# Properties

 Label 287.2.e.c Level $287$ Weight $2$ Character orbit 287.e Analytic conductor $2.292$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$287 = 7 \cdot 41$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 287.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.29170653801$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - x^{9} + 7 x^{8} + 4 x^{7} + 32 x^{6} + 3 x^{5} + 30 x^{4} - 7 x^{3} + 26 x^{2} - 5 x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - x^{9} + 7 x^{8} + 4 x^{7} + 32 x^{6} + 3 x^{5} + 30 x^{4} - 7 x^{3} + 26 x^{2} - 5 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$911 \nu^{9} + 318 \nu^{8} - 583 \nu^{7} + 16404 \nu^{6} - 265 \nu^{5} + 8533 \nu^{4} - 159430 \nu^{3} + 8480 \nu^{2} - 1643 \nu + 31405$$$$)/160271$$ $$\beta_{3}$$ $$=$$ $$($$$$-4571 \nu^{9} + 29016 \nu^{8} - 53196 \nu^{7} + 154140 \nu^{6} - 24180 \nu^{5} + 778596 \nu^{4} + 122097 \nu^{3} + 773760 \nu^{2} - 149916 \nu + 319366$$$$)/160271$$ $$\beta_{4}$$ $$=$$ $$($$$$4889 \nu^{9} - 36294 \nu^{8} + 66539 \nu^{7} - 199961 \nu^{6} + 30245 \nu^{5} - 973889 \nu^{4} + 52190 \nu^{3} - 967840 \nu^{2} + 187519 \nu - 672224$$$$)/160271$$ $$\beta_{5}$$ $$=$$ $$($$$$-5191 \nu^{9} + 12966 \nu^{8} - 23771 \nu^{7} + 9622 \nu^{6} - 10805 \nu^{5} + 347921 \nu^{4} + 288305 \nu^{3} + 345760 \nu^{2} - 66991 \nu + 388420$$$$)/160271$$ $$\beta_{6}$$ $$=$$ $$($$$$-31405 \nu^{9} + 32316 \nu^{8} - 219517 \nu^{7} - 126203 \nu^{6} - 988556 \nu^{5} - 94480 \nu^{4} - 933617 \nu^{3} + 60405 \nu^{2} - 808050 \nu + 155382$$$$)/160271$$ $$\beta_{7}$$ $$=$$ $$($$$$-62396 \nu^{9} + 33989 \nu^{8} - 409567 \nu^{7} - 433019 \nu^{6} - 2138559 \nu^{5} - 984502 \nu^{4} - 1969946 \nu^{3} - 1829 \nu^{2} - 1431066 \nu - 15562$$$$)/160271$$ $$\beta_{8}$$ $$=$$ $$($$$$-91217 \nu^{9} + 91837 \nu^{8} - 622469 \nu^{7} - 394293 \nu^{6} - 2774426 \nu^{5} - 287026 \nu^{4} - 2305835 \nu^{3} + 472311 \nu^{2} - 1943642 \nu + 373160$$$$)/160271$$ $$\beta_{9}$$ $$=$$ $$($$$$-128866 \nu^{9} + 127955 \nu^{8} - 902380 \nu^{7} - 514881 \nu^{6} - 4140116 \nu^{5} - 386333 \nu^{4} - 3874513 \nu^{3} + 1061492 \nu^{2} - 3358996 \nu + 645973$$$$)/160271$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{9} - \beta_{7} - 2 \beta_{6} - \beta_{3} - \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 5 \beta_{2} - 1$$ $$\nu^{4}$$ $$=$$ $$-7 \beta_{9} + \beta_{8} + 8 \beta_{7} + 9 \beta_{6} - \beta_{5} - 7 \beta_{4} - 10 \beta_{1} - 9$$ $$\nu^{5}$$ $$=$$ $$-18 \beta_{9} + 7 \beta_{8} + 19 \beta_{7} + 14 \beta_{6} + 19 \beta_{3} + 34 \beta_{2} - 34 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$12 \beta_{5} + 53 \beta_{4} + 60 \beta_{3} + 85 \beta_{2} + 57$$ $$\nu^{7}$$ $$=$$ $$145 \beta_{9} - 48 \beta_{8} - 157 \beta_{7} - 129 \beta_{6} + 48 \beta_{5} + 145 \beta_{4} + 255 \beta_{1} + 129$$ $$\nu^{8}$$ $$=$$ $$412 \beta_{9} - 109 \beta_{8} - 460 \beta_{7} - 413 \beta_{6} - 460 \beta_{3} - 686 \beta_{2} + 686 \beta_{1}$$ $$\nu^{9}$$ $$=$$ $$-351 \beta_{5} - 1146 \beta_{4} - 1255 \beta_{3} - 1971 \beta_{2} - 1069$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/287\mathbb{Z}\right)^\times$$.

