# Properties

 Label 165.2.p.a Level $165$ Weight $2$ Character orbit 165.p Analytic conductor $1.318$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$165 = 3 \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 165.p (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.31753163335$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - x^{14} + 15 x^{12} - 59 x^{10} + 104 x^{8} - 59 x^{6} + 15 x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$5^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{3} - \beta_{5} ) q^{2} + ( \beta_{2} - \beta_{5} + \beta_{6} - \beta_{8} - \beta_{10} ) q^{3} + ( -1 - 2 \beta_{2} + \beta_{6} + \beta_{7} - \beta_{12} ) q^{4} + \beta_{5} q^{5} + ( \beta_{1} + \beta_{3} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{11} + \beta_{13} - \beta_{14} ) q^{6} + ( 1 - \beta_{1} - \beta_{2} - 2 \beta_{6} - \beta_{7} + 2 \beta_{9} - \beta_{12} ) q^{7} + ( 3 \beta_{5} + \beta_{8} + 2 \beta_{10} + 5 \beta_{14} + \beta_{15} ) q^{8} + ( -2 + 2 \beta_{2} - 2 \beta_{5} - 2 \beta_{8} - \beta_{9} - 2 \beta_{10} - 2 \beta_{14} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{3} - \beta_{5} ) q^{2} + ( \beta_{2} - \beta_{5} + \beta_{6} - \beta_{8} - \beta_{10} ) q^{3} + ( -1 - 2 \beta_{2} + \beta_{6} + \beta_{7} - \beta_{12} ) q^{4} + \beta_{5} q^{5} + ( \beta_{1} + \beta_{3} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{11} + \beta_{13} - \beta_{14} ) q^{6} + ( 1 - \beta_{1} - \beta_{2} - 2 \beta_{6} - \beta_{7} + 2 \beta_{9} - \beta_{12} ) q^{7} + ( 3 \beta_{5} + \beta_{8} + 2 \beta_{10} + 5 \beta_{14} + \beta_{15} ) q^{8} + ( -2 + 2 \beta_{2} - 2 \beta_{5} - 2 \beta_{8} - \beta_{9} - 2 \beta_{10} - 2 \beta_{14} ) q^{9} + ( \beta_{2} + \beta_{12} ) q^{10} + ( \beta_{3} - \beta_{5} + \beta_{10} + 2 \beta_{13} + \beta_{15} ) q^{11} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + \beta_{10} + \beta_{12} - \beta_{14} ) q^{12} + ( -1 + \beta_{2} + \beta_{6} - \beta_{7} - \beta_{9} + \beta_{11} ) q^{13} + ( -2 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} + 5 \beta_{8} + \beta_{10} + \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{14} + ( -1 + \beta_{5} + \beta_{6} + \beta_{8} + \beta_{14} ) q^{15} + ( -4 \beta_{6} - \beta_{7} + \beta_{9} + 2 \beta_{11} + 2 \beta_{12} ) q^{16} + ( -2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{8} - 2 \beta_{13} ) q^{17} + ( 2 \beta_{1} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{18} + ( 3 - 2 \beta_{1} + \beta_{2} - \beta_{6} - \beta_{7} - 3 \beta_{9} - \beta_{11} - \beta_{12} ) q^{19} + ( \beta_{4} - \beta_{5} - \beta_{8} - \beta_{10} - \beta_{14} - \beta_{15} ) q^{20} + ( 1 - 4 \beta_{2} - \beta_{3} - \beta_{5} - 2 \beta_{6} + \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{21} + ( -1 + \beta_{1} + 2 \beta_{2} + 2 \beta_{6} + 2 \beta_{7} - 5 \beta_{9} + \beta_{12} ) q^{22} + ( \beta_{3} - \beta_{5} - 2 \beta_{8} - 3 \beta_{10} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{23} + ( -4 - \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - 