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Atiyah–Patodi–Singer rho invariants of lens spaces
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Numbers
$p$
$q$
$u$ 
$\rho(L(p,q),\chi_u)$
2
1
1:
0
3
1
1:
1/3
3
2
1:
-1/3
4
1
1:
1/2
4
1
2:
1
4
3
1:
-1/2
4
3
2:
-1
5
1
1:
3/5
5
1
2:
7/5
5
2
1:
-1/5
5
2
2:
1/5
5
4
1:
-3/5
5
4
2:
-7/5
6
1
1:
2/3
6
1
2:
5/3
6
1
3:
2
6
5
1:
-2/3
6
5
2:
-5/3
6
5
3:
-2
7
1
1:
5/7
7
1
2:
13/7
7
1
3:
17/7
7
2
1:
-1/7
7
2
2:
3/7
7
2
3:
5/7
7
3
1:
-3/7
7
3
2:
-5/7
7
3
3:
1/7
7
6
1:
-5/7
7
6
2:
-13/7
7
6
3:
-17/7
8
1
1:
3/4
8
1
2:
2
8
1
3:
11/4
8
1
4:
3
8
3
1:
1/4
8
3
2:
0
8
3
3:
1/4
8
3
4:
1
8
5
1:
-1/4
8
5
2:
0
8
5
3:
-1/4
8
5
4:
-1
8
7
1:
-3/4
8
7
2:
-2
8
7
3:
-11/4
8
7
4:
-3
9
1
1:
7/9
9
1
2:
19/9
9
1
3:
3
9
1
4:
31/9
9
2
1:
-1/9
9
2
2:
5/9
9
2
3:
1
9
2
4:
11/9
9
4
1:
-5/9
9
4
2:
-11/9
9
4
3:
-1
9
4
4:
1/9
9
8
1:
-7/9
9
8
2:
-19/9
9
8
3:
-3
9
8
4:
-31/9
10
1
1:
4/5
10
1
2:
11/5
10
1
3:
16/5
10
1
4:
19/5
10
1
5:
4
10
3
1:
-2/5
10
3
2:
-3/5
10
3
3:
2/5
10
3
4:
3/5
10
3
5:
0
10
9
1:
-4/5
10
9
2:
-11/5
10
9
3:
-16/5
10
9
4:
-19/5
10
9
5:
-4
11
1
1:
9/11
11
1
2:
25/11
11
1
3:
37/11
11
1
4:
45/11
11
1
5:
49/11
11
2
1:
-1/11
11
2
2:
7/11
11
2
3:
13/11
11
2
4:
17/11
11
2
5:
19/11
11
3
1:
3/11
11
3
2:
1/11
11
3
3:
5/11
11
3
4:
15/11
11
3
5:
9/11
11
5
1:
-7/11
11
5
2:
-17/11
11
5
3:
-19/11
11
5
4:
-13/11
11
5
5:
1/11
11
7
1:
-5/11
11
7
2:
-9/11
11
7
3:
-1/11
11
7
4:
-3/11
11
7
5:
-15/11
11
10
1:
-9/11
11
10
2:
-25/11
11
10
3:
-37/11
11
10
4:
-45/11
11
10
5:
-49/11
12
1
1:
5/6
12
1
2:
7/3
12
1
3:
7/2
12
1
4:
13/3
12
1
5:
29/6
12
1
6:
5
12
5
1:
1/6
12
5
2:
-1/3
12
5
3:
-1/2
12
5
4:
-1/3
12
5
5:
1/6
12
5
6:
1
12
7
1:
-1/6
12
7
2:
1/3
12
7
3:
1/2
12
7
4:
1/3
12
7
5:
-1/6
12
7
6:
-1
12
11
1:
-5/6
12
11
2:
-7/3
12
11
3:
-7/2
12
11
4:
-13/3
12
11
5:
-29/6
12
11
6:
-5
13
1
1:
11/13
13
1
2:
31/13
13
1
3:
47/13
13
1
4:
59/13
13
1
5:
67/13
13
1
6:
71/13
13
2
1:
-1/13
13
2
2:
9/13
13
2
3:
17/13
13
2
4:
23/13
13
2
5:
27/13
13
2
6:
29/13
13
3
1:
-5/13
13
3
2:
-7/13
13
3
3:
7/13
13
3
4:
11/13
13
3
5:
5/13
13
3
6:
15/13
13
4
1:
-7/13
13
4
2:
-15/13
13
4
3:
-11/13
13
4
4:
5/13
13
4
5:
7/13
13
4
6:
-5/13
13
5
