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Properties

Label 840.2.bk
Level $840$
Weight $2$
Character orbit 840.bk
Rep. character $\chi_{840}(113,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $72$
Newform subspaces $4$
Sturm bound $384$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 840.bk (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 15 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 4 \)
Sturm bound: \(384\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(840, [\chi])\).

Total New Old
Modular forms 416 72 344
Cusp forms 352 72 280
Eisenstein series 64 0 64

Trace form

\( 72 q + O(q^{10}) \) \( 72 q - 16 q^{13} + 24 q^{15} + 8 q^{25} + 24 q^{27} + 16 q^{31} + 24 q^{33} + 56 q^{37} - 72 q^{51} - 48 q^{55} - 24 q^{57} - 8 q^{63} - 48 q^{67} + 16 q^{73} - 24 q^{75} + 32 q^{81} + 104 q^{85} + 40 q^{87} + 64 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(840, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
840.2.bk.a 840.bk 15.e $4$ $6.707$ \(\Q(\zeta_{8})\) None \(0\) \(-4\) \(4\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-\zeta_{8}^{2}+\zeta_{8}^{3})q^{3}+(1+2\zeta_{8}^{2}+\cdots)q^{5}+\cdots\)
840.2.bk.b 840.bk 15.e $4$ $6.707$ \(\Q(\zeta_{8})\) None \(0\) \(4\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1-\zeta_{8}-\zeta_{8}^{2})q^{3}+(-1+2\zeta_{8}^{2}+\cdots)q^{5}+\cdots\)
840.2.bk.c 840.bk 15.e $32$ $6.707$ None \(0\) \(-4\) \(8\) \(0\) $\mathrm{SU}(2)[C_{4}]$
840.2.bk.d 840.bk 15.e $32$ $6.707$ None \(0\) \(4\) \(-8\) \(0\) $\mathrm{SU}(2)[C_{4}]$

Decomposition of \(S_{2}^{\mathrm{old}}(840, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(840, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(420, [\chi])\)\(^{\oplus 2}\)