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Properties

Label 208.2.s
Level $208$
Weight $2$
Character orbit 208.s
Rep. character $\chi_{208}(99,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $52$
Newform subspaces $2$
Sturm bound $56$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 208.s (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 208 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 2 \)
Sturm bound: \(56\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(208, [\chi])\).

Total New Old
Modular forms 60 60 0
Cusp forms 52 52 0
Eisenstein series 8 8 0

Trace form

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

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(208, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
208.2.s.a 208.s 208.s $2$ $1.661$ \(\Q(\sqrt{-1}) \) None \(2\) \(2\) \(0\) \(2\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+i)q^{2}+(1-i)q^{3}+2iq^{4}+2q^{6}+\cdots\)
208.2.s.b 208.s 208.s $50$ $1.661$ None \(-4\) \(-6\) \(-4\) \(-6\) $\mathrm{SU}(2)[C_{4}]$