# Properties

 Label 201.4.a.d Level 201 Weight 4 Character orbit 201.a Self dual Yes Analytic conductor 11.859 Analytic rank 0 Dimension 9 CM No Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$201 = 3 \cdot 67$$ Weight: $$k$$ = $$4$$ Character orbit: $$[\chi]$$ = 201.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$11.8593839112$$ Analytic rank: $$0$$ Dimension: $$9$$ Coefficient field: $$\mathbb{Q}[x]/(x^{9} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{8}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -3 q^{3} + ( 5 + \beta_{1} + \beta_{2} ) q^{4} + ( 1 + \beta_{1} + \beta_{2} + \beta_{7} ) q^{5} -3 \beta_{1} q^{6} + ( \beta_{1} + \beta_{3} - \beta_{5} + 2 \beta_{8} ) q^{7} + ( 5 + 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{8} + 9 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} -3 q^{3} + ( 5 + \beta_{1} + \beta_{2} ) q^{4} + ( 1 + \beta_{1} + \beta_{2} + \beta_{7} ) q^{5} -3 \beta_{1} q^{6} + ( \beta_{1} + \beta_{3} - \beta_{5} + 2 \beta_{8} ) q^{7} + ( 5 + 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{8} + 9 q^{9} + ( 4 + 8 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{6} - 3 \beta_{8} ) q^{10} + ( 7 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} ) q^{11} + ( -15 - 3 \beta_{1} - 3 \beta_{2} ) q^{12} + ( -23 + 4 \beta_{1} - 3 \beta_{2} + \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} ) q^{13} + ( 11 - 2 \beta_{1} - \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - \beta_{5} - 5 \beta_{7} - \beta_{8} ) q^{14} + ( -3 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{7} ) q^{15} + ( 18 + 13 \beta_{1} + \beta_{2} - 6 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - 4 \beta_{8} ) q^{16} + ( 13 - 3 \beta_{2} + 6 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} + 6 \beta_{8} ) q^{17} + 9 \beta_{1} q^{18} + ( 15 + 10 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} + 5 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} - 2 \beta_{8} ) q^{19} + ( 84 + 16 \beta_{1} + 3 \beta_{2} - 7 \beta_{3} - 8 \beta_{4} + 8 \beta_{5} + \beta_{6} + 4 \beta_{7} - 3 \beta_{8} ) q^{20} + ( -3 \beta_{1} - 3 \beta_{3} + 3 \beta_{5} - 6 \beta_{8} ) q^{21} + ( 23 + 10 \beta_{1} + 9 \beta_{4} - \beta_{5} + 2 \beta_{6} - 4 \beta_{7} + 6 \beta_{8} ) q^{22} + ( 51 + \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} - 3 \beta_{6} + 2 \beta_{7} + 4 \beta_{8} ) q^{23} + ( -15 - 15 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{24} + ( 51 + 15 \beta_{1} - 7 \beta_{4} + 9 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} ) q^{25} + ( 61 - 26 \beta_{1} + 4 \beta_{2} + 7 \beta_{3} + 18 \beta_{4} - 11 \beta_{5} - \beta_{6} - 9 \beta_{7} + 6 \beta_{8} ) q^{26} -27 q^{27} + ( -28 - 2 \beta_{1} - 10 \beta_{2} - 11 \beta_{5} - 12 \beta_{6} + 11 \beta_{7} + 7 \beta_{8} ) q^{28} + ( 64 - 22 \beta_{1} + 2 \beta_{2} - 7 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 9 \beta_{6} - 9 \beta_{7} ) q^{29} + ( -12 - 24 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 9 \beta_{5} - 3 \beta_{6} + 9 \beta_{8} ) q^{30} + ( 25 - 9 \beta_{1} - 7 \beta_{2} + 11 \beta_{3} - 11 \beta_{4} + 5 \beta_{5} - 6 \beta_{6} + 8 \beta_{7} - 8 \beta_{8} ) q^{31} + ( 138 + 9 \beta_{1} + 10 \beta_{2} - 8 \beta_{3} - 10 \beta_{4} + 4 \beta_{5} + 8 \beta_{6} - 4 \beta_{8} ) q^{32} + ( -21 - 6 \beta_{1} + 3 \beta_{2} - 6 \beta_{3} + 9 \beta_{4} + 6 \beta_{5} ) q^{33} + ( 11 - 22 \beta_{1} - 14 \beta_{2} + 28 \beta_{3} - \beta_{4} - 13 \beta_{5} - 6 \beta_{6} - 6 \beta_{7} + 4 \beta_{8} ) q^{34} + ( 24 - 41 \beta_{1} + 2 \beta_{2} + 13 \beta_{4} - 2 \beta_{5} - 10 \beta_{6} + 13 \beta_{7} - 6 \beta_{8} ) q^{35} + ( 45 + 9 \beta_{1} + 9 \beta_{2} ) q^{36} + ( -24 - 19 \beta_{1} + 8 \beta_{3} + 11 \beta_{4} - 11 \beta_{5} + 9 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} ) q^{37} + ( 129 + 22 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} - 29 \beta_{4} + 13 \beta_{5} - 4 \beta_{6} + 20 \beta_{7} - 24 \beta_{8} ) q^{38} + ( 69 - 12 \beta_{1} + 9 \beta_{2} - 3 \beta_{3} + 6 \beta_{4} + 12 \beta_{5} + 9 \beta_{6} - 9 \beta_{7} ) q^{39} + ( 167 + 64 \beta_{1} + 24 \beta_{2} - 27 \beta_{3} - 8 \beta_{4} + 26 \beta_{5} + 25 \beta_{6} - 12 \beta_{7} - 7 \beta_{8} ) q^{40} + ( 44 - 33 \beta_{1} - 10 \beta_{2} - 4 \beta_{3} - 9 \beta_{4} - 8 \beta_{5} - 12 \beta_{6} + 21 \beta_{7} - 2 \beta_{8} ) q^{41} + ( -33 + 6 \beta_{1} + 3 \beta_{2} - 12 \beta_{3} - 6 \beta_{4} + 3 \beta_{5} + 15 \beta_{7} + 3 \beta_{8} ) q^{42} + ( 23 + \beta_{1} + 13 \beta_{2} - 8 \beta_{3} + 4 \beta_{4} - 9 \beta_{5} + \beta_{6} - 5 \beta_{7} ) q^{43} + ( 97 - 14 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} - \beta_{4} - 3 \beta_{5} - 18 \beta_{6} + 4 \beta_{7} - 2 \beta_{8} ) q^{44} + ( 9 + 9 \beta_{1} + 9 \beta_{2} + 9 \beta_{7} ) q^{45} + ( 28 + 46 \beta_{1} + 10 \beta_{2} + 10 \beta_{4} + 3 \beta_{5} + 8 \beta_{6} - 23 \beta_{7} - 11 \beta_{8} ) q^{46} + ( 61 - 34 \beta_{1} + \beta_{2} + 9 \beta_{3} + 28 \beta_{4} + 6 \beta_{5} - 3 \beta_{6} - 11 \beta_{7} + 6 \beta_{8} ) q^{47} + ( -54 - 39 \beta_{1} - 3 \beta_{2} + 18 \beta_{3} + 9 \beta_{4} - 9 \beta_{5} + 12 \beta_{8} ) q^{48} + ( 74 - 39 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} + 8 \beta_{4} + 9 \beta_{5} + 13 \beta_{6} - 29 \beta_{7} + 2 \beta_{8} ) q^{49} + ( 199 + 61 \beta_{1} + 25 \beta_{2} - 15 \beta_{3} - 7 \beta_{4} + 43 \beta_{5} + 31 \beta_{6} - 14 \beta_{7} - 19 \beta_{8} ) q^{50} + ( -39 + 9 \beta_{2} - 18 \beta_{3} - 9 \beta_{4} + 6 \beta_{5} + 6 \beta_{7} - 18 \beta_{8} ) q^{51} + ( -183 + 6 \beta_{1} - 32 \beta_{2} + 31 \beta_{3} + 6 \beta_{4} - 47 \beta_{5} - 37 \beta_{6} - \beta_{7} + 46 \beta_{8} ) q^{52} + ( -34 - 21 \beta_{1} - 8 \beta_{2} - 10 \beta_{3} + 11 \beta_{4} + 4 \beta_{5} + 8 \beta_{6} - 33 \beta_{7} + 20 \beta_{8} ) q^{53} -27 \beta_{1} q^{54} + ( -51 - 18 \beta_{1} - 7 \beta_{2} - 2 \beta_{3} - 9 \beta_{4} + 8 \beta_{5} - 34 \beta_{6} + 34 \beta_{7} - 28 \beta_{8} ) q^{55} + ( -62 - 62 \beta_{1} + 9 \beta_{2} + 6 \beta_{3} + 33 \beta_{4} - 30 \beta_{5} - 4 \beta_{6} - 5 \beta_{7} + 7 \beta_{8} ) q^{56} + ( -45 - 30 \beta_{1} - 9 \beta_{2} + 12 \beta_{3} - 15 \beta_{4} - 6 \beta_{5} - 12 \beta_{6} - 12 \beta_{7} + 6 \beta_{8} ) q^{57} + ( -250 + 46 \beta_{1} - 30 \beta_{2} - 33 \beta_{3} - 33 \beta_{4} + 2 \beta_{5} - 7 \beta_{6} + 5 \beta_{7} + 6 \beta_{8} ) q^{58} + ( 98 - 43 \beta_{1} - 24 \beta_{2} - 18 \beta_{3} - 39 \beta_{4} - 8 \beta_{5} + 14 \beta_{6} + 17 \beta_{7} + 4 \beta_{8} ) q^{59} + ( -252 - 48 \beta_{1} - 9 \beta_{2} + 21 \beta_{3} + 24 \beta_{4} - 24 \beta_{5} - 3 \beta_{6} - 12 \beta_{7} + 9 \beta_{8} ) q^{60} + ( -89 + 14 \beta_{1} + 13 \beta_{2} - 37 \beta_{3} - 34 \beta_{4} + 2 \beta_{5} + 7 \beta_{6} + 19 \beta_{7} - 8 \beta_{8} ) q^{61} + ( -144 + 10 \beta_{1} + \beta_{2} + 14 \beta_{3} + 45 \beta_{4} + 26 \beta_{5} + 32 \beta_{6} + \beta_{7} + 5 \beta_{8} ) q^{62} + ( 9 \beta_{1} + 9 \beta_{3} - 9 \beta_{5} + 18 \beta_{8} ) q^{63} + ( -73 + 113 \beta_{1} + 17 \beta_{2} - 12 \beta_{3} + 10 \beta_{4} + 30 \beta_{5} + 24 \beta_{6} + 8 \beta_{8} ) q^{64} + ( -51 - 84 \beta_{1} - 49 \beta_{2} + 46 \beta_{3} - 19 \beta_{4} - 16 \beta_{5} - 14 \beta_{6} - 30 \beta_{7} + 6 \beta_{8} ) q^{65} + ( -69 - 30 \beta_{1} - 27 \beta_{4} + 3 \beta_{5} - 6 \beta_{6} + 12 \beta_{7} - 18 \beta_{8} ) q^{66} -67 q^{67} + ( -393 - 78 \beta_{1} - 40 \beta_{2} + 32 \beta_{3} + 29 \beta_{4} - 55 \beta_{5} - 26 \beta_{6} + 18 \beta_{7} + 28 \beta_{8} ) q^{68} + ( -153 - 3 \beta_{1} + 3 \beta_{2} + 9 \beta_{3} + 3 \beta_{4} + 9 \beta_{6} - 6 \beta_{7} - 12 \beta_{8} ) q^{69} + ( -575 + 2 \beta_{1} - 34 \beta_{2} + 29 \beta_{3} + 18 \beta_{4} - 30 \beta_{5} - 23 \beta_{6} + 24 \beta_{7} - 3 \beta_{8} ) q^{70} + ( 353 - 26 \beta_{1} + 17 \beta_{2} - 9 \beta_{3} + 10 \beta_{4} + 6 \beta_{5} + 23 \beta_{6} + 3 \beta_{7} + 6 \beta_{8} ) q^{71} + ( 45 + 45 \beta_{1} + 18 \beta_{2} - 18 \beta_{3} + 9 \beta_{4} + 9 \beta_{5} ) q^{72} + ( -123 - 21 \beta_{1} + 7 \beta_{2} + 20 \beta_{3} - 6 \beta_{4} + 19 \beta_{5} + 15 \beta_{6} - 51 \beta_{7} - 4 \beta_{8} ) q^{73} + ( -251 - 88 \beta_{1} - 67 \beta_{2} + 43 \beta_{3} - 39 \beta_{4} - 29 \beta_{5} - 39 \beta_{6} + 42 \beta_{7} - \beta_{8} ) q^{74} + ( -153 - 45 \beta_{1} + 21 \beta_{4} - 27 \beta_{5} - 9 \beta_{6} + 9 \beta_{7} - 6 \beta_{8} ) q^{75} + ( 45 + 198 \beta_{1} + 56 \beta_{2} - 46 \beta_{3} + 25 \beta_{4} + 95 \beta_{5} + 48 \beta_{6} - 28 \beta_{7} - 16 \beta_{8} ) q^{76} + ( 155 + 52 \beta_{1} - 51 \beta_{2} + 22 \beta_{3} - 47 \beta_{4} + 12 \beta_{5} - 26 \beta_{6} - 22 \beta_{7} + 6 \beta_{8} ) q^{77} + ( -183 + 78 \beta_{1} - 12 \beta_{2} - 21 \beta_{3} - 54 \beta_{4} + 33 \beta_{5} + 3 \beta_{6} + 27 \beta_{7} - 18 \beta_{8} ) q^{78} + ( 294 - 24 \beta_{1} + 14 \beta_{2} - 39 \beta_{3} - 21 \beta_{4} - 10 \beta_{5} + 5 \beta_{6} + 5 \beta_{7} + 16 \beta_{8} ) q^{79} + ( 123 + 212 \beta_{1} + 75 \beta_{2} - 81 \beta_{3} - 49 \beta_{4} + 73 \beta_{5} + 41 \beta_{6} - 4 \beta_{7} - 55 \beta_{8} ) q^{80} + 81 q^{81} + ( -389 - 14 \beta_{1} - 14 \beta_{2} + 13 \beta_{3} + 60 \beta_{4} + 21 \beta_{6} - 40 \beta_{7} - 23 \beta_{8} ) q^{82} + ( 132 - 49 \beta_{1} + 4 \beta_{2} - 12 \beta_{3} + 17 \beta_{4} + 26 \beta_{6} + 17 \beta_{7} - 40 \beta_{8} ) q^{83} + ( 84 + 6 \beta_{1} + 30 \beta_{2} + 33 \beta_{5} + 36 \beta_{6} - 33 \beta_{7} - 21 \beta_{8} ) q^{84} + ( -329 - 108 \beta_{1} + 11 \beta_{2} + 58 \beta_{3} + 87 \beta_{4} - 52 \beta_{5} - 30 \beta_{6} + 60 \beta_{7} ) q^{85} + ( -48 + 118 \beta_{1} + 7 \beta_{2} - 39 \beta_{3} + 2 \beta_{4} - 30 \beta_{5} - 31 \beta_{6} + 2 \beta_{7} + 23 \beta_{8} ) q^{86} + ( -192 + 66 \beta_{1} - 6 \beta_{2} + 21 \beta_{3} + 9 \beta_{4} + 6 \beta_{5} - 27 \beta_{6} + 27 \beta_{7} ) q^{87} + ( -457 + 90 \beta_{1} + 20 \beta_{2} + 13 \beta_{4} - 25 \beta_{5} - 18 \beta_{6} + 4 \beta_{7} - 10 \beta_{8} ) q^{88} + ( 85 - 2 \beta_{1} - 31 \beta_{2} - 36 \beta_{3} - 3 \beta_{4} - 18 \beta_{5} - 38 \beta_{6} + 34 \beta_{7} + 26 \beta_{8} ) q^{89} + ( 36 + 72 \beta_{1} + 18 \beta_{2} - 9 \beta_{3} + 9 \beta_{4} + 27 \beta_{5} + 9 \beta_{6} - 27 \beta_{8} ) q^{90} + ( 235 + 20 \beta_{1} + 53 \beta_{2} + 8 \beta_{3} + \beta_{4} + 30 \beta_{5} - 18 \beta_{6} - 26 \beta_{7} - 82 \beta_{8} ) q^{91} + ( 148 + 134 \beta_{1} + 41 \beta_{2} - 10 \beta_{3} - 37 \beta_{4} - 42 \beta_{5} - 24 \beta_{6} + 53 \beta_{7} + 37 \beta_{8} ) q^{92} + ( -75 + 27 \beta_{1} + 21 \beta_{2} - 33 \beta_{3} + 33 \beta_{4} - 15 \beta_{5} + 18 \beta_{6} - 