1:
-3/13
13
5
2:
1/13
13
5
3:
-1/13
13
5
4:
-9/13
13
5
5:
3/13
13
5
6:
9/13
13
6
1:
-9/13
13
6
2:
-23/13
13
6
3:
-29/13
13
6
4:
-27/13
13
6
5:
-17/13
13
6
6:
1/13
13
12
1:
-11/13
13
12
2:
-31/13
13
12
3:
-47/13
13
12
4:
-59/13
13
12
5:
-67/13
13
12
6:
-71/13
14
1
1:
6/7
14
1
2:
17/7
14
1
3:
26/7
14
1
4:
33/7
14
1
5:
38/7
14
1
6:
41/7
14
1
7:
6
14
3
1:
2/7
14
3
2:
1/7
14
3
3:
4/7
14
3
4:
11/7
14
3
5:
8/7
14
3
6:
9/7
14
3
7:
2
14
9
1:
-4/7
14
9
2:
-9/7
14
9
3:
-8/7
14
9
4:
-1/7
14
9
5:
-2/7
14
9
6:
-11/7
14
9
7:
-2
14
13
1:
-6/7
14
13
2:
-17/7
14
13
3:
-26/7
14
13
4:
-33/7
14
13
5:
-38/7
14
13
6:
-41/7
14
13
7:
-6
15
1
1:
13/15
15
1
2:
37/15
15
1
3:
19/5
15
1
4:
73/15
15
1
5:
17/3
15
1
6:
31/5
15
1
7:
97/15
15
2
1:
-1/15
15
2
2:
11/15
15
2
3:
7/5
15
2
4:
29/15
15
2
5:
7/3
15
2
6:
13/5
15
2
7:
41/15
15
4
1:
7/15
15
4
2:
13/15
15
4
3:
1/5
15
4
4:
7/15
15
4
5:
5/3
15
4
6:
9/5
15
4
7:
13/15
15
7
1:
-11/15
15
7
2:
-29/15
15
7
3:
-13/5
15
7
4:
-41/15
15
7
5:
-7/3
15
7
6:
-7/5
15
7
7:
1/15
15
11
1:
-7/15
15
11
2:
-13/15
15
11
3:
-1/5
15
11
4:
-7/15
15
11
5:
-5/3
15
11
6:
-9/5
15
11
7:
-13/15
15
14
1:
-13/15
15
14
2:
-37/15
15
14
3:
-19/5
15
14
4:
-73/15
15
14
5:
-17/3
15
14
6:
-31/5
15
14
7:
-97/15
16
1
1:
7/8
16
1
2:
5/2
16
1
3:
31/8
16
1
4:
5
16
1
5:
47/8
16
1
6:
13/2
16
1
7:
55/8
16
1
8:
7
16
3
1:
-3/8
16
3
2:
-1/2
16
3
3:
5/8
16
3
4:
1
16
3
5:
5/8
16
3
6:
3/2
16
3
7:
13/8
16
3
8:
1
16
5
1:
-5/8
16
5
2:
-3/2
16
5
3:
-13/8
16
5
4:
-1
16
5
5:
3/8
16
5
6:
1/2
16
5
7:
-5/8
16
5
8:
-1
16
7
1:
1/8
16
7
2:
-1/2
16
7
3:
-7/8
16
7
4:
-1
16
7
5:
-7/8
16
7
6:
-1/2
16
7
7:
1/8
16
7
8:
1
16
9
1:
-1/8
16
9
2:
1/2
16
9
3:
7/8
16
9
4:
1
16
9
5:
7/8
16
9
6:
1/2
16
9
7:
-1/8
16
9
8:
-1
16
15
1:
-7/8
16
15
2:
-5/2
16
15
3:
-31/8
16
15
4:
-5
16
15
5:
-47/8
16
15
6:
-13/2
16
15
7:
-55/8
16
15
8:
-7
17
1
1:
15/17
17
1
2:
43/17
17
1
3:
67/17
17
1
4:
87/17
17
1
5:
103/17
17
1
6:
115/17
17
1
7:
123/17
17
1
8:
127/17
17
2
1:
-1/17
17
2
2:
13/17
17
2
3:
25/17
17
2
4:
35/17
17
2
5:
43/17
17
2
6:
49/17
17
2
7:
53/17
17
2
8:
55/17
17
3
1:
5/17
17
3
2:
3/17
17
3
3:
11/17
17
3
4:
29/17
17
3
5:
23/17
17
3
6:
27/17
17
3
7:
41/17
17
3
8:
31/17
17
4
1:
-9/17
