# Properties

 Label 441.2.s.b Level $441$ Weight $2$ Character orbit 441.s Analytic conductor $3.521$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 441.s (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.52140272914$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{6})$$ Coefficient field: 10.0.288778218147.1 Defining polynomial: $$x^{10} - x^{9} + 7 x^{8} - 4 x^{7} + 34 x^{6} - 19 x^{5} + 64 x^{4} - x^{3} + 64 x^{2} - 21 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - x^{9} + 7 x^{8} - 4 x^{7} + 34 x^{6} - 19 x^{5} + 64 x^{4} - x^{3} + 64 x^{2} - 21 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-339 \nu^{9} + 1348 \nu^{8} - 4381 \nu^{7} + 7882 \nu^{6} - 19883 \nu^{5} + 36059 \nu^{4} - 75410 \nu^{3} + 44484 \nu^{2} - 15165 \nu + 29709$$$$)/72795$$ $$\beta_{3}$$ $$=$$ $$($$$$658 \nu^{9} + 2394 \nu^{8} + 4352 \nu^{7} + 10326 \nu^{6} + 25351 \nu^{5} + 51907 \nu^{4} + 47450 \nu^{3} + 30472 \nu^{2} + 130790 \nu + 98232$$$$)/72795$$ $$\beta_{4}$$ $$=$$ $$($$$$-4192 \nu^{9} - 796 \nu^{8} - 21678 \nu^{7} - 20279 \nu^{6} - 85319 \nu^{5} - 118353 \nu^{4} - 2560 \nu^{3} - 414508 \nu^{2} + 81750 \nu - 398583$$$$)/218385$$ $$\beta_{5}$$ $$=$$ $$($$$$8236 \nu^{9} - 9272 \nu^{8} + 54399 \nu^{7} - 28438 \nu^{6} + 233822 \nu^{5} - 150966 \nu^{4} + 361225 \nu^{3} + 82264 \nu^{2} + 31515 \nu - 336546$$$$)/218385$$ $$\beta_{6}$$ $$=$$ $$($$$$3301 \nu^{9} - 2962 \nu^{8} + 21759 \nu^{7} - 8823 \nu^{6} + 104352 \nu^{5} - 42836 \nu^{4} + 175205 \nu^{3} + 72109 \nu^{2} + 166780 \nu - 54156$$$$)/72795$$ $$\beta_{7}$$ $$=$$ $$($$$$-840 \nu^{9} + 248 \nu^{8} - 5659 \nu^{7} - 998 \nu^{6} - 27923 \nu^{5} - 3072 \nu^{4} - 51488 \nu^{3} - 30640 \nu^{2} - 51320 \nu + 11514$$$$)/14559$$ $$\beta_{8}$$ $$=$$ $$($$$$3085 \nu^{9} - 1373 \nu^{8} + 17808 \nu^{7} + 1181 \nu^{6} + 84554 \nu^{5} + 5736 \nu^{4} + 111910 \nu^{3} + 124546 \nu^{2} + 106440 \nu + 17856$$$$)/43677$$ $$\beta_{9}$$ $$=$$ $$($$$$-18476 \nu^{9} + 18997 \nu^{8} - 128469 \nu^{7} + 65033 \nu^{6} - 601717 \nu^{5} + 295851 \nu^{4} - 1019855 \nu^{3} - 222374 \nu^{2} - 668685 \nu + 178101$$$$)/218385$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{9} - \beta_{8} + 3 \beta_{6} - \beta_{4} - \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$\beta_{9} + \beta_{8} - 3 \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$-5 \beta_{9} + 5 \beta_{8} + \beta_{7} - 12 \beta_{6} - 5 \beta_{5} - \beta_{3} - 5 \beta_{2} + 5 \beta_{1} - 12$$ $$\nu^{5}$$ $$=$$ $$-5 \beta_{9} - \beta_{8} - \beta_{7} - 7 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} + 11 \beta_{2} - 11 