Space of modular forms of level 315, weight 4, and trivial character

• Label: 315.4.a
• Level: $315$
• Weight: $4$
• Character orbit: 315.a
• Rep. character: $\chi_{315}(1,\cdot)$
• Character field: $\Q$
• Dimension: $30$
• Newform subspaces: $16$
• Sturm bound: $192$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(192\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(315))\).

	Total	New	Old
Modular forms	152	30	122
Cusp forms	136	30	106
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
\(+\)	\(+\)	\(+\)	\(+\)		\(22\)	\(2\)	\(20\)		\(20\)	\(2\)	\(18\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)		\(18\)	\(4\)	\(14\)		\(16\)	\(4\)	\(12\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(16\)	\(2\)	\(14\)		\(14\)	\(2\)	\(12\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(20\)	\(4\)	\(16\)		\(18\)	\(4\)	\(14\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(19\)	\(4\)	\(15\)		\(17\)	\(4\)	\(13\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(19\)	\(5\)	\(14\)		\(17\)	\(5\)	\(12\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(19\)	\(6\)	\(13\)		\(17\)	\(6\)	\(11\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(19\)	\(3\)	\(16\)		\(17\)	\(3\)	\(14\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(80\)	\(17\)	\(63\)		\(72\)	\(17\)	\(55\)		\(8\)	\(0\)	\(8\)
Minus space			\(-\)		\(72\)	\(13\)	\(59\)		\(64\)	\(13\)	\(51\)		\(8\)	\(0\)	\(8\)

Trace form

30 * q - 2 * q^2 + 98 * q^4 + 14 * q^7 - 54 * q^8 + 60 * q^10 + 116 * q^11 + 16 * q^13 - 70 * q^14 + 466 * q^16 + 260 * q^17 + 196 * q^19 + 36 * q^22 - 400 * q^23 + 750 * q^25 + 468 * q^26 + 70 * q^28 - 16 * q^29 - 432 * q^31 - 54 * q^32 + 608 * q^34 - 140 * q^35 - 116 * q^37 + 748 * q^38 + 600 * q^40 - 356 * q^41 - 872 * q^43 + 1944 * q^44 - 1064 * q^46 + 272 * q^47 + 1470 * q^49 - 50 * q^50 + 648 * q^52 - 1900 * q^53 + 360 * q^55 - 546 * q^56 - 928 * q^58 + 2276 * q^59 - 1024 * q^61 + 1680 * q^62 + 3178 * q^64 - 40 * q^65 - 1456 * q^67 + 604 * q^68 + 140 * q^70 - 40 * q^71 + 1548 * q^73 - 2772 * q^74 + 2228 * q^76 - 1568 * q^77 + 564 * q^79 + 3200 * q^80 - 6516 * q^82 + 484 * q^83 + 940 * q^85 - 1728 * q^86 - 1024 * q^88 + 4236 * q^89 + 1092 * q^91 - 4896 * q^92 - 2356 * q^94 - 500 * q^95 + 124 * q^97 - 98 * q^98

