# Properties

 Label 170.2.k.b Level $170$ Weight $2$ Character orbit 170.k Analytic conductor $1.357$ Analytic rank $0$ Dimension $16$ 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: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{8})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 28 x^{14} + 286 x^{12} + 1412 x^{10} + 3709 x^{8} + 5264 x^{6} + 3780 x^{4} + 1072 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 28 x^{14} + 286 x^{12} + 1412 x^{10} + 3709 x^{8} + 5264 x^{6} + 3780 x^{4} + 1072 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$3 \nu^{14} + 72 \nu^{12} + 553 \nu^{10} + 1599 \nu^{8} + 1178 \nu^{6} - 1415 \nu^{4} - 1496 \nu^{2} - 150$$$$)/136$$ $$\beta_{3}$$ $$=$$ $$($$$$7 \nu^{15} + 202 \nu^{13} + 2146 \nu^{11} + 10990 \nu^{9} + 29161 \nu^{7} + 39204 \nu^{5} + 23630 \nu^{3} + 4512 \nu$$$$)/272$$ $$\beta_{4}$$ $$=$$ $$($$$$-606 \nu^{15} + 2681 \nu^{14} - 15156 \nu^{13} + 70430 \nu^{12} - 124847 \nu^{11} + 645475 \nu^{10} - 398393 \nu^{9} + 2683097 \nu^{8} - 268947 \nu^{7} + 5426998 \nu^{6} + 992877 \nu^{5} + 5200301 \nu^{4} + 1813186 \nu^{3} + 1930112 \nu^{2} + 795538 \nu + 27314$$$$)/52496$$ $$\beta_{5}$$ $$=$$ $$($$$$606 \nu^{15} + 2681 \nu^{14} + 15156 \nu^{13} + 70430 \nu^{12} + 124847 \nu^{11} + 645475 \nu^{10} + 398393 \nu^{9} + 2683097 \nu^{8} + 268947 \nu^{7} + 5426998 \nu^{6} - 992877 \nu^{5} + 5200301 \nu^{4} - 1813186 \nu^{3} + 1930112 \nu^{2} - 795538 \nu + 27314$$$$)/52496$$ $$\beta_{6}$$ $$=$$ $$($$$$-553 \nu^{15} + 2940 \nu^{14} - 14870 \nu^{13} + 78142 \nu^{12} - 141535 \nu^{11} + 730079 \nu^{10} - 619755 \nu^{9} + 3121007 \nu^{8} - 1312806 \nu^{7} + 6530061 \nu^{6} - 1165423 \nu^{5} + 6402955 \nu^{4} - 33592 \nu^{3} + 2271370 \nu^{2} + 326578 \nu - 40002$$$$)/52496$$ $$\beta_{7}$$ $$=$$ $$($$$$-553 \nu^{15} - 2940 \nu^{14} - 14870 \nu^{13} - 78142 \nu^{12} - 141535 \nu^{11} - 730079 \nu^{10} - 619755 \nu^{9} - 3121007 \nu^{8} - 1312806 \nu^{7} - 6530061 \nu^{6} - 1165423 \nu^{5} - 6402955 \nu^{4} - 33592 \nu^{3} - 2271370 \nu^{2} + 326578 \nu + 40002$$$$)/52496$$ $$\beta_{8}$$ $$=$$ $$($$$$2681 \nu^{15} + 1812 \nu^{14} + 70430 \nu^{13} + 48469 \nu^{12} + 645475 \nu^{11} + 457279 \nu^{10} + 2683097 \nu^{9} + 1978707 \nu^{8} + 5426998 \nu^{7} + 4182861 \nu^{6} + 5200301 \nu^{5} + 4103866 \nu^{4} + 1930112 \nu^{3} + 1445170 \nu^{2} + 27314 \nu + 2424$$$$)/52496$$ $$\beta_{9}$$ $$=$$ $$($$$$-2448 \nu^{15} + 2536 \nu^{14} - 65025 \nu^{13} + 68038 \nu^{12} - 608090 \nu^{11} + 645754 \nu^{10} - 2619768 \nu^{9} + 2828070 \nu^{8} - 5650018 \nu^{7} + 6124374 \nu^{6} - 6128857 \nu^{5} + 6300400 \nu^{4} - 3123580 \nu^{3} + 2458676 \nu^{2} - 615366 \nu + 46328$$$$)/52496$$ $$\beta_{10}$$ $$=$$ $$($$$$-2448 \nu^{15} - 2536 \nu^{14} - 65025 \nu^{13} - 68038 \nu^{12} - 608090 \nu^{11} - 645754 \nu^{10} - 2619768 \nu^{9} - 2828070 \nu^{8} - 5650018 \nu^{7} - 6124374 \nu^{6} - 6128857 \nu^{5} - 6300400 \nu^{4} - 3123580 \nu^{3} - 2458676 \nu^{2} - 615366 \nu - 46328$$$$)/52496$$ $$\beta_{11}$$ $$=$$ $$($$$$-3546 \nu^{15} - 2067 \nu^{14} - 93298 \nu^{13} - 53807 \nu^{12} - 854926 \nu^{11} - 484394 \nu^{10} - 3519400 \nu^{9} - 1944826 \nu^{8} - 6799008 \nu^{7} - 3681429 \nu^{6} - 5410078 \nu^{5} - 3143553 \nu^{4} - 458184 \nu^{3} - 1010718 \nu^{2} + 835540 \nu - 25102$$$$)/52496$$ $$\beta_{12}$$ $$=$$ $$($$$$2536 \nu^{15} - 3519 \nu^{14} + 68038 \nu^{13} - 92038 \nu^{12} + 645754 \nu^{11} - 836808 \nu^{10} + 2828070 \nu^{9} - 3429614 \nu^{8} + 6124374 \nu^{7} - 6757415 \nu^{6} + 6300400 \nu^{5} - 6129860 \nu^{4} + 2458676 \nu^{3} - 2008890 \nu^{2} + 46328 \nu - 62288$$$$)/52496$$ $$\beta_{13}$$ $$=$$ $$($$$$-2536 \nu^{15} - 3519 \nu^{14} - 68038 \nu^{13} - 92038 \nu^{12} - 645754 \nu^{11} - 836808 \nu^{10} - 2828070 \nu^{9} - 3429614 \nu^{8} - 6124374 \nu^{7} - 6757415 \nu^{6} - 6300400 \nu^{5} - 6129860 \nu^{4} - 2458676 \nu^{3} - 2008890 \nu^{2} - 46328 \nu - 9792$$$$)/52496$$ $$\beta_{14}$$ $$=$$ $$($$$$3546 \nu^{15} + 3295 \nu^{14} + 93298 \nu^{13} + 87053 \nu^{12} + 854926 \nu^{11} + 806556 \nu^{10} + 3519400 \nu^{9} + 3421368 \nu^{8} + 6799008 \nu^{7} + 7172567 \nu^{6} + 5410078 \nu^{5} + 7257049 \nu^{4} + 458184 \nu^{3} + 2849506 \nu^{2} - 835540 \nu + 29526$$$$)/52496$$ $$\beta_{15}$$ $$=$$ $$($$$$-2681 \nu^{15} + 7692 \nu^{14} - 70430 \nu^{13} + 204753 \nu^{12} - 645475 \nu^{11} + 1917437 \nu^{10} - 2683097 \nu^{9} + 8220721 \nu^{8} - 5426998 \nu^{7} + 17242983 \nu^{6} - 5200301 \nu^{5} + 16909776 \nu^{4} - 1930112 \nu^{3} + 5987910 \nu^{2} - 27314 \nu - 77580$$$$)/52496$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{13} - \beta_{12} + \beta_{7} - \beta_{6} - \beta_{2} - 4$$ $$\nu^{3}$$ $$=$$ $$-\beta_{14} + 2 \beta_{13} - 2 \beta_{12} + \beta_{11} - 2 \beta_{10} - 2 \beta_{9} - \beta_{7} - \beta_{6} + 4 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 5 \beta_{1} - 2$$ $$\nu^{4}$$ $$=$$ $$-4 \beta_{15} - 2 \beta_{14} + 9 \beta_{13} + 9 \beta_{12} - 2 \beta_{11} - 5 \beta_{10} + 5 \beta_{9} - 4 \beta_{8} - 13 \beta_{7} + 13 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + 11 \beta_{2} + 32$$ $$\nu^{5}$$ $$=$$ $$6 \beta_{15} + 19 \beta_{14} - 25 \beta_{13} + 25 \beta_{12} - 19 \beta_{11} + 24 \beta_{10} + 24 \beta_{9} - 6 \beta_{8} + 19 \beta_{7} + 7 \beta_{6} - 66 \beta_{5} + 28 \beta_{4} - 30 \beta_{3} + 41 \beta_{1} + 25$$ $$\nu^{6}$$ $$=$$ $$72 \beta_{15} + 32 \beta_{14} - 90 \beta_{13} - 90 \beta_{12} + 32 \beta_{11} + 75 \beta_{10} - 75 \beta_{9} + 72 \beta_{8} + 176 \beta_{7} - 176 \beta_{6} + 37 \beta_{5} + 37 \beta_{4} - 121 \beta_{2} - 331$$ $$\nu^{7}$$ $$=$$ $$-109 \beta_{15} - 280 \beta_{14} + 286 \beta_{13} - 286 \beta_{12} + 280 \beta_{11} - 284 \beta_{10} - 284 \beta_{9} + 109 \beta_{8} - 263 \beta_{7} - 45 \beta_{6} + 915 \beta_{5} - 355 \beta_{4} + 386 \beta_{3} - 421 \beta_{1} - 286$$ $$\nu^{8}$$ $$=$$ $$-1024 \beta_{15} - 434 \beta_{14} + 991 \beta_{13} + 991 \beta_{12} - 