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$6 \beta_{9} - 8 \beta_{8} - 14 \beta_{7} + 16 \beta_{5} + 9 \beta_{4} - 7 \beta_{3} + 22 \beta_{2} + 51$$ $$\nu^{7}$$ $$=$$ $$\beta_{9} - 31 \beta_{8} - 8 \beta_{7} + 31 \beta_{5} - 30 \beta_{4} + 8 \beta_{3} + \beta_{2} + 43 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$75 \beta_{9} - 66 \beta_{8} + 38 \beta_{7} + 222 \beta_{6} + 47 \beta_{5} - 37 \beta_{4} + 76 \beta_{3} + 8 \beta_{2} - 112 \beta_{1}$$ $$\nu^{9}$$ $$=$$ $$95 \beta_{9} + 189 \beta_{8} + 94 \beta_{7} + 37 \beta_{5} + 84 \beta_{4} + 47 \beta_{3} - 194 \beta_{2} - 3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$1 + \beta_{6}$$ $$-\beta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
362.1
 1.07065 − 1.85442i 0.187540 − 0.324828i −0.539982 + 0.935277i −1.04536 + 1.81062i 0.827154 − 1.43267i 1.07065 + 1.85442i 0.187540 + 0.324828i −0.539982 − 0.935277i −1.04536 − 1.81062i 0.827154 + 1.43267i
−2.24607 + 1.29677i −1.61958 + 0.613974i 2.36322 4.09323i −1.25299 2.84151 3.47925i 0 7.07116i 2.24607 1.98876i 2.81429 1.62483i
362.2 −0.621951 + 0.359083i −1.34572 1.09042i −0.742118 + 1.28539i 1.44755 1.22853 + 0.194963i 0 2.50226i 0.621951 + 2.93482i −0.900304 + 0.519791i
362.3 −0.254498 + 0.146935i 1.27564 1.17164i −0.956820 + 1.65726i −3.06027 −0.152492 + 0.485617i 0 1.15010i 0.254498 2.98919i 0.778834 0.449660i
362.4 1.30778 0.755047i 0.919842 + 1.46761i 0.140193 0.242822i 0.775876 2.31107 + 1.22479i 0 2.59678i −1.30778 + 2.69995i 1.01468 0.585823i
362.5 1.81474 1.04774i 0.769825 1.55157i 1.19552 2.07070i 2.08983 −0.228612 3.62227i 0 0.819421i −1.81474 2.38887i 3.79250 2.18960i
374.1 −2.24607 1.29677i −1.61958 0.613974i 2.36322 + 4.09323i −1.25299 2.84151 + 3.47925i 0 7.07116i 2.24607 + 1.98876i 2.81429 + 1.62483i
374.2 −0.621951 0.359083i −1.34572 + 1.09042i −0.742118 1.28539i 1.44755 1.22853 0.194963i 0 2.50226i 0.621951 2.93482i −0.900304 0.519791i
374.3 −0.254498 0.146935i 1.27564 + 1.17164i −0.956820 1.65726i −3.06027 −0.152492 0.485617i 0 1.15010i 0.254498 + 2.98919i 0.778834 + 0.449660i
374.4 1.30778 + 0.755047i 0.919842 1.46761i 0.140193 + 0.242822i 0.775876 2.31107 1.22479i 0 2.59678i −1.30778 2.69995i 1.01468 + 0.585823i
374.5 1.81474 + 1.04774i 0.769825 + 1.55157i 1.19552 + 2.07070i 2.08983 −0.228612 + 3.62227i 0 0.819421i −1.81474 + 2.38887i 3.79250 + 2.18960i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 374.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.s even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.s.b 10
3.b odd 2 1 1323.2.s.b 10
7.b odd 2 1 63.2.s.b yes 10
7.c even 3 1 63.2.i.b 10
