# Properties

 Label 8019.2.a.a Level 8019 Weight 2 Character orbit 8019.a Self dual yes Analytic conductor 64.032 Analytic rank 1 Dimension 3 CM no Inner twists 1

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$8019 = 3^{6} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8019.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.0320373809$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{18})^+$$ Defining polynomial: $$x^{3} - 3 x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 297) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} + ( -2 + \beta^{2} ) q^{4} + ( 3 + \beta - 2 \beta^{2} ) q^{5} + ( -3 + \beta^{2} ) q^{7} + ( -1 + \beta ) q^{8} +O(q^{10})$$ $$q -\beta q^{2} + ( -2 + \beta^{2} ) q^{4} + ( 3 + \beta - 2 \beta^{2} ) q^{5} + ( -3 + \beta^{2} ) q^{7} + ( -1 + \beta ) q^{8} + ( 2 + 3 \beta - \beta^{2} ) q^{10} - q^{11} + ( -7 - 3 \beta + 2 \beta^{2} ) q^{13} - q^{14} + ( 4 + \beta - 3 \beta^{2} ) q^{16} + ( 4 + \beta ) q^{17} + ( 6 - 2 \beta - \beta^{2} ) q^{19} + ( -5 - \beta + \beta^{2} ) q^{20} + \beta q^{22} + ( -4 + 2 \beta^{2} ) q^{23} + ( -2 \beta + \beta^{2} ) q^{25} + ( -2 + \beta + 3 \beta^{2} ) q^{26} + ( 6 + \beta - 2 \beta^{2} ) q^{28} + ( 6 - 4 \beta - 2 \beta^{2} ) q^{29} + ( -3 + 2 \beta + 2 \beta^{2} ) q^{31} + ( 5 + 3 \beta - \beta^{2} ) q^{32} + ( -4 \beta - \beta^{2} ) q^{34} + ( -8 - 2 \beta + 3 \beta^{2} ) q^{35} + ( 8 + 2 \beta - 4 \beta^{2} ) q^{37} + ( 1 - 3 \beta + 2 \beta^{2} ) q^{38} + ( -5 - 4 \beta + 3 \beta^{2} ) q^{40} + ( -5 + \beta + 3 \beta^{2} ) q^{41} + ( 7 + 6 \beta - 4 \beta^{2} ) q^{43} + ( 2 - \beta^{2} ) q^{44} + ( -2 - 2 \beta ) q^{46} + ( 7 - 2 \beta - \beta^{2} ) q^{47} + ( 2 + \beta - 3 \beta^{2} ) q^{49} + ( -1 - 3 \beta + 2 \beta^{2} ) q^{50} + ( 11 - \beta - 5 \beta^{2} ) q^{52} + ( -11 + 2 \beta^{2} ) q^{53} + ( -3 - \beta + 2 \beta^{2} ) q^{55} + ( 4 - \beta^{2} ) q^{56} + ( 2 + 4 \beta^{2} ) q^{58} + ( 7 + 2 \beta - 2 \beta^{2} ) q^{59} + ( -6 - 2 \beta + 2 \beta^{2} ) q^{61} + ( -2 - 3 \beta - 2 \beta^{2} ) q^{62} + ( -7 - 4 \beta + 3 \beta^{2} ) q^{64} + ( -13 + 4 \beta + 5 \beta^{2} ) q^{65} + ( 15 + \beta - 8 \beta^{2} ) q^{67} + ( -7 + \beta + 4 \beta^{2} ) q^{68} + ( -3 - \beta + 2 \beta^{2} ) q^{70} + ( -5 - \beta + 6 \beta^{2} ) q^{71} + ( 5 + 7 \beta - 2 \beta^{2} ) q^{73} + ( 4 + 4 \beta - 2 \beta^{2} ) q^{74} + ( -14 - 3 \beta + 5 \beta^{2} ) q^{76} + ( 3 - \beta^{2} ) q^{77} + ( 3 + \beta - 5 \beta^{2} ) q^{79} + ( 7 - 2 \beta + 2 \beta^{2} ) q^{80} + ( -3 - 4 \beta - \beta^{2} ) q^{82} + ( -17 - 4 \beta + 9 \beta^{2} ) q^{83} + ( 10 + \beta - 7 \beta^{2} ) q^{85} + ( 4 + 5 \beta - 6 \beta^{2} ) q^{86} + ( 1 - \beta ) q^{88} + ( -12 - 6 \beta + 9 \beta^{2} ) q^{89} + ( 18 + 2 \beta - 7 \beta^{2} ) q^{91} + ( 8 + 2 \beta - 2 \beta^{2} ) q^{92} + ( 1 - 4 \beta + 2 \beta^{2} ) q^{94} + ( 21 + 11 \beta - 11 \beta^{2} ) q^{95} + ( 17 - 2 \beta - 4 \beta^{2} ) q^{97} + ( 3 + 7 \beta - \beta^{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 3q^{5} - 3q^{7} - 3q^{8} + O(q^{10})$$ $$3q - 3q^{5} - 3q^{7} - 3q^{8} - 3q^{11} - 9q^{13} - 3q^{14} - 6q^{16} + 12q^{17} + 12q^{19} - 9q^{20} + 6q^{25} + 12q^{26} + 6q^{28} + 6q^{29} + 3q^{31} + 9q^{32} - 6q^{34} - 6q^{35} + 15q^{38} + 3q^{40} + 3q^{41} - 3q^{43} - 6q^{46} + 15q^{47} - 12q^{49} + 9q^{50} + 3q^{52} - 21q^{53} + 3q^{55} + 6q^{56} + 30q^{58} + 9q^{59} - 6q^{61} - 18q^{62} - 3q^{64} - 9q^{65} - 3q^{67} + 3q^{68} + 3q^{70} + 21q^{71} + 3q^{73} - 12q^{76} + 3q^{77} - 21q^{79} + 33q^{80} - 15q^{82} + 3q^{83} - 12q^{85} - 24q^{86} + 3q^{88} + 18q^{89} + 12q^{91} + 12q^{92} + 15q^{94} - 3q^{95} + 27q^{97} + 3q^{98} + O(q^{100})$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 1.87939 −0.347296 −1.53209
−1.87939 0 1.53209 −2.18479 0 0.532089 0.879385 0 4.10607
1.2 0.347296 0 −1.87939 2.41147 0 −2.87939 −1.34730 0 0.837496
1.3 1.53209 0 0.347296 −3.22668 0 −0.652704 −2.53209 0 −4.94356
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$11$$ $$1$$

