# Properties

 Label 216.2.n.b Level 216 Weight 2 Character orbit 216.n 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.n (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: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}\cdot 3^{4}$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - x^{15} + x^{14} + 2 x^{12} - 4 x^{11} - 8 x^{9} + 4 x^{8} - 16 x^{7} - 32 x^{5} + 32 x^{4} + 64 x^{2} - 128 x + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{15} + \nu^{14} + 3 \nu^{13} + 4 \nu^{12} + 8 \nu^{11} + 6 \nu^{10} + 8 \nu^{9} + 4 \nu^{8} - 12 \nu^{7} - 24 \nu^{6} - 32 \nu^{5} - 72 \nu^{4} - 96 \nu^{3} - 128 \nu^{2} - 96 \nu - 256$$$$)/192$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{15} + \nu^{14} - \nu^{13} + 2 \nu^{12} + 2 \nu^{11} - 8 \nu^{9} - 8 \nu^{8} - 12 \nu^{7} - 8 \nu^{6} - 32 \nu^{5} - 32 \nu^{4} - 32 \nu^{3} + 64 \nu^{2} + 64 \nu$$$$)/128$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{15} - \nu^{14} + \nu^{13} + 2 \nu^{11} - 4 \nu^{10} - 8 \nu^{8} + 4 \nu^{7} - 16 \nu^{6} - 32 \nu^{4} + 32 \nu^{3} + 64 \nu - 128$$$$)/128$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{14} - \nu^{13} - 2 \nu^{12} + \nu^{11} - 5 \nu^{10} - 4 \nu^{9} - 10 \nu^{8} + 8 \nu^{7} + 4 \nu^{6} + 32 \nu^{5} + 20 \nu^{4} + 64 \nu^{3} + 80 \nu^{2} + 80 \nu - 32$$$$)/96$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{15} + 3 \nu^{14} + 5 \nu^{13} + 12 \nu^{12} + 18 \nu^{11} + 28 \nu^{10} + 24 \nu^{9} + 24 \nu^{8} + 20 \nu^{7} - 64 \nu^{6} - 96 \nu^{5} - 160 \nu^{4} - 256 \nu^{3} - 384 \nu^{2} - 448 \nu - 512$$$$)/384$$ $$\beta_{8}$$ $$=$$ $$($$$$-\nu^{15} - \nu^{14} - 2 \nu^{13} - 3 \nu^{12} - 5 \nu^{11} - 4 \nu^{10} - 6 \nu^{9} + 8 \nu^{8} + 4 \nu^{7} + 16 \nu^{6} + 20 \nu^{5} + 64 \nu^{4} + 64 \nu^{3} + 80 \nu^{2} + 64 \nu + 128$$$$)/96$$ $$\beta_{9}$$ $$=$$ $$($$$$-2 \nu^{15} + \nu^{14} - 5 \nu^{13} - 5 \nu^{12} - 16 \nu^{11} - 10 \nu^{10} - 28 \nu^{9} - 8 \nu^{8} - 32 \nu^{7} + 12 \nu^{6} + 64 \nu^{5} + 160 \nu^{4} + 96 \nu^{3} + 256 \nu^{2} + 256 \nu + 704$$$$)/192$$ $$\beta_{10}$$ $$=$$ $$($$$$-2 \nu^{15} - \nu^{14} - 3 \nu^{13} - 3 \nu^{12} - 8 \nu^{11} - 6 \nu^{10} + 8 \nu^{8} + 20 \nu^{6} + 32 \nu^{5} + 96 \nu^{4} + 80 \nu^{3} + 128 \nu^{2} + 256$$$$)/96$$ $$\beta_{11}$$ $$=$$ $$($$$$4 \nu^{15} + \nu^{14} + 3 \nu^{13} + \nu^{12} + 8 \nu^{11} + 6 \nu^{10} + 8 \nu^{9} - 8 \nu^{8} - 12 \nu^{6} - 32 \nu^{5} - 96 \nu^{4} - 144 \nu^{3} - 128 \nu^{2} - 448$$$$)/192$$ $$\beta_{12}$$ $$=$$ $$($$$$-9 \nu^{15} - 7 \nu^{14} - 9 \nu^{13} - 16 \nu^{12} - 50 \nu^{11} - 12 \nu^{10} - 32 \nu^{9} + 8 \nu^{8} - 36 \nu^{7} + 80 \nu^{6} + 128 \nu^{5} + 480 \nu^{4} + 224 \nu^{3} + 512 \nu^{2} + 192 \nu + 1664$$$$)/384$$ $$\beta_{13}$$ $$=$$ $$($$$$-11 \nu^{15} - 3 \nu^{14} - 13 \nu^{13} - 14 \nu^{12} - 54 \nu^{11} - 32 \nu^{10} - 40 \nu^{9} + 24 \nu^{8} - 28 \nu^{7} + 120 \nu^{6} + 96 \nu^{5} + 800 \nu^{4} + 480 \nu^{3} + 576 \nu^{2} + 320 \nu + 2048$$$$)/384$$ $$\beta_{14}$$ $$=$$ $$($$$$-13 \nu^{15} - 7 \nu^{14} - 17 \nu^{13} - 28 \nu^{12} - 74 \nu^{11} - 28 \nu^{10} - 56 \nu^{9} + 8 \nu^{8} - 68 \nu^{7} + 192 \nu^{6} + 224 \nu^{5} + 736 \nu^{4} + 768 \nu^{3} + 896 \nu^{2} + 448 \nu + 2560$$$$)/384$$ $$\beta_{15}$$ $$=$$ $$($$$$15 \nu^{15} + \nu^{14} + 15 \nu^{13} + 32 \nu^{12} + 62 \nu^{11} + 36 \nu^{10} + 64 \nu^{9} - 8 \nu^{8} + 60 \nu^{7} - 112 \nu^{6} - 128 \nu^{5} - 864 \nu^{4} - 544 \nu^{3} - 704 \nu^{2} - 576 \nu - 2944$$$$)/384$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2}$$ $$\nu^{3}$$ $$=$$ $$\beta_{14} - \beta_{12} + \beta_{7} + \beta_{5}$$ $$\nu^{4}$$ $$=$$ $$\beta_{13} + \beta_{11} + \beta_{7} + \beta_{6} + \beta_{3}$$ $$\nu^{5}$$ $$=$$ $$\beta_{15} + 2 \beta_{9} + \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{3} + 1$$ $$\nu^{6}$$ $$=$$ $$\beta_{15} + \beta_{13} - \beta_{11} - 2 \beta_{10} - 2 \beta_{9} - 5 \beta_{7} - 5 \beta_{5} + 1$$ $$\nu^{7}$$ $$=$$ $$-2 \beta_{14} + \beta_{13} - \beta_{11} - 2 \beta_{10} + \beta_{7} - \beta_{6} - 2 \beta_{4} - 9 \beta_{3} + 2 \beta_{2}$$ $$\nu^{8}$$ $$=$$ $$\beta_{15} - 2 \beta_{12} + 2 \beta_{9} + 8 \beta_{8} - 3 \beta_{6} + \beta_{5} - 2 \beta_{4} + 5 \beta_{3} + 2 \beta_{2} + 5$$ $$\nu^{9}$$ $$=$$ $$-\beta_{15} + 2 \beta_{14} - \beta_{13} - 2 \beta_{12} + 5 \beta_{11} + 10 \beta_{10} - 2 \beta_{9} - 8 \beta_{8} + 3 \beta_{7} + 3 \beta_{5} + 8 \beta_{1} + 3$$ $$\nu^{10}$$ $$=$$ $$10 \beta_{14} - 3 \beta_{13} + 7 \beta_{11} - 2 \beta_{10} + 21 \beta_{7} + 7 \beta_{6} + 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + 8 \beta_{1}$$ $$\nu^{11}$$ $$=$$ $$-3 \beta_{15} - 2 \beta_{12} - 6 \beta_{9} + 8 \beta_{8} + 17 \beta_{6} - 11 \beta_{5} + 6 \beta_{4} + 25 \beta_{3} - 2 \beta_{2} + 25$$ $$\nu^{12}$$ $$=$$ $$19 \beta_{15} - 2 \beta_{14} + 19 \beta_{13} + 2 \beta_{12} - 15 \beta_{11} - 6 \beta_{10} + 6 \beta_{9} - 8 \beta_{8} - 5 \beta_{7} - 5 \beta_{5} + 8 \beta_{1} + 7$$ $$\nu^{13}$$ $$=$$ $$-6 \beta_{14} + 13 \beta_{13} - 17 \beta_{11} - 42 \beta_{10} - 35 \beta_{7} - 17 \beta_{6} - 38 \beta_{4} + 7 \beta_{3} + 42 \beta_{2} - 8 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$-11 \beta_{15} - 42 \beta_{12} + 26 \beta_{9} - 24 \beta_{8} - 23 \beta_{6} - 11 \beta_{5} - 26 \beta_{4} - 47 \beta_{3} + 22 \beta_{2} - 47$$ $$\nu^{15}$$ $$=$$ $$19 \beta_{15} + 22 \beta_{14} + 19 \beta_{13} - 22 \beta_{12} + 65 \beta_{11} + 10 \beta_{10} + 22 \beta_{9} + 24 \beta_{8} + 35 \beta_{7} + 35 \beta_{5} - 24 \beta_{1} + 87$$