 $$n$$ $$206$$ $$211$$ $$\chi(n)$$ $$-1 + \beta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
165.1
 0.440981 + 0.763802i −0.580000 − 1.00459i −0.863288 − 1.49526i 0.100998 + 0.174933i 1.40131 + 2.42714i 0.440981 − 0.763802i −0.580000 + 1.00459i −0.863288 + 1.49526i 0.100998 − 0.174933i 1.40131 − 2.42714i
−0.985135 1.70630i −0.257781 + 0.446490i −0.940981 + 1.62983i 0.257781 + 0.446490i 1.01580 1.91822 1.82221i −0.232565 1.36710 + 2.36788i 0.507899 0.879706i
165.2 −0.678233 1.17473i 1.11395 1.92942i 0.0800004 0.138565i −1.11395 1.92942i −3.02207 2.60927 + 0.437865i −2.92997 −0.981768 1.70047i −1.51103 + 2.61719i
165.3 0.564230 + 0.977276i −1.46472 + 2.53697i 0.363288 0.629233i 1.46472 + 2.53697i −3.30576 1.22536 + 2.34489i 3.07683 −2.79081 4.83382i −1.65288 + 2.86287i
165.4 0.894706 + 1.54968i 1.16033 2.00974i −0.600998 + 1.04096i −1.16033 2.00974i 4.15260 −1.87001 1.87165i 1.42796 −1.19271 2.06584i 2.07630 3.59626i
165.5 1.20443 + 2.08614i 0.448226 0.776350i −1.90131 + 3.29316i −0.448226 0.776350i 2.15943 0.117164 + 2.64316i −4.34226 1.09819 + 1.90212i 1.07971 1.87012i
247.1 −0.985135 + 1.70630i −0.257781 0.446490i −0.940981 1.62983i 0.257781 0.446490i 1.01580 1.91822 + 1.82221i −0.232565 1.36710 2.36788i 0.507899 + 0.879706i
247.2 −0.678233 + 1.17473i 1.11395 + 1.92942i 0.0800004 + 0.138565i −1.11395 + 1.92942i −3.02207 2.60927 0.437865i −2.92997 −0.981768 + 1.70047i −1.51103 2.61719i
247.3 0.564230 0.977276i −1.46472 2.53697i 0.363288 + 0.629233i 1.46472 2.53697i −3.30576 1.22536 2.34489i 3.07683 −2.79081 + 4.83382i −1.65288 2.86287i
247.4 0.894706 1.54968i 1.16033 + 2.00974i −0.600998 1.04096i −1.16033 + 2.00974i 4.15260 −1.87001 + 1.87165i 1.42796 −1.19271 + 2.06584i 2.07630 + 3.59626i
247.5 1.20443 2.08614i 0.448226 + 0.776350i −1.90131 3.29316i −0.448226 + 0.776350i 2.15943 0.117164 2.64316i −4.34226 1.09819 1.90212i 1.07971 + 1.87012i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 247.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 287.2.e.c 10
7.c even 3 1 inner 287.2.e.c 10
7.c even 3 1 2009.2.a.l 5
7.d odd 6 1 2009.2.a.m 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.2.e.c 10 1.a even 1 1 trivial
287.2.e.c 10 7.c even 3 1 inner
2009.2.a.l 5 7.c even 3 1
2009.2.a.m 5 7.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{10} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(287, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$169 - 104 T + 207 T^{2} - 68 T^{3} + 143 T^{4} - 47 T^{5} + 50 T^{6} - 10 T^{7} + 10 T^{8} - 2 T^{9} + T^{10}$$
$3$ $$49 + 14 T + 137 T^{2} - 150 T^{3} + 331 T^{4} - 167 T^{5} + 104 T^{6} - 22 T^{7} + 12 T^{8} - 2 T^{9} + T^{10}$$
