Properties

Label 21.10.a
Level 21
Weight 10
Character orbit a
Rep. character \(\chi_{21}(1,\cdot)\)
Character field \(\Q\)
Dimension 8
Newforms 4
Sturm bound 26
Trace bound 2

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Defining parameters

Level: \( N \) = \( 21 = 3 \cdot 7 \)
Weight: \( k \) = \( 10 \)
Character orbit: \([\chi]\) = 21.a (trivial)
Character field: \(\Q\)
Newforms: \( 4 \)
Sturm bound: \(26\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(21))\).

Total New Old
Modular forms 26 8 18
Cusp forms 22 8 14
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.
\(+\)\(+\)\(+\)\(2\)
\(+\)\(-\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(3\)
\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(3\)
Minus space\(-\)\(5\)

Trace form

\(8q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 2050q^{4} \) \(\mathstrut +\mathstrut 4552q^{5} \) \(\mathstrut -\mathstrut 6156q^{6} \) \(\mathstrut -\mathstrut 4802q^{7} \) \(\mathstrut +\mathstrut 9390q^{8} \) \(\mathstrut +\mathstrut 52488q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 2050q^{4} \) \(\mathstrut +\mathstrut 4552q^{5} \) \(\mathstrut -\mathstrut 6156q^{6} \) \(\mathstrut -\mathstrut 4802q^{7} \) \(\mathstrut +\mathstrut 9390q^{8} \) \(\mathstrut +\mathstrut 52488q^{9} \) \(\mathstrut +\mathstrut 16108q^{10} \) \(\mathstrut -\mathstrut 78140q^{11} \) \(\mathstrut +\mathstrut 115992q^{12} \) \(\mathstrut +\mathstrut 279176q^{13} \) \(\mathstrut +\mathstrut 24010q^{14} \) \(\mathstrut -\mathstrut 3564q^{15} \) \(\mathstrut +\mathstrut 214546q^{16} \) \(\mathstrut -\mathstrut 378912q^{17} \) \(\mathstrut +\mathstrut 13122q^{18} \) \(\mathstrut +\mathstrut 586184q^{19} \) \(\mathstrut +\mathstrut 1411100q^{20} \) \(\mathstrut -\mathstrut 388962q^{21} \) \(\mathstrut +\mathstrut 2396416q^{22} \) \(\mathstrut +\mathstrut 725724q^{23} \) \(\mathstrut -\mathstrut 1495908q^{24} \) \(\mathstrut -\mathstrut 2577248q^{25} \) \(\mathstrut +\mathstrut 957428q^{26} \) \(\mathstrut -\mathstrut 4057690q^{28} \) \(\mathstrut +\mathstrut 8576440q^{29} \) \(\mathstrut +\mathstrut 3227688q^{30} \) \(\mathstrut -\mathstrut 8356584q^{31} \) \(\mathstrut -\mathstrut 45299498q^{32} \) \(\mathstrut +\mathstrut 5409504q^{33} \) \(\mathstrut +\mathstrut 28069404q^{34} \) \(\mathstrut -\mathstrut 6204184q^{35} \) \(\mathstrut +\mathstrut 13450050q^{36} \) \(\mathstrut -\mathstrut 37232136q^{37} \) \(\mathstrut -\mathstrut 30025936q^{38} \) \(\mathstrut -\mathstrut 14262480q^{39} \) \(\mathstrut +\mathstrut 52787556q^{40} \) \(\mathstrut +\mathstrut 13951376q^{41} \) \(\mathstrut -\mathstrut 6223392q^{42} \) \(\mathstrut +\mathstrut 50242720q^{43} \) \(\mathstrut -\mathstrut 42073024q^{44} \) \(\mathstrut +\mathstrut 29865672q^{45} \) \(\mathstrut -\mathstrut 98920632q^{46} \) \(\mathstrut +\mathstrut 62344392q^{47} \) \(\mathstrut +\mathstrut 92730096q^{48} \) \(\mathstrut +\mathstrut 46118408q^{49} \) \(\mathstrut +\mathstrut 73994302q^{50} \) \(\mathstrut +\mathstrut 