Space of modular forms of level 147, weight 8, and Character 147.e

• Label: 147.8.e
• Level: $147$
• Weight: $8$
• Character orbit: 147.e
• Rep. character: $\chi_{147}(67,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $94$
• Newform subspaces: $16$
• Sturm bound: $149$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	147.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(149\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(147, [\chi])\).

	Total	New	Old
Modular forms	278	94	184
Cusp forms	246	94	152
Eisenstein series	32	0	32

Trace form

94 * q + 28 * q^2 - 27 * q^3 - 3436 * q^4 - 2 * q^5 + 864 * q^6 - 6876 * q^8 - 34263 * q^9 - 10266 * q^10 + 3242 * q^11 - 3456 * q^12 + 10134 * q^13 - 21276 * q^15 - 293764 * q^16 - 46200 * q^17 + 20412 * q^18 - 36147 * q^19 + 391448 * q^20 - 496556 * q^22 - 322456 * q^23 - 132678 * q^24 - 547541 * q^25 + 36794 * q^26 + 39366 * q^27 - 643088 * q^29 + 75816 * q^30 - 492987 * q^31 + 634122 * q^32 - 336366 * q^33 - 12936 * q^34 + 5009688 * q^36 + 1122357 * q^37 + 763226 * q^38 - 176553 * q^39 - 309162 * q^40 - 726076 * q^41 + 3806082 * q^43 + 1367576 * q^44 - 1458 * q^45 - 1513228 * q^46 + 521298 * q^47 + 339984 * q^48 - 11073880 * q^50 + 1211220 * q^51 + 4103724 * q^52 - 4389688 * q^53 - 314928 * q^54 - 2247288 * q^55 + 9482346 * q^57 - 1914346 * q^58 + 3018948 * q^59 - 9018 * q^60 - 4334166 * q^61 - 16674372 * q^62 + 28773352 * q^64 - 13833486 * q^65 - 1360692 * q^66 + 4379809 * q^67 + 10603740 * q^68 + 4218480 * q^69 - 8791276 * q^71 + 2506302 * q^72 - 11502039 * q^73 - 972922 * q^74 - 5783373 * q^75 + 1036920 * q^76 - 18010188 * q^78 + 10846805 * q^79 - 10668412 * q^80 - 24977727 * q^81 + 7833540 * q^82 + 3911364 * q^83 + 12397584 * q^85 + 13530790 * q^86 - 11002932 * q^87 + 37506006 * q^88 - 13577072 * q^89 + 14967828 * q^90 + 49066024 * q^92 + 21714291 * q^93 + 105324 * q^94 + 28848922 * q^95 - 5227146 * q^96 + 11558244 * q^97 - 4726836 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(147, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
147.8.e.a	$147$	$8$	147.e	7.c	$3$	$2$	$1$	$45.921$	\(\Q(\sqrt{-3}) \)	None					3.8.a.a	$2$	$0$	\(-6\)	\(-27\)	\(390\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-6\zeta_{6}q^{2}+(-3^{3}+3^{3}\zeta_{6})q^{3}+(92+\cdots)q^{4}+\cdots\)
147.8.e.b	$147$	$8$	147.e	7.c	$3$	$2$	$1$	$45.921$	\(\Q(\sqrt{-3}) \)	None					3.8.a.a	$2$	$0$	\(-6\)	\(27\)	\(-390\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-6\zeta_{6}q^{2}+(3^{3}-3^{3}\zeta_{6})q^{3}+(92-92\zeta_{6})q^{4}+\cdots\)
147.8.e.c	$147$	$8$	147.e	7.c	$3$	$2$	$1$	$45.921$	\(\Q(\sqrt{-3}) \)	None					21.8.a.a	$2$	$0$	\(-2\)	\(-27\)	\(278\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-2\zeta_{6}q^{2}+(-3^{3}+3^{3}\zeta_{6})q^{3}+(124+\cdots)q^{4}+\cdots\)
147.8.e.d	$147$	$8$	147.e	7.c	$3$	$2$	$1$	$45.921$	\(\Q(\sqrt{-3}) \)	None					21.8.a.a	$2$	$0$	\(-2\)	\(27\)	\(-278\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-2\zeta_{6}q^{2}+(3^{3}-3^{3}\zeta_{6})q^{3}+(124-124\zeta_{6})q^{4}+\cdots\)
147.8.e.e	$147$	$8$	147.e	7.c	$3$	$4$	$2$	$45.921$	\(\Q(\sqrt{-3}, \sqrt{67})\)	None					21.8.a.c	$2$	$0$	\(-12\)	\(-54\)	\(-24\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-6+\beta _{1}-6\beta _{2})q^{2}+3^{3}\beta _{2}q^{3}+\cdots\)
