# Properties

 Label 1638.2.bj.d Level $1638$ Weight $2$ Character orbit 1638.bj Analytic conductor $13.079$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1638,2,Mod(127,1638)]

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, 0, 5]))

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

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

chi := DirichletCharacter("1638.127");

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.bj (of order $$6$$, 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 546) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + (\zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{5} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{7} + \zeta_{12}^{3} q^{8}+O(q^{10})$$ q + z * q^2 + z^2 * q^4 + (z^3 - 2*z^2 + 1) * q^5 + (-z^3 + z) * q^7 + z^3 * q^8 $$q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + (\zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{5} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{7} + \zeta_{12}^{3} q^{8} + ( - 2 \zeta_{12}^{3} + \zeta_{12}^{2} + \cdots - 1) q^{10}+ \cdots + ( - \zeta_{12}^{3} + \zeta_{12}) q^{98}+O(q^{100})$$ q + z * q^2 + z^2 * q^4 + (z^3 - 2*z^2 + 1) * q^5 + (-z^3 + z) * q^7 + z^3 * q^8 + (-2*z^3 + z^2 + z - 1) * q^10 + (z^2 + 2*z + 1) * q^11 + (-3*z^3 - z) * q^13 + q^14 + (z^2 - 1) * q^16 + (z^3 - 2*z^2 + z) * q^17 + (-z^3 - 2*z^2 + z + 4) * q^19 + (z^3 - z^2 - z + 2) * q^20 + (z^3 + 2*z^2 + z) * q^22 + (4*z^3 - 2*z) * q^23 + (-2*z^3 + 4*z + 1) * q^25 + (-4*z^2 + 3) * q^26 + z * q^28 + (3*z^2 - 3) * q^29 + (z^3 - 10*z^2 + 5) * q^31 + (z^3 - z) * q^32 + (-2*z^3 + 2*z^2 - 1) * q^34 + (-z^3 + z^2 - z) * q^35 + (-z^2 + 3*z - 1) * q^37 + (-2*z^3 + 4*z + 1) * q^38 + (-z^3 + 2*z - 1) * q^40 + 7*z * q^41 + (z^3 - z^2 + z) * q^43 + (2*z^3 + 2*z^2 - 1) * q^44 + (2*z^2 - 4) * q^46 + (-z^3 - 4*z^2 + 2) * q^47 + (-z^2 + 1) * q^49 + (2*z^2 + z + 2) * q^50 + (-4*z^3 + 3*z) * q^52 + (-2*z^3 + 4*z + 7) * q^53 + (-2*z^3 - z^2 + z + 1) * q^55 + z^2 * q^56 + (3*z^3 - 3*z) * q^58 + (-6*z^3 + 4*z^2 + 6*z - 8) * q^59 + (z^3 + 10*z^2 + z) * q^61 + (-10*z^3 + z^2 + 5*z - 1) * q^62 - q^64 + (5*z^3 - z^2 - 7*z + 4) * q^65 + (-4*z^2 - 6*z - 4) * q^67 + (2*z^3 - 2*z^2 - z + 2) * q^68 + (z^3 - 2*z^2 + 1) * q^70 + (-3*z^3 + 3*z^2 + 3*z - 6) * q^71 + (10*z^3 - 4*z^2 + 2) * q^73 + (-z^3 + 3*z^2 - z) * q^74 + (2*z^2 + z + 2) * q^76 + (-z^3 + 2*z + 2) * q^77 + (-8*z^3 + 16*z - 3) * q^79 + (z^2 - z + 1) * q^80 + 7*z^2 * q^82 + (-3*z^3 + 10*z^2 - 5) * q^83 + (-5*z^3 + 3*z^2 + 5*z - 6) * q^85 + (-z^3 + 2*z^2 - 1) * q^86 + (2*z^3 + 2*z^2 - z - 2) * q^88 + (-2*z^2 - 3*z - 2) * q^89 + (-3*z^2 - 1) * q^91 + (2*z^3 - 4*z) * q^92 + (-4*z^3 - z^2 + 2*z + 1) * q^94 + (z^3 - 5*z^2 + z) * q^95 + (z^3 - z^2 - z + 2) * q^97 + (-z^3 + z) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4}+O(q^{10})$$ 4 * q + 2 * q^4 $$4 q + 2 q^{4} - 2 q^{10} + 6 q^{11} + 4 q^{14} - 2 q^{16} - 4 q^{17} + 12 q^{19} + 6 q^{20} + 4 q^{22} + 4 q^{25} + 4 q^{26} - 6 q^{29} + 2 q^{35} - 6 q^{37} + 4 q^{38} - 4 q^{40} - 2 q^{43} - 12 q^{46} + 2 q^{49} + 12 q^{50} + 28 q^{53} + 2 q^{55} + 2 q^{56} - 24 q^{59} + 20 q^{61} - 2 q^{62} - 4 q^{64} + 14 q^{65} - 24 q^{67} + 4 q^{68} - 18 q^{71} + 6 q^{74} + 12 q^{76} + 8 q^{77} - 12 q^{79} + 6 q^{80} + 14 q^{82} - 18 q^{85} - 4 q^{88} - 12 q^{89} - 10 q^{91} + 2 q^{94} - 10 q^{95} + 6 q^{97}+O(q^{100})$$ 4 * q + 2 * q^4 - 2 * q^10 + 6 * q^11 + 4 * q^14 - 2 * q^16 - 4 * q^17 + 12 * q^19 + 6 * q^20 + 4 * q^22 + 4 * q^25 + 4 * q^26 - 6 * q^29 + 2 * q^35 - 6 * q^37 + 4 * q^38 - 4 * q^40 - 2 * q^43 - 12 * q^46 + 2 * q^49 + 12 * q^50 + 28 * q^53 + 2 * q^55 + 2 * q^56 - 24 * q^59 + 20 * q^61 - 2 * q^62 - 4 * q^64 + 14 * q^65 - 24 * q^67 + 4 * q^68 - 18 * q^71 + 6 * q^74 + 12 * q^76 + 8 * q^77 - 12 * q^79 + 6 * q^80 + 14 * q^82 - 18 * q^85 - 4 * q^88 - 12 * q^89 - 10 * q^91 + 2 * q^94 - 10 * q^95 + 6 * q^97

