# Properties

 Label 170.2.k.a Level $170$ Weight $2$ Character orbit 170.k Analytic conductor $1.357$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$170 = 2 \cdot 5 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 170.k (of order $$8$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.35745683436$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{8})$$ Coefficient field: $$\Q(\zeta_{16})$$ Defining polynomial: $$x^{8} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{16}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{16}^{2} q^{2} + ( \zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{3} + \zeta_{16}^{4} + \zeta_{16}^{5} ) q^{3} + \zeta_{16}^{4} q^{4} -\zeta_{16}^{7} q^{5} + ( \zeta_{16}^{3} + \zeta_{16}^{4} - \zeta_{16}^{5} + \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{6} + ( -\zeta_{16}^{3} - \zeta_{16}^{4} + \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{7} + \zeta_{16}^{6} q^{8} + ( 1 - 2 \zeta_{16} + 2 \zeta_{16}^{3} - \zeta_{16}^{4} + 2 \zeta_{16}^{6} ) q^{9} +O(q^{10})$$ $$q + \zeta_{16}^{2} q^{2} + ( \zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{3} + \zeta_{16}^{4} + \zeta_{16}^{5} ) q^{3} + \zeta_{16}^{4} q^{4} -\zeta_{16}^{7} q^{5} + ( \zeta_{16}^{3} + \zeta_{16}^{4} - \zeta_{16}^{5} + \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{6} + ( -\zeta_{16}^{3} - \zeta_{16}^{4} + \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{7} + \zeta_{16}^{6} q^{8} + ( 1 - 2 \zeta_{16} + 2 \zeta_{16}^{3} - \zeta_{16}^{4} + 2 \zeta_{16}^{6} ) q^{9} + \zeta_{16} q^{10} + ( 2 - 2 \zeta_{16}^{2} - \zeta_{16}^{3} + \zeta_{16}^{4} - 2 \zeta_{16}^{5} + \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{11} + ( -1 - \zeta_{16} + \zeta_{16}^{5} + \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{12} + ( -\zeta_{16} + \zeta_{16}^{3} - 3 \zeta_{16}^{4} + \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{13} + ( -1 - \zeta_{16} - \zeta_{16}^{5} - \zeta_{16}^{6} ) q^{14} + ( 1 + \zeta_{16} - \zeta_{16}^{2} + \zeta_{16}^{3} + \zeta_{16}^{4} ) q^{15} - q^{16} + ( -2 \zeta_{16} - 2 \zeta_{16}^{5} + 3 \zeta_{16}^{7} ) q^{17} + ( -2 + \zeta_{16}^{2} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - \zeta_{16}^{6} ) q^{18} + ( -2 + 2 \zeta_{16} + 3 \zeta_{16}^{2} + 2 \zeta_{16}^{3} - 2 \zeta_{16}^{4} + \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{19} + \zeta_{16}^{3} q^{20} + ( \zeta_{16} - 2 \zeta_{16}^{3} - 2 \zeta_{16}^{4} - 2 \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{21} + ( -1 + \zeta_{16} + 2 \zeta_{16}^{2} - 2 \zeta_{16}^{4} - \zeta_{16}^{5} + \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{22} + ( 3 - 3 \zeta_{16}^{2} + 2 \zeta_{16}^{3} + \zeta_{16}^{4} + 4 \zeta_{16}^{5} + \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{23} + ( -1 + \zeta_{16} - \zeta_{16}^{2} - \zeta_{16}^{3} + \zeta_{16}^{7} ) q^{24} -\zeta_{16}^{6} q^{25} + ( \zeta_{16} - \zeta_{16}^{3} + \zeta_{16}^{5} - 