Properties

Label 21.4.e
Level 21
Weight 4
Character orbit e
Rep. character \(\chi_{21}(4,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 8
Newforms 2
Sturm bound 10
Trace bound 1

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Defining parameters

Level: \( N \) = \( 21 = 3 \cdot 7 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 21.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newforms: \( 2 \)
Sturm bound: \(10\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(21, [\chi])\).

Total New Old
Modular forms 20 8 12
Cusp forms 12 8 4
Eisenstein series 8 0 8

Trace form

\(8q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 6q^{3} \) \(\mathstrut -\mathstrut 26q^{4} \) \(\mathstrut -\mathstrut 8q^{5} \) \(\mathstrut -\mathstrut 24q^{6} \) \(\mathstrut -\mathstrut 20q^{7} \) \(\mathstrut +\mathstrut 120q^{8} \) \(\mathstrut -\mathstrut 36q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 6q^{3} \) \(\mathstrut -\mathstrut 26q^{4} \) \(\mathstrut -\mathstrut 8q^{5} \) \(\mathstrut -\mathstrut 24q^{6} \) \(\mathstrut -\mathstrut 20q^{7} \) \(\mathstrut +\mathstrut 120q^{8} \) \(\mathstrut -\mathstrut 36q^{9} \) \(\mathstrut +\mathstrut 46q^{10} \) \(\mathstrut -\mathstrut 20q^{11} \) \(\mathstrut +\mathstrut 72q^{12} \) \(\mathstrut -\mathstrut 4q^{13} \) \(\mathstrut -\mathstrut 242q^{14} \) \(\mathstrut -\mathstrut 84q^{15} \) \(\mathstrut -\mathstrut 170q^{16} \) \(\mathstrut -\mathstrut 132q^{17} \) \(\mathstrut +\mathstrut 18q^{18} \) \(\mathstrut +\mathstrut 218q^{19} \) \(\mathstrut +\mathstrut 872q^{20} \) \(\mathstrut +\mathstrut 108q^{21} \) \(\mathstrut +\mathstrut 76q^{22} \) \(\mathstrut -\mathstrut 132q^{23} \) \(\mathstrut +\mathstrut 54q^{24} \) \(\mathstrut -\mathstrut 14q^{25} \) \(\mathstrut -\mathstrut 466q^{26} \) \(\mathstrut -\mathstrut 108q^{27} \) \(\mathstrut -\mathstrut 166q^{28} \) \(\mathstrut -\mathstrut 488q^{29} \) \(\mathstrut -\mathstrut 192q^{30} \) \(\mathstrut +\mathstrut 348q^{31} \) \(\mathstrut -\mathstrut 728q^{32} \) \(\mathstrut +\mathstrut 150q^{33} \) \(\mathstrut -\mathstrut 552q^{34} \) \(\mathstrut +\mathstrut 140q^{35} \) \(\mathstrut +\mathstrut 468q^{36} \) \(\mathstrut +\mathstrut 54q^{37} \) \(\mathstrut +\mathstrut 350q^{38} \) \(\mathstrut +\mathstrut 378q^{39} \) \(\mathstrut +\mathstrut 42q^{40} \) \(\mathstrut +\mathstrut 1208q^{41} \) \(\mathstrut -\mathstrut 156q^{42} \) \(\mathstrut +\mathstrut 772q^{43} \) \(\mathstrut +\mathstrut 920q^{44} \) \(\mathstrut -\mathstrut 72q^{45} \) \(\mathstrut +\mathstrut 804q^{46} \) \(\mathstrut +\mathstrut 240q^{47} \) \(\mathstrut -\mathstrut 1872q^{48} \) \(\mathstrut -\mathstrut 940q^{49} \) \(\mathstrut -\mathstrut 2060q^{50} \) \(\mathstrut -\mathstrut 108q^{51} \) \(\mathstrut -\mathstrut 260q^{52} \) \(\mathstrut -\mathstrut 756q^{53} \) \(\mathstrut +\mathstrut 108q^{54} \) \(\mathstrut -\mathstrut 1972q^{55} \) \(\mathstrut +\mathstrut 1152q^{56} \) \(\mathstrut +\mathstrut 1116q^{57} \) \(\mathstrut +\mathstrut 358q^{58} \) \(\mathstrut -\mathstrut 1128q^{59} \) \(\mathstrut +\mathstrut 1326q^{60} \) \(\mathstrut +\mathstrut 188q^{61} \) \(\mathstrut +\mathstrut 3636q^{62} \) \(\mathstrut -\mathstrut 126q^{63} \) \(\mathstrut +\mathstrut 68q^{64} \) \(\mathstrut +\mathstrut 280q^{65} \) \(\mathstrut -\mathstrut 156q^{66} \) \(\mathstrut +\mathstrut 998q^{67} \) \(\mathstrut -\mathstrut 2028q^{68} \) \(\mathstrut -\mathstrut 1800q^{69} \) \(\mathstrut +\mathstrut 3566q^{70} \) \(\mathstrut -\mathstrut 48q^{71} \) \(\mathstrut -\mathstrut 540q^{72} \) \(\mathstrut -\mathstrut 1350q^{73} \) \(\mathstrut -\mathstrut 1950q^{74} \) \(\mathstrut +\mathstrut 738q^{75} \) \(\mathstrut -\mathstrut 4712q^{76} \) \(\mathstrut +\mathstrut 548q^{77} \) \(\mathstrut -\mathstrut 492q^{78} \) \(\mathstrut -\mathstrut 1328q^{79} \) \(\mathstrut -\mathstrut 388q^{80} \) \(\mathstrut -\mathstrut 324q^{81} \) \(\mathstrut +\mathstrut 956q^{82} \) \(\mathstrut +\mathstrut 1992q^{83} \) \(\mathstrut +\mathstrut 1536q^{84} \) \(\mathstrut +\mathstrut 3096q^{85} \) \(\mathstrut +\mathstrut 3286q^{86} \) \(\mathstrut +\mathstrut 1050q^{87} \) \(\mathstrut +\mathstrut 1206q^{88} \) \(\mathstrut -\mathstrut 2672q^{89} \) \(\mathstrut -\mathstrut 828q^{90} \) \(\mathstrut -\mathstrut 206q^{91} \) \(\mathstrut -\mathstrut 1512q^{92} \) \(\mathstrut +\mathstrut 474q^{93} \) \(\mathstrut +\mathstrut 3204q^{94} \) \(\mathstrut +\mathstrut 688q^{95} \) \(\mathstrut +\mathstrut 1914q^{96} \) \(\mathstrut +\mathstrut 1044q^{97} \) \(\mathstrut -\mathstrut 5884q^{98} \) \(\mathstrut +\mathstrut 360q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(21, [\chi])\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
21.4.e.a \(2\) \(1.239\) \(\Q(\sqrt{-3}) \) None \(3\) \(-3\) \(3\) \(-7\) \(q+(3-3\zeta_{6})q^{2}-3\zeta_{6}q^{3}-\zeta_{6}q^{4}+(3+\cdots)q^{5}+\cdots\)
21.4.e.b \(6\) \(1.239\) 6.0.9924270768.1 None \(-1\) \(9\) \(-11\) \(-13\) \(q-\beta _{1}q^{2}+(3-3\beta _{4})q^{3}+(-8+\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(21, [\chi])\) into lower level spaces

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