# Properties

 Label 370.2.g.d Level $370$ Weight $2$ Character orbit 370.g Analytic conductor $2.954$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$370 = 2 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 370.g (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.95446487479$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ Defining polynomial: $$x^{10} + 3 x^{8} - 8 x^{7} - 26 x^{6} + 12 x^{5} + 24 x^{4} + 166 x^{3} + 113 x^{2} - 152 x + 160$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 3 x^{8} - 8 x^{7} - 26 x^{6} + 12 x^{5} + 24 x^{4} + 166 x^{3} + 113 x^{2} - 152 x + 160$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$105 \nu^{9} + 3868 \nu^{8} - 17854 \nu^{7} + 34822 \nu^{6} - 123604 \nu^{5} + 73205 \nu^{4} + 120257 \nu^{3} - 211635 \nu^{2} + 840473 \nu - 881880$$$$)/887851$$ $$\beta_{2}$$ $$=$$ $$($$$$4830421 \nu^{9} + 27279422 \nu^{8} + 39766871 \nu^{7} + 36230609 \nu^{6} - 248412067 \nu^{5} - 422935188 \nu^{4} - 51534513 \nu^{3} + 4194704142 \nu^{2} + 5881796255 \nu - 1481360957$$$$)/ 5161077863$$ $$\beta_{3}$$ $$=$$ $$($$$$-4830421 \nu^{9} - 27279422 \nu^{8} - 39766871 \nu^{7} - 36230609 \nu^{6} + 248412067 \nu^{5} + 422935188 \nu^{4} + 51534513 \nu^{3} - 4194704142 \nu^{2} - 720718392 \nu + 1481360957$$$$)/ 5161077863$$ $$\beta_{4}$$ $$=$$ $$($$$$-12470333 \nu^{9} - 79772050 \nu^{8} - 195227485 \nu^{7} + 26574182 \nu^{6} + 577196282 \nu^{5} + 3464752928 \nu^{4} + 731490770 \nu^{3} - 9133030804 \nu^{2} - 20817063063 \nu - 18775197608$$$$)/ 10322155726$$ $$\beta_{5}$$ $$=$$ $$($$$$-40939147 \nu^{9} + 281415368 \nu^{8} - 506483273 \nu^{7} + 1711086296 \nu^{6} - 2353824822 \nu^{5} - 3136611584 \nu^{4} + 3432795828 \nu^{3} - 11350242598 \nu^{2} + 18569839845 \nu + 1358343716$$$$)/ 20644311452$$ $$\beta_{6}$$ $$=$$ $$($$$$22047 \nu^{9} + 420 \nu^{8} + 81613 \nu^{7} - 247792 \nu^{6} - 433934 \nu^{5} - 229852 \nu^{4} + 821948 \nu^{3} + 4140830 \nu^{2} + 1644771 \nu + 10748$$$$)/3551404$$ $$\beta_{7}$$ $$=$$ $$($$$$-78590417 \nu^{9} - 53270874 \nu^{8} - 432258509 \nu^{7} + 565223046 \nu^{6} + 1835553924 \nu^{5} + 1991880712 \nu^{4} + 1196070998 \nu^{3} - 17048329972 \nu^{2} - 23630326501 \nu - 13883295960$$$$)/ 10322155726$$ $$\beta_{8}$$ $$=$$ $$($$$$-106312323 \nu^{9} - 59569470 \nu^{8} - 234376097 \nu^{7} + 621147016 \nu^{6} + 3952005236 \nu^{5} - 910662862 \nu^{4} - 3262671660 \nu^{3} - 19246167516 \nu^{2} - 28243314479 \nu + 27078850384$$$$)/ 10322155726$$ $$\beta_{9}$$ $$=$$ $$($$$$-151550159 \nu^{9} + 32728460 \nu^{8} - 412250275 \nu^{7} + 1324404682 \nu^{6} + 3598338044 \nu^{5} - 2322899252 \nu^{4} - 5878785400 \nu^{3} - 22346975474 \nu^{2} - 6371750919 \nu + 25888295120$$$$)/ 10322155726$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{3} + \beta_{2}$$ $$\nu^{2}$$ $$=$$ $$\beta_{8} + \beta_{6} - \beta_{5} - \beta_{3} + \beta_{2} + 2 \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$-4 \beta_{9} + 4 \beta_{7} - 5 \beta_{6} + \beta_{5} - 5 \beta_{4} + 2 \beta_{3} + \beta_{2} - 3 \beta_{1} + 3$$ $$\nu^{4}$$ $$=$$ $$\beta_{9} - 7 \beta_{8} + 2 \beta_{7} - 5 \beta_{6} + 3 \beta_{5} + 4 \beta_{4} + 3 \beta_{3} + 4 \beta_{2} - 11 \beta_{1} + 15$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{9} + 12 \beta_{8} - 9 \beta_{7} + 16 \beta_{6} + 4 \beta_{5} + 17 \beta_{4} + 17 \beta_{3} + 31 \beta_{2} + 4 \beta_{1} - 10$$ $$\nu^{6}$$ $$=$$ $$-44 \beta_{9} + 54 \beta_{8} - 11 \beta_{7} - 41 \beta_{6} - 11 \beta_{5} - 30 \beta_{4} - 5 \beta_{3} + 13 \beta_{2} + 51 \beta_{1} - 44$$ $$\nu^{7}$$ $$=$$ $$-79 \beta_{9} - 72 \beta_{8} + 70 \beta_{7} - 221 \beta_{6} + 27 \beta_{5} - 123 \beta_{4} - 18 \beta_{3} - 96 \beta_{2} - 157 \beta_{1} + 78$$ $$\nu^{8}$$ $$=$$ $$201 \beta_{9} - 271 \beta_{8} - 9 \beta_{7} + 144 \beta_{6} + 208 \beta_{5} + 307 \beta_{4} + 103 \beta_{3} + 138 \beta_{2} - 439 \beta_{1} + 313$$ $$\nu^{9}$$ $$=$$ $$-7 \beta_{9} + 854 \beta_{8} - 688 \beta_{7} + 777 \beta_{6} + 33 \beta_{5} + 675 \beta_{4} + 413 \beta_{3} + 851 \beta_{2} + 863 \beta_{1} - 942$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/370\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$297$$ $$\chi(n)$$ $$-\beta_{6}$$ $$-\beta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1
 0.465873 + 0.587826i 2.03431 − 0.602710i −0.551861 − 1.73844i −1.66045 + 0.156295i −0.287871 + 2.59703i 0.465873 − 0.587826i 2.03431 + 0.602710i −0.551861 + 1.73844i −1.66045 − 0.156295i −0.287871 − 2.59703i
1.00000i −2.18458 + 2.18458i −1.00000 1.71871 + 1.43040i 2.18458 + 2.18458i −3.43742 + 3.43742i 1.00000i 6.54482i 1.43040 1.71871i
43.2 1.00000i −1.78347 + 1.78347i −1.00000 −0.250846 2.22195i 1.78347 + 1.78347i 0.501691 0.501691i 1.00000i 3.36152i −2.22195 + 0.250846i
43.3 1.00000i 0.215231 0.215231i −1.00000 0.336630 + 2.21058i −0.215231 0.215231i −0.673260 + 0.673260i 1.00000i 2.90735i 2.21058 0.336630i
43.4 1.00000i 1.05122 1.05122i −1.00000 0.609231 2.15147i −1.05122 1.05122i −1.21846 + 1.21846i 1.00000i 0.789858i −2.15147 0.609231i
43.5 1.00000i 1.70160 1.70160i −1.00000 −1.41373 + 1.73245i −1.70160 1.70160i 2.82745 2.82745i 1.00000i 2.79087i 1.73245 + 1.41373i
327.1 1.00000i −2.18458 2.18458i −1.00000 1.71871 1.43040i 2.18458 2.18458i −3.43742 3.43742i 1.00000i 6.54482i 1.43040 + 1.71871i
327.2 1.00000i −1.78347 1.78347i −1.00000 −0.250846 + 2.22195i 1.78347 1.78347i 0.501691 + 0.501691i 1.00000i 3.36152i −2.22195 0.250846i
327.3 1.00000i 0.215231 + 0.215231i −1.00000 0.336630 2.21058i −0.215231 + 0.215231i −0.673260 0.673260i 1.00000i 2.90735i 2.21058 + 0.336630i
327.4 1.00000i 1.05122 + 1.05122i −1.00000 0.609231 + 2.15147i −1.05122 + 1.05122i −1.21846 1.21846i 1.00000i 0.789858i −2.15147 + 0.609231i
327.5 1.00000i 1.70160 + 1.70160i −1.00000 −1.41373 1.73245i −1.70160 + 1.70160i 2.82745 + 2.82745i 1.00000i 2.79087i 1.73245 1.41373i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 327.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.f even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.g.d 10
5.c odd 4 1 370.2.h.d yes 10
37.d odd 4 1 370.2.h.d yes 10
185.f even 4 1 inner 370.2.g.d 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.g.d 10 1.a even 1 1 trivial
370.2.g.d 10 185.f even 4 1 inner
