# Properties

 Label 1143.2.a.j Level $1143$ Weight $2$ Character orbit 1143.a Self dual yes Analytic conductor $9.127$ Analytic rank $0$ Dimension $9$ 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: $$9$$ Coefficient field: $$\mathbb{Q}[x]/(x^{9} - \cdots)$$ Defining polynomial: $$x^{9} - 2 x^{8} - 14 x^{7} + 26 x^{6} + 59 x^{5} - 99 x^{4} - 66 x^{3} + 102 x^{2} - 24 x + 1$$ 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 $$1,\beta_1,\ldots,\beta_{8}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 2 + \beta_{2} ) q^{4} + \beta_{5} q^{5} + ( 1 - \beta_{8} ) q^{7} + ( 2 \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 2 + \beta_{2} ) q^{4} + \beta_{5} q^{5} + ( 1 - \beta_{8} ) q^{7} + ( 2 \beta_{1} + \beta_{3} ) q^{8} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{8} ) q^{10} + ( -1 - \beta_{1} + \beta_{6} ) q^{11} + ( 2 + \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{13} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{14} + ( 4 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{16} + ( 2 + \beta_{2} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{17} + ( -\beta_{2} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{19} + ( 2 - 2 \beta_{1} + 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} ) q^{20} + ( -3 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{22} + 2 \beta_{7} q^{23} + ( 2 + 2 \beta_{4} + 2 \beta_{5} - \beta_{8} ) q^{25} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{6} + \beta_{7} - \beta_{8} ) q^{26} + ( 3 + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} - \beta_{8} ) q^{28} + ( 3 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} ) q^{29} + ( -\beta_{2} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{31} + ( 3 + 2 \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{7} + \beta_{8} ) q^{32} + ( 1 + 2 \beta_{1} + \beta_{2} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} ) q^{34} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{35} + ( 4 - \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{7} ) q^{37} + ( -1 - \beta_{2} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} ) q^{38} + ( -4 + 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} + 3 \beta_{8} ) q^{40} + ( -\beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} - \beta_{8} ) q^{41} + ( -\beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{8} ) q^{43} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} ) q^{44} + ( 2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} ) q^{46} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} ) q^{47} + ( 1 + 3 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{8} ) q^{49} + ( 1 + 2 \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{50} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - 4 \beta_{5} - 2 \beta_{7} - \beta_{8} ) q^{52} + ( 3 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{53} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{55} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{4} - 