# Properties

 Label 690.2.e.b Level $690$ Weight $2$ Character orbit 690.e Analytic conductor $5.510$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$690 = 2 \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 690.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.50967773947$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 2 x^{15} + 3 x^{14} - 12 x^{13} + 15 x^{12} - 4 x^{11} + 45 x^{10} - 66 x^{9} - 32 x^{8} - 198 x^{7} + 405 x^{6} - 108 x^{5} + 1215 x^{4} - 2916 x^{3} + 2187 x^{2} - 4374 x + 6561$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 2 x^{15} + 3 x^{14} - 12 x^{13} + 15 x^{12} - 4 x^{11} + 45 x^{10} - 66 x^{9} - 32 x^{8} - 198 x^{7} + 405 x^{6} - 108 x^{5} + 1215 x^{4} - 2916 x^{3} + 2187 x^{2} - 4374 x + 6561$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$5 \nu^{15} - 7 \nu^{14} + 252 \nu^{13} - 240 \nu^{12} - 69 \nu^{11} - 1271 \nu^{10} + 1266 \nu^{9} + 3288 \nu^{8} + 5744 \nu^{7} - 10806 \nu^{6} - 16713 \nu^{5} - 16119 \nu^{4} + 64800 \nu^{3} + 972 \nu^{2} + 83835 \nu - 303993$$$$)/23328$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{15} - 2 \nu^{14} + 3 \nu^{13} - 12 \nu^{12} + 15 \nu^{11} - 4 \nu^{10} + 45 \nu^{9} - 66 \nu^{8} - 32 \nu^{7} - 198 \nu^{6} + 405 \nu^{5} - 108 \nu^{4} + 1215 \nu^{3} - 2916 \nu^{2} + 2187 \nu - 4374$$$$)/2187$$ $$\beta_{4}$$ $$=$$ $$($$$$31 \nu^{15} - 143 \nu^{14} + 444 \nu^{13} - 264 \nu^{12} + 465 \nu^{11} - 1987 \nu^{10} + 666 \nu^{9} + 2544 \nu^{8} + 8728 \nu^{7} - 11322 \nu^{6} - 6615 \nu^{5} - 58023 \nu^{4} + 103680 \nu^{3} - 49572 \nu^{2} + 152361 \nu - 356481$$$$)/69984$$ $$\beta_{5}$$ $$=$$ $$($$$$4 \nu^{15} - 209 \nu^{14} + 396 \nu^{13} - 264 \nu^{12} + 1068 \nu^{11} - 2167 \nu^{10} - 582 \nu^{9} - 2028 \nu^{8} + 12868 \nu^{7} - 300 \nu^{6} + 846 \nu^{5} - 61317 \nu^{4} + 78408 \nu^{3} - 60264 \nu^{2} + 326592 \nu - 352107$$$$)/34992$$ $$\beta_{6}$$ $$=$$ $$($$$$-9 \nu^{15} + 32 \nu^{14} + 8 \nu^{13} + 60 \nu^{12} - 159 \nu^{11} - 294 \nu^{10} + 52 \nu^{9} + 1188 \nu^{8} + 516 \nu^{7} - 1798 \nu^{6} - 6597 \nu^{5} + 468 \nu^{4} + 216 \nu^{3} + 25920 \nu^{2} - 25515 \nu - 29160$$$$)/11664$$ $$\beta_{7}$$ $$=$$ $$($$$$71 \nu^{15} - 151 \nu^{14} + 564 \nu^{13} - 816 \nu^{12} + 1281 \nu^{11} - 2147 \nu^{10} + 2394 \nu^{9} + 624 \nu^{8} + 13064 \nu^{7} - 21978 \nu^{6} - 5391 \nu^{5} - 69471 \nu^{4} + 158760 \nu^{3} - 109836 \nu^{2} + 327321 \nu - 811377$$$$)/69984$$ $$\beta_{8}$$ $$=$$ $$($$$$55 \nu^{15} - 350 \nu^{14} + 384 \nu^{13} - 480 \nu^{12} + 1761 \nu^{11} - 1660 \nu^{10} - 1128 \nu^{9} - 5772 \nu^{8} + 12244 \nu^{7} + 6402 \nu^{6} + 18207 \nu^{5} - 82026 \nu^{4} + 73872 \nu^{3} - 140940 \nu^{2} + 435213 \nu - 284310$$$$)/34992$$ $$\beta_{9}$$ $$=$$ $$($$$$-139 \nu^{15} + 263 \nu^{14} - 396 \nu^{13} + 912 \nu^{12} - 1365 \nu^{11} + 763 \nu^{10} - 2442 \nu^{9} + 5376 \nu^{8} - 5416 \nu^{7} + 10290 \nu^{6} - 23877 \nu^{5} + 65151 \nu^{4} - 120528 \nu^{3} + 210924 \nu^{2} - 306909 \nu + 356481$$$$)/69984$$ $$\beta_{10}$$ $$=$$ $$($$$$163 \nu^{15} + 91 \nu^{14} - 300 \nu^{13} - 768 \nu^{12} - 291 \nu^{11} + 3443 \nu^{10} + 5046 \nu^{9} - 3432 \nu^{8} - 21344 \nu^{7} - 16026 \nu^{6} + 35145 \nu^{5} + 54027 \nu^{4} + 2592 \nu^{3} - 113724 \nu^{2} - 276291 \nu + 207765$$$$)/69984$$ $$\beta_{11}$$ $$=$$ $$($$$$43 \nu^{15} - 62 \nu^{14} - 282 \nu^{12} + 357 \nu^{11} + 674 \nu^{10} + 624 \nu^{9} - 2892 \nu^{8} - 2960 \nu^{7} - 534 \nu^{6} + 15255 \nu^{5} - 5130 \nu^{4} + 9072 \nu^{3} - 74358 \nu^{2} + 47385 \nu + 34992$$$$)/17496$$ $$\beta_{12}$$ $$=$$ $$($$$$-94 \nu^{15} + 137 \nu^{14} + 180 \nu^{13} + 336 \nu^{12} - 690 \nu^{11} - 2117 \nu^{10} - 570 \nu^{9} + 7572 \nu^{8} + 10100 \nu^{7} - 3192 \nu^{6} - 40464 \nu^{5} - 7587 \nu^{4} + 3240 \nu^{3} + 141912 \nu^{2} + 7290 \nu - 242757$$$$)/34992$$ $$\beta_{13}$$ $$=$$ $$($$$$31 \nu^{15} - 114 \nu^{14} + 56 \nu^{13} - 120 \nu^{12} + 657 \nu^{11} + 32 \nu^{10} - 728 \nu^{9} - 3876 \nu^{8} + 1180 \nu^{7} + 5642 \nu^{6} + 15807 \nu^{5} - 20934 \nu^{4} - 864 \nu^{3} - 70956 \nu^{2} + 106677 \nu + 39366$$$$)/11664$$ $$\beta_{14}$$ $$=$$ $$($$$$-247 \nu^{15} + 383 \nu^{14} + 84 \nu^{13} + 1128 \nu^{12} - 1833 \nu^{11} - 4349 \nu^{10} - 2922 \nu^{9} + 18048 \nu^{8} + 15608 \nu^{7} + 186 \nu^{6} - 96705 \nu^{5} + 7911 \nu^{4} - 12960 \nu^{3} + 395604 \nu^{2} - 88209 \nu - 378351$$$$)/69984$$ $$\beta_{15}$$ $$=$$ $$($$$$-361 \nu^{15} + 269 \nu^{14} - 708 \nu^{13} + 2280 \nu^{12} - 1383 \nu^{11} - 1247 \nu^{10} - 12030 \nu^{9} + 4512 \nu^{8} + 9752 \nu^{7} + 53142 \nu^{6} - 38871 \nu^{5} + 36261 \nu^{4} - 273456 \nu^{3} + 448092 \nu^{2} - 98415 \nu + 789507$$$$)/69984$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{10} + \beta_{9} + \beta_{5} - \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$\beta_{14} - \beta_{12} - \beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} - \beta_{3} - \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{13} - \beta_{12} - 2 \beta_{11} + \beta_{10} + 2 \beta_{9} - \beta_{6} - \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_{1} + 1$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{15} - \beta_{14} + 2 \beta_{12} - 2 \beta_{11} + 4 \beta_{10} - 2 \beta_{9} - \beta_{8} - 3 \beta_{6} + \beta_{5} + 4 \beta_{4} + 3 \beta_{3} - \beta_{2} + 2 \beta_{1} - 4$$ $$\nu^{6}$$ $$=$$ $$-2 \beta_{15} + 6 \beta_{14} + \beta_{13} + \beta_{12} + 3 \beta_{10} - 4 \beta_{9} - 2 \beta_{8} - 4 \beta_{7} - 7 \beta_{6} + 4 \beta_{5} - \beta_{4} - 4 \beta_{3} - 2 \beta_{2} - 6 \beta_{1} - 9$$ $$\nu^{7}$$ $$=$$ $$4 \beta_{15} - 7 \beta_{14} - 2 \beta_{13} + 4 \beta_{12} - 8 \beta_{11} - \beta_{10} - 13 \beta_{9} + \beta_{8} + 4 \beta_{7} - 5 \beta_{6} - 6 \beta_{5} - 4 \beta_{4} + 2 \beta_{3} + \beta_{2} - 8 \beta_{1} + 15$$ $$\nu^{8}$$ $$=$$ $$2 \beta_{15} + \beta_{14} - \beta_{13} + 8 \beta_{12} - 10 \beta_{11} + 7 \beta_{10} + \beta_{9} + 13 \beta_{8} + 4 \beta_{7} - 14 \beta_{6} - 21 \beta_{5} + 17 \beta_{4} + 19 \beta_{3} - 4 \beta_{2} + 14 \beta_{1} + 14$$ $$\nu^{9}$$ $$=$$ $$-20 \beta_{15} - \beta_{14} - 19 \beta_{13} + 27 \beta_{12} - 4 \beta_{11} - 3 \beta_{10} - 20 \beta_{9} + 11 \beta_{8} - 16 \beta_{7} - 2 \beta_{6} + 15 \beta_{5} - 9 \beta_{4} + 6 \beta_{3} - 45 \beta_{2} + 19 \beta_{1} + 20$$ $$\nu^{10}$$ $$=$$ $$-14 \beta_{15} - 9 \beta_{14} - 11 \beta_{13} + 5 \beta_{12} + 26 \beta_{11} - 27 \beta_{10} + 26 \beta_{9} - 9 \beta_{8} + 28 \beta_{7} - 22 \beta_{6} + 9 \beta_{5} - 5 \beta_{4} - 83 \beta_{3} - 13 \beta_{2} - 5 \beta_{1} + 3$$ $$\nu^{11}$$ $$=$$ $$-48 \beta_{15} + 41 \beta_{14} + 15 \beta_{13} - 3 \beta_{12} - 32 \beta_{11} + 15 \beta_{10} + 124 \beta_{9} + 41 \beta_{8} + 76 \beta_{7} - 14 \beta_{6} - 15 \beta_{5} - 23 \beta_{4} + 23 \beta_{3} - 55 \beta_{2} + 6 \beta_{1} + 320$$ $$\nu^{12}$$ $$=$$ $$-86 \beta_{15} - 3 \beta_{14} - 9 \beta_{13} - 25 \beta_{12} - 106 \beta_{11} - 116 \beta_{10} - 61 \beta_{9} - 43 \beta_{8} + 12 \beta_{7} + 10 \beta_{6} + 12 \beta_{5} + 29 \beta_{4} - 22 \beta_{3} - 135 \beta_{2} + 357 \beta_{1} - 42$$ $$\nu^{13}$$ $$=$$ $$-136 \beta_{15} + 56 \beta_{14} + 197 \beta_{13} + 46 \beta_{12} + 116 \beta_{11} + 149 \beta_{10} + 559 \beta_{9} - 234 \beta_{8} - 56 \beta_{7} - 111 \beta_{6} + 380 \beta_{5} + 71 \beta_{4} - 400 \beta_{3} + 99 \beta_{2} + 66 \beta_{1} - 183$$ $$\nu^{14}$$ $$=$$ $$46 \beta_{15} + 339 \beta_{14} + 230 \beta_{13} + 64 \beta_{12} + 96 \beta_{11} + 106 \beta_{10} - 512 \beta_{9} - 557 \beta_{8} + 236 \beta_{7} - 319 \beta_{6} + 237 \beta_{5} - 46 \beta_{4} - 202 \beta_{3} - 139 \beta_{2} + 4 \beta_{1} + 431$$ $$\nu^{15}$$ $$=$$ $$86 \beta_{15} + 88 \beta_{14} + 523 \beta_{13} - 309 \beta_{12} - 986 \beta_{11} + 603 \beta_{10} - 274 \beta_{9} + 40 \beta_{8} + 76 \beta_{7} - 121 \beta_{6} - 426 \beta_{5} - 159 \beta_{4} - 510 \beta_{3} + 596 \beta_{2} + 935 \beta_{1} - 272$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/690\mathbb{Z}\right)^\times$$.

