# Properties

 Label 1143.2.a.e Level $1143$ Weight $2$ Character orbit 1143.a Self dual yes Analytic conductor $9.127$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1143 = 3^{2} \cdot 127$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1143.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.12690095103$$ Analytic rank: $$0$$ 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 127) 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.

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

## 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
−0.879385 0 −1.22668 2.34730 0 −1.34730 2.83750 0 −2.06418
1.2 1.34730 0 −0.184793 3.53209 0 −2.53209 −2.94356 0 4.75877
1.3 2.53209 0 4.41147 0.120615 0 0.879385 6.10607 0 0.305407
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$127$$ $$1$$

## 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 1143.2.a.e 3
3.b odd 2 1 127.2.a.a 3
12.b even 2 1 2032.2.a.k 3
15.d odd 2 1 3175.2.a.h 3
21.c even 2 1 6223.2.a.e 3
24.f even 2 1 8128.2.a.w 3
24.h odd 2 1 8128.2.a.bd 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
127.2.a.a 3 3.b odd 2 1
1143.2.a.e 3 1.a even 1 1 trivial
2032.2.a.k 3 12.b even 2 1
3175.2.a.h 3 15.d odd 2 1
6223.2.a.e 3 21.c even 2 1
8128.2.a.w 3 24.f even 2 1
8128.2.a.bd 3 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1143))$$:

 $$T_{2}^{3} - 3 T_{2}^{2} + 3$$ $$T_{5}^{3} - 6 T_{5}^{2} + 9 T_{5} - 1$$ $$T_{7}^{3} + 3 T_{7}^{2} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$3 - 3 T^{2} + T^{3}$$
$3$ $$T^{3}$$
$5$ $$-1 + 9 T - 6 T^{2} + T^{3}$$
$7$ $$-3 + 3 T^{2} + T^{3}$$
$11$ $$37 - 21 T + T^{3}$$
$13$ $$-37 - 18 T + 3 T^{2} + T^{3}$$
$17$ $$-199 + 105 T - 18 T^{2} + T^{3}$$
$19$ $$1 - 3 T^{2} + T^{3}$$
$23$ $$9 + 18 T - 9 T^{2} + T^{3}$$
$29$ $$-3 - 18 T + 3 T^{2} + T^{3}$$
$31$ $$-17 + 27 T - 12 T^{2} + T^{3}$$
$37$ $$296 - 84 T + T^{3}$$
$41$ $$192 - 12 T^{2} + T^{3}$$
$43$ $$-513 - 81 T + 9 T^{2} + T^{3}$$
$47$ $$379 - 81 T - 3 T^{2} + T^{3}$$
$53$ $$-57 - 126 T + 3 T^{2} + T^{3}$$
$59$ $$-37 - 21 T + T^{3}$$
$61$ $$-307 - 153 T + 3 T^{2} + T^{3}$$
$67$ $$-1 + 3 T^{2} + T^{3}$$
$71$ $$-867 - 153 T + 3 T^{2} + T^{3}$$
$73$ $$269 - 114 T - 3 T^{2} + T^{3}$$
$79$ $$71 - 120 T - 9 T^{2} + T^{3}$$
$83$ $$-2649 - 225 T + 12 T^{2} + T^{3}$$
$89$ $$-597 + 306 T - 33 T^{2} + T^{3}$$
$97$ $$-37 - 6 T + 15 T^{2} + T^{3}$$