Space of modular forms of level 847, weight 2, and Character 847.f

• Label: 847.2.f
• Level: $847$
• Weight: $2$
• Character orbit: 847.f
• Rep. character: $\chi_{847}(148,\cdot)$
• Character field: $\Q(\zeta_{5})$
• Dimension: $216$
• Newform subspaces: $26$
• Sturm bound: $176$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	847.f (of order \(5\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 11 \)
Character field:			\(\Q(\zeta_{5})\)
Newform subspaces:			\( 26 \)
Sturm bound:			\(176\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(3\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(847, [\chi])\).

	Total	New	Old
Modular forms	400	216	184
Cusp forms	304	216	88
Eisenstein series	96	0	96

Trace form

216 * q + 4 * q^2 + 8 * q^3 - 52 * q^4 + 4 * q^5 - 6 * q^6 + 2 * q^7 + 2 * q^8 - 44 * q^9 + 16 * q^10 - 36 * q^12 + 2 * q^13 - 3 * q^14 + 22 * q^15 - 74 * q^16 + 16 * q^17 - 15 * q^18 - 10 * q^19 - 14 * q^20 - 12 * q^21 - 40 * q^23 + 14 * q^24 - 66 * q^25 + 10 * q^26 + 26 * q^27 - 11 * q^28 + 6 * q^29 - 12 * q^30 + 28 * q^31 - 12 * q^32 + 48 * q^34 + 8 * q^35 - 102 * q^36 - 4 * q^37 - 26 * q^38 - 6 * q^39 + 26 * q^40 + 32 * q^41 - 8 * q^43 - 200 * q^45 + 13 * q^46 + 28 * q^47 + 52 * q^48 - 54 * q^49 - 18 * q^50 + 26 * q^51 - 54 * q^52 + 14 * q^53 - 64 * q^54 + 18 * q^56 + 14 * q^57 + 35 * q^58 - 40 * q^59 + 108 * q^60 - 74 * q^62 + 10 * q^63 - 56 * q^64 - 16 * q^65 + 44 * q^67 - 6 * q^68 + 12 * q^69 + 32 * q^70 + 42 * q^71 - 3 * q^72 + 32 * q^73 + 38 * q^74 + 6 * q^75 - 4 * q^78 - 6 * q^79 + 160 * q^80 + 6 * q^81 - 30 * q^82 - 28 * q^83 - 12 * q^84 + 4 * q^85 + 101 * q^86 + 60 * q^87 - 128 * q^89 + 12 * q^90 - 18 * q^91 + 129 * q^92 - 10 * q^93 + 54 * q^94 - 56 * q^95 + 32 * q^96 - 32 * q^97 - 6 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(847, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
847.2.f.a	$847$	$2$	847.f	11.c	$5$	$4$	$1$	$6.763$	\(\Q(\zeta_{10})\)	None					77.2.a.d	$2$	$0$	\(-5\)	\(-6\)	\(2\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+(-1-\zeta_{10}+\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
847.2.f.b	$847$	$2$	847.f	11.c	$5$	$4$	$1$	$6.763$	\(\Q(\zeta_{10})\)	None					77.2.a.d	$2$	$0$	\(-5\)	\(4\)	\(2\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+(-1-\zeta_{10}+\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
847.2.f.c	$847$	$2$	847.f	11.c	$5$	$4$	$1$	$6.763$	\(\Q(\zeta_{10})\)	None		✓			847.2.a.d	$2$	$0$	\(-4\)	\(3\)	\(-1\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+(-2+\zeta_{10}-\zeta_{10}^{2}+2\zeta_{10}^{3})q^{2}+\cdots\)
847.2.f.d	$847$	$2$	847.f	11.c	$5$	$4$	$1$	$6.763$	\(\Q(\zeta_{10})\)	None		✓			847.2.a.d	$2$	$0$	\(-1\)	\(-2\)	\(-1\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+(1-2\zeta_{10}+2\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}+\cdots\)