3 \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} + 3 \beta_{14} + \beta_{15} ) q^{24} + ( 1 - \beta_{2} - \beta_{6} + \beta_{9} ) q^{25} + ( -\beta_{5} + 2 \beta_{8} - \beta_{14} ) q^{26} + ( \beta_{2} + 3 \beta_{5} - 3 \beta_{6} - \beta_{9} - \beta_{10} - \beta_{14} ) q^{27} + ( -6 - 5 \beta_{1} + 7 \beta_{2} + 3 \beta_{6} + \beta_{7} - 3 \beta_{9} - 5 \beta_{11} ) q^{28} + ( \beta_{3} + \beta_{4} + \beta_{5} + \beta_{8} + 2 \beta_{10} - \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{29} + ( -\beta_{1} + \beta_{2} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{13} + \beta_{15} ) q^{30} + ( 4 - 3 \beta_{1} - \beta_{2} - 2 \beta_{6} - \beta_{7} + 2 \beta_{9} - \beta_{11} + \beta_{12} ) q^{31} + ( -2 \beta_{3} - 2 \beta_{5} - \beta_{8} - \beta_{10} - 2 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{32} + ( 2 - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{7} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{33} + ( 8 + 2 \beta_{1} + \beta_{2} - 6 \beta_{6} - 2 \beta_{7} + 6 \beta_{9} + 2 \beta_{11} + \beta_{12} ) q^{34} + ( \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{8} + 2 \beta_{14} + \beta_{15} ) q^{35} + ( 3 + 2 \beta_{1} + \beta_{2} + 4 \beta_{3} - 2 \beta_{5} + \beta_{6} - \beta_{7} + 3 \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{12} + 2 \beta_{13} + 2 \beta_{15} ) q^{36} + ( 1 - \beta_{1} - 3 \beta_{2} + 2 \beta_{7} - 2 \beta_{11} - 2 \beta_{12} ) q^{37} + ( \beta_{4} + 2 \beta_{5} + \beta_{8} - \beta_{10} - 2 \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{38} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{10} + \beta_{11} - \beta_{13} ) q^{39} + ( 3 + \beta_{1} - \beta_{2} + 2 \beta_{6} + 2 \beta_{9} ) q^{40} + ( -3 \beta_{5} + 4 \beta_{8} + 2 \beta_{10} + 3 \beta_{13} + \beta_{14} ) q^{41} + ( 1 - \beta_{1} - \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + \beta_{6} - 2 \beta_{7} + 7 \beta_{8} + 3 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 3 \beta_{12} - 2 \beta_{13} + 7 \beta_{14} + 2 \beta_{15} ) q^{42} + ( 1 - 3 \beta_{2} - \beta_{6} + 2 \beta_{9} + \beta_{11} ) q^{43} + ( 4 \beta_{4} - 6 \beta_{5} - 6 \beta_{8} - 2 \beta_{10} - 3 \beta_{13} + 3 \beta_{14} ) q^{44} + ( -2 + 2 \beta_{8} + 2 \beta_{10} + \beta_{14} ) q^{45} + ( -1 - 3 \beta_{2} - 3 \beta_{6} - \beta_{7} + \beta_{9} + \beta_{11} ) q^{46} + ( -2 \beta_{4} - 3 \beta_{5} - 2 \beta_{8} + 4 \beta_{10} + \beta_{13} - \beta_{14} ) q^{47} + ( 2 + \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{11} + \beta_{12} + \beta_{13} + 3 \beta_{14} + 3 \beta_{15} ) q^{48} + ( 5 \beta_{1} + \beta_{2} + 6 \beta_{6} + 4 \beta_{7} - 3 \beta_{11} + 2 \beta_{12} ) q^{49} + ( -\beta_{4} + \beta_{8} ) q^{50} + ( 1 - 2 \beta_{1} + 2 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} - 2 \beta_{10} + 2 \beta_{12} - 2 \beta_{13} - \beta_{14} ) q^{51} + ( -2 - \beta_{2} + \beta_{6} + \beta_{7} + 2 \beta_{9} - \beta_{11} - \beta_{12} ) q^{52} + ( \beta_{3} - \beta_{4} - 