↑

Properties

Label 21.3.f
Level $21$
Weight $3$
Character orbit 21.f
Rep. character $\chi_{21}(10,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $6$
Newform subspaces $3$
Sturm bound $8$
Trace bound $2$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 21.f (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 3 \)
Sturm bound: \(8\)
Trace bound: \(2\)
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 6 8
Cusp forms 6 6 0
Eisenstein series 8 0 8

Trace form

\( 6 q - 2 q^{2} - 3 q^{3} - 2 q^{4} - 6 q^{5} - q^{7} + 8 q^{8} + 9 q^{9} + O(q^{10}) \) \( 6 q - 2 q^{2} - 3 q^{3} - 2 q^{4} - 6 q^{5} - q^{7} + 8 q^{8} + 9 q^{9} - 30 q^{10} - 14 q^{11} + 24 q^{12} + 22 q^{14} - 24 q^{15} + 22 q^{16} + 48 q^{17} + 6 q^{18} + 33 q^{19} - 36 q^{21} + 28 q^{22} - 68 q^{23} - 54 q^{24} - 9 q^{25} - 126 q^{26} - 34 q^{28} + 64 q^{29} + 48 q^{30} - 69 q^{31} + 12 q^{32} + 108 q^{33} + 114 q^{35} - 12 q^{36} + 43 q^{37} + 174 q^{38} + 9 q^{39} + 42 q^{40} - 48 q^{42} - 134 q^{43} - 108 q^{44} - 18 q^{45} + 52 q^{46} - 222 q^{47} - 27 q^{49} - 68 q^{50} + 36 q^{51} + 84 q^{52} - 32 q^{53} + 32 q^{56} - 90 q^{57} + 34 q^{58} + 84 q^{59} + 18 q^{60} + 216 q^{61} - 15 q^{63} - 28 q^{64} - 78 q^{65} - 108 q^{66} + 69 q^{67} + 36 q^{68} - 198 q^{70} + 244 q^{71} + 12 q^{72} - 129 q^{73} + 38 q^{74} + 39 q^{75} + 52 q^{77} + 204 q^{78} - 21 q^{79} + 48 q^{80} - 27 q^{81} - 144 q^{82} + 120 q^{84} - 96 q^{85} + 158 q^{86} + 54 q^{87} - 202 q^{88} - 60 q^{89} + 303 q^{91} - 168 q^{92} - 45 q^{93} - 48 q^{94} - 150 q^{95} - 234 q^{96} - 176 q^{98} - 84 q^{99} + O(q^{100}) \)

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.f.a 21.f 7.d $2$ $0.572$ \(\Q(\sqrt{-3}) \) None \(-3\) \(-3\) \(9\) \(13\) $\mathrm{SU}(2)[C_{6}]$ \(q-3\zeta_{6}q^{2}+(-1-\zeta_{6})q^{3}+(-5+5\zeta_{6})q^{4}+\cdots\)
21.3.f.b 21.f 7.d $2$ $0.572$ \(\Q(\sqrt{-3}) \) None \(-1\) \(3\) \(-9\) \(-7\) $\mathrm{SU}(2)[C_{6}]$ \(q-\zeta_{6}q^{2}+(1+\zeta_{6})q^{3}+(3-3\zeta_{6})q^{4}+\cdots\)
21.3.f.c 21.f 7.d $2$ $0.572$ \(\Q(\sqrt{-3}) \) None \(2\) \(-3\) \(-6\) \(-7\) $\mathrm{SU}(2)[C_{6}]$ \(q+2\zeta_{6}q^{2}+(-1-\zeta_{6})q^{3}+(-4+2\zeta_{6})q^{5}+\cdots\)