Inspired by Lehmer’s conjecture on the non-vanishing of the Ramanujan $$\tau $$ τ -function, one may ask whether an odd integer $$\alpha $$ α can be equal to $$\tau (n)$$ τ ( n ) or any coefficient of a newform f(z). Balakrishnan, Craig, Ono and Tsai used the theory of Lucas sequences and Diophantine analysis to characterize non-admissible values of newforms of even weight $$k\ge 4$$ k ≥ 4 . We use these methods for weight 2 and 3 newforms and apply our results to L-functions of modular elliptic curves and certain K3 surfaces with Picard number $$\ge 19$$ ≥ 19 . In particular, for the complete list of weight 3 newforms $$f_\lambda (z)=\sum a_\lambda (n)q^n$$ f λ ( z ) = ∑ a λ ( n ) q n that are $$\eta $$ η -products, and for $$N_\lambda $$ N λ the conductor of some elliptic curve $$E_\lambda $$ E λ , we show that if $$|a_\lambda (n)|<100$$ | a λ ( n ) | < 100 is odd with $$n>1$$ n > 1 and $$(n,2N_\lambda )=1$$ ( n , 2 N λ ) = 1 , then $$\begin{aligned} a_\lambda (n) \in&\{-5,9,\pm 11,25, \pm 41, \pm 43, -45,\pm 47,49, \pm 53,55, \pm 59, \pm 61,\\&\pm 67, -69,\pm 71,\pm 73,75, \pm 79,\pm 81, \pm 83, \pm 89,\pm 93 \pm 97, 99\}. \end{aligned}$$ a λ ( n ) ∈ { - 5 , 9 , ± 11 , 25 , ± 41 , ± 43 , - 45 , ± 47 , 49 , ± 53 , 55 , ± 59 , ± 61 , ± 67 , - 69 , ± 71 , ± 73 , 75 , ± 79 , ± 81 , ± 83 , ± 89 , ± 93 ± 97 , 99 } . Assuming the Generalized Riemann Hypothesis, we can rule out a few more possibilities leaving $$\begin{aligned} a_\lambda (n) \in \{-5,9,\pm 11,25,-45,49,55,-69,75,\pm 81,\pm 93, 99\}. \end{aligned}$$ a λ ( n ) ∈ { - 5 , 9 , ± 11 , 25 , - 45 , 49 , 55 , - 69 , 75 , ± 81 , ± 93 , 99 } .

Let $$f(z)=q+\sum _{n\ge 2}a(n)q^n$$ f ( z ) = q + ∑ n ≥ 2 a ( n ) q n be a weight k normalized newform with integer coefficients and trivial residual mod 2 Galois representation. We extend the results of Amir and Hong in Amir and Hong (On L-functions of modular elliptic curves and certain K3 surfaces, Ramanujan J, 2021) for $$k=2$$ k = 2 by ruling out or locating all odd prime values $$|\ell |<100$$ | ℓ | < 100 of their Fourier coefficients a(n) when n satisfies some congruences. We also study the case of odd weights $$k\ge 1$$ k ≥ 1 newforms where the nebentypus is given by a quadratic Dirichlet character.

IntroductionChronic pain is often unrecognized and/or undertreated, and as a result has significant impact on functional abilities, quality of life, societal participation and health care utilization. Medications remain a mainstay of treatment, but selection for any given patient remains a challenge when trying to predict efficacy and/or side effects. There is interest to see whether genetic analysis of how a given drug is processed for a patient can help with rational drug choices. This appears to have some early support in cardiac, psychiatric and acute pain studies. We studied whether genetic analysis of drug processing using the Pillcheck program could have helped in choosing the appropriate medications in a cohort of patients suffering from chronic pain. To our knowledge this type of study has not been completed in this environment and/or patient population.MethodsWe retrospectively studied a 31 patient cohort seen in the Canadian Forces Health Services Unit (Ottawa) Physiatry clinic. All patients suffered from a diagnosed chronic pain condition, completed the Pillcheck genetic drug processing analysis and filled-in questionnaires looking at efficacy and side effects of the drugs. We analyzed the correlation between the Pillcheck predictions and participants’ self-reported treatment efficacy and tolerability. The goal was to explore the clinical utility of Pillcheck results in guiding prescriptions for chronic pain patients.Results31 patients returned completed questionnaires and had samples taken. Forty eight percent of the participants were actively treated with one of the study pain medications, and 84% had been taking at least one of these medications and discontinued. Pillcheck scores did not correlate with self-reported efficacy of any of the medications, nor did it correlate with self-reported side effects. Furthermore, active medications were more likely to receive a score indicating caution should be exerted, than were medications which had been discontinued.DiscussionIn a small cohort of pain patients with comorbid psychiatric disorders, genetic profiling using Pillcheck did not seem to correlate with reported benefit or side effect profile of commonly prescribed pain medications. Furthermore, discontinued medications were no more likely to be marked as warranting caution than did actively used medications.ConclusionRetrospectively using pharmacogenetics to guide medication selection in Canadian Forces members with chronic pain did not correlate with response or side effects. A larger prospective controlled study, measuring numerous clinical and non- clinical outcomes would be worthwhile in the future before widespread adoption for patients living with chronic pain.