2 \beta_{5} - 3 \beta_{8} - \beta_{10} + 2 \beta_{13} - 3 \beta_{14} + \beta_{15} ) q^{53} + ( 4 \beta_{2} - 3 \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{6} + 3 \beta_{8} - \beta_{11} + 3 \beta_{12} - 4 \beta_{13} - 4 \beta_{15} ) q^{54} + ( -\beta_{1} + 2 \beta_{2} - 2 \beta_{7} - 2 \beta_{9} - \beta_{12} ) q^{55} + ( -4 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} - 11 \beta_{8} - 7 \beta_{10} + 2 \beta_{13} - 5 \beta_{14} - 2 \beta_{15} ) q^{56} + ( -2 \beta_{1} + 4 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + 3 \beta_{6} - \beta_{7} - 2 \beta_{8} - 3 \beta_{9} + \beta_{11} - 2 \beta_{12} + 2 \beta_{14} - \beta_{15} ) q^{57} + ( 2 - 2 \beta_{1} - 6 \beta_{2} - 6 \beta_{6} + \beta_{7} + 2 \beta_{9} + \beta_{11} - \beta_{12} ) q^{58} + ( 2 \beta_{3} + \beta_{4} - 4 \beta_{5} + 3 \beta_{8} + \beta_{13} - 3 \beta_{14} ) q^{59} + ( \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} + \beta_{11} + 2 \beta_{12} - 2 \beta_{14} - 2 \beta_{15} ) q^{60} + ( -6 + \beta_{1} + 4 \beta_{2} + 3 \beta_{6} - 3 \beta_{7} - 4 \beta_{9} + \beta_{11} ) q^{61} + ( -\beta_{3} - 2 \beta_{4} - 4 \beta_{5} - 5 \beta_{8} - 10 \beta_{10} + 2 \beta_{13} - 8 \beta_{14} - \beta_{15} ) q^{62} + ( 2 + 3 \beta_{1} - 2 \beta_{5} + \beta_{7} + 4 \beta_{8} - 2 \beta_{9} + 2 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} ) q^{63} + ( 4 + \beta_{1} - 5 \beta_{2} + \beta_{7} + 4 \beta_{9} + \beta_{11} + \beta_{12} ) q^{64} + ( \beta_{3} - \beta_{5} - \beta_{10} + \beta_{13} + \beta_{15} ) q^{65} + ( -1 + \beta_{1} + 7 \beta_{2} + 2 \beta_{3} - \beta_{4} + 3 \beta_{5} + 4 \beta_{6} + \beta_{7} - 3 \beta_{8} - 2 \beta_{9} + 3 \beta_{10} - \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + 4 \beta_{14} + 2 \beta_{15} ) q^{66} + ( -7 - 2 \beta_{1} + \beta_{2} + \beta_{6} - 2 \beta_{7} - 3 \beta_{9} - \beta_{12} ) q^{67} + ( 4 \beta_{3} - 3 \beta_{4} - 5 \beta_{5} + 9 \beta_{8} + \beta_{10} - 9 \beta_{14} - 3 \beta_{15} ) q^{68} + ( -3 + 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} - 3 \beta_{9} - 3 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{69} + ( 2 + 2 \beta_{1} - 3 \beta_{2} - 4 \beta_{6} - \beta_{7} + 4 \beta_{11} + \beta_{12} ) q^{70} + ( -\beta_{3} + \beta_{4} + 7 \beta_{5} + 2 \beta_{8} + 2 \beta_{10} - 3 \beta_{13} + \beta_{14} - 4 \beta_{15} ) q^{71} + ( -4 - 4 \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - \beta_{8} - 6 \beta_{9} + 5 \beta_{10} + 2 \beta_{13} - 7 \beta_{14} ) q^{72} + ( -6 + 3 \beta_{1} + 2 \beta_{2} + \beta_{6} + 2 \beta_{7} - 4 \beta_{9} + 2 \beta_{12} ) q^{73} + ( 7 \beta_{5} - \beta_{8} + 3 \beta_{10} + 9 \beta_{14} + 3 \beta_{15} ) q^{74} + ( 1 - \beta_{2} - \beta_{5} - \beta_{10} ) q^{75} + ( -3 + 3 \beta_{2} + 7 \beta_{6} + \beta_{9} - 3 \beta_{12} ) q^{76} + ( -2 \beta_{3} + 5 \beta_{5} + 4 \beta_{8} + 2 \beta_{10} - 2 \beta_{13} + 8 \beta_{14} + 3 \beta_{15} ) q^{77} + ( 4 - 3 \beta_{6} + 2 \beta_{8} + 3 \beta_{9} + 2 \beta_{10} + \beta_{14} ) q^{78} + ( -3 - 2 \beta_{1} - \beta_{2} - 2 \beta_{7} - 2 \beta_{9} + 2 \beta_{11} - 2 \beta_{12} ) q^{79} + ( 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - \beta_{8} + \beta_{10} + 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{80} + ( -\beta_{2} + 8 \beta_{5} + 4 \beta_{8} + 8 \beta_{14} ) q^{81} + ( -\beta_{1} + 4 \beta_{2} + 5 \beta_{6} + \beta_{7} - 6 \beta_{9} - 3 \beta_{11} - 4 \beta_{12} ) q^{82} + ( -\beta_{3} - 3 \beta_{4} + \beta_{5} + 4 \beta_{8} - \beta_{13} + \beta_{14} ) q^{83} + ( -3 - 9 \beta_{1} + \beta_{3} - 6 \beta_{4} + 5 \beta_{5} - 12 \beta_{6} - 4 \beta_{7} + 3 \beta_{8} + 6 \beta_{10} - 4 \beta_{12} + 6 \beta_{13} - 5 \beta_{14} + \beta_{15} ) q^{84} + ( -1 - \beta_{2} + \beta_{6} + 2 \beta_{7} + \beta_{9} - 2 \beta_{11} - 2 \beta_{12} ) q^{85} + ( -2 \beta_{4} - \beta_{5} + 3 \beta_{8} - \beta_{10} + 3 \beta_{14} + 2 \beta_{15} ) q^{86} + ( 1 - 3 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{8} + 3 \beta_{9} + \beta_{10} + \beta_{11} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{87} + ( 2 + 5 \beta_{1} - 3 \beta_{2} - 3 \beta_{6} - 3 \beta_{7} + 2 \beta_{9} + 5 \beta_{11} - 3 \beta_{12} ) q^{88} + ( 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} - 3 \beta_{8} - 5 \beta_{10} + 2 \beta_{13} - 9 \beta_{14} - 2 \beta_{15} ) q^{89} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{11} - 2 \beta_{12} ) q^{90} + ( \beta_{1} + \beta_{2} + \beta_{6} - \beta_{7} ) q^{91} + ( 5 \beta_{5} + 8 \beta_{8} + 5 \beta_{14} ) q^{92} + ( -4 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - 5 \beta_{5} - 3 \beta_{7} - \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} + 3 \beta_{13} - 3 \beta_{14} + 2 \beta_{15} ) q^{93} + ( -6 + \beta_{1} - 4 \beta_{2} + 3 \beta_{6} + 5 \beta_{7} - 4 \beta_{9} + \beta_{11} ) q^{94} + ( \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{95} + ( 2 + 3 \beta_{2} - 2 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 3 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - 4 \beta_{14} ) q^{96} + ( -5 + 3 \beta_{1} + 2 \beta_{2} - \beta_{6} + 4 \beta_{7} + 7 \beta_{9} + 4 \beta_{11} + 5 \beta_{12} ) q^{97} + ( 11 \beta_{3} + 6 \beta_{4} - 9 \beta_{5} - 17 \beta_{8} + 8 \beta_{10} + 5 \beta_{13} + \beta_{14} + 5 \beta_{15} ) q^{98} + ( 4 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} + 2 \beta_{9} - 3 \beta_{10} - 4 \beta_{11} - \beta_{13} - 4 \beta_{14} - 2 \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 8q^{3} - 12q^{4} + 10q^{6} - 10q^{7} - 20q^{9} + O(q^{10})$$ $$16q + 8q^{3} - 12q^{4} + 10q^{6} - 10q^{7} - 20q^{9} + 4q^{12} - 12q^{15} - 16q^{16} + 20q^{18} + 40q^{19} + 30q^{22} - 70q^{24} + 4q^{25} - 4q^{27} - 110q^{28} - 10q^{30} + 10q^{31} + 12q^{33} + 100q^{34} + 40q^{36} - 2q^{37} - 10q^{39} + 50q^{40} - 10q^{42} - 32q^{45} - 40q^{46} + 22q^{48} + 42q^{49} - 40q^{52} + 6q^{55} + 40q^{57} - 20q^{58} + 14q^{60} - 50q^{61} + 70q^{63} + 42q^{64} + 30q^{66} - 108q^{67} - 12q^{69} + 40q^{70} - 40q^{72} - 50q^{73} + 12q^{75} + 40q^{78} - 40q^{79} - 