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

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

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

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 5
15.6.a.a 15.a 1.a $1$ $2.406$ \(\Q\) None \(-2\) \(-9\) \(-25\) \(-132\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-9q^{3}-28q^{4}-5^{2}q^{5}+18q^{6}+\cdots\)
15.6.a.b 15.a 1.a $1$ $2.406$ \(\Q\) None \(7\) \(9\) \(-25\) \(12\) $-$ $+$ $\mathrm{SU}(2)$ \(q+7q^{2}+9q^{3}+17q^{4}-5^{2}q^{5}+63q^{6}+\cdots\)
15.6.a.c 15.a 1.a $2$ $2.406$ \(\Q(\sqrt{409}) \) None \(-1\) \(-18\) \(50\) \(-112\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}-9q^{3}+(70+\beta )q^{4}+5^{2}q^{5}+\cdots\)

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

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