7.c even 3 1 441.2.o.d 10
7.d odd 6 1 441.2.i.b 10
7.d odd 6 1 441.2.o.c 10
9.c even 3 1 1323.2.i.b 10
9.d odd 6 1 441.2.i.b 10
21.c even 2 1 189.2.s.b 10
21.g even 6 1 1323.2.i.b 10
21.g even 6 1 1323.2.o.d 10
21.h odd 6 1 189.2.i.b 10
21.h odd 6 1 1323.2.o.c 10
28.d even 2 1 1008.2.df.b 10
28.g odd 6 1 1008.2.ca.b 10
63.g even 3 1 189.2.s.b 10
63.h even 3 1 567.2.p.d 10
63.h even 3 1 1323.2.o.d 10
63.i even 6 1 441.2.o.d 10
63.j odd 6 1 441.2.o.c 10
63.j odd 6 1 567.2.p.c 10
63.k odd 6 1 1323.2.s.b 10
63.l odd 6 1 189.2.i.b 10
63.l odd 6 1 567.2.p.c 10
63.n odd 6 1 63.2.s.b yes 10
63.o even 6 1 63.2.i.b 10
63.o even 6 1 567.2.p.d 10
63.s even 6 1 inner 441.2.s.b 10
63.t odd 6 1 1323.2.o.c 10
84.h odd 2 1 3024.2.df.b 10
84.n even 6 1 3024.2.ca.b 10
252.o even 6 1 1008.2.df.b 10
252.s odd 6 1 1008.2.ca.b 10
252.bi even 6 1 3024.2.ca.b 10
252.bl odd 6 1 3024.2.df.b 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.i.b 10 7.c even 3 1
63.2.i.b 10 63.o even 6 1
63.2.s.b yes 10 7.b odd 2 1
63.2.s.b yes 10 63.n odd 6 1
189.2.i.b 10 21.h odd 6 1
189.2.i.b 10 63.l odd 6 1
189.2.s.b 10 21.c even 2 1
189.2.s.b 10 63.g even 3 1
441.2.i.b 10 7.d odd 6 1
441.2.i.b 10 9.d odd 6 1
441.2.o.c 10 7.d odd 6 1
441.2.o.c 10 63.j odd 6 1
441.2.o.d 10 7.c even 3 1
441.2.o.d 10 63.i even 6 1
441.2.s.b 10 1.a even 1 1 trivial
441.2.s.b 10 63.s even 6 1 inner
567.2.p.c 10 63.j odd 6 1
567.2.p.c 10 63.l odd 6 1
567.2.p.d 10 63.h even 3 1
567.2.p.d 10 63.o even 6 1
1008.2.ca.b 10 28.g odd 6 1
1008.2.ca.b 10 252.s odd 6 1
1008.2.df.b 10 28.d even 2 1
1008.2.df.b 10 252.o even 6 1
1323.2.i.b 10 9.c even 3 1
1323.2.i.b 10 21.g even 6 1
1323.2.o.c 10 21.h odd 6 1
1323.2.o.c 10 63.t odd 6 1
1323.2.o.d 10 21.g even 6 1
1323.2.o.d 10 63.h even 3 1
1323.2.s.b 10 3.b odd 2 1
1323.2.s.b 10 63.k odd 6 1
3024.2.ca.b 10 84.n even 6 1
3024.2.ca.b 10 252.bi even 6 1
3024.2.df.b 10 84.h odd 2 1
3024.2.df.b 10 252.bl odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{10} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(441, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$3 + 21 T + 52 T^{2} + 21 T^{3} - 46 T^{4} - 24 T^{5} + 42 T^{6} - 7 T^{8} + T^{10}$$
$3$ $$243 + 81 T^{3} + 36 T^{4} - 9 T^{5} + 12 T^{6} + 9 T^{7} + T^{10}$$
$5$ $$( -9 + 12 T + 6 T^{2} - 9 T^{3} + T^{5} )^{2}$$
$7$ $$T^{10}$$
$11$ $$2883 + 9130 T^{2} + 4331 T^{4} + 684 T^{6} + 44 T^{8} + T^{10}$$
$13$ $$3267 + 297 T - 2070 T^{2} - 189 T^{3} + 1098 T^{4} + 468 T^{5} - 48 T^{6} - 54 T^{7} + 3 T^{8} + 6 T^{9} + T^{10}$$
$17$ $$263169 - 258552 T + 207846 T^{2} - 79218 T^{3} + 30888 T^{4} - 8613 T^{5} + 2673 T^{6} - 576 T^{7} + 111 T^{8} - 12 T^{9} + T^{10}$$
