# Properties

 Label 420.2.l.h Level $420$ Weight $2$ Character orbit 420.l Analytic conductor $3.354$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$420 = 2^{2} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 420.l (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.35371688489$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 4 x^{15} + 9 x^{14} - 16 x^{13} + 18 x^{12} - 4 x^{11} - 36 x^{10} + 102 x^{9} - 170 x^{8} + 204 x^{7} - 144 x^{6} - 32 x^{5} + 288 x^{4} - 512 x^{3} + 576 x^{2} - 512 x + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{6}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 4 x^{15} + 9 x^{14} - 16 x^{13} + 18 x^{12} - 4 x^{11} - 36 x^{10} + 102 x^{9} - 170 x^{8} + 204 x^{7} - 144 x^{6} - 32 x^{5} + 288 x^{4} - 512 x^{3} + 576 x^{2} - 512 x + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-3 \nu^{15} - 10 \nu^{14} + 17 \nu^{13} + 2 \nu^{12} - 18 \nu^{11} + 72 \nu^{10} - 100 \nu^{9} + 134 \nu^{8} + 42 \nu^{7} - 368 \nu^{6} + 528 \nu^{5} - 704 \nu^{4} + 224 \nu^{3} + 192 \nu^{2} - 1472 \nu + 2048$$$$)/128$$ $$\beta_{2}$$ $$=$$ $$($$$$23 \nu^{15} - 78 \nu^{14} + 179 \nu^{13} - 266 \nu^{12} + 202 \nu^{11} + 152 \nu^{10} - 972 \nu^{9} + 2002 \nu^{8} - 2786 \nu^{7} + 2512 \nu^{6} - 528 \nu^{5} - 2752 \nu^{4} + 6496 \nu^{3} - 8256 \nu^{2} + 6976 \nu - 3712$$$$)/256$$ $$\beta_{3}$$ $$=$$ $$($$$$25 \nu^{15} - 98 \nu^{14} + 93 \nu^{13} + 58 \nu^{12} - 330 \nu^{11} + 744 \nu^{10} - 1012 \nu^{9} + 718 \nu^{8} + 866 \nu^{7} - 3056 \nu^{6} + 4432 \nu^{5} - 4352 \nu^{4} + 1568 \nu^{3} + 3392 \nu^{2} - 10048 \nu + 8576$$$$)/256$$ $$\beta_{4}$$ $$=$$ $$($$$$-9 \nu^{15} + 14 \nu^{14} + 15 \nu^{13} - 66 \nu^{12} + 134 \nu^{11} - 164 \nu^{10} + 92 \nu^{9} + 218 \nu^{8} - 762 \nu^{7} + 1136 \nu^{6} - 1104 \nu^{5} + 376 \nu^{4} + 848 \nu^{3} - 2528 \nu^{2} + 3392 \nu - 1472$$$$)/64$$ $$\beta_{5}$$ $$=$$ $$($$$$-49 \nu^{15} + 90 \nu^{14} + 107 \nu^{13} - 498 \nu^{12} + 954 \nu^{11} - 1192 \nu^{10} + 596 \nu^{9} + 1634 \nu^{8} - 5602 \nu^{7} + 8608 \nu^{6} - 8336 \nu^{5} + 3168 \nu^{4} + 6368 \nu^{3} - 18240 \nu^{2} + 25920 \nu - 13952$$$$)/256$$ $$\beta_{6}$$ $$=$$ $$($$$$4 \nu^{15} - 13 \nu^{14} + 19 \nu^{13} - 16 \nu^{12} - 9 \nu^{11} + 63 \nu^{10} - 142 \nu^{9} + 194 \nu^{8} - 142 \nu^{7} - 16 \nu^{6} + 252 \nu^{5} - 470 \nu^{4} + 572 \nu^{3} - 336 \nu^{2} - 144 \nu + 176$$$$)/16$$ $$\beta_{7}$$ $$=$$ $$($$$$-13 \nu^{15} + 58 \nu^{14} - 137 \nu^{13} + 202 \nu^{12} - 158 \nu^{11} - 108 \nu^{10} + 700 \nu^{9} - 1486 \nu^{8} + 2038 \nu^{7} - 1776 \nu^{6} + 328 \nu^{5} + 2040 \nu^{4} - 4640 \nu^{3} + 5856 \nu^{2} - 4608 \nu + 1856$$$$)/64$$ $$\beta_{8}$$ $$=$$ $$($$$$-3 \nu^{15} - 114 \nu^{14} + 593 \nu^{13} - 1254 \nu^{12} + 1582 \nu^{11} - 904 \nu^{10} - 1828 \nu^{9} + 7014 \nu^{8} - 13254 \nu^{7} + 15712 \nu^{6} - 10256 \nu^{5} - 3584 \nu^{4} + 23200 \nu^{3} - 40512 \nu^{2} + 43968 \nu - 22656$$$$)/256$$ $$\beta_{9}$$ $$=$$ $$($$$$13 \nu^{15} - 114 \nu^{14} + 361 \nu^{13} - 630 \nu^{12} + 662 \nu^{11} - 88 \nu^{10} - 1492 \nu^{9} + 4070 \nu^{8} - 6534 \nu^{7} + 6736 \nu^{6} - 3200 \nu^{5} - 4032 \nu^{4} + 12960 \nu^{3} - 19584 \nu^{2} + 18368 \nu - 8192$$$$)/128$$ $$\beta_{10}$$ $$=$$ $$($$$$3 \nu^{15} - 198 \nu^{14} + 783 \nu^{13} - 1458 \nu^{12} + 1682 \nu^{11} - 584 \nu^{10} - 2876 \nu^{9} + 8954 \nu^{8} - 15370 \nu^{7} + 16592 \nu^{6} - 9040 \nu^{5} - 7616 \nu^{4} + 29152 \nu^{3} - 46656 \nu^{2} + 45632 \nu - 19584$$$$)/256$$ $$\beta_{11}$$ $$=$$ $$($$$$-5 \nu^{15} + 52 \nu^{14} - 185 \nu^{13} + 336 \nu^{12} - 374 \nu^{11} + 100 \nu^{10} + 716 \nu^{9} - 2094 \nu^{8} + 3522 \nu^{7} - 3748 \nu^{6} + 1960 \nu^{5} + 1872 \nu^{4} - 6784 \nu^{3} + 10560 \nu^{2} - 10304 \nu + 4608$$$$)/64$$ $$\beta_{12}$$ $$=$$ $$($$$$-43 \nu^{15} + 262 \nu^{14} - 743 \nu^{13} + 1218 \nu^{12} - 1170 \nu^{11} - 104 \nu^{10} + 3324 \nu^{9} - 8202 \nu^{8} + 12506 \nu^{7} - 12112 \nu^{6} + 4560 \nu^{5} + 9376 \nu^{4} - 25888 \nu^{3} + 36544 \nu^{2} - 32064 \nu + 12928$$$$)/256$$ $$\beta_{13}$$ $$=$$ $$($$$$-63 \nu^{15} + 286 \nu^{14} - 699 \nu^{13} + 1098 \nu^{12} - 922 \nu^{11} - 392 \nu^{10} + 3468 \nu^{9} - 7778 \nu^{8} + 11154 \nu^{7} - 10384 \nu^{6} + 2928 \nu^{5} + 9824 \nu^{4} - 24544 \nu^{3} + 32704 \nu^{2} - 27712 \nu + 12928$$$$)/256$$ $$\beta_{14}$$ $$=$$ $$($$$$-97 \nu^{15} + 410 \nu^{14} - 789 \nu^{13} + 910 \nu^{12} - 342 \nu^{11} - 1512 \nu^{10} + 4756 \nu^{9} - 8254 \nu^{8} + 8958 \nu^{7} - 4736 \nu^{6} - 4208 \nu^{5} + 15840 \nu^{4} - 25056 \nu^{3} + 24512 \nu^{2} - 9920 \nu - 2176$$$$)/256$$ $$\beta_{15}$$ $$=$$ $$($$$$-79 \nu^{15} + 370 \nu^{14} - 803 \nu^{13} + 1086 \nu^{12} - 706 \nu^{11} - 992 \nu^{10} + 4396 \nu^{9} - 8658 \nu^{8} + 10890 \nu^{7} - 8200 \nu^{6} - 656 \nu^{5} + 13824 \nu^{4} - 26784 \nu^{3} + 31104 \nu^{2} - 20032 \nu + 5120$$$$)/128$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{15} + \beta_{14} - 