# Properties

 Label 1638.2.bj.e Level $1638$ Weight $2$ Character orbit 1638.bj Analytic conductor $13.079$ 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] = [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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{12}^{3} + \zeta_{12}) q^{2} + ( - \zeta_{12}^{2} + 1) q^{4} + ( - 4 \zeta_{12}^{2} + 2) q^{5} + \zeta_{12} q^{7} - \zeta_{12}^{3} q^{8} +O(q^{10})$$ q + (-z^3 + z) * q^2 + (-z^2 + 1) * q^4 + (-4*z^2 + 2) * q^5 + z * q^7 - z^3 * q^8 $$q + ( - \zeta_{12}^{3} + \zeta_{12}) q^{2} + ( - \zeta_{12}^{2} + 1) q^{4} + ( - 4 \zeta_{12}^{2} + 2) q^{5} + \zeta_{12} q^{7} - \zeta_{12}^{3} q^{8} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{10} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{11} + ( - \zeta_{12}^{3} - 3 \zeta_{12}) q^{13} + q^{14} - \zeta_{12}^{2} q^{16} + ( - 4 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 2 \zeta_{12} - 3) q^{17} + (2 \zeta_{12}^{2} - 4 \zeta_{12} + 2) q^{19} + ( - 2 \zeta_{12}^{2} - 2) q^{20} + ( - 4 \zeta_{12}^{2} + 4) q^{22} + (\zeta_{12}^{3} - 4 \zeta_{12}^{2} + \zeta_{12}) q^{23} - 7 q^{25} + ( - \zeta_{12}^{2} - 3) q^{26} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{28} + 8 \zeta_{12}^{2} q^{29} + ( - 3 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2) q^{31} - \zeta_{12} q^{32} + (3 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2) q^{34} + ( - 4 \zeta_{12}^{3} + 2 \zeta_{12}) q^{35} + (2 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2 \zeta_{12} - 8) q^{37} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12} - 4) q^{38} + (2 \zeta_{12}^{3} - 4 \zeta_{12}) q^{40} + ( - 4 \zeta_{12}^{2} + 8) q^{41} + (4 \zeta_{12}^{3} - \zeta_{12}^{2} - 2 \zeta_{12} + 1) q^{43} - 4 \zeta_{12}^{3} q^{44} + (\zeta_{12}^{2} - 4 \zeta_{12} + 1) q^{46} + ( - 4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2) q^{47} + \zeta_{12}^{2} q^{49} + (7 \zeta_{12}^{3} - 7 \zeta_{12}) q^{50} + (3 \zeta_{12}^{3} - 4 \zeta_{12}) q^{52} + ( - \zeta_{12}^{3} + 2 \zeta_{12} + 10) q^{53} + ( - 8 \zeta_{12}^{3} - 8 \zeta_{12}) q^{55} + ( - \zeta_{12}^{2} + 1) q^{56} + 8 \zeta_{12} q^{58} + ( - \zeta_{12}^{2} - 8 \zeta_{12} - 1) q^{59} + (6 \zeta_{12}^{3} - 3 \zeta_{12}) q^{61} + (2 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 2 \zeta_{12}) q^{62} - q^{64} + (14 \zeta_{12}^{3} - 10 \zeta_{12}) q^{65} + (2 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 2 \zeta_{12} + 6) q^{67} + ( - 2 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 2 \zeta_{12}) q^{68} + ( - 4 \zeta_{12}^{2} + 2) q^{70} + ( - 4 \zeta_{12}^{2} - 5 \zeta_{12} - 4) q^{71} + (4 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 4) q^{73} + (8 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 4 \zeta_{12} - 2) q^{74} + (4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 4 \zeta_{12} + 4) q^{76} + 4 q^{77} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12} - 6) q^{79} + (2 \zeta_{12}^{2} - 4) q^{80} + ( - 8 \zeta_{12}^{3} + 4 \zeta_{12}) q^{82} + (2 \zeta_{12}^{2} - 