3 \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{26} + ( \zeta_{16} - 2 \zeta_{16}^{3} + 5 \zeta_{16}^{4} - 5 \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{27} + ( 1 - \zeta_{16}^{2} - \zeta_{16}^{3} - \zeta_{16}^{7} ) q^{28} + ( 1 + \zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{4} - \zeta_{16}^{5} - \zeta_{16}^{6} - 3 \zeta_{16}^{7} ) q^{29} + ( \zeta_{16}^{2} + \zeta_{16}^{3} - \zeta_{16}^{4} + \zeta_{16}^{5} + \zeta_{16}^{6} ) q^{30} + ( -2 - 4 \zeta_{16} + \zeta_{16}^{2} + \zeta_{16}^{3} + \zeta_{16}^{4} - 4 \zeta_{16}^{5} - 2 \zeta_{16}^{6} ) q^{31} -\zeta_{16}^{2} q^{32} + ( -2 + 5 \zeta_{16} + 2 \zeta_{16}^{2} - 4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} - 2 \zeta_{16}^{6} - 5 \zeta_{16}^{7} ) q^{33} + ( -3 \zeta_{16} - 2 \zeta_{16}^{3} - 2 \zeta_{16}^{7} ) q^{34} + ( -\zeta_{16}^{2} - \zeta_{16}^{3} + \zeta_{16}^{5} + \zeta_{16}^{6} ) q^{35} + ( 1 - 2 \zeta_{16}^{2} + \zeta_{16}^{4} - 2 \zeta_{16}^{5} + 2 \zeta_{16}^{7} ) q^{36} + ( -1 - 4 \zeta_{16}^{2} - 4 \zeta_{16}^{4} - \zeta_{16}^{6} ) q^{37} + ( \zeta_{16} - 2 \zeta_{16}^{2} + 2 \zeta_{16}^{3} + 3 \zeta_{16}^{4} + 2 \zeta_{16}^{5} - 2 \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{38} + ( 4 + 3 \zeta_{16} - 3 \zeta_{16}^{2} + 3 \zeta_{16}^{4} - 3 \zeta_{16}^{5} - 4 \zeta_{16}^{6} + 5 \zeta_{16}^{7} ) q^{39} + \zeta_{16}^{5} q^{40} + ( -4 \zeta_{16} + 2 \zeta_{16}^{3} + \zeta_{16}^{4} - \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{41} + ( -\zeta_{16} + \zeta_{16}^{3} - 2 \zeta_{16}^{5} - 2 \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{42} + ( -1 - \zeta_{16} + \zeta_{16}^{3} + \zeta_{16}^{4} + 2 \zeta_{16}^{5} + 2 \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{43} + ( -1 + 2 \zeta_{16} - \zeta_{16}^{2} + \zeta_{16}^{3} + 2 \zeta_{16}^{4} - 2 \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{44} + ( -2 + 2 \zeta_{16}^{2} - \zeta_{16}^{3} + 2 \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{45} + ( -1 - 2 \zeta_{16} + 3 \zeta_{16}^{2} - 3 \zeta_{16}^{4} + 2 \zeta_{16}^{5} + \zeta_{16}^{6} + 4 \zeta_{16}^{7} ) q^{46} + ( -3 \zeta_{16} + \zeta_{16}^{2} + \zeta_{16}^{4} + \zeta_{16}^{6} - 3 \zeta_{16}^{7} ) q^{47} + ( -\zeta_{16} - \zeta_{16}^{2} + \zeta_{16}^{3} - \zeta_{16}^{4} - \zeta_{16}^{5} ) q^{48} + ( -1 + 2 \zeta_{16} - 3 \zeta_{16}^{2} + 2 \zeta_{16}^{3} - \zeta_{16}^{4} - 2 \zeta_{16}^{5} + 2 \zeta_{16}^{7} ) q^{49} + q^{50} + ( -5 - \zeta_{16} + 3 \zeta_{16}^{2} - 5 \zeta_{16}^{3} - \zeta_{16}^{4} - 2 \zeta_{16}^{5} - 4 \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{51} + ( 3 - \zeta_{16} + \zeta_{16}^{3} - \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{52} + ( -1 + 3 \zeta_{16} - \zeta_{16}^{2} + 3 \zeta_{16}^{3} - \zeta_{16}^{4} + \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{53} + ( 5 - 2 \zeta_{16} + \zeta_{16}^{3} - 