2012
DOI: 10.1112/plms/pds004
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Newforms and spectral multiplicity for Γ0 (9)

Abstract: The goal of this paper was to explain certain experimentally observed properties of the (cuspidal) spectrum and its associated automorphic forms (Maass waveforms) on the congruence subgroup Γ0(9). The first property is that the spectrum possesses multiplicities in the so-called new part, where it was previously believed to be simple. The second property is that the spectrum does not contain any 'genuinely new' eigenvalues, in the sense that all eigenvalues of Γ0(9) appear in the spectrum of some congruence sub… Show more

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Cited by 5 publications
(7 citation statements)
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References 39 publications
(88 reference statements)
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“…[1]. It is also consistent with the notion of genuinely new Maass waveforms introduced in [66]. By using the description of the setC(G , G) below it is also easy to verify that the definition of old space agrees with the definition above for congruence subgroups.…”
Section: Old and Newformssupporting
confidence: 74%
“…[1]. It is also consistent with the notion of genuinely new Maass waveforms introduced in [66]. By using the description of the setC(G , G) below it is also easy to verify that the definition of old space agrees with the definition above for congruence subgroups.…”
Section: Old and Newformssupporting
confidence: 74%
“…A subset of these eigenvalues are eigenvalues for the groups Γ 0 (5), Γ 0 (6), and Γ 3 . These are in good agreement with those computed in [22] and [33] C arccosh(2),0 C arccosh(3),0 C arccosh(9),0 C 2 arccosh( 3 2 ),1/2 C 2 arccosh(2),1/2 and and C arccosh(5),0 C 2 arccosh(3),1/2 t µ(t) t µ(t) t µ(t) t µ(t) t µ(t)…”
Section: = Arccosh(5) ≈ 229243supporting
confidence: 90%
“…Thus, unless there are some non-trivial "oldforms" showing up, our claim is that there is no order in the case N = 4pu 2 that gives the same correspondence as in Theorem 8.1, but we have no proof of it. There could for example be a phenomenon analogous to the one for Γ 0 (9), where all newforms actually are forms, or twists of forms, corresponding to groups strictly containing Γ 0 (9) [29]. Hence, we definitely do not exclude the possibility of finding a correspondence with an adjusted notion of newforms in the case N = 4pu 2 .…”
Section: The Correspondencementioning
confidence: 86%