# Properties

 Label 1638.2.a.r Level $1638$ Weight $2$ Character orbit 1638.a Self dual yes Analytic conductor $13.079$ Analytic rank $0$ 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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + q^{5} - q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + q^5 - q^7 + q^8 $$q + q^{2} + q^{4} + q^{5} - q^{7} + q^{8} + q^{10} + q^{11} + q^{13} - q^{14} + q^{16} + q^{17} + 7 q^{19} + q^{20} + q^{22} - 3 q^{23} - 4 q^{25} + q^{26} - q^{28} + 3 q^{29} + 8 q^{31} + q^{32} + q^{34} - q^{35} + 7 q^{37} + 7 q^{38} + q^{40} - 8 q^{41} + 7 q^{43} + q^{44} - 3 q^{46} - 8 q^{47} + q^{49} - 4 q^{50} + q^{52} + 10 q^{53} + q^{55} - q^{56} + 3 q^{58} - 4 q^{59} + 7 q^{61} + 8 q^{62} + q^{64} + q^{65} + 2 q^{67} + q^{68} - q^{70} - 4 q^{71} - q^{73} + 7 q^{74} + 7 q^{76} - q^{77} + 2 q^{79} + q^{80} - 8 q^{82} + 6 q^{83} + q^{85} + 7 q^{86} + q^{88} - 14 q^{89} - q^{91} - 3 q^{92} - 8 q^{94} + 7 q^{95} - 14 q^{97} + q^{98}+O(q^{100})$$ q + q^2 + q^4 + q^5 - q^7 + q^8 + q^10 + q^11 + q^13 - q^14 + q^16 + q^17 + 7 * q^19 + q^20 + q^22 - 3 * q^23 - 4 * q^25 + q^26 - q^28 + 3 * q^29 + 8 * q^31 + q^32 + q^34 - q^35 + 7 * q^37 + 7 * q^38 + q^40 - 8 * q^41 + 7 * q^43 + q^44 - 3 * q^46 - 8 * q^47 + q^49 - 4 * q^50 + q^52 + 10 * q^53 + q^55 - q^56 + 3 * q^58 - 4 * q^59 + 7 * q^61 + 8 * q^62 + q^64 + q^65 + 2 * q^67 + q^68 - q^70 - 4 * q^71 - q^73 + 7 * q^74 + 7 * q^76 - q^77 + 2 * q^79 + q^80 - 8 * q^82 + 6 * q^83 + q^85 + 7 * q^86 + q^88 - 14 * q^89 - q^91 - 3 * q^92 - 8 * q^94 + 7 * q^95 - 14 * 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 1.00000 0 −1.00000 1.00000 0 1.00000
 $$n$$: 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.r 1
3.b odd 2 1 546.2.a.a 1
12.b even 2 1 4368.2.a.s 1
21.c even 2 1 3822.2.a.n 1
39.d odd 2 1 7098.2.a.t 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.a.a 1 3.b odd 2 1
1638.2.a.r 1 1.a even 1 1 trivial
3822.2.a.n 1 21.c even 2 1
4368.2.a.s 1 12.b even 2 1
7098.2.a.t 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1638))$$:

 $$T_{5} - 1$$ T5 - 1 $$T_{11} - 1$$ T11 - 1 $$T_{17} - 1$$ T17 - 1 $$T_{19} - 7$$ T19 - 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T + 1$$
$11$ $$T - 1$$
$13$ $$T - 1$$
$17$ $$T - 1$$
$19$ $$T - 7$$
$23$ $$T + 3$$
$29$ $$T - 3$$
$31$ $$T - 8$$
$37$ $$T - 7$$
$41$ $$T + 8$$
$43$ $$T - 7$$
$47$ $$T + 8$$
$53$ $$T - 10$$
$59$ $$T + 4$$
$61$ $$T - 7$$
$67$ $$T - 2$$
$71$ $$T + 4$$
$73$ $$T + 1$$
$79$ $$T - 2$$
$83$ $$T - 6$$
$89$ $$T + 14$$
$97$ $$T + 14$$