Questions tagged [eisenstein-series]
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95 questions
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A coefficient in Dirichlet series associated with a cofinite subgroup of $\mathrm{SL}(2,\mathbb R)$
Let $\Gamma$ be a discrete subgroup of $\operatorname{SL}(2,\mathbb R)$, acting on the upper half-plane $\mathbb H$. Suppose that $\Gamma\backslash \mathbb H$ is non-compact and its compactification $\...
- 7,193
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67
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Why the residues of Siegel Eisenstein series becomes constant at certain point?
$\DeclareMathOperator\GL{GL} \DeclareMathOperator\SO{SO} \DeclareMathOperator\Ind{Ind}\DeclareMathOperator\B{B}$Let $F$ be a number field and $V$ be a $(2n+1)$-dimensional quadratic space over $F$. ...
- 1,019
3
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1
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Fourier expansion of half-integral weight Eisenstein series associated with Kohnen's plus space
The Eisenstein series associated with Kohnen's plus space in $\Gamma_{0}(4)$ is expressed as follows,
\begin{align}
\begin{split}
E_{k + \tfrac{1}{2}}^{\infty}(\tau) =& \sum\limits_{\...
- 449
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1
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Image of the intertwining operator for GL(2) is $K$-invariant at the "pole" $s=1$
I am taking a look at the residues of Eisenstein series and have a question about a local computation. Let $k$ be a local field, $G = \operatorname{GL}_2(k)$, and $P$ (resp. $K$) the standard ...
- 6,180
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1
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328
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References for the construction of Beilinson's motivic Eisenstein classes
According to some authors, it is built in A.A.Beilinson "Higher regulator of modular curves" a class $\mathbf{Eis}_{\phi}$ in the motivic cohomology of the modular curve where $\phi$ is a ...
- 1,270
2
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1
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390
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Fourier series of Eisenstein series — elegant and very good approximation
I played around with the Fourier series of the Eisenstein series resp. divisor sums and did some calculations, see below. Although the deduction is not rigorous / wrong (as the power series for the ...
- 406
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242
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Explicit expression of automorphic representations as automorphic forms
Let‘s take $G=GL_n$ over a number field $F$ for example.
It's already known that every irreducible automorphic representation $\pi$ is a irreducible component of a induced representation $I(G,P;\...
- 111
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189
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About the structure of smooth automorphic forms
Recently I read Prof. Cogdell's notes: Lectures on L-functions, Converse Theorems, and Functoriality for $GL_n$. (Co)
In chap.2.3, the conception of smooth automorphic forms is introdued. Explicitly, ...
- 111
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Non-holomorphicity of Hecke regularization of weight-2 level-1 Eisenstein series
Originally asked on stackexchange since I figured it was a very elementary and silly error, but since I haven't had any response or interest in a couple days, I thought I'd post it here.
I have a &...
- 2,054
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Extrema of real analytic Eisenstein series and more general modular functions
The real analytic Eisenstein series defined by the Poincare sum
$$E(s,z)=\sum_{(m,n)\neq (0,0)} {y^s\over |mz+n|^{2s}}$$
for $z\in{\mathbb H}$ and ${\rm Re}(s)>1$ is a manifestly $SL(2,{\mathbb Z})$...
- 21
1
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1
answer
189
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Analytic continuation of the Eisenstein series defined over Hecke and Fricke subgroups
It is well known that the (real analytic) Eisenstein series is defined, in the slash notation, as follows
$$E_{s}(\tau) = \sum\limits_{\gamma\in\Gamma_{\infty}\backslash\text{SL}(2,\mathbb{Z})}\left.y^...
- 449
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1
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568
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History of points of view on Eisenstein series
What is the history of Eisenstein series? Did the mathematician Eisenstein actually encounter them?
There are, as far as I know, two major perspectives on what Eisenstein series are. The first is in ...
- 6,180
4
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141
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The meaning of $L_{\chi}^2(G(\mathbb Q) \backslash G(\mathbb A)^1)$
I'm reading James Arthur's notes on the trace formula and am confused on a point on pages 65 and 66. For $G$ a reductive group over $\mathbb Q$ we are going over the decomposition of the space $L^2(G(...
