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Properties

Label 210.2.i
Level $210$
Weight $2$
Character orbit 210.i
Rep. character $\chi_{210}(121,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $8$
Newform subspaces $4$
Sturm bound $96$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 210.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 4 \)
Sturm bound: \(96\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(210, [\chi])\).

Total New Old
Modular forms 112 8 104
Cusp forms 80 8 72
Eisenstein series 32 0 32

Trace form

\( 8 q - 4 q^{4} - 4 q^{9} + O(q^{10}) \) \( 8 q - 4 q^{4} - 4 q^{9} - 4 q^{10} + 4 q^{11} + 16 q^{13} - 12 q^{14} - 4 q^{16} + 8 q^{17} + 4 q^{19} - 16 q^{22} - 4 q^{25} + 4 q^{26} - 32 q^{29} + 8 q^{31} - 8 q^{33} - 16 q^{34} - 12 q^{35} + 8 q^{36} - 8 q^{37} + 8 q^{38} - 16 q^{39} - 4 q^{40} + 24 q^{41} + 16 q^{42} + 48 q^{43} + 4 q^{44} - 4 q^{46} + 16 q^{47} + 8 q^{49} - 8 q^{52} + 24 q^{53} - 16 q^{55} + 32 q^{57} - 32 q^{59} - 8 q^{61} + 32 q^{62} + 8 q^{64} + 4 q^{65} - 8 q^{67} + 8 q^{68} - 32 q^{69} - 48 q^{71} + 8 q^{73} + 12 q^{74} - 8 q^{76} - 24 q^{77} + 16 q^{78} + 8 q^{79} - 4 q^{81} - 16 q^{82} + 16 q^{83} - 16 q^{85} + 8 q^{86} + 8 q^{88} - 8 q^{89} + 8 q^{90} + 32 q^{91} + 12 q^{94} + 8 q^{95} - 64 q^{97} - 8 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(210, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
210.2.i.a 210.i 7.c $2$ $1.677$ \(\Q(\sqrt{-3}) \) None \(-1\) \(-1\) \(-1\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
210.2.i.b 210.i 7.c $2$ $1.677$ \(\Q(\sqrt{-3}) \) None \(-1\) \(1\) \(-1\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
210.2.i.c 210.i 7.c $2$ $1.677$ \(\Q(\sqrt{-3}) \) None \(1\) \(-1\) \(1\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
210.2.i.d 210.i 7.c $2$ $1.677$ \(\Q(\sqrt{-3}) \) None \(1\) \(1\) \(1\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(210, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(210, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)