2 \beta_{13} - \beta_{12} + \beta_{9} - 6 \beta_{8} - \beta_{6} + 4 \beta_{5} - \beta_{4} - 2 \beta_{3} + 6$$$$)/12$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{15} - 5 \beta_{14} + 3 \beta_{12} + 2 \beta_{11} + 4 \beta_{10} - 5 \beta_{9} - 6 \beta_{7} + 3 \beta_{6} + \beta_{4} - 12 \beta_{3} - 8 \beta_{2} + 4 \beta_{1} - 6$$$$)/12$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{15} + \beta_{14} + 5 \beta_{12} - 6 \beta_{11} - 3 \beta_{9} - 2 \beta_{7} + \beta_{6} - \beta_{4} - 4 \beta_{2} + 2$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$-7 \beta_{15} - \beta_{14} + 12 \beta_{13} + 3 \beta_{12} - 2 \beta_{11} - 4 \beta_{10} + 11 \beta_{9} + 6 \beta_{7} - 9 \beta_{6} - 7 \beta_{4} - 4 \beta_{2} - 16 \beta_{1} + 18$$$$)/12$$ $$\nu^{5}$$ $$=$$ $$($$$$-11 \beta_{15} - \beta_{14} - 4 \beta_{13} - 5 \beta_{12} + 6 \beta_{11} - \beta_{9} - 12 \beta_{8} + 6 \beta_{7} - 29 \beta_{6} + 8 \beta_{5} - 11 \beta_{4} - 4 \beta_{3} - 12 \beta_{1} - 18$$$$)/12$$ $$\nu^{6}$$ $$=$$ $$($$$$3 \beta_{15} - 11 \beta_{14} - 4 \beta_{13} + 9 \beta_{12} + 10 \beta_{11} + 8 \beta_{10} - 7 \beta_{9} + 8 \beta_{8} - 14 \beta_{7} - 11 \beta_{6} - 8 \beta_{5} + 3 \beta_{4} - 8 \beta_{3} - 8 \beta_{2} - 2$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$-19 \beta_{15} + 7 \beta_{14} + 4 \beta_{13} + 35 \beta_{12} - 6 \beta_{11} - 24 \beta_{10} + 7 \beta_{9} + 12 \beta_{8} - 30 \beta_{7} - 49 \beta_{6} + 28 \beta_{5} - 31 \beta_{4} + 40 \beta_{3} - 12 \beta_{1} + 30$$$$)/12$$ $$\nu^{8}$$ $$=$$ $$($$$$-23 \beta_{15} - 65 \beta_{14} + 48 \beta_{13} + 15 \beta_{12} + 50 \beta_{11} - 32 \beta_{10} + 67 \beta_{9} + 36 \beta_{8} + 42 \beta_{7} - 93 \beta_{6} + 48 \beta_{5} - 95 \beta_{4} - 12 \beta_{3} + 16 \beta_{2} - 32 \beta_{1} - 6$$$$)/12$$ $$\nu^{9}$$ $$=$$ $$($$$$7 \beta_{15} - 43 \beta_{14} - 16 \beta_{13} + 37 \beta_{12} - 14 \beta_{11} - 3 \beta_{9} + 16 \beta_{8} + 2 \beta_{7} - 47 \beta_{6} - 4 \beta_{5} - 29 \beta_{4} - 4 \beta_{3} - 32 \beta_{2} - 12 \beta_{1} - 38$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$17 \beta_{15} - 145 \beta_{14} + 135 \beta_{12} + 10 \beta_{11} - 40 \beta_{10} + 11 \beta_{9} + 132 \beta_{8} - 54 \beta_{7} - 93 \beta_{6} - 96 \beta_{5} + 89 \beta_{4} - 36 \beta_{3} - 136 \beta_{2} - 88 \beta_{1} - 78$$$$)/12$$ $$\nu^{11}$$ $$=$$ $$($$$$-203 \beta_{15} + 191 \beta_{14} + 128 \beta_{13} - 41 \beta_{12} - 162 \beta_{11} - 