1) q^{83} + (6 \zeta_{12}^{2} - 12 \zeta_{12} + 6) q^{85} + ( - \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2) q^{86} - 4 \zeta_{12}^{2} q^{88} + (10 \zeta_{12}^{3} - \zeta_{12}^{2} - 10 \zeta_{12} + 2) q^{89} + ( - 4 \zeta_{12}^{2} + 1) q^{91} + ( - \zeta_{12}^{3} + 2 \zeta_{12} - 4) q^{92} + ( - 2 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 2 \zeta_{12}) q^{94} + (16 \zeta_{12}^{3} - 12 \zeta_{12}^{2} - 8 \zeta_{12} + 12) q^{95} + (6 \zeta_{12}^{2} - 2 \zeta_{12} + 6) q^{97} + \zeta_{12} q^{98} +O(q^{100})$$ q + (-z^3 + z) * q^2 + (-z^2 + 1) * q^4 + (-4*z^2 + 2) * q^5 + z * q^7 - z^3 * q^8 + (-2*z^3 - 2*z) * q^10 + (-4*z^3 + 4*z) * q^11 + (-z^3 - 3*z) * q^13 + q^14 - z^2 * q^16 + (-4*z^3 + 3*z^2 + 2*z - 3) * q^17 + (2*z^2 - 4*z + 2) * q^19 + (-2*z^2 - 2) * q^20 + (-4*z^2 + 4) * q^22 + (z^3 - 4*z^2 + z) * q^23 - 7 * q^25 + (-z^2 - 3) * q^26 + (-z^3 + z) * q^28 + 8*z^2 * q^29 + (-3*z^3 + 4*z^2 - 2) * q^31 - z * q^32 + (3*z^3 - 4*z^2 + 2) * q^34 + (-4*z^3 + 2*z) * q^35 + (2*z^3 + 4*z^2 - 2*z - 8) * q^37 + (-2*z^3 + 4*z - 4) * q^38 + (2*z^3 - 4*z) * q^40 + (-4*z^2 + 8) * q^41 + (4*z^3 - z^2 - 2*z + 1) * q^43 - 4*z^3 * q^44 + (z^2 - 4*z + 1) * q^46 + (-4*z^3 - 4*z^2 + 2) * q^47 + z^2 * q^49 + (7*z^3 - 7*z) * q^50 + (3*z^3 - 4*z) * q^52 + (-z^3 + 2*z + 10) * q^53 + (-8*z^3 - 8*z) * q^55 + (-z^2 + 1) * q^56 + 8*z * q^58 + (-z^2 - 8*z - 1) * q^59 + (6*z^3 - 3*z) * q^61 + (2*z^3 - 3*z^2 + 2*z) * q^62 - q^64 + (14*z^3 - 10*z) * q^65 + (2*z^3 - 3*z^2 - 2*z + 6) * q^67 + (-2*z^3 + 3*z^2 - 2*z) * q^68 + (-4*z^2 + 2) * q^70 + (-4*z^2 - 5*z - 4) * q^71 + (4*z^3 - 8*z^2 + 4) * q^73 + (8*z^3 + 2*z^2 - 4*z - 2) * q^74 + (4*z^3 - 2*z^2 - 4*z + 4) * q^76 + 4 * q^77 + (-2*z^3 + 4*z - 6) * q^79 + (2*z^2 - 4) * q^80 + (-8*z^3 + 4*z) * q^82 + (2*z^2 - 1) * q^83 + (6*z^2 - 12*z + 6) * q^85 + (-z^3 + 4*z^2 - 2) * q^86 - 4*z^2 * q^88 + (10*z^3 - z^2 - 10*z + 2) * q^89 + (-4*z^2 + 1) * q^91 + (-z^3 + 2*z - 4) * q^92 + (-2*z^3 - 4*z^2 - 2*z) * q^94 + (16*z^3 - 12*z^2 - 8*z + 12) * q^95 + (6*z^2 - 2*z + 6) * q^97 + z * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4}+O(q^{10})$$ 4 * q + 2 * q^4 $$4 q + 2 q^{4} + 4 q^{14} - 2 q^{16} - 6 q^{17} + 12 q^{19} - 12 q^{20} + 8 q^{22} - 8 q^{23} - 28 q^{25} - 14 q^{26} + 16 q^{29} - 24 q^{37} - 16 q^{38} + 24 q^{41} + 2 q^{43} + 6 q^{46} + 2 q^{49} + 40 q^{53} + 2 q^{56} - 6 q^{59} - 6 q^{62} - 4 q^{64} + 18 q^{67} + 6 q^{68} - 24 q^{71} - 4 q^{74} + 12 q^{76} + 16 q^{77} - 24 q^{79} - 12 q^{80} + 36 q^{85} - 8 q^{88} + 6 q^{89} - 4 q^{91} - 16 q^{92} - 8 q^{94} + 24 q^{95} + 36 q^{97}+O(q^{100})$$ 4 * q + 2 * q^4 + 4 * q^14 - 2 * q^16 - 6 * q^17 + 12 * q^19 - 12 * q^20 + 8 * q^22 - 8 * q^23 - 28 * q^25 - 14 * q^26 + 16 * q^29 - 24 * q^37 - 16 * q^38 + 24 * q^41 + 2 * q^43 + 6 * q^46 + 2 * q^49 + 40 * q^53 + 2 * q^56 - 6 * q^59 - 6 * q^62 - 4 * q^64 + 18 * q^67 + 6 * q^68 - 24 * q^71 - 4 * q^74 + 12 * q^76 + 16 * q^77 - 24 * q^79 - 12 * q^80 + 36 * q^85 - 8 * q^88 + 6 * q^89 - 4 * q^91 - 16 * q^92 - 8 * q^94 + 24 * q^95 + 36 * q^97

