Space of modular forms of level 147, weight 6, and Character 147.e

• Label: 147.6.e
• Level: $147$
• Weight: $6$
• Character orbit: 147.e
• Rep. character: $\chi_{147}(67,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $66$
• Newform subspaces: $17$
• Sturm bound: $112$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	147.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 17 \)
Sturm bound:			\(112\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(147, [\chi])\).

	Total	New	Old
Modular forms	202	66	136
Cusp forms	170	66	104
Eisenstein series	32	0	32

Trace form

66 * q - 4 * q^2 + 9 * q^3 - 556 * q^4 - 22 * q^5 - 144 * q^6 + 36 * q^8 - 2673 * q^9 + 1182 * q^10 - 1070 * q^11 + 288 * q^12 - 2158 * q^13 - 396 * q^15 - 7492 * q^16 + 1848 * q^17 - 324 * q^18 + 2657 * q^19 - 1448 * q^20 - 11356 * q^22 + 7912 * q^23 + 3402 * q^24 - 25947 * q^25 - 17030 * q^26 - 1458 * q^27 - 3952 * q^29 - 9720 * q^30 + 725 * q^31 + 23130 * q^32 + 11430 * q^33 + 1176 * q^34 + 90072 * q^36 - 50017 * q^37 - 47750 * q^38 - 9081 * q^39 + 3318 * q^40 + 50812 * q^41 - 84646 * q^43 + 48952 * q^44 - 1782 * q^45 + 34996 * q^46 - 17118 * q^47 - 63216 * q^48 + 232072 * q^50 - 18108 * q^51 - 5428 * q^52 - 100376 * q^53 + 5832 * q^54 - 187944 * q^55 - 78570 * q^57 + 167662 * q^58 + 67764 * q^59 + 103158 * q^60 + 51854 * q^61 + 183420 * q^62 + 713352 * q^64 + 238302 * q^65 + 2988 * q^66 + 71125 * q^67 - 66612 * q^68 - 154224 * q^69 - 507308 * q^71 - 1458 * q^72 + 65927 * q^73 + 1078 * q^74 + 48807 * q^75 - 222184 * q^76 + 444852 * q^78 + 11397 * q^79 - 97676 * q^80 - 216513 * q^81 + 5268 * q^82 + 430740 * q^83 - 667056 * q^85 - 595258 * q^86 + 3060 * q^87 - 79722 * q^88 - 163120 * q^89 - 191484 * q^90 - 920248 * q^92 - 74079 * q^93 - 5988 * q^94 - 330934 * q^95 + 176886 * q^96 + 86924 * q^97 + 173340 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(147, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
147.6.e.a	$147$	$6$	147.e	7.c	$3$	$2$	$1$	$23.576$	\(\Q(\sqrt{-3}) \)	None					21.6.a.d	$2$	$0$	\(-10\)	\(-9\)	\(106\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-10\zeta_{6}q^{2}+(-9+9\zeta_{6})q^{3}+(-68+\cdots)q^{4}+\cdots\)
147.6.e.b	$147$	$6$	147.e	7.c	$3$	$2$	$1$	$23.576$	\(\Q(\sqrt{-3}) \)	None					21.6.a.d	$2$	$1$	\(-10\)	\(9\)	\(-106\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-10\zeta_{6}q^{2}+(9-9\zeta_{6})q^{3}+(-68+68\zeta_{6})q^{4}+\cdots\)
147.6.e.c	$147$	$6$	147.e	7.c	$3$	$2$	$1$	$23.576$	\(\Q(\sqrt{-3}) \)	None					21.6.a.c	$2$	$0$	\(-5\)	\(-9\)	\(-94\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-5\zeta_{6}q^{2}+(-9+9\zeta_{6})q^{3}+(7-7\zeta_{6})q^{4}+\cdots\)
147.6.e.d	$147$	$6$	147.e	7.c	$3$	$2$	$1$	$23.576$	\(\Q(\sqrt{-3}) \)	None					21.6.a.c	$2$	$0$	\(-5\)	\(9\)	\(94\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-5\zeta_{6}q^{2}+(9-9\zeta_{6})q^{3}+(7-7\zeta_{6})q^{4}+\cdots\)
147.6.e.e	$147$	$6$	147.e	7.c	$3$	$2$	$1$	$23.576$	\(\Q(\sqrt{-3}) \)	None					21.6.a.b	$2$	$0$	\(-1\)	\(-9\)	\(-34\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\zeta_{6}q^{2}+(-9+9\zeta_{6})q^{3}+(31-31\zeta_{6})q^{4}+\cdots\)