2 \zeta_{16}^{5} + 5 \zeta_{16}^{6} ) q^{54} + ( -2 \zeta_{16} - \zeta_{16}^{2} + \zeta_{16}^{3} - 2 \zeta_{16}^{4} + \zeta_{16}^{5} - \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{55} + ( \zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{4} - \zeta_{16}^{5} ) q^{56} + ( 2 - 2 \zeta_{16}^{2} + 8 \zeta_{16}^{3} + 2 \zeta_{16}^{4} - 3 \zeta_{16}^{5} + 2 \zeta_{16}^{6} + 8 \zeta_{16}^{7} ) q^{57} + ( 1 + 3 \zeta_{16} + \zeta_{16}^{2} + \zeta_{16}^{3} + \zeta_{16}^{4} - \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{58} + ( -5 - 2 \zeta_{16} + 2 \zeta_{16}^{3} + 5 \zeta_{16}^{4} - \zeta_{16}^{5} + 3 \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{59} + ( -1 + \zeta_{16}^{4} + \zeta_{16}^{5} - \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{60} + ( -3 - 3 \zeta_{16} - 3 \zeta_{16}^{2} - 2 \zeta_{16}^{3} - 5 \zeta_{16}^{4} + 5 \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{61} + ( 2 - 2 \zeta_{16}^{2} - 4 \zeta_{16}^{3} + \zeta_{16}^{4} + \zeta_{16}^{5} + \zeta_{16}^{6} - 4 \zeta_{16}^{7} ) q^{62} + ( 1 + \zeta_{16}^{2} - \zeta_{16}^{4} - \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{63} -\zeta_{16}^{4} q^{64} + ( -1 + \zeta_{16}^{2} - 3 \zeta_{16}^{3} + \zeta_{16}^{4} - \zeta_{16}^{6} ) q^{65} + ( 2 + 5 \zeta_{16} - 2 \zeta_{16}^{2} + 5 \zeta_{16}^{3} + 2 \zeta_{16}^{4} - 4 \zeta_{16}^{5} + 4 \zeta_{16}^{7} ) q^{66} + ( 3 \zeta_{16} + 2 \zeta_{16}^{2} + \zeta_{16}^{3} - \zeta_{16}^{5} - 2 \zeta_{16}^{6} - 3 \zeta_{16}^{7} ) q^{67} + ( 2 \zeta_{16} - 3 \zeta_{16}^{3} - 2 \zeta_{16}^{5} ) q^{68} + ( -2 - 3 \zeta_{16} - 9 \zeta_{16}^{3} + 9 \zeta_{16}^{5} + 3 \zeta_{16}^{7} ) q^{69} + ( -1 - \zeta_{16}^{4} - \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{70} + ( 3 + \zeta_{16} + 4 \zeta_{16}^{2} - 5 \zeta_{16}^{3} + 4 \zeta_{16}^{4} + \zeta_{16}^{5} + 3 \zeta_{16}^{6} ) q^{71} + ( -2 \zeta_{16} + \zeta_{16}^{2} - 2 \zeta_{16}^{4} + \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{72} + ( -2 + 2 \zeta_{16} + 2 \zeta_{16}^{2} - 2 \zeta_{16}^{4} - 2 \zeta_{16}^{5} + 2 \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{73} + ( 1 - \zeta_{16}^{2} - 4 \zeta_{16}^{4} - 4 \zeta_{16}^{6} ) q^{74} + ( 1 - \zeta_{16} + \zeta_{16}^{2} + \zeta_{16}^{3} - \zeta_{16}^{7} ) q^{75} + ( 2 - \zeta_{16} + \zeta_{16}^{3} - 2 \zeta_{16}^{4} + 2 \zeta_{16}^{5} + 3 \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{76} + ( 1 + 2 \zeta_{16} - 2 \zeta_{16}^{3} - \zeta_{16}^{4} + 2 \zeta_{16}^{5} + 6 \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{77} + ( 4 - 5 \zeta_{16} + 4 \zeta_{16}^{2} + 3 \zeta_{16}^{3} - 3 \zeta_{16}^{4} + 3 \zeta_{16}^{6} - 3 \zeta_{16}^{7} ) q^{78} + ( -1 + \zeta_{16}^{2} - 5 \zeta_{16}^{3} - 3 \zeta_{16}^{4} + 2 \zeta_{16}^{5} - 3 \zeta_{16}^{6} - 5 \zeta_{16}^{7} ) q^{79} + \zeta_{16}^{7} q^{80} + ( -6 \zeta_{16} + 5 \zeta_{16}^{2} + 4 \zeta_{16}^{3} + \zeta_{16}^{4} + 