# Properties

 Label 1143.2.a.i Level $1143$ Weight $2$ Character orbit 1143.a Self dual yes Analytic conductor $9.127$ Analytic rank $1$ Dimension $7$ 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: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Defining polynomial: $$x^{7} - 2 x^{6} - 8 x^{5} + 15 x^{4} + 17 x^{3} - 28 x^{2} - 11 x + 15$$ 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 $$1,\beta_1,\ldots,\beta_{6}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -1 + \beta_{4} ) q^{5} + ( -1 - \beta_{3} - \beta_{5} ) q^{7} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -1 + \beta_{4} ) q^{5} + ( -1 - \beta_{3} - \beta_{5} ) q^{7} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{8} + ( -2 + 2 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{10} + ( \beta_{1} - \beta_{4} + \beta_{5} ) q^{11} + ( -\beta_{1} + \beta_{3} - 2 \beta_{6} ) q^{13} + ( \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{14} + ( -1 + \beta_{3} ) q^{16} + ( -4 + \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} ) q^{17} + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{19} + ( -2 - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{20} + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{5} ) q^{22} + ( \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{23} + ( 1 + 2 \beta_{1} - \beta_{2} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{25} + ( \beta_{1} - 3 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{26} + ( -5 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{28} + ( 1 + 2 \beta_{1} - \beta_{3} + \beta_{5} + 2 \beta_{6} ) q^{29} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{5} - \beta_{6} ) q^{31} + ( -1 + 3 \beta_{2} - 3 \beta_{3} + \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{32} + ( -2 + 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{34} + ( 1 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{35} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{37} + ( -5 + 2 \beta_{1} - 3 \beta_{2} - \beta_{4} + 2 \beta_{6} ) q^{38} + ( -1 + \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{40} + ( -1 - \beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} ) q^{41} + ( \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{43} + ( 3 + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} ) q^{44} + ( 1 - 3 \beta_{2} + \beta_{3} + 3 \beta_{5} + 3 \beta_{6} ) q^{46} + ( -3 + 3 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} - \beta_{4} + 4 \beta_{5} + \beta_{6} ) q^{47} + ( 1 - 2 \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{49} + ( 1 - 3 \beta_{1} + 4 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{50} + ( 2 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{52} + ( -3 - 4 \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} ) q^{53} + ( -3 - 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{55} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{56} + ( -2 - 2 \beta_{1} - \beta_{2} + 4 \beta_{3} + \beta_{6} ) q^{58} + ( 1 - 2 \beta_{1} + 5 \beta_{2} - 4 \beta_{3} + 5 \beta_{4} - 4 \beta_{5} - 3 \beta_{6} ) q^{59} + ( 3 \beta_{1} - 3 \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{61} + ( -1 + 2 \beta_{2} + \beta_{3} - \beta_{5} - 3 \beta_{6} ) q^{62} + ( -1 - \beta_{1} - 3 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{64} + ( -2 + 2 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} - \beta_{4} - 3 \beta_{5} + \beta_{6} ) q^{65} + ( -3 - 4 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + \beta_{5} + 4 \beta_{6} ) q^{67} + ( -9 - 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{68} + ( 9 + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} ) q^{70} + ( -3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - \beta_{5} - \beta_{6} ) q^{71} + ( 2 - 5 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} - 3 \beta_{6} ) q^{73} + ( -1 - 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + \beta_{6} ) q^{74} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} ) q^{76} + ( -4 + 4 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{77} + ( -3 - 3 \beta_{1} + 4 \beta_{2} - \beta_{3} + 3 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} ) q^{79} + ( 1 + 3 \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{80} + ( 5 + \beta_{1} - \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} ) q^{82} + ( -3 - 2 \beta_{1} - 3 \beta_{2} + \beta_{4} + 2 \beta_{5} - 3 \beta_{6} ) q^{83} + ( 3 - 4 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{85} + ( -\beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{86} + ( 4 - 3 \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{88} + ( -2 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{89} + ( -5 - 4 \beta_{2} - \beta_{4} + 4 \beta_{5} - 2 \beta_{6} ) q^{91} + ( 5 + 3 \beta_{2} + \beta_{4} - \beta_{6} ) q^{92} + ( -5 + 5 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{94} + ( 7 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{95} + ( 2 \beta_{1} - 7 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 5 \beta_{5} + \beta_{6} ) q^{97} + ( 3 + \beta_{1} - 5 \beta_{2} + \beta_{3} - 4 \beta_{4} + 5 \beta_{5} + 5 \beta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7q - 2q^{2} + 6q^{4} - 8q^{5} - 3q^{7} - 3q^{8} + O(q^{10})$$ $$7q - 2q^{2} + 6q^{4} - 8q^{5} - 3q^{7} - 3q^{8} - 5q^{10} - q^{13} + 4q^{14} - 8q^{16} - 24q^{17} - 5q^{19} - 11q^{20} - 9q^{22} + q^{23} + 7q^{25} + 4q^{26} - 26q^{28} + 7q^{29} - 8q^{31} + 2q^{32} - q^{34} - 4q^{35} - 6q^{37} - 29q^{38} - 3q^{40} - 14q^{41} - q^{43} + 21q^{44} - 3q^{46} - 25q^{47} - 10q^{50} + 6q^{52} - 29q^{53} - 23q^{55} - 9q^{56} - 22q^{58} + 12q^{59} + 7q^{61} - 4q^{62} - 3q^{64} - 3q^{65} - 25q^{67} - 53q^{68} + 51q^{70} - 7q^{71} + 13q^{73} - 11q^{74} + 12q^{76} - 19q^{77} - 23q^{79} + 14q^{80} + 26q^{82} - 26q^{83} + 15q^{85} - 5q^{86} + 25q^{88} - 13q^{89} - 40q^{91} + 32q^{92} - 19q^{94} + 40q^{95} - 5q^{97} + 11q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - 2 x^{6} - 8 x^{5} + 15 x^{4} + 17 x^{3} - 28 x^{2} - 11 x + 15$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{4} - 6 \nu^{2} + 5$$ $$\beta_{4}$$ $$=$$ $$\nu^{6} - \nu^{5} - 8 \nu^{4} + 6 \nu^{3} + 16 \nu^{2} - 5 \nu - 8$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} - 2 \nu^{4} - 7 \nu^{3} + 13 \nu^{2} + 9 \nu - 14$$ $$\beta_{6}$$ $$=$$ $$\nu^{6} - 3 \nu^{5} - 5 \nu^{4} + 19 \nu^{3} - 2 \nu^{2} - 20 \nu + 9$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$-\beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} + 3 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{3} + 6 \beta_{2} + 13$$ $$\nu^{5}$$ $$=$$ $$-7 \beta_{6} - 13 \beta_{5} + 7 \beta_{4} - 5 \beta_{3} + 13 \beta_{2} + 12 \beta_{1} + 1$$ $$\nu^{6}$$ $$=$$ $$-\beta_{6} - \beta_{5} + 2 \beta_{4} + 9 \beta_{3} + 33 \beta_{2} - \beta_{1} + 65$$

