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Properties

Label 209.2.e
Level $209$
Weight $2$
Character orbit 209.e
Rep. character $\chi_{209}(45,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $36$
Newform subspaces $2$
Sturm bound $40$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 209 = 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 209.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 19 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 2 \)
Sturm bound: \(40\)
Trace bound: \(2\)
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 36 8
Cusp forms 36 36 0
Eisenstein series 8 0 8

Trace form

\( 36 q - 4 q^{3} - 18 q^{4} + 4 q^{6} - 12 q^{7} - 12 q^{8} - 22 q^{9} + O(q^{10}) \) \( 36 q - 4 q^{3} - 18 q^{4} + 4 q^{6} - 12 q^{7} - 12 q^{8} - 22 q^{9} + 10 q^{10} + 24 q^{12} + 10 q^{13} - 2 q^{14} + 6 q^{15} - 18 q^{16} + 8 q^{17} + 4 q^{18} + 10 q^{19} - 40 q^{20} - 2 q^{22} + 10 q^{23} + 14 q^{24} - 22 q^{25} - 24 q^{26} + 20 q^{27} + 6 q^{28} + 40 q^{30} - 16 q^{31} + 22 q^{32} - 4 q^{33} + 4 q^{34} - 6 q^{35} - 12 q^{36} - 8 q^{37} + 16 q^{39} - 12 q^{40} - 12 q^{41} + 2 q^{42} + 16 q^{43} - 12 q^{45} - 8 q^{46} - 16 q^{47} - 34 q^{48} - 8 q^{50} + 10 q^{51} + 28 q^{52} - 20 q^{53} + 16 q^{54} - 44 q^{56} - 50 q^{57} + 8 q^{58} - 6 q^{59} + 26 q^{60} - 12 q^{61} + 6 q^{62} + 26 q^{63} + 96 q^{64} - 16 q^{65} + 10 q^{66} - 16 q^{67} - 4 q^{68} + 48 q^{69} - 32 q^{70} - 20 q^{71} + 38 q^{72} - 8 q^{73} + 42 q^{74} + 100 q^{75} - 92 q^{76} - 6 q^{78} + 30 q^{79} + 54 q^{80} - 74 q^{81} + 38 q^{82} + 20 q^{83} - 104 q^{84} + 20 q^{85} + 22 q^{86} - 104 q^{87} - 12 q^{88} - 14 q^{89} + 26 q^{90} - 48 q^{91} + 42 q^{92} + 12 q^{93} - 112 q^{94} + 12 q^{95} - 40 q^{96} - 28 q^{97} - 40 q^{98} + O(q^{100}) \)

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.e.a 209.e 19.c $18$ $1.669$ \(\mathbb{Q}[x]/(x^{18} - \cdots)\) None \(-1\) \(-4\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}-\beta _{7}q^{3}+(-1-\beta _{4}-\beta _{9}+\cdots)q^{4}+\cdots\)
209.2.e.b 209.e 19.c $18$ $1.669$ \(\mathbb{Q}[x]/(x^{18} - \cdots)\) None \(1\) \(0\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{2}+(\beta _{4}-\beta _{12})q^{3}+(-1+\beta _{3}+\cdots)q^{4}+\cdots\)