## Character values

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

 $$n$$ $$55$$ $$109$$ $$137$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1 - \beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 −0.722180 + 1.21592i −1.34532 + 0.436011i 0.587625 + 1.28635i 0.820200 + 1.15207i −1.12494 − 0.857038i 1.41411 − 0.0174668i −0.179748 − 1.40274i 1.05026 − 0.947078i −0.722180 − 1.21592i −1.34532 − 0.436011i 0.587625 − 1.28635i 0.820200 − 1.15207i −1.12494 + 0.857038i 1.41411 + 0.0174668i −0.179748 + 1.40274i 1.05026 + 0.947078i
−1.41411 + 0.0174668i 0 1.99939 0.0493999i 3.17262 + 1.83171i 0 −0.191926 0.332426i −2.82649 + 0.104780i 0 −4.51841 2.53482i
37.2 −1.05026 + 0.947078i 0 0.206086 1.98935i −0.602794 0.348023i 0 0.795065 + 1.37709i 1.66763 + 2.28452i 0 0.962695 0.205379i
37.3 −0.820200 1.15207i 0 −0.654545 + 1.88986i −1.97542 1.14051i 0 −0.907824 1.57240i 2.71411 0.795980i 0 0.306290 + 3.21128i
37.4 −0.587625 1.28635i 0 −1.30939 + 1.51178i 1.97542 + 1.14051i 0 −0.907824 1.57240i 2.71411 + 0.795980i 0 0.306290 3.21128i
37.5 0.179748 + 1.40274i 0 −1.93538 + 0.504281i 1.19115 + 0.687709i 0 1.80469 + 3.12581i −1.05526 2.62420i 0 −0.750573 + 1.79449i
37.6 0.722180 1.21592i 0 −0.956913 1.75622i −3.17262 1.83171i 0 −0.191926 0.332426i −2.82649 0.104780i 0 −4.51841 + 2.53482i
37.7 1.12494 + 0.857038i 0 0.530970 + 1.92823i −1.19115 0.687709i 0 1.80469 + 3.12581i −1.05526 + 2.62420i 0 −0.750573 1.79449i
37.8 1.34532 0.436011i 0 1.61979 1.17315i 0.602794 + 0.348023i 0 0.795065 + 1.37709i 1.66763 2.28452i 0 0.962695 + 0.205379i
181.1 −1.41411 0.0174668i 0 1.99939 + 0.0493999i 3.17262 1.83171i 0 −0.191926 + 0.332426i −2.82649 0.104780i 0 −4.51841 + 2.53482i
181.2 −1.05026 0.947078i 0 0.206086 + 1.98935i −0.602794 + 0.348023i 0 0.795065 1.37709i 1.66763 2.28452i 0 0.962695 + 0.205379i
181.3 −0.820200 + 1.15207i 0 −0.654545 1.88986i −1.97542 + 1.14051i 0 −0.907824 + 1.57240i 2.71411 + 0.795980i 0 0.306290 3.21128i
181.4 −0.587625 + 1.28635i 0 −1.30939 1.51178i 1.97542 1.14051i 0 −0.907824 + 1.57240i 2.71411 0.795980i 0 0.306290 + 3.21128i
181.5 0.179748 1.40274i 0 −1.93538 0.504281i 1.19115 0.687709i 0 1.80469 3.12581i −1.05526 + 2.62420i 0 −0.750573 1.79449i
