Properties

 Label 1638.2.j.j Level $1638$ Weight $2$ Character orbit 1638.j Analytic conductor $13.079$ Analytic rank $0$ Dimension $2$ Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1638,2,Mod(235,1638)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1638, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4, 0]))

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

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

chi := DirichletCharacter("1638.235");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1638.j (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: $$13.0794958511$$ 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 546) 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} + 3 \zeta_{6} q^{5} + ( - 3 \zeta_{6} + 1) q^{7} - q^{8}+O(q^{10})$$ q + z * q^2 + (z - 1) * q^4 + 3*z * q^5 + (-3*z + 1) * q^7 - q^8 $$q + \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} + 3 \zeta_{6} q^{5} + ( - 3 \zeta_{6} + 1) q^{7} - q^{8} + (3 \zeta_{6} - 3) q^{10} + ( - 3 \zeta_{6} + 3) q^{11} + q^{13} + ( - 2 \zeta_{6} + 3) q^{14} - \zeta_{6} q^{16} + ( - 6 \zeta_{6} + 6) q^{17} + 4 \zeta_{6} q^{19} - 3 q^{20} + 3 q^{22} + 6 \zeta_{6} q^{23} + (4 \zeta_{6} - 4) q^{25} + \zeta_{6} q^{26} + (\zeta_{6} + 2) q^{28} + 9 q^{29} + (5 \zeta_{6} - 5) q^{31} + ( - \zeta_{6} + 1) q^{32} + 6 q^{34} + ( - 6 \zeta_{6} + 9) q^{35} + 4 \zeta_{6} q^{37} + (4 \zeta_{6} - 4) q^{38} - 3 \zeta_{6} q^{40} + 12 q^{41} - 4 q^{43} + 3 \zeta_{6} q^{44} + (6 \zeta_{6} - 6) q^{46} - 12 \zeta_{6} q^{47} + (3 \zeta_{6} - 8) q^{49} - 4 q^{50} + (\zeta_{6} - 1) q^{52} + (9 \zeta_{6} - 9) q^{53} + 9 q^{55} + (3 \zeta_{6} - 1) q^{56} + 9 \zeta_{6} q^{58} + (9 \zeta_{6} - 9) q^{59} - 8 \zeta_{6} q^{61} - 5 q^{62} + q^{64} + 3 \zeta_{6} q^{65} + ( - 4 \zeta_{6} + 4) q^{67} + 6 \zeta_{6} q^{68} + (3 \zeta_{6} + 6) q^{70} - 6 q^{71} + (14 \zeta_{6} - 14) q^{73} + (4 \zeta_{6} - 4) q^{74} - 4 q^{76} + ( - 3 \zeta_{6} - 6) q^{77} + \zeta_{6} q^{79} + ( - 3 \zeta_{6} + 3) q^{80} + 12 \zeta_{6} q^{82} - 3 q^{83} + 18 q^{85} - 4 \zeta_{6} q^{86} + (3 \zeta_{6} - 3) q^{88} + ( - 3 \zeta_{6} + 1) q^{91} - 6 q^{92} + ( - 12 \zeta_{6} + 12) q^{94} + (12 \zeta_{6} - 12) q^{95} + 5 q^{97} + ( - 5 \zeta_{6} - 3) q^{98} +O(q^{100})$$ q + z * q^2 + (z - 1) * q^4 + 3*z * q^5 + (-3*z + 1) * q^7 - q^8 + (3*z - 3) * q^10 + (-3*z + 3) * q^11 + q^13 + (-2*z + 3) * q^14 - z * q^16 + (-6*z + 6) * q^17 + 4*z * q^19 - 3 * q^20 + 3 * q^22 + 6*z * q^23 + (4*z - 4) * q^25 + z * q^26 + (z + 2) * q^28 + 9 * q^29 + (5*z - 5) * q^31 + (-z + 1) * q^32 + 6 * q^34 + (-6*z + 9) * q^35 + 4*z * q^37 + (4*z - 4) * q^38 - 3*z * q^40 + 12 * q^41 - 4 * q^43 + 3*z * q^44 + (6*z - 6) * q^46 - 12*z * q^47 + (3*z - 8) * q^49 - 4 * q^50 + (z - 1) * q^52 + (9*z - 9) * q^53 + 9 * q^55 + (3*z - 1) * q^56 + 9*z * q^58 + (9*z - 9) * q^59 - 8*z * q^61 - 5 * q^62 + q^64 + 3*z * q^65 + (-4*z + 4) * q^67 + 6*z * q^68 + (3*z + 6) * q^70 - 6 * q^71 + (14*z - 14) * q^73 + (4*z - 4) * q^74 - 4 * q^76 + (-3*z - 6) * q^77 + z * q^79 + (-3*z + 3) * q^80 + 12*z * q^82 - 3 * q^83 + 18 * q^85 - 4*z * q^86 + (3*z - 3) * q^88 + (-3*z + 1) * q^91 - 6 * q^92 + (-12*z + 12) * q^94 + (12*z - 12) * q^95 + 5 * q^97 + (-5*z - 3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - q^{4} + 3 q^{5} - q^{7} - 2 q^{8}+O(q^{10})$$ 2 * q + q^2 - q^4 + 3 * q^5 - q^7 - 2 * q^8 $$2 q + q^{2} - q^{4} + 3 q^{5} - q^{7} - 2 q^{8} - 3 q^{10} + 3 q^{11} + 2 q^{13} + 4 q^{14} - q^{16} + 6 q^{17} + 4 q^{19} - 6 q^{20} + 6 q^{22} + 6 q^{23} - 4 q^{25} + q^{26} + 5 q^{28} + 18 q^{29} - 5 q^{31} + q^{32} + 12 q^{34} + 12 q^{35} + 4 q^{37} - 4 q^{38} - 3 q^{40} + 24 q^{41} - 8 q^{43} + 3 q^{44} - 6 q^{46} - 12 q^{47} - 13 q^{49} - 8 q^{50} - q^{52} - 9 q^{53} + 18 q^{55} + q^{56} + 9 q^{58} - 9 q^{59} - 8 q^{61} - 10 q^{62} + 2 q^{64} + 3 q^{65} + 4 q^{67} + 6 q^{68} + 15 q^{70} - 12 q^{71} - 14 q^{73} - 4 q^{74} - 8 q^{76} - 15 q^{77} + q^{79} + 3 q^{80} + 12 q^{82} - 6 q^{83} + 36 q^{85} - 4 q^{86} - 3 q^{88} - q^{91} - 12 q^{92} + 12 q^{94} - 12 q^{95} + 10 q^{97} - 11 q^{98}+O(q^{100})$$ 2 * q + q^2 - q^4 + 3 * q^5 - q^7 - 2 * q^8 - 3 * q^10 + 3 * q^11 + 2 * q^13 + 4 * q^14 - q^16 + 6 * q^17 + 4 * q^19 - 6 * q^20 + 6 * q^22 + 6 * q^23 - 4 * q^25 + q^26 + 5 * q^28 + 18 * q^29 - 5 * q^31 + q^32 + 12 * q^34 + 12 * q^35 + 4 * q^37 - 4 * q^38 - 3 * q^40 + 24 * q^41 - 8 * q^43 + 3 * q^44 - 6 * q^46 - 12 * q^47 - 13 * q^49 - 8 * q^50 - q^52 - 9 * q^53 + 18 * q^55 + q^56 + 9 * q^58 - 9 * q^59 - 8 * q^61 - 10 * q^62 + 2 * q^64 + 3 * q^65 + 4 * q^67 + 6 * q^68 + 15 * q^70 - 12 * q^71 - 14 * q^73 - 4 * q^74 - 8 * q^76 - 15 * q^77 + q^79 + 3 * q^80 + 12 * q^82 - 6 * q^83 + 36 * q^85 - 4 * q^86 - 3 * q^88 - q^91 - 12 * q^92 + 12 * q^94 - 12 * q^95 + 10 * q^97 - 11 * q^98

