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Properties

Label 170.2.k
Level $170$
Weight $2$
Character orbit 170.k
Rep. character $\chi_{170}(111,\cdot)$
Character field $\Q(\zeta_{8})$
Dimension $24$
Newform subspaces $2$
Sturm bound $54$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 170 = 2 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 170.k (of order \(8\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 17 \)
Character field: \(\Q(\zeta_{8})\)
Newform subspaces: \( 2 \)
Sturm bound: \(54\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(170, [\chi])\).

Total New Old
Modular forms 128 24 104
Cusp forms 96 24 72
Eisenstein series 32 0 32

Trace form

\( 24 q + 8 q^{9} + O(q^{10}) \) \( 24 q + 8 q^{9} + 8 q^{11} - 8 q^{12} - 16 q^{14} + 16 q^{15} - 24 q^{16} - 8 q^{18} - 16 q^{19} - 16 q^{22} + 32 q^{23} - 8 q^{24} - 24 q^{27} + 16 q^{29} + 16 q^{31} + 16 q^{34} + 16 q^{35} + 8 q^{36} - 16 q^{37} - 32 q^{41} + 32 q^{42} - 24 q^{43} - 32 q^{45} - 32 q^{46} - 16 q^{49} - 8 q^{50} - 48 q^{51} + 16 q^{52} - 48 q^{53} + 40 q^{54} - 24 q^{59} - 16 q^{60} - 48 q^{61} + 48 q^{62} + 64 q^{63} - 16 q^{65} + 8 q^{66} + 16 q^{67} - 32 q^{69} + 32 q^{71} + 8 q^{75} + 16 q^{76} + 32 q^{77} + 64 q^{78} + 32 q^{79} + 24 q^{82} + 40 q^{83} + 32 q^{84} - 16 q^{86} + 24 q^{88} + 16 q^{91} + 16 q^{92} - 64 q^{93} + 32 q^{94} + 32 q^{95} - 8 q^{97} + 16 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(170, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
170.2.k.a 170.k 17.d $8$ $1.357$ \(\Q(\zeta_{16})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{8}]$ \(q+\zeta_{16}^{2}q^{2}+(\zeta_{16}+\zeta_{16}^{2}-\zeta_{16}^{3}+\cdots)q^{3}+\cdots\)
170.2.k.b 170.k 17.d $16$ $1.357$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{8}]$ \(q+\beta _{9}q^{2}+(-\beta _{6}+\beta _{15})q^{3}-\beta _{3}q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(170, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(170, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(17, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(34, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(85, [\chi])\)\(^{\oplus 2}\)