# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$420 = 2^{2} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 420.c (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: $$x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 3 x^{12} + 2 x^{11} - 7 x^{10} + 12 x^{9} - 28 x^{8} + 24 x^{7} - 28 x^{6} + 16 x^{5} + 48 x^{4} - 128 x^{3} + 192 x^{2} - 256 x + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{8}$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 3 x^{12} + 2 x^{11} - 7 x^{10} + 12 x^{9} - 28 x^{8} + 24 x^{7} - 28 x^{6} + 16 x^{5} + 48 x^{4} - 128 x^{3} + 192 x^{2} - 256 x + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{15} + 4 \nu^{14} + 5 \nu^{13} - 6 \nu^{12} - 7 \nu^{11} - 20 \nu^{10} - 33 \nu^{9} - 10 \nu^{8} + 136 \nu^{6} - 52 \nu^{5} + 24 \nu^{4} - 32 \nu^{3} + 192 \nu^{2} + 704 \nu + 384$$$$)/512$$ $$\beta_{4}$$ $$=$$ $$($$$$-3 \nu^{14} - 5 \nu^{12} + 2 \nu^{11} + 7 \nu^{10} + 17 \nu^{8} - 18 \nu^{7} + 20 \nu^{6} + 8 \nu^{5} + 68 \nu^{4} + 72 \nu^{3} - 112 \nu^{2} + 32 \nu - 384$$$$)/128$$ $$\beta_{5}$$ $$=$$ $$($$$$-5 \nu^{15} + 4 \nu^{14} - 15 \nu^{13} + 18 \nu^{12} + 5 \nu^{11} - 4 \nu^{10} + 3 \nu^{9} - 34 \nu^{8} + 56 \nu^{7} - 40 \nu^{6} + 60 \nu^{5} - 8 \nu^{4} - 448 \nu + 640$$$$)/512$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{15} + 5 \nu^{14} + \nu^{13} + 5 \nu^{12} - 13 \nu^{11} - 21 \nu^{10} - 5 \nu^{9} - 9 \nu^{8} + 28 \nu^{7} - 48 \nu^{6} - 52 \nu^{5} - 68 \nu^{4} + 16 \nu^{3} + 352 \nu^{2} + 256 \nu + 576$$$$)/128$$ $$\beta_{7}$$ $$=$$ $$($$$$-5 \nu^{15} + 16 \nu^{14} - 15 \nu^{13} + 38 \nu^{12} - 3 \nu^{11} - 32 \nu^{10} + 3 \nu^{9} - 102 \nu^{8} + 128 \nu^{7} - 120 \nu^{6} + 28 \nu^{5} - 280 \nu^{4} - 800 \nu^{3} + 448 \nu^{2} - 1088 \nu + 2176$$$$)/512$$ $$\beta_{8}$$ $$=$$ $$($$$$-5 \nu^{15} - 28 \nu^{14} + \nu^{13} - 14 \nu^{12} + 53 \nu^{11} + 60 \nu^{10} + 51 \nu^{9} + 62 \nu^{8} - 56 \nu^{7} + 312 \nu^{6} + 124 \nu^{5} + 568 \nu^{4} + 64 \nu^{3} - 1280 \nu^{2} - 704 \nu - 2944$$$$)/512$$ $$\beta_{9}$$ $$=$$ $$($$$$\nu^{15} - 2 \nu^{14} + 3 \nu^{13} - 4 \nu^{12} + 3 \nu^{11} + 2 \nu^{10} - 7 \nu^{9} + 12 \nu^{8} - 28 \nu^{7} + 24 \nu^{6} - 28 \nu^{5} + 16 \nu^{4} + 48 \nu^{3} - 128 \nu^{2} + 192 \nu - 256$$$$)/64$$ $$\beta_{10}$$ $$=$$ $$($$$$3 \nu^{15} + 5 \nu^{13} - 2 \nu^{12} - 7 \nu^{11} - 17 \nu^{9} + 18 \nu^{8} - 20 \nu^{7} - 8 \nu^{6} - 68 \nu^{5} - 72 \nu^{4} + 112 \nu^{3} - 32 \nu^{2} + 384 \nu$$$$)/128$$ $$\beta_{11}$$ $$=$$ $$($$$$-11 \nu^{15} - 8 \nu^{14} - 9 \nu^{13} + 18 \nu^{12} + 43 \nu^{11} + 8 \nu^{10} + 21 \nu^{9} - 50 \nu^{8} + 72 \nu^{7} + 72 \nu^{6} + 356 \nu^{5} + 56 \nu^{4} - 512 \nu^{3} - 640 \nu^{2} - 