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(315))\) 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
315.4.a.a	$315$	$4$	315.a	1.a	$1$	$1$	$1$	$18.586$	\(\Q\)	None	✓				105.4.a.b	$1$	$0$	\(-5\)	\(0\)	\(-5\)	\(7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{2}+17q^{4}-5q^{5}+7q^{7}-45q^{8}+\cdots\)
315.4.a.b	$315$	$4$	315.a	1.a	$1$	$1$	$1$	$18.586$	\(\Q\)	None	✓	✓			315.4.a.b	$1$	$1$	\(-3\)	\(0\)	\(-5\)	\(7\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+q^{4}-5q^{5}+7q^{7}+21q^{8}+\cdots\)
315.4.a.c	$315$	$4$	315.a	1.a	$1$	$1$	$1$	$18.586$	\(\Q\)	None	✓				35.4.a.a	$1$	$1$	\(-1\)	\(0\)	\(5\)	\(7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-7q^{4}+5q^{5}+7q^{7}+15q^{8}+\cdots\)
315.4.a.d	$315$	$4$	315.a	1.a	$1$	$1$	$1$	$18.586$	\(\Q\)	None	✓				105.4.a.a	$1$	$0$	\(0\)	\(0\)	\(-5\)	\(7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{4}-5q^{5}+7q^{7}-42q^{11}+20q^{13}+\cdots\)
315.4.a.e	$315$	$4$	315.a	1.a	$1$	$1$	$1$	$18.586$	\(\Q\)	None	✓	✓			315.4.a.b	$1$	$0$	\(3\)	\(0\)	\(5\)	\(7\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+q^{4}+5q^{5}+7q^{7}-21q^{8}+\cdots\)
315.4.a.f	$315$	$4$	315.a	1.a	$1$	$2$	$2$	$18.586$	\(\Q(\sqrt{2}) \)	None	✓				35.4.a.b	$1$	$0$	\(-8\)	\(0\)	\(10\)	\(-14\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-4+\beta )q^{2}+(10-8\beta )q^{4}+5q^{5}+\cdots\)
315.4.a.g	$315$	$4$	315.a	1.a	$1$	$2$	$2$	$18.586$	\(\Q(\sqrt{41}) \)	None	✓		✓		105.4.a.g	$1$	$1$	\(-3\)	\(0\)	\(10\)	\(14\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+(3+3\beta )q^{4}+5q^{5}+\cdots\)
315.4.a.h	$315$	$4$	315.a	1.a	$1$	$2$	$2$	$18.586$	\(\Q(\sqrt{17}) \)	None	✓	✓	✓	✓	315.4.a.h	$1$	$0$	\(-1\)	\(0\)	\(-10\)	\(-14\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-4+\beta )q^{4}-5q^{5}-7q^{7}+\cdots\)
315.4.a.i	$315$	$4$	315.a	1.a	$1$	$2$	$2$	$18.586$	\(\Q(\sqrt{65}) \)	None	✓				105.4.a.f	$1$	$1$	\(-1\)	\(0\)	\(-10\)	\(-14\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(8+\beta )q^{4}-5q^{5}-7q^{7}+(-2^{4}+\cdots)q^{8}+\cdots\)
315.4.a.j	$315$	$4$	315.a	1.a	$1$	$2$	$2$	$18.586$	\(\Q(\sqrt{17}) \)	None	✓	✓	✓	✓	315.4.a.h	$1$	$1$	\(1\)	\(0\)	\(10\)	\(-14\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-4+\beta )q^{4}+5q^{5}-7q^{7}+\cdots\)
315.4.a.k	$315$	$4$	315.a	1.a	$1$	$2$	$2$	$18.586$	\(\Q(\sqrt{2}) \)	None	✓				105.4.a.e	$1$	$1$	\(2\)	\(0\)	\(-10\)	\(-14\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(1+2\beta )q^{4}-5q^{5}-7q^{7}+\cdots\)
315.4.a.l	$315$	$4$	315.a	1.a	$1$	$2$	$2$	$18.586$	\(\Q(\sqrt{5}) \)	None	✓				105.4.a.d	$1$	$0$	\(4\)	\(0\)	\(10\)	\(-14\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{2}+(1+4\beta )q^{4}+5q^{5}-7q^{7}+\cdots\)
315.4.a.m	$315$	$4$	315.a	1.a	$1$	$2$	$2$	$18.586$	\(\Q(\sqrt{17}) \)	None	✓				105.4.a.c	$1$	$0$	\(7\)	\(0\)	\(10\)	\(-14\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(4-\beta )q^{2}+(12-7\beta )q^{4}+5q^{5}-7q^{7}+\cdots\)
315.4.a.n	$315$	$4$	315.a	1.a	$1$	$3$	$3$	$18.586$	3.3.22952.1	None	✓	✓	✓		315.4.a.n	$1$	$1$	\(-2\)	\(0\)	\(-15\)	\(21\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(5-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
315.4.a.o	$315$	$4$	315.a	1.a	$1$	$3$	$3$	$18.586$	3.3.22952.1	None	✓	✓	✓		315.4.a.n	$1$	$0$	\(2\)	\(0\)	\(15\)	\(21\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(5-2\beta _{1}+\beta _{2})q^{4}+5q^{5}+\cdots\)
315.4.a.p	$315$	$4$	315.a	1.a	$1$	$3$	$3$	$18.586$	3.3.14360.1	None	✓		✓		35.4.a.c	$1$	$0$	\(3\)	\(0\)	\(-15\)	\(21\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(4-\beta _{1}+\beta _{2})q^{4}-5q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(315))\) into lower level spaces

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