Properties

 Label 1386.2.k.j Level $1386$ Weight $2$ Character orbit 1386.k Analytic conductor $11.067$ Analytic rank $0$ Dimension $2$ 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: $$2$$ 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 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} + 3 \zeta_{6} q^{5} + ( - \zeta_{6} - 2) q^{7} + q^{8} +O(q^{10})$$ q - z * q^2 + (z - 1) * q^4 + 3*z * q^5 + (-z - 2) * q^7 + q^8 $$q - \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} + 3 \zeta_{6} q^{5} + ( - \zeta_{6} - 2) q^{7} + q^{8} + ( - 3 \zeta_{6} + 3) q^{10} + (\zeta_{6} - 1) q^{11} + 6 q^{13} + (3 \zeta_{6} - 1) q^{14} - \zeta_{6} q^{16} + (5 \zeta_{6} - 5) q^{17} - 6 \zeta_{6} q^{19} - 3 q^{20} + q^{22} + 5 \zeta_{6} q^{23} + (4 \zeta_{6} - 4) q^{25} - 6 \zeta_{6} q^{26} + ( - 2 \zeta_{6} + 3) q^{28} + 6 q^{29} + (4 \zeta_{6} - 4) q^{31} + (\zeta_{6} - 1) q^{32} + 5 q^{34} + ( - 9 \zeta_{6} + 3) q^{35} + 2 \zeta_{6} q^{37} + (6 \zeta_{6} - 6) q^{38} + 3 \zeta_{6} q^{40} - 5 q^{41} - 10 q^{43} - \zeta_{6} q^{44} + ( - 5 \zeta_{6} + 5) q^{46} + 9 \zeta_{6} q^{47} + (5 \zeta_{6} + 3) q^{49} + 4 q^{50} + (6 \zeta_{6} - 6) q^{52} + ( - 2 \zeta_{6} + 2) q^{53} - 3 q^{55} + ( - \zeta_{6} - 2) q^{56} - 6 \zeta_{6} q^{58} + (12 \zeta_{6} - 12) q^{59} + 5 \zeta_{6} q^{61} + 4 q^{62} + q^{64} + 18 \zeta_{6} q^{65} + (5 \zeta_{6} - 5) q^{67} - 5 \zeta_{6} q^{68} + (6 \zeta_{6} - 9) q^{70} - 4 q^{71} + (12 \zeta_{6} - 12) q^{73} + ( - 2 \zeta_{6} + 2) q^{74} + 6 q^{76} + ( - 2 \zeta_{6} + 3) q^{77} + \zeta_{6} q^{79} + ( - 3 \zeta_{6} + 3) q^{80} + 5 \zeta_{6} q^{82} - q^{83} - 15 q^{85} + 10 \zeta_{6} q^{86} + (\zeta_{6} - 1) q^{88} + 6 \zeta_{6} q^{89} + ( - 6 \zeta_{6} - 12) q^{91} - 5 q^{92} + ( - 9 \zeta_{6} + 9) q^{94} + ( - 18 \zeta_{6} + 18) q^{95} + 9 q^{97} + ( - 8 \zeta_{6} + 5) q^{98} +O(q^{100})$$ q - z * q^2 + (z - 1) * q^4 + 3*z * q^5 + (-z - 2) * q^7 + q^8 + (-3*z + 3) * q^10 + (z - 1) * q^11 + 6 * q^13 + (3*z - 1) * q^14 - z * q^16 + (5*z - 5) * q^17 - 6*z * q^19 - 3 * q^20 + q^22 + 5*z * q^23 + (4*z - 4) * q^25 - 6*z * q^26 + (-2*z + 3) * q^28 + 6 * q^29 + (4*z - 4) * q^31 + (z - 1) * q^32 + 5 * q^34 + (-9*z + 3) * q^35 + 2*z * q^37 + (6*z - 6) * q^38 + 3*z * q^40 - 5 * q^41 - 10 * q^43 - z * q^44 + (-5*z + 5) * q^46 + 9*z * q^47 + (5*z + 3) * q^49 + 4 * q^50 + (6*z - 6) * q^52 + (-2*z + 2) * q^53 - 3 * q^55 + (-z - 2) * q^56 - 6*z * q^58 + (12*z - 12) * q^59 + 5*z * q^61 + 4 * q^62 + q^64 + 18*z * q^65 + (5*z - 5) * q^67 - 5*z * q^68 + (6*z - 9) * q^70 - 4 * q^71 + (12*z - 12) * q^73 + (-2*z + 2) * q^74 + 6 * q^76 + (-2*z + 3) * q^77 + z * q^79 + (-3*z + 3) * q^80 + 5*z * q^82 - q^83 - 15 * q^85 + 10*z * q^86 + (z - 1) * q^88 + 6*z * q^89 + (-6*z - 12) * q^91 - 5 * q^92 + (-9*z + 9) * q^94 + (-18*z + 18) * q^95 + 9 * q^97 + (-8*z + 5) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{4} + 3 q^{5} - 5 q^{7} + 2 q^{8}+O(q^{10})$$ 2 * q - q^2 - q^4 + 3 * q^5 - 5 * q^7 + 2 * q^8 $$2 q - q^{2} - q^{4} + 3 q^{5} - 5 q^{7} + 2 q^{8} + 3 q^{10} - q^{11} + 12 q^{13} + q^{14} - q^{16} - 5 q^{17} - 6 q^{19} - 6 q^{20} + 2 q^{22} + 5 q^{23} - 4 q^{25} - 6 q^{26} + 4 q^{28} + 12 q^{29} - 4 q^{31} - q^{32} + 10 q^{34} - 3 q^{35} + 2 q^{37} - 6 q^{38} + 3 q^{40} - 10 q^{41} - 20 q^{43} - q^{44} + 5 q^{46} + 9 q^{47} + 11 q^{49} + 8 q^{50} - 6 q^{52} + 2 q^{53} - 6 q^{55} - 5 q^{56} - 6 q^{58} - 12 q^{59} + 5 q^{61} + 8 q^{62} + 2 q^{64} + 18 q^{65} - 5 q^{67} - 5 q^{68} - 12 q^{70} - 8 q^{71} - 12 q^{73} + 2 q^{74} + 12 q^{76} + 4 q^{77} + q^{79} + 3 q^{80} + 5 q^{82} - 2 q^{83} - 30 q^{85} + 10 q^{86} - q^{88} + 6 q^{89} - 30 q^{91} - 10 q^{92} + 9 q^{94} + 18 q^{95} + 18 q^{97} + 2 q^{98}+O(q^{100})$$ 2 * q - q^2 - q^4 + 3 * q^5 - 5 * q^7 + 2 * q^8 + 3 * q^10 - q^11 + 12 * q^13 + q^14 - q^16 - 5 * q^17 - 6 * q^19 - 6 * q^20 + 2 * q^22 + 5 * q^23 - 4 * q^25 - 6 * q^26 + 4 * q^28 + 12 * q^29 - 4 * q^31 - q^32 + 10 * q^34 - 3 * q^35 + 2 * q^37 - 6 * q^38 + 3 * q^40 - 10 * q^41 - 20 * q^43 - q^44 + 5 * q^46 + 9 * q^47 + 11 * q^49 + 8 * q^50 - 6 * q^52 + 2 * q^53 - 6 * q^55 - 5 * q^56 - 6 * q^58 - 12 * q^59 + 5 * q^61 + 8 * q^62 + 2 * q^64 + 18 * q^65 - 5 * q^67 - 5 * q^68 - 12 * q^70 - 8 * q^71 - 12 * q^73 + 2 * q^74 + 12 * q^76 + 4 * q^77 + q^79 + 3 * q^80 + 5 * q^82 - 2 * q^83 - 30 * q^85 + 10 * q^86 - q^88 + 6 * q^89 - 30 * q^91 - 10 * q^92 + 9 * q^94 + 18 * q^95 + 18 * 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 1.50000 2.59808i 0 −2.50000 + 0.866025i 1.00000 0 1.50000 + 2.59808i
991.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.50000 + 2.59808i 0 −2.50000 0.866025i 1.00000 0 1.50000 2.59808i
 $$n$$: 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.j 2
3.b odd 2 1 462.2.i.a 2
7.c even 3 1 inner 1386.2.k.j 2
7.c even 3 1 9702.2.a.be 1
7.d odd 6 1 9702.2.a.ce 1
21.g even 6 1 3234.2.a.a 1
21.h odd 6 1 462.2.i.a 2
21.h odd 6 1 3234.2.a.o 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.a 2 3.b odd 2 1
462.2.i.a 2 21.h odd 6 1
1386.2.k.j 2 1.a even 1 1 trivial
1386.2.k.j 2 7.c even 3 1 inner
3234.2.a.a 1 21.g even 6 1
3234.2.a.o 1 21.h odd 6 1
9702.2.a.be 1 7.c even 3 1
9702.2.a.ce 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 $$S_{2}^{\mathrm{new}}(1386, [\chi])$$:

 $$T_{5}^{2} - 3T_{5} + 9$$ T5^2 - 3*T5 + 9 $$T_{13} - 6$$ T13 - 6 $$T_{17}^{2} + 5T_{17} + 25$$ T17^2 + 5*T17 + 25 $$T_{23}^{2} - 5T_{23} + 25$$ T23^2 - 5*T23 + 25

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 3T + 9$$
$7$ $$T^{2} + 5T + 7$$
$11$ $$T^{2} + T + 1$$
$13$ $$(T - 6)^{2}$$
$17$ $$T^{2} + 5T + 25$$
$19$ $$T^{2} + 6T + 36$$
$23$ $$T^{2} - 5T + 25$$
$29$ $$(T - 6)^{2}$$
$31$ $$T^{2} + 4T + 16$$
$37$ $$T^{2} - 2T + 4$$
$41$ $$(T + 5)^{2}$$
$43$ $$(T + 10)^{2}$$
$47$ $$T^{2} - 9T + 81$$
$53$ $$T^{2} - 2T + 4$$
$59$ $$T^{2} + 12T + 144$$
$61$ $$T^{2} - 5T + 25$$
$67$ $$T^{2} + 5T + 25$$
$71$ $$(T + 4)^{2}$$
$73$ $$T^{2} + 12T + 144$$
$79$ $$T^{2} - T + 1$$
$83$ $$(T + 1)^{2}$$
$89$ $$T^{2} - 6T + 36$$
$97$ $$(T - 9)^{2}$$