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Properties

Label 510.2.l
Level $510$
Weight $2$
Character orbit 510.l
Rep. character $\chi_{510}(137,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $64$
Newform subspaces $7$
Sturm bound $216$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 510.l (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 15 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 7 \)
Sturm bound: \(216\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(510, [\chi])\).

Total New Old
Modular forms 232 64 168
Cusp forms 200 64 136
Eisenstein series 32 0 32

Trace form

\( 64 q + 8 q^{3} + 8 q^{6} + 8 q^{7} + O(q^{10}) \) \( 64 q + 8 q^{3} + 8 q^{6} + 8 q^{7} - 8 q^{10} - 8 q^{12} - 8 q^{13} + 8 q^{15} - 64 q^{16} - 16 q^{18} - 8 q^{22} + 48 q^{25} + 8 q^{27} + 8 q^{28} + 32 q^{31} + 8 q^{33} - 8 q^{36} - 48 q^{37} - 32 q^{42} + 8 q^{43} + 32 q^{45} - 48 q^{46} - 8 q^{48} + 8 q^{52} - 48 q^{55} + 16 q^{57} + 56 q^{58} + 8 q^{60} + 16 q^{61} - 24 q^{63} + 32 q^{66} + 40 q^{67} + 8 q^{70} + 16 q^{72} - 8 q^{73} - 72 q^{75} - 32 q^{76} - 104 q^{81} + 64 q^{87} - 8 q^{88} - 32 q^{90} - 64 q^{91} + 88 q^{93} - 8 q^{96} + 56 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(510, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
510.2.l.a 510.l 15.e $4$ $4.072$ \(\Q(\zeta_{8})\) None \(0\) \(4\) \(0\) \(-12\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{8}q^{2}+(1+\zeta_{8}^{2}+\zeta_{8}^{3})q^{3}+\zeta_{8}^{2}q^{4}+\cdots\)
510.2.l.b 510.l 15.e $4$ $4.072$ \(\Q(\zeta_{8})\) None \(0\) \(4\) \(0\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{8}q^{2}+(1+\zeta_{8}^{2}+\zeta_{8}^{3})q^{3}+\zeta_{8}^{2}q^{4}+\cdots\)
510.2.l.c 510.l 15.e $4$ $4.072$ \(\Q(\zeta_{8})\) None \(0\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{8}q^{2}+(1+\zeta_{8}^{2}-\zeta_{8}^{3})q^{3}+\zeta_{8}^{2}q^{4}+\cdots\)
510.2.l.d 510.l 15.e $8$ $4.072$ 8.0.4030726144.1 None \(0\) \(-8\) \(4\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{6}q^{2}+(-1-\beta _{4}+\beta _{6})q^{3}-\beta _{2}q^{4}+\cdots\)
510.2.l.e 510.l 15.e $8$ $4.072$ 8.0.4030726144.1 None \(0\) \(0\) \(-4\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{4}q^{2}+(-\beta _{2}+\beta _{4}+\beta _{6})q^{3}+\beta _{2}q^{4}+\cdots\)
510.2.l.f 510.l 15.e $8$ $4.072$ \(\Q(\zeta_{24})\) None \(0\) \(4\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{4}]$ \(q+(\zeta_{24}-\zeta_{24}^{5})q^{2}+(1-\zeta_{24}^{2}-\zeta_{24}^{3}+\cdots)q^{3}+\cdots\)
510.2.l.g 510.l 15.e $28$ $4.072$ None \(0\) \(0\) \(0\) \(4\) $\mathrm{SU}(2)[C_{4}]$

Decomposition of \(S_{2}^{\mathrm{old}}(510, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(510, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 2}\)