264 \beta_{10} + 71 \beta_{9} - 72 \beta_{8} + 102 \beta_{7} - 53 \beta_{6} + 116 \beta_{5} + 25 \beta_{4} + 44 \beta_{3} + 96 \beta_{2} - 36 \beta_{1} + 6$$$$)/12$$ $$\nu^{12}$$ $$=$$ $$($$$$-5 \beta_{15} - 67 \beta_{14} + 48 \beta_{13} + 37 \beta_{12} + 30 \beta_{11} + 32 \beta_{10} + 25 \beta_{9} + 4 \beta_{8} - 18 \beta_{7} - 63 \beta_{6} - 48 \beta_{5} - 13 \beta_{4} - 116 \beta_{3} + 16 \beta_{2} - 2$$$$)/4$$ $$\nu^{13}$$ $$=$$ $$($$$$245 \beta_{15} - 425 \beta_{14} - 128 \beta_{13} + 431 \beta_{12} + 30 \beta_{11} + 264 \beta_{10} - 89 \beta_{9} + 144 \beta_{8} - 570 \beta_{7} - 469 \beta_{6} - 236 \beta_{5} - 103 \beta_{4} + 52 \beta_{3} - 288 \beta_{2} - 156 \beta_{1} - 594$$$$)/12$$ $$\nu^{14}$$ $$=$$ $$($$$$49 \beta_{15} - 593 \beta_{14} + 336 \beta_{13} + 135 \beta_{12} + 578 \beta_{11} - 176 \beta_{10} + 235 \beta_{9} + 660 \beta_{8} - 174 \beta_{7} - 597 \beta_{6} + 120 \beta_{5} - 191 \beta_{4} + 324 \beta_{3} + 112 \beta_{2} - 440 \beta_{1} - 846$$$$)/12$$ $$\nu^{15}$$ $$=$$ $$($$$$-73 \beta_{15} - 59 \beta_{14} + 72 \beta_{13} - 27 \beta_{12} - 102 \beta_{11} - 248 \beta_{10} + 109 \beta_{9} + 96 \beta_{8} + 210 \beta_{7} - 295 \beta_{6} + 164 \beta_{5} - 277 \beta_{4} + 244 \beta_{3} + 192 \beta_{2} - 108 \beta_{1} + 50$$$$)/4$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/420\mathbb{Z}\right)^\times$$.

 $$n$$ $$211$$ $$241$$ $$281$$ $$337$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
239.1
 −1.41329 − 0.0512263i −1.41329 + 0.0512263i 1.37874 + 0.314767i 1.37874 − 0.314767i 0.545199 + 1.30490i 0.545199 − 1.30490i −0.172292 + 1.40368i −0.172292 − 1.40368i −0.556656 − 1.30005i −0.556656 + 1.30005i 1.40708 + 0.141881i 1.40708 − 0.141881i −0.167106 + 1.40431i −0.167106 − 1.40431i 0.978323 + 1.02122i 0.978323 − 1.02122i
−1.38390 0.291259i 0.751690 + 1.56044i 1.83034 + 0.806145i −1.60157 1.56044i −0.585769 2.37842i 1.00000 −2.29820 1.64872i −1.86993 + 2.34593i 1.76192 + 2.62595i
239.2 −1.38390 + 0.291259i 0.751690 1.56044i 1.83034 0.806145i −1.60157 + 1.56044i −0.585769 + 2.37842i 1.00000 −2.29820 + 1.64872i −1.86993 2.34593i 1.76192 2.62595i
239.3 −1.25945 0.643263i 1.48716 0.887900i 1.17242 + 1.62032i 2.05223 + 0.887900i −2.44415 + 0.161632i 1.00000 −0.434319 2.79488i 1.42327 2.64089i −2.01352 2.43839i
239.4 −1.25945 + 0.643263i 1.48716 + 0.887900i 1.17242 1.62032i 2.05223 0.887900i −2.44415 0.161632i 1.00000 −0.434319 + 2.79488i 1.42327 + 2.64089i −2.01352 + 2.43839i