4 \zeta_{16}^{5} + 5 \zeta_{16}^{6} - 6 \zeta_{16}^{7} ) q^{81} + ( 1 + 2 \zeta_{16} - 4 \zeta_{16}^{3} + 2 \zeta_{16}^{5} + \zeta_{16}^{6} ) q^{82} + ( 1 + 2 \zeta_{16} + 4 \zeta_{16}^{2} + 2 \zeta_{16}^{3} + \zeta_{16}^{4} - \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{83} + ( 2 + 2 \zeta_{16} - \zeta_{16}^{3} + \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{84} + ( -2 - 2 \zeta_{16}^{4} + 3 \zeta_{16}^{6} ) q^{85} + ( -2 - 2 \zeta_{16} - \zeta_{16}^{2} - \zeta_{16}^{3} + \zeta_{16}^{5} + \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{86} + ( 4 + 5 \zeta_{16} + \zeta_{16}^{2} + 5 \zeta_{16}^{3} + 4 \zeta_{16}^{4} ) q^{87} + ( 2 + \zeta_{16} - \zeta_{16}^{2} + 2 \zeta_{16}^{3} - \zeta_{16}^{4} + \zeta_{16}^{5} + 2 \zeta_{16}^{6} ) q^{88} + ( 4 \zeta_{16} + 4 \zeta_{16}^{2} + \zeta_{16}^{4} + 4 \zeta_{16}^{6} + 4 \zeta_{16}^{7} ) q^{89} + ( \zeta_{16} - 2 \zeta_{16}^{2} + 2 \zeta_{16}^{4} - \zeta_{16}^{5} + 2 \zeta_{16}^{7} ) q^{90} + ( -1 + \zeta_{16}^{2} + \zeta_{16}^{3} + 2 \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{91} + ( -1 - 4 \zeta_{16} - \zeta_{16}^{2} - 2 \zeta_{16}^{3} + 3 \zeta_{16}^{4} - 3 \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{92} + ( -4 - \zeta_{16} + \zeta_{16}^{3} + 4 \zeta_{16}^{4} - 5 \zeta_{16}^{5} - 7 \zeta_{16}^{6} - 5 \zeta_{16}^{7} ) q^{93} + ( -1 + 3 \zeta_{16} - 3 \zeta_{16}^{3} + \zeta_{16}^{4} + \zeta_{16}^{6} ) q^{94} + ( 2 + 3 \zeta_{16} + 2 \zeta_{16}^{2} - 2 \zeta_{16}^{3} + \zeta_{16}^{4} - \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{95} + ( -\zeta_{16}^{3} - \zeta_{16}^{4} + \zeta_{16}^{5} - \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{96} + ( -4 + 8 \zeta_{16} - 2 \zeta_{16}^{2} + 2 \zeta_{16}^{4} - 8 \zeta_{16}^{5} + 4 \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{97} + ( -2 \zeta_{16} - \zeta_{16}^{2} + 2 \zeta_{16}^{3} - 3 \zeta_{16}^{4} + 2 \zeta_{16}^{5} - \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{98} + ( 9 - 6 \zeta_{16} - \zeta_{16}^{2} + 10 \zeta_{16}^{3} - \zeta_{16}^{4} - 6 \zeta_{16}^{5} + 9 \zeta_{16}^{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 8q^{9} + O(q^{10})$$ $$8q + 8q^{9} + 16q^{11} - 8q^{12} - 8q^{14} + 8q^{15} - 8q^{16} - 16q^{18} - 16q^{19} - 8q^{22} + 24q^{23} - 8q^{24} + 8q^{28} + 8q^{29} - 16q^{31} - 16q^{33} + 8q^{36} - 8q^{37} + 32q^{39} - 8q^{43} - 8q^{44} - 16q^{45} - 8q^{46} - 8q^{49} + 8q^{50} - 40q^{51} + 24q^{52} - 8q^{53} + 40q^{54} + 16q^{57} + 8q^{58} - 40q^{59} - 8q^{60} - 24q^{61} + 16q^{62} + 8q^{63} - 8q^{65} + 16q^{66} - 16q^{69} - 8q^{70} + 24q^{71} - 16q^{73} + 8q^{74} + 8q^{75} + 16q^{76} + 8q^{77} + 32q^{78} - 8q^{79} + 8q^{82} + 8q^{83} + 16q^{84} - 16q^{85} - 16q^{86} + 32q^{87} + 16q^{88} - 8q^{91} - 8q^{92} - 32q^{93} - 8q^{94} + 16q^{95} - 32q^{97} + 72q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/170\mathbb{Z}\right)^\times$$.

