# Properties

 Label 1638.2.a.d Level $1638$ Weight $2$ Character orbit 1638.a Self dual yes Analytic conductor $13.079$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1638,2,Mod(1,1638)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1638, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1638.1");

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.a (trivial)

## 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: yes Analytic conductor: $$13.0794958511$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 546) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−1.00000 0 1.00000 −2.00000 0 1.00000 −1.00000 0 2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$+1$$
$$3$$ $$-1$$
$$7$$ $$-1$$
$$13$$ $$+1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.a.d 1
3.b odd 2 1 546.2.a.g 1
12.b even 2 1 4368.2.a.k 1
21.c even 2 1 3822.2.a.t 1
39.d odd 2 1 7098.2.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.a.g 1 3.b odd 2 1
1638.2.a.d 1 1.a even 1 1 trivial
3822.2.a.t 1 21.c even 2 1
4368.2.a.k 1 12.b even 2 1
7098.2.a.j 1 39.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1638))$$:

 $$T_{5} + 2$$ T5 + 2 $$T_{11} - 4$$ T11 - 4 $$T_{17} + 6$$ T17 + 6 $$T_{19} + 4$$ T19 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T + 2$$
$7$ $$T - 1$$
$11$ $$T - 4$$
$13$ $$T + 1$$
$17$ $$T + 6$$
$19$ $$T + 4$$
$23$ $$T$$
$29$ $$T - 6$$
$31$ $$T + 8$$
$37$ $$T - 10$$
$41$ $$T - 6$$
$43$ $$T - 4$$
$47$ $$T + 4$$
$53$ $$T + 10$$
$59$ $$T + 4$$
$61$ $$T + 6$$
$67$ $$T + 8$$
$71$ $$T$$
$73$ $$T + 10$$
$79$ $$T + 8$$
$83$ $$T + 4$$
$89$ $$T - 6$$
$97$ $$T + 2$$