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Properties

Label 209.2.a
Level $209$
Weight $2$
Character orbit 209.a
Rep. character $\chi_{209}(1,\cdot)$
Character field $\Q$
Dimension $15$
Newform subspaces $4$
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.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(40\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(209))\).

Total New Old
Modular forms 22 15 7
Cusp forms 19 15 4
Eisenstein series 3 0 3

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(11\)\(19\)FrickeDim
\(+\)\(+\)\(+\)\(2\)
\(+\)\(-\)\(-\)\(7\)
\(-\)\(+\)\(-\)\(5\)
\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(3\)
Minus space\(-\)\(12\)

Trace form

\( 15 q + q^{2} + 2 q^{3} + 19 q^{4} - 8 q^{5} - 8 q^{6} + 8 q^{7} - 3 q^{8} + 13 q^{9} + O(q^{10}) \) \( 15 q + q^{2} + 2 q^{3} + 19 q^{4} - 8 q^{5} - 8 q^{6} + 8 q^{7} - 3 q^{8} + 13 q^{9} + 6 q^{10} - 3 q^{11} - 12 q^{12} - 2 q^{13} - 12 q^{14} + 14 q^{15} + 31 q^{16} + 2 q^{17} - 3 q^{18} + q^{19} - 6 q^{20} + 3 q^{22} + 10 q^{23} - 8 q^{24} + 11 q^{25} - 2 q^{26} - 22 q^{27} + 24 q^{28} - 18 q^{29} - 32 q^{30} + 18 q^{31} - 35 q^{32} + 2 q^{33} - 6 q^{34} + 16 q^{35} + 7 q^{36} - 3 q^{38} + 20 q^{39} - 18 q^{40} - 2 q^{41} - 48 q^{42} + 24 q^{43} - 11 q^{44} - 26 q^{45} - 8 q^{46} - 56 q^{48} + 27 q^{49} - 65 q^{50} - 8 q^{51} - 58 q^{52} + 2 q^{53} - 32 q^{54} - 8 q^{55} - 4 q^{56} + 4 q^{57} - 26 q^{58} - 10 q^{59} + 8 q^{60} - 14 q^{61} + 4 q^{62} + 24 q^{63} + 63 q^{64} - 16 q^{65} + 4 q^{66} + 10 q^{67} + 6 q^{68} - 2 q^{69} - 12 q^{70} + 26 q^{71} + 25 q^{72} + 10 q^{73} + 54 q^{74} + 20 q^{75} + 7 q^{76} - 4 q^{77} - 4 q^{78} + 48 q^{79} - 34 q^{80} - q^{81} - 22 q^{82} - 20 q^{83} + 16 q^{84} - 28 q^{85} + 44 q^{86} + 44 q^{87} + 15 q^{88} - 28 q^{89} + 38 q^{90} + 24 q^{91} + 52 q^{92} - 18 q^{93} + 24 q^{94} + 6 q^{95} + 56 q^{96} - 8 q^{97} + 25 q^{98} - 9 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(209))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 11 19
209.2.a.a 209.a 1.a $1$ $1.669$ \(\Q\) None 209.2.a.a \(0\) \(1\) \(-3\) \(-4\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{4}-3q^{5}-4q^{7}-2q^{9}+\cdots\)
209.2.a.b 209.a 1.a $2$ $1.669$ \(\Q(\sqrt{2}) \) None 209.2.a.b \(0\) \(-2\) \(-2\) \(-4\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(-1-\beta )q^{3}-q^{5}+(-2-\beta )q^{6}+\cdots\)
209.2.a.c 209.a 1.a $5$ $1.669$ 5.5.246832.1 None 209.2.a.c \(2\) \(1\) \(-5\) \(6\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{2}+\beta _{3}q^{3}+(\beta _{1}+\beta _{2}+\beta _{3}+\beta _{4})q^{4}+\cdots\)
209.2.a.d 209.a 1.a $7$ $1.669$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 209.2.a.d \(-1\) \(2\) \(2\) \(10\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-\beta _{2}q^{3}+(2+\beta _{2}+\beta _{3})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(209))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(209)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 2}\)