# Properties

 Label 1386.4.a.g 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

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 4 q^{4} + 14 q^{5} + 7 q^{7} - 8 q^{8} + O(q^{10})$$ $$q - 2 q^{2} + 4 q^{4} + 14 q^{5} + 7 q^{7} - 8 q^{8} - 28 q^{10} + 11 q^{11} - 16 q^{13} - 14 q^{14} + 16 q^{16} - 108 q^{17} + 116 q^{19} + 56 q^{20} - 22 q^{22} - 68 q^{23} + 71 q^{25} + 32 q^{26} + 28 q^{28} - 122 q^{29} - 262 q^{31} - 32 q^{32} + 216 q^{34} + 98 q^{35} + 130 q^{37} - 232 q^{38} - 112 q^{40} - 204 q^{41} - 396 q^{43} + 44 q^{44} + 136 q^{46} - 166 q^{47} + 49 q^{49} - 142 q^{50} - 64 q^{52} - 442 q^{53} + 154 q^{55} - 56 q^{56} + 244 q^{58} - 702 q^{59} + 196 q^{61} + 524 q^{62} + 64 q^{64} - 224 q^{65} - 416 q^{67} - 432 q^{68} - 196 q^{70} - 492 q^{71} + 408 q^{73} - 260 q^{74} + 464 q^{76} + 77 q^{77} + 600 q^{79} + 224 q^{80} + 408 q^{82} + 1212 q^{83} - 1512 q^{85} + 792 q^{86} - 88 q^{88} - 1146 q^{89} - 112 q^{91} - 272 q^{92} + 332 q^{94} + 1624 q^{95} - 482 q^{97} - 98 q^{98} + O(q^{100})$$

## 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 14.0000 0 7.00000 −8.00000 0 −28.0000
 $$n$$: 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.g 1
3.b odd 2 1 154.4.a.c 1
12.b even 2 1 1232.4.a.i 1
21.c even 2 1 1078.4.a.h 1
33.d even 2 1 1694.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.4.a.c 1 3.b odd 2 1
1078.4.a.h 1 21.c even 2 1
1232.4.a.i 1 12.b even 2 1
1386.4.a.g 1 1.a even 1 1 trivial
1694.4.a.a 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1386))$$:

 $$T_{5} - 14$$ $$T_{13} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + T$$
$3$ $$T$$
$5$ $$-14 + T$$
$7$ $$-7 + T$$
$11$ $$-11 + T$$
$13$ $$16 + T$$
$17$ $$108 + T$$
$19$ $$-116 + T$$
$23$ $$68 + T$$
$29$ $$122 + T$$
$31$ $$262 + T$$
$37$ $$-130 + T$$
$41$ $$204 + T$$
$43$ $$396 + T$$
$47$ $$166 + T$$
$53$ $$442 + T$$
$59$ $$702 + T$$
$61$ $$-196 + T$$
$67$ $$416 + T$$
$71$ $$492 + T$$
$73$ $$-408 + T$$
$79$ $$-600 + T$$
$83$ $$-1212 + T$$
$89$ $$1146 + T$$
$97$ $$482 + T$$