# Properties

 Label 1386.2.k.t Level $1386$ Weight $2$ Character orbit 1386.k Analytic conductor $11.067$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 154) 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + ( - \beta_{2} - 1) q^{4} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{5} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{7} - q^{8}+O(q^{10})$$ q - b2 * q^2 + (-b2 - 1) * q^4 + (-b3 + 2*b2 - b1) * q^5 + (-b3 + b2 + b1) * q^7 - q^8 $$q - \beta_{2} q^{2} + ( - \beta_{2} - 1) q^{4} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{5} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{7} - q^{8} + (2 \beta_{2} - \beta_1 + 2) q^{10} + (\beta_{2} + 1) q^{11} + (2 \beta_{3} - 1) q^{13} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{14} + \beta_{2} q^{16} + ( - 2 \beta_{2} + 4 \beta_1 - 2) q^{17} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{19} + (\beta_{3} + 2) q^{20} + q^{22} + (3 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{23} + ( - \beta_{2} + 4 \beta_1 - 1) q^{25} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{26} + ( - \beta_{3} - 2 \beta_1 + 1) q^{28} + (4 \beta_{3} + 3) q^{29} + (4 \beta_{2} + 4) q^{31} + (\beta_{2} + 1) q^{32} + ( - 4 \beta_{3} - 2) q^{34} + (4 \beta_{3} - 4 \beta_{2} + 3 \beta_1) q^{35} + ( - \beta_{3} - 8 \beta_{2} - \beta_1) q^{37} + ( - 2 \beta_{2} + \beta_1 - 2) q^{38} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{40} + ( - \beta_{3} + 4) q^{41} + 4 \beta_{3} q^{43} - \beta_{2} q^{44} + (2 \beta_{2} + 3 \beta_1 + 2) q^{46} + (6 \beta_{3} + 2 \beta_{2} + 6 \beta_1) q^{47} + (4 \beta_{3} + 5 \beta_{2} + 2 \beta_1 + 5) q^{49} + ( - 4 \beta_{3} - 1) q^{50} + (\beta_{2} + 2 \beta_1 + 1) q^{52} + ( - 2 \beta_{2} + 7 \beta_1 - 2) q^{53} + ( - \beta_{3} - 2) q^{55} + (\beta_{3} - \beta_{2} - \beta_1) q^{56} + (4 \beta_{3} - 3 \beta_{2} + 4 \beta_1) q^{58} + ( - 7 \beta_{2} + \beta_1 - 7) q^{59} + (2 \beta_{3} + 9 \beta_{2} + 2 \beta_1) q^{61} + 4 q^{62} + q^{64} + ( - 3 \beta_{3} + 2 \beta_{2} - 3 \beta_1) q^{65} + ( - 7 \beta_{2} + 3 \beta_1 - 7) q^{67} + ( - 4 \beta_{3} + 2 \beta_{2} - 4 \beta_1) q^{68} + (\beta_{3} - 4 \beta_{2} + 4 \beta_1 - 4) q^{70} + ( - 5 \beta_{3} + 4) q^{71} + (8 \beta_{2} + \beta_1 + 