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ID: INPUT{id.yaml}

Title: >
  Spectral parameter of Maass forms of level 1

Definition: >
  A Maass form (of weight 0) of level 1 is a
  smooth, square-integrable, automorphic eigenform 
  of the Laplace-Beltrami operator $\Delta$
  that is invariant under $SL(2,\mathbb{Z})$.
  The Laplace eigenvalue of a Maass form has the form 
  $\lambda = 1/4 + R^2$,
  where $R > 0$ is called the spectral parameter.
  
Parameters:
  R':
    display: $R'$
    title: spectral parameter (truncated)
    type: Z
   
Comments:
    
Formulas:

Programs:

References:
  BoStVe06:
    bib: >
      A. Booker, A. Strömbergsson, A. Venkatesh, 
      "Effective Computation of Maass Cusp Forms".
      IMRN 2006, Article ID 71281, 34 pages.
      (HREF{http://www2.math.uu.se/~astrombe/emaass/emaass.html}[data files])
    url: http://www2.math.uu.se/~astrombe/papers/bsv27april06.pdf
  Str12:
    bib: >
      Strömberg, Fredrik,
      "Maass waveforms on (Γ0(N),χ) (computational aspects)".
      Hyperbolic geometry and applications in quantum chaos and cosmology, 187–228,
      London Math. Soc. Lecture Note Ser., 397, Cambridge Univ. Press, 2012
    mr: 2895234

Links:
  BoStVeData:
    title: "A. Booker, A. Strömbergsson, A. Venkatesh: data files"
    url: http://www2.math.uu.se/~astrombe/emaass/emaass.html
  LMFDB:
    title: "LMFDB: Maass forms of level 1"
    url: https://www.lmfdb.org/ModularForm/GL2/Q/Maass/?level=1
  LMFDBreliability:
    title: "LMFDB: Maass forms: Reliability"
    url: https://www.lmfdb.org/ModularForm/GL2/Q/Maass/Reliability
  WikiBSD:
    title: "Wikipedia: Birch and Swinnerton-Dyer conjecture"
    url: https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture
    
Similar tables:
  
Keywords:
  
Tags:
- number theory
- modular form
- eigenvalue

Data properties:
  type: R
  complete: no
  sources:
  - CITE{BoStVeData} (higher precision available, from Booker, Strömbergsson, Venkatesh CITE{BoStVe06})
  - CITE{LMFDB} (computed by Fredrik Stroemberg CITE{Str12} and Holger Then)
  reliability: >
    For heuristics see CITE{BoStVeData} and CITE{LMFDBreliability}.

Display properties:
  number-header: $R$

Numbers: INPUT{numbers.yaml}

— preview —

Spectral parameter of Maass forms of level 1

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Numbers

$R'$

$R$

INPUT{numbers.yaml} (not shown in preview)

Definition

A Maass form (of weight 0) of level 1 is a smooth, square-integrable, automorphic eigenform of the Laplace-Beltrami operator $\Delta$ that is invariant under $SL(2,\mathbb{Z})$. The Laplace eigenvalue of a Maass form has the form $\lambda = 1/4 + R^2$, where $R > 0$ is called the spectral parameter.

Parameters

$R'$

— spectral parameter (truncated)

References

[1]

A. Booker, A. Strömbergsson, A. Venkatesh, "Effective Computation of Maass Cusp Forms". IMRN 2006, Article ID 71281, 34 pages. (data files) http://www2.math.uu.se/~astrombe/papers/bsv27april06.pdf

[2]

Strömberg, Fredrik, "Maass waveforms on (Γ0(N),χ) (computational aspects)". Hyperbolic geometry and applications in quantum chaos and cosmology, 187–228, London Math. Soc. Lecture Note Ser., 397, Cambridge Univ. Press, 2012

Links

[3]

A. Booker, A. Strömbergsson, A. Venkatesh: data files

[4]

LMFDB: Maass forms of level 1

[5]

LMFDB: Maass forms: Reliability

[6]

Wikipedia: Birch and Swinnerton-Dyer conjecture

Data properties

Entries are of type: real number

Table is complete: no

Sources of data: [3] (higher precision available, from Booker, Strömbergsson, Venkatesh [1]), [4] (computed by Fredrik Stroemberg [2] and Holger Then)

Reliability: For heuristics see [3] and [5].