Space of modular forms of level 847, weight 4, and trivial character

• Label: 847.4.a
• Level: $847$
• Weight: $4$
• Character orbit: 847.a
• Rep. character: $\chi_{847}(1,\cdot)$
• Character field: $\Q$
• Dimension: $163$
• Newform subspaces: $18$
• Sturm bound: $352$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	847.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 18 \)
Sturm bound:			\(352\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(847))\).

	Total	New	Old
Modular forms	276	163	113
Cusp forms	252	163	89
Eisenstein series	24	0	24

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

\(7\)	\(11\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(72\)	\(41\)	\(31\)		\(66\)	\(41\)	\(25\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)		\(67\)	\(40\)	\(27\)		\(61\)	\(40\)	\(21\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(66\)	\(37\)	\(29\)		\(60\)	\(37\)	\(23\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(71\)	\(45\)	\(26\)		\(65\)	\(45\)	\(20\)		\(6\)	\(0\)	\(6\)
Plus space		\(+\)		\(143\)	\(86\)	\(57\)		\(131\)	\(86\)	\(45\)		\(12\)	\(0\)	\(12\)
Minus space		\(-\)		\(133\)	\(77\)	\(56\)		\(121\)	\(77\)	\(44\)		\(12\)	\(0\)	\(12\)

Trace form

163 * q + 5 * q^2 + 2 * q^3 + 631 * q^4 - 38 * q^6 + 7 * q^7 + 45 * q^8 + 1463 * q^9 + 128 * q^10 + 142 * q^12 + 140 * q^13 - 63 * q^14 - 176 * q^15 + 2559 * q^16 - 206 * q^17 + 37 * q^18 + 22 * q^19 + 12 * q^20 - 14 * q^21 - 300 * q^23 + 226 * q^24 + 4349 * q^25 + 616 * q^26 - 364 * q^27 - 49 * q^28 + 470 * q^29 + 192 * q^30 + 516 * q^31 + 245 * q^32 + 374 * q^34 + 168 * q^35 + 5175 * q^36 - 502 * q^37 + 822 * q^38 + 536 * q^39 + 144 * q^40 + 2 * q^41 - 14 * q^42 - 424 * q^43 - 112 * q^45 + 808 * q^46 - 876 * q^47 + 2382 * q^48 + 7987 * q^49 + 71 * q^50 + 524 * q^51 + 8 * q^52 + 302 * q^53 - 3364 * q^54 - 567 * q^56 - 2892 * q^57 - 146 * q^58 + 1118 * q^59 + 2780 * q^60 + 328 * q^61 - 184 * q^62 + 791 * q^63 + 10307 * q^64 - 344 * q^65 - 1292 * q^67 - 1434 * q^68 - 728 * q^69 - 392 * q^70 - 2532 * q^71 + 1757 * q^72 + 1062 * q^73 - 46 * q^74 - 3178 * q^75 + 742 * q^76 + 5084 * q^78 + 344 * q^79 - 2896 * q^80 + 12471 * q^81 - 1662 * q^82 - 690 * q^83 - 882 * q^84 + 664 * q^85 + 708 * q^86 - 988 * q^87 + 2878 * q^89 + 11176 * q^90 + 308 * q^91 - 1516 * q^92 - 5280 * q^93 - 1240 * q^94 + 1184 * q^95 + 250 * q^96 + 114 * q^97 + 245 * q^98

