# Properties

 Label 819.2.ct.a Level $819$ Weight $2$ Character orbit 819.ct Analytic conductor $6.540$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$819 = 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 819.ct (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.53974792554$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: 12.0.58891012706304.1 Defining polynomial: $$x^{12} - 5 x^{10} - 2 x^{9} + 15 x^{8} + 2 x^{7} - 30 x^{6} + 4 x^{5} + 60 x^{4} - 16 x^{3} - 80 x^{2} + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 5 x^{10} - 2 x^{9} + 15 x^{8} + 2 x^{7} - 30 x^{6} + 4 x^{5} + 60 x^{4} - 16 x^{3} - 80 x^{2} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{11} - 5 \nu^{9} - 2 \nu^{8} + 15 \nu^{7} + 2 \nu^{6} - 30 \nu^{5} + 4 \nu^{4} + 60 \nu^{3} - 16 \nu^{2} - 80 \nu$$$$)/32$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{11} + 4 \nu^{10} + 7 \nu^{9} - 6 \nu^{8} - 13 \nu^{7} + 30 \nu^{6} - 6 \nu^{5} - 28 \nu^{4} - 4 \nu^{3} + 16 \nu^{2} - 48 \nu + 96$$$$)/32$$ $$\beta_{4}$$ $$=$$ $$($$$$3 \nu^{11} + 2 \nu^{10} - 11 \nu^{9} - 8 \nu^{8} + 21 \nu^{7} + 4 \nu^{6} - 42 \nu^{5} + 84 \nu^{3} + 24 \nu^{2} - 64 \nu - 64$$$$)/32$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{11} + 3 \nu^{10} - 3 \nu^{9} - 13 \nu^{8} + 7 \nu^{7} + 23 \nu^{6} - 26 \nu^{5} - 38 \nu^{4} + 60 \nu^{3} + 68 \nu^{2} - 56 \nu - 64$$$$)/16$$ $$\beta_{6}$$ $$=$$ $$($$$$-5 \nu^{11} + 4 \nu^{10} + 13 \nu^{9} - 10 \nu^{8} - 23 \nu^{7} + 42 \nu^{6} + 10 \nu^{5} - 68 \nu^{4} - 20 \nu^{3} + 48 \nu^{2} - 32 \nu + 64$$$$)/32$$ $$\beta_{7}$$ $$=$$ $$($$$$-5 \nu^{11} + 2 \nu^{10} + 17 \nu^{9} - 8 \nu^{8} - 39 \nu^{7} + 44 \nu^{6} + 50 \nu^{5} - 88 \nu^{4} - 68 \nu^{3} + 120 \nu^{2} + 16 \nu - 32$$$$)/32$$ $$\beta_{8}$$ $$=$$ $$($$$$-\nu^{11} + \nu^{10} + 5 \nu^{9} - 3 \nu^{8} - 13 \nu^{7} + 13 \nu^{6} + 20 \nu^{5} - 34 \nu^{4} - 28 \nu^{3} + 52 \nu^{2} + 24 \nu - 32$$$$)/8$$ $$\beta_{9}$$ $$=$$ $$($$$$-\nu^{11} + 3 \nu^{10} + 5 \nu^{9} - 13 \nu^{8} - 13 \nu^{7} + 35 \nu^{6} + 12 \nu^{5} - 70 \nu^{4} + 8 \nu^{3} + 108 \nu^{2} - 16 \nu - 80$$$$)/16$$ $$\beta_{10}$$ $$=$$ $$($$$$-3 \nu^{11} + 15 \nu^{9} - 2 \nu^{8} - 37 \nu^{7} + 18 \nu^{6} + 66 \nu^{5} - 68 \nu^{4} - 92 \nu^{3} + 96 \nu^{2} + 96 \nu - 64$$$$)/16$$ $$\beta_{11}$$ $$=$$ $$($$$$-\nu^{11} + 4 \nu^{10} + 9 \nu^{9} - 18 \nu^{8} - 27 \nu^{7} + 50 \nu^{6} + 34 \nu^{5} - 116 \nu^{4} - 20 \nu^{3} + 192 \nu^{2} + 16 \nu - 160$$$$)/16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{11} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + 1$$ $$\nu^{3}$$ $$=$$ $$-\beta_{11} + \beta_{10} + \beta_{9} - \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-\beta_{11} + \beta_{9} + \beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 