8) q^{73} + ( - 8 \beta_{2} - \beta_1 - 8) q^{74} + ( - \beta_{3} - 2) q^{76} + (\beta_{3} + 2 \beta_1 - 1) q^{77} + ( - 3 \beta_{3} - 9 \beta_{2} - 3 \beta_1) q^{79} + ( - 2 \beta_{2} + \beta_1 - 2) q^{80} + ( - \beta_{3} - 4 \beta_{2} - \beta_1) q^{82} + ( - 10 \beta_{3} - 2) q^{83} + (10 \beta_{3} + 12) q^{85} + (4 \beta_{3} + 4 \beta_1) q^{86} + ( - \beta_{2} - 1) q^{88} + ( - 6 \beta_{3} - 4 \beta_{2} - 6 \beta_1) q^{89} + ( - \beta_{3} - 5 \beta_{2} - 3 \beta_1 - 8) q^{91} + ( - 3 \beta_{3} + 2) q^{92} + (2 \beta_{2} + 6 \beta_1 + 2) q^{94} + (6 \beta_{2} - 4 \beta_1 + 6) q^{95} + (2 \beta_{3} - 1) q^{97} + (2 \beta_{3} + 4 \beta_1 + 5) q^{98}+O(q^{100})$$ q - b2 * q^2 + (-b2 - 1) * q^4 + (-b3 + 2*b2 - b1) * q^5 + (-b3 + b2 + b1) * q^7 - q^8 + (2*b2 - b1 + 2) * q^10 + (b2 + 1) * q^11 + (2*b3 - 1) * q^13 + (-2*b3 + b2 - b1 + 1) * q^14 + b2 * q^16 + (-2*b2 + 4*b1 - 2) * q^17 + (b3 - 2*b2 + b1) * q^19 + (b3 + 2) * q^20 + q^22 + (3*b3 + 2*b2 + 3*b1) * q^23 + (-b2 + 4*b1 - 1) * q^25 + (2*b3 + b2 + 2*b1) * q^26 + (-b3 - 2*b1 + 1) * q^28 + (4*b3 + 3) * q^29 + (4*b2 + 4) * q^31 + (b2 + 1) * q^32 + (-4*b3 - 2) * q^34 + (4*b3 - 4*b2 + 3*b1) * q^35 + (-b3 - 8*b2 - b1) * q^37 + (-2*b2 + b1 - 2) * q^38 + (b3 - 2*b2 + b1) * q^40 + (-b3 + 4) * q^41 + 4*b3 * q^43 - b2 * q^44 + (2*b2 + 3*b1 + 2) * q^46 + (6*b3 + 2*b2 + 6*b1) * q^47 + (4*b3 + 5*b2 + 2*b1 + 5) * q^49 + (-4*b3 - 1) * q^50 + (b2 + 2*b1 + 1) * q^52 + (-2*b2 + 7*b1 - 2) * q^53 + (-b3 - 2) * q^55 + (b3 - b2 - b1) * q^56 + (4*b3 - 3*b2 + 4*b1) * q^58 + (-7*b2 + b1 - 7) * q^59 + (2*b3 + 9*b2 + 2*b1) * q^61 + 4 * q^62 + q^64 + (-3*b3 + 2*b2 - 3*b1) * q^65 + (-7*b2 + 3*b1 - 7) * q^67 + (-4*b3 + 2*b2 - 4*b1) * q^68 + (b3 - 4*b2 + 4*b1 - 4) * q^70 + (-5*b3 + 4) * q^71 + (8*b2 + b1 + 8) * q^73 + (-8*b2 - b1 - 8) * q^74 + (-b3 - 2) * q^76 + (b3 + 2*b1 - 1) * q^77 + (-3*b3 - 9*b2 - 3*b1) * q^79 + (-2*b2 + b1 - 2) * q^80 + (-b3 - 4*b2 - b1) * q^82 + (-10*b3 - 2) * q^83 + (10*b3 + 12) * q^85 + (4*b3 + 4*b1) * q^86 + (-b2 - 1) * q^88 + (-6*b3 - 4*b2 - 6*b1) * q^89 + (-b3 - 5*b2 - 3*b1 - 8) * q^91 + (-3*b3 + 2) * q^92 + (2*b2 + 6*b1 + 2) * q^94 + (6*b2 - 4*b1 + 6) * q^95 + (2*b3 - 1) * q^97 + (2*b3 + 4*b1 + 5) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 2 q^{4} - 4 