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!



Modular Forms on $\Gamma_0(4)$ with Nebentypus
**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](http://www.math.jussieu.fr/~merel/M70.pdf), 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 $f$.


