# Properties

 Label 1386.2.a.r Level $1386$ Weight $2$ Character orbit 1386.a Self dual yes Analytic conductor $11.067$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1386.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$11.0672657201$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.1304.1 Defining polynomial: $$x^{3} - 11 x - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 11 x - 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{2} + 3 \nu - 8$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{2} - \nu - 8$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$3 \beta_{2} + \beta_{1} + 16$$$$)/2$$

## 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
 −0.182370 3.40405 −3.22168
1.00000 0 1.00000 −2.89219 0 1.00000 1.00000 0 −2.89219
1.2 1.00000 0 1.00000 1.09174 0 1.00000 1.00000 0 1.09174
1.3 1.00000 0 1.00000 3.80044 0 1.00000 1.00000 0 3.80044
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

## 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 1386.2.a.r yes 3
3.b odd 2 1 1386.2.a.q 3
7.b odd 2 1 9702.2.a.dy 3
21.c even 2 1 9702.2.a.dx 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1386.2.a.q 3 3.b odd 2 1
1386.2.a.r yes 3 1.a even 1 1 trivial
9702.2.a.dx 3 21.c even 2 1
9702.2.a.dy 3 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1386))$$:

 $$T_{5}^{3} - 2 T_{5}^{2} - 10 T_{5} + 12$$ $$T_{13}^{3} - 44 T_{13} - 16$$ $$T_{17}^{3} - 2 T_{17}^{2} - 10 T_{17} + 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{3}$$
$3$ $$T^{3}$$
$5$ $$12 - 10 T - 2 T^{2} + T^{3}$$
$7$ $$( -1 + T )^{3}$$
$11$ $$( -1 + T )^{3}$$
$13$ $$-16 - 44 T + T^{3}$$
$17$ $$12 - 10 T - 2 T^{2} + T^{3}$$
$19$ $$188 - 6 T - 10 T^{2} + T^{3}$$
$23$ $$48 - 56 T - 2 T^{2} + T^{3}$$
$29$ $$-96 - 32 T + 6 T^{2} + T^{3}$$
$31$ $$-128 - 102 T + T^{3}$$
$37$ $$152 - 12 T - 10 T^{2} + T^{3}$$
$41$ $$-12 - 10 T + 2 T^{2} + T^{3}$$
$43$ $$976 - 56 T - 14 T^{2} + T^{3}$$
$47$ $$-396 - 38 T + 10 T^{2} + T^{3}$$
$53$ $$72 + 20 T - 14 T^{2} + T^{3}$$
$59$ $$-1152 - 136 T + 8 T^{2} + T^{3}$$
$61$ $$1168 - 140 T - 8 T^{2} + T^{3}$$
$67$ $$64 - 52 T - 4 T^{2} + T^{3}$$
$71$ $$-864 - 160 T + 2 T^{2} + T^{3}$$
$73$ $$44 + 54 T + 14 T^{2} + T^{3}$$
$79$ $$144 - 36 T - 8 T^{2} + T^{3}$$
$83$ $$216 + 122 T + 20 T^{2} + T^{3}$$
$89$ $$-144 + 28 T + 16 T^{2} + T^{3}$$
$97$ $$968 - 68 T - 18 T^{2} + T^{3}$$