1$$ $$\nu^{5}$$ $$=$$ $$-2 \beta_{11} + 4 \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{1} + 1$$ $$\nu^{6}$$ $$=$$ $$2 \beta_{11} - 2 \beta_{10} + \beta_{9} - \beta_{8} - \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + \beta_{3} + \beta_{2} + 4 \beta_{1} - 2$$ $$\nu^{7}$$ $$=$$ $$-5 \beta_{11} - 3 \beta_{10} + 5 \beta_{9} + 7 \beta_{8} - \beta_{7} - 2 \beta_{6} + 3 \beta_{5} - 7 \beta_{4} - 3 \beta_{3} + \beta_{2} + 1$$ $$\nu^{8}$$ $$=$$ $$-3 \beta_{11} + 7 \beta_{8} + \beta_{7} - 5 \beta_{6} - \beta_{5} + 2 \beta_{4} + 4 \beta_{3} + 10 \beta_{2} + 8 \beta_{1} - 3$$ $$\nu^{9}$$ $$=$$ $$-7 \beta_{11} + 3 \beta_{10} + 3 \beta_{9} + 6 \beta_{8} - 4 \beta_{7} - 11 \beta_{6} + 12 \beta_{5} - 17 \beta_{4} + 7 \beta_{3} + 7 \beta_{2} + \beta_{1} - 8$$ $$\nu^{10}$$ $$=$$ $$-17 \beta_{11} + 4 \beta_{10} + 13 \beta_{9} + 18 \beta_{8} - 9 \beta_{7} - 8 \beta_{6} + 3 \beta_{5} + 7 \beta_{4} + 9 \beta_{3} + 3 \beta_{2} - 6 \beta_{1} - 11$$ $$\nu^{11}$$ $$=$$ $$18 \beta_{11} + 4 \beta_{10} - 6 \beta_{9} - 13 \beta_{8} - 21 \beta_{7} - 11 \beta_{6} - \beta_{5} - 26 \beta_{4} + 22 \beta_{3} + 14 \beta_{2} + 3 \beta_{1} - 7$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/819\mathbb{Z}\right)^\times$$.

 $$n$$ $$92$$ $$379$$ $$703$$ $$\chi(n)$$ $$1$$ $$1 - \beta_{9}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 −1.30089 + 0.554694i 1.40744 + 0.138282i −1.08105 + 0.911778i 1.34408 + 0.439820i 0.759479 − 1.19298i −1.12906 − 0.851598i −1.30089 − 0.554694i 1.40744 − 0.138282i −1.08105 − 0.911778i 1.34408 − 0.439820i 0.759479 + 1.19298i −1.12906 + 0.851598i
−1.82678 + 1.05469i 0 1.22476 2.12135i 3.60178i 0 −0.866025 0.500000i 0.948212i 0 3.79878 + 6.57967i
127.2 −1.10554 + 0.638282i 0 −0.185192 + 0.320762i 1.81487i 0 −0.866025 0.500000i 3.02595i 0 −1.15840 2.00641i
127.3 −0.713220 + 0.411778i 0 −0.660878 + 1.14467i 3.16209i 0 0.866025 + 0.500000i 2.73565i 0 −1.30208 2.25527i
127.4 0.104235 0.0601799i 0 −0.992757 + 1.71951i 1.68817i 0 0.866025 + 0.500000i 0.479696i 0 −0.101594 0.175965i
127.5 1.20027 0.692976i 0 −0.0395678 + 0.0685334i 0.518957i 0 −0.866025 0.500000i 2.88158i 0 0.359625 + 0.622889i
127.6 2.34104 1.35160i 0 2.65363 4.59623i 3.25812i 0 0.866025 + 0.500000i 8.94020i 0 4.40367 + 7.62739i
316.1 −1.82678 1.05469i 0 1.22476 + 2.12135i 3.60178i 0 −0.866025 + 0.500000i 0.948212i 0 3.79878 6.57967i
316.2 −1.10554 0.638282i 0 −0.185192 0.320762i 1.81487i 0 −0.866025 + 0.500000i 3.02595i 0 −1.15840 + 2.00641i
316.3 −0.713220 0.411778i 0 −0.660878 1.14467i 3.16209i 0 0.866025 0.500000i 2.73565i 0 −1.30208 + 2.25527i
