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Properties

Label 399.2.k
Level $399$
Weight $2$
Character orbit 399.k
Rep. character $\chi_{399}(64,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $40$
Newform subspaces $4$
Sturm bound $106$
Trace bound $2$

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Defining parameters

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

Dimensions

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

Total New Old
Modular forms 112 40 72
Cusp forms 96 40 56
Eisenstein series 16 0 16

Trace form

\( 40 q - 24 q^{4} - 4 q^{6} - 20 q^{9} + O(q^{10}) \) \( 40 q - 24 q^{4} - 4 q^{6} - 20 q^{9} - 4 q^{10} + 8 q^{13} + 4 q^{15} - 32 q^{16} - 4 q^{17} + 16 q^{19} - 80 q^{20} + 4 q^{21} + 44 q^{22} + 8 q^{23} - 32 q^{25} + 56 q^{26} + 4 q^{29} + 8 q^{30} - 16 q^{31} + 20 q^{32} - 4 q^{33} - 16 q^{34} - 4 q^{35} - 24 q^{36} - 44 q^{38} - 48 q^{40} - 8 q^{41} + 4 q^{43} + 36 q^{44} + 16 q^{46} - 28 q^{47} - 16 q^{48} + 40 q^{49} + 128 q^{50} + 8 q^{51} - 4 q^{52} + 12 q^{53} - 4 q^{54} + 4 q^{55} + 40 q^{58} + 20 q^{59} + 20 q^{60} + 32 q^{61} - 16 q^{62} + 48 q^{64} + 48 q^{65} - 8 q^{66} + 8 q^{67} + 40 q^{68} - 56 q^{69} - 8 q^{70} - 32 q^{71} - 4 q^{73} + 24 q^{74} + 16 q^{75} - 12 q^{76} + 16 q^{77} - 24 q^{78} + 36 q^{79} - 20 q^{80} - 20 q^{81} + 8 q^{82} - 8 q^{83} - 24 q^{84} - 16 q^{85} - 48 q^{87} - 144 q^{88} - 40 q^{89} - 4 q^{90} + 16 q^{91} - 20 q^{92} + 32 q^{93} - 88 q^{95} + 88 q^{96} - 44 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
399.2.k.a 399.k 19.c $8$ $3.186$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-1\) \(-4\) \(2\) \(8\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}+\beta _{4}q^{3}+(-1-\beta _{3}-\beta _{4}+\cdots)q^{4}+\cdots\)
399.2.k.b 399.k 19.c $8$ $3.186$ 8.0.310217769.2 None \(1\) \(4\) \(2\) \(-8\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{4}-\beta _{6})q^{2}+(1-\beta _{5})q^{3}+(-\beta _{1}+\cdots)q^{4}+\cdots\)
399.2.k.c 399.k 19.c $12$ $3.186$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-1\) \(-6\) \(0\) \(-12\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}-\beta _{7}q^{3}+(-1+\beta _{7}-\beta _{10}+\cdots)q^{4}+\cdots\)
399.2.k.d 399.k 19.c $12$ $3.186$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(1\) \(6\) \(-4\) \(12\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{2}+\beta _{4}q^{3}+(-1+\beta _{3}+\beta _{4}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(399, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(399, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(57, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(133, [\chi])\)\(^{\oplus 2}\)