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Properties

Label 840.2.z
Level $840$
Weight $2$
Character orbit 840.z
Rep. character $\chi_{840}(811,\cdot)$
Character field $\Q$
Dimension $64$
Newform subspaces $4$
Sturm bound $384$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 840.z (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 56 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(384\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(11\), \(13\)

Dimensions

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

Total New Old
Modular forms 200 64 136
Cusp forms 184 64 120
Eisenstein series 16 0 16

Trace form

\( 64 q - 4 q^{2} - 4 q^{4} - 4 q^{8} - 64 q^{9} + O(q^{10}) \) \( 64 q - 4 q^{2} - 4 q^{4} - 4 q^{8} - 64 q^{9} + 16 q^{11} + 8 q^{14} - 20 q^{16} + 4 q^{18} - 16 q^{22} + 64 q^{25} - 8 q^{28} + 36 q^{32} + 4 q^{36} - 12 q^{42} + 16 q^{43} - 8 q^{44} - 40 q^{46} + 16 q^{49} - 4 q^{50} - 56 q^{56} + 32 q^{57} - 56 q^{58} + 44 q^{64} - 16 q^{67} + 4 q^{70} + 4 q^{72} + 24 q^{78} + 64 q^{81} - 12 q^{84} + 24 q^{86} + 112 q^{88} + 80 q^{91} - 16 q^{92} - 44 q^{98} - 16 q^{99} + O(q^{100}) \)

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.z.a 840.z 56.e $4$ $6.707$ \(\Q(i, \sqrt{6})\) None \(-4\) \(0\) \(-4\) \(4\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-1-\beta _{1})q^{2}+\beta _{1}q^{3}+2\beta _{1}q^{4}+\cdots\)
840.2.z.b 840.z 56.e $4$ $6.707$ \(\Q(i, \sqrt{6})\) None \(-4\) \(0\) \(4\) \(-4\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-1-\beta _{1})q^{2}-\beta _{1}q^{3}+2\beta _{1}q^{4}+\cdots\)
840.2.z.c 840.z 56.e $28$ $6.707$ None \(2\) \(0\) \(-28\) \(-4\) $\mathrm{SU}(2)[C_{2}]$
840.2.z.d 840.z 56.e $28$ $6.707$ None \(2\) \(0\) \(28\) \(4\) $\mathrm{SU}(2)[C_{2}]$

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

\( S_{2}^{\mathrm{old}}(840, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(280, [\chi])\)\(^{\oplus 2}\)