316.4 0.104235 + 0.0601799i 0 −0.992757 1.71951i 1.68817i 0 0.866025 0.500000i 0.479696i 0 −0.101594 + 0.175965i
316.5 1.20027 + 0.692976i 0 −0.0395678 0.0685334i 0.518957i 0 −0.866025 + 0.500000i 2.88158i 0 0.359625 0.622889i
316.6 2.34104 + 1.35160i 0 2.65363 + 4.59623i 3.25812i 0 0.866025 0.500000i 8.94020i 0 4.40367 7.62739i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 316.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.ct.a 12
3.b odd 2 1 91.2.q.a 12
12.b even 2 1 1456.2.cc.c 12
13.e even 6 1 inner 819.2.ct.a 12
21.c even 2 1 637.2.q.h 12
21.g even 6 1 637.2.k.g 12
21.g even 6 1 637.2.u.i 12
21.h odd 6 1 637.2.k.h 12
21.h odd 6 1 637.2.u.h 12
39.h odd 6 1 91.2.q.a 12
39.h odd 6 1 1183.2.c.i 12
39.i odd 6 1 1183.2.c.i 12
39.k even 12 1 1183.2.a.m 6
39.k even 12 1 1183.2.a.p 6
156.r even 6 1 1456.2.cc.c 12
273.u even 6 1 637.2.q.h 12
273.x odd 6 1 637.2.k.h 12
273.y even 6 1 637.2.k.g 12
273.bp odd 6 1 637.2.u.h 12
273.br even 6 1 637.2.u.i 12
273.ca odd 12 1 8281.2.a.by 6
273.ca odd 12 1 8281.2.a.ch 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.q.a 12 3.b odd 2 1
91.2.q.a 12 39.h odd 6 1
637.2.k.g 12 21.g even 6 1
637.2.k.g 12 273.y even 6 1
637.2.k.h 12 21.h odd 6 1
637.2.k.h 12 273.x odd 6 1
637.2.q.h 12 21.c even 2 1
637.2.q.h 12 273.u even 6 1
637.2.u.h 12 21.h odd 6 1
637.2.u.h 12 273.bp odd 6 1
637.2.u.i 12 21.g even 6 1
637.2.u.i 12 273.br even 6 1
819.2.ct.a 12 1.a even 1 1 trivial
819.2.ct.a 12 13.e even 6 1 inner
1183.2.a.m 6 39.k even 12 1
1183.2.a.p 6 39.k even 12 1
1183.2.c.i 12 39.h odd 6 1
1183.2.c.i 12 39.i odd 6 1
1456.2.cc.c 12 12.b even 2 1
1456.2.cc.c 12 156.r even 6 1
8281.2.a.by 6 273.ca odd 12 1
8281.2.a.ch 6 273.ca odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(819, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 12 T + 36 T^{2} + 144 T^{3} + 112 T^{4} - 72 T^{5} - 82 T^{6} + 36 T^{7} + 52 T^{8} - 8 T^{10} + T^{12}$$
$3$ $$T^{12}$$
$5$ $$3481 + 16148 T^{2} + 13040 T^{4} + 4146 T^{6} + 600 T^{8} + 40 T^{10} + T^{12}$$
$7$ $$( 1 - T^{2} + T^{4} )^{3}$$
$11$ $$256 - 768 T - 80 T^{2} + 2544 T^{3} + 2025 T^{4} - 1494 T^{5} - 771 T^{6} + 522 T^{7} + 248 T^{8} - 114 T^{9} - 7 T^{10} + 6 T^{11} + T^{12}$$
$13$ $$4826809 - 1485172 T + 599781 T^{2} - 70304 T^{3} - 23998 T^{4} + 12012 T^{5} - 6587 T^{6} + 924 T^{7} - 142 T^{8} - 32 T^{9} + 21 T^{10} - 4 T^{11} + T^{12}$$
$17$ $$241081 + 109984 T + 132173 T^{2} + 21512 T^{3} + 31018 T^{4} + 2576 T^{5} + 5229 T^{6} - 148 T^{7} + 514 T^{8} - 36 T^{9} + 37 T^{10} - 4 T^{11} + T^{12}$$
$19$ $$55696 - 138768 T + 137196 T^{2} - 54684 T^{3} - 4499 T^{4} + 9486 T^{5} + 299 T^{6} - 2370 T^{7} + 748 T^{8} - 29 T^{10} + T^{12}$$
