Space of modular forms of level 315, weight 10, and trivial character

• Label: 315.10.a
• Level: $315$
• Weight: $10$
• Character orbit: 315.a
• Rep. character: $\chi_{315}(1,\cdot)$
• Character field: $\Q$
• Dimension: $90$
• Newform subspaces: $17$
• Sturm bound: $480$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 17 \)
Sturm bound:			\(480\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(315))\).

	Total	New	Old
Modular forms	440	90	350
Cusp forms	424	90	334
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
\(+\)	\(+\)	\(+\)	$+$	\(10\)
\(+\)	\(+\)	\(-\)	$-$	\(8\)
\(+\)	\(-\)	\(+\)	$-$	\(10\)
\(+\)	\(-\)	\(-\)	$+$	\(8\)
\(-\)	\(+\)	\(+\)	$-$	\(14\)
\(-\)	\(+\)	\(-\)	$+$	\(13\)
\(-\)	\(-\)	\(+\)	$+$	\(12\)
\(-\)	\(-\)	\(-\)	$-$	\(15\)
Plus space			\(+\)	\(43\)
Minus space			\(-\)	\(47\)

Trace form

\( 90 q + 66 q^{2} + 22186 q^{4} - 4802 q^{7} + 76698 q^{8} + O(q^{10}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(315))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5	7
315.10.a.a	$315$	$10$	315.a	1.a	$1$	$1$	$1$	$162.236$	\(\Q\)	None	✓	$1$	$1$	\(-28\)	\(0\)	\(-625\)	\(2401\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-28q^{2}+272q^{4}-5^{4}q^{5}+7^{4}q^{7}+\cdots\)
315.10.a.b	$315$	$10$	315.a	1.a	$1$	$2$	$2$	$162.236$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(24\)	\(0\)	\(-1250\)	\(4802\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(12+\beta )q^{2}+(-360+24\beta )q^{4}-5^{4}q^{5}+\cdots\)
315.10.a.c	$315$	$10$	315.a	1.a	$1$	$4$	$4$	$162.236$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(-8\)	\(0\)	\(2500\)	\(-9604\)		$-$	$-$	$+$	$+$	$2^{6}\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{2}+(106-\beta _{1}+\beta _{3})q^{4}+\cdots\)
315.10.a.d	$315$	$10$	315.a	1.a	$1$	$4$	$4$	$162.236$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(-5\)	\(0\)	\(2500\)	\(-9604\)		$-$	$-$	$+$	$+$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}+(237+3\beta _{1}+\beta _{3})q^{4}+\cdots\)
315.10.a.e	$315$	$10$	315.a	1.a	$1$	$4$	$4$	$162.236$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(13\)	\(0\)	\(2500\)	\(9604\)		$-$	$-$	$-$	$-$	$2^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1})q^{2}+(123+8\beta _{1}+\beta _{2})q^{4}+\cdots\)
315.10.a.f	$315$	$10$	315.a	1.a	$1$	$4$	$4$	$162.236$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(17\)	\(0\)	\(-2500\)	\(-9604\)		$-$	$+$	$+$	$-$	$2^{2}\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+(4+\beta _{1})q^{2}+(235+9\beta _{1}+\beta _{3})q^{4}+\cdots\)
315.10.a.g	$315$	$10$	315.a	1.a	$1$	$4$	$4$	$162.236$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(19\)	\(0\)	\(2500\)	\(-9604\)		$-$	$-$	$+$	$+$	$2\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+(5+\beta _{1})q^{2}+(435+9\beta _{1}+\beta _{2}+\beta _{3})q^{4}+\cdots\)
315.10.a.h	$315$	$10$	315.a	1.a	$1$	$4$	$4$	$162.236$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(20\)	\(0\)	\(-2500\)	\(-9604\)		$-$	$+$	$+$	$-$	$2^{5}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(5-\beta _{1})q^{2}+(106-8\beta _{1}+2\beta _{2})q^{4}+\cdots\)
315.10.a.i	$315$	$10$	315.a	1.a	$1$	$4$	$4$	$162.236$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(41\)	\(0\)	\(-2500\)	\(9604\)		$-$	$+$	$-$	$+$	$2^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(10-\beta _{1})q^{2}+(120-21\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
315.10.a.j	$315$	$10$	315.a	1.a	$1$	$5$	$5$	$162.236$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(-2\)	\(0\)	\(3125\)	\(12005\)		$-$	$-$	$-$	$-$	$2^{3}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(168-4\beta _{1}+\beta _{3})q^{4}+5^{4}q^{5}+\cdots\)
315.10.a.k	$315$	$10$	315.a	1.a	$1$	$6$	$6$	$162.236$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(-26\)	\(0\)	\(-3750\)	\(14406\)		$-$	$+$	$-$	$+$	$2^{7}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(-4-\beta _{1})q^{2}+(426+3\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
315.10.a.l	$315$	$10$	315.a	1.a	$1$	$6$	$6$	$162.236$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(-15\)	\(0\)	\(-3750\)	\(-14406\)		$-$	$+$	$+$	$-$	$2^{3}\cdot 3^{4}\cdot 5$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1})q^{2}+(503-3\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
315.10.a.m	$315$	$10$	315.a	1.a	$1$	$6$	$6$	$162.236$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(16\)	\(0\)	\(3750\)	\(14406\)		$-$	$-$	$-$	$-$	$2^{7}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(3-\beta _{1})q^{2}+(428-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
315.10.a.n	$315$	$10$	315.a	1.a	$1$	$8$	$8$	$162.236$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(-25\)	\(0\)	\(5000\)	\(19208\)		$+$	$-$	$-$	$+$	$2^{10}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{1})q^{2}+(233+5\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
315.10.a.o	$315$	$10$	315.a	1.a	$1$	$8$	$8$	$162.236$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(25\)	\(0\)	\(-5000\)	\(19208\)		$+$	$+$	$-$	$-$	$2^{10}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1})q^{2}+(233+5\beta _{1}+\beta _{2})q^{4}+\cdots\)
315.10.a.p	$315$	$10$	315.a	1.a	$1$	$10$	$10$	$162.236$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$1$	\(-7\)	\(0\)	\(-6250\)	\(-24010\)		$+$	$+$	$+$	$+$	$2^{11}\cdot 3^{12}\cdot 5^{4}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(223-\beta _{1}+\beta _{2})q^{4}+\cdots\)
315.10.a.q	$315$	$10$	315.a	1.a	$1$	$10$	$10$	$162.236$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$0$	\(7\)	\(0\)	\(6250\)	\(-24010\)		$+$	$-$	$+$	$-$	$2^{11}\cdot 3^{12}\cdot 5^{4}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(223-\beta _{1}+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(315))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(315)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 2}\)