4q^{81} + 50q^{82} - 150q^{84} - 20q^{85} + 70q^{88} - 20q^{90} + 10q^{91} - 50q^{94} + 40q^{96} - 58q^{97} + 12q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - x^{14} + 15 x^{12} - 59 x^{10} + 104 x^{8} - 59 x^{6} + 15 x^{4} - x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$27 \nu^{14} + 50 \nu^{12} + 355 \nu^{10} - 484 \nu^{8} - 1348 \nu^{6} + 4523 \nu^{4} - 750 \nu^{2} - 365$$$$)/384$$ $$\beta_{2}$$ $$=$$ $$($$$$21 \nu^{14} - 22 \nu^{12} + 329 \nu^{10} - 1260 \nu^{8} + 2436 \nu^{6} - 1995 \nu^{4} + 1498 \nu^{2} - 119$$$$)/384$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{15} - 35 \nu^{13} + 40 \nu^{11} - 560 \nu^{9} + 1972 \nu^{7} - 3075 \nu^{5} + 1041 \nu^{3} - 12 \nu$$$$)/96$$ $$\beta_{4}$$ $$=$$ $$($$$$12 \nu^{15} + \nu^{13} + 181 \nu^{11} - 524 \nu^{9} + 688 \nu^{7} - 128 \nu^{5} + 717 \nu^{3} - 231 \nu$$$$)/96$$ $$\beta_{5}$$ $$=$$ $$($$$$9 \nu^{15} - 23 \nu^{13} + 148 \nu^{11} - 736 \nu^{9} + 1748 \nu^{7} - 1867 \nu^{5} + 685 \nu^{3} + 16 \nu$$$$)/96$$ $$\beta_{6}$$ $$=$$ $$($$$$9 \nu^{14} - 17 \nu^{12} + 138 \nu^{10} - 648 \nu^{8} + 1332 \nu^{6} - 1107 \nu^{4} + 155 \nu^{2} + 30$$$$)/96$$ $$\beta_{7}$$ $$=$$ $$($$$$-14 \nu^{14} + 7 \nu^{12} - 195 \nu^{10} + 724 \nu^{8} - 920 \nu^{6} - 210 \nu^{4} + 555 \nu^{2} - 23$$$$)/96$$ $$\beta_{8}$$ $$=$$ $$($$$$63 \nu^{15} - 42 \nu^{13} + 947 \nu^{11} - 3428 \nu^{9} + 5644 \nu^{7} - 2945 \nu^{5} + 1990 \nu^{3} - 685 \nu$$$$)/384$$ $$\beta_{9}$$ $$=$$ $$($$$$-17 \nu^{14} + 5 \nu^{12} - 246 \nu^{10} + 824 \nu^{8} - 1108 \nu^{6} - 101 \nu^{4} + 201 \nu^{2} - 58$$$$)/96$$ $$\beta_{10}$$ $$=$$ $$($$$$-77 \nu^{15} + 134 \nu^{13} - 1185 \nu^{11} + 5388 \nu^{9} - 10980 \nu^{7} + 9107 \nu^{5} - 2666 \nu^{3} + 543 \nu$$$$)/384$$ $$\beta_{11}$$ $$=$$ $$($$$$7 \nu^{14} - 17 \nu^{12} + 112 \nu^{10} - 560 \nu^{8} + 1276 \nu^{6} - 1269 \nu^{4} + 411 \nu^{2} - 12$$$$)/48$$ $$\beta_{12}$$ $$=$$ $$($$$$-15 \nu^{14} + 9 \nu^{12} - 224 \nu^{10} + 800 \nu^{8} - 1276 \nu^{6} + 557 \nu^{4} - 307 \nu^{2} + 28$$$$)/48$$ $$\beta_{13}$$ $$=$$ $$($$$$-155 \nu^{15} + 166 \nu^{13} - 2315 \nu^{11} + 9284 \nu^{9} - 16444 \nu^{7} + 9013 \nu^{5} - 410 \nu^{3} - 1499 \nu$$$$)/384$$ $$\beta_{14}$$ $$=$$ $$($$$$-91 \nu^{15} + 70 \nu^{13} - 1355 \nu^{11} + 5060 \nu^{9} - 8380 \nu^{7} + 3765 \nu^{5} - 890 \nu^{3} - 155 \nu$$$$)/192$$ $$\beta_{15}$$ $$=$$ $$($$$$375 \nu^{15} - 270 \nu^{13} + 5559 \nu^{11} - 20564 \nu^{9} + 33388 \nu^{7} - 13145 \nu^{5} + 2290 \nu^{3} + 391 \nu$$$$)/384$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{15} + 3 \beta_{14} - 4 \beta_{13} + 3 \beta_{8} + 2 \beta_{5} + \beta_{4} - 2 \beta_{3}$$$$)/5$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{12} - 2 \beta_{11} - 2 \beta_{7} + \beta_{6} + 11 \beta_{2} - \beta_{1}$$$$)/5$$ $$\nu^{3}$$ $$=$$ $$\beta_{15} + \beta_{14} - 3 \beta_{8} - 3 \beta_{5} + 2 \beta_{4} + 2 \beta_{3}$$ $$\nu^{4}$$ $$=$$ $$($$$$-12 \beta_{12} + 7 \beta_{11} + 5 \beta_{9} - 13 \beta_{7} - 41 \beta_{6} - 11 \beta_{2} - 19 \beta_{1}$$$$)/5$$ $$\nu^{5}$$ $$=$$ $$($$$$-24 \beta_{15} - 73 \beta_{14} + 24 \beta_{13} - 20 \beta_{10} - 38 \beta_{8} + 18 \beta_{5} - 6 \beta_{4} - 18 \beta_{3}$$$$)/5$$ $$\nu^{6}$$ $$=$$ $$5 \beta_{12} + 11 \beta_{11} - 6 \beta_{9} + 11 \beta_{7} + 6 \beta_{6} - 23 \beta_{2} + 17 \beta_{1} + 6$$ $$\nu^{7}$$ $$=$$ $$($$$$-36 \beta_{15} + 153 \beta_{14} - 109 \beta_{13} + 333 \beta_{8} + 262 \beta_{5} - 109 \beta_{4} - 182 \beta_{3}$$$$)/5$$ $$\nu^{8}$$ $$=$$ $$($$$$237 \beta_{12} - 237 \beta_{11} - 110 \beta_{9} + 98 \beta_{7} + 631 \beta_{6} + 631 \beta_{2} + 139 \beta_{1} - 110$$$$)/5$$ $$\nu^{9}$$ $$=$$ $$124 \beta_{15} + 226 \beta_{14} - 35 \beta_{13} + 67 \beta_{10} - 62 \beta_{8} - 195 \beta_{5} + 97 \beta_{4} + 159 \beta_{3}$$ $$\nu^{10}$$ $$=$$ $$($$$$-1032 \beta_{12} - 408 \beta_{11} + 660 \beta_{9} - 1248 \beta_{7} - 2316 \beta_{6} + 624 \beta_{2} - 2064 \beta_{1} - 445$$$$)/5$$ $$\nu^{11}$$ $$=$$ $$($$$$-799 \beta_{15} - 5763 \beta_{14} + 2684 \beta_{13} - 840 \beta_{10} - 5763 \beta_{8} - 2182 \beta_{5} + 799 \beta_{4} + 1342 \beta_{3}$$$$)/5$$ $$\nu^{12}$$ $$=$$ $$-350 \beta_{12} + 1106 \beta_{11} + 350 \beta_{7} - 1141 \beta_{6} - 2771 \beta_{2} + 553 \beta_{1} + 588$$ $$\nu^{13}$$ $$=$$ $$($$$$-9421 \beta_{15} - 5857 \beta_{14} - 3564 \beta_{13} - 3780 \beta_{10} + 21543 \beta_{8} + 25107 \beta_{5} - 11714 \beta_{4} - 18842 \beta_{3}$$$$)/5$$ $$\nu^{14}$$ $$=$$ $$($$$$24372 \beta_{12} - 7607 \beta_{11} - 12985 \beta_{9} + 19793 \beta_{7} + 60721 \beta_{6} + 25171 \beta_{2} + 31979 \beta_{1}$$$$)/5$$ $$\nu^{15}$$ $$=$$ $$8280 \beta_{15} + 25393 \beta_{14} - 8280 \beta_{13} + 5480 \beta_{10} + 10610 \beta_{8} - 5130 \beta_{5} + 3150 \beta_{4} + 5130 \beta_{3}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/165\mathbb{Z}\right)^\times$$.

 $$n$$ $$46$$ $$56$$ $$67$$ $$\chi(n)$$ $$\beta_{2}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
41.1
 1.23158 + 1.69513i 0.280526 + 0.386111i −0.280526 − 0.386111i −1.23158 − 1.69513i −1.28932 − 0.418926i −0.701538 − 0.227943i 0.701538 + 0.227943i 1.28932 + 0.418926i −1.28932 + 0.418926i −0.701538 + 0.227943i 0.701538 − 0.227943i 1.28932 − 0.418926i 1.23158 − 1.69513i 0.280526 − 0.386111i −0.280526 + 0.386111i −1.23158 + 1.69513i
−1.99274 + 1.44781i −0.0877853 + 1.72982i 1.25683 3.86812i 0.587785 0.809017i −2.32953 3.57419i −4.90846 1.59485i 1.57346 + 4.84261i −2.98459 0.303706i 2.46317i
41.2 −0.453901 + 0.329779i 1.08779 + 1.34786i −0.520762 + 1.60274i −0.587785 + 0.809017i −0.938242 0.253067i −0.254663 0.0827449i −0.638924 1.96641i −0.633446 + 2.93236i 0.561053i
41.3 0.453901 0.329779i −0.0877853 + 1.72982i −0.520762 + 1.60274i 0.587785 0.809017i 0.530613 + 0.814119i −0.254663 0.0827449i 0.638924 + 1.96641i −2.98459 0.303706i 0.561053i
41.4 1.99274 1.44781i 1.08779 + 1.34786i 1.25683 3.86812i −0.587785 + 0.809017i 4.11912 + 1.11103i −4.90846 1.59485i −1.57346 4.84261i −0.633446 + 2.93236i 2.46317i
101.1 −0.796845 + 2.45244i −0.451057 1.67229i −3.76145 2.73286i 0.951057 0.309017i 4.46061 + 0.226367i 2.05478 2.82816i 5.52712 4.01569i −2.59310 + 1.50859i 2.57865i