23031540q^{51} \) \(\mathstrut +\mathstrut 231982372q^{52} \) \(\mathstrut -\mathstrut 217936680q^{53} \) \(\mathstrut -\mathstrut 40389516q^{54} \) \(\mathstrut +\mathstrut 130559768q^{55} \) \(\mathstrut +\mathstrut 63083874q^{56} \) \(\mathstrut +\mathstrut 5276664q^{57} \) \(\mathstrut -\mathstrut 443463044q^{58} \) \(\mathstrut +\mathstrut 69912216q^{59} \) \(\mathstrut -\mathstrut 29254608q^{60} \) \(\mathstrut -\mathstrut 85792600q^{61} \) \(\mathstrut -\mathstrut 308107200q^{62} \) \(\mathstrut -\mathstrut 31505922q^{63} \) \(\mathstrut -\mathstrut 56923702q^{64} \) \(\mathstrut +\mathstrut 246963088q^{65} \) \(\mathstrut +\mathstrut 35066520q^{66} \) \(\mathstrut +\mathstrut 179326184q^{67} \) \(\mathstrut -\mathstrut 853640772q^{68} \) \(\mathstrut +\mathstrut 547698672q^{69} \) \(\mathstrut +\mathstrut 291106844q^{70} \) \(\mathstrut +\mathstrut 151882212q^{71} \) \(\mathstrut +\mathstrut 61607790q^{72} \) \(\mathstrut +\mathstrut 781362720q^{73} \) \(\mathstrut -\mathstrut 758102628q^{74} \) \(\mathstrut -\mathstrut 638703792q^{75} \) \(\mathstrut -\mathstrut 219853280q^{76} \) \(\mathstrut +\mathstrut 467349848q^{77} \) \(\mathstrut -\mathstrut 408004128q^{78} \) \(\mathstrut +\mathstrut 75820360q^{79} \) \(\mathstrut -\mathstrut 225047332q^{80} \) \(\mathstrut +\mathstrut 344373768q^{81} \) \(\mathstrut -\mathstrut 1050294052q^{82} \) \(\mathstrut -\mathstrut 167217024q^{83} \) \(\mathstrut -\mathstrut 298722816q^{84} \) \(\mathstrut -\mathstrut 2054101752q^{85} \) \(\mathstrut +\mathstrut 3203864920q^{86} \) \(\mathstrut -\mathstrut 472571496q^{87} \) \(\mathstrut +\mathstrut 1195436784q^{88} \) \(\mathstrut +\mathstrut 473386912q^{89} \) \(\mathstrut +\mathstrut 105684588q^{90} \) \(\mathstrut -\mathstrut 720175148q^{91} \) \(\mathstrut +\mathstrut 3191894664q^{92} \) \(\mathstrut +\mathstrut 89676072q^{93} \) \(\mathstrut +\mathstrut 130819488q^{94} \) \(\mathstrut +\mathstrut 1028816560q^{95} \) \(\mathstrut -\mathstrut 374635692q^{96} \) \(\mathstrut +\mathstrut 2488240800q^{97} \) \(\mathstrut +\mathstrut 11529602q^{98} \) \(\mathstrut -\mathstrut 512676540q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(21))\) 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
21.10.a.a \(1\) \(10.816\) \(\Q\) None \(-24\) \(81\) \(-144\) \(2401\) \(-\) \(-\) \(q-24q^{2}+3^{4}q^{3}+2^{6}q^{4}-12^{2}q^{5}+\cdots\)
21.10.a.b \(2\) \(10.816\) \(\Q(\sqrt{2353}) \) None \(9\) \(-162\) \(1170\) \(-4802\) \(+\) \(+\) \(q+(5-\beta )q^{2}-3^{4}q^{3}+(101-9\beta )q^{4}+\cdots\)
21.10.a.c \(2\) \(10.816\) \(\Q(\sqrt{345}) \) None \(30\) \(-162\) \(1128\) \(4802\) \(+\) \(-\) \(q+(15-\beta )q^{2}-3^{4}q^{3}+(58-30\beta )q^{4}+\cdots\)
21.10.a.d \(3\) \(10.816\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-13\) \(243\) \(2398\) \(-7203\) \(-\) \(+\) \(q+(-4-\beta _{1})q^{2}+3^{4}q^{3}+(555+7\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(21))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(21)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)