# Properties

 Label 1143.2.a.g Level $1143$ Weight $2$ Character orbit 1143.a Self dual yes Analytic conductor $9.127$ Analytic rank $1$ Dimension $5$ 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: $$1$$ Dimension: $$5$$ Coefficient field: 5.5.246832.1 Defining polynomial: $$x^{5} - 2 x^{4} - 5 x^{3} + 6 x^{2} + 7 x - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 381) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2 x^{4} - 5 x^{3} + 6 x^{2} + 7 x - 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 2 \nu^{2} - 3 \nu + 3$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - 3 \nu^{3} - 2 \nu^{2} + 7 \nu + 1$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 2 \beta_{2} + 5 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + 3 \beta_{3} + 8 \beta_{2} + 10 \beta_{1} + 6$$

## 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
 2.71457 −1.51908 −1.15351 1.71250 0.245526
−2.65432 0 5.04540 −0.121872 0 −0.0602522 −8.08346 0 0.323487
1.2 −1.82669 0 1.33679 0.563416 0 3.34577 1.21147 0 −1.02918
1.3 −0.484093 0 −1.76565 −2.26452 0 1.63760 1.82293 0 1.09624
1.4 0.779856 0 −1.39182 2.98063 0 −2.49235 −2.64513 0 2.32446
1.5 2.18524 0 2.77529 −2.15766 0 −2.43077 1.69419 0 −4.71500
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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.g 5
3.b odd 2 1 381.2.a.d 5
12.b even 2 1 6096.2.a.bf 5
15.d odd 2 1 9525.2.a.j 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
381.2.a.d 5 3.b odd 2 1
1143.2.a.g 5 1.a even 1 1 trivial
6096.2.a.bf 5 12.b even 2 1
9525.2.a.j 5 15.d odd 2 1

## Hecke kernels

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

 $$T_{2}^{5} + 2 T_{2}^{4} - 6 T_{2}^{3} - 10 T_{2}^{2} + 5 T_{2} + 4$$ $$T_{5}^{5} + T_{5}^{4} - 9 T_{5}^{3} - 11 T_{5}^{2} + 7 T_{5} + 1$$ $$T_{7}^{5} - 13 T_{7}^{3} - 4 T_{7}^{2} + 33 T_{7} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 5 T - 10 T^{2} - 6 T^{3} + 2 T^{4} + T^{5}$$
$3$ $$T^{5}$$
$5$ $$1 + 7 T - 11 T^{2} - 9 T^{3} + T^{4} + T^{5}$$
$7$ $$2 + 33 T - 4 T^{2} - 13 T^{3} + T^{5}$$
$11$ $$-8 + 33 T + 96 T^{2} + 63 T^{3} + 14 T^{4} + T^{5}$$
$13$ $$19 - 9 T - 33 T^{2} - 3 T^{3} + 5 T^{4} + T^{5}$$
$17$ $$-64 + 304 T - 64 T^{2} - 32 T^{3} + 4 T^{4} + T^{5}$$
$19$ $$-256 + 192 T + 64 T^{2} - 40 T^{3} - 4 T^{4} + T^{5}$$
$23$ $$-592 - 448 T + 4 T^{2} + 64 T^{3} + 15 T^{4} + T^{5}$$
$29$ $$2279 + 29 T - 347 T^{2} - 31 T^{3} + 9 T^{4} + T^{5}$$
$31$ $$1840 + 1424 T - 4 T^{2} - 100 T^{3} - 3 T^{4} + T^{5}$$
$37$ $$907 - 669 T - 653 T^{2} - 91 T^{3} + 5 T^{4} + T^{5}$$
$41$ $$2368 + 1424 T - 320 T^{2} - 116 T^{3} + 4 T^{4} + T^{5}$$
$43$ $$-11272 + 1661 T + 746 T^{2} - 89 T^{3} - 10 T^{4} + T^{5}$$
$47$ $$-152 + 581 T + 114 T^{2} - 49 T^{3} - 4 T^{4} + T^{5}$$
$53$ $$-211 - 475 T - 305 T^{2} - 55 T^{3} + 3 T^{4} + T^{5}$$
$59$ $$1520 - 2496 T - 284 T^{2} + 128 T^{3} + 23 T^{4} + T^{5}$$
$61$ $$119213 + 7395 T - 2891 T^{2} - 191 T^{3} + 15 T^{4} + T^{5}$$
$67$ $$9836 - 18595 T + 3982 T^{2} - 137 T^{3} - 18 T^{4} + T^{5}$$
$71$ $$106 - 135 T - 106 T^{2} + 19 T^{3} + 12 T^{4} + T^{5}$$
$73$ $$1369 + 9735 T + 4317 T^{2} + 665 T^{3} + 43 T^{4} + T^{5}$$
$79$ $$-256 - 464 T + 64 T^{2} + 64 T^{3} - 16 T^{4} + T^{5}$$
$83$ $$-13120 - 10976 T - 2744 T^{2} - 176 T^{3} + 11 T^{4} + T^{5}$$
$89$ $$-13159 + 7699 T - 671 T^{2} - 153 T^{3} + 9 T^{4} + T^{5}$$
$97$ $$-5120 - 6512 T - 1280 T^{2} + 28 T^{3} + 20 T^{4} + T^{5}$$