Properties

Label 10.6.a
Level 10
Weight 6
Character orbit a
Rep. character \(\chi_{10}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newforms 3
Sturm bound 9
Trace bound 3

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Defining parameters

Level: \( N \) = \( 10 = 2 \cdot 5 \)
Weight: \( k \) = \( 6 \)
Character orbit: \([\chi]\) = 10.a (trivial)
Character field: \(\Q\)
Newforms: \( 3 \)
Sturm bound: \(9\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(10))\).

Total New Old
Modular forms 9 3 6
Cusp forms 5 3 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.

\(2\)\(5\)FrickeDim.
\(+\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(1\)
Plus space\(+\)\(1\)
Minus space\(-\)\(2\)

Trace form

\(3q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 48q^{4} \) \(\mathstrut -\mathstrut 25q^{5} \) \(\mathstrut +\mathstrut 32q^{6} \) \(\mathstrut -\mathstrut 312q^{7} \) \(\mathstrut -\mathstrut 64q^{8} \) \(\mathstrut +\mathstrut 559q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 48q^{4} \) \(\mathstrut -\mathstrut 25q^{5} \) \(\mathstrut +\mathstrut 32q^{6} \) \(\mathstrut -\mathstrut 312q^{7} \) \(\mathstrut -\mathstrut 64q^{8} \) \(\mathstrut +\mathstrut 559q^{9} \) \(\mathstrut -\mathstrut 100q^{10} \) \(\mathstrut -\mathstrut 444q^{11} \) \(\mathstrut +\mathstrut 64q^{12} \) \(\mathstrut +\mathstrut 114q^{13} \) \(\mathstrut +\mathstrut 304q^{14} \) \(\mathstrut +\mathstrut 1100q^{15} \) \(\mathstrut +\mathstrut 768q^{16} \) \(\mathstrut +\mathstrut 918q^{17} \) \(\mathstrut -\mathstrut 3892q^{18} \) \(\mathstrut -\mathstrut 1140q^{19} \) \(\mathstrut -\mathstrut 400q^{20} \) \(\mathstrut -\mathstrut 4264q^{21} \) \(\mathstrut +\mathstrut 3312q^{22} \) \(\mathstrut +\mathstrut 3144q^{23} \) \(\mathstrut +\mathstrut 512q^{24} \) \(\mathstrut +\mathstrut 1875q^{25} \) \(\mathstrut +\mathstrut 8392q^{26} \) \(\mathstrut -\mathstrut 5480q^{27} \) \(\mathstrut -\mathstrut 4992q^{28} \) \(\mathstrut +\mathstrut 2490q^{29} \) \(\mathstrut -\mathstrut 5600q^{30} \) \(\mathstrut -\mathstrut 8544q^{31} \) \(\mathstrut -\mathstrut 1024q^{32} \) \(\mathstrut +\mathstrut 24288q^{33} \) \(\mathstrut +\mathstrut 2424q^{34} \) \(\mathstrut -\mathstrut 800q^{35} \) \(\mathstrut +\mathstrut 8944q^{36} \) \(\mathstrut -\mathstrut 16902q^{37} \) \(\mathstrut -\mathstrut 17360q^{38} \) \(\mathstrut -\mathstrut 14872q^{39} \) \(\mathstrut -\mathstrut 1600q^{40} \) \(\mathstrut +\mathstrut 2406q^{41} \) \(\mathstrut +\mathstrut 11392q^{42} \) \(\mathstrut +\mathstrut 13164q^{43} \) \(\mathstrut -\mathstrut 7104q^{44} \) \(\mathstrut +\mathstrut 2675q^{45} \) \(\mathstrut -\mathstrut 48q^{46} \) \(\mathstrut +\mathstrut 44448q^{47} \) \(\mathstrut +\mathstrut 