370.2.h.d yes 10 5.c odd 4 1
370.2.h.d yes 10 37.d odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{10} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(370, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{5}$$
$3$ $$72 - 372 T + 961 T^{2} - 442 T^{3} + 74 T^{4} + 60 T^{5} + 59 T^{6} - 12 T^{7} + 2 T^{8} + 2 T^{9} + T^{10}$$
$5$ $$3125 - 1250 T + 2000 T^{2} - 650 T^{3} + 580 T^{4} - 164 T^{5} + 116 T^{6} - 26 T^{7} + 16 T^{8} - 2 T^{9} + T^{10}$$
$7$ $$512 + 128 T + 16 T^{2} + 352 T^{3} + 1440 T^{4} + 960 T^{5} + 316 T^{6} - 16 T^{7} + 8 T^{8} + 4 T^{9} + T^{10}$$
$11$ $$266256 + 161833 T^{2} + 30378 T^{4} + 2395 T^{6} + 82 T^{8} + T^{10}$$
$13$ $$16 + 185 T^{2} + 398 T^{4} + 227 T^{6} + 38 T^{8} + T^{10}$$
$17$ $$( 344 - 268 T - 112 T^{2} + 26 T^{3} + 12 T^{4} + T^{5} )^{2}$$
$19$ $$12800 + 58240 T + 132496 T^{2} + 129504 T^{3} + 71968 T^{4} + 20368 T^{5} + 2844 T^{6} + 8 T^{7} + 32 T^{8} + 8 T^{9} + T^{10}$$
$23$ $$250000 + 201129 T^{2} + 39414 T^{4} + 3019 T^{6} + 94 T^{8} + T^{10}$$
$29$ $$4222418 - 1729070 T + 354025 T^{2} - 192838 T^{3} + 204384 T^{4} - 98882 T^{5} + 27759 T^{6} - 4744 T^{7} + 512 T^{8} - 32 T^{9} + T^{10}$$
$31$ $$10952 + 47804 T + 104329 T^{2} + 118290 T^{3} + 83514 T^{4} + 38688 T^{5} + 12123 T^{6} + 2520 T^{7} + 338 T^{8} + 26 T^{9} + T^{10}$$
$37$ $$69343957 - 22489932 T + 6736849 T^{2} - 1577088 T^{3} + 300662 T^{4} - 52584 T^{5} + 8126 T^{6} - 1152 T^{7} + 133 T^{8} - 12 T^{9} + T^{10}$$
$41$ $$57600 + 350761 T^{2} + 76634 T^{4} + 5403 T^{6} + 130 T^{8} + T^{10}$$
$43$ $$350464 + 444176 T^{2} + 146368 T^{4} + 7852 T^{6} + 152 T^{8} + T^{10}$$
$47$ $$18678272 - 32784768 T + 28772496 T^{2} - 14855232 T^{3} + 4981376 T^{4} - 1100576 T^{5} + 165672 T^{6} - 16912 T^{7} + 1152 T^{8} - 48 T^{9} + T^{10}$$
$53$ $$4310048 - 2712864 T + 853776 T^{2} - 70080 T^{3} + 20664 T^{4} - 12328 T^{5} + 4236 T^{6} - 328 T^{7} + 2 T^{8} + 2 T^{9} + T^{10}$$
$59$ $$1340377088 - 102930688 T + 3952144 T^{2} - 1124192 T^{3} + 2512928 T^{4} - 238800 T^{5} + 11400 T^{6} - 184 T^{7} + 200 T^{8} - 20 T^{9} + T^{10}$$
$61$ $$4418 + 4042 T + 1849 T^{2} - 218 T^{3} + 23064 T^{4} + 19690 T^{5} + 8367 T^{6} + 1980 T^{7} + 288 T^{8} + 24 T^{9} + T^{10}$$
$67$ $$18072072 + 16995924 T + 7991929 T^{2} + 1652486 T^{3} + 130058 T^{4} - 14768 T^{5} + 4147 T^{6} + 616 T^{7} + 50 T^{8} - 10 T^{9} + T^{10}$$
$71$ $$( -576 - 880 T - 384 T^{2} - 36 T^{3} + 8 T^{4} + T^{5} )^{2}$$
$73$ $$369430562 + 122563638 T + 20331081 T^{2} - 184758 T^{3} + 505104 T^{4} + 150898 T^{5} + 22311 T^{6} + 400 T^{7} + 8 T^{8} + 4 T^{9} + T^{10}$$
$79$ $$23970888 + 19435668 T + 7879249 T^{2} + 768938 T^{3} + 13898 T^{4} - 2380 T^{5} + 5835 T^{6} + 204 T^{7} + 2 T^{8} - 2 T^{9} + T^{10}$$
$83$ $$46851200 + 42166080 T + 18974736 T^{2} + 2799456 T^{3} + 121248 T^{4} - 21008 T^{5} + 15624 T^{6} + 952 T^{7} + 32 T^{8} - 8 T^{9} + T^{10}$$
$89$ $$3591200 + 7729120 T + 8317456 T^{2} + 745984 T^{3} + 14608 T^{4} - 14864 T^{5} + 11656 T^{6} + 128 T^{7} + 2 T^{8} - 2 T^{9} + T^{10}$$
$97$ $$( -48 + 124 T - 20 T^{2} - 22 T^{3} + 2 T^{4} + T^{5} )^{2}$$