17
4
2:
-19/17
17
4
3:
-13/17
17
4
4:
9/17
17
4
5:
13/17
17
4
6:
-1/17
17
4
7:
1/17
17
4
8:
19/17
17
5
1:
3/17
17
5
2:
-5/17
17
5
3:
-7/17
17
5
4:
-3/17
17
5
5:
7/17
17
5
6:
23/17
17
5
7:
11/17
17
5
8:
5/17
17
8
1:
-13/17
17
8
2:
-35/17
17
8
3:
-49/17
17
8
4:
-55/17
17
8
5:
-53/17
17
8
6:
-43/17
17
8
7:
-25/17
17
8
8:
1/17
17
10
1:
-7/17
17
10
2:
-11/17
17
10
3:
5/17
17
10
4:
7/17
17
10
5:
-5/17
17
10
6:
3/17
17
10
7:
-3/17
17
10
8:
-23/17
17
11
1:
-11/17
17
11
2:
-27/17
17
11
3:
-31/17
17
11
4:
-23/17
17
11
5:
-3/17
17
11
6:
-5/17
17
11
7:
-29/17
17
11
8:
-41/17
17
16
1:
-15/17
17
16
2:
-43/17
17
16
3:
-67/17
17
16
4:
-87/17
17
16
5:
-103/17
17
16
6:
-115/17
17
16
7:
-123/17
17
16
8:
-127/17
18
1
1:
8/9
18
1
2:
23/9
18
1
3:
4
18
1
4:
47/9
18
1
5:
56/9
18
1
6:
7
18
1
7:
68/9
18
1
8:
71/9
18
1
9:
8
18
5
1:
-2/9
18
5
2:
1/9
18
5
3:
0
18
5
4:
-5/9
18
5
5:
4/9
18
5
6:
1
18
5
7:
10/9
18
5
8:
7/9
18
5
9:
0
18
7
1:
-4/9
18
7
2:
-7/9
18
7
3:
0
18
7
4:
-1/9
18
7
5:
-10/9
18
7
6:
-1
18
7
7:
2/9
18
7
8:
5/9
18
7
9:
0
18
17
1:
-8/9
18
17
2:
-23/9
18
17
3:
-4
18
17
4:
-47/9
18
17
5:
-56/9
18
17
6:
-7
18
17
7:
-68/9
18
17
8:
-71/9
18
17
9:
-8
19
1
1:
17/19
19
1
2:
49/19
19
1
3:
77/19
19
1
4:
101/19
19
1
5:
121/19
19
1
6:
137/19
19
1
7:
149/19
19
1
8:
157/19
19
1
9:
161/19
19
2
1:
-1/19
19
2
2:
15/19
19
2
3:
29/19
19
2
4:
41/19
19
2
5:
51/19
19
2
6:
59/19
19
2
7:
65/19
19
2
8:
69/19
19
2
9:
71/19
19
3
1:
-7/19
19
3
2:
-9/19
19
3
3:
13/19
19
3
4:
21/19
19
3
5:
15/19
19
3
6:
33/19
19
3
7:
37/19
19
3
8:
27/19
19
3
9:
41/19
19
4
1:
9/19
19
4
2:
17/19
19
4
3:
5/19
19
4
4:
11/19
19
4
5:
35/19
19
4
6:
39/19
19
4
7:
23/19
19
4
8:
25/19
19
4
9:
45/19
19
6
1:
-13/19
19
6
2:
-33/19
19
6
3:
-41/19
19
6
4:
-37/19
19
6
5:
-21/19
19
6
6:
7/19
19
6
7:
9/19
19
6
8:
-15/19
19
6
9:
-27/19
19
7
1:
-3/19
19
7
2:
7/19
19
7
3:
11/19
19
7
4:
9/19
19
7
5:
1/19
19
7
6:
-13/19
19
7
7:
5/19
19
7
8:
17/19
19
7
9:
23/19
19
8
1:
-5/19
19
8
2:
-1/19
19
8
3:
-7/19
19
8
4:
-23/19
19
8
5:
-11/19
19
8
6:
-9/19
19
8
7:
-17/19
19
8
8:
3/19
19
8
9:
13/19
19
9
1:
-15/19
19
9
2:
-41/19
19
9
3:
-59/19
19
9
4:
-69/19
19
9
5:
-71/19
19
9
6:
-65/19
19
9
7:
-51/19
19
9
8:
-29/19
19
9
9:
1/19
19
14
1:
-11/19
19
14
2:
-25/19
19
14
3:
-23/19
19
14
4:
-5/19
19
14
5:
-9/19
19
14
6:
-35/19
19
14
7:
-45/19
19
14
8:
-39/19
19
14
9:
-17/19
19
18
1:
-17/19
19
18
2:
-49/19
19
18
3:
-77/19
19
18
4:
-101/19
19
18
5:
-121/19
19
18
6:
-137/19
19
18
7:
-149/19
19
18
8:
-157/19
19
18
9:
-161/19
20
1
1:
9/10
20
1
2:
13/5
20
1
3:
41/10
20
1
4:
27/5
20
1
5:
13/2
20
1
6:
37/5
20
1
7:
81/10
20
1
8:
43/5
20
1
9:
89/10
20
1
10:
9
20
3
1:
3/10
20
3
2:
1/5
20
3
3:
7/10
20
3
4:
9/5
20
3
5:
3/2
20
3
6:
9/5
20
3
7:
27/10
20
3
8:
11/5
20
3
9:
23/10
20
3
10:
3
20
9
1:
1/10
20
9
2:
-3/5
20
9
3:
-11/10
20
9
4:
-7/5
20
9
5:
-3/2
20
9
6:
-7/5
20
9
7:
-11/10
20
9
8:
-3/5
20
9
9:
1/10
20
9
10:
1
20
11
1:
-1/10
20
11
2:
3/5
20
11
3:
11/10
20
11
4:
7/5
20
11
5:
3/2
20
11
6:
7/5
20
11
7:
11/10
20
11
8:
3/5
20
11
9:
-1/10
20
11
10:
-1
20
13
1:
-7/10
20
13
2:
-9/5
20
13
3:
-23/10
20
13
4:
-11/5
20
13
5:
-3/2
20
13
6:
-1/5
20
13
7:
-3/10
20
13
8:
-9/5
20
13
9:
-27/10
20
13
10:
-3
20
19
1:
-9/10
20
19
2:
-13/5
20
19
3:
-41/10
20
19
4:
-27/5
20
19
5:
-13/2
20
19
6:
-37/5
20
19
7:
-81/10
20
19
8:
-43/5
20
19
9:
-89/10
20
19
10:
-9
Definition
The invariant $\rho_u(p,q)=\eta_u(p,q)-\eta_0(p,q)$ is the difference of the unreduced odd signature eta invariants of $L(p,q)$ twisted by $\chi_u$ and by the trivial character [4].
Parameters
$p$
—   order of the fundamental group ($p\ge2$)
$q$
—   quotient-action exponent ($1\le q<p$ and $\gcd(p,q)=1$)
$u$
—   character exponent ($1\le u<p$)
Formulas
(1)
$\eta_0(p,q)=-\frac1p\sum_{j=1}^{p-1}\cot(\pi j/p)\cot(\pi qj/p)$.
(2)
$\rho_u(p,q)=-4\sum_{j=1}^{k-1}((qj/p))-2((qk/p))$, where $0\le k<p$ and $qk\equiv u\pmod p$. Here $((x))=x-\lfloor x\rfloor-\tfrac12$ for $x\notin\mathbb Z$ and $((x))=0$ for $x\in\mathbb Z$ [1], Proposition 2.10.
(3)
$\rho_u(p,q)=\frac{2}{p}\sum_{j=1}^{p-1}\cot(\pi j/p)\cot(\pi qj/p)\sin^2(\pi uj/p)$.
(4)
$\rho_{(-u)\bmod p}(p,q)=\rho_u(p,q)$.
(5)
$\rho_u(p,p-q)=-\rho_u(p,q)$.
(6)
$\rho_u(p,q)=\rho_{ur\bmod p}(p,r)$, where $1\le r<p$ and $qr\equiv1\pmod p$ [2], Remark 3.3.23.
(7)
$\rho_u(p,1)=\frac{2u(p-u)}p-1$ for $1\le u<p$; the trivial character has $\rho_0(p,q)=0$.
