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Properties

Label 406.2.e
Level $406$
Weight $2$
Character orbit 406.e
Rep. character $\chi_{406}(233,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $40$
Newform subspaces $4$
Sturm bound $120$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 406 = 2 \cdot 7 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 406.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 4 \)
Sturm bound: \(120\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(406, [\chi])\).

Total New Old
Modular forms 128 40 88
Cusp forms 112 40 72
Eisenstein series 16 0 16

Trace form

\( 40 q - 20 q^{4} - 4 q^{5} - 4 q^{7} - 24 q^{9} + O(q^{10}) \) \( 40 q - 20 q^{4} - 4 q^{5} - 4 q^{7} - 24 q^{9} + 4 q^{10} - 8 q^{11} + 8 q^{13} + 4 q^{14} - 16 q^{15} - 20 q^{16} + 16 q^{17} + 8 q^{20} + 16 q^{21} - 16 q^{22} + 4 q^{23} - 24 q^{25} + 24 q^{27} - 4 q^{28} + 4 q^{30} + 12 q^{31} - 28 q^{33} - 4 q^{35} + 48 q^{36} + 20 q^{37} - 4 q^{39} + 4 q^{40} - 24 q^{41} + 16 q^{43} - 8 q^{44} + 4 q^{45} + 4 q^{46} - 4 q^{47} - 20 q^{49} + 16 q^{50} - 20 q^{51} - 4 q^{52} - 20 q^{53} - 40 q^{55} - 8 q^{56} - 48 q^{57} + 8 q^{59} + 8 q^{60} + 16 q^{61} + 48 q^{62} - 4 q^{63} + 40 q^{64} - 20 q^{65} + 16 q^{66} + 16 q^{68} + 72 q^{69} + 8 q^{70} - 64 q^{71} + 4 q^{73} - 4 q^{74} + 44 q^{75} - 32 q^{77} - 16 q^{78} - 16 q^{79} - 4 q^{80} - 12 q^{81} - 24 q^{82} + 88 q^{83} - 32 q^{84} + 16 q^{85} + 12 q^{86} + 8 q^{88} + 28 q^{89} - 32 q^{90} - 12 q^{91} - 8 q^{92} - 28 q^{93} - 36 q^{94} + 28 q^{95} + 72 q^{97} - 24 q^{98} + 80 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(406, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
406.2.e.a 406.e 7.c $10$ $3.242$ 10.0.\(\cdots\).1 None \(-5\) \(-3\) \(-7\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{4})q^{2}+(-\beta _{3}-\beta _{4})q^{3}-\beta _{4}q^{4}+\cdots\)
406.2.e.b 406.e 7.c $10$ $3.242$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(-5\) \(3\) \(7\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1-\beta _{6})q^{2}+(-\beta _{1}-\beta _{6})q^{3}+\beta _{6}q^{4}+\cdots\)
406.2.e.c 406.e 7.c $10$ $3.242$ 10.0.\(\cdots\).1 None \(5\) \(-3\) \(-5\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{6}q^{2}+(-\beta _{1}-\beta _{7})q^{3}+(-1+\beta _{6}+\cdots)q^{4}+\cdots\)
406.2.e.d 406.e 7.c $10$ $3.242$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(5\) \(3\) \(1\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{6}q^{2}+(1-\beta _{5}+\beta _{6})q^{3}+(-1-\beta _{6}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(406, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(406, [\chi]) \cong \)