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Properties

Label 840.2.bg
Level $840$
Weight $2$
Character orbit 840.bg
Rep. character $\chi_{840}(121,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $32$
Newform subspaces $10$
Sturm bound $384$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 840.bg (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 10 \)
Sturm bound: \(384\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(11\)

Dimensions

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

Total New Old
Modular forms 416 32 384
Cusp forms 352 32 320
Eisenstein series 64 0 64

Trace form

\( 32 q - 4 q^{3} - 4 q^{7} - 16 q^{9} + O(q^{10}) \) \( 32 q - 4 q^{3} - 4 q^{7} - 16 q^{9} - 4 q^{11} - 24 q^{13} - 8 q^{17} + 8 q^{21} - 16 q^{25} + 8 q^{27} + 16 q^{29} + 4 q^{31} + 8 q^{33} + 4 q^{35} - 4 q^{37} + 4 q^{39} - 24 q^{41} - 24 q^{43} + 12 q^{49} + 8 q^{51} + 24 q^{53} - 8 q^{57} + 8 q^{59} + 16 q^{61} - 4 q^{63} + 12 q^{65} + 12 q^{67} + 16 q^{69} - 16 q^{71} + 20 q^{73} - 4 q^{75} + 12 q^{79} - 16 q^{81} + 32 q^{83} - 40 q^{89} - 20 q^{91} + 4 q^{93} + 8 q^{95} + 8 q^{99} + O(q^{100}) \)

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.bg.a 840.bg 7.c $2$ $6.707$ \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(-1\) \(-5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}-\zeta_{6}q^{5}+(-2-\zeta_{6})q^{7}+\cdots\)
840.2.bg.b 840.bg 7.c $2$ $6.707$ \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(-1\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}-\zeta_{6}q^{5}+(-2+3\zeta_{6})q^{7}+\cdots\)
840.2.bg.c 840.bg 7.c $2$ $6.707$ \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(-1\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}-\zeta_{6}q^{5}+(3-2\zeta_{6})q^{7}+\cdots\)
840.2.bg.d 840.bg 7.c $2$ $6.707$ \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(1\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}+\zeta_{6}q^{5}+(-3+2\zeta_{6})q^{7}+\cdots\)
840.2.bg.e 840.bg 7.c $2$ $6.707$ \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(1\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}+\zeta_{6}q^{5}+(2-3\zeta_{6})q^{7}+\cdots\)
840.2.bg.f 840.bg 7.c $2$ $6.707$ \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(1\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}+\zeta_{6}q^{5}+(2+\zeta_{6})q^{7}+\cdots\)
840.2.bg.g 840.bg 7.c $4$ $6.707$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(-2\) \(-2\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1-\beta _{2})q^{3}+\beta _{2}q^{5}+(-\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
840.2.bg.h 840.bg 7.c $4$ $6.707$ \(\Q(\sqrt{-3}, \sqrt{7})\) None \(0\) \(-2\) \(2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1-\beta _{2})q^{3}-\beta _{2}q^{5}+(\beta _{1}+\beta _{3})q^{7}+\cdots\)
840.2.bg.i 840.bg 7.c $6$ $6.707$ 6.0.38363328.2 None \(0\) \(-3\) \(-3\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1-\beta _{3})q^{3}+\beta _{3}q^{5}+(-\beta _{1}+\beta _{4}+\cdots)q^{7}+\cdots\)
840.2.bg.j 840.bg 7.c $6$ $6.707$ 6.0.29428272.1 None \(0\) \(-3\) \(3\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{3}q^{3}+(1+\beta _{3})q^{5}+(\beta _{1}+\beta _{5})q^{7}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(840, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 2}\)