q^{5} - 2 q^{7} - 4 q^{8}+O(q^{10})$$ 4 * q + 2 * q^2 - 2 * q^4 - 4 * q^5 - 2 * q^7 - 4 * q^8 $$4 q + 2 q^{2} - 2 q^{4} - 4 q^{5} - 2 q^{7} - 4 q^{8} + 4 q^{10} + 2 q^{11} - 4 q^{13} + 2 q^{14} - 2 q^{16} - 4 q^{17} + 4 q^{19} + 8 q^{20} + 4 q^{22} - 4 q^{23} - 2 q^{25} - 2 q^{26} + 4 q^{28} + 12 q^{29} + 8 q^{31} + 2 q^{32} - 8 q^{34} + 8 q^{35} + 16 q^{37} - 4 q^{38} + 4 q^{40} + 16 q^{41} + 2 q^{44} + 4 q^{46} - 4 q^{47} + 10 q^{49} - 4 q^{50} + 2 q^{52} - 4 q^{53} - 8 q^{55} + 2 q^{56} + 6 q^{58} - 14 q^{59} - 18 q^{61} + 16 q^{62} + 4 q^{64} - 4 q^{65} - 14 q^{67} - 4 q^{68} - 8 q^{70} + 16 q^{71} + 16 q^{73} - 16 q^{74} - 8 q^{76} - 4 q^{77} + 18 q^{79} - 4 q^{80} + 8 q^{82} - 8 q^{83} + 48 q^{85} - 2 q^{88} + 8 q^{89} - 22 q^{91} + 8 q^{92} + 4 q^{94} + 12 q^{95} - 4 q^{97} + 20 q^{98}+O(q^{100})$$ 4 * q + 2 * q^2 - 2 * q^4 - 4 * q^5 - 2 * q^7 - 4 * q^8 + 4 * q^10 + 2 * q^11 - 4 * q^13 + 2 * q^14 - 2 * q^16 - 4 * q^17 + 4 * q^19 + 8 * q^20 + 4 * q^22 - 4 * q^23 - 2 * q^25 - 2 * q^26 + 4 * q^28 + 12 * q^29 + 8 * q^31 + 2 * q^32 - 8 * q^34 + 8 * q^35 + 16 * q^37 - 4 * q^38 + 4 * q^40 + 16 * q^41 + 2 * q^44 + 4 * q^46 - 4 * q^47 + 10 * q^49 - 4 * q^50 + 2 * q^52 - 4 * q^53 - 8 * q^55 + 2 * q^56 + 6 * q^58 - 14 * q^59 - 18 * q^61 + 16 * q^62 + 4 * q^64 - 4 * q^65 - 14 * q^67 - 4 * q^68 - 8 * q^70 + 16 * q^71 + 16 * q^73 - 16 * q^74 - 8 * q^76 - 4 * q^77 + 18 * q^79 - 4 * q^80 + 8 * q^82 - 8 * q^83 + 48 * q^85 - 2 * q^88 + 8 * q^89 - 22 * q^91 + 8 * q^92 + 4 * q^94 + 12 * q^95 - 4 * q^97 + 20 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## 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
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
0.500000 0.866025i 0 −0.500000 0.866025i −1.70711 + 2.95680i 0 −2.62132 0.358719i −1.00000 0 1.70711 + 2.95680i
793.2 0.500000 0.866025i 0 −0.500000 0.866025i −0.292893 + 0.507306i 0 1.62132 + 2.09077i −1.00000 0 0.292893 + 0.507306i
991.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.70711 2.95680i 0 −2.62132 + 0.358719i −1.00000 0 1.70711 2.95680i
991.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.292893 0.507306i 0 1.62132 2.09077i −1.00000 0 0.292893 0.507306i
 $$n$$: e.g. 2-40 or 990-1000 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.t 4