6 \beta_{5} - 3 \beta_{6} - \beta_{7} - 3 \beta_{8} ) q^{56} + ( -5 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{7} - \beta_{8} ) q^{58} + ( -2 - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{59} + ( 1 + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{61} + ( 3 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{62} + ( \beta_{1} + 2 \beta_{3} + 3 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} + \beta_{8} ) q^{64} + ( -3 + \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{65} + ( 2 - 3 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} ) q^{67} + ( 8 + \beta_{1} + 4 \beta_{2} - \beta_{3} - 2 \beta_{4} - 6 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} ) q^{68} + ( -9 - 3 \beta_{1} - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{70} + ( -5 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{71} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{73} + ( -5 + 6 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{74} + ( -4 - \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + 6 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} ) q^{76} + ( -2 \beta_{2} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 4 \beta_{8} ) q^{77} + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{7} - \beta_{8} ) q^{79} + ( 6 - 2 \beta_{1} + 2 \beta_{3} + 4 \beta_{4} + 9 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} ) q^{80} + ( 2 - 3 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} ) q^{82} + ( 4 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{5} - 2 \beta_{7} ) q^{83} + ( -4 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{5} ) q^{85} + ( -6 - 2 \beta_{1} - 4 \beta_{2} + 3 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} ) q^{86} + ( -6 - 5 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{7} ) q^{88} + ( 4 - 2 \beta_{1} + 2 \beta_{4} + \beta_{5} ) q^{89} + ( 1 + 3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} - 2 \beta_{7} - 4 \beta_{8} ) q^{91} + ( 6 + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 4 \beta_{8} ) q^{92} + ( -6 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - 4 \beta_{5} - 3 \beta_{7} ) q^{94} + ( 4 + 2 \beta_{1} + 2 \beta_{2} ) q^{95} + ( 2 \beta_{3} + 2 \beta_{5} ) q^{97} + ( 10 + \beta_{1} + 4 \beta_{2} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$9q + 2q^{2} + 14q^{4} + 4q^{5} + 10q^{7} + 6q^{8} + O(q^{10})$$ $$9q + 2q^{2} + 14q^{4} + 4q^{5} + 10q^{7} + 6q^{8} - 4q^{10} - 8q^{11} + 14q^{13} - 4q^{14} + 32q^{16} + 6q^{17} + 12q^{19} + 28q^{20} - 18q^{22} + 4q^{23} + 21q^{25} + 14q^{26} + 8q^{29} + 4q^{31} + 29q^{32} - 3q^{34} - 6q^{35} + 22q^{37} + 7q^{38} + 2q^{41} + 6q^{43} - 17q^{44} - 10q^{46} + 2q^{47} + 23q^{49} + 20q^{50} - 9q^{52} + 12q^{53} - 22q^{55} - 18q^{56} - 28q^{58} + 6q^{59} + 2q^{61} + 15q^{62} + 24q^{64} - 4q^{65} + 18q^{67} + 24q^{68} - 72q^{70} - 24q^{71} + 14q^{73} - 3q^{74} + 4q^{76} + 18q^{77} + 12q^{79} + 86q^{80} + 4q^{82} + 