# Properties

 Label 216.2.l.b Level $216$ Weight $2$ Character orbit 216.l Analytic conductor $1.725$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$216 = 2^{3} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 216.l (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.72476868366$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 3 x^{15} + 7 x^{14} - 12 x^{13} + 16 x^{12} - 12 x^{11} - 8 x^{10} + 36 x^{9} - 68 x^{8} + 72 x^{7} - 32 x^{6} - 96 x^{5} + 256 x^{4} - 384 x^{3} + 448 x^{2} - 384 x + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 72) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 3 x^{15} + 7 x^{14} - 12 x^{13} + 16 x^{12} - 12 x^{11} - 8 x^{10} + 36 x^{9} - 68 x^{8} + 72 x^{7} - 32 x^{6} - 96 x^{5} + 256 x^{4} - 384 x^{3} + 448 x^{2} - 384 x + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{15} - 13 \nu^{14} + 11 \nu^{13} + 4 \nu^{12} - 38 \nu^{11} + 60 \nu^{10} - 104 \nu^{9} + 68 \nu^{8} + 148 \nu^{7} - 344 \nu^{6} + 440 \nu^{5} - 240 \nu^{4} - 32 \nu^{3} + 608 \nu^{2} - 1152 \nu + 1280$$$$)/896$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{15} + \nu^{14} + 25 \nu^{13} - 38 \nu^{12} + 46 \nu^{11} - 24 \nu^{10} + 8 \nu^{9} + 68 \nu^{8} - 244 \nu^{7} + 272 \nu^{6} - 8 \nu^{5} - 128 \nu^{4} + 416 \nu^{3} - 1184 \nu^{2} + 1088 \nu - 512$$$$)/896$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{15} + 6 \nu^{14} - 11 \nu^{13} + 24 \nu^{12} - 39 \nu^{11} + 10 \nu^{10} + 20 \nu^{9} - 124 \nu^{8} + 160 \nu^{7} - 160 \nu^{6} - 20 \nu^{5} + 240 \nu^{4} - 528 \nu^{3} + 848 \nu^{2} - 640 \nu + 512$$$$)/448$$ $$\beta_{5}$$ $$=$$ $$($$$$3 \nu^{15} - 11 \nu^{14} + 47 \nu^{13} - 58 \nu^{12} + 40 \nu^{11} + 40 \nu^{10} - 144 \nu^{9} + 316 \nu^{8} - 340 \nu^{7} - 80 \nu^{6} + 592 \nu^{5} - 1168 \nu^{4} + 1248 \nu^{3} - 1088 \nu^{2} + 128 \nu + 1152$$$$)/896$$ $$\beta_{6}$$ $$=$$ $$($$$$-5 \nu^{15} + 23 \nu^{14} - 13 \nu^{13} - 6 \nu^{12} + 50 \nu^{11} - 132 \nu^{10} + 240 \nu^{9} - 116 \nu^{8} - 236 \nu^{7} + 656 \nu^{6} - 856 \nu^{5} + 752 \nu^{4} - 288 \nu^{3} - 1696 \nu^{2} + 2176 \nu - 1920$$$$)/896$$ $$\beta_{7}$$ $$=$$ $$($$$$-3 \nu^{15} + 4 \nu^{14} - 5 \nu^{13} + 2 \nu^{12} + 23 \nu^{11} - 40 \nu^{10} + 32 \nu^{9} + 20 \nu^{8} - 24 \nu^{7} + 80 \nu^{6} - 60 \nu^{5} + 104 \nu^{4} + 208 \nu^{3} - 144 \nu^{2} - 352 \nu + 192$$$$)/448$$ $$\beta_{8}$$ $$=$$ $$($$$$3 \nu^{15} + 3 \nu^{14} - 9 \nu^{13} + 26 \nu^{12} - 30 \nu^{11} + 12 \nu^{10} + 52 \nu^{9} - 132 \nu^{8} + 164 \nu^{7} - 80 \nu^{6} - 136 \nu^{5} + 400 \nu^{4} - 656 \nu^{3} + 704 \nu^{2} - 320 \nu - 192$$$$)/448$$ $$\beta_{9}$$ $$=$$ $$($$$$-4 \nu^{15} + 17 \nu^{14} - 37 \nu^{13} + 19 \nu^{12} + 12 \nu^{11} - 86 \nu^{10} + 192 \nu^{9} - 216 \nu^{8} + 52 \nu^{7} + 340 \nu^{6} - 584 \nu^{5} + 792 \nu^{4} - 544 \nu^{3} - 192 \nu^{2} + 1248 \nu - 1088$$$$)/448$$ $$\beta_{10}$$ $$=$$ $$($$$$5 \nu^{15} - 9 \nu^{14} + 27 \nu^{13} - 36 \nu^{12} + 34 \nu^{11} - 8 \nu^{10} - 72 \nu^{9} + 172 \nu^{8} - 156 \nu^{7} + 72 \nu^{6} + 184 \nu^{5} - 640 \nu^{4} + 736 \nu^{3} - 768 \nu^{2} + 512 \nu - 320$$$$)/448$$ $$\beta_{11}$$ $$=$$ $$($$$$5 \nu^{15} - 2 \nu^{14} - 8 \nu^{13} + 27 \nu^{12} - 36 \nu^{11} + 48 \nu^{10} - 16 \nu^{9} - 108 \nu^{8} + 208 \nu^{7} - 236 \nu^{6} + 72 \nu^{5} + 144 \nu^{4} - 496 \nu^{3} + 800 \nu^{2} - 384 \nu + 128$$$$)/448$$ $$\beta_{12}$$ $$=$$ $$($$$$-6 \nu^{15} + 15 \nu^{14} - 24 \nu^{13} + 18 \nu^{12} - 3 \nu^{11} - 66 \nu^{10} + 120 \nu^{9} - 128 \nu^{8} - 20 \nu^{7} + 160 \nu^{6} - 260 \nu^{5} + 432 \nu^{4} - 480 \nu^{3} + 48 \nu^{2} + 192 \nu + 384$$$$)/448$$ $$\beta_{13}$$ $$=$$ $$($$$$5 \nu^{15} - 16 \nu^{14} + 48 \nu^{13} - 71 \nu^{12} + 76 \nu^{11} - 22 \nu^{10} - 100 \nu^{9} + 284 \nu^{8} - 408 \nu^{7} + 212 \nu^{6} + 184 \nu^{5} - 920 \nu^{4} + 1520 \nu^{3} - 1888 \nu^{2} + 1632 \nu - 768$$$$)/448$$ $$\beta_{14}$$ $$=$$ $$($$$$3 \nu^{14} - 7 \nu^{13} + 15 \nu^{12} - 22 \nu^{11} + 16 \nu^{10} + 4 \nu^{9} - 56 \nu^{8} + 92 \nu^{7} - 116 \nu^{6} + 32 \nu^{5} + 144 \nu^{4} - 352 \nu^{3} + 512 \nu^{2} - 448 \nu + 384$$$$)/64$$ $$\beta_{15}$$ $$=$$ $$($$$$3 \nu^{15} - 18 \nu^{14} + 47 \nu^{13} - 100 \nu^{12} + 117 \nu^{11} - 100 \nu^{10} - 60 \nu^{9} + 316 \nu^{8} - 592 \nu^{7} + 648 \nu^{6} - 52 \nu^{5} - 888 \nu^{4} + 2144 \nu^{3} - 2768 \nu^{2} + 3040 \nu - 1536$$$$)/448$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{11} + \beta_{8} - \beta_{6} - \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$\beta_{15} - \beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} + \beta_{7} + \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{3} - 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{15} + \beta_{14} - \beta_{12} - \beta_{10} + \beta_{7} + \beta_{6} - \beta_{4} + \beta_{2} - 1$$ $$\nu^{5}$$ $$=$$ $$\beta_{15} - 2 \beta_{13} + \beta_{12} + 2 \beta_{11} + \beta_{7} + 2 \beta_{5} - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + \beta_{1} - 5$$ $$\nu^{6}$$ $$=$$ $$2 \beta_{15} - \beta_{14} - 6 \beta_{13} + 2 \beta_{11} + \beta_{10} - 2 \beta_{9} + \beta_{6} + 2 \beta_{5} + 2 \beta_{3} - \beta_{2} + \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$-\beta_{15} - \beta_{14} - 2 \beta_{13} - 3 \beta_{12} + 3 \beta_{10} + 2 \beta_{9} - 2 \beta_{8} + \beta_{7} + \beta_{6} + 3 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + 2 \beta_{1} + 3$$ $$\nu^{8}$$ $$=$$ $$\beta_{15} + 2 \beta_{14} - 2 \beta_{13} - 3 \beta_{12} - 4 \beta_{11} + 2 \beta_{10} + 2 \beta_{8} + \beta_{7} + 4 \beta_{6} + 2 \beta_{5} - 5 \beta_{4} - 6 \beta_{3} + 2 \beta_{2} + 5 \beta_{1} - 1$$ $$\nu^{9}$$ $$=$$ $$-2 \beta_{15} - 3 \beta_{14} - 4 \beta_{11} - 3 \beta_{10} + 6 \beta_{9} + 12 \beta_{8} - 3 \beta_{6} + 14 \beta_{5} - 4 \beta_{3} + 5 \beta_{2} + \beta_{1} - 4$$ $$\nu^{10}$$ $$=$$ $$-\beta_{15} - 3 \beta_{14} - 6 \beta_{13} - 7 \beta_{12} - 7 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} - 3 \beta_{7} - \beta_{6} + 12 \beta_{5} - \beta_{4} + 2 \beta_{3} - 7 \beta_{2} + 7$$ $$\nu^{11}$$ $$=$$ $$-17 \beta_{15} - 4 \beta_{14} + 18 \beta_{13} + 7 \beta_{12} - 8 \beta_{11} + 4 \beta_{10} + 4 \beta_{9} + 2 \beta_{8} + 3 \beta_{7} - 26 \beta_{6} - 22 \beta_{5} - 7 \beta_{4} + 22 \beta_{3} - 16 \beta_{2} + 9 \beta_{1} + 21$$ $$\nu^{12}$$ $$=$$ $$8 \beta_{15} - 11 \beta_{14} - 24 \beta_{13} + 18 \beta_{12} + 40 \beta_{11} - 7 \beta_{10} - 34 \beta_{9} + 18 \beta_{7} + 21 \beta_{6} + 10 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} - 11 \beta_{2} + 35 \beta_{1} - 22$$ $$\nu^{13}$$ $$=$$ $$-5 \beta_{15} - 13 \beta_{14} - 6 \beta_{13} + \beta_{12} - \beta_{10} - 18 \beta_{9} + 18 \beta_{8} + 13 \beta_{7} + 5 \beta_{6} + 31 \beta_{4} + 26 \beta_{3} - \beta_{2} + 38 \beta_{1} - 1$$ $$\nu^{14}$$ $$=$$ $$17 \beta_{15} + 18 \beta_{14} - 34 \beta_{13} - 43 \beta_{12} + 16 \beta_{11} + 18 \beta_{10} - 42 \beta_{8} + 25 \beta_{7} + 48 \beta_{6} + 34 \beta_{5} + 27 \beta_{4} + 2 \beta_{3} - 26 \beta_{2} + 17 \beta_{1} - 25$$ $$\nu^{15}$$ $$=$$ $$50 \beta_{15} + 45 \beta_{14} - 20 \beta_{13} - 20 \beta_{12} + 48 \beta_{11} + 25 \beta_{10} + 6 \beta_{9} + 12 \beta_{8} + 20 \beta_{7} + 49 \beta_{6} + 38 \beta_{5} - 60 \beta_{4} - 56 \beta_{3} - 11 \beta_{2} - 79 \beta_{1} + 8$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/216\mathbb{Z}\right)^\times$$.

