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Properties

Label 209.6.a
Level $209$
Weight $6$
Character orbit 209.a
Rep. character $\chi_{209}(1,\cdot)$
Character field $\Q$
Dimension $74$
Newform subspaces $4$
Sturm bound $120$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 209 = 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 209.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(120\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(209))\).

Total New Old
Modular forms 102 74 28
Cusp forms 98 74 24
Eisenstein series 4 0 4

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
\(+\)\(+\)$+$\(17\)
\(+\)\(-\)$-$\(22\)
\(-\)\(+\)$-$\(20\)
\(-\)\(-\)$+$\(15\)
Plus space\(+\)\(32\)
Minus space\(-\)\(42\)

Trace form

\( 74 q + 8 q^{2} - 2 q^{3} + 1152 q^{4} - 10 q^{5} - 68 q^{6} + 1236 q^{8} + 6368 q^{9} + O(q^{10}) \) \( 74 q + 8 q^{2} - 2 q^{3} + 1152 q^{4} - 10 q^{5} - 68 q^{6} + 1236 q^{8} + 6368 q^{9} - 592 q^{10} - 484 q^{11} - 736 q^{12} + 1624 q^{13} + 940 q^{14} + 1070 q^{15} + 18168 q^{16} - 3656 q^{17} + 10312 q^{18} + 3032 q^{20} - 8960 q^{21} - 9350 q^{23} + 11404 q^{24} + 52140 q^{25} - 22732 q^{26} - 254 q^{27} - 22432 q^{28} + 7944 q^{29} - 46296 q^{30} + 2794 q^{31} + 35780 q^{32} - 7502 q^{33} + 31608 q^{34} + 36416 q^{35} + 107004 q^{36} - 30458 q^{37} - 8664 q^{38} + 32076 q^{39} - 27528 q^{40} + 1300 q^{41} + 67476 q^{42} - 24660 q^{43} - 24200 q^{44} - 7408 q^{45} - 12184 q^{46} - 51968 q^{47} - 125728 q^{48} + 170070 q^{49} - 57608 q^{50} + 74192 q^{51} + 41936 q^{52} - 24876 q^{53} - 108404 q^{54} - 10406 q^{55} + 22004 q^{56} + 108036 q^{58} - 23766 q^{59} + 487500 q^{60} + 250724 q^{61} - 48148 q^{62} + 114544 q^{63} + 345488 q^{64} - 201772 q^{65} - 34848 q^{66} + 34294 q^{67} - 67236 q^{68} - 157470 q^{69} + 550916 q^{70} + 107354 q^{71} + 699220 q^{72} - 195848 q^{73} + 259404 q^{74} - 117624 q^{75} - 17908 q^{77} - 286404 q^{78} + 59872 q^{79} + 49424 q^{80} + 168306 q^{81} + 398984 q^{82} + 475608 q^{83} - 868928 q^{84} - 130488 q^{85} - 406768 q^{86} - 173372 q^{87} + 418414 q^{89} + 320464 q^{90} + 372072 q^{91} - 220292 q^{92} + 158434 q^{93} + 296936 q^{94} + 177420 q^{96} - 70670 q^{97} + 732408 q^{98} - 2662 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(209))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 11 19
209.6.a.a 209.a 1.a $15$ $33.520$ \(\mathbb{Q}[x]/(x^{15} - \cdots)\) None \(-8\) \(-34\) \(-74\) \(-478\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(-2+\beta _{4})q^{3}+(8+\cdots)q^{4}+\cdots\)
209.6.a.b 209.a 1.a $17$ $33.520$ \(\mathbb{Q}[x]/(x^{17} - \cdots)\) None \(4\) \(-3\) \(-31\) \(-306\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+\beta _{4}q^{3}+(13+\beta _{1}+\beta _{2})q^{4}+\cdots\)
209.6.a.c 209.a 1.a $20$ $33.520$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(12\) \(2\) \(26\) \(404\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}-\beta _{3}q^{3}+(18-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
209.6.a.d 209.a 1.a $22$ $33.520$ None \(0\) \(33\) \(69\) \(380\) $+$ $-$ $\mathrm{SU}(2)$

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(209))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(209)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 2}\)