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Properties

Label 21.3.h
Level $21$
Weight $3$
Character orbit 21.h
Rep. character $\chi_{21}(2,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $6$
Newform subspaces $2$
Sturm bound $8$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 21.h (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 21 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(8\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(21, [\chi])\).

Total New Old
Modular forms 14 14 0
Cusp forms 6 6 0
Eisenstein series 8 8 0

Trace form

\( 6 q - q^{3} - 2 q^{4} - 20 q^{6} + q^{7} - 7 q^{9} + O(q^{10}) \) \( 6 q - q^{3} - 2 q^{4} - 20 q^{6} + q^{7} - 7 q^{9} - 10 q^{10} + 16 q^{12} + 38 q^{13} + 20 q^{15} + 22 q^{16} + 40 q^{18} - 43 q^{19} + 22 q^{21} - 100 q^{22} + 30 q^{24} - 65 q^{25} - 142 q^{27} - 6 q^{28} - 20 q^{30} + 19 q^{31} + 50 q^{33} + 240 q^{34} + 76 q^{36} + 49 q^{37} + 77 q^{39} - 30 q^{40} - 70 q^{42} + 54 q^{43} - 40 q^{45} - 60 q^{46} - 248 q^{48} - 27 q^{49} - 120 q^{51} - 96 q^{52} + 70 q^{54} + 100 q^{55} + 62 q^{57} - 70 q^{58} + 10 q^{60} - 22 q^{61} + 127 q^{63} - 36 q^{64} + 100 q^{66} - 91 q^{67} + 120 q^{69} + 70 q^{70} + 120 q^{72} + 61 q^{73} - 5 q^{75} + 24 q^{76} + 40 q^{78} + 147 q^{79} + 77 q^{81} - 140 q^{82} - 76 q^{84} - 240 q^{85} - 70 q^{87} + 150 q^{88} - 20 q^{90} - 327 q^{91} - 27 q^{93} + 60 q^{94} - 70 q^{96} - 368 q^{97} - 400 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(21, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
21.3.h.a 21.h 21.h $2$ $0.572$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(3\) \(0\) \(-13\) $\mathrm{U}(1)[D_{6}]$ \(q+3\zeta_{6}q^{3}-4\zeta_{6}q^{4}+(-5-3\zeta_{6})q^{7}+\cdots\)
21.3.h.b 21.h 21.h $4$ $0.572$ \(\Q(\sqrt{-3}, \sqrt{-5})\) None \(0\) \(-4\) \(0\) \(14\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{1}q^{2}+(-\beta _{1}-2\beta _{2}+\beta _{3})q^{3}+\beta _{2}q^{4}+\cdots\)