20q^{83} - 24q^{85} - 16q^{86} - 55q^{88} + 30q^{89} + 14q^{91} + 46q^{92} - 66q^{94} + 32q^{95} + 12q^{97} + 62q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{9} - 2 x^{8} - 14 x^{7} + 26 x^{6} + 59 x^{5} - 99 x^{4} - 66 x^{3} + 102 x^{2} - 24 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 6 \nu$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{8} + \nu^{7} + 15 \nu^{6} - 11 \nu^{5} - 70 \nu^{4} + 29 \nu^{3} + 99 \nu^{2} - 3 \nu - 7$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{8} + 3 \nu^{7} + 14 \nu^{6} - 40 \nu^{5} - 58 \nu^{4} + 157 \nu^{3} + 59 \nu^{2} - 170 \nu + 32$$$$)/6$$ $$\beta_{6}$$ $$=$$ $$($$$$-5 \nu^{8} + 9 \nu^{7} + 73 \nu^{6} - 113 \nu^{5} - 338 \nu^{4} + 401 \nu^{3} + 499 \nu^{2} - 337 \nu - 5$$$$)/12$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{8} - 2 \nu^{7} - 14 \nu^{6} + 25 \nu^{5} + 60 \nu^{4} - 89 \nu^{3} - 74 \nu^{2} + 80 \nu - 11$$$$)/2$$ $$\beta_{8}$$ $$=$$ $$($$$$3 \nu^{8} - 5 \nu^{7} - 43 \nu^{6} + 65 \nu^{5} + 190 \nu^{4} - 247 \nu^{3} - 247 \nu^{2} + 251 \nu - 27$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 6 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-\beta_{6} + \beta_{5} + \beta_{4} + 7 \beta_{2} + \beta_{1} + 24$$ $$\nu^{5}$$ $$=$$ $$\beta_{8} - \beta_{7} + \beta_{4} + 10 \beta_{3} + 38 \beta_{1} + 3$$ $$\nu^{6}$$ $$=$$ $$\beta_{8} + 3 \beta_{7} - 8 \beta_{6} + 14 \beta_{5} + 13 \beta_{4} + 2 \beta_{3} + 46 \beta_{2} + 11 \beta_{1} + 152$$ $$\nu^{7}$$ $$=$$ $$15 \beta_{8} - 13 \beta_{7} + 2 \beta_{6} + 4 \beta_{5} + 13 \beta_{4} + 82 \beta_{3} + \beta_{2} + 250 \beta_{1} + 36$$ $$\nu^{8}$$ $$=$$ $$19 \beta_{8} + 43 \beta_{7} - 48 \beta_{6} + 144 \beta_{5} + 123 \beta_{4} + 31 \beta_{3} + 300 \beta_{2} + 98 \beta_{1} + 992$$

## 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
 −2.63041 −2.14900 −1.44437 0.0532857 0.247918 0.682747 1.92789 2.55353 2.75841
−2.63041 0 4.91907 3.66812 0 1.33794 −7.67834 0 −9.64866
1.2 −2.14900 0 2.61819 −0.492551 0 −0.251160 −1.32848 0 1.05849
1.3 −1.44437 0 0.0862103 −0.766977 0 3.56571 2.76422 0 1.10780
1.4 0.0532857 0 −1.99716 3.85537 0 4.59060 −0.212992 0 0.205436
1.5 0.247918 0 −1.93854 −0.730098 0 −3.26267 −0.976436 0 −0.181005
1.6 0.682747 0 −1.53386 −3.92611 0 1.75230 −2.41273 0 −2.68054
1.7 1.92789 0 1.71676 1.15841 0 2.35752 −0.546061 0 2.23329
1.8 2.55353 0 4.52051 −2.44899 0 3.89705 6.43620 0 −6.25358
1.9 2.75841 0 5.60882 3.68284 0 −3.98729 9.95461 0 10.1588
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.9 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.j 9
3.b odd 2 1 381.2.a.e 9
12.b even 2 1 6096.2.a.bk 9
15.d odd 2 1 9525.2.a.p 9

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
381.2.a.e 9 3.b odd 2 1
1143.2.a.j 9 1.a even 1 1 trivial
6096.2.a.bk 9 12.b even 2 1
9525.2.a.p 9 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1143))$$:

 $$T_{2}^{9} - \cdots$$ $$T_{5}^{9} - \cdots$$ $$T_{7}^{9} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 24 T + 102 T^{2} - 66 T^{3} - 99 T^{4} + 59 T^{5} + 26 T^{6} - 14 T^{7} - 2 T^{8} + T^{9}$$
