# Properties

 Label 1638.2.j.i 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} + ( - \zeta_{6} - 2) q^{7} - q^{8}+O(q^{10})$$ q + z * q^2 + (z - 1) * q^4 + (-z - 2) * q^7 - q^8 $$q + \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} + ( - \zeta_{6} - 2) q^{7} - q^{8} + (5 \zeta_{6} - 5) q^{11} - q^{13} + ( - 3 \zeta_{6} + 1) q^{14} - \zeta_{6} q^{16} + ( - 7 \zeta_{6} + 7) q^{17} - 7 \zeta_{6} q^{19} - 5 q^{22} + 2 \zeta_{6} q^{23} + ( - 5 \zeta_{6} + 5) q^{25} - \zeta_{6} q^{26} + ( - 2 \zeta_{6} + 3) q^{28} + 9 q^{29} + ( - \zeta_{6} + 1) q^{32} + 7 q^{34} - 4 \zeta_{6} q^{37} + ( - 7 \zeta_{6} + 7) q^{38} - 4 q^{41} + 2 q^{43} - 5 \zeta_{6} q^{44} + (2 \zeta_{6} - 2) q^{46} - 3 \zeta_{6} q^{47} + (5 \zeta_{6} + 3) q^{49} + 5 q^{50} + ( - \zeta_{6} + 1) q^{52} + ( - \zeta_{6} + 1) q^{53} + (\zeta_{6} + 2) q^{56} + 9 \zeta_{6} q^{58} + ( - 7 \zeta_{6} + 7) q^{59} - 13 \zeta_{6} q^{61} + q^{64} + (3 \zeta_{6} - 3) q^{67} + 7 \zeta_{6} q^{68} - 9 q^{71} + ( - 10 \zeta_{6} + 10) q^{73} + ( - 4 \zeta_{6} + 4) q^{74} + 7 q^{76} + ( - 10 \zeta_{6} + 15) q^{77} - 14 \zeta_{6} q^{79} - 4 \zeta_{6} q^{82} - 16 q^{83} + 2 \zeta_{6} q^{86} + ( - 5 \zeta_{6} + 5) q^{88} - 12 \zeta_{6} q^{89} + (\zeta_{6} + 2) q^{91} - 2 q^{92} + ( - 3 \zeta_{6} + 3) q^{94} + 6 q^{97} + (8 \zeta_{6} - 5) q^{98} +O(q^{100})$$ q + z * q^2 + (z - 1) * q^4 + (-z - 2) * q^7 - q^8 + (5*z - 5) * q^11 - q^13 + (-3*z + 1) * q^14 - z * q^16 + (-7*z + 7) * q^17 - 7*z * q^19 - 5 * q^22 + 2*z * q^23 + (-5*z + 5) * q^25 - z * q^26 + (-2*z + 3) * q^28 + 9 * q^29 + (-z + 1) * q^32 + 7 * q^34 - 4*z * q^37 + (-7*z + 7) * q^38 - 4 * q^41 + 2 * q^43 - 5*z * q^44 + (2*z - 2) * q^46 - 3*z * q^47 + (5*z + 3) * q^49 + 5 * q^50 + (-z + 1) * q^52 + (-z + 1) * q^53 + (z + 2) * q^56 + 9*z * q^58 + (-7*z + 7) * q^59 - 13*z * q^61 + q^64 + (3*z - 3) * q^67 + 7*z * q^68 - 9 * q^71 + (-10*z + 10) * q^73 + (-4*z + 4) * q^74 + 7 * q^76 + (-10*z + 15) * q^77 - 14*z * q^79 - 4*z * q^82 - 16 * q^83 + 2*z * q^86 + (-5*z + 5) * q^88 - 12*z * q^89 + (z + 2) * q^91 - 2 * q^92 + (-3*z + 3) * q^94 + 6 * q^97 + (8*z - 5) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - q^{4} - 5 q^{7} - 2 q^{8}+O(q^{10})$$ 2 * q + q^2 - q^4 - 5 * q^7 - 2 * q^8 $$2 q + q^{2} - q^{4} - 5 q^{7} - 2 q^{8} - 5 q^{11} - 