# First order p-adic deformations of weight one newforms

## TL;DR

This paper investigates the initial p-adic deformations of weight one newforms, linking their Fourier coefficients to p-adic logarithms of algebraic numbers in the related Galois field.

## Contribution

It introduces a novel connection between first-order p-adic deformations of weight one newforms and p-adic logarithms of algebraic numbers.

## Key findings

- Established a relationship between Fourier coefficients and p-adic logarithms.
- Provided a framework for understanding first-order p-adic deformations.
- Linked deformations to algebraic number theory and Galois representations.

## Abstract

This article studies the first-order $p$-adic deformations of classical weight one newforms, relating their fourier coefficients to the $p$-adic logarithms of algebraic numbers in the field cut out by the associated projective Galois representation.