 $$n$$ $$55$$ $$109$$ $$137$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1 - \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
35.1
 −0.409484 + 1.35363i 0.867527 + 1.11687i −1.37702 − 0.322193i 1.12063 + 0.862658i 1.40985 + 0.111062i −0.533474 − 1.30973i −0.186766 − 1.40183i 0.608741 − 1.27649i −0.409484 − 1.35363i 0.867527 − 1.11687i −1.37702 + 0.322193i 1.12063 − 0.862658i 1.40985 − 0.111062i −0.533474 + 1.30973i −0.186766 + 1.40183i 0.608741 + 1.27649i
−1.37702 + 0.322193i 0 1.79238 0.887333i −0.565188 0.978934i 0 −3.71499 2.14485i −2.18226 + 1.79937i 0 1.09368 + 1.16591i
35.2 −0.533474 + 1.30973i 0 −1.43081 1.39742i 0.895377 + 1.55084i 0 2.08793 + 1.20546i 2.59355 1.12850i 0 −2.50885 + 0.345375i
35.3 −0.409484 1.35363i 0 −1.66465 + 1.10858i 0.565188 + 0.978934i 0 3.71499 + 2.14485i 2.18226 + 1.79937i 0 1.09368 1.16591i
35.4 −0.186766 + 1.40183i 0 −1.93024 0.523628i −1.60936 2.78750i 0 −1.82223 1.05206i 1.09454 2.60806i 0 4.20817 1.73544i
35.5 0.608741 + 1.27649i 0 −1.25887 + 1.55411i 1.74322 + 3.01934i 0 −1.80802 1.04386i −2.75013 0.660890i 0 −2.79300 + 4.06320i
35.6 0.867527 1.11687i 0 −0.494795 1.93783i −0.895377 1.55084i 0 −2.08793 1.20546i −2.59355 1.12850i 0 −2.50885 0.345375i
35.7 1.12063 0.862658i 0 0.511643 1.93345i 1.60936 + 2.78750i 0 1.82223 + 1.05206i −1.09454 2.60806i 0 4.20817 + 1.73544i
35.8 1.40985 0.111062i 0 1.97533 0.313160i −1.74322 3.01934i 0 1.80802 + 1.04386i 2.75013 0.660890i 0 −2.79300 4.06320i
179.1 −1.37702 0.322193i 0 1.79238 + 0.887333i −0.565188 + 0.978934i 0 −3.71499 + 2.14485i −2.18226 1.79937i 0 1.09368 1.16591i
179.2 −0.533474 1.30973i 0 −1.43081 + 1.39742i 0.895377 1.55084i 0 2.08793 1.20546i 2.59355 + 1.12850i 0 −2.50885 0.345375i
179.3 −0.409484 + 1.35363i 0 −1.66465 1.10858i 0.565188 0.978934i 0 3.71499 2.14485i 2.18226 1.79937i 0 1.09368 + 1.16591i
179.4 −0.186766 1.40183i 0 −1.93024 + 0.523628i −1.60936 + 2.78750i 0 −1.82223 + 1.05206i 1.09454 + 2.60806i 0 4.20817 + 1.73544i
179.5 0.608741 1.27649i 0 −1.25887 1.55411i 1.74322 3.01934i 0 −1.80802 + 1.04386i −2.75013 + 0.660890i 0 −2.79300 4.06320i
179.6 0.867527 + 1.11687i 0 −0.494795 + 1.93783i −0.895377 + 1.55084i 0 −2.08793 + 1.20546i −2.59355 + 1.12850i 0 −2.50885 + 0.345375i