 $$n$$ $$71$$ $$137$$ $$\chi(n)$$ $$\zeta_{16}^{6}$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
111.1
 −0.382683 − 0.923880i 0.382683 + 0.923880i −0.382683 + 0.923880i 0.382683 − 0.923880i −0.923880 − 0.382683i 0.923880 + 0.382683i −0.923880 + 0.382683i 0.923880 − 0.382683i
−0.707107 + 0.707107i −2.93755 1.21677i 1.00000i −0.382683 + 0.923880i 2.93755 1.21677i 0.165911 + 0.400544i 0.707107 + 0.707107i 5.02734 + 5.02734i −0.382683 0.923880i
111.2 −0.707107 + 0.707107i 1.52334 + 0.630986i 1.00000i 0.382683 0.923880i −1.52334 + 0.630986i 1.24830 + 3.01367i 0.707107 + 0.707107i −0.198912 0.198912i 0.382683 + 0.923880i
121.1 −0.707107 0.707107i −2.93755 + 1.21677i 1.00000i −0.382683 0.923880i 2.93755 + 1.21677i 0.165911 0.400544i 0.707107 0.707107i 5.02734 5.02734i −0.382683 + 0.923880i
121.2 −0.707107 0.707107i 1.52334 0.630986i 1.00000i 0.382683 + 0.923880i −1.52334 0.630986i 1.24830 3.01367i 0.707107 0.707107i −0.198912 + 0.198912i 0.382683 0.923880i
151.1 0.707107 + 0.707107i 0.548594 + 1.32442i 1.00000i −0.923880 + 0.382683i −0.548594 + 1.32442i 0.599456 + 0.248303i −0.707107 + 0.707107i 0.668179 0.668179i −0.923880 0.382683i
151.2 0.707107 + 0.707107i 0.865619 + 2.08979i 1.00000i 0.923880 0.382683i −0.865619 + 2.08979i −2.01367 0.834089i −0.707107 + 0.707107i −1.49661 + 1.49661i 0.923880 + 0.382683i
161.1 0.707107 0.707107i 0.548594 1.32442i 1.00000i −0.923880 0.382683i −0.548594 1.32442i 0.599456 0.248303i −0.707107 0.707107i 0.668179 + 0.668179i −0.923880 + 0.382683i
161.2 0.707107 0.707107i 0.865619 2.08979i 1.00000i 0.923880 + 0.382683i −0.865619 2.08979i −2.01367 + 0.834089i −0.707107 0.707107i −1.49661 1.49661i 0.923880 0.382683i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 161.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 170.2.k.a 8
5.b even 2 1 850.2.l.d 8
5.c odd 4 1 850.2.o.c 8
5.c odd 4 1 850.2.o.f 8
17.d even 8 1 inner 170.2.k.a 8
17.e odd 16 1 2890.2.a.bc 4
17.e odd 16 1 2890.2.a.bf 4
17.e odd 16 2 2890.2.b.p 8
85.k odd 8 1 850.2.o.f 8
85.m even 8 1 850.2.l.d 8
85.n odd 8 1 850.2.o.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.k.a 8 1.a even 1 1 trivial
170.2.k.a 8 17.d even 8 1 inner
850.2.l.d 8 5.b even 2 1
850.2.l.d 8 85.m even 8 1
850.2.o.c 8 5.c odd 4 1
850.2.o.c 8 85.n odd 8 1
850.2.o.f 8 5.c odd 4 1
850.2.o.f 8 85.k odd 8 1
2890.2.a.bc 4 17.e odd 16 1
2890.2.a.bf 4 17.e odd 16 1
2890.2.b.p 8 17.e odd 16 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} - 4 T_{3}^{6} + 16 T_{3}^{5} + 8 T_{3}^{4} - 136 T_{3}^{3} + 332 T_{3}^{2} - 408 T_{3} + 289$$ acting on $$S_{2}^{\mathrm{new}}(170, [\chi])$$.