# Properties

 Label 1386.2.k.v Level $1386$ Weight $2$ Character orbit 1386.k Analytic conductor $11.067$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1386,2,Mod(793,1386)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1386, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 0]))

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

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

chi := DirichletCharacter("1386.793");

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.k (of order $$3$$, degree $$2$$, minimal)

## 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: no Analytic conductor: $$11.0672657201$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.21870000.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 3x^{5} + 24x^{4} - 43x^{3} + 138x^{2} - 117x + 73$$ x^6 - 3*x^5 + 24*x^4 - 43*x^3 + 138*x^2 - 117*x + 73 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 462) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + ( - \beta_{2} - 1) q^{4} + ( - \beta_{4} + \beta_1) q^{5} + (\beta_{3} + \beta_{2} + 1) q^{7} - q^{8}+O(q^{10})$$ q - b2 * q^2 + (-b2 - 1) * q^4 + (-b4 + b1) * q^5 + (b3 + b2 + 1) * q^7 - q^8 $$q - \beta_{2} q^{2} + ( - \beta_{2} - 1) q^{4} + ( - \beta_{4} + \beta_1) q^{5} + (\beta_{3} + \beta_{2} + 1) q^{7} - q^{8} + ( - \beta_{3} + \beta_1 - 1) q^{10} + ( - \beta_{2} - 1) q^{11} + (\beta_{5} - \beta_{4} + \beta_{3} + 1) q^{14} + \beta_{2} q^{16} + ( - 2 \beta_{5} - 2 \beta_{3} + \beta_{2} + 1) q^{17} + ( - \beta_{5} + \beta_{4} - 2 \beta_{2} + \beta_1) q^{19} + (\beta_{4} - \beta_{3} - 1) q^{20} - q^{22} + (\beta_{5} - 2 \beta_{4} - 2 \beta_{2}) q^{23} + (\beta_{5} - \beta_{3} - 5 \beta_{2} + 2 \beta_1 - 7) q^{25} + (\beta_{5} - \beta_{4} - \beta_{2}) q^{28} + ( - 2 \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{29} + ( - 2 \beta_{5} - 2 \beta_{3} + 2 \beta_{2} + 2) q^{31} + (\beta_{2} + 1) q^{32} + ( - 4 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 1) q^{34} + ( - 3 \beta_{5} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 4) q^{35} + (\beta_{5} - \beta_{4} + 6 \beta_{2} - \beta_1) q^{37} + (\beta_{5} + \beta_{3} - 3 \beta_{2} - 3) q^{38} + (\beta_{4} - \beta_1) q^{40} + ( - 2 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} + \beta_{2} + \beta_1 - 6) q^{41} + ( - 6 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 1) q^{43} + \beta_{2} q^{44} + ( - \beta_{5} - 2 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{46} + (\beta_{5} - 2 \beta_{2} - 2 \beta_1) q^{47} + ( - \beta_{5} - \beta_{4} + 3 \beta_{2} - \beta_1 + 1) q^{49} + (2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 7) q^{50} + (8 \beta_{2} + 8) q^{53} + (\beta_{4} - \beta_{3} - 1) q^{55} + ( - \beta_{3} - \beta_{2} - 1) q^{56} + ( - \beta_{5} - \beta_{4} + 4 \beta_{2} + 3 \beta_1) q^{58} + (\beta_{5} - \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 