Properties

Label 15.6.a
Level 15
Weight 6
Character orbit a
Rep. character \(\chi_{15}(1,\cdot)\)
Character field \(\Q\)
Dimension 4
Newforms 3
Sturm bound 12
Trace bound 2

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Defining parameters

Level: \( N \) = \( 15 = 3 \cdot 5 \)
Weight: \( k \) = \( 6 \)
Character orbit: \([\chi]\) = 15.a (trivial)
Character field: \(\Q\)
Newforms: \( 3 \)
Sturm bound: \(12\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(15))\).

Total New Old
Modular forms 12 4 8
Cusp forms 8 4 4
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.

\(3\)\(5\)FrickeDim.
\(+\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(1\)
Plus space\(+\)\(1\)
Minus space\(-\)\(3\)

Trace form

\(4q \) \(\mathstrut +\mathstrut 4q^{2} \) \(\mathstrut -\mathstrut 18q^{3} \) \(\mathstrut +\mathstrut 130q^{4} \) \(\mathstrut +\mathstrut 90q^{6} \) \(\mathstrut -\mathstrut 232q^{7} \) \(\mathstrut -\mathstrut 228q^{8} \) \(\mathstrut +\mathstrut 324q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 4q^{2} \) \(\mathstrut -\mathstrut 18q^{3} \) \(\mathstrut +\mathstrut 130q^{4} \) \(\mathstrut +\mathstrut 90q^{6} \) \(\mathstrut -\mathstrut 232q^{7} \) \(\mathstrut -\mathstrut 228q^{8} \) \(\mathstrut +\mathstrut 324q^{9} \) \(\mathstrut -\mathstrut 150q^{10} \) \(\mathstrut +\mathstrut 832q^{11} \) \(\mathstrut -\mathstrut 864q^{12} \) \(\mathstrut -\mathstrut 784q^{13} \) \(\mathstrut -\mathstrut 2868q^{14} \) \(\mathstrut -\mathstrut 450q^{15} \) \(\mathstrut +\mathstrut 2962q^{16} \) \(\mathstrut +\mathstrut 2656q^{17} \) \(\mathstrut +\mathstrut 324q^{18} \) \(\mathstrut +\mathstrut 704q^{19} \) \(\mathstrut +\mathstrut 3800q^{20} \) \(\mathstrut +\mathstrut 2304q^{21} \) \(\mathstrut -\mathstrut 6828q^{22} \) \(\mathstrut +\mathstrut 264q^{23} \) \(\mathstrut +\mathstrut 162q^{24} \) \(\mathstrut +\mathstrut 2500q^{25} \) \(\mathstrut -\mathstrut 2612q^{26} \) \(\mathstrut -\mathstrut 1458q^{27} \) \(\mathstrut -\mathstrut 724q^{28} \) \(\mathstrut -\mathstrut 2032q^{29} \) \(\mathstrut -\mathstrut 1800q^{30} \) \(\mathstrut +\mathstrut 4256q^{31} \) \(\mathstrut -\mathstrut 27052q^{32} \) \(\mathstrut -\mathstrut 5472q^{33} \) \(\mathstrut +\mathstrut 1020q^{34} \) \(\mathstrut +\mathstrut 200q^{35} \) \(\mathstrut +\mathstrut 10530q^{36} \) \(\mathstrut +\mathstrut 15008q^{37} \) \(\mathstrut +\mathstrut 56104q^{38} \) \(\mathstrut -\mathstrut 10476q^{39} \) \(\mathstrut -\mathstrut 6450q^{40} \) \(\mathstrut -\mathstrut 18424q^{41} \) \(\mathstrut +\mathstrut 27324q^{42} \) \(\mathstrut -\mathstrut 48520q^{43} \) \(\mathstrut +\mathstrut 12716q^{44} \) \(\mathstrut +\mathstrut 13776q^{46} \) \(\mathstrut -\mathstrut 1160q^{47} \) \(\mathstrut -\mathstrut 