# Properties

 Label 799.1.c.a Level $799$ Weight $1$ Character orbit 799.c Self dual yes Analytic conductor $0.399$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -47, -799, 17 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [799,1,Mod(798,799)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(799, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("799.798");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$799 = 17 \cdot 47$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 799.c (of order $$2$$, degree $$1$$, 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: yes Analytic conductor: $$0.398752945094$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{17}, \sqrt{-47})$$ Artin image: $D_4$ Artin field: Galois closure of 4.2.13583.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 3 q^{4} - 4 q^{8} + q^{9}+O(q^{10})$$ q - 2 * q^2 + 3 * q^4 - 4 * q^8 + q^9 $$q - 2 q^{2} + 3 q^{4} - 4 q^{8} + q^{9} + 5 q^{16} + q^{17} - 2 q^{18} - q^{25} - 6 q^{32} - 2 q^{34} + 3 q^{36} + q^{47} + q^{49} + 2 q^{50} + 2 q^{53} - 2 q^{59} + 7 q^{64} + 3 q^{68} - 4 q^{72} + q^{81} + 2 q^{83} + 2 q^{89} - 2 q^{94} - 2 q^{98}+O(q^{100})$$ q - 2 * q^2 + 3 * q^4 - 4 * q^8 + q^9 + 5 * q^16 + q^17 - 2 * q^18 - q^25 - 6 * q^32 - 2 * q^34 + 3 * q^36 + q^47 + q^49 + 2 * q^50 + 2 * q^53 - 2 * q^59 + 7 * q^64 + 3 * q^68 - 4 * q^72 + q^81 + 2 * q^83 + 2 * q^89 - 2 * q^94 - 2 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/799\mathbb{Z}\right)^\times$$.

 $$n$$ $$52$$ $$377$$ $$\chi(n)$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
798.1
 0
−2.00000 0 3.00000 0 0 0 −4.00000 1.00000 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
17.b even 2 1 RM by $$\Q(\sqrt{17})$$
47.b odd 2 1 CM by $$\Q(\sqrt{-47})$$
799.c odd 2 1 CM by $$\Q(\sqrt{-799})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 799.1.c.a 1
17.b even 2 1 RM 799.1.c.a 1
47.b odd 2 1 CM 799.1.c.a 1
799.c odd 2 1 CM 799.1.c.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
799.1.c.a 1 1.a even 1 1 trivial
799.1.c.a 1 17.b even 2 1 RM
799.1.c.a 1 47.b odd 2 1 CM
799.1.c.a 1 799.c odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 2$$ acting on $$S_{1}^{\mathrm{new}}(799, [\chi])$$.

## Hecke characteristic polynomials

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