5) q^{59} + (2 \beta_{5} - \beta_{4} - 4 \beta_{2} - 3 \beta_1) q^{61} + ( - 4 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{62} + q^{64} + ( - \beta_{5} - 3 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 4) q^{67} + ( - 2 \beta_{5} + 2 \beta_{4} + \beta_{2} + 2 \beta_1) q^{68} + ( - 2 \beta_{5} + \beta_{4} - \beta_{3} - 5 \beta_{2} + 3 \beta_1 - 4) q^{70} + ( - 6 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 3) q^{71} + (2 \beta_{5} + 4 \beta_{3} - 2 \beta_1 + 2) q^{73} + ( - \beta_{5} - \beta_{3} + 7 \beta_{2} + 7) q^{74} + (2 \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 3) q^{76} + (\beta_{5} - \beta_{4} - \beta_{2}) q^{77} + ( - \beta_{4} + 4 \beta_{2} + \beta_1) q^{79} + (\beta_{3} - \beta_1 + 1) q^{80} + ( - \beta_{5} + 3 \beta_{4} + 5 \beta_{2} - \beta_1) q^{82} + (2 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} - \beta_{2} - \beta_1 - 4) q^{83} + (8 \beta_{5} - 7 \beta_{4} + 7 \beta_{3} - 4 \beta_{2} - 4 \beta_1 + 3) q^{85} + ( - 3 \beta_{5} + 3 \beta_{4} + 2 \beta_{2} + 3 \beta_1) q^{86} + (\beta_{2} + 1) q^{88} + (2 \beta_{4} + 4 \beta_{2} - 2 \beta_1) q^{89} + ( - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 - 2) q^{92} + ( - \beta_{5} - \beta_{2} - \beta_1) q^{94} + ( - 2 \beta_{5} - 6 \beta_{3} + 4 \beta_1 - 4) q^{95} + 7 q^{97} + ( - 3 \beta_{5} - \beta_{3} + 5 \beta_{2} + 2 \beta_1 + 4) q^{98}+O(q^{100})$$ q - b2 * q^2 + (-b2 - 1) * q^4 + (-b4 + b1) * q^5 + (b3 + b2 + 1) * q^7 - q^8 + (-b3 + b1 - 1) * q^10 + (-b2 - 1) * q^11 + (b5 - b4 + b3 + 1) * q^14 + b2 * q^16 + (-2*b5 - 2*b3 + b2 + 1) * q^17 + (-b5 + b4 - 2*b2 + b1) * q^19 + (b4 - b3 - 1) * q^20 - q^22 + (b5 - 2*b4 - 2*b2) * q^23 + (b5 - b3 - 5*b2 + 2*b1 - 7) * q^25 + (b5 - b4 - b2) * q^28 + (-2*b5 - b4 + b3 + b2 + b1 - 1) * q^29 + (-2*b5 - 2*b3 + 2*b2 + 2) * q^31 + (b2 + 1) * q^32 + (-4*b5 + 2*b4 - 2*b3 + 2*b2 + 2*b1 + 1) * q^34 + (-3*b5 - b3 - 2*b2 + 2*b1 + 4) * q^35 + (b5 - b4 + 6*b2 - b1) * q^37 + (b5 + b3 - 3*b2 - 3) * q^38 + (b4 - b1) * q^40 + (-2*b5 + 3*b4 - 3*b3 + b2 + b1 - 6) * q^41 + (-6*b5 + 3*b4 - 3*b3 + 3*b2 + 3*b1 + 1) * q^43 + b2 * q^44 + (-b5 - 2*b3 - b2 + b1 - 2) * q^46 + (b5 - 2*b2 - 2*b1) * q^47 + (-b5 - b4 + 3*b2 - b1 + 1) * q^49 + (2*b5 + b4 - b3 - b2 - b1 - 7) * q^50 + (8*b2 + 8) * q^53 + (b4 - b3 - 1) * q^55 + (-b3 - b2 - 1) * q^56 + (-b5 - b4 + 4*b2 + 3*b1) * q^58 + (b5 - b3 - 3*b2 + 2*b1 - 5) * q^59 + (2*b5 - b4 - 4*b2 - 3*b1) * q^61 + (-4*b5 + 2*b4 - 2*b3 + 2*b2 + 2*b1 + 2) * q^62 + q^64 + (-b5 - 3*b3 - 2*b2 + 2*b1 - 4) * q^67 + (-2*b5 + 2*b4 + b2 + 2*b1) * q^68 + (-2*b5 + b4 - b3 - 5*b2 + 3*b1 - 4) * q^70 + (-6*b5 + 3*b4 - 3*b3 + 3*b2 + 3*b1 - 3) * q^71 + (2*b5 + 4*b3 - 2*b1 + 2) * q^73 + (-b5 - b3 + 7*b2 + 7) * q^74 + (2*b5 - b4 + b3 - b2 - b1 - 3) * q^76 + (b5 - b4 - b2) * q^77 + (-b4 + 4*b2 + b1) * q^79 + (b3 - b1 + 1) * q^80 + (-b5 + 3*b4 + 5*b2 - b1) * q^82 + (2*b5 - 3*b4 + 3*b3 - b2 - b1 - 4) * q^83 + (8*b5 - 7*b4 + 7*b3 - 4*b2 - 4*b1 + 3) * q^85 + (-3*b5 + 3*b4 + 2*b2 + 3*b1) * q^86 + (b2 + 1) * q^88 + (2*b4 + 4*b2 - 2*b1) * q^89 + (-2*b5 + 2*b4 - 2*b3 + b2 + b1 - 2) * q^92 + (-b5 - b2 - b1) * q^94 + (-2*b5 - 6*b3 + 4*b1 - 4) * q^95 + 7 * q^97 + (-3*b5 - b3 + 5*b2 + 2*b1 + 4) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 3 q^{2} - 3 q^{4} - 6 q^{8}+O(q^{10})$$ 6 * q + 3 * q^2 - 3 * q^4 - 6 * q^8 $$6 q + 3 q^{2} - 3 q^{4} - 6 q^{8} - 3 q^{11} + 3 q^{14} - 3 q^{16} + 3 q^{17} + 9 q^{19} - 6 q^{22} + 3 q^{23} - 15 q^{25} + 3 q^{28} - 18 q^{29} + 6 q^{31} + 3 q^{32} + 6 q^{34} + 30 q^{35} - 21 q^{37} - 9 q^{38} - 24 q^{41} + 6 q^{43} - 3 q^{44} - 3 q^{46} + 3 q^{47} - 12 q^{49} - 30 q^{50} + 24 q^{53} - 9 q^{58} - 9 q^{59} + 6 q^{61} + 12 q^{62} + 6 q^{64} - 6 q^{67} + 3 q^{68} - 18 q^{71} + 21 q^{74} - 18 q^{76} + 3 q^{77} - 12 q^{79} - 12 q^{82} - 36 q^{83} + 3 q^{86} + 3 q^{88} - 12 q^{89} - 6 q^{92} - 3 q^{94} + 42 q^{97} + 9 q^{98}+O(q^{100})$$ 6 * q + 3 * q^2 - 3 * q^4 - 6 * q^8 - 3 * q^11 + 3 * q^14 - 3 * q^16 + 3 * q^17 + 9 * q^19 - 6 * q^22 + 3 * q^23 - 15 * q^25 + 3 * q^28 - 18 * q^29 + 6 * q^31 + 3 * q^32 + 6 * q^34 + 30 * q^35 - 21 * q^37 - 9 * q^38 - 24 * q^41 + 6 * q^43 - 3 * q^44 - 3 * q^46 + 3 * q^47 - 12 * q^49 - 30 * q^50 + 24 * q^53 - 9 * q^58 - 9 * q^59 + 6 * q^61 + 12 * q^62 + 6 * q^64 - 6 * q^67 + 3 * q^68 - 18 * q^71 + 21 * q^74 - 18 * q^76 + 3 * q^77 - 12 * q^79 - 12 * q^82 - 36 * q^83 + 3 * q^86 + 3 * q^88 - 12 * q^89 - 6 * q^92 - 3 * q^94 + 42 * q^97 + 9 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3x^{5} + 24x^{4} - 43x^{3} + 138x^{2} - 117x + 73$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -2\nu^{5} + 5\nu^{4} - 30\nu^{3} + 40\nu^{2} - 70\nu + 13 ) / 31$$ (-2*v^5 + 5*v^4 - 30*v^3 + 40*v^2 - 70*v + 13) / 31 $$\beta_{3}$$ $$=$$ $$( -\nu^{5} + 18\nu^{4} - 15\nu^{3} + 144\nu^{2} + 151\nu - 164 ) / 62$$ (-v^5 + 18*v^4 - 15*v^3 + 144*v^2 + 151*v - 164) / 62 $$\beta_{4}$$ $$=$$ $$( -\nu^{5} - 13\nu^{4} + 47\nu^{3} - 197\nu^{2} + 461\nu - 133 ) / 62$$ (-v^5 - 13*v^4 + 47*v^3 - 197*v^2 + 461*v - 133) / 62 $$\beta_{5}$$ $$=$$ $$( -\nu^{5} - 13\nu^{4} + 16\nu^{3} - 166\nu^{2} + 151\nu - 102 ) / 31$$ (-v^5 - 13*v^4 + 16*v^3 - 166*v^2 + 151*v - 102) / 31
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-2\beta_{5} + 2\beta_{4} - 2\beta_{3} + \beta_{2} + 2\beta _1 - 8$$ -2*b5 + 2*b4 - 2*b3 + b2 + 2*b1 - 8 $$\nu^{3}$$ $$=$$ $$-3\beta_{5} + 4\beta_{4} - 2\beta_{3} + \beta_{2} - 8\beta _1 - 7$$ -3*b5 + 4*b4 - 2*b3 + b2 - 8*b1 - 7 $$\nu^{4}$$ $$=$$ $$16\beta_{5} - 16\beta_{4} + 20\beta_{3} - 9\beta_{2} - 28\beta _1 + 75$$ 16*b5 - 16*b4 + 20*b3 - 9*b2 - 28*b1 + 75 $$\nu^{5}$$ $$=$$ $$45\beta_{5} - 60\beta_{4} + 40\beta_{3} - 33\beta_{2} + 55\beta _1 + 139$$ 45*b5 - 60*b4 + 40*b3 - 33*b2 + 55*b1 + 139

