# Properties

 Label 6010.2.a.c Level 6010 Weight 2 Character orbit 6010.a Self dual yes Analytic conductor 47.990 Analytic rank 1 Dimension 16 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$6010 = 2 \cdot 5 \cdot 601$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 6010.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$47.9900916148$$ Analytic rank: $$1$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 8 x^{15} + 9 x^{14} + 75 x^{13} - 178 x^{12} - 232 x^{11} + 872 x^{10} + 228 x^{9} - 1986 x^{8} + 164 x^{7} + 2332 x^{6} - 440 x^{5} - 1344 x^{4} + 244 x^{3} + 295 x^{2} - 41 x - 10$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ $$\beta_{3}$$ $$=$$ $$($$$$25986 \nu^{15} - 3039938 \nu^{14} + 22373644 \nu^{13} - 24822547 \nu^{12} - 180887926 \nu^{11} + 419917416 \nu^{10} + 422463844 \nu^{9} - 1628277055 \nu^{8} - 171959962 \nu^{7} + 2649268970 \nu^{6} - 438850745 \nu^{5} - 1879892501 \nu^{4} + 414738405 \nu^{3} + 462320623 \nu^{2} - 89999479 \nu - 17499053$$$$)/3115183$$ $$\beta_{4}$$ $$=$$ $$($$$$-75052 \nu^{15} - 416320 \nu^{14} + 7114944 \nu^{13} - 14421854 \nu^{12} - 48394711 \nu^{11} + 157537117 \nu^{10} + 79579696 \nu^{9} - 536624577 \nu^{8} + 59905022 \nu^{7} + 793919965 \nu^{6} - 225824922 \nu^{5} - 522240622 \nu^{4} + 145436552 \nu^{3} + 127539985 \nu^{2} - 24830756 \nu - 2133268$$$$)/3115183$$ $$\beta_{5}$$ $$=$$ $$($$$$180376 \nu^{15} - 2419781 \nu^{14} + 9497505 \nu^{13} + 2661981 \nu^{12} - 91910225 \nu^{11} + 118706510 \nu^{10} + 274101206 \nu^{9} - 560118954 \nu^{8} - 327354943 \nu^{7} + 988201962 \nu^{6} + 157843278 \nu^{5} - 763974869 \nu^{4} - 43939251 \nu^{3} + 233958171 \nu^{2} + 10010483 \nu - 20383548$$$$)/3115183$$ $$\beta_{6}$$ $$=$$ $$($$$$284786 \nu^{15} - 1819192 \nu^{14} - 441924 \nu^{13} + 20718600 \nu^{12} - 13794772 \nu^{11} - 100540960 \nu^{10} + 74146068 \nu^{9} + 279407436 \nu^{8} - 157995531 \nu^{7} - 442765003 \nu^{6} + 150259400 \nu^{5} + 331067185 \nu^{4} - 39287261 \nu^{3} - 67060795 \nu^{2} - 9102667 \nu + 706474$$$$)/3115183$$ $$\beta_{7}$$ $$=$$ $$($$$$362678 \nu^{15} - 2892660 \nu^{14} + 4353048 \nu^{13} + 18928481 \nu^{12} - 57805768 \nu^{11} - 10924214 \nu^{10} + 182827079 \nu^{9} - 139100342 \nu^{8} - 176603303 \nu^{7} + 347372878 \nu^{6} - 81890064 \nu^{5} - 319570964 \nu^{4} + 201791930 \nu^{3} + 116648156 \nu^{2} - 61280117 \nu - 6358506$$$$)/3115183$$ $$\beta_{8}$$ $$=$$ $$($$$$-384044 \nu^{15} + 3421318 \nu^{14} - 6474594 \nu^{13} - 22475282 \nu^{12} + 82865319 \nu^{11} + 16650207 \nu^{10} - 294055924 \nu^{9} + 129035112 \nu^{8} + 429564885 \nu^{7} - 280058413 \nu^{6} - 230548103 \nu^{5} + 