Properties

Label 252.2.bf
Level $252$
Weight $2$
Character orbit 252.bf
Rep. character $\chi_{252}(19,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $36$
Newform subspaces $7$
Sturm bound $96$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 252.bf (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 28 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 7 \)
Sturm bound: \(96\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(11\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(252, [\chi])\).

Total New Old
Modular forms 112 44 68
Cusp forms 80 36 44
Eisenstein series 32 8 24

Trace form

\( 36 q + 6 q^{5} + O(q^{10}) \) \( 36 q + 6 q^{5} + 6 q^{10} + 20 q^{14} + 4 q^{16} + 6 q^{17} - 20 q^{22} + 4 q^{25} - 18 q^{26} + 24 q^{28} + 16 q^{29} - 20 q^{32} - 14 q^{37} - 36 q^{38} - 48 q^{40} - 26 q^{46} - 20 q^{49} - 12 q^{50} - 36 q^{52} + 6 q^{53} - 56 q^{56} - 4 q^{58} - 18 q^{61} - 24 q^{64} - 20 q^{65} + 36 q^{68} - 30 q^{70} - 42 q^{73} + 44 q^{74} - 6 q^{77} + 96 q^{80} + 24 q^{82} - 20 q^{85} + 10 q^{86} + 56 q^{88} - 54 q^{89} + 48 q^{92} + 78 q^{94} + 94 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(252, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
252.2.bf.a 252.bf 28.f $4$ $2.012$ \(\Q(\sqrt{-3}, \sqrt{-7})\) None \(-1\) \(0\) \(6\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{3}q^{2}+(1+\beta _{1}-\beta _{2})q^{4}+(2-\beta _{2}+\cdots)q^{5}+\cdots\)
252.2.bf.b 252.bf 28.f $4$ $2.012$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{3}q^{2}-2q^{4}-2\beta _{1}q^{5}+(1-3\beta _{2}+\cdots)q^{7}+\cdots\)
252.2.bf.c 252.bf 28.f $4$ $2.012$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(2\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{1}q^{2}+2\beta _{2}q^{4}+(2\beta _{1}-2\beta _{3})q^{5}+\cdots\)
252.2.bf.d 252.bf 28.f $4$ $2.012$ \(\Q(\sqrt{-3}, \sqrt{-7})\) None \(1\) \(0\) \(-6\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{3}q^{2}+(1+\beta _{1}-\beta _{2})q^{4}+(-2+\beta _{2}+\cdots)q^{5}+\cdots\)
252.2.bf.e 252.bf 28.f $4$ $2.012$ \(\Q(\zeta_{12})\) None \(2\) \(0\) \(6\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+2\zeta_{12}q^{4}+\cdots\)
252.2.bf.f 252.bf 28.f $8$ $2.012$ 8.0.562828176.1 None \(-1\) \(0\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{5}q^{2}+(-\beta _{4}+\beta _{6}-\beta _{7})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots\)
252.2.bf.g 252.bf 28.f $8$ $2.012$ 8.0.562828176.1 None \(-1\) \(0\) \(0\) \(2\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{1}-\beta _{5})q^{2}+(-\beta _{2}-\beta _{6}+\beta _{7})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(252, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(252, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 2}\)