847.2.f.e	$847$	$2$	847.f	11.c	$5$	$4$	$1$	$6.763$	\(\Q(\zeta_{10})\)	None					77.2.a.c	$4$	$0$	\(-1\)	\(-2\)	\(2\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+(-1+\zeta_{10}-\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
847.2.f.f	$847$	$2$	847.f	11.c	$5$	$4$	$1$	$6.763$	\(\Q(\zeta_{10})\)	None					77.2.a.b	$4$	$0$	\(0\)	\(-1\)	\(-3\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+\zeta_{10}^{2}q^{3}+2\zeta_{10}^{3}q^{4}-3\zeta_{10}q^{5}+\cdots\)
847.2.f.g	$847$	$2$	847.f	11.c	$5$	$4$	$1$	$6.763$	\(\Q(\zeta_{10})\)	None					77.2.a.b	$4$	$0$	\(0\)	\(-1\)	\(-3\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+\zeta_{10}^{2}q^{3}+2\zeta_{10}^{3}q^{4}-3\zeta_{10}q^{5}+\cdots\)
847.2.f.h	$847$	$2$	847.f	11.c	$5$	$4$	$1$	$6.763$	\(\Q(\zeta_{10})\)	None					77.2.a.a	$4$	$0$	\(0\)	\(3\)	\(1\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q-3\zeta_{10}^{2}q^{3}+2\zeta_{10}^{3}q^{4}+\zeta_{10}q^{5}+\cdots\)
847.2.f.i	$847$	$2$	847.f	11.c	$5$	$4$	$1$	$6.763$	\(\Q(\zeta_{10})\)	None					77.2.a.a	$4$	$0$	\(0\)	\(3\)	\(1\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q-3\zeta_{10}^{2}q^{3}+2\zeta_{10}^{3}q^{4}+\zeta_{10}q^{5}+\cdots\)
847.2.f.j	$847$	$2$	847.f	11.c	$5$	$4$	$1$	$6.763$	\(\Q(\zeta_{10})\)	None		✓			847.2.a.d	$2$	$0$	\(1\)	\(-2\)	\(-1\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+(-1+2\zeta_{10}-2\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
847.2.f.k	$847$	$2$	847.f	11.c	$5$	$4$	$1$	$6.763$	\(\Q(\zeta_{10})\)	None					77.2.a.c	$4$	$0$	\(1\)	\(-2\)	\(2\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+(1-\zeta_{10}+\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}+2\zeta_{10}^{2}q^{3}+\cdots\)
847.2.f.l	$847$	$2$	847.f	11.c	$5$	$4$	$1$	$6.763$	\(\Q(\zeta_{10})\)	None		✓			847.2.a.d	$2$	$0$	\(4\)	\(3\)	\(-1\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+(2-\zeta_{10}+\zeta_{10}^{2}-2\zeta_{10}^{3})q^{2}+(\zeta_{10}+\cdots)q^{3}+\cdots\)
847.2.f.m	$847$	$2$	847.f	11.c	$5$	$4$	$1$	$6.763$	\(\Q(\zeta_{10})\)	None					77.2.a.d	$2$	$0$	\(5\)	\(-6\)	\(2\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+(1+\zeta_{10}-\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}+(-2\zeta_{10}+\cdots)q^{3}+\cdots\)
847.2.f.n	$847$	$2$	847.f	11.c	$5$	$4$	$1$	$6.763$	\(\Q(\zeta_{10})\)	None					77.2.a.d	$2$	$0$	\(5\)	\(4\)	\(2\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+(1+\zeta_{10}-\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}+(2\zeta_{10}+\cdots)q^{3}+\cdots\)
847.2.f.o	$847$	$2$	847.f	11.c	$5$	$8$	$2$	$6.763$	8.0.446265625.1	None		✓			847.2.a.e	$4$	$0$	\(-1\)	\(1\)	\(0\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+\beta _{5}q^{2}+(-\beta _{1}+\beta _{2}+\beta _{5}+\beta _{6})q^{3}+\cdots\)
