# Properties

 Label 1386.2.a.l Level $1386$ Weight $2$ Character orbit 1386.a Self dual yes Analytic conductor $11.067$ Analytic rank $0$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 154) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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
 0
1.00000 0 1.00000 4.00000 0 −1.00000 1.00000 0 4.00000
 $$n$$: 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.l 1
3.b odd 2 1 154.2.a.a 1
7.b odd 2 1 9702.2.a.ba 1
12.b even 2 1 1232.2.a.e 1
15.d odd 2 1 3850.2.a.u 1
15.e even 4 2 3850.2.c.j 2
21.c even 2 1 1078.2.a.d 1
21.g even 6 2 1078.2.e.i 2
21.h odd 6 2 1078.2.e.j 2
24.f even 2 1 4928.2.a.w 1
24.h odd 2 1 4928.2.a.v 1
33.d even 2 1 1694.2.a.g 1
84.h odd 2 1 8624.2.a.r 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.a 1 3.b odd 2 1
1078.2.a.d 1 21.c even 2 1
1078.2.e.i 2 21.g even 6 2
1078.2.e.j 2 21.h odd 6 2
1232.2.a.e 1 12.b even 2 1
1386.2.a.l 1 1.a even 1 1 trivial
1694.2.a.g 1 33.d even 2 1
3850.2.a.u 1 15.d odd 2 1
3850.2.c.j 2 15.e even 4 2
4928.2.a.v 1 24.h odd 2 1
4928.2.a.w 1 24.f even 2 1
8624.2.a.r 1 84.h odd 2 1
9702.2.a.ba 1 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1386))$$:

 $$T_{5} - 4$$ $$T_{13} - 2$$ $$T_{17} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$T$$
$5$ $$-4 + T$$
$7$ $$1 + T$$
$11$ $$-1 + T$$
$13$ $$-2 + T$$
$17$ $$-4 + T$$
$19$ $$6 + T$$
$23$ $$4 + T$$
$29$ $$-2 + T$$
$31$ $$2 + T$$
$37$ $$-10 + T$$
$41$ $$4 + T$$
$43$ $$8 + T$$
$47$ $$2 + T$$
$53$ $$6 + T$$
$59$ $$-12 + T$$
$61$ $$14 + T$$
$67$ $$12 + T$$
$71$ $$-8 + T$$
$73$ $$-4 + T$$
$79$ $$T$$
$83$ $$-6 + T$$
$89$ $$-6 + T$$
$97$ $$14 + T$$