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Properties

Label 209.2.g
Level $209$
Weight $2$
Character orbit 209.g
Rep. character $\chi_{209}(65,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $36$
Newform subspaces $2$
Sturm bound $40$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 209 = 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 209.g (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 209 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(40\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(209, [\chi])\).

Total New Old
Modular forms 44 44 0
Cusp forms 36 36 0
Eisenstein series 8 8 0

Trace form

\( 36 q - 12 q^{3} - 18 q^{4} - 4 q^{5} + 6 q^{9} + O(q^{10}) \) \( 36 q - 12 q^{3} - 18 q^{4} - 4 q^{5} + 6 q^{9} - 8 q^{11} + 18 q^{14} - 18 q^{15} - 18 q^{16} - 8 q^{20} + 10 q^{23} + 2 q^{25} - 8 q^{26} + 12 q^{34} + 16 q^{36} - 4 q^{38} + 10 q^{42} - 2 q^{44} - 4 q^{45} + 6 q^{48} - 40 q^{49} + 48 q^{53} + 6 q^{55} - 128 q^{58} + 66 q^{59} + 66 q^{60} + 22 q^{66} - 60 q^{67} + 96 q^{70} - 48 q^{71} + 64 q^{77} - 78 q^{78} + 34 q^{80} - 26 q^{81} + 34 q^{82} + 18 q^{86} - 42 q^{89} - 12 q^{91} + 42 q^{92} - 16 q^{93} - 96 q^{97} - 56 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(209, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
209.2.g.a 209.g 209.g $16$ $1.669$ 16.0.\(\cdots\).1 None \(0\) \(-12\) \(8\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{1}+\beta _{5})q^{2}+(-1+\beta _{6})q^{3}+(-1+\cdots)q^{4}+\cdots\)
209.2.g.b 209.g 209.g $20$ $1.669$ \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None \(0\) \(0\) \(-12\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\beta _{1}-\beta _{6})q^{2}-\beta _{12}q^{3}+(-1+\beta _{3}+\cdots)q^{4}+\cdots\)