Properties

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 $5$

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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

\( 66 q - 4 q^{2} + 9 q^{3} - 556 q^{4} - 22 q^{5} - 144 q^{6} + 36 q^{8} - 2673 q^{9} + O(q^{10}) \) \( 66 q - 4 q^{2} + 9 q^{3} - 556 q^{4} - 22 q^{5} - 144 q^{6} + 36 q^{8} - 2673 q^{9} + 1182 q^{10} - 1070 q^{11} + 288 q^{12} - 2158 q^{13} - 396 q^{15} - 7492 q^{16} + 1848 q^{17} - 324 q^{18} + 2657 q^{19} - 1448 q^{20} - 11356 q^{22} + 7912 q^{23} + 3402 q^{24} - 25947 q^{25} - 17030 q^{26} - 1458 q^{27} - 3952 q^{29} - 9720 q^{30} + 725 q^{31} + 23130 q^{32} + 11430 q^{33} + 1176 q^{34} + 90072 q^{36} - 50017 q^{37} - 47750 q^{38} - 9081 q^{39} + 3318 q^{40} + 50812 q^{41} - 84646 q^{43} + 48952 q^{44} - 1782 q^{45} + 34996 q^{46} - 17118 q^{47} - 63216 q^{48} + 232072 q^{50} - 18108 q^{51} - 5428 q^{52} - 100376 q^{53} + 5832 q^{54} - 187944 q^{55} - 78570 q^{57} + 167662 q^{58} + 67764 q^{59} + 103158 q^{60} + 51854 q^{61} + 183420 q^{62} + 713352 q^{64} + 238302 q^{65} + 2988 q^{66} + 71125 q^{67} - 66612 q^{68} - 154224 q^{69} - 507308 q^{71} - 1458 q^{72} + 65927 q^{73} + 1078 q^{74} + 48807 q^{75} - 222184 q^{76} + 444852 q^{78} + 11397 q^{79} - 97676 q^{80} - 216513 q^{81} + 5268 q^{82} + 430740 q^{83} - 667056 q^{85} - 595258 q^{86} + 3060 q^{87} - 79722 q^{88} - 163120 q^{89} - 191484 q^{90} - 920248 q^{92} - 74079 q^{93} - 5988 q^{94} - 330934 q^{95} + 176886 q^{96} + 86924 q^{97} + 173340 q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(147, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
147.6.e.a 147.e 7.c $2$ $23.576$ \(\Q(\sqrt{-3}) \) None \(-10\) \(-9\) \(106\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-10\zeta_{6}q^{2}+(-9+9\zeta_{6})q^{3}+(-68+\cdots)q^{4}+\cdots\)
147.6.e.b 147.e 7.c $2$ $23.576$ \(\Q(\sqrt{-3}) \) None \(-10\) \(9\) \(-106\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-10\zeta_{6}q^{2}+(9-9\zeta_{6})q^{3}+(-68+68\zeta_{6})q^{4}+\cdots\)
147.6.e.c 147.e 7.c $2$ $23.576$ \(\Q(\sqrt{-3}) \) None \(-5\) \(-9\) \(-94\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-5\zeta_{6}q^{2}+(-9+9\zeta_{6})q^{3}+(7-7\zeta_{6})q^{4}+\cdots\)
147.6.e.d 147.e 7.c $2$ $23.576$ \(\Q(\sqrt{-3}) \) None \(-5\) \(9\) \(94\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-5\zeta_{6}q^{2}+(9-9\zeta_{6})q^{3}+(7-7\zeta_{6})q^{4}+\cdots\)
147.6.e.e 147.e 7.c $2$ $23.576$ \(\Q(\sqrt{-3}) \) None \(-1\) \(-9\) \(-34\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{6}q^{2}+(-9+9\zeta_{6})q^{3}+(31-31\zeta_{6})q^{4}+\cdots\)
147.6.e.f 147.e 7.c $2$ $23.576$ \(\Q(\sqrt{-3}) \) None \(-1\) \(9\) \(34\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{6}q^{2}+(9-9\zeta_{6})q^{3}+(31-31\zeta_{6})q^{4}+\cdots\)
147.6.e.g 147.e 7.c $2$ $23.576$ \(\Q(\sqrt{-3}) \) None \(2\) \(-9\) \(11\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\zeta_{6}q^{2}+(-9+9\zeta_{6})q^{3}+(28-28\zeta_{6})q^{4}+\cdots\)
147.6.e.h 147.e 7.c $2$ $23.576$ \(\Q(\sqrt{-3}) \) None \(6\) \(-9\) \(-6\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+6\zeta_{6}q^{2}+(-9+9\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
147.6.e.i 147.e 7.c $2$ $23.576$ \(\Q(\sqrt{-3}) \) None \(6\) \(-9\) \(78\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+6\zeta_{6}q^{2}+(-9+9\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
147.6.e.j 147.e 7.c $2$ $23.576$ \(\Q(\sqrt{-3}) \) None \(6\) \(9\) \(-78\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+6\zeta_{6}q^{2}+(9-9\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
147.6.e.k 147.e 7.c $2$ $23.576$ \(\Q(\sqrt{-3}) \) None \(6\) \(9\) \(6\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+6\zeta_{6}q^{2}+(9-9\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
147.6.e.l 147.e 7.c $4$ $23.576$ \(\Q(\sqrt{-3}, \sqrt{-83})\) None \(3\) \(-18\) \(-33\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{1}-\beta _{3})q^{2}+9\beta _{1}q^{3}+(31\beta _{1}+\cdots)q^{4}+\cdots\)
147.6.e.m 147.e 7.c $4$ $23.576$ \(\Q(\sqrt{-3}, \sqrt{193})\) None \(3\) \(-18\) \(72\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{1}+\beta _{2})q^{2}+(-9+9\beta _{2})q^{3}+(-20+\cdots)q^{4}+\cdots\)
147.6.e.n 147.e 7.c $4$ $23.576$ \(\Q(\sqrt{-3}, \sqrt{193})\) None \(3\) \(18\) \(-72\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{1}+\beta _{2})q^{2}+(9-9\beta _{2})q^{3}+(-20+\cdots)q^{4}+\cdots\)
147.6.e.o 147.e 7.c $8$ $23.576$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-3\) \(36\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{1}-\beta _{2})q^{2}-9\beta _{2}q^{3}+(-\beta _{1}+\cdots)q^{4}+\cdots\)
147.6.e.p 147.e 7.c $12$ $23.576$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-2\) \(-54\) \(-100\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}+(-9-9\beta _{2})q^{3}+(-5^{2}-5^{2}\beta _{2}+\cdots)q^{4}+\cdots\)
147.6.e.q 147.e 7.c $12$ $23.576$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-2\) \(54\) \(100\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}+(9+9\beta _{2})q^{3}+(-5^{2}-5^{2}\beta _{2}+\cdots)q^{4}+\cdots\)

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

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