Properties

Label 208.2.w
Level $208$
Weight $2$
Character orbit 208.w
Rep. character $\chi_{208}(17,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $12$
Newform subspaces $3$
Sturm bound $56$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 208.w (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 3 \)
Sturm bound: \(56\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

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

Total New Old
Modular forms 68 16 52
Cusp forms 44 12 32
Eisenstein series 24 4 20

Trace form

\( 12 q - q^{3} + 9 q^{7} - 5 q^{9} + O(q^{10}) \) \( 12 q - q^{3} + 9 q^{7} - 5 q^{9} + 3 q^{11} - q^{13} - 6 q^{15} + 3 q^{19} - 5 q^{23} - 6 q^{25} + 26 q^{27} - 2 q^{29} - 3 q^{33} - 16 q^{35} - 18 q^{37} - 5 q^{39} - 6 q^{41} + 7 q^{43} + 27 q^{45} - 9 q^{49} - 50 q^{51} + 2 q^{53} - 16 q^{55} - 39 q^{59} - 30 q^{63} + 5 q^{65} + 33 q^{67} - 3 q^{69} + 39 q^{71} + 21 q^{75} - 42 q^{77} - 32 q^{79} - 10 q^{81} + 15 q^{85} + 25 q^{87} - 3 q^{89} + 49 q^{91} + 30 q^{93} - 22 q^{95} + 21 q^{97} + O(q^{100}) \)

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.w.a 208.w 13.e $2$ $1.661$ \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(0\) \(3\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-1+\zeta_{6})q^{3}+(2-\zeta_{6})q^{7}+2\zeta_{6}q^{9}+\cdots\)
208.2.w.b 208.w 13.e $2$ $1.661$ \(\Q(\sqrt{-3}) \) None \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(2-2\zeta_{6})q^{3}+(1-2\zeta_{6})q^{5}-\zeta_{6}q^{9}+\cdots\)
208.2.w.c 208.w 13.e $8$ $1.661$ 8.0.195105024.2 None \(0\) \(-2\) \(0\) \(6\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\beta _{2}-\beta _{4}+\beta _{5})q^{3}+(\beta _{1}+\beta _{3}+\beta _{5}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(208, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(208, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(52, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(104, [\chi])\)\(^{\oplus 2}\)