- 6,180
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Eisenstein series evaluated at $2i$
Consider the real analytic Eisenstein series defined by
$$
E(z,s) := \sum_{\gamma\in\Gamma_\infty\setminus\Gamma} Im(\gamma z)^s
$$
where as usual, $\Gamma=SL(2,\mathbb{Z})$ and $\Gamma_\infty$ is the ...
- 589
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242
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Constant coefficient of Eisenstein series
Let $\chi$ be a character of $\mathbb{Q}^{\times}\setminus\mathbb{A}^{\times}$. We define an induced representation of $Mp_2(\mathbb{A})\simeq SL_2(\mathbb{A})\times \mathbb{C}^1$,
$$I(s,\chi) := \{\...
- 103
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Density of the Mellin transform inside the direct integral of induced representations
I'm trying to better understand the continuous spectrum of $G = \operatorname{GL}_2(\mathbb A_{\mathbb Q})$, which is the direct integral of induced representations $\mathbf H(s) = \operatorname{Ind}_{...
- 6,180
3
votes
1
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183
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Calculating the residue of Eisenstein series from the residue of the intertwining operator
I've been reading the article Forms of $\operatorname{GL}(2)$ from the analytic point of view by Gelbart and Jacquet in Corvallis and am confused on a particular claim (equation 5.17 on page 232).
The ...
- 6,180
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79
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How to see the surjectivity of $L^2_{\text{cont}}$ onto the direct integral of Hilbert space representations?
I've been reading the article Forms of $\operatorname{GL}(2)$ from the analytic point of view by Gelbart and Jacquet in Corvallis and am confused on one point. Let $G = \operatorname{GL}_2$, and $V = ...
- 6,180
9
votes
1
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225
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Why are characters orthogonal to cusp forms?
Let $G = \operatorname{GL}_2$, and let $V = L^2(Z(\mathbb A)G(\mathbb Q) \backslash G(\mathbb A),\omega)$, for $\omega$ a character of the ideles $\mathbb A^{\ast}$, identified with a central ...
- 6,180
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Is there some precise way in which modular cocycle $c/(c\tau +d)$ "is" the generator of $H^1(\mathcal{M}_{ell}, \omega^2) \cong \mathbb{Z}/12$?
It is semiclassical that the extension class $\text{Ext}^1(\omega^{-1},\omega)\cong H^1(\mathcal{M}_{ell},\omega^2)$ over the modular stack $\mathcal{M}_{ell}/\mathbb{Z}$ is nonvanishing, being cyclic ...
- 2,054
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1
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229
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Index and weight of Weierstrass $\sigma$-function as a Jacobi form, versus a statement in a note by Zagier
Let
$$\sigma_L(w; \tau):=\frac{w}{\exp\left(\sum_{k\ge 2} 2G_k(q)\frac{w^k}{k!}\right)}$$
be the version of the Weierstrass $\sigma$-function which is used to orient $\text{tmf}$; here $w=2\pi i z$, $...
- 2,054
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176
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Factorizing classical Eisenstein series
In the course of my research, I found some surprising (for me) factorizations
of Eisenstein series in levels $1$, $2$, $3$, and $4$. For instance, in level $1$
set with standard modular form notation
$...
- 13.1k
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92
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The kernel $K(x,y)$ as an integral over Eisenstein series for $\operatorname{GL}_2$
Let $G = \operatorname{GL}_2$, $f \in C_c^{\infty}(G(\mathbb A)/Z(\mathbb A))$, and $V = L^2( G(\mathbb Q)Z(\mathbb A)\backslash G(\mathbb A))$ (trivial central character). Then the operator $R(f)$ ...
- 6,180
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296
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Eisenstein Series at CM points
Suppose that $L= \mathbb{Z}\tau + \mathbb{Z}$ is a lattice with CM. Consider the Eisenstein sum
$$ G_{2k}(L) = \sum_{(m,n)\neq (0,0)} \frac{1}{(m\tau+n)^{2k}}$$
where $k$ is a positive integer greater ...
- 901
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Is there a similar formula like Ramanunjan's Eisenstein series identity for $\sum_{k=1}^{n-1}k^2 \sigma(k)\sigma(n-k)$?
