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Properties

Label 399.2.w
Level $399$
Weight $2$
Character orbit 399.w
Rep. character $\chi_{399}(170,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $100$
Newform subspaces $4$
Sturm bound $106$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 399 = 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 399.w (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 399 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 4 \)
Sturm bound: \(106\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(5\), \(31\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(399, [\chi])\).

Total New Old
Modular forms 116 116 0
Cusp forms 100 100 0
Eisenstein series 16 16 0

Trace form

\( 100 q - 52 q^{4} - 2 q^{7} - 2 q^{9} + O(q^{10}) \) \( 100 q - 52 q^{4} - 2 q^{7} - 2 q^{9} - 56 q^{16} - 3 q^{19} - 8 q^{24} + 42 q^{25} + 8 q^{28} - 24 q^{36} + 2 q^{39} - 104 q^{42} + 20 q^{43} - 44 q^{45} + 10 q^{49} - 44 q^{54} - 88 q^{55} - 6 q^{57} + 4 q^{58} + 4 q^{61} + 2 q^{63} + 104 q^{64} + 28 q^{66} + 6 q^{73} + 124 q^{76} + 46 q^{81} - 92 q^{82} - 72 q^{85} - 48 q^{87} - 46 q^{93} + 100 q^{96} + 104 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(399, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
399.2.w.a 399.w 399.w $2$ $3.186$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) 399.2.w.a \(0\) \(-3\) \(0\) \(1\) $\mathrm{U}(1)[D_{6}]$ \(q+(-2+\zeta_{6})q^{3}+2\zeta_{6}q^{4}+(-1+3\zeta_{6})q^{7}+\cdots\)
399.2.w.b 399.w 399.w $2$ $3.186$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) 399.2.w.a \(0\) \(3\) \(0\) \(1\) $\mathrm{U}(1)[D_{6}]$ \(q+(2-\zeta_{6})q^{3}+2\zeta_{6}q^{4}+(-1+3\zeta_{6})q^{7}+\cdots\)
399.2.w.c 399.w 399.w $8$ $3.186$ 8.0.3317760000.8 None 399.2.w.c \(0\) \(0\) \(0\) \(-16\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{1}q^{3}+2\beta _{4}q^{4}+\beta _{6}q^{5}+(-1-2\beta _{4}+\cdots)q^{7}+\cdots\)
399.2.w.d 399.w 399.w $88$ $3.186$ None 399.2.w.d \(0\) \(0\) \(0\) \(12\) $\mathrm{SU}(2)[C_{6}]$