$5$ $$49 - 14 T + 137 T^{2} + 150 T^{3} + 331 T^{4} + 167 T^{5} + 104 T^{6} + 22 T^{7} + 12 T^{8} + 2 T^{9} + T^{10}$$
$7$ $$16807 - 19208 T + 12348 T^{2} - 5635 T^{3} + 2114 T^{4} - 783 T^{5} + 302 T^{6} - 115 T^{7} + 36 T^{8} - 8 T^{9} + T^{10}$$
$11$ $$2809 - 371 T + 2434 T^{2} - 3 T^{3} + 1728 T^{4} - 104 T^{5} + 272 T^{6} - 72 T^{7} + 39 T^{8} - 6 T^{9} + T^{10}$$
$13$ $$( -7 - 2 T + 19 T^{2} - 8 T^{3} - 2 T^{4} + T^{5} )^{2}$$
$17$ $$25921 - 17388 T + 19875 T^{2} - 2542 T^{3} + 4818 T^{4} - 788 T^{5} + 670 T^{6} - 27 T^{7} + 34 T^{8} - 3 T^{9} + T^{10}$$
$19$ $$405769 + 26117 T + 91498 T^{2} + 14603 T^{3} + 16078 T^{4} + 2319 T^{5} + 1202 T^{6} + 170 T^{7} + 65 T^{8} + 7 T^{9} + T^{10}$$
$23$ $$36481 + 38582 T + 44433 T^{2} + 10678 T^{3} + 8037 T^{4} + 913 T^{5} + 1242 T^{6} + 38 T^{7} + 38 T^{8} + T^{10}$$
$29$ $$( 49 - 294 T - 136 T^{2} + 9 T^{3} + 10 T^{4} + T^{5} )^{2}$$
$31$ $$90601 + 288358 T + 848835 T^{2} + 256706 T^{3} + 113643 T^{4} - 2401 T^{5} + 4260 T^{6} - 86 T^{7} + 98 T^{8} - 6 T^{9} + T^{10}$$
$37$ $$61009 - 176358 T + 454962 T^{2} - 190618 T^{3} + 100140 T^{4} - 11027 T^{5} + 8935 T^{6} - 1614 T^{7} + 259 T^{8} - 18 T^{9} + T^{10}$$
$41$ $$( 1 + T )^{10}$$
$43$ $$( -2573 - 3928 T - 1178 T^{2} - 53 T^{3} + 14 T^{4} + T^{5} )^{2}$$
$47$ $$1432809 + 1027026 T + 801999 T^{2} + 161088 T^{3} + 74080 T^{4} + 834 T^{5} + 6876 T^{6} - 151 T^{7} + 96 T^{8} + 3 T^{9} + T^{10}$$
$53$ $$2647129 - 4739451 T + 6251698 T^{2} - 3609069 T^{3} + 1550212 T^{4} - 215567 T^{5} + 29670 T^{6} - 1666 T^{7} + 201 T^{8} - 9 T^{9} + T^{10}$$
$59$ $$324396121 + 15201284 T + 22559679 T^{2} - 1131838 T^{3} + 1126628 T^{4} - 53722 T^{5} + 23900 T^{6} - 2369 T^{7} + 364 T^{8} - 19 T^{9} + T^{10}$$
$61$ $$159997201 - 11295557 T + 20416048 T^{2} + 1789811 T^{3} + 2100386 T^{4} + 53245 T^{5} + 35036 T^{6} - 3470 T^{7} + 513 T^{8} - 23 T^{9} + T^{10}$$
$67$ $$372374209 - 45309356 T + 31428975 T^{2} - 2056826 T^{3} + 1908362 T^{4} - 148946 T^{5} + 30650 T^{6} - 1201 T^{7} + 256 T^{8} - 11 T^{9} + T^{10}$$
$71$ $$( -7241 + 5192 T + 61 T^{2} - 150 T^{3} + T^{5} )^{2}$$
$73$ $$9186961 + 15188341 T + 19999855 T^{2} + 7781726 T^{3} + 2330789 T^{4} + 312715 T^{5} + 39029 T^{6} + 1942 T^{7} + 279 T^{8} + 13 T^{9} + T^{10}$$
$79$ $$3108174001 - 474998520 T + 191061275 T^{2} - 43555606 T^{3} + 11512976 T^{4} - 1818014 T^{5} + 227204 T^{6} - 18423 T^{7} + 1128 T^{8} - 41 T^{9} + T^{10}$$
$83$ $$( 2317 + 1244 T - 144 T^{2} - 77 T^{3} + 2 T^{4} + T^{5} )^{2}$$
$89$ $$2081093161 + 13685700 T + 166234398 T^{2} + 21808138 T^{3} + 12700798 T^{4} + 951361 T^{5} + 113689 T^{6} + 3770 T^{7} + 447 T^{8} + 14 T^{9} + T^{10}$$
$97$ $$( -56763 - 37896 T - 5393 T^{2} - 37 T^{3} + 27 T^{4} + T^{5} )^{2}$$