3.b odd 2 1 154.2.e.e 4
7.c even 3 1 inner 1386.2.k.t 4
7.c even 3 1 9702.2.a.cx 2
7.d odd 6 1 9702.2.a.ch 2
12.b even 2 1 1232.2.q.f 4
21.c even 2 1 1078.2.e.m 4
21.g even 6 1 1078.2.a.x 2
21.g even 6 1 1078.2.e.m 4
21.h odd 6 1 154.2.e.e 4
21.h odd 6 1 1078.2.a.t 2
84.j odd 6 1 8624.2.a.bh 2
84.n even 6 1 1232.2.q.f 4
84.n even 6 1 8624.2.a.cc 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.e 4 3.b odd 2 1
154.2.e.e 4 21.h odd 6 1
1078.2.a.t 2 21.h odd 6 1
1078.2.a.x 2 21.g even 6 1
1078.2.e.m 4 21.c even 2 1
1078.2.e.m 4 21.g even 6 1
1232.2.q.f 4 12.b even 2 1
1232.2.q.f 4 84.n even 6 1
1386.2.k.t 4 1.a even 1 1 trivial
1386.2.k.t 4 7.c even 3 1 inner
8624.2.a.bh 2 84.j odd 6 1
8624.2.a.cc 2 84.n even 6 1
9702.2.a.ch 2 7.d odd 6 1
9702.2.a.cx 2 7.c even 3 1

## Hecke kernels

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

 $$T_{5}^{4} + 4T_{5}^{3} + 14T_{5}^{2} + 8T_{5} + 4$$ T5^4 + 4*T5^3 + 14*T5^2 + 8*T5 + 4 $$T_{13}^{2} + 2T_{13} - 7$$ T13^2 + 2*T13 - 7 $$T_{17}^{4} + 4T_{17}^{3} + 44T_{17}^{2} - 112T_{17} + 784$$ T17^4 + 4*T17^3 + 44*T17^2 - 112*T17 + 784 $$T_{23}^{4} + 4T_{23}^{3} + 30T_{23}^{2} - 56T_{23} + 196$$ T23^4 + 4*T23^3 + 30*T23^2 - 56*T23 + 196

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 4 T^{3} + 14 T^{2} + 8 T + 4$$
$7$ $$T^{4} + 2 T^{3} - 3 T^{2} + 14 T + 49$$
$11$ $$(T^{2} - T + 1)^{2}$$
$13$ $$(T^{2} + 2 T - 7)^{2}$$
$17$ $$T^{4} + 4 T^{3} + 44 T^{2} - 112 T + 784$$
$19$ $$T^{4} - 4 T^{3} + 14 T^{2} - 8 T + 4$$
$23$ $$T^{4} + 4 T^{3} + 30 T^{2} - 56 T + 196$$
$29$ $$(T^{2} - 6 T - 23)^{2}$$
$31$ $$(T^{2} - 4 T + 16)^{2}$$
$37$ $$T^{4} - 16 T^{3} + 194 T^{2} + \cdots + 3844$$
$41$ $$(T^{2} - 8 T + 14)^{2}$$
$43$ $$(T^{2} - 32)^{2}$$
$47$ $$T^{4} + 4 T^{3} + 84 T^{2} + \cdots + 4624$$
$53$ $$T^{4} + 4 T^{3} + 110 T^{2} + \cdots + 8836$$
$59$ $$T^{4} + 14 T^{3} + 149 T^{2} + \cdots + 2209$$
$61$ $$T^{4} + 18 T^{3} + 251 T^{2} + \cdots + 5329$$
$67$ $$T^{4} + 14 T^{3} + 165 T^{2} + \cdots + 961$$
$71$ $$(T^{2} - 8 T - 34)^{2}$$
$73$ $$T^{4} - 16 T^{3} + 194 T^{2} + \cdots + 3844$$
$79$ $$T^{4} - 18 T^{3} + 261 T^{2} + \cdots + 3969$$
$83$ $$(T^{2} + 4 T - 196)^{2}$$
$89$ $$T^{4} - 8 T^{3} + 120 T^{2} + \cdots + 3136$$
$97$ $$(T^{2} + 2 T - 7)^{2}$$