# Properties

 Label 1386.2.a.o Level $1386$ Weight $2$ Character orbit 1386.a Self dual yes Analytic conductor $11.067$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{10})$$ Defining polynomial: $$x^{2} - 10$$ 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 $$\beta = \sqrt{10}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −3.16228 3.16228
1.00000 0 1.00000 −3.16228 0 −1.00000 1.00000 0 −3.16228
1.2 1.00000 0 1.00000 3.16228 0 −1.00000 1.00000 0 3.16228
 $$n$$: 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.o yes 2
3.b odd 2 1 1386.2.a.n 2
7.b odd 2 1 9702.2.a.dc 2
21.c even 2 1 9702.2.a.cn 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1386.2.a.n 2 3.b odd 2 1
1386.2.a.o yes 2 1.a even 1 1 trivial
9702.2.a.cn 2 21.c even 2 1
9702.2.a.dc 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1386))$$:

 $$T_{5}^{2} - 10$$ $$T_{13} - 2$$ $$T_{17}^{2} - 8 T_{17} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$-10 + T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$( -2 + T )^{2}$$
$17$ $$6 - 8 T + T^{2}$$
$19$ $$-6 - 4 T + T^{2}$$
$23$ $$-40 + T^{2}$$
$29$ $$( -4 + T )^{2}$$
$31$ $$26 - 12 T + T^{2}$$
$37$ $$-36 - 4 T + T^{2}$$
$41$ $$6 + 8 T + T^{2}$$
$43$ $$-40 + T^{2}$$
$47$ $$-86 - 4 T + T^{2}$$
$53$ $$-36 + 4 T + T^{2}$$
$59$ $$24 + 16 T + T^{2}$$
$61$ $$( -10 + T )^{2}$$
$67$ $$-36 + 4 T + T^{2}$$
$71$ $$( 8 + T )^{2}$$
$73$ $$-74 - 8 T + T^{2}$$
$79$ $$-36 - 4 T + T^{2}$$
$83$ $$-6 + 4 T + T^{2}$$
$89$ $$60 - 20 T + T^{2}$$
$97$ $$-156 - 4 T + T^{2}$$