This question is related to the last question about van der Pol's identity for the sum of divisors.
In Touchard (1953) it is mentioned that the sum of divisors $\sigma(n)$ satisfies the following ...
- 3,134
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217
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Maass--Selberg for any Eisenstein series on higher rank
Does there exist a Maass--Selberg relation for any Langlands Eisensein series on $\mathrm{GL}(n)$? By any I mean an Eisenstein series which is induced from any standard parabolic with any discrete ...
- 1,692
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183
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Multidimensional series: an application of quantum field theory
While computing the quantum vacuum energy of a real scalar field defined on $\mathbb{R}\times \mathbb{T}^3$, I encountered the following sum:
$$ \sum_{n_1^2+n_2^2+n_3^2\geq 1}^{\infty} \frac{1}{(n_1^2+...
- 19
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1
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301
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Question on the residual representation
Let $G=SO_n$ and fix a borel subgroup $P_0$ of $G$. Let $P=MN$ be a standard maximal parabolic subgroup $G$ and $\sigma$ a cuspidal representation of $M$
Consider the normalized parabolic induced ...
- 1,759
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Intertwining operator is not an isomorphism?
Let $F$ be a number field and $G$ a symplectic group over $F$.
Let $P=MN$ is a maximal parabolic subgroup of $G$ and $W_M=N_G(M)/M$. Since $P$ is maximal, $W_M \simeq S_2$. Let $w$ be a non-trivial ...
- 1,759
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Quantum ergodicity of Eisenstein series on arithmetic quotients of hyperbolic space
Let $E(z,1/2+it)$ be the Eisenstein series furnishing the continuous spectrum of the Laplace operator $\Delta$ on $X=PSL_2(\mathbb{Z})\setminus H^2$ and $dV(z)=y^{-2} \,dx \,dy$ be the volume element ...
- 218
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An explicit formula for a cuspidal form of weight $2$ and arbitrarily large prime level
In Miyake's book on modular forms explicit formulas for the $q$-expansions of a basis of the space Eisenstein series of arbitrary level and weight were given. I guess similar formulas for a basis of ...
- 61
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101
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Decomposing functions on the fundamental domain of the torus into cusp forms, eisenstein series
I am trying to do some elementary calculations to understand the properties of the following spectral resolution on $H/SL(2,\mathbf{Z})$. (Half plane mod modular group; fundamental domain of the torus)...
- 242
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0
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291
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Derivatives of Eisenstein series
A book of Moeglin-Waldspurger says that there was a conjecture that every automorphic form arises as the derivative of an Eisenstein series which is proved there for function field case and proved by ...
- 1,024
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0
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355
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What is definition of Cohen–Eisenstein series?
I only find Cohen–Eisenstein series of weight 3/2 (in the paper A note on the Fourier coefficients of a Cohen-Eisenstein series). I founded some general definition in "Modular Forms with Integral and ...
- 743
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Hypergeometric representation of Eisenstein series
It is well known (Fricke ?) that $E_4^{1/4}$ and $E_6^{1/6}$ can be represented as Gauss hypergeometric functions of $1728/j$ and $1728/(1728-j)$ respectively.
The same result is true in levels $2$, $...
- 13.1k
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Why can there be holomorphic modular forms of negative half integral weight?
In Shimura's paper "ON THE HOLOMORPHY OF CERTAIN DIRICHLET SERIES", he constructed a family of Eisenstein series $E(z,s)$ by summing factor of automorphy. $E(z,s)$ is of negative half-integral weight, ...
- 41
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What is meant by "roots in $Lie(N)$" in root space decomposition of Lie algebras?
Let $G = GL_n$ and $T$ the invertible diagonal matrices and $N$ the upper triangular matrices with only $1$'s on the diagonal. Then the Lie algebra $\mathcal{G}$ has the roots space decomposition
$$
\...
- 3,625
3
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What is meant by singular hyperplane of $c(w, \cdot)$? (global intertwining operator related to Eisenstein series)
Let $P_0$ be a minimal $\mathbb{Q}$-parabolic subgroup of $G$, a semisimple linear algebraic group over $\mathbb{Q}$. Then $P_0 = M_0 N_0$ where $M_0$ is a Levi subgroup of $P_0$. Let $E^G_{P_0}$ be ...