434 \beta_{11} - 961 \beta_{10} + 961 \beta_{9} - 1024 \beta_{8} - 2291 \beta_{7} + 2291 \beta_{6} - 504 \beta_{5} - 504 \beta_{4} + 1379 \beta_{2} + 3772$$ $$\nu^{9}$$ $$=$$ $$1528 \beta_{15} + 3749 \beta_{14} - 3331 \beta_{13} + 3331 \beta_{12} - 3749 \beta_{11} + 3440 \beta_{10} + 3440 \beta_{9} - 1528 \beta_{8} + 3387 \beta_{7} + 331 \beta_{6} - 11916 \beta_{5} + 4418 \beta_{4} - 4806 \beta_{3} + 4763 \beta_{1} + 3331$$ $$\nu^{10}$$ $$=$$ $$13444 \beta_{15} + 5608 \beta_{14} - 11534 \beta_{13} - 11534 \beta_{12} + 5608 \beta_{11} + 11939 \beta_{10} - 11939 \beta_{9} + 13444 \beta_{8} + 29036 \beta_{7} - 29036 \beta_{6} + 6387 \beta_{5} + 6387 \beta_{4} - 16231 \beta_{2} - 44789$$ $$\nu^{11}$$ $$=$$ $$-19831 \beta_{15} - 48088 \beta_{14} + 39704 \beta_{13} - 39704 \beta_{12} + 48088 \beta_{11} - 42120 \beta_{10} - 42120 \beta_{9} + 19831 \beta_{8} - 42581 \beta_{7} - 2919 \beta_{6} + 150829 \beta_{5} - 54653 \beta_{4} + 59350 \beta_{3} - 56323 \beta_{1} - 39704$$ $$\nu^{12}$$ $$=$$ $$-170660 \beta_{15} - 70838 \beta_{14} + 138147 \beta_{13} + 138147 \beta_{12} - 70838 \beta_{11} - 147327 \beta_{10} + 147327 \beta_{9} - 170660 \beta_{8} - 362863 \beta_{7} + 362863 \beta_{6} - 79366 \beta_{5} - 79366 \beta_{4} + 195081 \beta_{2} + 542050$$ $$\nu^{13}$$ $$=$$ $$250026 \beta_{15} + 604361 \beta_{14} - 480555 \beta_{13} + 480555 \beta_{12} - 604361 \beta_{11} + 517792 \beta_{10} + 517792 \beta_{9} - 250026 \beta_{8} + 529849 \beta_{7} + 29797 \beta_{6} - 1883270 \beta_{5} + 674548 \beta_{4} - 731482 \beta_{3} + 680197 \beta_{1} + 480555$$ $$\nu^{14}$$ $$=$$ $$2133296 \beta_{15} + 884184 \beta_{14} - 1678544 \beta_{13} - 1678544 \beta_{12} + 884184 \beta_{11} + 1815497 \beta_{10} - 1815497 \beta_{9} + 2133296 \beta_{8} + 4502770 \beta_{7} - 4502770 \beta_{6} + 980607 \beta_{5} + 980607 \beta_{4} - 2372789 \beta_{2} - 6620449$$ $$\nu^{15}$$ $$=$$ $$-3113903 \beta_{15} - 7520250 \beta_{14} + 5866830 \beta_{13} - 5866830 \beta_{12} + 7520250 \beta_{11} - 6374568 \beta_{10} - 6374568 \beta_{9} + 3113903 \beta_{8} - 6560815 \beta_{7} - 333009 \beta_{6} + 23358425 \beta_{5} - 8317925 \beta_{4} + 9012170 \beta_{3} - 8298993 \beta_{1} - 5866830$$

## Character values

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

 $$n$$ $$71$$ $$137$$ $$\chi(n)$$ $$-\beta_{10}$$ $$1$$

## 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
 − 3.51034i 0.0614939i − 2.47571i 1.09612i 3.51034i − 0.0614939i 2.47571i − 1.09612i − 1.33738i 1.46868i 0.923170i − 1.88289i 1.33738i − 1.46868i − 0.923170i 1.88289i
0.707107 0.707107i −2.31925 0.960664i 1.00000i −0.382683 + 0.923880i −2.31925 + 0.960664i −1.57873 3.81140i −0.707107 0.707107i 2.33472 + 2.33472i 0.382683 + 0.923880i
111.2 0.707107 0.707107i −0.980692 0.406216i 1.00000i 0.382683 0.923880i −0.980692 + 0.406216i −0.758398 1.83094i −0.707107 0.707107i −1.32457 1.32457i −0.382683 0.923880i
