Properties

Label 21.8.a
Level 21
Weight 8
Character orbit a
Rep. character \(\chi_{21}(1,\cdot)\)
Character field \(\Q\)
Dimension 8
Newforms 4
Sturm bound 21
Trace bound 2

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Defining parameters

Level: \( N \) = \( 21 = 3 \cdot 7 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 21.a (trivial)
Character field: \(\Q\)
Newforms: \( 4 \)
Sturm bound: \(21\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(21))\).

Total New Old
Modular forms 20 8 12
Cusp forms 16 8 8
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.
\(+\)\(+\)\(+\)\(2\)
\(+\)\(-\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(1\)
\(-\)\(-\)\(+\)\(3\)
Plus space\(+\)\(5\)
Minus space\(-\)\(3\)

Trace form

\(8q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 770q^{4} \) \(\mathstrut -\mathstrut 776q^{5} \) \(\mathstrut -\mathstrut 108q^{6} \) \(\mathstrut +\mathstrut 686q^{7} \) \(\mathstrut +\mathstrut 1854q^{8} \) \(\mathstrut +\mathstrut 5832q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 770q^{4} \) \(\mathstrut -\mathstrut 776q^{5} \) \(\mathstrut -\mathstrut 108q^{6} \) \(\mathstrut +\mathstrut 686q^{7} \) \(\mathstrut +\mathstrut 1854q^{8} \) \(\mathstrut +\mathstrut 5832q^{9} \) \(\mathstrut +\mathstrut 3852q^{10} \) \(\mathstrut +\mathstrut 1432q^{11} \) \(\mathstrut -\mathstrut 15336q^{12} \) \(\mathstrut +\mathstrut 12112q^{13} \) \(\mathstrut -\mathstrut 8918q^{14} \) \(\mathstrut -\mathstrut 216q^{15} \) \(\mathstrut +\mathstrut 92882q^{16} \) \(\mathstrut -\mathstrut 40968q^{17} \) \(\mathstrut +\mathstrut 1458q^{18} \) \(\mathstrut -\mathstrut 97424q^{19} \) \(\mathstrut +\mathstrut 572q^{20} \) \(\mathstrut +\mathstrut 18522q^{21} \) \(\mathstrut -\mathstrut 68400q^{22} \) \(\mathstrut +\mathstrut 134808q^{23} \) \(\mathstrut -\mathstrut 54756q^{24} \) \(\mathstrut +\mathstrut 138632q^{25} \) \(\mathstrut -\mathstrut 651436q^{26} \) \(\mathstrut +\mathstrut 107702q^{28} \) \(\mathstrut -\mathstrut 439328q^{29} \) \(\mathstrut +\mathstrut 367848q^{30} \) \(\mathstrut +\mathstrut 442432q^{31} \) \(\mathstrut +\mathstrut 523654q^{32} \) \(\mathstrut +\mathstrut 190296q^{33} \) \(\mathstrut -\mathstrut 935460q^{34} \) \(\mathstrut -\mathstrut 58996q^{35} \) \(\mathstrut +\mathstrut 561330q^{36} \) \(\mathstrut -\mathstrut 518624q^{37} \) \(\mathstrut +\mathstrut 444368q^{38} \) \(\mathstrut -\mathstrut 30672q^{39} \) \(\mathstrut +\mathstrut 1448676q^{40} \) \(\mathstrut +\mathstrut 1751048q^{41} \) \(\mathstrut +\mathstrut 148176q^{42} \) \(\mathstrut +\mathstrut 271216q^{43} \) \(\mathstrut -\mathstrut 3091024q^{44} \) \(\mathstrut -\mathstrut 565704q^{45} \) \(\mathstrut +\mathstrut 37512q^{46} \) \(\mathstrut +\mathstrut 1356144q^{47} \) \(\mathstrut -\mathstrut 1884816q^{48} \) \(\mathstrut +\mathstrut 941192q^{49} \) \(\mathstrut -\mathstrut 2099378q^{50} \) \(\mathstrut +\mathstrut 2017224q^{51} \) \(\mathstrut -\mathstrut 696572q^{52} \) \(\mathstrut -\mathstrut 1085904q^{53} \) \(\mathstrut -\mathstrut 78732q^{54} \) \(\mathstrut -\mathstrut 3055728q^{55} \) \(\mathstrut -\mathstrut 3825822q^{56} \) \(\mathstrut +\mathstrut 2204928q^{57} \) \(\mathstrut -\mathstrut 2019924q^{58} \) \(\mathstrut -\mathstrut 1151088q^{59} \) \(\mathstrut -\mathstrut 4629744q^{60} \) \(\mathstrut +\mathstrut 5183200q^{61} \) \(\mathstrut +\mathstrut 12950208q^{62} \) \(\mathstrut +\mathstrut 500094q^{63} \) \(\mathstrut +\mathstrut 8451050q^{64} \) \(\mathstrut +\mathstrut 3820624q^{65} \) \(\mathstrut -\mathstrut 4485672q^{66} \) \(\mathstrut -\mathstrut 6863648q^{67} \) \(\mathstrut -\mathstrut 6501060q^{68} \) \(\mathstrut -\mathstrut 4491288q^{69} \) \(\mathstrut -\mathstrut 1140132q^{70} \) \(\mathstrut -\mathstrut 3994680q^{71} \) \(\mathstrut +\mathstrut 1351566q^{72} \) \(\mathstrut -\mathstrut 10351520q^{73} \) \(\mathstrut +\mathstrut 12127020q^{74} \) \(\mathstrut +\mathstrut 3748896q^{75} \) \(\mathstrut +\mathstrut 832192q^{76} \) \(\mathstrut +\mathstrut 2118368q^{77} \) \(\mathstrut +\mathstrut 1252800q^{78} \) \(\mathstrut +\mathstrut 13872064q^{79} \) \(\mathstrut +\mathstrut 28655804q^{80} \) \(\mathstrut +\mathstrut 4251528q^{81} \) \(\mathstrut -\mathstrut 21988356q^{82} \) \(\mathstrut -\mathstrut 9046368q^{83} \) \(\mathstrut +\mathstrut 3556224q^{84} \) \(\mathstrut -\mathstrut 16470384q^{85} \) \(\mathstrut -\mathstrut 4645208q^{86} \) \(\mathstrut +\mathstrut 271296q^{87} \) \(\mathstrut +\mathstrut 14529696q^{88} \) \(\mathstrut -\mathstrut 353816q^{89} \) \(\mathstrut +\mathstrut 2808108q^{90} \) \(\mathstrut +\mathstrut 9888004q^{91} \) \(\mathstrut -\mathstrut 15951096q^{92} \) \(\mathstrut -\mathstrut 10010736q^{93} \) \(\mathstrut +\mathstrut 7381056q^{94} \) \(\mathstrut -\mathstrut 21731744q^{95} \) \(\mathstrut -\mathstrut 16331436q^{96} \) \(\mathstrut -\mathstrut 5445920q^{97} \) \(\mathstrut +\mathstrut 235298q^{98} \) \(\mathstrut +\mathstrut 1043928q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(21))\) 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
21.8.a.a \(1\) \(6.560\) \(\Q\) None \(2\) \(27\) \(-278\) \(-343\) \(-\) \(+\) \(q+2q^{2}+3^{3}q^{3}-124q^{4}-278q^{5}+\cdots\)
21.8.a.b \(2\) \(6.560\) \(\Q(\sqrt{1065}) \) None \(-9\) \(-54\) \(-360\) \(686\) \(+\) \(-\) \(q+(-4-\beta )q^{2}-3^{3}q^{3}+(154+9\beta )q^{4}+\cdots\)
21.8.a.c \(2\) \(6.560\) \(\Q(\sqrt{67}) \) None \(12\) \(-54\) \(-24\) \(-686\) \(+\) \(+\) \(q+(6+\beta )q^{2}-3^{3}q^{3}+(176+12\beta )q^{4}+\cdots\)
21.8.a.d \(3\) \(6.560\) 3.3.2910828.1 None \(-3\) \(81\) \(-114\) \(1029\) \(-\) \(-\) \(q+(-1+\beta _{1})q^{2}+3^{3}q^{3}+(75+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(21))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(21)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)