147.8.e.f	$147$	$8$	147.e	7.c	$3$	$4$	$2$	$45.921$	\(\Q(\sqrt{-3}, \sqrt{67})\)	None					21.8.a.c	$2$	$0$	\(-12\)	\(54\)	\(24\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-6+\beta _{1}-6\beta _{2})q^{2}-3^{3}\beta _{2}q^{3}+\cdots\)
147.8.e.g	$147$	$8$	147.e	7.c	$3$	$4$	$2$	$45.921$	\(\Q(\sqrt{-3}, \sqrt{-355})\)	None					21.8.a.b	$2$	$0$	\(9\)	\(-54\)	\(-360\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(4+4\beta _{1}-\beta _{3})q^{2}+3^{3}\beta _{1}q^{3}+(154\beta _{1}+\cdots)q^{4}+\cdots\)
147.8.e.h	$147$	$8$	147.e	7.c	$3$	$4$	$2$	$45.921$	\(\Q(\sqrt{-3}, \sqrt{-355})\)	None					21.8.a.b	$2$	$0$	\(9\)	\(54\)	\(360\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(4+4\beta _{1}-\beta _{3})q^{2}-3^{3}\beta _{1}q^{3}+(154\beta _{1}+\cdots)q^{4}+\cdots\)
147.8.e.i	$147$	$8$	147.e	7.c	$3$	$6$	$3$	$45.921$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None					21.8.a.d	$2$	$0$	\(3\)	\(-81\)	\(114\)	\(0\)	$2^{2}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\beta _{1}-\beta _{3})q^{2}-3^{3}\beta _{1}q^{3}+(-75\beta _{1}+\cdots)q^{4}+\cdots\)
147.8.e.j	$147$	$8$	147.e	7.c	$3$	$6$	$3$	$45.921$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None					21.8.a.d	$2$	$0$	\(3\)	\(81\)	\(-114\)	\(0\)	$2^{2}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\beta _{1}-\beta _{3})q^{2}+3^{3}\beta _{1}q^{3}+(-75\beta _{1}+\cdots)q^{4}+\cdots\)
147.8.e.k	$147$	$8$	147.e	7.c	$3$	$8$	$4$	$45.921$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					21.8.e.a	$2$	$0$	\(1\)	\(108\)	\(196\)	\(0\)	$2^{2}\cdot 3^{3}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{6}q^{2}-3^{3}\beta _{4}q^{3}+(-2\beta _{1}+5^{2}\beta _{4}+\cdots)q^{4}+\cdots\)
147.8.e.l	$147$	$8$	147.e	7.c	$3$	$8$	$4$	$45.921$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		✓			147.8.a.f	$2$	$0$	\(15\)	\(-108\)	\(-504\)	\(0\)	$2^{4}\cdot 3^{5}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(4-4\beta _{1}+\beta _{3})q^{2}-3^{3}\beta _{1}q^{3}+(-110\beta _{1}+\cdots)q^{4}+\cdots\)
147.8.e.m	$147$	$8$	147.e	7.c	$3$	$8$	$4$	$45.921$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		✓			147.8.a.f	$2$	$0$	\(15\)	\(108\)	\(504\)	\(0\)	$2^{4}\cdot 3^{5}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(4-4\beta _{1}+\beta _{3})q^{2}+3^{3}\beta _{1}q^{3}+(-110\beta _{1}+\cdots)q^{4}+\cdots\)
147.8.e.n	$147$	$8$	147.e	7.c	$3$	$10$	$5$	$45.921$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None					21.8.e.b	$2$	$0$	\(-15\)	\(-135\)	\(-198\)	\(0\)	$2^{2}\cdot 3^{3}\cdot 7^{4}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-3-\beta _{1}-3\beta _{4})q^{2}+3^{3}\beta _{4}q^{3}+\cdots\)
147.8.e.o	$147$	$8$	147.e	7.c	$3$	$12$	$6$	$45.921$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		✓			147.8.a.l	$2$	$0$	\(14\)	\(-162\)	\(500\)	\(0\)	$2^{8}\cdot 7^{6}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2+2\beta _{1}-\beta _{5})q^{2}+3^{3}\beta _{1}q^{3}+(72\beta _{1}+\cdots)q^{4}+\cdots\)
147.8.e.p	$147$	$8$	147.e	7.c	$3$	$12$	$6$	$45.921$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		✓			147.8.a.l	$2$	$0$	\(14\)	\(162\)	\(-500\)	\(0\)	$2^{8}\cdot 7^{6}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2+2\beta _{1}-\beta _{5})q^{2}-3^{3}\beta _{1}q^{3}+(72\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(147, [\chi])\) into lower level spaces

\( S_{8}^{\mathrm{old}}(147, [\chi]) \simeq \) \(S_{8}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 2}\)