# Properties

 Label 1386.2.a.e 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 462) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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.e 1
3.b odd 2 1 462.2.a.e 1
7.b odd 2 1 9702.2.a.b 1
12.b even 2 1 3696.2.a.p 1
21.c even 2 1 3234.2.a.v 1
33.d even 2 1 5082.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.a.e 1 3.b odd 2 1
1386.2.a.e 1 1.a even 1 1 trivial
3234.2.a.v 1 21.c even 2 1
3696.2.a.p 1 12.b even 2 1
5082.2.a.a 1 33.d even 2 1
9702.2.a.b 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1386))$$:

 $$T_{5} - 4$$ $$T_{13} + 6$$ $$T_{17} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$T$$
$5$ $$-4 + T$$
$7$ $$-1 + T$$
$11$ $$-1 + T$$
$13$ $$6 + T$$
$17$ $$-4 + T$$
$19$ $$2 + T$$
$23$ $$-8 + T$$
$29$ $$-6 + T$$
$31$ $$-6 + T$$
$37$ $$6 + T$$
$41$ $$12 + T$$
$43$ $$-4 + T$$
$47$ $$6 + T$$
$53$ $$2 + T$$
$59$ $$T$$
$61$ $$-10 + T$$
$67$ $$-4 + T$$
$71$ $$-12 + T$$
$73$ $$T$$
$79$ $$16 + T$$
$83$ $$-14 + T$$
$89$ $$-14 + T$$
$97$ $$14 + T$$