239.5 −0.979286 1.02029i −1.71822 0.218455i −0.0819979 + 1.99832i −2.22537 + 0.218455i 1.45974 + 1.96702i 1.00000 2.11917 1.87326i 2.90455 + 0.750707i 2.40216 + 2.05660i
239.6 −0.979286 + 1.02029i −1.71822 + 0.218455i −0.0819979 1.99832i −2.22537 0.218455i 1.45974 1.96702i 1.00000 2.11917 + 1.87326i 2.90455 0.750707i 2.40216 2.05660i
239.7 −0.538162 1.30782i −0.520627 1.65195i −1.42076 + 1.40763i 1.50700 + 1.65195i −1.88027 + 1.56990i 1.00000 2.60553 + 1.10056i −2.45790 + 1.72010i 1.34944 2.85990i
239.8 −0.538162 + 1.30782i −0.520627 + 1.65195i −1.42076 1.40763i 1.50700 1.65195i −1.88027 1.56990i 1.00000 2.60553 1.10056i −2.45790 1.72010i 1.34944 + 2.85990i
239.9 0.538162 1.30782i −0.520627 1.65195i −1.42076 1.40763i −1.50700 + 1.65195i −2.44063 0.208135i 1.00000 −2.60553 + 1.10056i −2.45790 + 1.72010i 1.34944 + 2.85990i
239.10 0.538162 + 1.30782i −0.520627 + 1.65195i −1.42076 + 1.40763i −1.50700 1.65195i −2.44063 + 0.208135i 1.00000 −2.60553 1.10056i −2.45790 1.72010i 1.34944 2.85990i
239.11 0.979286 1.02029i −1.71822 0.218455i −0.0819979 1.99832i 2.22537 + 0.218455i −1.90552 + 1.53916i 1.00000 −2.11917 1.87326i 2.90455 + 0.750707i 2.40216 2.05660i
239.12 0.979286 + 1.02029i −1.71822 + 0.218455i −0.0819979 + 1.99832i 2.22537 0.218455i −1.90552 1.53916i 1.00000 −2.11917 + 1.87326i 2.90455 0.750707i 2.40216 + 2.05660i
239.13 1.25945 0.643263i 1.48716 0.887900i 1.17242 1.62032i −2.05223 + 0.887900i 1.30184 2.07490i 1.00000 0.434319 2.79488i 1.42327 2.64089i −2.01352 + 2.43839i
239.14 1.25945 + 0.643263i 1.48716 + 0.887900i 1.17242 + 1.62032i −2.05223 0.887900i 1.30184 + 2.07490i 1.00000 0.434319 + 2.79488i 1.42327 + 2.64089i −2.01352 2.43839i
239.15 1.38390 0.291259i 0.751690 + 1.56044i 1.83034 0.806145i 1.60157 1.56044i 1.49475 + 1.94055i 1.00000 2.29820 1.64872i −1.86993 + 2.34593i 1.76192 2.62595i
239.16 1.38390 + 0.291259i 0.751690 1.56044i 1.83034 + 0.806145i 1.60157 + 1.56044i 1.49475 1.94055i 1.00000 2.29820 + 1.64872i −1.86993 2.34593i 1.76192 + 2.62595i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 239.16 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
20.d odd 2 1 inner
60.h even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 420.2.l.h yes 16
3.b odd 2 1 inner 420.2.l.h yes 16
4.b odd 2 1 420.2.l.g 16
5.b even 2 1 420.2.l.g 16
12.b even 2 1 420.2.l.g 16
15.d odd 2 1 420.2.l.g 16
20.d odd 2 1 inner 420.2.l.h yes 16