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times$$.

 $$n$$ $$379$$ $$703$$ $$911$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
235.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.50000 + 2.59808i 0 −0.500000 2.59808i −1.00000 0 −1.50000 + 2.59808i
1171.1 0.500000 0.866025i 0 −0.500000 0.866025i 1.50000 2.59808i 0 −0.500000 + 2.59808i −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 1638.2.j.j 2
3.b odd 2 1 546.2.i.a 2
7.c even 3 1 inner 1638.2.j.j 2
21.g even 6 1 3822.2.a.s 1
21.h odd 6 1 546.2.i.a 2
21.h odd 6 1 3822.2.a.bh 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.i.a 2 3.b odd 2 1
546.2.i.a 2 21.h odd 6 1
1638.2.j.j 2 1.a even 1 1 trivial
1638.2.j.j 2 7.c even 3 1 inner
3822.2.a.s 1 21.g even 6 1
3822.2.a.bh 1 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1638, [\chi])$$:

 $$T_{5}^{2} - 3T_{5} + 9$$ T5^2 - 3*T5 + 9 $$T_{11}^{2} - 3T_{11} + 9$$ T11^2 - 3*T11 + 9 $$T_{17}^{2} - 6T_{17} + 36$$ T17^2 - 6*T17 + 36

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 3T + 9$$
$7$ $$T^{2} + T + 7$$
$11$ $$T^{2} - 3T + 9$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2} - 6T + 36$$
$19$ $$T^{2} - 4T + 16$$
$23$ $$T^{2} - 6T + 36$$
$29$ $$(T - 9)^{2}$$
$31$ $$T^{2} + 5T + 25$$
$37$ $$T^{2} - 4T + 16$$
$41$ $$(T - 12)^{2}$$
$43$ $$(T + 4)^{2}$$
$47$ $$T^{2} + 12T + 144$$
$53$ $$T^{2} + 9T + 81$$
$59$ $$T^{2} + 9T + 81$$
$61$ $$T^{2} + 8T + 64$$
$67$ $$T^{2} - 4T + 16$$
$71$ $$(T + 6)^{2}$$
$73$ $$T^{2} + 14T + 196$$
$79$ $$T^{2} - T + 1$$
$83$ $$(T + 3)^{2}$$
$89$ $$T^{2}$$
$97$ $$(T - 5)^{2}$$