1088 \nu + 128$$$$)/512$$ $$\beta_{12}$$ $$=$$ $$($$$$3 \nu^{15} - 2 \nu^{14} + 3 \nu^{13} - 8 \nu^{12} - \nu^{11} + 14 \nu^{10} + \nu^{9} + 32 \nu^{8} - 34 \nu^{7} - 4 \nu^{6} - 20 \nu^{5} + 64 \nu^{4} + 168 \nu^{3} - 176 \nu^{2} + 96 \nu - 576$$$$)/128$$ $$\beta_{13}$$ $$=$$ $$($$$$-3 \nu^{15} + 2 \nu^{14} - 5 \nu^{13} + 8 \nu^{12} + 3 \nu^{11} - 2 \nu^{10} + 9 \nu^{9} - 32 \nu^{8} + 40 \nu^{7} - 8 \nu^{6} + 60 \nu^{5} + 16 \nu^{4} - 176 \nu^{3} + 96 \nu^{2} - 384 \nu + 384$$$$)/64$$ $$\beta_{14}$$ $$=$$ $$($$$$-3 \nu^{15} + 6 \nu^{14} - 3 \nu^{13} + 12 \nu^{12} + \nu^{11} - 10 \nu^{10} + 7 \nu^{9} - 36 \nu^{8} + 50 \nu^{7} - 36 \nu^{6} + 44 \nu^{5} - 64 \nu^{4} - 280 \nu^{3} + 240 \nu^{2} - 352 \nu + 704$$$$)/64$$ $$\beta_{15}$$ $$=$$ $$($$$$-33 \nu^{15} + 40 \nu^{14} - 43 \nu^{13} + 86 \nu^{12} + 17 \nu^{11} - 104 \nu^{10} + 79 \nu^{9} - 374 \nu^{8} + 520 \nu^{7} - 232 \nu^{6} + 428 \nu^{5} - 24 \nu^{4} - 2240 \nu^{3} + 2304 \nu^{2} - 3520 \nu + 5504$$$$)/512$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{7} + \beta_{5} - \beta_{4} - \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{13} + \beta_{12} + \beta_{9} + \beta_{6} + \beta_{4} + 1$$ $$\nu^{5}$$ $$=$$ $$-\beta_{15} + \beta_{13} + \beta_{12} + 2 \beta_{11} - \beta_{10} - \beta_{8} + \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} - 1$$ $$\nu^{6}$$ $$=$$ $$-\beta_{15} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{4} + 3 \beta_{3} + \beta_{2} - \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$4 \beta_{15} - 2 \beta_{14} - 4 \beta_{13} + 2 \beta_{12} + 2 \beta_{11} + 4 \beta_{10} - 4 \beta_{9} + 2 \beta_{8} + \beta_{7} - 2 \beta_{6} + \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + 3 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$2 \beta_{14} - 5 \beta_{13} + \beta_{12} + 2 \beta_{11} - 3 \beta_{9} + 2 \beta_{8} + 4 \beta_{7} + \beta_{6} + 2 \beta_{5} + 7 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 7$$ $$\nu^{9}$$ $$=$$ $$-3 \beta_{15} + 2 \beta_{14} - \beta_{13} - 3 \beta_{12} - 4 \beta_{11} - 7 \beta_{10} - 2 \beta_{9} + 3 \beta_{8} - 2 \beta_{7} + \beta_{6} - 3 \beta_{5} - 6 \beta_{4} - 3 \beta_{3} - 5 \beta_{2} + 6 \beta_{1} + 5$$ $$\nu^{10}$$ $$=$$ $$\beta_{15} + 2 \beta_{14} + 2 \beta_{13} - 6 \beta_{11} + 9 \beta_{10} - 3 \beta_{9} + \beta_{8} - 5 \beta_{7} - 8 \beta_{6} + 2 \beta_{5} - \beta_{3} + 3 \beta_{2} + 9 \beta_{1} + 6$$ $$\nu^{11}$$ $$=$$ $$10 \beta_{15} - 4 \beta_{14} - 14 \beta_{13} + 10 \beta_{12} + 8 \beta_{11} - 2 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} + 3 \beta_{7} - 6 \beta_{6} + 15 \beta_{5} - 21 \beta_{4} + 2 \beta_{3} + 18 \beta_{2} + 9 \beta_{1} + 14$$ $$\nu^{12}$$ $$=$$ $$-14 \beta_{15} + 8 \beta_{14} + 17 \beta_{13} + \beta_{12} + 2 \beta_{10} + 11 \beta_{9} + 6 \beta_{8} - 2 \beta_{7} + 17 \beta_{6} + 8 \beta_{5} + 5 \beta_{4} - 6 \beta_{3} + 8 \beta_{2} - 6 \beta_{1} - 15$$ $$\nu^{13}$$ $$=$$ $$-\beta_{15} + 12 \beta_{14} + 9 \beta_{13} - 3 \beta_{12} - 2 \beta_{11} + 15 \beta_{10} + 8 \beta_{9} + 3 \beta_{8} - 36 \beta_{7} + 5 \beta_{6} - 9 \beta_{5} - 4 \beta_{4} - 11 \beta_{3} - 15 \beta_{2} - 32 \beta_{1} - 1$$ $$\nu^{14}$$ $$=$$ $$-\beta_{15} + 12 \beta_{14} - 12 \beta_{13} + 24 \beta_{12} - 4 \beta_{11} - 17 \beta_{10} - \beta_{9} - 3 \beta_{8} - 7 \beta_{7} - 8 \beta_{6} + 28 \beta_{5} - 22 \beta_{4} + 31 \beta_{3} - 11 \beta_{2} - \beta_{1} - 20$$ $$\nu^{15}$$ $$=$$ $$-38 \beta_{14} + 8 \beta_{13} + 66 \beta_{12} + 46 \beta_{11} - 24 \beta_{10} + 2 \beta_{8} + 77 \beta_{7} + 22 \beta_{6} - 27 \beta_{5} - 79 \beta_{4} + 34 \beta_{3} + 68 \beta_{2} + 31 \beta_{1} + 12$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
391.1
 −1.39396 − 0.238466i −1.39396 + 0.238466i −0.947441 − 1.04993i −0.947441 + 1.04993i −0.449546 − 1.34086i −0.449546 + 1.34086i −0.102186 − 1.41052i −0.102186 + 1.41052i 0.309204 − 1.38000i 0.309204 + 1.38000i 1.07312 − 0.921096i 1.07312 + 0.921096i 1.10145 − 0.887017i 1.10145 + 0.887017i 1.40936 − 0.117062i 1.40936 + 0.117062i
−1.39396 0.238466i 1.00000 1.88627 + 0.664826i 1.00000i −1.39396 0.238466i −2.37694 + 1.16196i −2.47085 1.37655i 1.00000 −0.238466 + 1.39396i
391.2 −1.39396 + 0.238466i 1.00000 1.88627 0.664826i 1.00000i −1.39396 + 0.238466i −2.37694 1.16196i −2.47085 + 1.37655i 1.00000 −0.238466 1.39396i
391.3 −0.947441 1.04993i 1.00000 −0.204711 + 1.98950i 1.00000i −0.947441 1.04993i 2.29670 + 1.31346i 2.28279 1.67000i 1.00000 −1.04993 + 0.947441i
391.4 −0.947441 + 1.04993i 1.00000 −0.204711 1.98950i 1.00000i −0.947441 + 1.04993i 2.29670 1.31346i 2.28279 + 1.67000i 1.00000 −1.04993 0.947441i
391.5 −0.449546 1.34086i 1.00000 −1.59582 + 1.20556i 1.00000i −0.449546 1.34086i −1.40015 2.24490i 2.33388 + 1.59781i 1.00000 1.34086 0.449546i
391.6 −0.449546 + 1.34086i 1.00000 −1.59582 1.20556i 1.00000i −0.449546 + 1.34086i −1.40015 + 2.24490i 2.33388 1.59781i 1.00000 1.34086 + 0.449546i
391.7 −0.102186 1.41052i 1.00000 −1.97912 + 0.288270i 1.00000i −0.102186 1.41052i 0.178143 + 2.63975i 0.608847 + 2.76212i 1.00000 1.41052 0.102186i
391.8 −0.102186 + 1.41052i 1.00000 −1.97912 0.288270i 1.00000i −0.102186 + 1.41052i 0.178143 2.63975i 0.608847 2.76212i 1.00000 1.41052 + 0.102186i
391.9 0.309204 1.38000i 1.00000 −1.80879 0.853401i 1.00000i 0.309204 1.38000i 2.64459 0.0785232i −1.73698 + 2.23224i 1.00000 −1.38000 0.309204i