101.2 −0.433574 + 1.33440i 1.45106 + 0.945746i 0.0253869 + 0.0184446i −0.951057 + 0.309017i −1.89115 + 1.52624i 0.608337 0.837304i −2.30584 + 1.67529i 1.21113 + 2.74466i 1.40308i
101.3 0.433574 1.33440i −0.451057 1.67229i 0.0253869 + 0.0184446i 0.951057 0.309017i −2.42707 0.123169i 0.608337 0.837304i 2.30584 1.67529i −2.59310 + 1.50859i 1.40308i
101.4 0.796845 2.45244i 1.45106 + 0.945746i −3.76145 2.73286i −0.951057 + 0.309017i 3.47565 2.80501i 2.05478 2.82816i −5.52712 + 4.01569i 1.21113 + 2.74466i 2.57865i
116.1 −0.796845 2.45244i −0.451057 + 1.67229i −3.76145 + 2.73286i 0.951057 + 0.309017i 4.46061 0.226367i 2.05478 + 2.82816i 5.52712 + 4.01569i −2.59310 1.50859i 2.57865i
116.2 −0.433574 1.33440i 1.45106 0.945746i 0.0253869 0.0184446i −0.951057 0.309017i −1.89115 1.52624i 0.608337 + 0.837304i −2.30584 1.67529i 1.21113 2.74466i 1.40308i
116.3 0.433574 + 1.33440i −0.451057 + 1.67229i 0.0253869 0.0184446i 0.951057 + 0.309017i −2.42707 + 0.123169i 0.608337 + 0.837304i 2.30584 + 1.67529i −2.59310 1.50859i 1.40308i
116.4 0.796845 + 2.45244i 1.45106 0.945746i −3.76145 + 2.73286i −0.951057 0.309017i 3.47565 + 2.80501i 2.05478 + 2.82816i −5.52712 4.01569i 1.21113 2.74466i 2.57865i
161.1 −1.99274 1.44781i −0.0877853 1.72982i 1.25683 + 3.86812i 0.587785 + 0.809017i −2.32953 + 3.57419i −4.90846 + 1.59485i 1.57346 4.84261i −2.98459 + 0.303706i 2.46317i
161.2 −0.453901 0.329779i 1.08779 1.34786i −0.520762 1.60274i −0.587785 0.809017i −0.938242 + 0.253067i −0.254663 + 0.0827449i −0.638924 + 1.96641i −0.633446 2.93236i 0.561053i
161.3 0.453901 + 0.329779i −0.0877853 1.72982i −0.520762 1.60274i 0.587785 + 0.809017i 0.530613 0.814119i −0.254663 + 0.0827449i 0.638924 1.96641i −2.98459 + 0.303706i 0.561053i
161.4 1.99274 + 1.44781i 1.08779 1.34786i 1.25683 + 3.86812i −0.587785 0.809017i 4.11912 1.11103i −4.90846 + 1.59485i −1.57346 + 4.84261i −0.633446 2.93236i 2.46317i
 $$n$$: 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
3.b odd 2 1 inner
11.d odd 10 1 inner
33.f even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.2.p.a 16
3.b odd 2 1 inner 165.2.p.a 16
5.b even 2 1 825.2.bi.d 16
5.c odd 4 1 825.2.bs.e 16
5.c odd 4 1 825.2.bs.f 16
11.d odd 10 1 inner 165.2.p.a 16
15.d odd 2 1 825.2.bi.d 16
15.e even 4 1 825.2.bs.e 16
15.e even 4 1 825.2.bs.f 16
33.f even 10 1 inner 165.2.p.a 16
55.h odd 10 1 825.2.bi.d 16
55.l even 20 1 825.2.bs.e 16
55.l even 20 1 825.2.bs.f 16
165.r even 10 1 825.2.bi.d 16
165.u odd 20 1 825.2.bs.e 16
165.u odd 20 1 825.2.bs.f 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.p.a 16 1.a even 1 1 trivial
165.2.p.a 16 3.b odd 2 1 inner
165.2.p.a 16 11.d odd 10 1 inner
165.2.p.a 16 33.f even 10 1 inner
825.2.bi.d 16 5.b even 2 1
825.2.bi.d 16 15.d odd 2 1
825.2.bi.d 16 55.h odd 10 1
825.2.bi.d 16 165.r even 10 1
825.2.bs.e 16 5.c odd 4 1
825.2.bs.e 16 15.e even 4 1
825.2.bs.e 16 55.l even 20 1
825.2.bs.e 16 165.u odd 20 1
825.2.bs.f 16 5.c odd 4 1
825.2.bs.f 16 15.e even 4 1