$19$ $$2187 - 9477 T + 14175 T^{2} - 2106 T^{3} - 2970 T^{4} + 567 T^{5} + 594 T^{6} - 81 T^{7} - 24 T^{8} + 3 T^{9} + T^{10}$$
$23$ $$27 + 8722 T^{2} + 8093 T^{4} + 1299 T^{6} + 65 T^{8} + T^{10}$$
$29$ $$186003 + 1621737 T + 5085994 T^{2} + 3249987 T^{3} + 902708 T^{4} + 105537 T^{5} - 414 T^{6} - 1050 T^{7} + 5 T^{8} + 15 T^{9} + T^{10}$$
$31$ $$16875 - 84375 T + 170325 T^{2} - 148500 T^{3} + 65322 T^{4} - 13293 T^{5} + 276 T^{6} + 297 T^{7} - 6 T^{8} - 9 T^{9} + T^{10}$$
$37$ $$369664 + 233472 T + 385792 T^{2} - 213760 T^{3} + 130048 T^{4} - 25600 T^{5} + 5440 T^{6} - 472 T^{7} + 88 T^{8} - 6 T^{9} + T^{10}$$
$41$ $$40487769 - 9869013 T + 6127956 T^{2} - 238005 T^{3} + 424548 T^{4} - 31095 T^{5} + 11814 T^{6} - 360 T^{7} + 171 T^{8} - 9 T^{9} + T^{10}$$
$43$ $$12243001 + 6707583 T + 4255723 T^{2} + 570524 T^{3} + 281512 T^{4} + 36083 T^{5} + 13714 T^{6} + 713 T^{7} + 136 T^{8} - 3 T^{9} + T^{10}$$
$47$ $$321489 + 472878 T + 564579 T^{2} + 236880 T^{3} + 94392 T^{4} + 15444 T^{5} + 5820 T^{6} + 1047 T^{7} + 186 T^{8} + 15 T^{9} + T^{10}$$
$53$ $$871563 + 1039731 T + 212941 T^{2} - 239196 T^{3} + 5912 T^{4} + 18843 T^{5} + 1266 T^{6} - 495 T^{7} - 28 T^{8} + 9 T^{9} + T^{10}$$
$59$ $$4100625 - 2490750 T + 1373175 T^{2} - 425070 T^{3} + 144531 T^{4} - 36459 T^{5} + 9528 T^{6} - 1650 T^{7} + 240 T^{8} - 18 T^{9} + T^{10}$$
$61$ $$826875 - 897750 T + 197325 T^{2} + 138510 T^{3} - 32157 T^{4} - 23211 T^{5} + 10806 T^{6} - 1224 T^{7} - 54 T^{8} + 12 T^{9} + T^{10}$$
$67$ $$361 - 1520 T + 5526 T^{2} - 5694 T^{3} + 6546 T^{4} + 819 T^{5} + 3189 T^{6} - 438 T^{7} + 153 T^{8} + 10 T^{9} + T^{10}$$
$71$ $$46216875 + 17533075 T^{2} + 1560557 T^{4} + 40104 T^{6} + 359 T^{8} + T^{10}$$
$73$ $$789507 - 1343547 T + 646704 T^{2} + 196425 T^{3} - 75870 T^{4} - 22761 T^{5} + 11016 T^{6} - 324 T^{7} - 105 T^{8} + 3 T^{9} + T^{10}$$
$79$ $$1067655625 - 584555750 T + 226863000 T^{2} - 48342930 T^{3} + 8053914 T^{4} - 799857 T^{5} + 76611 T^{6} - 4884 T^{7} + 441 T^{8} - 20 T^{9} + T^{10}$$
$83$ $$340734681 + 31509513 T + 28498023 T^{2} - 4359474 T^{3} + 1551933 T^{4} - 144513 T^{5} + 25413 T^{6} - 1962 T^{7} + 279 T^{8} - 15 T^{9} + T^{10}$$
$89$ $$32455809 - 13946256 T + 10914912 T^{2} + 1123794 T^{3} + 822744 T^{4} + 48033 T^{5} + 30753 T^{6} + 3816 T^{7} + 489 T^{8} + 24 T^{9} + T^{10}$$
$97$ $$9687627 + 15202620 T + 8518455 T^{2} + 888300 T^{3} - 502227 T^{4} - 63981 T^{5} + 32406 T^{6} - 1116 T^{7} - 174 T^{8} + 6 T^{9} + T^{10}$$
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