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Properties

Label 21.6.e
Level $21$
Weight $6$
Character orbit 21.e
Rep. character $\chi_{21}(4,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $14$
Newform subspaces $3$
Sturm bound $16$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 21.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 3 \)
Sturm bound: \(16\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(21, [\chi])\).

Total New Old
Modular forms 30 14 16
Cusp forms 22 14 8
Eisenstein series 8 0 8

Trace form

\( 14 q + 2 q^{2} - 9 q^{3} - 106 q^{4} + 22 q^{5} + 144 q^{6} + 167 q^{7} - 264 q^{8} - 567 q^{9} + O(q^{10}) \) \( 14 q + 2 q^{2} - 9 q^{3} - 106 q^{4} + 22 q^{5} + 144 q^{6} + 167 q^{7} - 264 q^{8} - 567 q^{9} - 1182 q^{10} + 466 q^{11} - 288 q^{12} + 2158 q^{13} + 4558 q^{14} + 396 q^{15} - 3034 q^{16} - 1848 q^{17} + 162 q^{18} - 2657 q^{19} + 1448 q^{20} - 4068 q^{21} + 4836 q^{22} - 5232 q^{23} - 3402 q^{24} - 205 q^{25} + 17030 q^{26} + 1458 q^{27} - 9358 q^{28} + 1480 q^{29} + 5544 q^{30} - 725 q^{31} - 11384 q^{32} - 11430 q^{33} - 1176 q^{34} - 19234 q^{35} + 17172 q^{36} + 15187 q^{37} + 47750 q^{38} + 1395 q^{39} - 3318 q^{40} - 50812 q^{41} - 16344 q^{42} + 37066 q^{43} - 26632 q^{44} + 1782 q^{45} - 27588 q^{46} + 17118 q^{47} + 63216 q^{48} - 36763 q^{49} - 61292 q^{50} + 11664 q^{51} + 5428 q^{52} + 47652 q^{53} - 5832 q^{54} + 187944 q^{55} + 80784 q^{56} - 29466 q^{57} - 45198 q^{58} - 67764 q^{59} - 78426 q^{60} - 51854 q^{61} - 183420 q^{62} + 14013 q^{63} - 228412 q^{64} - 40370 q^{65} - 2988 q^{66} + 7309 q^{67} + 66612 q^{68} + 154224 q^{69} + 149490 q^{70} + 306180 q^{71} + 10692 q^{72} - 65927 q^{73} + 240570 q^{74} - 48807 q^{75} + 222184 q^{76} - 204544 q^{77} - 238428 q^{78} + 48997 q^{79} + 97676 q^{80} - 45927 q^{81} - 5268 q^{82} - 430740 q^{83} + 47160 q^{84} - 132912 q^{85} - 137330 q^{86} - 3060 q^{87} - 332442 q^{88} + 163120 q^{89} + 191484 q^{90} - 136925 q^{91} + 833112 q^{92} + 121905 q^{93} + 5988 q^{94} + 284842 q^{95} - 176886 q^{96} - 86924 q^{97} - 40636 q^{98} - 75492 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(21, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
21.6.e.a 21.e 7.c $2$ $3.368$ \(\Q(\sqrt{-3}) \) None \(2\) \(9\) \(-11\) \(259\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{2}+9\zeta_{6}q^{3}+28\zeta_{6}q^{4}+\cdots\)
21.6.e.b 21.e 7.c $4$ $3.368$ \(\Q(\sqrt{-3}, \sqrt{-83})\) None \(3\) \(18\) \(33\) \(-350\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{1}-\beta _{3})q^{2}-9\beta _{1}q^{3}+(31\beta _{1}+\cdots)q^{4}+\cdots\)
21.6.e.c 21.e 7.c $8$ $3.368$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-3\) \(-36\) \(0\) \(258\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{1}-\beta _{2})q^{2}+9\beta _{2}q^{3}+(-\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(21, [\chi])\) into lower level spaces

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