111.3 0.707107 0.707107i 1.36338 + 0.564729i 1.00000i 0.382683 0.923880i 1.36338 0.564729i 1.35785 + 3.27815i −0.707107 0.707107i −0.581445 0.581445i −0.382683 0.923880i
111.4 0.707107 0.707107i 1.93656 + 0.802151i 1.00000i −0.382683 + 0.923880i 1.93656 0.802151i −0.434936 1.05003i −0.707107 0.707107i 0.985516 + 0.985516i 0.382683 + 0.923880i
121.1 0.707107 + 0.707107i −2.31925 + 0.960664i 1.00000i −0.382683 0.923880i −2.31925 0.960664i −1.57873 + 3.81140i −0.707107 + 0.707107i 2.33472 2.33472i 0.382683 0.923880i
121.2 0.707107 + 0.707107i −0.980692 + 0.406216i 1.00000i 0.382683 + 0.923880i −0.980692 0.406216i −0.758398 + 1.83094i −0.707107 + 0.707107i −1.32457 + 1.32457i −0.382683 + 0.923880i
121.3 0.707107 + 0.707107i 1.36338 0.564729i 1.00000i 0.382683 + 0.923880i 1.36338 + 0.564729i 1.35785 3.27815i −0.707107 + 0.707107i −0.581445 + 0.581445i −0.382683 + 0.923880i
121.4 0.707107 + 0.707107i 1.93656 0.802151i 1.00000i −0.382683 0.923880i 1.93656 + 0.802151i −0.434936 + 1.05003i −0.707107 + 0.707107i 0.985516 0.985516i 0.382683 0.923880i
151.1 −0.707107 0.707107i −0.894478 2.15946i 1.00000i −0.923880 + 0.382683i −0.894478 + 2.15946i −3.32333 1.37657i 0.707107 0.707107i −1.74186 + 1.74186i 0.923880 + 0.382683i
151.2 −0.707107 0.707107i −0.179356 0.433004i 1.00000i 0.923880 0.382683i −0.179356 + 0.433004i 1.29538 + 0.536563i 0.707107 0.707107i 1.96600 1.96600i −0.923880 0.382683i
151.3 −0.707107 0.707107i −0.0294014 0.0709814i 1.00000i −0.923880 + 0.382683i −0.0294014 + 0.0709814i 3.48924 + 1.44529i 0.707107 0.707107i 2.11715 2.11715i 0.923880 + 0.382683i
151.4 −0.707107 0.707107i 1.10324 + 2.66345i 1.00000i 0.923880 0.382683i 1.10324 2.66345i −0.0470744 0.0194989i 0.707107 0.707107i −3.75550 + 3.75550i −0.923880 0.382683i
161.1 −0.707107 + 0.707107i −0.894478 + 2.15946i 1.00000i −0.923880 0.382683i −0.894478 2.15946i −3.32333 + 1.37657i 0.707107 + 0.707107i −1.74186 1.74186i 0.923880 0.382683i
161.2 −0.707107 + 0.707107i −0.179356 + 0.433004i 1.00000i 0.923880 + 0.382683i −0.179356 0.433004i 1.29538 0.536563i 0.707107 + 0.707107i 1.96600 + 1.96600i −0.923880 + 0.382683i
161.3 −0.707107 + 0.707107i −0.0294014 + 0.0709814i 1.00000i −0.923880 0.382683i −0.0294014 0.0709814i 3.48924 1.44529i 0.707107 + 0.707107i 2.11715 + 2.11715i 0.923880 0.382683i
161.4 −0.707107 + 0.707107i 1.10324 2.66345i 1.00000i 0.923880 + 0.382683i 1.10324 + 2.66345i −0.0470744 + 0.0194989i 0.707107 + 0.707107i −3.75550 3.75550i −0.923880 + 0.382683i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 161.4 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.b 16
5.b even 2 1 850.2.l.e 16
5.c odd 4 1 850.2.o.g 16
5.c odd 4 1 850.2.o.j 16
17.d even 8 1 inner 170.2.k.b 16
17.e odd 16 1 2890.2.a.bi 8
17.e odd 16 1 2890.2.a.bj 8
17.e odd 16 2 2890.2.b.r 16
85.k odd 8 1 850.2.o.g 16
85.m even 8 1 850.2.l.e 16
85.n odd 8 1 850.2.o.j 16

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{16} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(170, [\chi])$$.