1024q^{48} \) \(\mathstrut -\mathstrut 6429q^{49} \) \(\mathstrut -\mathstrut 2500q^{50} \) \(\mathstrut -\mathstrut 10584q^{51} \) \(\mathstrut +\mathstrut 1824q^{52} \) \(\mathstrut -\mathstrut 44406q^{53} \) \(\mathstrut +\mathstrut 320q^{54} \) \(\mathstrut +\mathstrut 17700q^{55} \) \(\mathstrut +\mathstrut 4864q^{56} \) \(\mathstrut -\mathstrut 33040q^{57} \) \(\mathstrut +\mathstrut 37320q^{58} \) \(\mathstrut +\mathstrut 1380q^{59} \) \(\mathstrut +\mathstrut 17600q^{60} \) \(\mathstrut +\mathstrut 12306q^{61} \) \(\mathstrut -\mathstrut 20768q^{62} \) \(\mathstrut -\mathstrut 42376q^{63} \) \(\mathstrut +\mathstrut 12288q^{64} \) \(\mathstrut -\mathstrut 50150q^{65} \) \(\mathstrut -\mathstrut 87936q^{66} \) \(\mathstrut +\mathstrut 10788q^{67} \) \(\mathstrut +\mathstrut 14688q^{68} \) \(\mathstrut +\mathstrut 146568q^{69} \) \(\mathstrut +\mathstrut 26800q^{70} \) \(\mathstrut +\mathstrut 88056q^{71} \) \(\mathstrut -\mathstrut 62272q^{72} \) \(\mathstrut -\mathstrut 9666q^{73} \) \(\mathstrut +\mathstrut 23464q^{74} \) \(\mathstrut +\mathstrut 2500q^{75} \) \(\mathstrut -\mathstrut 18240q^{76} \) \(\mathstrut -\mathstrut 28464q^{77} \) \(\mathstrut +\mathstrut 112576q^{78} \) \(\mathstrut -\mathstrut 81360q^{79} \) \(\mathstrut -\mathstrut 6400q^{80} \) \(\mathstrut +\mathstrut 28243q^{81} \) \(\mathstrut -\mathstrut 12648q^{82} \) \(\mathstrut -\mathstrut 220476q^{83} \) \(\mathstrut -\mathstrut 68224q^{84} \) \(\mathstrut -\mathstrut 34050q^{85} \) \(\mathstrut -\mathstrut 72128q^{86} \) \(\mathstrut +\mathstrut 233880q^{87} \) \(\mathstrut +\mathstrut 52992q^{88} \) \(\mathstrut -\mathstrut 14130q^{89} \) \(\mathstrut +\mathstrut 30700q^{90} \) \(\mathstrut +\mathstrut 33216q^{91} \) \(\mathstrut +\mathstrut 50304q^{92} \) \(\mathstrut +\mathstrut 117968q^{93} \) \(\mathstrut -\mathstrut 72816q^{94} \) \(\mathstrut +\mathstrut 53500q^{95} \) \(\mathstrut +\mathstrut 8192q^{96} \) \(\mathstrut -\mathstrut 224682q^{97} \) \(\mathstrut +\mathstrut 2652q^{98} \) \(\mathstrut -\mathstrut 328332q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(10))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5
10.6.a.a \(1\) \(1.604\) \(\Q\) None \(-4\) \(-26\) \(-25\) \(-22\) \(+\) \(+\) \(q-4q^{2}-26q^{3}+2^{4}q^{4}-5^{2}q^{5}+104q^{6}+\cdots\)
10.6.a.b \(1\) \(1.604\) \(\Q\) None \(-4\) \(24\) \(25\) \(-172\) \(+\) \(-\) \(q-4q^{2}+24q^{3}+2^{4}q^{4}+5^{2}q^{5}-96q^{6}+\cdots\)
10.6.a.c \(1\) \(1.604\) \(\Q\) None \(4\) \(6\) \(-25\) \(-118\) \(-\) \(+\) \(q+4q^{2}+6q^{3}+2^{4}q^{4}-5^{2}q^{5}+24q^{6}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(10))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(10)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)