## Character values

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

 $$n$$ $$379$$ $$703$$ $$911$$ $$\chi(n)$$ $$\zeta_{12}^{2}$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−0.866025 + 0.500000i 0 0.500000 0.866025i 2.73205i 0 −0.866025 0.500000i 1.00000i 0 −1.36603 2.36603i
127.2 0.866025 0.500000i 0 0.500000 0.866025i 0.732051i 0 0.866025 + 0.500000i 1.00000i 0 0.366025 + 0.633975i
1135.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 2.73205i 0 −0.866025 + 0.500000i 1.00000i 0 −1.36603 + 2.36603i
1135.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.732051i 0 0.866025 0.500000i 1.00000i 0 0.366025 0.633975i
 $$n$$: 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
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.bj.d 4
3.b odd 2 1 546.2.s.d 4
13.e even 6 1 inner 1638.2.bj.d 4
39.h odd 6 1 546.2.s.d 4
39.k even 12 1 7098.2.a.bj 2
39.k even 12 1 7098.2.a.bs 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.s.d 4 3.b odd 2 1
546.2.s.d 4 39.h odd 6 1
1638.2.bj.d 4 1.a even 1 1 trivial
1638.2.bj.d 4 13.e even 6 1 inner
7098.2.a.bj 2 39.k even 12 1
7098.2.a.bs 2 39.k even 12 1

## Hecke kernels

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

 $$T_{5}^{4} + 8T_{5}^{2} + 4$$ T5^4 + 8*T5^2 + 4 $$T_{11}^{4} - 6T_{11}^{3} + 11T_{11}^{2} + 6T_{11} + 1$$ T11^4 - 6*T11^3 + 11*T11^2 + 6*T11 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 1$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 8T^{2} + 4$$
$7$ $$T^{4} - T^{2} + 1$$
$11$ $$T^{4} - 6 T^{3} + \cdots + 1$$
$13$ $$T^{4} + 23T^{2} + 169$$
$17$ $$T^{4} + 4 T^{3} + \cdots + 1$$
$19$ $$T^{4} - 12 T^{3} + \cdots + 121$$
$23$ $$T^{4} + 12T^{2} + 144$$
$29$ $$(T^{2} + 3 T + 9)^{2}$$
$31$ $$T^{4} + 152T^{2} + 5476$$
$37$ $$T^{4} + 6 T^{3} + \cdots + 36$$
$41$ $$T^{4} - 49T^{2} + 2401$$
$43$ $$T^{4} + 2 T^{3} + \cdots + 4$$
$47$ $$T^{4} + 26T^{2} + 121$$
$53$ $$(T^{2} - 14 T + 37)^{2}$$
$59$ $$T^{4} + 24 T^{3} + \cdots + 144$$
$61$ $$T^{4} - 20 T^{3} + \cdots + 9409$$
$67$ $$T^{4} + 24 T^{3} + \cdots + 144$$
$71$ $$T^{4} + 18 T^{3} + \cdots + 324$$
$73$ $$T^{4} + 224T^{2} + 7744$$
$79$ $$(T^{2} + 6 T - 183)^{2}$$
$83$ $$T^{4} + 168T^{2} + 4356$$
$89$ $$T^{4} + 12 T^{3} + \cdots + 9$$
$97$ $$T^{4} - 6 T^{3} + \cdots + 4$$