Properties

Label 21.6.a
Level 21
Weight 6
Character orbit a
Rep. character \(\chi_{21}(1,\cdot)\)
Character field \(\Q\)
Dimension 4
Newforms 4
Sturm bound 16
Trace bound 2

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Defining parameters

Level: \( N \) = \( 21 = 3 \cdot 7 \)
Weight: \( k \) = \( 6 \)
Character orbit: \([\chi]\) = 21.a (trivial)
Character field: \(\Q\)
Newforms: \( 4 \)
Sturm bound: \(16\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(21))\).

Total New Old
Modular forms 16 4 12
Cusp forms 12 4 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.
\(+\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(2\)
Plus space\(+\)\(1\)
Minus space\(-\)\(3\)

Trace form

\(4q \) \(\mathstrut +\mathstrut 10q^{2} \) \(\mathstrut +\mathstrut 34q^{4} \) \(\mathstrut +\mathstrut 32q^{5} \) \(\mathstrut +\mathstrut 180q^{6} \) \(\mathstrut -\mathstrut 98q^{7} \) \(\mathstrut +\mathstrut 270q^{8} \) \(\mathstrut +\mathstrut 324q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 10q^{2} \) \(\mathstrut +\mathstrut 34q^{4} \) \(\mathstrut +\mathstrut 32q^{5} \) \(\mathstrut +\mathstrut 180q^{6} \) \(\mathstrut -\mathstrut 98q^{7} \) \(\mathstrut +\mathstrut 270q^{8} \) \(\mathstrut +\mathstrut 324q^{9} \) \(\mathstrut -\mathstrut 1092q^{10} \) \(\mathstrut +\mathstrut 248q^{11} \) \(\mathstrut +\mathstrut 792q^{12} \) \(\mathstrut -\mathstrut 88q^{13} \) \(\mathstrut -\mathstrut 1078q^{14} \) \(\mathstrut -\mathstrut 504q^{15} \) \(\mathstrut +\mathstrut 466q^{16} \) \(\mathstrut -\mathstrut 3168q^{17} \) \(\mathstrut +\mathstrut 810q^{18} \) \(\mathstrut +\mathstrut 4400q^{19} \) \(\mathstrut -\mathstrut 6500q^{20} \) \(\mathstrut -\mathstrut 882q^{21} \) \(\mathstrut -\mathstrut 1824q^{22} \) \(\mathstrut -\mathstrut 744q^{23} \) \(\mathstrut +\mathstrut 540q^{24} \) \(\mathstrut +\mathstrut 14812q^{25} \) \(\mathstrut +\mathstrut 5956q^{26} \) \(\mathstrut -\mathstrut 1274q^{28} \) \(\mathstrut -\mathstrut 7048q^{29} \) \(\mathstrut -\mathstrut 792q^{30} \) \(\mathstrut +\mathstrut 7616q^{31} \) \(\mathstrut +\mathstrut 9590q^{32} \) \(\mathstrut +\mathstrut 360q^{33} \) \(\mathstrut -\mathstrut 12372q^{34} \) \(\mathstrut +\mathstrut 6076q^{35} \) \(\mathstrut +\mathstrut 2754q^{36} \) \(\mathstrut +\mathstrut 1208q^{37} \) \(\mathstrut -\mathstrut 15632q^{38} \) \(\mathstrut -\mathstrut 1008q^{39} \) \(\mathstrut -\mathstrut 41244q^{40} \) \(\mathstrut +\mathstrut 17872q^{41} \) \(\mathstrut -\mathstrut 3528q^{42} \) \(\mathstrut -\mathstrut 27040q^{43} \) \(\mathstrut +\mathstrut 18208q^{44} \) \(\mathstrut +\mathstrut 2592q^{45} \) \(\mathstrut -\mathstrut 43032q^{46} \) \(\mathstrut -\mathstrut 11184q^{47} \) \(\mathstrut +\mathstrut 7920q^{48} \) \(\mathstrut +\mathstrut 9604q^{49} \) \(\mathstrut +\mathstrut 89942q^{50} \) \(\mathstrut -\mathstrut 11880q^{51} \) \(\mathstrut +\mathstrut 35108q^{52} \) \(\mathstrut -\mathstrut 42840q^{53} \) \(\mathstrut +\mathstrut 14580q^{54} \) \(\mathstrut +\mathstrut 41328q^{55} \) \(\mathstrut +\mathstrut 3234q^{56} \) \(\mathstrut -\mathstrut 24768q^{57} \) \(\mathstrut +\mathstrut 63180q^{58} \) \(\mathstrut +\mathstrut 6576q^{59} \) \(\mathstrut -\mathstrut 83088q^{60} \) \(\mathstrut +\mathstrut 7832q^{61} \) \(\mathstrut +\mathstrut 65664q^{62} \) \(\mathstrut -\mathstrut 7938q^{63} \) \(\mathstrut +\mathstrut 19018q^{64} \) \(\mathstrut -\mathstrut 193312q^{65} \) \(\mathstrut +\mathstrut 37656q^{66} \) \(\mathstrut +\mathstrut 5360q^{67} \) \(\mathstrut +\mathstrut 23292q^{68} \) \(\mathstrut -\mathstrut 24840q^{69} \) \(\mathstrut +\mathstrut 7644q^{70} \) \(\mathstrut +\mathstrut 4008q^{71} \) \(\mathstrut +\mathstrut 21870q^{72} \) \(\mathstrut +\mathstrut 101816q^{73} \) \(\mathstrut +\mathstrut 47532q^{74} \) \(\mathstrut +\mathstrut 115488q^{75} \) \(\mathstrut -\mathstrut 90784q^{76} \) \(\mathstrut +\mathstrut 31360q^{77} \) \(\mathstrut -\mathstrut 2304q^{78} \) \(\mathstrut -\mathstrut 58720q^{79} \) \(\mathstrut -\mathstrut 341732q^{80} \) \(\mathstrut +\mathstrut 26244q^{81} \) \(\mathstrut -\mathstrut 110868q^{82} \) \(\mathstrut -\mathstrut 6912q^{83} \) \(\mathstrut -\mathstrut 42336q^{84} \) \(\mathstrut -\mathstrut 149232q^{85} \) \(\mathstrut +\mathstrut 42008q^{86} \) \(\mathstrut +\mathstrut 182880q^{87} \) \(\mathstrut +\mathstrut 118992q^{88} \) \(\mathstrut -\mathstrut 16624q^{89} \) \(\mathstrut -\mathstrut 88452q^{90} \) \(\mathstrut -\mathstrut 39004q^{91} \) \(\mathstrut +\mathstrut 39816q^{92} \) \(\mathstrut +\mathstrut 112032q^{93} \) \(\mathstrut +\mathstrut 26208q^{94} \) \(\mathstrut +\mathstrut 438080q^{95} \) \(\mathstrut +\mathstrut 7380q^{96} \) \(\mathstrut -\mathstrut 279592q^{97} \) \(\mathstrut +\mathstrut 24010q^{98} \) \(\mathstrut +\mathstrut 20088q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(21))\) 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
21.6.a.a \(1\) \(3.368\) \(\Q\) None \(-6\) \(-9\) \(78\) \(49\) \(+\) \(-\) \(q-6q^{2}-9q^{3}+4q^{4}+78q^{5}+54q^{6}+\cdots\)
21.6.a.b \(1\) \(3.368\) \(\Q\) None \(1\) \(-9\) \(-34\) \(-49\) \(+\) \(+\) \(q+q^{2}-9q^{3}-31q^{4}-34q^{5}-9q^{6}+\cdots\)
21.6.a.c \(1\) \(3.368\) \(\Q\) None \(5\) \(9\) \(94\) \(-49\) \(-\) \(+\) \(q+5q^{2}+9q^{3}-7q^{4}+94q^{5}+45q^{6}+\cdots\)
21.6.a.d \(1\) \(3.368\) \(\Q\) None \(10\) \(9\) \(-106\) \(-49\) \(-\) \(+\) \(q+10q^{2}+9q^{3}+68q^{4}-106q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(21))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(21)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)