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Properties

Label 65.2.f
Level $65$
Weight $2$
Character orbit 65.f
Rep. character $\chi_{65}(18,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $10$
Newform subspaces $2$
Sturm bound $14$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 65 = 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 65.f (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 65 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 2 \)
Sturm bound: \(14\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(65, [\chi])\).

Total New Old
Modular forms 18 18 0
Cusp forms 10 10 0
Eisenstein series 8 8 0

Trace form

\( 10 q - 4 q^{3} - 6 q^{4} - 6 q^{5} - 4 q^{6} - 4 q^{7} + O(q^{10}) \) \( 10 q - 4 q^{3} - 6 q^{4} - 6 q^{5} - 4 q^{6} - 4 q^{7} + 8 q^{10} + 4 q^{11} + 10 q^{13} - 4 q^{15} - 10 q^{16} + 14 q^{17} + 18 q^{18} - 4 q^{19} - 6 q^{20} - 16 q^{21} + 8 q^{22} - 20 q^{23} + 4 q^{24} - 6 q^{25} + 12 q^{26} + 20 q^{27} - 12 q^{28} - 16 q^{30} + 12 q^{31} - 2 q^{34} - 16 q^{35} - 44 q^{37} + 8 q^{38} - 4 q^{39} + 28 q^{40} + 2 q^{41} + 20 q^{42} - 4 q^{43} - 12 q^{44} + 24 q^{45} + 8 q^{46} + 28 q^{47} - 16 q^{48} + 18 q^{49} + 36 q^{50} - 42 q^{52} - 14 q^{53} + 12 q^{54} + 16 q^{55} + 24 q^{58} - 8 q^{59} - 16 q^{60} - 8 q^{61} - 40 q^{62} + 34 q^{64} + 2 q^{65} - 40 q^{66} + 2 q^{68} - 8 q^{69} - 72 q^{70} - 8 q^{71} - 22 q^{72} + 44 q^{75} + 16 q^{76} - 20 q^{77} + 4 q^{78} - 22 q^{80} - 10 q^{81} + 34 q^{82} + 60 q^{83} - 20 q^{84} + 38 q^{85} - 48 q^{86} + 16 q^{87} - 16 q^{88} + 18 q^{89} - 10 q^{90} + 28 q^{91} + 44 q^{92} - 20 q^{93} - 28 q^{95} + 40 q^{96} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(65, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
65.2.f.a 65.f 65.f $2$ $0.519$ \(\Q(\sqrt{-1}) \) None \(0\) \(2\) \(-4\) \(-4\) $\mathrm{SU}(2)[C_{4}]$ \(q+iq^{2}+(1-i)q^{3}+q^{4}+(-2-i)q^{5}+\cdots\)
65.2.f.b 65.f 65.f $8$ $0.519$ 8.0.619810816.2 None \(0\) \(-6\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{6}q^{2}+(-1+\beta _{2}+\beta _{4}-\beta _{5})q^{3}+\cdots\)