{5}+\cdots\)
315.8.a.e	$315$	$8$	315.a	1.a	$1$	$3$	$3$	$98.401$	3.3.2268428.1	None	✓				35.8.a.b	$1$	$1$	\(23\)	\(0\)	\(375\)	\(1029\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(8-\beta _{1})q^{2}+(80-8\beta _{1}+\beta _{2})q^{4}+\cdots\)
315.8.a.f	$315$	$8$	315.a	1.a	$1$	$4$	$4$	$98.401$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		105.8.a.i	$1$	$1$	\(-25\)	\(0\)	\(500\)	\(1372\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-6-\beta _{1})q^{2}+(2^{5}+12\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
315.8.a.g	$315$	$8$	315.a	1.a	$1$	$4$	$4$	$98.401$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				105.8.a.h	$1$	$0$	\(-11\)	\(0\)	\(-500\)	\(1372\)		$-$	$+$	$-$	$+$	$2^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{1})q^{2}+(6^{2}+5\beta _{1}+\beta _{2})q^{4}+\cdots\)
315.8.a.h	$315$	$8$	315.a	1.a	$1$	$4$	$4$	$98.401$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				105.8.a.g	$1$	$0$	\(-4\)	\(0\)	\(500\)	\(-1372\)		$-$	$-$	$+$	$+$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}+(26+\beta _{1}+\beta _{2})q^{4}+\cdots\)
315.8.a.i	$315$	$8$	315.a	1.a	$1$	$4$	$4$	$98.401$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				105.8.a.f	$1$	$1$	\(1\)	\(0\)	\(-500\)	\(-1372\)		$-$	$+$	$+$	$-$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(105+\beta _{2})q^{4}-5^{3}q^{5}-7^{3}q^{7}+\cdots\)
315.8.a.j	$315$	$8$	315.a	1.a	$1$	$4$	$4$	$98.401$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				35.8.a.c	$1$	$0$	\(2\)	\(0\)	\(500\)	\(-1372\)		$-$	$-$	$+$	$+$	$2\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+(69-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
315.8.a.k	$315$	$8$	315.a	1.a	$1$	$4$	$4$	$98.401$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				105.8.a.e	$1$	$0$	\(5\)	\(0\)	\(500\)	\(-1372\)		$-$	$-$	$+$	$+$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+(105+2\beta _{1}+\beta _{3})q^{4}+\cdots\)
315.8.a.l	$315$	$8$	315.a	1.a	$1$	$4$	$4$	$98.401$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				105.8.a.d	$1$	$1$	\(10\)	\(0\)	\(-500\)	\(-1372\)		$-$	$+$	$+$	$-$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(3-\beta _{1})q^{2}+(29-4\beta _{1}+\beta _{2})q^{4}+\cdots\)
315.8.a.m	$315$	$8$	315.a	1.a	$1$	$5$	$5$	$98.401$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		35.8.a.d	$1$	$0$	\(-11\)	\(0\)	\(-625\)	\(1715\)		$-$	$+$	$-$	$+$	$2\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{1})q^{2}+(111-\beta _{1}+\beta _{3})q^{4}+\cdots\)
315.8.a.n	$315$	$8$	315.a	1.a	$1$	$6$	$6$	$98.401$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	315.8.a.n	$1$	$1$	\(-21\)	\(0\)	\(750\)	\(-2058\)		$+$	$-$	$+$	$-$	$2^{6}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+(-4+\beta _{1})q^{2}+(77-3\beta _{1}+\beta _{2})q^{4}+\cdots\)
315.8.a.o	$315$	$8$	315.a	1.a	$1$	$6$	$6$	$98.401$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	315.8.a.n	$1$	$0$	\(21\)	\(0\)	\(-750\)	\(-2058\)		$+$	$+$	$+$	$+$	$2^{6}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+(4-\beta _{1})q^{2}+(77-3\beta _{1}+\beta _{2})q^{4}+\cdots\)
315.8.a.p	$315$	$8$	315.a	1.a	$1$	$8$	$8$	$98.401$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓	✓	315.8.a.p	$1$	$0$	\(-5\)	\(0\)	\(1000\)	\(2744\)		$+$	$-$	$-$	$+$	$2^{7}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(3^{4}+\beta _{2})q^{4}+5^{3}q^{5}+\cdots\)
315.8.a.q	$315$	$8$	315.a	1.a	$1$	$8$	$8$	$98.401$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓	✓	315.8.a.p	$1$	$1$	\(5\)	\(0\)	\(-1000\)	\(2744\)		$+$	$+$	$-$	$-$	$2^{7}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(3^{4}+\beta _{2})q^{4}-5^{3}q^{5}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(315))\) into lower level spaces

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