{
  "abstract": "Let $F\\in S_{2k}^{new}(M,χ^2)$ be a Hecke newform, and $D$ a fundamental discriminant with $(-1)χ(-1)D>0$. Via the Shimura Correspondence, we get associated modular forms of half-integral weight, and Waldspurger showed that for such $g=\\sum c_nq^n \\in S_{k+1/2}(4M',χ;F)$, if $c_{|D|}(g)\\neq 0$, then $|c_{|D|}(g)|^2 \\propto |D|^{k-1/2}L(F\\otimes \\overlineχ\\otimes χ_D,k)$. Kohnen refined this in the case that $f\\in S_{2k}^{new}(N)$ for $N$ odd and squarefree, making explicit $|c_{|D|}(g)|^2$ for $g \\in S^+_{k+1/2}(4N;f)$. However, $c_{|D|}(g)=0$ unless $\\left(\\frac{D}{p}\\right)=w_p(f)$ for all $p|N$. In this paper, we produce more generally $g \\in S^+_{k+1/2}(4\\ \\mathrm{lcm}(N,N_1^2),χ;f\\otimes χ)$, where $χ$ is a character of odd squarefree conductor $N_1$, using a family of generalised Shimura/Shintani maps $Ψ_χ^D,{Ψ_χ^D}^*$ due to Kojima and Tokuno. Extending their work (and Kohnen's), we compute its Petersson norm, and give an explicit formula for all the $|c_m(g)|^2$. The key technical ingredient is an explicit set of representatives for $L^0_{M_1,M}(M_1^2D^2)/Γ_0(M)$ of binary quadratic forms. We also use a new multiplicity-one result based on work of Kohnen, Ueda, and Waldspurger. In the case $N_1|N$, we reconstruct forms previously considered by Baruch-Mao and Prasanna, which change Kohnen's condition to $\\left(\\frac{D}{p}\\right)=-w_p(f)$ for all $p|N_1$, thus relating twisted $L$-values to squares of Fourier coefficients for all classes of discriminants.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.32470",
  "author_total": 1,
  "authors": [
    {
      "name": "Mark Chambers"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/anj6zszh32dgk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.32470",
  "fulltext_url": null,
  "id": "anj6zszh32dgk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.32470",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7223708646",
  "pdf_url": "https://arxiv.org/pdf/2610.32470",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/anj6zszh32dgk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.32470",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.32470"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Binary Quadratic Forms, Shintani Lifts, and Twisted L-Values",
  "title_text": "Binary Quadratic Forms, Shintani Lifts, and Twisted L-Values",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/anj6zszh32dgk",
  "venue": "arXiv",
  "versions": []
}