Given a modular form, what is the precise formulation of BSD (in particular, the residue formula for the $L$-function at special values)? And what about the special values if the $L$-function is twisted by some character? Does there exist a good reference?
[Math] BSD for modular forms
modular-formsnt.number-theory
Related Solutions
Edit: Here's a rather silly method that should work if SAGE is just giving you cusp forms: $\Gamma_0(4)$ has a single normalized cusp form of weight 6, given by $\eta(2\tau)^{12} = q - 12q^3 + 54q^5 - \dots$, so take your basis of cusp forms of weight $k/2 + 6$, and divide each element by this form to get a basis of the space of modular forms of weight $k/2$.
Edit in response to Buzzard: Thanks for pointing out that I should make this argument. Here is a proof that the cusp form has minimal vanishing at all cusps. $\Gamma_0(4)$ is conjugate to $\Gamma(2)$ by $\tau \mapsto 2\tau$, so it suffices to check that $\Delta(\tau)$, the square of $\eta(\tau)^{12}$, vanishes to twice the minimum order at each cusp of $\Gamma(2)$. The quotient $\Gamma(1)/\Gamma(2) \cong S_3$ acts transitively on the cusps of $X(2)$ with stabilizers of order 2, so the quotient map to $X(1)$ has ramification degree 2 at each cusp. $\Delta(\tau)$ is invariant under the weight 12 action of $\Gamma(1)$, and $\Delta(\tau)$ has minimal vanishing at infinity on $X(1)$.
Old answer: If you have a cusp form of weight $k/2$ for $\Gamma_0(4)$ (e.g., given to you by SAGE), you can multiply it by the modular function $\frac{\eta(\tau)^8}{\eta(4\tau)^8} = q^{-1} - 8 + 20q - 62q^3 + 216q^5 - \dots$ to get a modular form of the same weight, that is nonvanishing at one of the three cusps and vanishing at the other two. If you want a form that is nonzero at one of the other cusps, you can multiply by $\frac{\eta(4\tau)^8}{\eta(\tau)^8}$ (has a pole at zero) or by $\frac{\eta(\tau)^{16}\eta(4\tau)^8}{\eta(2\tau)^{24}}$ (pole at $1/2$). [Constant term $-8$ added Sept. 23, in response to an email correction from Michael Somos.]
If you are looking for examples of modular forms whose zeros can be described explicitly, then you probably want the zeros to be cusps or imaginary quadratic irrationals. In this case the Gross-Kohnen-Zagier theorem implicitly gives lots of examples, by describing the relations between Heegner points on modular elliptic curves. (Heegner points are closely related to imaginary quadratic numbers in the upper half plane.) Many examples of modular forms with zeros at imaginary quadratic irrationals can also be constructed explicitly as automorphic products.
Best Answer
My comments are getting too long, so here is a tentative answer.
First, a general statement: conjectures predicting special values of $L$-functions are formulated for all motives over number fields. Because normalized eigenforms (even twisted by finite order characters $\chi$) are attached to motives (or vice-versa), there exists conjectures predicting the special values of $L(f,s,\chi)$.
Now what do they look like? For simplicity and because you are especially interested in the value of the residue, I'll restrict to the case of a critical value (so $s=0,\cdots,k-1$ if $f$ is of weight $k$ EDIT: $s=1,\cdots,k-1$ Thanks to David Loeffler for pointing this out). The general conjectures then imply that there exists a special element which is a basis of the determinant of the motivic cohomology which is sent to $L(f,s,\chi)$ by the realization morphism from motivic cohomology to Betti cohomology and to a specific basis of the determinant of étale cohomology by the realization morphism to $p$-adic étale cohomology (for any $p$). But forget about this, because when $s≠k/2$ (which I assume henceforth), then $L(f,s,\chi)$ is non-zero (by Jacquet) and in this case, K.Kato has constructed a candidate $z$ for this conjectural element in his article $p$-adic Hodge theory and values of zeta functions of modular forms. I can't really say that this $z$ lives in the right space, simply because I am unsure whether motivic cohomology is properly defined in this case, but at least it lives in something that has all the property you would wish for motivic cohomology (namely the second $K$-theory group $K_{2}$ of the modular curves) and it is sent to the right value through the realization morphism to Betti cohomology.
So now the Tamagawa Number Conjecture predicts that it should be sent to a specific basis of the determinant of the $p$-adic étale cohmology for any $p$. Unraveling what it means in this case, you get that $H^2(\textrm{Spec}\ \mathbb Z[1/p],T)$ is a finite group and the following conjectural equality: \begin{equation} \sharp H^2_{et}(\textrm{Spec}\ \mathbb Z[1/p],T)=[H^1_{et}(\textrm{Spec}\ \mathbb Z[1/p],T):z] \end{equation} Here $T$ is any lattice in the Galois representation of your modular form and $[-:-]$ denotes generalized index (so $[\mathbb Z_{p}:1/p]=p^{-1}$).
Now you have a perfectly valid expression of a generalized BSD conjecture, and this is how I think of conjectures about special values in this case. However, you might want to express this conjecture in a way that recovers usual BSD when $f$ comes from an elliptic curve $E$. This again is a doable exercise (albeit one I find non-trivial) which is done for instance in Burns-Flach Math. Ann. 305 (section 1.7) or O.Venjakob London Math. Soc. Lecture Note Ser., 320 (section 3.1). I generally recommend the latter article because I learnt a lot from it myself but beware that, on this very specific question, there is a typo in the definition of one of the crucial objects, if memory serves well, so I rather recommend using both articles in parallel.