Properties

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

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Defining parameters

Level: \( N \) = \( 21 = 3 \cdot 7 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 21.a (trivial)
Character field: \(\Q\)
Newforms: \( 3 \)
Sturm bound: \(10\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(21))\).

Total New Old
Modular forms 10 4 6
Cusp forms 6 4 2
Eisenstein series 4 0 4

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(1\)
\(-\)\(-\)\(+\)\(2\)
Plus space\(+\)\(3\)
Minus space\(-\)\(1\)

Trace form

\(4q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 26q^{4} \) \(\mathstrut -\mathstrut 16q^{5} \) \(\mathstrut -\mathstrut 12q^{6} \) \(\mathstrut +\mathstrut 14q^{7} \) \(\mathstrut -\mathstrut 66q^{8} \) \(\mathstrut +\mathstrut 36q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 26q^{4} \) \(\mathstrut -\mathstrut 16q^{5} \) \(\mathstrut -\mathstrut 12q^{6} \) \(\mathstrut +\mathstrut 14q^{7} \) \(\mathstrut -\mathstrut 66q^{8} \) \(\mathstrut +\mathstrut 36q^{9} \) \(\mathstrut -\mathstrut 28q^{10} \) \(\mathstrut +\mathstrut 20q^{11} \) \(\mathstrut +\mathstrut 24q^{12} \) \(\mathstrut -\mathstrut 80q^{13} \) \(\mathstrut -\mathstrut 70q^{14} \) \(\mathstrut +\mathstrut 84q^{15} \) \(\mathstrut +\mathstrut 2q^{16} \) \(\mathstrut +\mathstrut 120q^{17} \) \(\mathstrut -\mathstrut 18q^{18} \) \(\mathstrut +\mathstrut 40q^{19} \) \(\mathstrut +\mathstrut 172q^{20} \) \(\mathstrut +\mathstrut 42q^{21} \) \(\mathstrut +\mathstrut 80q^{22} \) \(\mathstrut -\mathstrut 36q^{23} \) \(\mathstrut -\mathstrut 324q^{24} \) \(\mathstrut -\mathstrut 28q^{25} \) \(\mathstrut +\mathstrut 172q^{26} \) \(\mathstrut +\mathstrut 70q^{28} \) \(\mathstrut -\mathstrut 160q^{29} \) \(\mathstrut -\mathstrut 312q^{30} \) \(\mathstrut -\mathstrut 168q^{31} \) \(\mathstrut -\mathstrut 490q^{32} \) \(\mathstrut -\mathstrut 96q^{33} \) \(\mathstrut +\mathstrut 276q^{34} \) \(\mathstrut -\mathstrut 56q^{35} \) \(\mathstrut +\mathstrut 234q^{36} \) \(\mathstrut -\mathstrut 96q^{37} \) \(\mathstrut +\mathstrut 1360q^{38} \) \(\mathstrut +\mathstrut 336q^{39} \) \(\mathstrut -\mathstrut 924q^{40} \) \(\mathstrut -\mathstrut 1016q^{41} \) \(\mathstrut +\mathstrut 84q^{42} \) \(\mathstrut +\mathstrut 176q^{43} \) \(\mathstrut +\mathstrut 1264q^{44} \) \(\mathstrut -\mathstrut 144q^{45} \) \(\mathstrut +\mathstrut 792q^{46} \) \(\mathstrut +\mathstrut 552q^{47} \) \(\mathstrut +\mathstrut 816q^{48} \) \(\mathstrut +\mathstrut 196q^{49} \) \(\mathstrut -\mathstrut 1198q^{50} \) \(\mathstrut -\mathstrut 396q^{51} \) \(\mathstrut -\mathstrut 1420q^{52} \) \(\mathstrut -\mathstrut 1344q^{53} \) \(\mathstrut -\mathstrut 108q^{54} \) \(\mathstrut +\mathstrut 952q^{55} \) \(\mathstrut -\mathstrut 462q^{56} \) \(\mathstrut +\mathstrut 264q^{57} \) \(\mathstrut -\mathstrut 652q^{58} \) \(\mathstrut +\mathstrut 792q^{59} \) \(\mathstrut +\mathstrut 816q^{60} \) \(\mathstrut +\mathstrut 592q^{61} \) \(\mathstrut -\mathstrut 1824q^{62} \) \(\mathstrut +\mathstrut 126q^{63} \) \(\mathstrut +\mathstrut 1210q^{64} \) \(\mathstrut +\mathstrut 224q^{65} \) \(\mathstrut -\mathstrut 1896q^{66} \) \(\mathstrut +\mathstrut 1144q^{67} \) \(\mathstrut +\mathstrut 492q^{68} \) \(\mathstrut +\mathstrut 144q^{69} \) \(\mathstrut +\mathstrut 28q^{70} \) \(\mathstrut -\mathstrut 444q^{71} \) \(\mathstrut -\mathstrut 594q^{72} \) \(\mathstrut -\mathstrut 72q^{73} \) \(\mathstrut -\mathstrut 780q^{74} \) \(\mathstrut -\mathstrut 624q^{75} \) \(\mathstrut -\mathstrut 832q^{76} \) \(\mathstrut -\mathstrut 728q^{77} \) \(\mathstrut +\mathstrut 1392q^{78} \) \(\mathstrut -\mathstrut 856q^{79} \) \(\mathstrut +\mathstrut 3484q^{80} \) \(\mathstrut +\mathstrut 324q^{81} \) \(\mathstrut +\mathstrut 1492q^{82} \) \(\mathstrut +\mathstrut 2208q^{83} \) \(\mathstrut +\mathstrut 504q^{84} \) \(\mathstrut -\mathstrut 1224q^{85} \) \(\mathstrut +\mathstrut 1880q^{86} \) \(\mathstrut -\mathstrut 1032q^{87} \) \(\mathstrut -\mathstrut 1920q^{88} \) \(\mathstrut +\mathstrut 920q^{89} \) \(\mathstrut -\mathstrut 252q^{90} \) \(\mathstrut +\mathstrut 308q^{91} \) \(\mathstrut -\mathstrut 3192q^{92} \) \(\mathstrut +\mathstrut 744q^{93} \) \(\mathstrut -\mathstrut 1344q^{94} \) \(\mathstrut +\mathstrut 656q^{95} \) \(\mathstrut -\mathstrut 204q^{96} \) \(\mathstrut -\mathstrut 744q^{97} \) \(\mathstrut -\mathstrut 98q^{98} \) \(\mathstrut +\mathstrut 180q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(21))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 7
21.4.a.a \(1\) \(1.239\) \(\Q\) None \(-3\) \(-3\) \(-18\) \(7\) \(+\) \(-\) \(q-3q^{2}-3q^{3}+q^{4}-18q^{5}+9q^{6}+\cdots\)
21.4.a.b \(1\) \(1.239\) \(\Q\) None \(4\) \(-3\) \(-4\) \(-7\) \(+\) \(+\) \(q+4q^{2}-3q^{3}+8q^{4}-4q^{5}-12q^{6}+\cdots\)
21.4.a.c \(2\) \(1.239\) \(\Q(\sqrt{57}) \) None \(-3\) \(6\) \(6\) \(14\) \(-\) \(-\) \(q+(-1-\beta )q^{2}+3q^{3}+(7+3\beta )q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(21))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(21)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)