- 3,625
5
votes
1
answer
616
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Question on an application of Langlands' result on the constant term of Eisenstein series (Is this a typo?)
I would like to understand an argument in https://link.springer.com/content/pdf/10.1007/BF01393904.pdf, which uses Langlands' result on the constant term of Eisenstein series, but I'm not getting it ...
- 3,625
6
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1
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558
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Analogues of Hecke relations for Maass forms
If a (suitably normalised) holomorphic cusp newform has q-expansion
$$f(z) = \sum_n \lambda_f(n) e(nz),$$
then we know the Hecke relations for $(mn,q)=1$,
$$(\star) \qquad \lambda_f(m)\lambda_f(n) = \...
- 937
5
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1
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446
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An easier reference than "On the Functional Equations Satisfied by Eisenstein Series"?
I'd like to learn about Eisenstein series so I started reading "On the Functional Equations Satisfied by Eisenstein Series"by Langlands.
http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/pdf/...
- 3,625
7
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2
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Bound on an expression involving J-function coefficients
I would like to show that
$$(m+1)\,c_m- \sum_{n=1}^{m-1} c_n \,\sigma_{m-n} > 0$$
for all $m$, where $c_i$ is the $i$th coefficient of Klein's $J$-function
$$J(q)= \frac{1728 \;E_4^3(q) }{...
- 71
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0
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304
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Question about the Fourier expansion of adelic Eisenstein series for $\operatorname{GL}_2$
My reference is Daniel Bump's book, Automorphic Forms and Representations, Chapter 3.7. Let $k$ be a number field, $G = \operatorname{GL}_2$, $B$ and $T$ the usual Borel subgroup and maximal torus ...
- 6,180
10
votes
0
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304
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Generalized Jacobians and modular units
Let $X$ be a proper algebraic smooth curve over a characteristic zero field $k$ and let $J$ be the Jacobian variety of $X$. Let $K$ be the function field of $X$. Assume that we are given $n$ distinct ...
- 1,109
9
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1
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638
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Algebraic independence of $P,Q,R$ or $E_2,E_4,E_6$ over $\mathbb C(z)$
Let $P,Q,R$ be the Fourier series of the Eisenstein series $E_2,E_4,E_6$, that is,
$$
P(q)=1-24\sum_{n=1}^{\infty}\sigma_1(n)q^n,
$$
$$
Q(q)=1+240\sum_{n=1}^{\infty}\sigma_3(n)q^n,
$$
$$
R(q)=1-504\...
- 663
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0
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467
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Eisenstein series of Hilbert modular forms
I am reading Shimura's paper "The Special Values of the Zeta Functions Associated With Hilbert Modular Forms" and I do not exactly understand his definition of the Eisenstein series in section 3.
...
- 123
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1
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578
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Decategorification of Gaitsgory's strange functional equation?
Let $G$ a complex reductive algebraic group, $X$ be a smooth compact complex curve. The moduli stack $Bun_G$ of principal $G$-bundles on $X$ is generally speaking not quasi-compact (so we do not have ...
user74900
3
votes
0
answers
81
views
Spectral theory for Fuchsian groups of the first kind
There are tons of material on the spectral theory of $L^2(\Gamma\backslash G)$ for a lattice $\Gamma$ in $G=PSL_2({\mathbb R})$. There are also many papers on the case of $\Gamma$ being convex-...
user1688
6
votes
1
answer
600
views
How to compute Coefficients in Chudnovsky's Formula?
My aim is to understand all three coefficients arising in the Chudnovsky-Formula (see also Question 300385). Two of them are easily computed, but I failed with the third:
It is known that for all $\...
- 598
21
votes
1
answer
1k
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Why does this quasi-modular function have integral values?
It is a well-known result that the modular function $1728J(\tau) := \frac{1728E_4(\tau)^3}{E_4(\tau)^3-E_6(\tau)^2}$ has integral values if $\tau$ has class number 1 - for example at $\tau_{163}:=\...
- 598