60.h even 2 1 inner 420.2.l.h yes 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.l.g 16 4.b odd 2 1
420.2.l.g 16 5.b even 2 1
420.2.l.g 16 12.b even 2 1
420.2.l.g 16 15.d odd 2 1
420.2.l.h yes 16 1.a even 1 1 trivial
420.2.l.h yes 16 3.b odd 2 1 inner
420.2.l.h yes 16 20.d odd 2 1 inner
420.2.l.h yes 16 60.h even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(420, [\chi])$$:

 $$T_{11}^{8} - 66 T_{11}^{6} + 1400 T_{11}^{4} - 11360 T_{11}^{2} + 31104$$ $$T_{17}^{8} - 76 T_{17}^{6} + 1968 T_{17}^{4} - 19520 T_{17}^{2} + 55296$$ $$T_{43}^{4} + 4 T_{43}^{3} - 96 T_{43}^{2} - 256 T_{43} + 1616$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$256 - 192 T^{2} + 112 T^{4} - 52 T^{6} + 28 T^{8} - 13 T^{10} + 7 T^{12} - 3 T^{14} + T^{16}$$
$3$ $$( 81 + 12 T^{3} - 2 T^{4} + 4 T^{5} + T^{8} )^{2}$$
$5$ $$390625 - 250000 T^{2} + 97500 T^{4} - 28800 T^{6} + 6534 T^{8} - 1152 T^{10} + 156 T^{12} - 16 T^{14} + T^{16}$$
$7$ $$( -1 + T )^{16}$$
$11$ $$( 31104 - 11360 T^{2} + 1400 T^{4} - 66 T^{6} + T^{8} )^{2}$$
$13$ $$( 864 + 976 T^{2} + 328 T^{4} + 38 T^{6} + T^{8} )^{2}$$
$17$ $$( 55296 - 19520 T^{2} + 1968 T^{4} - 76 T^{6} + T^{8} )^{2}$$
$19$ $$( 864 + 976 T^{2} + 328 T^{4} + 38 T^{6} + T^{8} )^{2}$$
$23$ $$( 135424 + 46848 T^{2} + 3936 T^{4} + 112 T^{6} + T^{8} )^{2}$$
$29$ $$( 1048576 + 176576 T^{2} + 8880 T^{4} + 164 T^{6} + T^{8} )^{2}$$
$31$ $$( 13824 + 11008 T^{2} + 2240 T^{4} + 90 T^{6} + T^{8} )^{2}$$
$37$ $$( 884736 + 147904 T^{2} + 7472 T^{4} + 148 T^{6} + T^{8} )^{2}$$
$41$ $$( 5914624 + 593856 T^{2} + 19248 T^{4} + 244 T^{6} + T^{8} )^{2}$$
$43$ $$( 1616 - 256 T - 96 T^{2} + 4 T^{3} + T^{4} )^{4}$$
$47$ $$( 5914624 + 575232 T^{2} + 17792 T^{4} + 224 T^{6} + T^{8} )^{2}$$
$53$ $$( 3456 - 36320 T^{2} + 5048 T^{4} - 146 T^{6} + T^{8} )^{2}$$
$59$ $$( 13824 - 30224 T^{2} + 6560 T^{4} - 168 T^{6} + T^{8} )^{2}$$
$61$ $$( -712 + 308 T - 8 T^{2} - 10 T^{3} + T^{4} )^{4}$$
$67$ $$( -864 + 504 T - 36 T^{2} - 14 T^{3} + T^{4} )^{4}$$
$71$ $$( 4478976 - 551936 T^{2} + 22848 T^{4} - 342 T^{6} + T^{8} )^{2}$$
$73$ $$( 39567744 + 2250400 T^{2} + 43576 T^{4} + 350 T^{6} + T^{8} )^{2}$$
$79$ $$( 4990464 + 845632 T^{2} + 29008 T^{4} + 332 T^{6} + T^{8} )^{2}$$
$83$ $$( 92416 + 157424 T^{2} + 21696 T^{4} + 380 T^{6} + T^{8} )^{2}$$
$89$ $$( 7661824 + 2849024 T^{2} + 78560 T^{4} + 528 T^{6} + T^{8} )^{2}$$
$97$ $$( 59308416 + 3883936 T^{2} + 77688 T^{4} + 510 T^{6} + T^{8} )^{2}$$