## Character values

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

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

## 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 3.46410i 0 −0.866025 0.500000i 1.00000i 0 1.73205 + 3.00000i
127.2 0.866025 0.500000i 0 0.500000 0.866025i 3.46410i 0 0.866025 + 0.500000i 1.00000i 0 −1.73205 3.00000i
1135.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 3.46410i 0 −0.866025 + 0.500000i 1.00000i 0 1.73205 3.00000i
1135.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 3.46410i 0 0.866025 0.500000i 1.00000i 0 −1.73205 + 3.00000i
 $$n$$: 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.e 4
3.b odd 2 1 546.2.s.c 4
13.e even 6 1 inner 1638.2.bj.e 4
39.h odd 6 1 546.2.s.c 4
39.k even 12 1 7098.2.a.bn 2
39.k even 12 1 7098.2.a.bz 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.s.c 4 3.b odd 2 1
546.2.s.c 4 39.h odd 6 1
1638.2.bj.e 4 1.a even 1 1 trivial
1638.2.bj.e 4 13.e even 6 1 inner
7098.2.a.bn 2 39.k even 12 1
7098.2.a.bz 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 $$S_{2}^{\mathrm{new}}(1638, [\chi])$$:

 $$T_{5}^{2} + 12$$ T5^2 + 12 $$T_{11}^{4} - 16T_{11}^{2} + 256$$ T11^4 - 16*T11^2 + 256

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 1$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 12)^{2}$$
$7$ $$T^{4} - T^{2} + 1$$
$11$ $$T^{4} - 16T^{2} + 256$$
$13$ $$T^{4} - T^{2} + 169$$
$17$ $$T^{4} + 6 T^{3} + 39 T^{2} - 18 T + 9$$
$19$ $$T^{4} - 12 T^{3} + 44 T^{2} + 48 T + 16$$
$23$ $$T^{4} + 8 T^{3} + 51 T^{2} + 104 T + 169$$
$29$ $$(T^{2} - 8 T + 64)^{2}$$
$31$ $$T^{4} + 42T^{2} + 9$$
$37$ $$T^{4} + 24 T^{3} + 236 T^{2} + \cdots + 1936$$
$41$ $$(T^{2} - 12 T + 48)^{2}$$
$43$ $$T^{4} - 2 T^{3} + 15 T^{2} + 22 T + 121$$
$47$ $$T^{4} + 56T^{2} + 16$$
$53$ $$(T^{2} - 20 T + 97)^{2}$$
$59$ $$T^{4} + 6 T^{3} - 49 T^{2} + \cdots + 3721$$
$61$ $$T^{4} + 27T^{2} + 729$$
$67$ $$T^{4} - 18 T^{3} + 131 T^{2} + \cdots + 529$$
$71$ $$T^{4} + 24 T^{3} + 215 T^{2} + \cdots + 529$$
$73$ $$T^{4} + 128T^{2} + 1024$$
$79$ $$(T^{2} + 12 T + 24)^{2}$$
$83$ $$(T^{2} + 3)^{2}$$
$89$ $$T^{4} - 6 T^{3} - 85 T^{2} + \cdots + 9409$$
$97$ $$T^{4} - 36 T^{3} + 536 T^{2} + \cdots + 10816$$