181.6 0.722180 + 1.21592i 0 −0.956913 + 1.75622i −3.17262 + 1.83171i 0 −0.191926 + 0.332426i −2.82649 + 0.104780i 0 −4.51841 2.53482i
181.7 1.12494 0.857038i 0 0.530970 1.92823i −1.19115 + 0.687709i 0 1.80469 3.12581i −1.05526 2.62420i 0 −0.750573 + 1.79449i
181.8 1.34532 + 0.436011i 0 1.61979 + 1.17315i 0.602794 0.348023i 0 0.795065 1.37709i 1.66763 + 2.28452i 0 0.962695 0.205379i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 181.8 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.b even 2 1 inner
9.c even 3 1 inner
72.n 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.n.b 16
3.b odd 2 1 72.2.n.b 16
4.b odd 2 1 864.2.r.b 16
8.b even 2 1 inner 216.2.n.b 16
8.d odd 2 1 864.2.r.b 16
9.c even 3 1 inner 216.2.n.b 16
9.c even 3 1 648.2.d.k 8
9.d odd 6 1 72.2.n.b 16
9.d odd 6 1 648.2.d.j 8
12.b even 2 1 288.2.r.b 16
24.f even 2 1 288.2.r.b 16
24.h odd 2 1 72.2.n.b 16
36.f odd 6 1 864.2.r.b 16
36.f odd 6 1 2592.2.d.k 8
36.h even 6 1 288.2.r.b 16
36.h even 6 1 2592.2.d.j 8
72.j odd 6 1 72.2.n.b 16
72.j odd 6 1 648.2.d.j 8
72.l even 6 1 288.2.r.b 16
72.l even 6 1 2592.2.d.j 8
72.n even 6 1 inner 216.2.n.b 16
72.n even 6 1 648.2.d.k 8
72.p odd 6 1 864.2.r.b 16
72.p odd 6 1 2592.2.d.k 8

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

$p$ $F_p(T)$
$2$ $$1 + T + T^{2} + 2 T^{4} + 4 T^{5} + 8 T^{7} + 4 T^{8} + 16 T^{9} + 32 T^{11} + 32 T^{12} + 64 T^{14} + 128 T^{15} + 256 T^{16}$$
$3$ 
$5$ $$1 + 19 T^{2} + 176 T^{4} + 1031 T^{6} + 3893 T^{8} + 4928 T^{10} - 61926 T^{12} - 631122 T^{14} - 3717344 T^{16} - 15778050 T^{18} - 38703750 T^{20} + 77000000 T^{22} + 1520703125 T^{24} + 10068359375 T^{26} + 42968750000 T^{28} + 115966796875 T^{30} + 152587890625 T^{32}$$
$7$ $$( 1 - 3 T - 14 T^{2} + 39 T^{3} + 139 T^{4} - 252 T^{5} - 1208 T^{6} + 666 T^{7} + 9424 T^{8} + 4662 T^{9} - 59192 T^{10} - 86436 T^{11} + 333739 T^{12} + 655473 T^{13} - 1647086 T^{14} - 2470629 T^{15} + 5764801 T^{16} )^{2}$$
$11$ $$1 + 48 T^{2} + 1090 T^{4} + 17592 T^{6} + 242041 T^{8} + 2632140 T^{10} + 21031138 T^{12} + 158095260 T^{14} + 1558598596 T^{16} + 19129526460 T^{18} + 307916891458 T^{20} + 4662996570540 T^{22} + 51883637916121 T^{24} + 456291173580792 T^{26} + 3420886930625890 T^{28} + 18227992011995568 T^{30} + 45949729863572161 T^{32}$$
$13$ $$1 + 51 T^{2} + 1288 T^{4} + 16911 T^{6} + 73645 T^{8} - 1059264 T^{10} - 5456606 T^{12} + 411982806 T^{14} + 9084740848 T^{16} + 69625094214 T^{18} - 155846123966 T^{20} - 5112865008576 T^{22} + 60074488948045 T^{24} + 2331324955658439 T^{26} + 30007933637755528 T^{28} + 200806195670663739 T^{30} + 665416609183179841 T^{32}$$