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times$$.

 $$n$$ $$155$$ $$199$$ $$1135$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
793.1
 0.5 − 0.679547i 0.5 + 3.05087i 0.5 − 3.23735i 0.5 + 0.679547i 0.5 − 3.05087i 0.5 + 3.23735i
0.500000 0.866025i 0 −0.500000 0.866025i −1.40280 + 2.42972i 0 −1.40280 2.24325i −1.00000 0 1.40280 + 2.42972i
793.2 0.500000 0.866025i 0 −0.500000 0.866025i −0.806615 + 1.39710i 0 −0.806615 + 2.51980i −1.00000 0 0.806615 + 1.39710i
793.3 0.500000 0.866025i 0 −0.500000 0.866025i 2.20942 3.82682i 0 2.20942 + 1.45550i −1.00000 0 −2.20942 3.82682i
991.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.40280 2.42972i 0 −1.40280 + 2.24325i −1.00000 0 1.40280 2.42972i
991.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.806615 1.39710i 0 −0.806615 2.51980i −1.00000 0 0.806615 1.39710i
991.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 2.20942 + 3.82682i 0 2.20942 1.45550i −1.00000 0 −2.20942 + 3.82682i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 991.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1386.2.k.v 6
3.b odd 2 1 462.2.i.g 6
7.c even 3 1 inner 1386.2.k.v 6
7.c even 3 1 9702.2.a.dv 3
7.d odd 6 1 9702.2.a.dw 3
21.g even 6 1 3234.2.a.bh 3
21.h odd 6 1 462.2.i.g 6
21.h odd 6 1 3234.2.a.bf 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.g 6 3.b odd 2 1
462.2.i.g 6 21.h odd 6 1
1386.2.k.v 6 1.a even 1 1 trivial
1386.2.k.v 6 7.c even 3 1 inner
3234.2.a.bf 3 21.h odd 6 1
3234.2.a.bh 3 21.g even 6 1
9702.2.a.dv 3 7.c even 3 1
9702.2.a.dw 3 7.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1386, [\chi])$$:

 $$T_{5}^{6} + 15T_{5}^{4} + 40T_{5}^{3} + 225T_{5}^{2} + 300T_{5} + 400$$ T5^6 + 15*T5^4 + 40*T5^3 + 225*T5^2 + 300*T5 + 400 $$T_{13}$$ T13 $$T_{17}^{6} - 3T_{17}^{5} + 66T_{17}^{4} - 107T_{17}^{3} + 3666T_{17}^{2} - 7923T_{17} + 19321$$ T17^6 - 3*T17^5 + 66*T17^4 - 107*T17^3 + 3666*T17^2 - 7923*T17 + 19321 $$T_{23}^{6} - 3T_{23}^{5} + 36T_{23}^{4} - 97T_{23}^{3} + 996T_{23}^{2} - 2403T_{23} + 7921$$ T23^6 - 3*T23^5 + 36*T23^4 - 97*T23^3 + 996*T23^2 - 2403*T23 + 7921

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{3}$$
$3$ $$T^{6}$$
$5$ $$T^{6} + 15 T^{4} + 40 T^{3} + \cdots + 400$$
$7$ $$T^{6} + 6 T^{4} - 20 T^{3} + 42 T^{2} + \cdots + 343$$
$11$ $$(T^{2} + T + 1)^{3}$$
$13$ $$T^{6}$$
$17$ $$T^{6} - 3 T^{5} + 66 T^{4} + \cdots + 19321$$
$19$ $$T^{6} - 9 T^{5} + 69 T^{4} - 164 T^{3} + \cdots + 784$$
$23$ $$T^{6} - 3 T^{5} + 36 T^{4} + \cdots + 7921$$
$29$ $$(T^{3} + 9 T^{2} - 48 T - 348)^{2}$$
$31$ $$T^{6} - 6 T^{5} + 84 T^{4} + \cdots + 36864$$
$37$ $$T^{6} + 21 T^{5} + 309 T^{4} + \cdots + 51984$$
$41$ $$(T^{3} + 12 T^{2} - 27 T - 306)^{2}$$
$43$ $$(T^{3} - 3 T^{2} - 132 T + 404)^{2}$$
$47$ $$T^{6} - 3 T^{5} + 36 T^{4} + 63 T^{3} + \cdots + 81$$
$53$ $$(T^{2} - 8 T + 64)^{3}$$
$59$ $$T^{6} + 9 T^{5} + 129 T^{4} + \cdots + 2304$$
$61$ $$T^{6} - 6 T^{5} + 99 T^{4} + \cdots + 9604$$
$67$ $$T^{6} + 6 T^{5} + 99 T^{4} + \cdots + 44944$$
$71$ $$(T^{3} + 9 T^{2} - 108 T - 108)^{2}$$
$73$ $$T^{6} + 120 T^{4} - 960 T^{3} + \cdots + 230400$$
$79$ $$T^{6} + 12 T^{5} + 111 T^{4} + \cdots + 256$$
$83$ $$(T^{3} + 18 T^{2} + 33 T - 164)^{2}$$
$89$ $$T^{6} + 12 T^{5} + 156 T^{4} + \cdots + 256$$
$97$ $$(T - 7)^{6}$$