Properties

Label 15.6.a
Level $15$
Weight $6$
Character orbit 15.a
Rep. character $\chi_{15}(1,\cdot)$
Character field $\Q$
Dimension $4$
Newform subspaces $3$
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: \(\Q\)
Newform subspaces: \( 3 \)
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

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

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(15))\) into newform subspaces

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

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