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Properties

Label 164.2.f
Level $164$
Weight $2$
Character orbit 164.f
Rep. character $\chi_{164}(9,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $6$
Newform subspaces $1$
Sturm bound $42$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 164 = 2^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 164.f (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 41 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 1 \)
Sturm bound: \(42\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(164, [\chi])\).

Total New Old
Modular forms 48 6 42
Cusp forms 36 6 30
Eisenstein series 12 0 12

Trace form

\( 6 q + 2 q^{3} + O(q^{10}) \) \( 6 q + 2 q^{3} - 4 q^{11} - 2 q^{13} - 2 q^{15} - 2 q^{17} + 2 q^{19} + 4 q^{23} + 6 q^{25} + 2 q^{27} - 10 q^{29} + 8 q^{31} + 18 q^{35} - 36 q^{37} + 6 q^{41} - 20 q^{45} - 26 q^{47} + 12 q^{51} - 14 q^{53} - 6 q^{55} - 24 q^{57} - 4 q^{59} + 18 q^{63} + 20 q^{65} + 16 q^{67} + 8 q^{69} + 26 q^{71} - 10 q^{75} - 2 q^{79} + 38 q^{81} + 36 q^{83} + 44 q^{85} + 14 q^{89} - 28 q^{93} - 26 q^{95} - 6 q^{97} - 2 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(164, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
164.2.f.a 164.f 41.c $6$ $1.310$ 6.0.5089536.1 None \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{3}+(-\beta _{4}+\beta _{5})q^{5}+(\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(164, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(164, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(41, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(82, [\chi])\)\(^{\oplus 2}\)