 $$n$$ $$277$$ $$461$$ $$511$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
551.1
 −0.251873 + 1.71364i −1.14349 + 1.30094i 1.53677 + 0.798958i −1.62589 + 0.597052i 1.73162 + 0.0386882i 1.37342 − 1.05533i −0.462710 − 1.66910i −0.157845 − 1.72484i −0.251873 − 1.71364i −1.14349 − 1.30094i 1.53677 − 0.798958i −1.62589 − 0.597052i 1.73162 − 0.0386882i 1.37342 + 1.05533i −0.462710 + 1.66910i −0.157845 + 1.72484i
1.00000i −1.71364 0.251873i −1.00000 1.00000 −0.251873 + 1.71364i 0.194278i 1.00000i 2.87312 + 0.863238i 1.00000i
551.2 1.00000i −1.30094 1.14349i −1.00000 1.00000 −1.14349 + 1.30094i 1.02070i 1.00000i 0.384868 + 2.97521i 1.00000i
551.3 1.00000i −0.798958 + 1.53677i −1.00000 1.00000 1.53677 + 0.798958i 0.145234i 1.00000i −1.72333 2.45563i 1.00000i
551.4 1.00000i −0.597052 1.62589i −1.00000 1.00000 −1.62589 + 0.597052i 3.80421i 1.00000i −2.28706 + 1.94149i 1.00000i
551.5 1.00000i −0.0386882 + 1.73162i −1.00000 1.00000 1.73162 + 0.0386882i 2.83902i 1.00000i −2.99701 0.133986i 1.00000i
551.6 1.00000i 1.05533 + 1.37342i −1.00000 1.00000 1.37342 1.05533i 1.91110i 1.00000i −0.772562 + 2.89882i 1.00000i
551.7 1.00000i 1.66910 0.462710i −1.00000 1.00000 −0.462710 1.66910i 2.82123i 1.00000i 2.57180 1.54462i 1.00000i
551.8 1.00000i 1.72484 0.157845i −1.00000 1.00000 −0.157845 1.72484i 4.77029i 1.00000i 2.95017 0.544517i 1.00000i
551.9 1.00000i −1.71364 + 0.251873i −1.00000 1.00000 −0.251873 1.71364i 0.194278i 1.00000i 2.87312 0.863238i 1.00000i
551.10 1.00000i −1.30094 + 1.14349i −1.00000 1.00000 −1.14349 1.30094i 1.02070i 1.00000i 0.384868 2.97521i 1.00000i
551.11 1.00000i −0.798958 1.53677i −1.00000 1.00000 1.53677 0.798958i 0.145234i 1.00000i −1.72333 + 2.45563i 1.00000i
551.12 1.00000i −0.597052 + 1.62589i −1.00000 1.00000 −1.62589 0.597052i 3.80421i 1.00000i −2.28706 1.94149i 1.00000i
551.13 1.00000i −0.0386882 1.73162i −1.00000 1.00000 1.73162 0.0386882i 2.83902i 1.00000i −2.99701 + 0.133986i 1.00000i
551.14 1.00000i 1.05533 1.37342i −1.00000 1.00000 1.37342 + 1.05533i 1.91110i 1.00000i −0.772562 2.89882i 1.00000i
551.15 1.00000i 1.66910 + 0.462710i −1.00000 1.00000 −0.462710 + 1.66910i 2.82123i 1.00000i 2.57180 + 1.54462i 1.00000i
551.16 1.00000i 1.72484 + 0.157845i −1.00000 1.00000 −0.157845 + 1.72484i 4.77029i 1.00000i 2.95017 + 0.544517i 1.00000i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 551.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
69.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 690.2.e.b yes 16
3.b odd 2 1 690.2.e.a 16
23.b odd 2 1 690.2.e.a 16
69.c even 2 1 inner 690.2.e.b yes 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.e.a 16 3.b odd 2 1
690.2.e.a 16 23.b odd 2 1
690.2.e.b yes 16 1.a even 1 1 trivial
690.2.e.b yes 16 69.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{8} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(690, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{8}$$
$3$ $$6561 - 729 T^{2} - 1944 T^{3} - 405 T^{4} + 324 T^{5} + 153 T^{6} + 60 T^{7} - 56 T^{8} + 20 T^{9} + 17 T^{10} + 12 T^{11} - 5 T^{12} - 8 T^{13} - T^{14} + T^{16}$$
$5$ $$( -1 + T )^{16}$$
