Properties

Label 3.10.a
Level 3
Weight 10
Character orbit a
Rep. character \(\chi_{3}(1,\cdot)\)
Character field \(\Q\)
Dimension 2
Newforms 2
Sturm bound 3
Trace bound 2

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Defining parameters

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

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(3))\).

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

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

\(3\)Dim.
\(+\)\(1\)
\(-\)\(1\)

Trace form

\(2q \) \(\mathstrut -\mathstrut 18q^{2} \) \(\mathstrut +\mathstrut 596q^{4} \) \(\mathstrut -\mathstrut 2844q^{5} \) \(\mathstrut +\mathstrut 4374q^{6} \) \(\mathstrut +\mathstrut 4648q^{7} \) \(\mathstrut -\mathstrut 22392q^{8} \) \(\mathstrut +\mathstrut 13122q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 18q^{2} \) \(\mathstrut +\mathstrut 596q^{4} \) \(\mathstrut -\mathstrut 2844q^{5} \) \(\mathstrut +\mathstrut 4374q^{6} \) \(\mathstrut +\mathstrut 4648q^{7} \) \(\mathstrut -\mathstrut 22392q^{8} \) \(\mathstrut +\mathstrut 13122q^{9} \) \(\mathstrut +\mathstrut 19764q^{10} \) \(\mathstrut +\mathstrut 22608q^{11} \) \(\mathstrut -\mathstrut 78732q^{12} \) \(\mathstrut -\mathstrut 120308q^{13} \) \(\mathstrut +\mathstrut 325584q^{14} \) \(\mathstrut -\mathstrut 17496q^{15} \) \(\mathstrut -\mathstrut 179440q^{16} \) \(\mathstrut -\mathstrut 171180q^{17} \) \(\mathstrut -\mathstrut 118098q^{18} \) \(\mathstrut +\mathstrut 737104q^{19} \) \(\mathstrut -\mathstrut 742536q^{20} \) \(\mathstrut +\mathstrut 1102248q^{21} \) \(\mathstrut +\mathstrut 327240q^{22} \) \(\mathstrut -\mathstrut 2732976q^{23} \) \(\mathstrut -\mathstrut 227448q^{24} \) \(\mathstrut +\mathstrut 161246q^{25} \) \(\mathstrut +\mathstrut 6016644q^{26} \) \(\mathstrut -\mathstrut 5228384q^{28} \) \(\mathstrut -\mathstrut 1950012q^{29} \) \(\mathstrut -\mathstrut 6062364q^{30} \) \(\mathstrut +\mathstrut 6701512q^{31} \) \(\mathstrut +\mathstrut 10875168q^{32} \) \(\mathstrut +\mathstrut 1592136q^{33} \) \(\mathstrut -\mathstrut 8921988q^{34} \) \(\mathstrut -\mathstrut 8079120q^{35} \) \(\mathstrut +\mathstrut 3910356q^{36} \) \(\mathstrut +\mathstrut 1874668q^{37} \) \(\mathstrut -\mathstrut 18758664q^{38} \) \(\mathstrut +\mathstrut 14801616q^{39} \) \(\mathstrut +\mathstrut 32144688q^{40} \) \(\mathstrut -\mathstrut 11576988q^{41} \) \(\mathstrut +\mathstrut 244944q^{42} \) \(\mathstrut -\mathstrut 27506192q^{43} \) \(\mathstrut -\mathstrut 2815632q^{44} \) \(\mathstrut -\mathstrut 18659484q^{45} \) \(\mathstrut +\mathstrut 3158352q^{46} \) \(\mathstrut +\mathstrut 51281136q^{47} \) \(\mathstrut -\mathstrut 6613488q^{48} \) \(\mathstrut +\mathstrut 22683570q^{49} \) \(\mathstrut +\mathstrut 15134994q^{50} \) \(\mathstrut -\mathstrut 31387824q^{51} \) \(\mathstrut -\mathstrut 124661480q^{52} \) \(\mathstrut +\mathstrut 94588884q^{53} \) \(\mathstrut +\mathstrut 28697814q^{54} \) \(\mathstrut -\mathstrut 34271424q^{55} \) \(\mathstrut -\mathstrut 71144640q^{56} \) \(\mathstrut -\mathstrut 36374184q^{57} \) \(\mathstrut +\mathstrut 323595972q^{58} \) \(\mathstrut -\mathstrut 106386624q^{59} \) \(\mathstrut +\mathstrut 106743096q^{60} \) \(\mathstrut -\mathstrut 147126020q^{61} \) \(\mathstrut -\mathstrut 261185184q^{62} \) \(\mathstrut +\mathstrut 30495528q^{63} \) \(\mathstrut -\mathstrut 78156736q^{64} \) \(\mathstrut +\mathstrut 151342488q^{65} \) \(\mathstrut +\mathstrut 35114472q^{66} \) \(\mathstrut +\mathstrut 374059840q^{67} \) \(\mathstrut +\mathstrut 137315304q^{68} \) \(\mathstrut -\mathstrut 64315296q^{69} \) \(\mathstrut -\mathstrut 463307040q^{70} \) \(\mathstrut -\mathstrut 80513136q^{71} \) \(\mathstrut -\mathstrut 146913912q^{72} \) \(\mathstrut -\mathstrut 270650156q^{73} \) \(\mathstrut +\mathstrut 437254164q^{74} \) \(\mathstrut +\mathstrut 49758624q^{75} \) \(\mathstrut +\mathstrut 437902096q^{76} \) \(\mathstrut +\mathstrut 186280416q^{77} \) \(\mathstrut -\mathstrut 396328140q^{78} \) \(\mathstrut -\mathstrut 209879960q^{79} \) \(\mathstrut +\mathstrut 263981664q^{80} \) \(\mathstrut +\mathstrut 86093442q^{81} \) \(\mathstrut +\mathstrut 26522316q^{82} \) \(\mathstrut -\mathstrut 1020687984q^{83} \) \(\mathstrut +\mathstrut 145496736q^{84} \) \(\mathstrut +\mathstrut 285268392q^{85} \) \(\mathstrut -\mathstrut 259740792q^{86} \) \(\mathstrut +\mathstrut 918137592q^{87} \) \(\mathstrut -\mathstrut 280716192q^{88} \) \(\mathstrut +\mathstrut 90873684q^{89} \) \(\mathstrut +\mathstrut 129671604q^{90} \) \(\mathstrut +\mathstrut 963739952q^{91} \) \(\mathstrut -\mathstrut 428535072q^{92} \) \(\mathstrut -\mathstrut 602614728q^{93} \) \(\mathstrut -\mathstrut 605638944q^{94} \) \(\mathstrut -\mathstrut 999662976q^{95} \) \(\mathstrut -\mathstrut 216460512q^{96} \) \(\mathstrut +\mathstrut 339473380q^{97} \) \(\mathstrut +\mathstrut 1503597438q^{98} \) \(\mathstrut +\mathstrut 148331088q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(3))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3
3.10.a.a \(1\) \(1.545\) \(\Q\) None \(-36\) \(-81\) \(-1314\) \(-4480\) \(+\) \(q-6^{2}q^{2}-3^{4}q^{3}+28^{2}q^{4}-1314q^{5}+\cdots\)
3.10.a.b \(1\) \(1.545\) \(\Q\) None \(18\) \(81\) \(-1530\) \(9128\) \(-\) \(q+18q^{2}+3^{4}q^{3}-188q^{4}-1530q^{5}+\cdots\)