49680q^{48} \) \(\mathstrut +\mathstrut 8964q^{49} \) \(\mathstrut +\mathstrut 2500q^{50} \) \(\mathstrut +\mathstrut 15372q^{51} \) \(\mathstrut +\mathstrut 61136q^{52} \) \(\mathstrut +\mathstrut 19744q^{53} \) \(\mathstrut +\mathstrut 7290q^{54} \) \(\mathstrut -\mathstrut 8400q^{55} \) \(\mathstrut -\mathstrut 131100q^{56} \) \(\mathstrut +\mathstrut 19224q^{57} \) \(\mathstrut -\mathstrut 43452q^{58} \) \(\mathstrut +\mathstrut 8176q^{59} \) \(\mathstrut -\mathstrut 41850q^{60} \) \(\mathstrut -\mathstrut 3736q^{61} \) \(\mathstrut -\mathstrut 19704q^{62} \) \(\mathstrut -\mathstrut 18792q^{63} \) \(\mathstrut +\mathstrut 7018q^{64} \) \(\mathstrut +\mathstrut 63400q^{65} \) \(\mathstrut +\mathstrut 75564q^{66} \) \(\mathstrut -\mathstrut 31864q^{67} \) \(\mathstrut +\mathstrut 240728q^{68} \) \(\mathstrut +\mathstrut 55512q^{69} \) \(\mathstrut -\mathstrut 89100q^{70} \) \(\mathstrut -\mathstrut 146032q^{71} \) \(\mathstrut -\mathstrut 18468q^{72} \) \(\mathstrut +\mathstrut 61784q^{73} \) \(\mathstrut -\mathstrut 125548q^{74} \) \(\mathstrut -\mathstrut 11250q^{75} \) \(\mathstrut +\mathstrut 145856q^{76} \) \(\mathstrut +\mathstrut 29856q^{77} \) \(\mathstrut -\mathstrut 99216q^{78} \) \(\mathstrut +\mathstrut 102800q^{79} \) \(\mathstrut +\mathstrut 105200q^{80} \) \(\mathstrut +\mathstrut 26244q^{81} \) \(\mathstrut -\mathstrut 37224q^{82} \) \(\mathstrut +\mathstrut 20952q^{83} \) \(\mathstrut +\mathstrut 10188q^{84} \) \(\mathstrut +\mathstrut 35400q^{85} \) \(\mathstrut -\mathstrut 110408q^{86} \) \(\mathstrut -\mathstrut 56412q^{87} \) \(\mathstrut -\mathstrut 240468q^{88} \) \(\mathstrut -\mathstrut 20136q^{89} \) \(\mathstrut -\mathstrut 12150q^{90} \) \(\mathstrut -\mathstrut 22544q^{91} \) \(\mathstrut -\mathstrut 169632q^{92} \) \(\mathstrut -\mathstrut 140688q^{93} \) \(\mathstrut -\mathstrut 281976q^{94} \) \(\mathstrut +\mathstrut 55600q^{95} \) \(\mathstrut +\mathstrut 142794q^{96} \) \(\mathstrut +\mathstrut 84776q^{97} \) \(\mathstrut +\mathstrut 236084q^{98} \) \(\mathstrut +\mathstrut 67392q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(15))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 5
15.6.a.a \(1\) \(2.406\) \(\Q\) None \(-2\) \(-9\) \(-25\) \(-132\) \(+\) \(+\) \(q-2q^{2}-9q^{3}-28q^{4}-5^{2}q^{5}+18q^{6}+\cdots\)
15.6.a.b \(1\) \(2.406\) \(\Q\) None \(7\) \(9\) \(-25\) \(12\) \(-\) \(+\) \(q+7q^{2}+9q^{3}+17q^{4}-5^{2}q^{5}+63q^{6}+\cdots\)
15.6.a.c \(2\) \(2.406\) \(\Q(\sqrt{409}) \) None \(-1\) \(-18\) \(50\) \(-112\) \(+\) \(-\) \(q-\beta q^{2}-9q^{3}+(70+\beta )q^{4}+5^{2}q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(15))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(15)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)