# Properties

 Label 1386.4.a.a Level $1386$ Weight $4$ Character orbit 1386.a Self dual yes Analytic conductor $81.777$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1386.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$81.7766472680$$ Analytic rank: $$1$$ 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

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 4 q^{4} - 18 q^{5} + 7 q^{7} - 8 q^{8} + O(q^{10})$$ $$q - 2 q^{2} + 4 q^{4} - 18 q^{5} + 7 q^{7} - 8 q^{8} + 36 q^{10} + 11 q^{11} + 56 q^{13} - 14 q^{14} + 16 q^{16} - 36 q^{17} - 28 q^{19} - 72 q^{20} - 22 q^{22} - 180 q^{23} + 199 q^{25} - 112 q^{26} + 28 q^{28} + 54 q^{29} - 334 q^{31} - 32 q^{32} + 72 q^{34} - 126 q^{35} + 386 q^{37} + 56 q^{38} + 144 q^{40} + 444 q^{41} - 316 q^{43} + 44 q^{44} + 360 q^{46} + 402 q^{47} + 49 q^{49} - 398 q^{50} + 224 q^{52} + 486 q^{53} - 198 q^{55} - 56 q^{56} - 108 q^{58} + 282 q^{59} + 380 q^{61} + 668 q^{62} + 64 q^{64} - 1008 q^{65} + 176 q^{67} - 144 q^{68} + 252 q^{70} + 324 q^{71} + 800 q^{73} - 772 q^{74} - 112 q^{76} + 77 q^{77} - 1144 q^{79} - 288 q^{80} - 888 q^{82} - 468 q^{83} + 648 q^{85} + 632 q^{86} - 88 q^{88} + 870 q^{89} + 392 q^{91} - 720 q^{92} - 804 q^{94} + 504 q^{95} - 1330 q^{97} - 98 q^{98} + O(q^{100})$$

## 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
−2.00000 0 4.00000 −18.0000 0 7.00000 −8.00000 0 36.0000
 $$n$$: 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 1386.4.a.a 1
3.b odd 2 1 154.4.a.d 1
12.b even 2 1 1232.4.a.f 1
21.c even 2 1 1078.4.a.g 1
33.d even 2 1 1694.4.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.4.a.d 1 3.b odd 2 1
1078.4.a.g 1 21.c even 2 1
1232.4.a.f 1 12.b even 2 1
1386.4.a.a 1 1.a even 1 1 trivial
1694.4.a.c 1 33.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1386))$$:

 $$T_{5} + 18$$ $$T_{13} - 56$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + T$$
$3$ $$T$$
$5$ $$18 + T$$
$7$ $$-7 + T$$
$11$ $$-11 + T$$
$13$ $$-56 + T$$
$17$ $$36 + T$$
$19$ $$28 + T$$
$23$ $$180 + T$$
$29$ $$-54 + T$$
$31$ $$334 + T$$
$37$ $$-386 + T$$
$41$ $$-444 + T$$
$43$ $$316 + T$$
$47$ $$-402 + T$$
$53$ $$-486 + T$$
$59$ $$-282 + T$$
$61$ $$-380 + T$$
$67$ $$-176 + T$$
$71$ $$-324 + T$$
$73$ $$-800 + T$$
$79$ $$1144 + T$$
$83$ $$468 + T$$
$89$ $$-870 + T$$
$97$ $$1330 + T$$