$17$ $$( 1 - 7 T + 66 T^{2} - 309 T^{3} + 1702 T^{4} - 5253 T^{5} + 19074 T^{6} - 34391 T^{7} + 83521 T^{8} )^{4}$$
$19$ $$( 1 - 69 T^{2} + 2306 T^{4} - 53763 T^{6} + 1069146 T^{8} - 19408443 T^{10} + 300520226 T^{12} - 3246165789 T^{14} + 16983563041 T^{16} )^{2}$$
$23$ $$( 1 - 5 T - 32 T^{2} - 45 T^{3} + 1385 T^{4} + 3040 T^{5} - 11142 T^{6} - 48640 T^{7} - 123716 T^{8} - 1118720 T^{9} - 5894118 T^{10} + 36987680 T^{11} + 387579785 T^{12} - 289635435 T^{13} - 4737148448 T^{14} - 17024127235 T^{15} + 78310985281 T^{16} )^{2}$$
$29$ $$1 + 123 T^{2} + 7912 T^{4} + 340647 T^{6} + 10067965 T^{8} + 144056448 T^{10} - 3705324014 T^{12} - 328820871018 T^{14} - 12137186779472 T^{16} - 276538352526138 T^{18} - 2620705273945934 T^{20} + 85688134810823808 T^{22} + 5036463377066894365 T^{24} +$$$$14\!\cdots\!47$$$$T^{26} +$$$$27\!\cdots\!92$$$$T^{28} +$$$$36\!\cdots\!63$$$$T^{30} +$$$$25\!\cdots\!21$$$$T^{32}$$
$31$ $$( 1 + 5 T - 48 T^{2} - 155 T^{3} + 1121 T^{4} - 2040 T^{5} - 44678 T^{6} + 79040 T^{7} + 1805724 T^{8} + 2450240 T^{9} - 42935558 T^{10} - 60773640 T^{11} + 1035267041 T^{12} - 4437518405 T^{13} - 42600176688 T^{14} + 137563070555 T^{15} + 852891037441 T^{16} )^{2}$$
$37$ $$( 1 - 144 T^{2} + 12668 T^{4} - 733824 T^{6} + 31784838 T^{8} - 1004605056 T^{10} + 23741871548 T^{12} - 369464602896 T^{14} + 3512479453921 T^{16} )^{2}$$
$41$ $$( 1 - 4 T - 74 T^{2} - 288 T^{3} + 4517 T^{4} + 22796 T^{5} - 11538 T^{6} - 738968 T^{7} - 2462804 T^{8} - 30297688 T^{9} - 19395378 T^{10} + 1571123116 T^{11} + 12763962437 T^{12} - 33366585888 T^{13} - 351507713834 T^{14} - 779017095524 T^{15} + 7984925229121 T^{16} )^{2}$$
$43$ $$1 + 324 T^{2} + 58258 T^{4} + 7305024 T^{6} + 705172105 T^{8} + 54954910764 T^{10} + 3558033934834 T^{12} + 194436327001344 T^{14} + 9048929543300068 T^{16} + 359512768625485056 T^{18} + 12164209974444414034 T^{20} +$$$$34\!\cdots\!36$$$$T^{22} +$$$$82\!\cdots\!05$$$$T^{24} +$$$$15\!\cdots\!76$$$$T^{26} +$$$$23\!\cdots\!58$$$$T^{28} +$$$$23\!\cdots\!76$$$$T^{30} +$$$$13\!\cdots\!01$$$$T^{32}$$
$47$ $$( 1 + 3 T - 98 T^{2} - 219 T^{3} + 3523 T^{4} - 1116 T^{5} - 253856 T^{6} + 226218 T^{7} + 18909448 T^{8} + 10632246 T^{9} - 560767904 T^{10} - 115866468 T^{11} + 17191116163 T^{12} - 50226556533 T^{13} - 1056363102242 T^{14} + 1519869361389 T^{15} + 23811286661761 T^{16} )^{2}$$
$53$ $$( 1 - 264 T^{2} + 35516 T^{4} - 3123720 T^{6} + 194863110 T^{8} - 8774529480 T^{10} + 280238323196 T^{12} - 5851391338056 T^{14} + 62259690411361 T^{16} )^{2}$$