$3$ $$T^{9}$$
$5$ $$160 + 592 T + 384 T^{2} - 612 T^{3} - 524 T^{4} + 185 T^{5} + 94 T^{6} - 25 T^{7} - 4 T^{8} + T^{9}$$
$7$ $$1152 + 2352 T - 7544 T^{2} + 5304 T^{3} - 532 T^{4} - 707 T^{5} + 222 T^{6} + 7 T^{7} - 10 T^{8} + T^{9}$$
$11$ $$5644 - 3671 T - 7234 T^{2} + 1474 T^{3} + 2258 T^{4} - 29 T^{5} - 238 T^{6} - 18 T^{7} + 8 T^{8} + T^{9}$$
$13$ $$418 - 4575 T - 5728 T^{2} + 10198 T^{3} - 894 T^{4} - 1521 T^{5} + 344 T^{6} + 30 T^{7} - 14 T^{8} + T^{9}$$
$17$ $$-4768 + 4912 T + 4960 T^{2} - 3248 T^{3} - 1630 T^{4} + 601 T^{5} + 182 T^{6} - 43 T^{7} - 6 T^{8} + T^{9}$$
$19$ $$2816 - 6464 T - 3072 T^{2} + 7416 T^{3} - 572 T^{4} - 1499 T^{5} + 420 T^{6} + 5 T^{7} - 12 T^{8} + T^{9}$$
$23$ $$169984 + 365824 T + 125824 T^{2} - 99520 T^{3} - 15712 T^{4} + 6784 T^{5} + 472 T^{6} - 148 T^{7} - 4 T^{8} + T^{9}$$
$29$ $$-288 - 10032 T + 43448 T^{2} - 28620 T^{3} - 7370 T^{4} + 3485 T^{5} + 530 T^{6} - 97 T^{7} - 8 T^{8} + T^{9}$$
$31$ $$123392 + 262128 T + 106624 T^{2} - 53604 T^{3} - 16152 T^{4} + 4589 T^{5} + 504 T^{6} - 127 T^{7} - 4 T^{8} + T^{9}$$
$37$ $$136082 + 399893 T - 556200 T^{2} + 183010 T^{3} + 8018 T^{4} - 12845 T^{5} + 1648 T^{6} + 62 T^{7} - 22 T^{8} + T^{9}$$
$41$ $$30880 + 200624 T - 11768 T^{2} - 92500 T^{3} + 1890 T^{4} + 8613 T^{5} + 154 T^{6} - 175 T^{7} - 2 T^{8} + T^{9}$$
$43$ $$292288 - 375536 T + 74552 T^{2} + 72736 T^{3} - 32672 T^{4} + 1189 T^{5} + 1222 T^{6} - 145 T^{7} - 6 T^{8} + T^{9}$$
$47$ $$-1356304 + 1092589 T + 570452 T^{2} - 407462 T^{3} - 36662 T^{4} + 18583 T^{5} + 614 T^{6} - 262 T^{7} - 2 T^{8} + T^{9}$$
$53$ $$-633760 - 101824 T + 782712 T^{2} - 31448 T^{3} - 82170 T^{4} + 5497 T^{5} + 2258 T^{6} - 181 T^{7} - 12 T^{8} + T^{9}$$
$59$ $$-3599360 - 6511872 T - 3304704 T^{2} - 502976 T^{3} + 57632 T^{4} + 20320 T^{5} + 264 T^{6} - 236 T^{7} - 6 T^{8} + T^{9}$$
$61$ $$-6116062 + 19037393 T + 1699844 T^{2} - 1587054 T^{3} - 72782 T^{4} + 40731 T^{5} + 696 T^{6} - 358 T^{7} - 2 T^{8} + T^{9}$$
$67$ $$-1696960 + 5063952 T - 4424696 T^{2} + 1573888 T^{3} - 199976 T^{4} - 10699 T^{5} + 4126 T^{6} - 145 T^{7} - 18 T^{8} + T^{9}$$
$71$ $$165399656 - 20296923 T - 13163138 T^{2} + 898990 T^{3} + 405162 T^{4} - 6769 T^{5} - 5266 T^{6} - 98 T^{7} + 24 T^{8} + T^{9}$$
$73$ $$271190 + 103241 T - 808152 T^{2} + 612086 T^{3} - 139862 T^{4} + 55 T^{5} + 2996 T^{6} - 178 T^{7} - 14 T^{8} + T^{9}$$
$79$ $$-104192 - 581712 T + 759392 T^{2} - 87808 T^{3} - 83000 T^{4} + 8349 T^{5} + 1996 T^{6} - 175 T^{7} - 12 T^{8} + T^{9}$$
$83$ $$-226643968 - 14961664 T + 18408448 T^{2} + 404864 T^{3} - 519104 T^{4} + 8336 T^{5} + 5568 T^{6} - 208 T^{7} - 20 T^{8} + T^{9}$$
$89$ $$825248 - 57840 T - 479512 T^{2} + 124388 T^{3} + 45818 T^{4} - 19323 T^{5} + 1348 T^{6} + 207 T^{7} - 30 T^{8} + T^{9}$$
$97$ $$-13712896 + 197632 T + 3520896 T^{2} - 180480 T^{3} - 207712 T^{4} + 14896 T^{5} + 3616 T^{6} - 300 T^{7} - 12 T^{8} + T^{9}$$