$7$ $$64 + 4832 T^{2} + 87988 T^{4} + 131924 T^{6} + 61605 T^{8} + 12586 T^{10} + 1247 T^{12} + 58 T^{14} + T^{16}$$
$11$ $$( 68 + 920 T + 1560 T^{2} - 1704 T^{3} + 93 T^{4} + 202 T^{5} - 29 T^{6} - 6 T^{7} + T^{8} )^{2}$$
$13$ $$( -43184 + 57376 T - 16198 T^{2} - 4432 T^{3} + 2063 T^{4} + 84 T^{5} - 79 T^{6} + T^{8} )^{2}$$
$17$ $$( 90592 + 23696 T - 26102 T^{2} - 3344 T^{3} + 2607 T^{4} + 84 T^{5} - 91 T^{6} + T^{8} )^{2}$$
$19$ $$3625216 + 51262208 T^{2} + 38349152 T^{4} + 11273808 T^{6} + 1657521 T^{8} + 131802 T^{10} + 5675 T^{12} + 122 T^{14} + T^{16}$$
$23$ $$78310985281 + 13619301788 T + 4145004892 T^{2} + 25745372 T^{3} + 130965588 T^{4} + 19710540 T^{5} + 8483044 T^{6} - 611892 T^{7} - 81770 T^{8} - 26604 T^{9} + 16036 T^{10} + 1620 T^{11} + 468 T^{12} + 4 T^{13} + 28 T^{14} + 4 T^{15} + T^{16}$$
$29$ $$2517630976 + 2480144384 T^{2} + 875347968 T^{4} + 147873792 T^{6} + 13224704 T^{8} + 645440 T^{10} + 16804 T^{12} + 212 T^{14} + T^{16}$$
$31$ $$( 90304 + 11968 T - 55232 T^{2} + 680 T^{3} + 6485 T^{4} + 166 T^{5} - 161 T^{6} - 2 T^{7} + T^{8} )^{2}$$
$37$ $$484704256 + 3609493504 T^{2} + 1656529920 T^{4} + 285813760 T^{6} + 23943296 T^{8} + 1055232 T^{10} + 24388 T^{12} + 268 T^{14} + T^{16}$$
$41$ $$256000000 + 1761826816 T^{2} + 3251947840 T^{4} + 899415920 T^{6} + 75863193 T^{8} + 2587122 T^{10} + 42163 T^{12} + 330 T^{14} + T^{16}$$
$43$ $$27688960000 + 285733683200 T^{2} + 325838270464 T^{4} + 24369926144 T^{6} + 760866112 T^{8} + 12437472 T^{10} + 111828 T^{12} + 524 T^{14} + T^{16}$$
$47$ $$12845056 + 21672427520 T^{2} + 12726095872 T^{4} + 1793886208 T^{6} + 108829440 T^{8} + 3267072 T^{10} + 49456 T^{12} + 360 T^{14} + T^{16}$$
$53$ $$( 756224 + 274944 T - 120000 T^{2} - 33728 T^{3} + 7424 T^{4} + 1072 T^{5} - 200 T^{6} - 4 T^{7} + T^{8} )^{2}$$
$59$ $$32943702016 + 25300631552 T^{2} + 7545864192 T^{4} + 1094397952 T^{6} + 78781952 T^{8} + 2625280 T^{10} + 42944 T^{12} + 336 T^{14} + T^{16}$$
$61$ $$5217752377600 + 2395189614784 T^{2} + 403804093220 T^{4} + 31674839660 T^{6} + 1186612733 T^{8} + 20022006 T^{10} + 164975 T^{12} + 654 T^{14} + T^{16}$$
$67$ $$2824120582144 + 976608346112 T^{2} + 136187236352 T^{4} + 9770151424 T^{6} + 382302720 T^{8} + 8080640 T^{10} + 89508 T^{12} + 484 T^{14} + T^{16}$$
$71$ $$4296168198400 + 4622368947264 T^{2} + 1318628475780 T^{4} + 69623014388 T^{6} + 1657134917 T^{8} + 21441782 T^{10} + 156999 T^{12} + 614 T^{14} + T^{16}$$
$73$ $$( 51200 - 158720 T - 7168 T^{2} + 26624 T^{3} + 1664 T^{4} - 1232 T^{5} - 132 T^{6} + 8 T^{7} + T^{8} )^{2}$$
$79$ $$20404373094400 + 4884082376704 T^{2} + 461974520832 T^{4} + 22831236096 T^{6} + 647470464 T^{8} + 10779456 T^{10} + 102532 T^{12} + 508 T^{14} + T^{16}$$
$83$ $$( -51542912 + 13608704 T + 1778560 T^{2} - 616480 T^{3} + 3152 T^{4} + 6896 T^{5} - 266 T^{6} - 20 T^{7} + T^{8} )^{2}$$
$89$ $$( 278751040 + 113958144 T + 13819760 T^{2} - 259616 T^{3} - 176076 T^{4} - 10400 T^{5} + 240 T^{6} + 40 T^{7} + T^{8} )^{2}$$
$97$ $$487900948710400 + 96812045727616 T^{2} + 6204079484964 T^{4} + 195550678900 T^{6} + 3481583997 T^{8} + 36613546 T^{10} + 224463 T^{12} + 738 T^{14} + T^{16}$$