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Properties

Label 414.2.e
Level $414$
Weight $2$
Character orbit 414.e
Rep. character $\chi_{414}(139,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $44$
Newform subspaces $5$
Sturm bound $144$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 414.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 5 \)
Sturm bound: \(144\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(414, [\chi])\).

Total New Old
Modular forms 152 44 108
Cusp forms 136 44 92
Eisenstein series 16 0 16

Trace form

\( 44 q + 2 q^{2} + 2 q^{3} - 22 q^{4} + 2 q^{6} + 4 q^{7} - 4 q^{8} - 14 q^{9} + O(q^{10}) \) \( 44 q + 2 q^{2} + 2 q^{3} - 22 q^{4} + 2 q^{6} + 4 q^{7} - 4 q^{8} - 14 q^{9} - 10 q^{11} - 4 q^{12} + 4 q^{13} + 20 q^{15} - 22 q^{16} + 28 q^{17} - 4 q^{18} + 4 q^{19} - 16 q^{21} - 6 q^{22} + 2 q^{24} - 22 q^{25} - 24 q^{26} + 20 q^{27} - 8 q^{28} + 28 q^{29} - 12 q^{30} + 4 q^{31} + 2 q^{32} + 30 q^{33} - 6 q^{34} + 10 q^{36} - 8 q^{37} - 14 q^{38} - 48 q^{39} + 14 q^{41} - 20 q^{42} - 2 q^{43} + 20 q^{44} - 16 q^{45} - 20 q^{47} + 2 q^{48} - 18 q^{49} - 2 q^{50} + 22 q^{51} + 4 q^{52} - 80 q^{53} - 10 q^{54} - 24 q^{55} + 10 q^{57} + 12 q^{58} - 2 q^{59} - 16 q^{60} + 28 q^{61} - 8 q^{62} + 48 q^{63} + 44 q^{64} + 20 q^{65} + 22 q^{67} - 14 q^{68} + 12 q^{70} + 72 q^{71} + 2 q^{72} - 44 q^{73} - 2 q^{75} - 2 q^{76} - 16 q^{77} - 4 q^{78} - 8 q^{79} - 14 q^{81} - 36 q^{82} - 12 q^{83} - 4 q^{84} - 30 q^{86} + 52 q^{87} - 6 q^{88} + 72 q^{89} + 28 q^{90} - 16 q^{91} + 36 q^{93} - 12 q^{95} - 4 q^{96} + 22 q^{97} - 4 q^{98} + 12 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(414, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
414.2.e.a 414.e 9.c $2$ $3.306$ \(\Q(\sqrt{-3}) \) None \(1\) \(3\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}+(1+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
414.2.e.b 414.e 9.c $10$ $3.306$ 10.0.\(\cdots\).1 None \(-5\) \(1\) \(-5\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{3}q^{2}-\beta _{7}q^{3}+(-1+\beta _{3})q^{4}+(-1+\cdots)q^{5}+\cdots\)
414.2.e.c 414.e 9.c $10$ $3.306$ 10.0.\(\cdots\).1 None \(-5\) \(1\) \(5\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1-\beta _{4})q^{2}+(1+\beta _{5}-\beta _{6}-\beta _{9})q^{3}+\cdots\)
414.2.e.d 414.e 9.c $10$ $3.306$ 10.0.\(\cdots\).1 None \(5\) \(-3\) \(-1\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{7})q^{2}+\beta _{1}q^{3}+\beta _{7}q^{4}+(\beta _{2}+\cdots)q^{5}+\cdots\)
414.2.e.e 414.e 9.c $12$ $3.306$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(6\) \(0\) \(5\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{2}q^{2}-\beta _{1}q^{3}+(-1+\beta _{2})q^{4}+(1+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(414, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(414, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(207, [\chi])\)\(^{\oplus 2}\)