179.7 1.12063 + 0.862658i 0 0.511643 + 1.93345i 1.60936 2.78750i 0 1.82223 1.05206i −1.09454 + 2.60806i 0 4.20817 1.73544i
179.8 1.40985 + 0.111062i 0 1.97533 + 0.313160i −1.74322 + 3.01934i 0 1.80802 1.04386i 2.75013 + 0.660890i 0 −2.79300 + 4.06320i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 179.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
9.d odd 6 1 inner
72.l even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.2.l.b 16
3.b odd 2 1 72.2.l.b 16
4.b odd 2 1 864.2.p.b 16
8.b even 2 1 864.2.p.b 16
8.d odd 2 1 inner 216.2.l.b 16
9.c even 3 1 72.2.l.b 16
9.c even 3 1 648.2.f.b 16
9.d odd 6 1 inner 216.2.l.b 16
9.d odd 6 1 648.2.f.b 16
12.b even 2 1 288.2.p.b 16
24.f even 2 1 72.2.l.b 16
24.h odd 2 1 288.2.p.b 16
36.f odd 6 1 288.2.p.b 16
36.f odd 6 1 2592.2.f.b 16
36.h even 6 1 864.2.p.b 16
36.h even 6 1 2592.2.f.b 16
72.j odd 6 1 864.2.p.b 16
72.j odd 6 1 2592.2.f.b 16
72.l even 6 1 inner 216.2.l.b 16
72.l even 6 1 648.2.f.b 16
72.n even 6 1 288.2.p.b 16
72.n even 6 1 2592.2.f.b 16
72.p odd 6 1 72.2.l.b 16
72.p odd 6 1 648.2.f.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.l.b 16 3.b odd 2 1
72.2.l.b 16 9.c even 3 1
72.2.l.b 16 24.f even 2 1
72.2.l.b 16 72.p odd 6 1
216.2.l.b 16 1.a even 1 1 trivial
216.2.l.b 16 8.d odd 2 1 inner
216.2.l.b 16 9.d odd 6 1 inner
216.2.l.b 16 72.l even 6 1 inner
288.2.p.b 16 12.b even 2 1
288.2.p.b 16 24.h odd 2 1
288.2.p.b 16 36.f odd 6 1
288.2.p.b 16 72.n even 6 1
648.2.f.b 16 9.c even 3 1
648.2.f.b 16 9.d odd 6 1
648.2.f.b 16 72.l even 6 1
648.2.f.b 16 72.p odd 6 1
864.2.p.b 16 4.b odd 2 1
864.2.p.b 16 8.b even 2 1
864.2.p.b 16 36.h even 6 1
864.2.p.b 16 72.j odd 6 1
2592.2.f.b 16 36.f odd 6 1
2592.2.f.b 16 36.h even 6 1
2592.2.f.b 16 72.j odd 6 1
2592.2.f.b 16 72.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{16} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(216, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$256 - 384 T + 448 T^{2} - 384 T^{3} + 256 T^{4} - 96 T^{5} - 32 T^{6} + 72 T^{7} - 68 T^{8} + 36 T^{9} - 8 T^{10} - 12 T^{11} + 16 T^{12} - 12 T^{13} + 7 T^{14} - 3 T^{15} + T^{16}$$
$3$ $$T^{16}$$
$5$ $$266256 + 339012 T^{2} + 312453 T^{4} + 123903 T^{6} + 35106 T^{8} + 4923 T^{10} + 498 T^{12} + 27 T^{14} + T^{16}$$
$7$ $$4260096 - 2904048 T^{2} + 1279953 T^{4} - 340749 T^{6} + 66426 T^{8} - 8373 T^{10} + 750 T^{12} - 33 T^{14} + T^{16}$$
