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Properties

Label 52.4.f
Level $52$
Weight $4$
Character orbit 52.f
Rep. character $\chi_{52}(31,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $38$
Newform subspaces $2$
Sturm bound $28$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 52 = 2^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 52.f (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 52 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 2 \)
Sturm bound: \(28\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(52, [\chi])\).

Total New Old
Modular forms 46 46 0
Cusp forms 38 38 0
Eisenstein series 8 8 0

Trace form

\( 38 q - 2 q^{2} - 2 q^{5} - 16 q^{6} - 8 q^{8} - 278 q^{9} - 4 q^{13} + 204 q^{14} - 76 q^{16} + 222 q^{18} + 356 q^{20} + 224 q^{21} - 48 q^{22} - 496 q^{24} + 350 q^{26} - 356 q^{28} - 8 q^{29} + 548 q^{32}+ \cdots + 1854 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{4}^{\mathrm{new}}(52, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
52.4.f.a 52.f 52.f $2$ $3.068$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) 52.4.f.a \(-4\) \(0\) \(18\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+(-2 i-2)q^{2}+8 i q^{4}+(9 i+9)q^{5}+\cdots\)
52.4.f.b 52.f 52.f $36$ $3.068$ None 52.4.f.b \(2\) \(0\) \(-20\) \(0\) $\mathrm{SU}(2)[C_{4}]$