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Properties

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 $3$

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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

\( 94 q + 28 q^{2} - 27 q^{3} - 3436 q^{4} - 2 q^{5} + 864 q^{6} - 6876 q^{8} - 34263 q^{9} + O(q^{10}) \) \( 94 q + 28 q^{2} - 27 q^{3} - 3436 q^{4} - 2 q^{5} + 864 q^{6} - 6876 q^{8} - 34263 q^{9} - 10266 q^{10} + 3242 q^{11} - 3456 q^{12} + 10134 q^{13} - 21276 q^{15} - 293764 q^{16} - 46200 q^{17} + 20412 q^{18} - 36147 q^{19} + 391448 q^{20} - 496556 q^{22} - 322456 q^{23} - 132678 q^{24} - 547541 q^{25} + 36794 q^{26} + 39366 q^{27} - 643088 q^{29} + 75816 q^{30} - 492987 q^{31} + 634122 q^{32} - 336366 q^{33} - 12936 q^{34} + 5009688 q^{36} + 1122357 q^{37} + 763226 q^{38} - 176553 q^{39} - 309162 q^{40} - 726076 q^{41} + 3806082 q^{43} + 1367576 q^{44} - 1458 q^{45} - 1513228 q^{46} + 521298 q^{47} + 339984 q^{48} - 11073880 q^{50} + 1211220 q^{51} + 4103724 q^{52} - 4389688 q^{53} - 314928 q^{54} - 2247288 q^{55} + 9482346 q^{57} - 1914346 q^{58} + 3018948 q^{59} - 9018 q^{60} - 4334166 q^{61} - 16674372 q^{62} + 28773352 q^{64} - 13833486 q^{65} - 1360692 q^{66} + 4379809 q^{67} + 10603740 q^{68} + 4218480 q^{69} - 8791276 q^{71} + 2506302 q^{72} - 11502039 q^{73} - 972922 q^{74} - 5783373 q^{75} + 1036920 q^{76} - 18010188 q^{78} + 10846805 q^{79} - 10668412 q^{80} - 24977727 q^{81} + 7833540 q^{82} + 3911364 q^{83} + 12397584 q^{85} + 13530790 q^{86} - 11002932 q^{87} + 37506006 q^{88} - 13577072 q^{89} + 14967828 q^{90} + 49066024 q^{92} + 21714291 q^{93} + 105324 q^{94} + 28848922 q^{95} - 5227146 q^{96} + 11558244 q^{97} - 4726836 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(147, [\chi])\) into newform subspaces

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

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

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