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(847))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		7	11
847.4.a.a	$847$	$4$	847.a	1.a	$1$	$1$	$1$	$49.975$	\(\Q\)	None	✓				77.4.a.a	$1$	$1$	\(-3\)	\(4\)	\(12\)	\(-7\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+4q^{3}+q^{4}+12q^{5}-12q^{6}+\cdots\)
847.4.a.b	$847$	$4$	847.a	1.a	$1$	$1$	$1$	$49.975$	\(\Q\)	None	✓				7.4.a.a	$1$	$0$	\(1\)	\(-2\)	\(16\)	\(7\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-2q^{3}-7q^{4}+2^{4}q^{5}-2q^{6}+\cdots\)
847.4.a.c	$847$	$4$	847.a	1.a	$1$	$2$	$2$	$49.975$	\(\Q(\sqrt{2}) \)	None	✓				77.4.a.b	$1$	$1$	\(2\)	\(-4\)	\(-4\)	\(-14\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}-2q^{3}+(1+2\beta )q^{4}+(-2+\cdots)q^{5}+\cdots\)
847.4.a.d	$847$	$4$	847.a	1.a	$1$	$4$	$4$	$49.975$	4.4.522072.1	None	✓				77.4.a.d	$1$	$0$	\(2\)	\(14\)	\(10\)	\(28\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+(4+\beta _{1})q^{3}+(6-\beta _{2}-2\beta _{3})q^{4}+\cdots\)
847.4.a.e	$847$	$4$	847.a	1.a	$1$	$4$	$4$	$49.975$	4.4.509800.1	None	✓				77.4.a.c	$1$	$0$	\(4\)	\(-12\)	\(-18\)	\(28\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{2})q^{2}+(-3+\beta _{3})q^{3}+(6-\beta _{1}+\cdots)q^{4}+\cdots\)
847.4.a.f	$847$	$4$	847.a	1.a	$1$	$5$	$5$	$49.975$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				77.4.a.e	$1$	$1$	\(-1\)	\(2\)	\(-24\)	\(-35\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-\beta _{3}q^{3}+(9+\beta _{1}+\beta _{2})q^{4}+\cdots\)
847.4.a.g	$847$	$4$	847.a	1.a	$1$	$7$	$7$	$49.975$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓			847.4.a.g	$1$	$1$	\(-4\)	\(2\)	\(-18\)	\(49\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+\beta _{3}q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
847.4.a.h	$847$	$4$	847.a	1.a	$1$	$7$	$7$	$49.975$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓			847.4.a.g	$1$	$0$	\(4\)	\(2\)	\(-18\)	\(-49\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+\beta _{3}q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
847.4.a.i	$847$	$4$	847.a	1.a	$1$	$8$	$8$	$49.975$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓			847.4.a.i	$1$	$1$	\(-6\)	\(2\)	\(-2\)	\(-56\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}-\beta _{3}q^{3}+(4-\beta _{1}-\beta _{3}+\cdots)q^{4}+\cdots\)
847.4.a.j	$847$	$4$	847.a	1.a	$1$	$8$	$8$	$49.975$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓			847.4.a.j	$1$	$1$	\(-2\)	\(2\)	\(2\)	\(-56\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(\beta _{1}+\beta _{3})q^{3}+(4+\beta _{2})q^{4}+\cdots\)
847.4.a.k	$847$	$4$	847.a	1.a	$1$	$8$	$8$	$49.975$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓			847.4.a.j	$1$	$0$	\(2\)	\(2\)	\(2\)	\(56\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(\beta _{1}+\beta _{3})q^{3}+(4+\beta _{2})q^{4}+\cdots\)
847.4.a.l	$847$	$4$	847.a	1.a	$1$	$8$	$8$	$49.975$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓			847.4.a.i	$1$	$0$	\(6\)	\(2\)	\(-2\)	\(56\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-\beta _{3}q^{3}+(4-\beta _{1}-\beta _{3}+\cdots)q^{4}+\cdots\)
847.4.a.m	$847$	$4$	847.a	1.a	$1$	$14$	$14$	$49.975$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓	✓			847.4.a.m	$1$	$1$	\(-8\)	\(4\)	\(6\)	\(98\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+\beta _{5}q^{3}+(5-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
847.4.a.n	$847$	$4$	847.a	1.a	$1$	$14$	$14$	$49.975$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓	✓			847.4.a.m	$1$	$0$	\(8\)	\(4\)	\(6\)	\(-98\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+\beta _{5}q^{3}+(5-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
847.4.a.o	$847$	$4$	847.a	1.a	$1$	$16$	$16$	$49.975$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓		✓		77.4.f.a	$1$	$1$	\(-6\)	\(-21\)	\(-32\)	\(112\)		$-$	$+$	$-$	$2^{2}\cdot 11^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1+\beta _{4})q^{3}+(3+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
847.4.a.p	$847$	$4$	847.a	1.a	$1$	$16$	$16$	$49.975$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓		✓		77.4.f.a	$1$	$1$	\(6\)	\(-21\)	\(-32\)	\(-112\)		$+$	$-$	$-$	$2^{2}\cdot 11^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-1+\beta _{4})q^{3}+(3+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
847.4.a.q	$847$	$4$	847.a	1.a	$1$	$20$	$20$	$49.975$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓		✓		77.4.f.b	$1$	$0$	\(-1\)	\(11\)	\(48\)	\(-140\)		$+$	$+$	$+$	$2^{2}\cdot 11^{4}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1-\beta _{4})q^{3}+(5+\beta _{2})q^{4}+\cdots\)
847.4.a.r	$847$	$4$	847.a	1.a	$1$	$20$	$20$	$49.975$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓		✓		77.4.f.b	$1$	$0$	\(1\)	\(11\)	\(48\)	\(140\)		$-$	$-$	$+$	$2^{2}\cdot 11^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1-\beta _{4})q^{3}+(5+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(847))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(847)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 2}\)