Properties

 Label 1386.2.k.e Level $1386$ Weight $2$ Character orbit 1386.k Analytic conductor $11.067$ Analytic rank $1$ Dimension $2$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

Character values

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

 $$n$$ $$155$$ $$199$$ $$1135$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

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 − 0.866025i 0.5 + 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 −0.500000 2.59808i 1.00000 0 0
991.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 −0.500000 + 2.59808i 1.00000 0 0
 $$n$$: 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.e 2
3.b odd 2 1 154.2.e.c 2
7.c even 3 1 inner 1386.2.k.e 2
7.c even 3 1 9702.2.a.br 1
7.d odd 6 1 9702.2.a.bs 1
12.b even 2 1 1232.2.q.d 2
21.c even 2 1 1078.2.e.k 2
21.g even 6 1 1078.2.a.c 1
21.g even 6 1 1078.2.e.k 2
21.h odd 6 1 154.2.e.c 2
21.h odd 6 1 1078.2.a.e 1
84.j odd 6 1 8624.2.a.u 1
84.n even 6 1 1232.2.q.d 2
84.n even 6 1 8624.2.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.c 2 3.b odd 2 1
154.2.e.c 2 21.h odd 6 1
1078.2.a.c 1 21.g even 6 1
1078.2.a.e 1 21.h odd 6 1
1078.2.e.k 2 21.c even 2 1
1078.2.e.k 2 21.g even 6 1
1232.2.q.d 2 12.b even 2 1
1232.2.q.d 2 84.n even 6 1
1386.2.k.e 2 1.a even 1 1 trivial
1386.2.k.e 2 7.c even 3 1 inner
8624.2.a.k 1 84.n even 6 1
8624.2.a.u 1 84.j odd 6 1
9702.2.a.br 1 7.c even 3 1
9702.2.a.bs 1 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 $$S_{2}^{\mathrm{new}}(1386, [\chi])$$:

 $$T_{5}$$ T5 $$T_{13} + 1$$ T13 + 1 $$T_{17}^{2} + 6T_{17} + 36$$ T17^2 + 6*T17 + 36 $$T_{23}^{2} + 6T_{23} + 36$$ T23^2 + 6*T23 + 36

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + T + 7$$
$11$ $$T^{2} + T + 1$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} + 6T + 36$$
$19$ $$T^{2} + 2T + 4$$
$23$ $$T^{2} + 6T + 36$$
$29$ $$(T + 9)^{2}$$
$31$ $$T^{2} - 4T + 16$$
$37$ $$T^{2} + 2T + 4$$
$41$ $$(T - 6)^{2}$$
$43$ $$(T + 4)^{2}$$
$47$ $$T^{2} + 6T + 36$$
$53$ $$T^{2}$$
$59$ $$T^{2} + 3T + 9$$
$61$ $$T^{2} + 11T + 121$$
$67$ $$T^{2} + 11T + 121$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 2T + 4$$
$79$ $$T^{2} + 5T + 25$$
$83$ $$(T - 6)^{2}$$
$89$ $$T^{2} + 18T + 324$$
$97$ $$(T + 13)^{2}$$