# Properties

 Label 1638.2.a.c 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 182) 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 $$q - q^{2} + q^{4} - 2 q^{5} - q^{7} - q^{8} + 2 q^{10} - 4 q^{11} - q^{13} + q^{14} + q^{16} + 6 q^{17} - 2 q^{20} + 4 q^{22} - 8 q^{23} - q^{25} + q^{26} - q^{28} + 10 q^{29} - 8 q^{31} - q^{32} - 6 q^{34} + 2 q^{35} + 6 q^{37} + 2 q^{40} + 6 q^{41} + 4 q^{43} - 4 q^{44} + 8 q^{46} + 8 q^{47} + q^{49} + q^{50} - q^{52} - 6 q^{53} + 8 q^{55} + q^{56} - 10 q^{58} - 8 q^{59} + 10 q^{61} + 8 q^{62} + q^{64} + 2 q^{65} + 4 q^{67} + 6 q^{68} - 2 q^{70} + 8 q^{71} + 2 q^{73} - 6 q^{74} + 4 q^{77} + 8 q^{79} - 2 q^{80} - 6 q^{82} - 12 q^{85} - 4 q^{86} + 4 q^{88} - 18 q^{89} + q^{91} - 8 q^{92} - 8 q^{94} + 2 q^{97} - q^{98}+O(q^{100})$$ 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 - 2 * q^20 + 4 * q^22 - 8 * q^23 - q^25 + q^26 - q^28 + 10 * q^29 - 8 * q^31 - q^32 - 6 * q^34 + 2 * q^35 + 6 * q^37 + 2 * q^40 + 6 * q^41 + 4 * q^43 - 4 * q^44 + 8 * q^46 + 8 * q^47 + q^49 + q^50 - q^52 - 6 * q^53 + 8 * q^55 + q^56 - 10 * q^58 - 8 * q^59 + 10 * q^61 + 8 * q^62 + q^64 + 2 * q^65 + 4 * q^67 + 6 * q^68 - 2 * q^70 + 8 * q^71 + 2 * q^73 - 6 * q^74 + 4 * q^77 + 8 * q^79 - 2 * q^80 - 6 * q^82 - 12 * q^85 - 4 * q^86 + 4 * q^88 - 18 * q^89 + q^91 - 8 * q^92 - 8 * q^94 + 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.c 1
3.b odd 2 1 182.2.a.c 1
12.b even 2 1 1456.2.a.i 1
15.d odd 2 1 4550.2.a.g 1
21.c even 2 1 1274.2.a.l 1
21.g even 6 2 1274.2.f.g 2
21.h odd 6 2 1274.2.f.f 2
24.f even 2 1 5824.2.a.m 1
24.h odd 2 1 5824.2.a.l 1
39.d odd 2 1 2366.2.a.d 1
39.f even 4 2 2366.2.d.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.a.c 1 3.b odd 2 1
1274.2.a.l 1 21.c even 2 1
1274.2.f.f 2 21.h odd 6 2
1274.2.f.g 2 21.g even 6 2
1456.2.a.i 1 12.b even 2 1
1638.2.a.c 1 1.a even 1 1 trivial
2366.2.a.d 1 39.d odd 2 1
2366.2.d.d 2 39.f even 4 2
4550.2.a.g 1 15.d odd 2 1
5824.2.a.l 1 24.h odd 2 1
5824.2.a.m 1 24.f even 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 $$T_{19}$$ T19

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T + 2$$
$7$ $$T + 1$$
$11$ $$T + 4$$
$13$ $$T + 1$$
$17$ $$T - 6$$
$19$ $$T$$
$23$ $$T + 8$$
$29$ $$T - 10$$
$31$ $$T + 8$$
$37$ $$T - 6$$
$41$ $$T - 6$$
$43$ $$T - 4$$
$47$ $$T - 8$$
$53$ $$T + 6$$
$59$ $$T + 8$$
$61$ $$T - 10$$
$67$ $$T - 4$$
$71$ $$T - 8$$
$73$ $$T - 2$$
$79$ $$T - 8$$
$83$ $$T$$
$89$ $$T + 18$$
$97$ $$T - 2$$