Comments
(8)
Let $L(p,q)=S^3/\langle g\rangle$, where $S^3$ has unit radius and the boundary orientation of the unit ball in $\mathbb C^2$, $g(z_1,z_2)=(\zeta_p z_1,\zeta_p^q z_2)$, $\zeta_p=e^{2\pi i/p}$ and $\chi_u(g)=\zeta_p^u$. The quotient inherits the round metric and orientation [2], Appendix A.1.
(9)
The eta invariants use the odd signature operator on even-degree complex forms, with complex eigenspace multiplicities, excluding zero eigenvalues. Their sign is fixed by (1). The rho invariant is retained as a rational number, without reduction modulo integers or a kernel correction. It is independent of the metric [1], Section 2.2.
(10)
Oriented lens spaces with the same $p$ are diffeomorphic exactly when $q^\prime\equiv q^{\pm1}\pmod p$ [3]. Replacing $q$ by its inverse requires the character relabeling (6). Conjugate characters give equal invariants, and nonfaithful characters are included. Residual automorphisms can give equal values in distinct rows.
(11)
For these finite-image characters, the Casson–Gordon signature defect $\sigma(L(p,q),\chi_u)$ in the convention of [1], Proposition 2.8, equals $-\rho_u(p,q)$.
(12)
Ray–Singer torsion of lens spaces uses the same quotient generator and complex line characters, but takes derivatives of spectral zeta functions of Hodge Laplacians. Positive torsion is insensitive to orientation, while rho changes sign on reversing it.
Programs
(P1)
sage
import numberdb.sage as numberdb
from sage.rings.rational_field import QQ

def rho(p, q, u):
    k = (u * pow(q, -1, p)) % p
    def saw(n):
        r = n % p
        return QQ(0) if r == 0 else QQ(r)/QQ(p) - QQ(1)/QQ(2)
    return -4*sum((saw(q*j) for j in range(1, k)), QQ(0)) - 2*saw(q*k)

print(rho(7, 2, 1))
References
[1]
E. Toffoli, The Atiyah–Patodi–Singer rho invariant and signatures of links, Proceedings of the Edinburgh Mathematical Society 65(2) (2022), 404–440. See Sections 2.2–2.4, Proposition 2.10 and equation (2.4). (doi)
[2]
E. Toffoli, Rho invariants for manifolds with boundary and low-dimensional topology, PhD thesis, University of Regensburg (2019). See Section 3.3.4 and Appendix A.1; available in [5].
[3]
M. Sağlam, Holomorphic curves in the symplectizations of lens spaces: An elementary approach, Münster Journal of Mathematics 15 (2022), 389–440. See Theorem 1.3 and the following paragraph on p.392. (doi)
Links
Similar tables
Atiyah–Patodi–Singer eta invariants of lens spaces —   The twisted and untwisted spectral invariants whose difference is rho.
Ray–Singer torsion of lens spaces —   A different spectral invariant of the same complex line bundles, unchanged by orientation reversal.
Data properties
Entries are of type: rational number
How well the digits are known: exact
Table is complete: no (it holds 550 exact rho invariants on all 87 oriented lens-space presentations with $2\le p\le20$, using the smallest positive $q$ in each inversion orbit modulo $p$ and every $1\le u\le\lfloor p/2\rfloor$. This covers every oriented lens-space type and every nontrivial character up to conjugation within the bound, including nonfaithful characters. The cutoff in fundamental-group order is a practical, noncanonical reference extent; complete type and character coverage is preferred to extending only $L(p,1)$. Character labels transform by (6) when a presentation is replaced by its inverse)
How they were obtained:

Computed by (2) using exact Sage rational arithmetic. Every selected value agrees with direct weighted lattice-point counts from [1], equation (2.4), and with the difference of the twisted and untwisted eta values read from T622.

more

Independent Arb evaluation of (3) at 512 bits encloses every value; all balls are finite, their radii are less than $4.76\times10^{-151}$, and the nonzero values retain at least 498 bits of relative accuracy. Zero, positive and negative controls precede these checks. Conjugation, orientation reversal, coordinate exchange and the homogeneous formula are checked on all 1744 character labels through $p=20$, including reconstruction from the selected rows. The attached generator, checks, argument-selection note and numerical evidence reproduce the computation.



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