Properties

Label 208.6.f
Level $208$
Weight $6$
Character orbit 208.f
Rep. character $\chi_{208}(129,\cdot)$
Character field $\Q$
Dimension $34$
Newform subspaces $5$
Sturm bound $168$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 208.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(168\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(208, [\chi])\).

Total New Old
Modular forms 146 36 110
Cusp forms 134 34 100
Eisenstein series 12 2 10

Trace form

\( 34 q + 20 q^{3} + 2590 q^{9} + O(q^{10}) \) \( 34 q + 20 q^{3} + 2590 q^{9} - 62 q^{13} + 200 q^{17} + 3176 q^{23} - 17194 q^{25} + 10052 q^{27} - 4284 q^{29} + 8452 q^{35} + 1792 q^{39} + 25556 q^{43} - 69102 q^{49} - 105004 q^{51} + 28876 q^{53} - 73344 q^{55} - 52076 q^{61} + 6900 q^{65} - 63672 q^{69} - 36200 q^{75} + 44840 q^{77} + 99144 q^{79} + 293922 q^{81} - 237568 q^{87} - 151900 q^{91} + 92424 q^{95} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(208, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
208.6.f.a 208.f 13.b $2$ $33.360$ \(\Q(\sqrt{-1}) \) None 26.6.b.b \(0\) \(-8\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-4q^{3}+34iq^{5}-41iq^{7}-227q^{9}+\cdots\)
208.6.f.b 208.f 13.b $2$ $33.360$ \(\Q(\sqrt{-1}) \) None 26.6.b.a \(0\) \(26\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+13q^{3}+17iq^{5}-35iq^{7}-74q^{9}+\cdots\)
208.6.f.c 208.f 13.b $6$ $33.360$ \(\mathbb{Q}[x]/(x^{6} + \cdots)\) None 13.6.b.a \(0\) \(-16\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-3-\beta _{2})q^{3}+(-\beta _{1}+\beta _{3})q^{5}+(-3\beta _{1}+\cdots)q^{7}+\cdots\)
208.6.f.d 208.f 13.b $6$ $33.360$ \(\mathbb{Q}[x]/(x^{6} + \cdots)\) None 52.6.d.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{3}q^{3}-\beta _{1}q^{5}+\beta _{2}q^{7}+(118+\beta _{3}+\cdots)q^{9}+\cdots\)
208.6.f.e 208.f 13.b $18$ $33.360$ \(\mathbb{Q}[x]/(x^{18} + \cdots)\) None 104.6.f.a \(0\) \(18\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(1+\beta _{2})q^{3}+\beta _{8}q^{5}+\beta _{10}q^{7}+(90+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(208, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(208, [\chi]) \simeq \) \(S_{6}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(52, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(104, [\chi])\)\(^{\oplus 2}\)