391.10 0.309204 + 1.38000i 1.00000 −1.80879 + 0.853401i 1.00000i 0.309204 + 1.38000i 2.64459 + 0.0785232i −1.73698 2.23224i 1.00000 −1.38000 + 0.309204i
391.11 1.07312 0.921096i 1.00000 0.303166 1.97689i 1.00000i 1.07312 0.921096i 1.82575 + 1.91485i −1.49557 2.40068i 1.00000 0.921096 + 1.07312i
391.12 1.07312 + 0.921096i 1.00000 0.303166 + 1.97689i 1.00000i 1.07312 + 0.921096i 1.82575 1.91485i −1.49557 + 2.40068i 1.00000 0.921096 1.07312i
391.13 1.10145 0.887017i 1.00000 0.426402 1.95402i 1.00000i 1.10145 0.887017i −0.391948 2.61656i −1.26358 2.53049i 1.00000 −0.887017 1.10145i
391.14 1.10145 + 0.887017i 1.00000 0.426402 + 1.95402i 1.00000i 1.10145 + 0.887017i −0.391948 + 2.61656i −1.26358 + 2.53049i 1.00000 −0.887017 + 1.10145i
391.15 1.40936 0.117062i 1.00000 1.97259 0.329965i 1.00000i 1.40936 0.117062i −0.776136 + 2.52935i 2.74147 0.695955i 1.00000 −0.117062 1.40936i
391.16 1.40936 + 0.117062i 1.00000 1.97259 + 0.329965i 1.00000i 1.40936 + 0.117062i −0.776136 2.52935i 2.74147 + 0.695955i 1.00000 −0.117062 + 1.40936i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 391.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d 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.c.b yes 16
3.b odd 2 1 1260.2.c.e 16
4.b odd 2 1 420.2.c.a 16
7.b odd 2 1 420.2.c.a 16
12.b even 2 1 1260.2.c.d 16
21.c even 2 1 1260.2.c.d 16
28.d even 2 1 inner 420.2.c.b yes 16
84.h odd 2 1 1260.2.c.e 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.c.a 16 4.b odd 2 1
420.2.c.a 16 7.b odd 2 1
420.2.c.b yes 16 1.a even 1 1 trivial
420.2.c.b yes 16 28.d even 2 1 inner
1260.2.c.d 16 12.b even 2 1
1260.2.c.d 16 21.c even 2 1
1260.2.c.e 16 3.b odd 2 1
1260.2.c.e 16 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{19}^{8} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(420, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$256 - 256 T + 192 T^{2} - 128 T^{3} + 48 T^{4} + 16 T^{5} - 28 T^{6} + 24 T^{7} - 28 T^{8} + 12 T^{9} - 7 T^{10} + 2 T^{11} + 3 T^{12} - 4 T^{13} + 3 T^{14} - 2 T^{15} + T^{16}$$
$3$ $$( -1 + T )^{16}$$
$5$ $$( 1 + T^{2} )^{8}$$
$7$ $$5764801 - 3294172 T + 1882384 T^{2} - 1142876 T^{3} + 451388 T^{4} - 160524 T^{5} + 65072 T^{6} - 22988 T^{7} + 6854 T^{8} - 3284 T^{9} + 1328 T^{10} - 468 T^{11} + 188 T^{12} - 68 T^{13} + 16 T^{14} - 4 T^{15} + T^{16}$$
$11$ $$16384 + 172032 T^{2} + 627712 T^{4} + 947712 T^{6} + 549248 T^{8} + 75904 T^{10} + 4292 T^{12} + 108 T^{14} + T^{16}$$
$13$ $$1982464 + 60882944 T^{2} + 38963200 T^{4} + 10193408 T^{6} + 1398080 T^{8} + 108192 T^{10} + 4724 T^{12} + 108 T^{14} + T^{16}$$