825.2.bs.f 16 55.l even 20 1
825.2.bs.f 16 165.u odd 20 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(165, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$625 - 625 T^{2} + 5375 T^{4} + 5625 T^{6} + 2450 T^{8} + 375 T^{10} + 65 T^{12} + 10 T^{14} + T^{16}$$
$3$ $$( 81 - 108 T + 117 T^{2} - 90 T^{3} + 61 T^{4} - 30 T^{5} + 13 T^{6} - 4 T^{7} + T^{8} )^{2}$$
$5$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
$7$ $$( 25 + 150 T + 175 T^{2} - 225 T^{3} + 320 T^{4} + 15 T^{5} - 5 T^{6} + 5 T^{7} + T^{8} )^{2}$$
$11$ $$214358881 - 47832147 T^{2} + 7730448 T^{4} - 915849 T^{6} + 87575 T^{8} - 7569 T^{10} + 528 T^{12} - 27 T^{14} + T^{16}$$
$13$ $$( 25 + 25 T + 25 T^{2} - 75 T^{3} + 20 T^{4} + 15 T^{5} - 5 T^{6} + T^{8} )^{2}$$
$17$ $$625 - 353750 T^{2} + 524253375 T^{4} + 22410500 T^{6} + 2317325 T^{8} + 178100 T^{10} + 7335 T^{12} + 130 T^{14} + T^{16}$$
$19$ $$( 3025 + 4675 T - 675 T^{2} - 1725 T^{3} + 1830 T^{4} - 725 T^{5} + 165 T^{6} - 20 T^{7} + T^{8} )^{2}$$
$23$ $$( 1 + 2659 T^{2} + 741 T^{4} + 59 T^{6} + T^{8} )^{2}$$
$29$ $$81450625 + 16019375 T^{2} + 4290375 T^{4} + 1029625 T^{6} + 206450 T^{8} + 5575 T^{10} + 1065 T^{12} + 50 T^{14} + T^{16}$$
$31$ $$( 600625 + 135625 T + 104875 T^{2} - 4625 T^{3} + 150 T^{4} + 125 T^{5} + 45 T^{6} - 5 T^{7} + T^{8} )^{2}$$
$37$ $$( 22201 - 17731 T + 20477 T^{2} - 5503 T^{3} + 1430 T^{4} - 47 T^{5} - 13 T^{6} + T^{7} + T^{8} )^{2}$$
$41$ $$184062450625 + 38204676250 T^{2} + 6507197625 T^{4} + 1076651875 T^{6} + 219772650 T^{8} + 959725 T^{10} + 19055 T^{12} + 195 T^{14} + T^{16}$$
$43$ $$( 25 + 775 T^{2} + 430 T^{4} + 40 T^{6} + T^{8} )^{2}$$
$47$ $$1908029761 + 4215303862 T^{2} + 25302412353 T^{4} - 1296238621 T^{6} + 34246130 T^{8} - 607831 T^{10} + 14043 T^{12} - 173 T^{14} + T^{16}$$
$53$ $$492884401 - 35232987 T^{2} + 26672228 T^{4} - 6429129 T^{6} + 723955 T^{8} - 32829 T^{10} + 968 T^{12} - 27 T^{14} + T^{16}$$
$59$ $$5465500541281 - 802433531317 T^{2} + 51143793603 T^{4} - 1361619179 T^{6} + 23271680 T^{8} - 406799 T^{10} + 10563 T^{12} + 23 T^{14} + T^{16}$$
$61$ $$( 15625 + 15625 T + 12500 T^{2} - 9375 T^{3} - 875 T^{4} + 375 T^{5} + 200 T^{6} + 25 T^{7} + T^{8} )^{2}$$
$67$ $$( 281 + 768 T + 244 T^{2} + 27 T^{3} + T^{4} )^{4}$$
$71$ $$360750390625 + 28754921875 T^{2} + 9661171875 T^{4} - 120709375 T^{6} + 25890000 T^{8} - 1745375 T^{10} + 53875 T^{12} - 365 T^{14} + T^{16}$$
$73$ $$( 8673025 + 1339975 T - 206350 T^{2} - 76725 T^{3} - 4205 T^{4} + 1335 T^{5} + 290 T^{6} + 25 T^{7} + T^{8} )^{2}$$
$79$ $$( 483025 + 528200 T + 206300 T^{2} + 33700 T^{3} + 4270 T^{4} + 1120 T^{5} + 215 T^{6} + 20 T^{7} + T^{8} )^{2}$$
$83$ $$133974300625 + 12179481875 T^{2} + 2604267875 T^{4} + 421760625 T^{6} + 37734200 T^{8} + 1091625 T^{10} + 13115 T^{12} - 5 T^{14} + T^{16}$$
$89$ $$( 161925625 + 6417250 T^{2} + 86575 T^{4} + 490 T^{6} + T^{8} )^{2}$$
$97$ $$( 5958481 - 12251379 T + 9416242 T^{2} + 754953 T^{3} + 117905 T^{4} + 7707 T^{5} + 592 T^{6} + 29 T^{7} + T^{8} )^{2}$$