# Properties

 Label 1386.2.a.k 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1386,2,Mod(1,1386)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1386, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1386.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 2.00000 0 1.00000 1.00000 0 2.00000
 $$n$$: 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.k 1
3.b odd 2 1 462.2.a.a 1
7.b odd 2 1 9702.2.a.bf 1
12.b even 2 1 3696.2.a.s 1
21.c even 2 1 3234.2.a.n 1
33.d even 2 1 5082.2.a.q 1

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

 $$T_{5} - 2$$ T5 - 2 $$T_{13} - 2$$ T13 - 2 $$T_{17} - 6$$ T17 - 6

## Hecke characteristic polynomials

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