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Properties

Label 208.4.k
Level $208$
Weight $4$
Character orbit 208.k
Rep. character $\chi_{208}(31,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $42$
Newform subspaces $3$
Sturm bound $112$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 208.k (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 52 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 3 \)
Sturm bound: \(112\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(208, [\chi])\).

Total New Old
Modular forms 180 42 138
Cusp forms 156 42 114
Eisenstein series 24 0 24

Trace form

\( 42 q - 6 q^{5} - 378 q^{9} + O(q^{10}) \) \( 42 q - 6 q^{5} - 378 q^{9} - 120 q^{21} - 486 q^{37} - 1242 q^{41} + 270 q^{45} - 540 q^{53} + 2712 q^{57} - 2940 q^{61} - 1626 q^{65} - 6 q^{73} + 3402 q^{81} + 4764 q^{85} + 4542 q^{89} + 2184 q^{93} - 894 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(208, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
208.4.k.a 208.k 52.f $2$ $12.272$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(26\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+(13+13i)q^{5}+3^{3}q^{9}+(-46-9i)q^{13}+\cdots\)
208.4.k.b 208.k 52.f $12$ $12.272$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(0\) \(-28\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{3}+(-2+2\beta _{2}-\beta _{4}+\beta _{5}-\beta _{6}+\cdots)q^{5}+\cdots\)
208.4.k.c 208.k 52.f $28$ $12.272$ None \(0\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{4}]$

Decomposition of \(S_{4}^{\mathrm{old}}(208, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(208, [\chi]) \cong \)