2 q^{13} - q^{14} - q^{16} + 7 q^{17} - 7 q^{19} - 10 q^{22} + 2 q^{23} + 5 q^{25} - q^{26} + 4 q^{28} + 18 q^{29} + q^{32} + 14 q^{34} - 4 q^{37} + 7 q^{38} - 8 q^{41} + 4 q^{43} - 5 q^{44} - 2 q^{46} - 3 q^{47} + 11 q^{49} + 10 q^{50} + q^{52} + q^{53} + 5 q^{56} + 9 q^{58} + 7 q^{59} - 13 q^{61} + 2 q^{64} - 3 q^{67} + 7 q^{68} - 18 q^{71} + 10 q^{73} + 4 q^{74} + 14 q^{76} + 20 q^{77} - 14 q^{79} - 4 q^{82} - 32 q^{83} + 2 q^{86} + 5 q^{88} - 12 q^{89} + 5 q^{91} - 4 q^{92} + 3 q^{94} + 12 q^{97} - 2 q^{98}+O(q^{100})$$ 2 * q + q^2 - q^4 - 5 * q^7 - 2 * q^8 - 5 * q^11 - 2 * q^13 - q^14 - q^16 + 7 * q^17 - 7 * q^19 - 10 * q^22 + 2 * q^23 + 5 * q^25 - q^26 + 4 * q^28 + 18 * q^29 + q^32 + 14 * q^34 - 4 * q^37 + 7 * q^38 - 8 * q^41 + 4 * q^43 - 5 * q^44 - 2 * q^46 - 3 * q^47 + 11 * q^49 + 10 * q^50 + q^52 + q^53 + 5 * q^56 + 9 * q^58 + 7 * q^59 - 13 * q^61 + 2 * q^64 - 3 * q^67 + 7 * q^68 - 18 * q^71 + 10 * q^73 + 4 * q^74 + 14 * q^76 + 20 * q^77 - 14 * q^79 - 4 * q^82 - 32 * q^83 + 2 * q^86 + 5 * q^88 - 12 * q^89 + 5 * q^91 - 4 * q^92 + 3 * q^94 + 12 * q^97 - 2 * 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 0 0 −2.50000 0.866025i −1.00000 0 0
1171.1 0.500000 0.866025i 0 −0.500000 0.866025i 0 0 −2.50000 + 0.866025i −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 1638.2.j.i 2
3.b odd 2 1 546.2.i.d 2
7.c even 3 1 inner 1638.2.j.i 2
21.g even 6 1 3822.2.a.bf 1
21.h odd 6 1 546.2.i.d 2
21.h odd 6 1 3822.2.a.u 1

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

## Hecke kernels

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

 $$T_{5}$$ T5 $$T_{11}^{2} + 5T_{11} + 25$$ T11^2 + 5*T11 + 25 $$T_{17}^{2} - 7T_{17} + 49$$ T17^2 - 7*T17 + 49

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 5T + 7$$
$11$ $$T^{2} + 5T + 25$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} - 7T + 49$$
$19$ $$T^{2} + 7T + 49$$
$23$ $$T^{2} - 2T + 4$$
$29$ $$(T - 9)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 4T + 16$$
$41$ $$(T + 4)^{2}$$
$43$ $$(T - 2)^{2}$$
$47$ $$T^{2} + 3T + 9$$
$53$ $$T^{2} - T + 1$$
$59$ $$T^{2} - 7T + 49$$
$61$ $$T^{2} + 13T + 169$$
$67$ $$T^{2} + 3T + 9$$
$71$ $$(T + 9)^{2}$$
$73$ $$T^{2} - 10T + 100$$
$79$ $$T^{2} + 14T + 196$$
$83$ $$(T + 16)^{2}$$
$89$ $$T^{2} + 12T + 144$$
$97$ $$(T - 6)^{2}$$