# Properties

 Label 1143.2.a.h Level $1143$ Weight $2$ Character orbit 1143.a Self dual yes Analytic conductor $9.127$ Analytic rank $0$ 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: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.81509.1 Defining polynomial: $$x^{5} - x^{4} - 5 x^{3} + 3 x^{2} + 5 x - 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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.

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

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

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

## 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
 −1.71377 −1.14113 0.370865 1.21568 2.26835
−1.71377 0 0.936991 4.13449 0 0.967471 1.82175 0 −7.08555
1.2 −1.14113 0 −0.697826 −0.866045 0 −4.61870 3.07857 0 0.988269
1.3 0.370865 0 −1.86246 0.926151 0 4.31895 −1.43245 0 0.343477
1.4 1.21568 0 −0.522133 −1.83044 0 −2.18527 −3.06610 0 −2.22522
1.5 2.26835 0 3.14543 2.63584 0 1.51754 2.59823 0 5.97902
 $$n$$: 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.h 5
3.b odd 2 1 381.2.a.c 5
12.b even 2 1 6096.2.a.be 5
15.d odd 2 1 9525.2.a.k 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
381.2.a.c 5 3.b odd 2 1
1143.2.a.h 5 1.a even 1 1 trivial
6096.2.a.be 5 12.b even 2 1
9525.2.a.k 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1143))$$:

 $$T_{2}^{5} - T_{2}^{4} - 5 T_{2}^{3} + 3 T_{2}^{2} + 5 T_{2} - 2$$ $$T_{5}^{5} - 5 T_{5}^{4} - 2 T_{5}^{3} + 24 T_{5}^{2} - 16$$ $$T_{7}^{5} - 24 T_{7}^{3} + 8 T_{7}^{2} + 80 T_{7} - 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 + 5 T + 3 T^{2} - 5 T^{3} - T^{4} + T^{5}$$
$3$ $$T^{5}$$
$5$ $$-16 + 24 T^{2} - 2 T^{3} - 5 T^{4} + T^{5}$$
$7$ $$-64 + 80 T + 8 T^{2} - 24 T^{3} + T^{5}$$
$11$ $$-2 + 193 T - 220 T^{2} + 91 T^{3} - 16 T^{4} + T^{5}$$
$13$ $$-1 - 41 T + 47 T^{2} - 11 T^{3} - 3 T^{4} + T^{5}$$
$17$ $$-162 + 9 T + 62 T^{2} - 7 T^{3} - 6 T^{4} + T^{5}$$
$19$ $$1504 + 173 T - 248 T^{2} - 27 T^{3} + 8 T^{4} + T^{5}$$
$23$ $$-16 - 24 T + 40 T^{2} - 9 T^{4} + T^{5}$$
$29$ $$-1936 - 880 T + 236 T^{2} + 60 T^{3} - 17 T^{4} + T^{5}$$
$31$ $$4261 + 129 T - 519 T^{2} - 55 T^{3} + 9 T^{4} + T^{5}$$
$37$ $$-17 - T + 191 T^{2} - 67 T^{3} + T^{4} + T^{5}$$
$41$ $$-6506 + 3185 T + 530 T^{2} - 151 T^{3} - 2 T^{4} + T^{5}$$
$43$ $$-256 + 112 T + 120 T^{2} - 64 T^{3} + 4 T^{4} + T^{5}$$
$47$ $$-974 - 895 T + 1060 T^{2} - 121 T^{3} - 8 T^{4} + T^{5}$$
$53$ $$8208 - 5832 T + 1248 T^{2} - 32 T^{3} - 15 T^{4} + T^{5}$$
$59$ $$-2224 - 1120 T + 392 T^{2} + 62 T^{3} - 19 T^{4} + T^{5}$$
$61$ $$317 + 383 T - 71 T^{2} - 63 T^{3} - T^{4} + T^{5}$$
$67$ $$-3424 + 2720 T + 328 T^{2} - 168 T^{3} + 2 T^{4} + T^{5}$$
$71$ $$45502 + 9981 T - 652 T^{2} - 213 T^{3} + T^{5}$$
$73$ $$51089 + 21287 T + 1409 T^{2} - 235 T^{3} - 13 T^{4} + T^{5}$$
$79$ $$10008 - 4755 T - 524 T^{2} + 185 T^{3} + 28 T^{4} + T^{5}$$
$83$ $$-6416 + 4016 T + 136 T^{2} - 240 T^{3} - T^{4} + T^{5}$$
$89$ $$11344 + 9296 T - 12 T^{2} - 268 T^{3} + T^{4} + T^{5}$$
$97$ $$-321696 - 7008 T + 6064 T^{2} - 80 T^{3} - 28 T^{4} + T^{5}$$