$59$ $$1 + 364 T^{2} + 70130 T^{4} + 9549968 T^{6} + 1028964713 T^{8} + 92991430988 T^{10} + 7291973853618 T^{12} + 506327334867240 T^{14} + 31497451193778532 T^{16} + 1762525452672862440 T^{18} + 88359479586850462098 T^{20} +$$$$39\!\cdots\!08$$$$T^{22} +$$$$15\!\cdots\!73$$$$T^{24} +$$$$48\!\cdots\!68$$$$T^{26} +$$$$12\!\cdots\!30$$$$T^{28} +$$$$22\!\cdots\!04$$$$T^{30} +$$$$21\!\cdots\!41$$$$T^{32}$$
$61$ $$1 + 323 T^{2} + 52152 T^{4} + 6035887 T^{6} + 584536925 T^{8} + 49803739968 T^{10} + 3801157073266 T^{12} + 264102902456582 T^{14} + 16823476283802768 T^{16} + 982726900040941622 T^{18} + 52630216452466386706 T^{20} +$$$$25\!\cdots\!48$$$$T^{22} +$$$$11\!\cdots\!25$$$$T^{24} +$$$$43\!\cdots\!87$$$$T^{26} +$$$$13\!\cdots\!92$$$$T^{28} +$$$$31\!\cdots\!43$$$$T^{30} +$$$$36\!\cdots\!61$$$$T^{32}$$
$67$ $$1 + 296 T^{2} + 39450 T^{4} + 3796936 T^{6} + 362561921 T^{8} + 33090030780 T^{10} + 2631719645962 T^{12} + 193036342550180 T^{14} + 13423375489686516 T^{16} + 866540141707758020 T^{18} + 53032101023857423402 T^{20} +$$$$29\!\cdots\!20$$$$T^{22} +$$$$14\!\cdots\!61$$$$T^{24} +$$$$69\!\cdots\!64$$$$T^{26} +$$$$32\!\cdots\!50$$$$T^{28} +$$$$10\!\cdots\!84$$$$T^{30} +$$$$16\!\cdots\!81$$$$T^{32}$$
$71$ $$( 1 + 18 T + 332 T^{2} + 3582 T^{3} + 36198 T^{4} + 254322 T^{5} + 1673612 T^{6} + 6442398 T^{7} + 25411681 T^{8} )^{4}$$
$73$ $$( 1 + 11 T + 278 T^{2} + 2325 T^{3} + 29966 T^{4} + 169725 T^{5} + 1481462 T^{6} + 4279187 T^{7} + 28398241 T^{8} )^{4}$$
$79$ $$( 1 + 15 T - 80 T^{2} - 1305 T^{3} + 13273 T^{4} + 59400 T^{5} - 1907150 T^{6} - 913560 T^{7} + 190569148 T^{8} - 72171240 T^{9} - 11902523150 T^{10} + 29286516600 T^{11} + 516984425113 T^{12} - 4015558600695 T^{13} - 19446996441680 T^{14} + 288058634792385 T^{15} + 1517108809906561 T^{16} )^{2}$$
$83$ $$1 + 559 T^{2} + 168260 T^{4} + 35967119 T^{6} + 6055752221 T^{8} + 843016361960 T^{10} + 99713806040442 T^{12} + 10182388433060610 T^{14} + 904920089638581976 T^{16} + 70146473915354542290 T^{18} +$$$$47\!\cdots\!82$$$$T^{20} +$$$$27\!\cdots\!40$$$$T^{22} +$$$$13\!\cdots\!61$$$$T^{24} +$$$$55\!\cdots\!31$$$$T^{26} +$$$$17\!\cdots\!60$$$$T^{28} +$$$$41\!\cdots\!11$$$$T^{30} +$$$$50\!\cdots\!81$$$$T^{32}$$
$89$ $$( 1 + 16 T + 408 T^{2} + 4116 T^{3} + 56278 T^{4} + 366324 T^{5} + 3231768 T^{6} + 11279504 T^{7} + 62742241 T^{8} )^{4}$$
$97$ $$( 1 - 254 T^{2} + 1560 T^{3} + 35209 T^{4} - 273780 T^{5} - 2055806 T^{6} + 15575820 T^{7} + 104325124 T^{8} + 1510854540 T^{9} - 19343078654 T^{10} - 249871613940 T^{11} + 3117027454729 T^{12} + 13396250800920 T^{13} - 211574889251966 T^{14} + 7837433594376961 T^{16} )^{2}$$