Properties

Label 21.6.g
Level $21$
Weight $6$
Character orbit 21.g
Rep. character $\chi_{21}(5,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $22$
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.g (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 21 \)
Character field: \(\Q(\zeta_{6})\)
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 30 0
Cusp forms 22 22 0
Eisenstein series 8 8 0

Trace form

\( 22 q - 3 q^{3} + 126 q^{4} + 85 q^{7} - 81 q^{9} + O(q^{10}) \) \( 22 q - 3 q^{3} + 126 q^{4} + 85 q^{7} - 81 q^{9} - 750 q^{10} + 1536 q^{12} + 984 q^{15} - 2122 q^{16} - 3702 q^{18} + 1095 q^{19} - 4500 q^{21} - 7076 q^{22} + 9558 q^{24} + 2695 q^{25} + 29982 q^{28} + 2082 q^{30} - 21909 q^{31} + 28674 q^{33} - 10380 q^{36} - 4915 q^{37} - 14121 q^{39} - 61590 q^{40} - 66990 q^{42} + 47126 q^{43} - 26478 q^{45} + 19000 q^{46} + 12181 q^{49} + 48528 q^{51} + 124992 q^{52} + 114534 q^{54} - 6246 q^{57} - 85022 q^{58} + 56394 q^{60} - 219216 q^{61} - 74409 q^{63} + 68188 q^{64} - 338202 q^{66} - 67027 q^{67} + 204330 q^{70} + 343920 q^{72} + 376755 q^{73} + 365025 q^{75} - 400728 q^{78} - 384343 q^{79} + 158787 q^{81} - 540288 q^{82} - 191904 q^{84} + 239016 q^{85} - 616896 q^{87} - 329866 q^{88} + 149025 q^{91} + 419337 q^{93} + 1023252 q^{94} + 1249542 q^{96} - 140304 q^{99} + O(q^{100}) \)

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.g.a 21.g 21.g $2$ $3.368$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(27\) \(0\) \(211\) $\mathrm{U}(1)[D_{6}]$ \(q+(9+9\zeta_{6})q^{3}+(-2^{5}+2^{5}\zeta_{6})q^{4}+\cdots\)
21.6.g.b 21.g 21.g $4$ $3.368$ \(\Q(\sqrt{-3}, \sqrt{-17})\) None \(0\) \(-48\) \(0\) \(490\) $\mathrm{SU}(2)[C_{6}]$ \(q+2\beta _{1}q^{2}+(-2^{4}+\beta _{1}+8\beta _{2}+\beta _{3})q^{3}+\cdots\)
21.6.g.c 21.g 21.g $16$ $3.368$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(18\) \(0\) \(-616\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{1}q^{2}+(1-\beta _{1}+\beta _{2}-\beta _{6})q^{3}+(12+\cdots)q^{4}+\cdots\)