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8019.2.a.a 3
3.b odd 2 1 8019.2.a.b 3
27.e even 9 2 297.2.j.a 6
27.f odd 18 2 891.2.j.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
297.2.j.a 6 27.e even 9 2
891.2.j.a 6 27.f odd 18 2
8019.2.a.a 3 1.a even 1 1 trivial
8019.2.a.b 3 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 3 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8019))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T^{2} + T^{3} + 6 T^{4} + 8 T^{6}$$
$3$ 1
$5$ $$1 + 3 T + 9 T^{2} + 13 T^{3} + 45 T^{4} + 75 T^{5} + 125 T^{6}$$
$7$ $$1 + 3 T + 21 T^{2} + 41 T^{3} + 147 T^{4} + 147 T^{5} + 343 T^{6}$$
$11$ $$( 1 + T )^{3}$$
$13$ $$1 + 9 T + 45 T^{2} + 161 T^{3} + 585 T^{4} + 1521 T^{5} + 2197 T^{6}$$
$17$ $$1 - 12 T + 96 T^{2} - 461 T^{3} + 1632 T^{4} - 3468 T^{5} + 4913 T^{6}$$
$19$ $$1 - 12 T + 84 T^{2} - 399 T^{3} + 1596 T^{4} - 4332 T^{5} + 6859 T^{6}$$
$23$ $$1 + 57 T^{2} + 8 T^{3} + 1311 T^{4} + 12167 T^{6}$$
$29$ $$1 - 6 T + 15 T^{2} + 108 T^{3} + 435 T^{4} - 5046 T^{5} + 24389 T^{6}$$
$31$ $$1 - 3 T + 60 T^{2} - 223 T^{3} + 1860 T^{4} - 2883 T^{5} + 29791 T^{6}$$
$37$ $$1 + 75 T^{2} - 72 T^{3} + 2775 T^{4} + 50653 T^{6}$$
$41$ $$1 - 3 T + 87 T^{2} - 227 T^{3} + 3567 T^{4} - 5043 T^{5} + 68921 T^{6}$$
$43$ $$1 + 3 T + 48 T^{2} + 471 T^{3} + 2064 T^{4} + 5547 T^{5} + 79507 T^{6}$$
$47$ $$1 - 15 T + 195 T^{2} - 1393 T^{3} + 9165 T^{4} - 33135 T^{5} + 103823 T^{6}$$
$53$ $$1 + 21 T + 294 T^{2} + 2493 T^{3} + 15582 T^{4} + 58989 T^{5} + 148877 T^{6}$$
$59$ $$1 - 9 T + 192 T^{2} - 1045 T^{3} + 11328 T^{4} - 31329 T^{5} + 205379 T^{6}$$
$61$ $$1 + 6 T + 183 T^{2} + 708 T^{3} + 11163 T^{4} + 22326 T^{5} + 226981 T^{6}$$
$67$ $$1 + 3 T + 33 T^{2} - 425 T^{3} + 2211 T^{4} + 13467 T^{5} + 300763 T^{6}$$
$71$ $$1 - 21 T + 267 T^{2} - 2385 T^{3} + 18957 T^{4} - 105861 T^{5} + 357911 T^{6}$$
$73$ $$1 - 3 T + 105 T^{2} - 169 T^{3} + 7665 T^{4} - 15987 T^{5} + 389017 T^{6}$$
$79$ $$1 + 21 T + 321 T^{2} + 3049 T^{3} + 25359 T^{4} + 131061 T^{5} + 493039 T^{6}$$
$83$ $$1 - 3 T + 69 T^{2} + 585 T^{3} + 5727 T^{4} - 20667 T^{5} + 571787 T^{6}$$
$89$ $$1 - 18 T + 186 T^{2} - 1827 T^{3} + 16554 T^{4} - 142578 T^{5} + 704969 T^{6}$$
$97$ $$1 - 27 T + 450 T^{2} - 5075 T^{3} + 43650 T^{4} - 254043 T^{5} + 912673 T^{6}$$
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