Properties

Label 13.6.a
Level 13
Weight 6
Character orbit a
Rep. character \(\chi_{13}(1,\cdot)\)
Character field \(\Q\)
Dimension 5
Newforms 2
Sturm bound 7
Trace bound 1

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Defining parameters

Level: \( N \) = \( 13 \)
Weight: \( k \) = \( 6 \)
Character orbit: \([\chi]\) = 13.a (trivial)
Character field: \(\Q\)
Newforms: \( 2 \)
Sturm bound: \(7\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(13))\).

Total New Old
Modular forms 7 5 2
Cusp forms 5 5 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.

\(13\)Dim.
\(+\)\(2\)
\(-\)\(3\)

Trace form

\(5q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 20q^{3} \) \(\mathstrut +\mathstrut 78q^{4} \) \(\mathstrut +\mathstrut 14q^{5} \) \(\mathstrut -\mathstrut 180q^{6} \) \(\mathstrut -\mathstrut 96q^{7} \) \(\mathstrut +\mathstrut 552q^{8} \) \(\mathstrut +\mathstrut 21q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(5q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 20q^{3} \) \(\mathstrut +\mathstrut 78q^{4} \) \(\mathstrut +\mathstrut 14q^{5} \) \(\mathstrut -\mathstrut 180q^{6} \) \(\mathstrut -\mathstrut 96q^{7} \) \(\mathstrut +\mathstrut 552q^{8} \) \(\mathstrut +\mathstrut 21q^{9} \) \(\mathstrut -\mathstrut 846q^{10} \) \(\mathstrut +\mathstrut 180q^{11} \) \(\mathstrut -\mathstrut 234q^{12} \) \(\mathstrut +\mathstrut 169q^{13} \) \(\mathstrut -\mathstrut 1298q^{14} \) \(\mathstrut +\mathstrut 520q^{15} \) \(\mathstrut +\mathstrut 3250q^{16} \) \(\mathstrut -\mathstrut 1722q^{17} \) \(\mathstrut +\mathstrut 1874q^{18} \) \(\mathstrut -\mathstrut 164q^{19} \) \(\mathstrut -\mathstrut 1564q^{20} \) \(\mathstrut +\mathstrut 5464q^{21} \) \(\mathstrut -\mathstrut 3184q^{22} \) \(\mathstrut +\mathstrut 1000q^{23} \) \(\mathstrut -\mathstrut 14436q^{24} \) \(\mathstrut +\mathstrut 10255q^{25} \) \(\mathstrut +\mathstrut 2028q^{26} \) \(\mathstrut -\mathstrut 9008q^{27} \) \(\mathstrut -\mathstrut 2912q^{28} \) \(\mathstrut -\mathstrut 9570q^{29} \) \(\mathstrut +\mathstrut 1658q^{30} \) \(\mathstrut +\mathstrut 5112q^{31} \) \(\mathstrut +\mathstrut 29696q^{32} \) \(\mathstrut +\mathstrut 7720q^{33} \) \(\mathstrut +\mathstrut 21476q^{34} \) \(\mathstrut -\mathstrut 18696q^{35} \) \(\mathstrut -\mathstrut 16376q^{36} \) \(\mathstrut -\mathstrut 37490q^{37} \) \(\mathstrut +\mathstrut 34804q^{38} \) \(\mathstrut +\mathstrut 6084q^{39} \) \(\mathstrut -\mathstrut 62174q^{40} \) \(\mathstrut -\mathstrut 3158q^{41} \) \(\mathstrut +\mathstrut 12666q^{42} \) \(\mathstrut +\mathstrut 4452q^{43} \) \(\mathstrut +\mathstrut 61444q^{44} \) \(\mathstrut +\mathstrut 38750q^{45} \) \(\mathstrut +\mathstrut 15188q^{46} \) \(\mathstrut +\mathstrut 44232q^{47} \) \(\mathstrut -\mathstrut 101858q^{48} \) \(\mathstrut -\mathstrut 35887q^{49} \) \(\mathstrut -\mathstrut 99898q^{50} \) \(\mathstrut +\mathstrut 56816q^{51} \) \(\mathstrut +\mathstrut 27716q^{52} \) \(\mathstrut -\mathstrut 56290q^{53} \) \(\mathstrut +\mathstrut 54420q^{54} \) \(\mathstrut +\mathstrut 14648q^{55} \) \(\mathstrut -\mathstrut 28942q^{56} \) \(\mathstrut +\mathstrut 4656q^{57} \) \(\mathstrut -\mathstrut 16688q^{58} \) \(\mathstrut +\mathstrut 25012q^{59} \) \(\mathstrut +\mathstrut 160128q^{60} \) \(\mathstrut -\mathstrut 10770q^{61} \) \(\mathstrut -\mathstrut 63448q^{62} \) \(\mathstrut -\mathstrut 105424q^{63} \) \(\mathstrut +\mathstrut 120610q^{64} \) \(\mathstrut +\mathstrut 16562q^{65} \) \(\mathstrut -\mathstrut 116640q^{66} \) \(\mathstrut -\mathstrut 13796q^{67} \) \(\mathstrut +\mathstrut 12810q^{68} \) \(\mathstrut -\mathstrut 55592q^{69} \) \(\mathstrut +\mathstrut 105284q^{70} \) \(\mathstrut +\mathstrut 144976q^{71} \) \(\mathstrut -\mathstrut 5904q^{72} \) \(\mathstrut +\mathstrut 31386q^{73} \) \(\mathstrut -\mathstrut 105014q^{74} \) \(\mathstrut -\mathstrut 6020q^{75} \) \(\mathstrut -\mathstrut 94120q^{76} \) \(\mathstrut +\mathstrut 35696q^{77} \) \(\mathstrut -\mathstrut 36842q^{78} \) \(\mathstrut -\mathstrut 117168q^{79} \) \(\mathstrut -\mathstrut 146948q^{80} \) \(\mathstrut -\mathstrut 15459q^{81} \) \(\mathstrut +\mathstrut 106068q^{82} \) \(\mathstrut +\mathstrut 22380q^{83} \) \(\mathstrut -\mathstrut 29108q^{84} \) \(\mathstrut -\mathstrut 43516q^{85} \) \(\mathstrut -\mathstrut 35844q^{86} \) \(\mathstrut +\mathstrut 147992q^{87} \) \(\mathstrut +\mathstrut 144984q^{88} \) \(\mathstrut +\mathstrut 150202q^{89} \) \(\mathstrut -\mathstrut 24360q^{90} \) \(\mathstrut -\mathstrut 4056q^{91} \) \(\mathstrut +\mathstrut 297100q^{92} \) \(\mathstrut +\mathstrut 20160q^{93} \) \(\mathstrut -\mathstrut 34282q^{94} \) \(\mathstrut -\mathstrut 180488q^{95} \) \(\mathstrut -\mathstrut 202148q^{96} \) \(\mathstrut +\mathstrut 41578q^{97} \) \(\mathstrut -\mathstrut 54926q^{98} \) \(\mathstrut -\mathstrut 220140q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(13))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 13
13.6.a.a \(2\) \(2.085\) \(\Q(\sqrt{17}) \) None \(-5\) \(-28\) \(-42\) \(-36\) \(+\) \(q+(-2-\beta )q^{2}+(-17+6\beta )q^{3}+(-24+\cdots)q^{4}+\cdots\)
13.6.a.b \(3\) \(2.085\) 3.3.168897.1 None \(7\) \(8\) \(56\) \(-60\) \(-\) \(q+(2+\beta _{1})q^{2}+(3-\beta _{1}-\beta _{2})q^{3}+(40+\cdots)q^{4}+\cdots\)