847.2.f.p	$847$	$2$	847.f	11.c	$5$	$8$	$2$	$6.763$	8.0.159390625.1	None					77.2.f.a	$2$	$0$	\(-1\)	\(6\)	\(3\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q-\beta _{1}q^{2}+(1-\beta _{3})q^{3}+(\beta _{1}-\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots\)
847.2.f.q	$847$	$2$	847.f	11.c	$5$	$8$	$2$	$6.763$	8.0.159390625.1	None					77.2.f.a	$2$	$0$	\(1\)	\(-4\)	\(3\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q-\beta _{4}q^{2}+(-\beta _{2}-\beta _{3})q^{3}+(1-\beta _{1}+\cdots)q^{4}+\cdots\)
847.2.f.r	$847$	$2$	847.f	11.c	$5$	$8$	$2$	$6.763$	8.0.446265625.1	None		✓			847.2.a.e	$4$	$0$	\(1\)	\(1\)	\(0\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q-\beta _{5}q^{2}+(-\beta _{1}+\beta _{2}+\beta _{5}+\beta _{6})q^{3}+\cdots\)
847.2.f.s	$847$	$2$	847.f	11.c	$5$	$8$	$2$	$6.763$	8.0.159390625.1	None					77.2.f.a	$2$	$0$	\(1\)	\(6\)	\(3\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+\beta _{1}q^{2}+(1-\beta _{3})q^{3}+(\beta _{1}-\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots\)
847.2.f.t	$847$	$2$	847.f	11.c	$5$	$12$	$3$	$6.763$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		✓			847.2.a.i	$4$	$0$	\(-2\)	\(1\)	\(-1\)	\(-3\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+(-1-\beta _{1}-\beta _{2}-\beta _{3}+\beta _{6}+\beta _{7}+\cdots)q^{2}+\cdots\)
847.2.f.u	$847$	$2$	847.f	11.c	$5$	$12$	$3$	$6.763$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		✓			847.2.a.i	$4$	$0$	\(2\)	\(1\)	\(-1\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+(1+\beta _{1}+\beta _{2}+\beta _{3}-\beta _{6}-\beta _{7}-\beta _{8}+\cdots)q^{2}+\cdots\)
847.2.f.v	$847$	$2$	847.f	11.c	$5$	$16$	$4$	$6.763$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None					77.2.f.b	$2$	$0$	\(-2\)	\(-2\)	\(-5\)	\(4\)	$5$	$\mathrm{SU}(2)[C_{5}]$	\(q-\beta _{6}q^{2}+(-\beta _{10}+\beta _{11}-\beta _{14}-\beta _{15})q^{3}+\cdots\)
847.2.f.w	$847$	$2$	847.f	11.c	$5$	$16$	$4$	$6.763$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None					77.2.f.b	$2$	$0$	\(2\)	\(-2\)	\(-5\)	\(-4\)	$5$	$\mathrm{SU}(2)[C_{5}]$	\(q+\beta _{6}q^{2}+(-\beta _{10}+\beta _{11}-\beta _{14}-\beta _{15})q^{3}+\cdots\)
847.2.f.x	$847$	$2$	847.f	11.c	$5$	$16$	$4$	$6.763$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None					77.2.f.b	$2$	$0$	\(3\)	\(-2\)	\(-5\)	\(4\)	$5$	$\mathrm{SU}(2)[C_{5}]$	\(q+(\beta _{5}+\beta _{6})q^{2}+(\beta _{9}-\beta _{11}+\beta _{13})q^{3}+\cdots\)
847.2.f.y	$847$	$2$	847.f	11.c	$5$	$24$	$6$	$6.763$		None		✓			847.2.a.m	$4$	$0$	\(-4\)	\(2\)	\(4\)	\(-6\)		$\mathrm{SU}(2)[C_{5}]$
847.2.f.z	$847$	$2$	847.f	11.c	$5$	$24$	$6$	$6.763$		None		✓			847.2.a.m	$4$	$0$	\(4\)	\(2\)	\(4\)	\(6\)		$\mathrm{SU}(2)[C_{5}]$

Decomposition of \(S_{2}^{\mathrm{old}}(847, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(847, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(77, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(121, [\chi])\)\(^{\oplus 2}\)