## 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
 2.41605 2.06395 1.24403 0.818322 −1.09124 −1.20613 −2.24499
−2.41605 0 3.83730 0.919173 0 −4.13370 −4.43902 0 −2.22077
1.2 −2.06395 0 2.25990 −3.73266 0 1.84347 −0.536426 0 7.70403
1.3 −1.24403 0 −0.452382 2.65960 0 −1.13710 3.05084 0 −3.30863
1.4 −0.818322 0 −1.33035 −2.74338 0 −0.135055 2.72530 0 2.24497
1.5 1.09124 0 −0.809198 −0.395790 0 3.35479 −3.06551 0 −0.431901
1.6 1.20613 0 −0.545241 −1.52027 0 1.05834 −3.06990 0 −1.83365
1.7 2.24499 0 3.03996 −3.18668 0 −3.85075 2.33471 0 −7.15405
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.7 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.i 7
3.b odd 2 1 127.2.a.b 7
12.b even 2 1 2032.2.a.p 7
15.d odd 2 1 3175.2.a.j 7
21.c even 2 1 6223.2.a.h 7
24.f even 2 1 8128.2.a.bj 7
24.h odd 2 1 8128.2.a.bi 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
127.2.a.b 7 3.b odd 2 1
1143.2.a.i 7 1.a even 1 1 trivial
2032.2.a.p 7 12.b even 2 1
3175.2.a.j 7 15.d odd 2 1
6223.2.a.h 7 21.c even 2 1
8128.2.a.bi 7 24.h odd 2 1
8128.2.a.bj 7 24.f even 2 1

