Suppose $f$ is a weight-5 modular form that is labeled as 27.5.b.a in LMFDB. Let $\chi_3$ be the Dirichlet character \begin{equation} \chi_3(p) =\left(\frac{3}{p} \right). \end{equation} I want to numerically compute the special values of the $L$-function $L(f \otimes \chi_3, s)$ at $s=1,2,3$. I am trying to use the Dokchitser’s $L$-functions Calculator in Sage. The modular form $f \otimes \chi_3$ should be associated to a pure motive of weight-4 and rank-2, and its Hodge type should be $(4,0)+(0,4)$. So the gamma factor should be $[0,1]$. But I do not know the other invariants of $f \otimes \chi_3$ that are needed by the Dokchitser’s $L$-functions Calculator. Does anyone know how to compute these special values to a very high precision, say the first 100 digits?
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1$\begingroup$ The LMFDB lists the twist of $f$ by $\chi_{3}$ as the weight 5 modular form with label 432.5.e.a and the $L$-function for it is here. So the weight should be 5, the conductor is 432, and the sign is 1. $\endgroup$– Jeremy RouseCommented Dec 4, 2020 at 1:40
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$\begingroup$ In Sage, I tried "L = Dokchitser(conductor=432, gammaV=[0,1], weight=5, eps=1,init='1', prec=200)". But the value of L at $s=5/2$ is 2.530840607. However the value of the $L$-function of 432.5.e.a at $s=5/2$ is 1.264584993. Is there something that I have not understood? 432 is the level of this modular form, is the conductor the same as the level? $\endgroup$– WenzheCommented Dec 4, 2020 at 3:57
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$\begingroup$ Sorry, I think I understand where is the problem. I should include more coefficients of 432.5.e.a in my computation. $\endgroup$– WenzheCommented Dec 4, 2020 at 6:48
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1$\begingroup$ In Pari/GP you can compute the twist of a newform using the command mftwist(f,D). Then you can compute the L-function, it's built-in in Pari/GP. $\endgroup$– François BrunaultCommented Dec 4, 2020 at 9:39
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