147.6.e.f	$147$	$6$	147.e	7.c	$3$	$2$	$1$	$23.576$	\(\Q(\sqrt{-3}) \)	None					21.6.a.b	$2$	$0$	\(-1\)	\(9\)	\(34\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\zeta_{6}q^{2}+(9-9\zeta_{6})q^{3}+(31-31\zeta_{6})q^{4}+\cdots\)
147.6.e.g	$147$	$6$	147.e	7.c	$3$	$2$	$1$	$23.576$	\(\Q(\sqrt{-3}) \)	None					21.6.e.a	$2$	$0$	\(2\)	\(-9\)	\(11\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\zeta_{6}q^{2}+(-9+9\zeta_{6})q^{3}+(28-28\zeta_{6})q^{4}+\cdots\)
147.6.e.h	$147$	$6$	147.e	7.c	$3$	$2$	$1$	$23.576$	\(\Q(\sqrt{-3}) \)	None					3.6.a.a	$2$	$0$	\(6\)	\(-9\)	\(-6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+6\zeta_{6}q^{2}+(-9+9\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
147.6.e.i	$147$	$6$	147.e	7.c	$3$	$2$	$1$	$23.576$	\(\Q(\sqrt{-3}) \)	None					21.6.a.a	$2$	$0$	\(6\)	\(-9\)	\(78\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+6\zeta_{6}q^{2}+(-9+9\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
147.6.e.j	$147$	$6$	147.e	7.c	$3$	$2$	$1$	$23.576$	\(\Q(\sqrt{-3}) \)	None					21.6.a.a	$2$	$0$	\(6\)	\(9\)	\(-78\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+6\zeta_{6}q^{2}+(9-9\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
147.6.e.k	$147$	$6$	147.e	7.c	$3$	$2$	$1$	$23.576$	\(\Q(\sqrt{-3}) \)	None					3.6.a.a	$2$	$0$	\(6\)	\(9\)	\(6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+6\zeta_{6}q^{2}+(9-9\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
147.6.e.l	$147$	$6$	147.e	7.c	$3$	$4$	$2$	$23.576$	\(\Q(\sqrt{-3}, \sqrt{-83})\)	None					21.6.e.b	$2$	$0$	\(3\)	\(-18\)	\(-33\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{1}-\beta _{3})q^{2}+9\beta _{1}q^{3}+(31\beta _{1}+\cdots)q^{4}+\cdots\)
147.6.e.m	$147$	$6$	147.e	7.c	$3$	$4$	$2$	$23.576$	\(\Q(\sqrt{-3}, \sqrt{193})\)	None		✓			147.6.a.h	$2$	$0$	\(3\)	\(-18\)	\(72\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{1}+\beta _{2})q^{2}+(-9+9\beta _{2})q^{3}+(-20+\cdots)q^{4}+\cdots\)
147.6.e.n	$147$	$6$	147.e	7.c	$3$	$4$	$2$	$23.576$	\(\Q(\sqrt{-3}, \sqrt{193})\)	None		✓			147.6.a.h	$2$	$0$	\(3\)	\(18\)	\(-72\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{1}+\beta _{2})q^{2}+(9-9\beta _{2})q^{3}+(-20+\cdots)q^{4}+\cdots\)
147.6.e.o	$147$	$6$	147.e	7.c	$3$	$8$	$4$	$23.576$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					21.6.e.c	$2$	$0$	\(-3\)	\(36\)	\(0\)	\(0\)	$3\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\beta _{1}-\beta _{2})q^{2}-9\beta _{2}q^{3}+(-\beta _{1}+\cdots)q^{4}+\cdots\)
147.6.e.p	$147$	$6$	147.e	7.c	$3$	$12$	$6$	$23.576$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		✓			147.6.a.n	$2$	$0$	\(-2\)	\(-54\)	\(-100\)	\(0\)	$2^{8}\cdot 7^{6}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{1}q^{2}+(-9-9\beta _{2})q^{3}+(-5^{2}-5^{2}\beta _{2}+\cdots)q^{4}+\cdots\)
147.6.e.q	$147$	$6$	147.e	7.c	$3$	$12$	$6$	$23.576$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		✓			147.6.a.n	$2$	$0$	\(-2\)	\(54\)	\(100\)	\(0\)	$2^{8}\cdot 7^{6}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{1}q^{2}+(9+9\beta _{2})q^{3}+(-5^{2}-5^{2}\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(147, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(147, [\chi]) \simeq \) \(S_{6}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 2}\)