178775499 \nu^{4} - 8887973 \nu^{3} - 34425448 \nu^{2} + 12448056 \nu - 714592$$$$)/3115183$$ $$\beta_{9}$$ $$=$$ $$($$$$-39704 \nu^{15} + 366746 \nu^{14} - 785482 \nu^{13} - 2133030 \nu^{12} + 9493309 \nu^{11} - 971276 \nu^{10} - 32730627 \nu^{9} + 23795599 \nu^{8} + 47347456 \nu^{7} - 46199287 \nu^{6} - 29093811 \nu^{5} + 29632797 \nu^{4} + 6564984 \nu^{3} - 4266261 \nu^{2} - 556740 \nu - 43793$$$$)/163957$$ $$\beta_{10}$$ $$=$$ $$($$$$-800612 \nu^{15} + 5318060 \nu^{14} - 244087 \nu^{13} - 58445943 \nu^{12} + 60827688 \nu^{11} + 255794130 \nu^{10} - 323031488 \nu^{9} - 600603130 \nu^{8} + 709158898 \nu^{7} + 816274892 \nu^{6} - 731389267 \nu^{5} - 601061929 \nu^{4} + 323396742 \nu^{3} + 181790025 \nu^{2} - 48865179 \nu - 7297457$$$$)/3115183$$ $$\beta_{11}$$ $$=$$ $$($$$$-972224 \nu^{15} + 8826787 \nu^{14} - 18013961 \nu^{13} - 54357008 \nu^{12} + 225954995 \nu^{11} + 3659483 \nu^{10} - 818923620 \nu^{9} + 503191029 \nu^{8} + 1324407527 \nu^{7} - 1099582230 \nu^{6} - 1049225189 \nu^{5} + 852144102 \nu^{4} + 392375534 \nu^{3} - 211092724 \nu^{2} - 44134004 \nu + 4252139$$$$)/3115183$$ $$\beta_{12}$$ $$=$$ $$($$$$-1048336 \nu^{15} + 8878763 \nu^{14} - 14466751 \nu^{13} - 64724264 \nu^{12} + 206543323 \nu^{11} + 92532283 \nu^{10} - 804630339 \nu^{9} + 238488877 \nu^{8} + 1382719669 \nu^{7} - 757194522 \nu^{6} - 1110390251 \nu^{5} + 658088239 \nu^{4} + 363077413 \nu^{3} - 173968012 \nu^{2} - 25743241 \nu + 5288765$$$$)/3115183$$ $$\beta_{13}$$ $$=$$ $$($$$$1124047 \nu^{15} - 8847625 \nu^{14} + 11444848 \nu^{13} + 67320977 \nu^{12} - 173501774 \nu^{11} - 128681409 \nu^{10} + 645278405 \nu^{9} - 69804760 \nu^{8} - 1000042516 \nu^{7} + 424081214 \nu^{6} + 650190464 \nu^{5} - 392052900 \nu^{4} - 119306620 \nu^{3} + 112016908 \nu^{2} - 10778289 \nu - 1358606$$$$)/3115183$$ $$\beta_{14}$$ $$=$$ $$($$$$-1618763 \nu^{15} + 13952863 \nu^{14} - 25482926 \nu^{13} - 88167362 \nu^{12} + 323120564 \nu^{11} + 34111348 \nu^{10} - 1108457254 \nu^{9} + 654630562 \nu^{8} + 1555714470 \nu^{7} - 1396057392 \nu^{6} - 862892792 \nu^{5} + 958014404 \nu^{4} + 97238878 \nu^{3} - 170186194 \nu^{2} + 27985975 \nu - 7526881$$$$)/3115183$$ $$\beta_{15}$$ $$=$$ $$($$$$1717836 \nu^{15} - 13768724 \nu^{14} + 19404207 \nu^{13} + 100594394 \nu^{12} - 280173236 \nu^{11} - 160180219 \nu^{10} + 1017753578 \nu^{9} - 243563866 \nu^{8} - 1527505049 \nu^{7} + 858348612 \nu^{6} + 939591930 \nu^{5} - 739948253 \nu^{4} - 146521530 \nu^{3} + 214322353 \nu^{2} - 16102888 \nu - 10162035$$$$)/3115183$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 2$$ $$\nu^{3}$$ $$=$$ $$-\beta_{15} + 2 \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + 5 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-3 \beta_{15} + 6 \beta_{13} - 2 \beta_{12} + 2 \beta_{11} + 3 \beta_{10} - 3 \beta_{9} + 2 \beta_{8} + \beta_{7} - 2 \beta_{6} + 7 \beta_{4} - 4 \beta_{3} + 10 \beta_{2} + 10 \beta_{1} + 5$$ $$\nu^{5}$$ $$=$$ $$-13 \beta_{15} + 28 \beta_{13} - 12 \beta_{12} + 13 \beta_{11} + 13 \beta_{10} - 12 \beta_{9} + 9 \beta_{8} - 6 \beta_{6} + \beta_{5} + 31 \beta_{4} - 16 \beta_{3} + 28 \beta_{2} + 34 \beta_{1} - 3$$ $$\nu^{6}$$ $$=$$ $$-40 \beta_{15} - 3 \beta_{14} + 87 \beta_{13} - 33 \beta_{12} + 36 \beta_{11} + 42 \beta_{10} - 32 \beta_{9} + 19 \beta_{8} + 2 \beta_{7} - 31 \beta_{6} + 3 \beta_{5} + 103 \beta_{4} - 56 \beta_{3} + 109 \beta_{2} + 84 \beta_{1} + 6$$ $$\nu^{7}$$ $$=$$ $$-140 \beta_{15} - 13 \beta_{14} + 319 \beta_{13} - 132 \beta_{12} + 149 \beta_{11} + 148 \beta_{10} - 100 \beta_{9} + 60 \beta_{8} - 21 \beta_{7} - 102 \beta_{6} + 15 \beta_{5} + 368 \beta_{4} - 191 \beta_{3} + 344 \beta_{2} + 259 \beta_{1} - 49$$ $$\nu^{8}$$ $$=$$ $$-443 \beta_{15} - 71 \beta_{14} + 1025 \beta_{13} - 403 \beta_{12} + 461 \beta_{11} + 487 \beta_{10} - 262 \beta_{9} + 121 \beta_{8} - 84 \beta_{7} - 396 \beta_{6} + 50 \beta_{5} + 1209 \beta_{4} - 642 \beta_{3} + 1221 \beta_{2} + 701 \beta_{1} - 107$$ $$\nu^{9}$$ $$=$$ $$-1476 \beta_{15} - 276 \beta_{14} + 3512 \beta_{13} - 1416 \beta_{12} + 1652 \beta_{11} + 1638 \beta_{10} - 748 \beta_{9} + 301 \beta_{8} - 467 \beta_{7} - 1330 \beta_{6} + 192 \beta_{5} + 4075 \beta_{4} - 2123 \beta_{3} + 3998 \beta_{2} + 2110 \beta_{1} - 629$$ $$\nu^{10}$$ $$=$$ $$-4758 \beta_{15} - 1126 \beta_{14} + 11468 \beta_{13} - 4492 \beta_{12} + 5313 \beta_{11} + 5404 \beta_{10} - 1938 \beta_{9} + 438 \beta_{8} - 1772 \beta_{7} - 4716 \beta_{6} + 659 \beta_{5} + 13342 \beta_{4} - 7027 \beta_{3} + 13650 \beta_{2} + 6014 \beta_{1} - 1896$$ $$\nu^{11}$$ $$=$$ $$-15673 \beta_{15} - 4126 \beta_{14} + 38386 \beta_{13} - 15058 \beta_{12} + 18058 \beta_{11} + 17929 \beta_{10} - 5242 \beta_{9} + 396 \beta_{8} - 7120 \beta_{7} - 15845 \beta_{6} + 2354 \beta_{5} + 44113 \beta_{4} - 23081 \beta_{3} + 45129 \beta_{2} + 18116 \beta_{1} - 7473$$ $$\nu^{12}$$ $$=$$ $$-51058 \beta_{15} - 15215 \beta_{14} + 126139 \beta_{13} - 48520 \beta_{12} + 58871 \beta_{11} + 59106 \beta_{10} - 13264 \beta_{9} - 2769 \beta_{8} - 25673 \beta_{7} - 54176 \beta_{6} + 8095 \beta_{5} + 144315 \beta_{4} - 75954 \beta_{3} + 151396 \beta_{2} + 53466 \beta_{1} - 24198$$ $$\nu^{13}$$ $$=$$ $$-167860 \beta_{15} - 53769 \beta_{14} + 418395 \beta_{13} - 159874 \beta_{12} + 196018 \beta_{11} + 195119 \beta_{10} - 33961 \beta_{9} - 18334 \beta_{8} - 93348 \beta_{7} - 181236 \beta_{6} + 28096 \beta_{5} + 474039 \beta_{4} - 249156 \beta_{3} + 501159 \beta_{2} + 162718 \beta_{1} - 85607$$ $$\nu^{14}$$ $$=$$ $$-549791 \beta_{15} - 189315 \beta_{14} + 1377611 \beta_{13} - 518871 \beta_{12} + 642097 \beta_{11} + 642570 \beta_{10} - 81445 \beta_{9} - 90804 \beta_{8} - 325797 \beta_{7} - 609317 \beta_{6} + 95967 \beta_{5} + 1551184 \beta_{4} - 818281 \beta_{3} + 1666889 \beta_{2} + 492790 \beta_{1} - 282370$$ $$\nu^{15}$$ $$=$$ $$-1807643 \beta_{15} - 653596 \beta_{14} + 4551335 \beta_{13} - 1699915 \beta_{12} + 2120493 \beta_{11} + 2116648 \beta_{10} - 189849 \beta_{9} - 374428 \beta_{8} - 1134451 \beta_{7} - 2029069 \beta_{6} + 327799 \beta_{5} + 5085340 \beta_{4} - 2684287 \beta_{3} + 5511955 \beta_{2} + 1519951 \beta_{1} - 960385$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.06059 −1.45194 −1.28136 −1.23485 −0.981279 −0.624714 −0.136067 0.293003 0.630852 1.27770 1.52769 1.74152 1.84416 2.41044 2.75851 3.28691
1.00000 −3.06059 1.00000 1.00000 −3.06059 0.302669 1.00000 6.36721 1.00000
1.2 1.00000 −2.45194 1.00000 1.00000 −2.45194 1.53125 1.00000 3.01199 1.00000
1.3 1.00000 −2.28136 1.00000 1.00000 −2.28136 1.99437 1.00000 2.20462 1.00000
1.4 1.00000 −2.23485 1.00000 1.00000 −2.23485 −0.971978 1.00000 1.99454 1.00000
1.5 1.00000 −1.98128 1.00000 1.00000 −1.98128 −4.22080 1.00000 0.925466 1.00000
1.6 1.00000 −1.62471 1.00000 1.00000 −1.62471 −0.333284 1.00000 −0.360304 1.00000
1.7 1.00000 −1.13607 1.00000 1.00000 −1.13607 −1.31630 1.00000 −1.70935 1.00000
1.8 1.00000 −0.706997 1.00000 1.00000 −0.706997 −2.05365 1.00000 −2.50015 1.00000
1.9 1.00000 −0.369148 1.00000 1.00000 −0.369148 4.16284 1.00000 −2.86373 1.00000
1.10 1.00000 0.277700 1.00000 1.00000 0.277700 2.55233 1.00000 −2.92288 1.00000
1.11 1.00000 0.527686 1.00000 1.00000 0.527686 −1.45681 1.00000 −2.72155 1.00000
1.12 1.00000 0.741521 1.00000 1.00000 0.741521 −2.92320 1.00000 −2.45015 1.00000
1.13 1.00000 0.844163 1.00000 1.00000 0.844163 −0.532701 1.00000 −2.28739 1.00000
1.14 1.00000 1.41044 1.00000 1.00000 1.41044 −3.15104 1.00000 −1.01066 1.00000
1.15 1.00000 1.75851 1.00000 1.00000 1.75851 −0.115571 1.00000 0.0923708 1.00000
1.16 1.00000 2.28691 1.00000 1.00000 2.28691 −3.46813 1.00000 2.22998 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6010.2.a.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6010.2.a.c 16 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$
$$601$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{16} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6010))$$.