24 \beta_{7} + 24 \beta_{8} ) q^{93} + ( -447 - 20 \beta_{1} - 58 \beta_{2} + 67 \beta_{3} - 40 \beta_{4} - 59 \beta_{5} - 49 \beta_{6} + 41 \beta_{7} + 24 \beta_{8} ) q^{94} + ( 655 + 218 \beta_{1} + 57 \beta_{2} - 12 \beta_{3} + 67 \beta_{4} + 82 \beta_{5} + 66 \beta_{6} - 62 \beta_{7} + 18 \beta_{8} ) q^{95} + ( -414 - 27 \beta_{1} - 30 \beta_{2} + 24 \beta_{3} + 30 \beta_{4} - 12 \beta_{5} - 24 \beta_{6} + 12 \beta_{8} ) q^{96} + ( -181 + 4 \beta_{1} + 31 \beta_{2} + 32 \beta_{3} + 49 \beta_{4} - 30 \beta_{5} + 16 \beta_{6} - 18 \beta_{7} + 104 \beta_{8} ) q^{97} + ( -482 + 17 \beta_{1} - 65 \beta_{2} - 9 \beta_{3} - 70 \beta_{4} - 16 \beta_{5} - 25 \beta_{6} + 40 \beta_{7} + 43 \beta_{8} ) q^{98} + ( 63 + 18 \beta_{1} - 9 \beta_{2} + 18 \beta_{3} - 27 \beta_{4} - 18 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$9q + 3q^{2} - 27q^{3} + 45q^{4} + 12q^{5} - 9q^{6} + 8q^{7} + 51q^{8} + 81q^{9} + O(q^{10})$$ $$9q + 3q^{2} - 27q^{3} + 45q^{4} + 12q^{5} - 9q^{6} + 8q^{7} + 51q^{8} + 81q^{9} + 45q^{10} + 72q^{11} - 135q^{12} - 166q^{13} + 93q^{14} - 36q^{15} + 173q^{16} + 146q^{17} + 27q^{18} + 154q^{19} + 763q^{20} - 24q^{21} + 244q^{22} + 476q^{23} - 153q^{24} + 465q^{25} + 502q^{26} - 243q^{27} - 141q^{28} + 432q^{29} - 135q^{30} + 248q^{31} + 1171q^{32} - 216q^{33} + 146q^{34} + 178q^{35} + 405q^{36} - 240q^{37} + 1182q^{38} + 498q^{39} + 1409q^{40} + 406q^{41} - 279q^{42} + 154q^{43} + 892q^{44} + 108q^{45} + 273q^{46} + 494q^{47} - 519q^{48} + 431q^{49} + 1658q^{50} - 438q^{51} - 1258q^{52} - 450q^{53} - 81q^{54} - 346q^{55} - 659q^{56} - 462q^{57} - 2114q^{58} + 732q^{59} - 2289q^{60} - 914q^{61} - 1265q^{62} + 72q^{63} - 467q^{64} - 536q^{65} - 732q^{66} - 603q^{67} - 3314q^{68} - 1428q^{69} - 4805q^{70} + 2990q^{71} + 459q^{72} - 1384q^{73} - 2043q^{74} - 1395q^{75} + 450q^{76} + 1660q^{77} - 1506q^{78} + 2438q^{79} + 995q^{80} + 729q^{81} - 3561q^{82} + 972q^{83} + 423q^{84} - 2706q^{85} - 21q^{86} - 1296q^{87} - 3796q^{88} + 1034q^{89} + 405q^{90} + 1898q^{91} + 1827q^{92} - 744q^{93} - 3502q^{94} + 6040q^{95} - 3513q^{96} - 1516q^{97} - 3996q^{98} + 648q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{9} - 3 x^{8} - 54 x^{7} + 138 x^{6} + 949 x^{5} - 2039 x^{4} - 5472 x^{3} + 10352 x^{2} + 3808 x - 6656$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 13$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{8} - 6 \nu^{7} - 16 \nu^{6} + 146 \nu^{5} - 289 \nu^{4} - 492 \nu^{3} + 4784 \nu^{2} - 800 \nu - 7552$$$$)/640$$ $$\beta_{4}$$ $$=$$ $$($$$$3 \nu^{8} - 18 \nu^{7} - 88 \nu^{6} + 558 \nu^{5} + 533 \nu^{4} - 3996 \nu^{3} + 832 \nu^{2} + 1920 \nu + 1024$$$$)/320$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{8} + 6 \nu^{7} + 36 \nu^{6} - 206 \nu^{5} - 411 \nu^{4} + 1912 \nu^{3} + 1656 \nu^{2} - 4400 \nu - 928$$$$)/160$$ $$\beta_{6}$$ $$=$$ $$($$$$3 \nu^{8} - 18 \nu^{7} - 88 \nu^{6} + 598 \nu^{5} + 493 \nu^{4} - 5316 \nu^{3} + 1512 \nu^{2} + 10880 \nu - 2496$$$$)/320$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{8} + 16 \nu^{7} - 24 \nu^{6} - 486 \nu^{5} + 1409 \nu^{4} + 3382 \nu^{3} - 11144 \nu^{2} - 640 \nu + 8192$$$$)/320$$ $$\beta_{8}$$ $$=$$ $$($$$$-9 \nu^{8} + 54 \nu^{7} + 264 \nu^{6} - 1674 \nu^{5} - 1759 \nu^{4} + 12468 \nu^{3} + 544 \nu^{2} - 12960 \nu - 2432$$$$)/640$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 13$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 21 \beta_{1} + 5$$ $$\nu^{4}$$ $$=$$ $$-4 \beta_{8} + 3 \beta_{5} - 3 \beta_{4} - 6 \beta_{3} + 25 \beta_{2} + 37 \beta_{1} + 266$$ $$\nu^{5}$$ $$=$$ $$-4 \beta_{8} + 8 \beta_{6} + 36 \beta_{5} + 22 \beta_{4} - 72 \beta_{3} + 74 \beta_{2} + 489 \beta_{1} + 298$$ $$\nu^{6}$$ $$=$$ $$-152 \beta_{8} + 24 \beta_{6} + 150 \beta_{5} - 110 \beta_{4} - 252 \beta_{3} + 633 \beta_{2} + 1209 \beta_{1} + 6087$$ $$\nu^{7}$$ $$=$$ $$-296 \beta_{8} + 32 \beta_{7} + 368 \beta_{6} + 1199 \beta_{5} + 355 \beta_{4} - 2142 \beta_{3} + 2306 \beta_{2} + 12029 \beta_{1} + 11447$$ $$\nu^{8}$$ $$=$$ $$-4780 \beta_{8} + 192 \beta_{7} + 1424 \beta_{6} + 5697 \beta_{5} - 3217 \beta_{4} - 8450 \beta_{3} + 16585 \beta_{2} + 37165 \beta_{1} + 147260$$

## 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
 −4.46691 −4.44127 −3.19321 −0.874823 0.837618 1.74668 3.11334 4.92710 5.35147
−4.46691 −3.00000 11.9533 −5.77925 13.4007 −31.2412 −17.6590 9.00000 25.8154
1.2 −4.44127 −3.00000 11.7249 14.0824 13.3238 12.5267 −16.5432 9.00000 −62.5436
1.3 −3.19321 −3.00000 2.19659 10.4341 9.57963 21.8629 18.5315 9.00000 −33.3182
1.4 −0.874823 −3.00000 −7.23468 −14.3103 2.62447 −20.3511 13.3277 9.00000 12.5190
1.5 0.837618 −3.00000 −7.29840 −4.06696 −2.51285 −13.0667 −12.8142 9.00000 −3.40656
1.6 1.74668 −3.00000 −4.94910 −20.3979 −5.24005 28.2703 −22.6180 9.00000 −35.6287
1.7 3.11334 −3.00000 1.69289 13.7648 −9.34002 11.2339 −19.6362 9.00000 42.8546
1.8 4.92710 −3.00000 16.2764 −2.16819 −14.7813 13.4538 40.7785 9.00000 −10.6829
1.9 5.35147 −3.00000 20.6382 20.4413 −16.0544 −14.6886 67.6328 9.00000 109.391
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.9 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$67$$ $$1$$

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{2}^{9} - \cdots$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(201))$$.