Second order vanishing for real quadratic twists of the weight 4 level 7 Hecke cuspform

[Data] for quadratic twists of the weight 4 level 7 Hecke cuspform $f$.

This form can be given as $f=q-q^2-2q^3-7q^4+16q^5+2q^6-7q7+\dots.$

Here are the first few normalized values of the twisted $L$-series. This should be cmpared with Table 3.2 of the Rosson Tornaria paper which starts on page 315 of the Newton Proceedings. Note, that in the data below the discriminants divisible by 7 have been omitted, and also the entries have been scaled back by a factor of 7. And the entry corresdponding to $d=1$ is also not present. Below are the beginnings of the data. (The actual data set has millions of entries.)

8 1
29 -4
37 0
44 4
53 4
57 -2
60 -6
65 2
85 -4
88 6
92 0
93 -4
109 4
113 2
120 -2
137 -4
141 12
149 0
156 14
165 -12
172 -4
177 6
184 -12
193 6
197 8

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