$17$ $$589824 + 3407872 T^{2} + 4907008 T^{4} + 2729984 T^{6} + 641792 T^{8} + 71296 T^{10} + 3952 T^{12} + 104 T^{14} + T^{16}$$
$19$ $$( 1408 + 6784 T + 1696 T^{2} - 2400 T^{3} - 352 T^{4} + 288 T^{5} + 6 T^{6} - 12 T^{7} + T^{8} )^{2}$$
$23$ $$110166016 + 264994816 T^{2} + 196538368 T^{4} + 59332608 T^{6} + 7611136 T^{8} + 463744 T^{10} + 13792 T^{12} + 192 T^{14} + T^{16}$$
$29$ $$( -475392 + 357376 T + 36736 T^{2} - 45056 T^{3} + 2800 T^{4} + 1264 T^{5} - 132 T^{6} - 8 T^{7} + T^{8} )^{2}$$
$31$ $$( 435072 - 44288 T - 101408 T^{2} + 9088 T^{3} + 6616 T^{4} - 352 T^{5} - 150 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$37$ $$( -843008 + 540928 T + 21504 T^{2} - 52800 T^{3} + 2352 T^{4} + 1504 T^{5} - 108 T^{6} - 12 T^{7} + T^{8} )^{2}$$
$41$ $$186292371456 + 317850714112 T^{2} + 145118531584 T^{4} + 12630601728 T^{6} + 463714816 T^{8} + 8764288 T^{10} + 89680 T^{12} + 472 T^{14} + T^{16}$$
$43$ $$32480690176 + 89542098944 T^{2} + 28063666176 T^{4} + 2828883968 T^{6} + 132512000 T^{8} + 3322752 T^{10} + 46176 T^{12} + 336 T^{14} + T^{16}$$
$47$ $$( 1243904 - 411392 T - 177344 T^{2} + 41984 T^{3} + 9264 T^{4} - 1152 T^{5} - 176 T^{6} + 8 T^{7} + T^{8} )^{2}$$
$53$ $$( 311424 - 756608 T + 129344 T^{2} + 72352 T^{3} - 5288 T^{4} - 2704 T^{5} - 106 T^{6} + 16 T^{7} + T^{8} )^{2}$$
$59$ $$( -323072 - 815104 T - 52032 T^{2} + 97280 T^{3} + 14640 T^{4} - 1440 T^{5} - 256 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$61$ $$334139490304 + 337266016256 T^{2} + 80774565888 T^{4} + 8235839488 T^{6} + 408406528 T^{8} + 9941632 T^{10} + 110800 T^{12} + 552 T^{14} + T^{16}$$
$67$ $$1665379926016 + 1333371076608 T^{2} + 235939749888 T^{4} + 16515217408 T^{6} + 559284992 T^{8} + 10131840 T^{10} + 100080 T^{12} + 504 T^{14} + T^{16}$$
$71$ $$57232008953856 + 36387631611904 T^{2} + 3186748889088 T^{4} + 120727758336 T^{6} + 2458985024 T^{8} + 28845536 T^{10} + 194196 T^{12} + 692 T^{14} + T^{16}$$
$73$ $$34021064704 + 39953014784 T^{2} + 16064015360 T^{4} + 2638006272 T^{6} + 177219264 T^{8} + 5348384 T^{10} + 74260 T^{12} + 460 T^{14} + T^{16}$$
$79$ $$57415827456 + 81825366016 T^{2} + 29610151936 T^{4} + 4426000384 T^{6} + 291090176 T^{8} + 7517952 T^{10} + 89072 T^{12} + 488 T^{14} + T^{16}$$
$83$ $$( -30015488 - 23134208 T - 2523136 T^{2} + 457728 T^{3} + 65920 T^{4} - 2592 T^{5} - 468 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$89$ $$278378643456 + 1212059189248 T^{2} + 286828687360 T^{4} + 23481739264 T^{6} + 804129536 T^{8} + 13741952 T^{10} + 123744 T^{12} + 560 T^{14} + T^{16}$$
$97$ $$970650566656 + 5353125814272 T^{2} + 2353403653120 T^{4} + 174997760000 T^{6} + 4227116736 T^{8} + 48005920 T^{10} + 283540 T^{12} + 844 T^{14} + T^{16}$$