$23$ $$38539264 - 9138176 T + 12198912 T^{2} - 5170176 T^{3} + 3630592 T^{4} - 1115904 T^{5} + 332096 T^{6} - 52416 T^{7} + 9312 T^{8} - 976 T^{9} + 164 T^{10} - 12 T^{11} + T^{12}$$
$29$ $$10042561 + 1064784 T + 4587524 T^{2} + 3112876 T^{3} + 2323356 T^{4} + 854112 T^{5} + 261878 T^{6} + 47832 T^{7} + 7876 T^{8} + 780 T^{9} + 108 T^{10} + 8 T^{11} + T^{12}$$
$31$ $$913936 + 1285560 T^{2} + 568225 T^{4} + 96896 T^{6} + 5854 T^{8} + 136 T^{10} + T^{12}$$
$37$ $$1755945216 - 633588480 T - 260200512 T^{2} + 121383360 T^{3} + 65323584 T^{4} + 8644320 T^{5} - 396036 T^{6} - 159840 T^{7} + 8109 T^{8} + 5670 T^{9} + 723 T^{10} + 42 T^{11} + T^{12}$$
$41$ $$884705536 + 421175040 T - 8238656 T^{2} - 35739840 T^{3} - 872384 T^{4} + 2742240 T^{5} + 452252 T^{6} - 35760 T^{7} - 11651 T^{8} + 450 T^{9} + 315 T^{10} + 30 T^{11} + T^{12}$$
$43$ $$2408704 + 5860352 T + 10500784 T^{2} + 8862336 T^{3} + 5690569 T^{4} + 1037954 T^{5} + 282645 T^{6} + 3650 T^{7} + 9640 T^{8} + 38 T^{9} + 113 T^{10} - 2 T^{11} + T^{12}$$
$47$ $$9461776 + 47561752 T^{2} + 10113609 T^{4} + 722232 T^{6} + 21782 T^{8} + 272 T^{10} + T^{12}$$
$53$ $$( -2339 - 7302 T - 3353 T^{2} + 700 T^{3} + 91 T^{4} - 22 T^{5} + T^{6} )^{2}$$
$59$ $$4571923456 + 3919023360 T + 1190986848 T^{2} + 61031880 T^{3} - 34227911 T^{4} - 3548430 T^{5} + 1035110 T^{6} + 185580 T^{7} - 2237 T^{8} - 1980 T^{9} - 2 T^{10} + 18 T^{11} + T^{12}$$
$61$ $$5607424 - 3788800 T + 7030784 T^{2} - 3685376 T^{3} + 6036160 T^{4} - 2984960 T^{5} + 1823136 T^{6} - 177656 T^{7} + 29281 T^{8} - 1614 T^{9} + 283 T^{10} - 14 T^{11} + T^{12}$$
$67$ $$613651984 - 1248211536 T + 790406444 T^{2} + 113725716 T^{3} - 37845559 T^{4} - 4849356 T^{5} + 1364502 T^{6} + 185988 T^{7} - 11949 T^{8} - 2352 T^{9} + 94 T^{10} + 24 T^{11} + T^{12}$$
$71$ $$46895104 - 26296320 T - 3083264 T^{2} + 4485120 T^{3} + 367104 T^{4} - 620544 T^{5} + 82752 T^{6} + 17280 T^{7} - 3808 T^{8} - 480 T^{9} + 212 T^{10} - 24 T^{11} + T^{12}$$
$73$ $$1386221824 + 513361280 T^{2} + 55965104 T^{4} + 2238456 T^{6} + 40473 T^{8} + 334 T^{10} + T^{12}$$
$79$ $$( -512 + 1664 T - 1584 T^{2} + 192 T^{3} + 212 T^{4} + 28 T^{5} + T^{6} )^{2}$$
$83$ $$141324544 + 454322976 T^{2} + 48190849 T^{4} + 1905008 T^{6} + 35086 T^{8} + 304 T^{10} + T^{12}$$
$89$ $$1834580224 - 1271596416 T - 422744080 T^{2} + 496650552 T^{3} + 291555473 T^{4} + 41341974 T^{5} - 2811261 T^{6} - 792798 T^{7} + 66288 T^{8} + 3660 T^{9} - 257 T^{10} - 12 T^{11} + T^{12}$$
$97$ $$53465344 + 190755456 T + 211235504 T^{2} - 55750056 T^{3} - 14421439 T^{4} + 4289964 T^{5} + 938031 T^{6} - 349092 T^{7} + 28032 T^{8} + 1110 T^{9} - 173 T^{10} - 6 T^{11} + T^{12}$$