$11$ $$( 1 - 12 T + 44 T^{2} + 48 T^{3} - 9 T^{4} - 24 T^{5} + 8 T^{6} + 6 T^{7} + T^{8} )^{2}$$
$13$ $$266256 - 1772460 T^{2} + 11382813 T^{4} - 2713221 T^{6} + 454938 T^{8} - 39129 T^{10} + 2442 T^{12} - 57 T^{14} + T^{16}$$
$17$ $$( 784 + 992 T^{2} + 360 T^{4} + 35 T^{6} + T^{8} )^{2}$$
$19$ $$( 16 - 8 T - 12 T^{2} + T^{3} + T^{4} )^{4}$$
$23$ $$639280656 + 454125924 T^{2} + 255771909 T^{4} + 42464691 T^{6} + 5182026 T^{8} + 225735 T^{10} + 7158 T^{12} + 99 T^{14} + T^{16}$$
$29$ $$17449353216 + 10074829824 T^{2} + 5081053545 T^{4} + 389228679 T^{6} + 20607630 T^{8} + 599547 T^{10} + 12654 T^{12} + 135 T^{14} + T^{16}$$
$31$ $$279189651456 - 45038923776 T^{2} + 4696155729 T^{4} - 290875401 T^{6} + 13147422 T^{8} - 398493 T^{10} + 8826 T^{12} - 117 T^{14} + T^{16}$$
$37$ $$( 74304 + 40176 T^{2} + 4896 T^{4} + 156 T^{6} + T^{8} )^{2}$$
$41$ $$( 7921 - 8010 T + 920 T^{2} + 1800 T^{3} - 51 T^{4} - 360 T^{5} + 128 T^{6} - 18 T^{7} + T^{8} )^{2}$$
$43$ $$( 6889 + 11786 T + 15184 T^{2} + 7856 T^{3} + 3115 T^{4} + 524 T^{5} + 76 T^{6} - 4 T^{7} + T^{8} )^{2}$$
$47$ $$266256 + 1701252 T^{2} + 10094661 T^{4} + 4840839 T^{6} + 1892526 T^{8} + 160239 T^{10} + 10818 T^{12} + 111 T^{14} + T^{16}$$
$53$ $$( 297216 - 331344 T^{2} + 15408 T^{4} - 228 T^{6} + T^{8} )^{2}$$
$59$ $$( 528529 - 165756 T - 33562 T^{2} + 15960 T^{3} + 3717 T^{4} - 420 T^{5} - 58 T^{6} + 6 T^{7} + T^{8} )^{2}$$
$61$ $$1192149524736 - 215024661360 T^{2} + 26804642049 T^{4} - 1747852317 T^{6} + 82050270 T^{8} - 1679649 T^{10} + 24750 T^{12} - 189 T^{14} + T^{16}$$
$67$ $$( 582169 - 209062 T + 125434 T^{2} + 5876 T^{3} + 5785 T^{4} + 20 T^{5} + 130 T^{6} + 8 T^{7} + T^{8} )^{2}$$
$71$ $$( 74304 - 28944 T^{2} + 3744 T^{4} - 168 T^{6} + T^{8} )^{2}$$
$73$ $$( 172 - 224 T - 78 T^{2} + T^{3} + T^{4} )^{4}$$
$79$ $$74509345296 - 36110680524 T^{2} + 13340118429 T^{4} - 1880575641 T^{6} + 199135626 T^{8} - 3530925 T^{10} + 46758 T^{12} - 249 T^{14} + T^{16}$$
$83$ $$( 432964 - 487578 T + 162629 T^{2} + 22971 T^{3} - 6366 T^{4} - 837 T^{5} + 212 T^{6} + 27 T^{7} + T^{8} )^{2}$$
$89$ $$( 891136 + 689456 T^{2} + 23808 T^{4} + 272 T^{6} + T^{8} )^{2}$$
$97$ $$( 1018081 + 904064 T + 978382 T^{2} - 147832 T^{3} + 32851 T^{4} - 1096 T^{5} + 190 T^{6} - 4 T^{7} + T^{8} )^{2}$$