Space of modular forms of level 312, weight 4, and trivial character

• Label: 312.4.a
• Level: $312$
• Weight: $4$
• Character orbit: 312.a
• Rep. character: $\chi_{312}(1,\cdot)$
• Character field: $\Q$
• Dimension: $18$
• Newform subspaces: $8$
• Sturm bound: $224$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	312.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(224\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(312))\).

	Total	New	Old
Modular forms	176	18	158
Cusp forms	160	18	142
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.

\(2\)	\(3\)	\(13\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(26\)	\(2\)	\(24\)		\(24\)	\(2\)	\(22\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)		\(18\)	\(2\)	\(16\)		\(16\)	\(2\)	\(14\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(21\)	\(2\)	\(19\)		\(19\)	\(2\)	\(17\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(23\)	\(3\)	\(20\)		\(21\)	\(3\)	\(18\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(22\)	\(2\)	\(20\)		\(20\)	\(2\)	\(18\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(22\)	\(3\)	\(19\)		\(20\)	\(3\)	\(17\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(19\)	\(2\)	\(17\)		\(17\)	\(2\)	\(15\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(25\)	\(2\)	\(23\)		\(23\)	\(2\)	\(21\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(90\)	\(10\)	\(80\)		\(82\)	\(10\)	\(72\)		\(8\)	\(0\)	\(8\)
Minus space			\(-\)		\(86\)	\(8\)	\(78\)		\(78\)	\(8\)	\(70\)		\(8\)	\(0\)	\(8\)

Trace form

18 * q + 12 * q^7 + 162 * q^9 + 26 * q^13 - 84 * q^15 + 4 * q^17 - 228 * q^19 + 84 * q^21 - 168 * q^23 + 270 * q^25 - 140 * q^29 - 276 * q^31 + 168 * q^33 + 1192 * q^35 + 100 * q^37 + 200 * q^41 + 112 * q^43 + 688 * q^47 + 994 * q^49 - 676 * q^53 + 2080 * q^55 - 468 * q^57 - 216 * q^59 + 60 * q^61 + 108 * q^63 - 556 * q^67 - 600 * q^69 - 712 * q^71 + 1564 * q^73 - 1216 * q^77 + 904 * q^79 + 1458 * q^81 + 1752 * q^83 - 3808 * q^85 - 744 * q^87 + 1568 * q^89 - 1092 * q^91 - 1404 * q^93 - 3592 * q^95 - 964 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(312))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	13
312.4.a.a	$312$	$4$	312.a	1.a	$1$	$2$	$2$	$18.409$	\(\Q(\sqrt{3}) \)	None	✓	✓	✓	✓	312.4.a.a	$1$	$1$	\(0\)	\(-6\)	\(-4\)	\(4\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-2+\beta )q^{5}+(2-3\beta )q^{7}+\cdots\)
312.4.a.b	$312$	$4$	312.a	1.a	$1$	$2$	$2$	$18.409$	\(\Q(\sqrt{55}) \)	None	✓	✓	✓	✓	312.4.a.b	$1$	$0$	\(0\)	\(-6\)	\(-4\)	\(20\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-2+\beta )q^{5}+(10+\beta )q^{7}+\cdots\)
312.4.a.c	$312$	$4$	312.a	1.a	$1$	$2$	$2$	$18.409$	\(\Q(\sqrt{113}) \)	None	✓	✓	✓	✓	312.4.a.c	$1$	$1$	\(0\)	\(-6\)	\(6\)	\(-10\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}+(3+\beta )q^{5}+(-5-\beta )q^{7}+9q^{9}+\cdots\)
312.4.a.d	$312$	$4$	312.a	1.a	$1$	$2$	$2$	$18.409$	\(\Q(\sqrt{17}) \)	None	✓	✓	✓	✓	312.4.a.d	$1$	$1$	\(0\)	\(6\)	\(-18\)	\(-10\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-9-\beta )q^{5}+(-5+5\beta )q^{7}+\cdots\)
312.4.a.e	$312$	$4$	312.a	1.a	$1$	$2$	$2$	$18.409$	\(\Q(\sqrt{7}) \)	None	✓	✓	✓	✓	312.4.a.e	$1$	$1$	\(0\)	\(6\)	\(-4\)	\(-20\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-2+\beta )q^{5}+(-10-3\beta )q^{7}+\cdots\)
312.4.a.f	$312$	$4$	312.a	1.a	$1$	$2$	$2$	$18.409$	\(\Q(\sqrt{43}) \)	None	✓	✓	✓	✓	312.4.a.f	$1$	$0$	\(0\)	\(6\)	\(12\)	\(44\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+(6+\beta )q^{5}+(22+\beta )q^{7}+9q^{9}+\cdots\)
312.4.a.g	$312$	$4$	312.a	1.a	$1$	$3$	$3$	$18.409$	3.3.36248.1	None	✓	✓	✓	✓	312.4.a.g	$1$	$0$	\(0\)	\(-9\)	\(16\)	\(-22\)		$-$	$+$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(5+\beta _{2})q^{5}+(-8+\beta _{1}+\beta _{2})q^{7}+\cdots\)
312.4.a.h	$312$	$4$	312.a	1.a	$1$	$3$	$3$	$18.409$	3.3.13916.1	None	✓	✓	✓	✓	312.4.a.h	$1$	$0$	\(0\)	\(9\)	\(-4\)	\(6\)		$+$	$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-1+\beta _{2})q^{5}+(2+\beta _{1}+\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(312))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(312)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(156))\)\(^{\oplus 2}\)