# Properties

 Label 1143.2.a.f Level $1143$ Weight $2$ Character orbit 1143.a Self dual yes Analytic conductor $9.127$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# 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: $$4$$ Coefficient field: $$\Q(\sqrt{3}, \sqrt{7})$$ Defining polynomial: $$x^{4} - 5 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5 x^{2} + 1$$:

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

## 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
 0.456850 −2.18890 2.18890 −0.456850
−1.73205 0 1.00000 −0.456850 0 −3.79129 1.73205 0 0.791288
1.2 −1.73205 0 1.00000 2.18890 0 0.791288 1.73205 0 −3.79129
1.3 1.73205 0 1.00000 −2.18890 0 0.791288 −1.73205 0 −3.79129
1.4 1.73205 0 1.00000 0.456850 0 −3.79129 −1.73205 0 0.791288
 $$n$$: 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1143.2.a.f 4
3.b odd 2 1 inner 1143.2.a.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1143.2.a.f 4 1.a even 1 1 trivial
1143.2.a.f 4 3.b odd 2 1 inner

## Hecke kernels

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

 $$T_{2}^{2} - 3$$ $$T_{5}^{4} - 5 T_{5}^{2} + 1$$ $$T_{7}^{2} + 3 T_{7} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -3 + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$1 - 5 T^{2} + T^{4}$$
$7$ $$( -3 + 3 T + T^{2} )^{2}$$
$11$ $$1 - 5 T^{2} + T^{4}$$
$13$ $$( 7 + 7 T + T^{2} )^{2}$$
$17$ $$16 - 20 T^{2} + T^{4}$$
$19$ $$( -20 + 2 T + T^{2} )^{2}$$
$23$ $$T^{4}$$
$29$ $$225 - 33 T^{2} + T^{4}$$
$31$ $$( -20 + 2 T + T^{2} )^{2}$$
$37$ $$( -5 + T + T^{2} )^{2}$$
$41$ $$3600 - 132 T^{2} + T^{4}$$
$43$ $$( -45 + 3 T + T^{2} )^{2}$$
$47$ $$1 - 5 T^{2} + T^{4}$$
$53$ $$225 - 33 T^{2} + T^{4}$$
$59$ $$256 - 80 T^{2} + T^{4}$$
$61$ $$( -5 + T + T^{2} )^{2}$$
$67$ $$( 7 + 7 T + T^{2} )^{2}$$
$71$ $$9 - 69 T^{2} + T^{4}$$
$73$ $$( -5 + 13 T + T^{2} )^{2}$$
$79$ $$( -20 - 2 T + T^{2} )^{2}$$
$83$ $$144 - 276 T^{2} + T^{4}$$
$89$ $$9 - 69 T^{2} + T^{4}$$
$97$ $$( 8 + T )^{4}$$