Let $f$ be a modular form on $\Gamma_0(4)$ of weight $k\in\tfrac 12\mathbb{Z}$ with trivial Nebentypus.
Is it true that if you twist $f$ by $\tfrac 12$, i.e. look at the function $g$ with $g(\tau)=f(\tau+\tfrac 12)$, this is again a modular form of the same weight $k$ on $\Gamma_0(4)$, but this time with Nebentypus $\chi$, the non-trivial Dirichlet character modulo 4?
Edit: You can show that $g$ is a modular form with trivial character on $\Gamma_0(16)$:
Consider the operators $V_m$ and $U_m$. If $f(\tau)=\sum\limits_{n=0}^\infty \alpha_f(n)q^n$ is the Fourier expansion of $f$ then these operators have the following effect:
$$ (V_mf)(\tau)=\sum \alpha_f(n)q^{mn}\quad\text{and}\quad (U_mf)(\tau)=\sum\alpha_f(mn)q^n.$$
They both send modular forms on $\Gamma_0(N)$ to modular forms on $\Gamma_0(mN)$ of the same weight. (cf. http://people.mpim-bonn.mpg.de/zagier/files/scanned/IntroductionToModularForms/fulltext.pdf, page 251 in the book, 14 in the pdf). If you apply $U_2$ and $V_2$ to $f$ you get the form $f_{ev}(\tau)=\sum\alpha_f(2n)q^{2n}$ on $\Gamma_0(16)$ and claearly we have $g(\tau)=2f_{ev}(\tau)-f(\tau)$.
I suppose there should either be a reference for this, which I have not been able to find, or a simple proof that I don't see right now.
Thank you very much for your help!
Best Answer
EDIT : In what follows the weight $k$ is assumed to be an integer.
The matrix $\begin{pmatrix} 1 & 1/2 \\ 0 & 1 \end{pmatrix}$ normalizes the group $\Gamma_0(4)$, so in your case $g$ is still a modular form on $\Gamma_0(4)$ with trivial Nebentypus.
More generally if you start with $f$ on $\Gamma_0(N)$ and twist it by $1/m$ then you get a form on $\Gamma_0(N') \cap \Gamma_1(m)$ with $N'=\operatorname{lcm}(N,m^2)$.
In general, when you consider $g(z)=f(z+1/m)$, you are twisting $f$ by the additive character $\alpha(n)=\exp(2\pi i n/m)$. You can always write $\alpha$ as a linear combination of (not necessarily primitive) Dirichlet characters $\chi$ of level dividing $m$. Thus you can write $g$ as a linear combination of twists $f \otimes \chi$ for such $\chi$ (up to bad Euler factors). If you want to do this completely explicitly then the formulas are quite complicated in general, see Merel, Symboles de Manin et valeurs de fonctions L, Section 2.5. At some point, it may be useful to switch to the adelic language and work with the automorphic representation associated to the newform $f$.