## Hecke kernels

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

 $$T_{2}^{7} + 2 T_{2}^{6} - 8 T_{2}^{5} - 15 T_{2}^{4} + 17 T_{2}^{3} + 28 T_{2}^{2} - 11 T_{2} - 15$$ $$T_{5}^{7} + 8 T_{5}^{6} + 11 T_{5}^{5} - 53 T_{5}^{4} - 146 T_{5}^{3} - 32 T_{5}^{2} + 128 T_{5} + 48$$ $$T_{7}^{7} + 3 T_{7}^{6} - 20 T_{7}^{5} - 41 T_{7}^{4} + 114 T_{7}^{3} + 64 T_{7}^{2} - 112 T_{7} - 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-15 - 11 T + 28 T^{2} + 17 T^{3} - 15 T^{4} - 8 T^{5} + 2 T^{6} + T^{7}$$
$3$ $$T^{7}$$
$5$ $$48 + 128 T - 32 T^{2} - 146 T^{3} - 53 T^{4} + 11 T^{5} + 8 T^{6} + T^{7}$$
$7$ $$-16 - 112 T + 64 T^{2} + 114 T^{3} - 41 T^{4} - 20 T^{5} + 3 T^{6} + T^{7}$$
$11$ $$-3 - 5 T + 37 T^{2} + 88 T^{3} + 17 T^{4} - 28 T^{5} + T^{7}$$
$13$ $$5383 - 10416 T + 52 T^{2} + 1515 T^{3} - 38 T^{4} - 69 T^{5} + T^{6} + T^{7}$$
$17$ $$-38235 - 45913 T - 19593 T^{2} - 2678 T^{3} + 467 T^{4} + 200 T^{5} + 24 T^{6} + T^{7}$$
$19$ $$853 - 2664 T + 1582 T^{2} + 685 T^{3} - 206 T^{4} - 51 T^{5} + 5 T^{6} + T^{7}$$
$23$ $$-8016 + 12376 T - 6344 T^{2} + 812 T^{3} + 279 T^{4} - 74 T^{5} - T^{6} + T^{7}$$
$29$ $$5520 - 5368 T - 2512 T^{2} + 1612 T^{3} + 359 T^{4} - 72 T^{5} - 7 T^{6} + T^{7}$$
$31$ $$-2845 - 229 T + 3651 T^{2} + 648 T^{3} - 465 T^{4} - 68 T^{5} + 8 T^{6} + T^{7}$$
$37$ $$-920 + 16084 T + 11180 T^{2} + 981 T^{3} - 550 T^{4} - 81 T^{5} + 6 T^{6} + T^{7}$$
$41$ $$-4032 - 9296 T - 8072 T^{2} - 3199 T^{3} - 494 T^{4} + 23 T^{5} + 14 T^{6} + T^{7}$$
$43$ $$10096 + 2296 T - 6236 T^{2} + 1374 T^{3} + 287 T^{4} - 99 T^{5} + T^{6} + T^{7}$$
$47$ $$1046391 + 439559 T + 12320 T^{2} - 16340 T^{3} - 1920 T^{4} + 100 T^{5} + 25 T^{6} + T^{7}$$
$53$ $$755376 + 283688 T - 43804 T^{2} - 28158 T^{3} - 2659 T^{4} + 142 T^{5} + 29 T^{6} + T^{7}$$
$59$ $$339120 + 572048 T - 206960 T^{2} + 6446 T^{3} + 3351 T^{4} - 233 T^{5} - 12 T^{6} + T^{7}$$
$61$ $$3625 - 9711 T - 6956 T^{2} + 2454 T^{3} + 522 T^{4} - 96 T^{5} - 7 T^{6} + T^{7}$$
$67$ $$-64784 + 534864 T + 90672 T^{2} - 15628 T^{3} - 3183 T^{4} + 26 T^{5} + 25 T^{6} + T^{7}$$
$71$ $$84633 + 161143 T + 79912 T^{2} + 9756 T^{3} - 1424 T^{4} - 228 T^{5} + 7 T^{6} + T^{7}$$
$73$ $$17401 + 8644 T - 58764 T^{2} + 2483 T^{3} + 2198 T^{4} - 161 T^{5} - 13 T^{6} + T^{7}$$
$79$ $$1841711 + 916400 T + 84554 T^{2} - 19855 T^{3} - 3470 T^{4} - 7 T^{5} + 23 T^{6} + T^{7}$$
$83$ $$-16464 + 542920 T + 111104 T^{2} - 20636 T^{3} - 4299 T^{4} - 9 T^{5} + 26 T^{6} + T^{7}$$
$89$ $$-432 + 1184 T + 2296 T^{2} + 62 T^{3} - 431 T^{4} - 12 T^{5} + 13 T^{6} + T^{7}$$
$97$ $$-12656 - 172648 T + 41452 T^{2} + 14750 T^{3} - 1263 T^{4} - 280 T^{5} + 5 T^{6} + T^{7}$$