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Atiyah–Patodi–Singer eta invariants of lens spaces
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Numbers
$p$
$q$
$u$ 
$\eta(L(p,q),\chi_u)$
2
1
0:
0
2
1
1:
0
3
1
0:
-2/9
3
1
1:
1/9
3
2
0:
2/9
3
2
1:
-1/9
4
1
0:
-1/2
4
1
1:
0
4
1
2:
1/2
4
3
0:
1/2
4
3
1:
0
4
3
2:
-1/2
5
1
0:
-4/5
5
1
1:
-1/5
5
1
2:
3/5
5
2
0:
0
5
2
1:
-1/5
5
2
2:
1/5
5
4
0:
4/5
5
4
1:
1/5
5
4
2:
-3/5
6
1
0:
-10/9
6
1
1:
-4/9
6
1
2:
5/9
6
1
3:
8/9
6
5
0:
10/9
6
5
1:
4/9
6
5
2:
-5/9
6
5
3:
-8/9
7
1
0:
-10/7
7
1
1:
-5/7
7
1
2:
3/7
7
1
3:
1
7
2
0:
-2/7
7
2
1:
-3/7
7
2
2:
1/7
7
2
3:
3/7
7
3
0:
2/7
7
3
1:
-1/7
7
3
2:
-3/7
7
3
3:
3/7
7
6
0:
10/7
7
6
1:
5/7
7
6
2:
-3/7
7
6
3:
-1
8
1
0:
-7/4
8
1
1:
-1
8
1
2:
1/4
8
1
3:
1
8
1
4:
5/4
8
3
0:
-1/4
8
3
1:
0
8
3
2:
-1/4
8
3
3:
0
8
3
4:
3/4
8
5
0:
1/4
8
5
1:
0
8
5
2:
1/4
8
5
3:
0
8
5
4:
-3/4
8
7
0:
7/4
8
7
1:
1
8
7
2:
-1/4
8
7
3:
-1
8
7
4:
-5/4
9
1
0:
-56/27
9
1
1:
-35/27
9
1
2:
1/27
9
1
3:
25/27
9
1
4:
37/27
9
2
0:
-16/27
9
2
1:
-19/27
9
2
2:
-1/27
9
2
3:
11/27
9
2
4:
17/27
9
4
0:
16/27
9
4
1:
1/27
9
4
2:
-17/27
9
4
3:
-11/27
9
4
4:
19/27
9
8
0:
56/27
9
8
1:
35/27
9
8
2:
-1/27
9
8
3:
-25/27
9
8
4:
-37/27
10
1
0:
-12/5
10
1
1:
-8/5
10
1
2:
-1/5
10
1
3:
4/5
10
1
4:
7/5
10
1
5:
8/5
10
3
0:
0
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
0:
12/5
10
9
1:
8/5
10
9
2:
1/5
10
9
3:
-4/5
10
9
4:
-7/5
10
9
5:
-8/5
11
1
0:
-30/11
11
1
1:
-21/11
11
1
2:
-5/11
11
1
3:
7/11
11
1
4:
15/11
11
1
5:
19/11
11
2
0:
-10/11
11
2
1:
-1
11
2
2:
-3/11
11
2
3:
3/11
11
2
4:
7/11
11
2
5:
9/11
11
3
0:
-6/11
11
3
1:
-3/11
11
3
2:
-5/11
11
3
3:
-1/11
11
3
4:
9/11
11
3
5:
3/11
11
5
0:
10/11
11
5
1:
3/11
11
5
2:
-7/11
11
5
3:
-9/11
11
5
4:
-3/11
11
5
5:
1
11
7
0:
6/11
11
7
1:
1/11
11
7
2:
-3/11
11
7
3:
5/11
11
7
4:
3/11
11
7
5:
-9/11
11
10
0:
30/11
11
10
1:
21/11
11
10
2:
5/11
11
10
3:
-7/11
11
10
4:
-15/11
11
10
5:
-19/11
12
1
0:
-55/18
12
1
1:
-20/9
12
1
2:
-13/18
12
1
3:
4/9
12
1
4:
23/18
12
1
5:
16/9
12
1
6:
35/18
12
5
0:
1/18
12
5
1:
2/9
12
5
2:
-5/18
12
5
3:
-4/9
12
5
4:
-5/18
12
5
5:
2/9
12
5
6:
19/18
12
7
0:
-1/18
12
7
1:
-2/9
12
7
2:
5/18
12
7
3:
4/9
12
7
4:
5/18
12
7
5:
-2/9
12
7
6:
-19/18
12
11
0:
55/18
12
11
1:
20/9
12
11
2:
13/18
12
11
3:
-4/9
12
11
4:
-23/18
12
11
5:
-16/9
12
11
6:
-35/18
13
1
0:
-44/13
13
1
1:
-33/13
13
1
2:
-1
13
1
3:
3/13
13
1
4:
15/13
13
1
5:
23/13
13
1
6:
27/13
13
2
0:
-16/13
13
2
1:
-17/13
13
2
2:
-7/13
13
2
3:
1/13
13
2
4:
7/13
13
2
5:
11/13
13
2
6:
1
13
3
0:
-4/13
13
3
1:
-9/13
13
3
2:
-11/13
13
3
3:
3/13
13
3
4:
7/13
13
3
5:
1/13
13
3
6:
11/13
13
4
0:
4/13
13
4
1:
-3/13
13
4
2:
-11/13
13
4
3:
-7/13
13
4
4:
9/13
13
4
5:
11/13
13
4
6:
-1/13
13
5
0:
0
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
0:
16/13
13
6
1:
7/13
13
6
2:
-7/13
13
6
3:
-1
13
6
4:
-11/13
13
6
5:
-1/13
13
6
6:
17/13
13
12
0:
44/13
13
12
1:
33/13
13
12
2:
1
13
12
3:
-3/13
13
12
4:
-15/13
13
12
5:
-23/13
13
12
6:
-27/13
14
1
0:
-26/7
14
1
1:
-20/7
14
1
2:
-9/7
14
1
3:
0
14
1
4:
1
14
1
5:
12/7
14
1
6:
15/7
14
1
7:
16/7
14
3
0:
-6/7
14
3
1:
-4/7
14
3
2:
-5/7
14
3
3:
-2/7
14
3
4:
5/7
14
3
5:
2/7
14
3
6:
3/7
14
3
7:
8/7
14
9
0:
6/7
14
9
1:
2/7
14
9
2:
-3/7
14
9
3:
-2/7
14
9
4:
5/7
14
9
5:
4/7
14
9
6:
-5/7
14
9
7:
-8/7
14
13
0:
26/7
14
13
1:
20/7
14
13
2:
9/7
14
13
3:
0
14
13
4:
-1
14
13
5:
-12/7
14
13
6:
-15/7
14
13
7:
-16/7
15
1
0:
-182/45
15
1
1:
-143/45
15
1
2:
-71/45
15
1
3:
-11/45
15
1
4:
37/45
15
1
5:
73/45
15
1
6:
97/45
15
1
7:
109/45
15
2
0:
-14/9
15
2
1:
-73/45
15
2
2:
-37/45
15
2
3:
-7/45
15
2
4:
17/45
15
2
5:
7/9
15
2
6:
47/45
15
2
7:
53/45
15
4
0:
-38/45
15
4
1:
-17/45
15
4
2:
1/45
15
4
3:
-29/45
15
4
4:
-17/45
15
4
5:
37/45
15
4
6:
43/45
15
4
7:
1/45
15
7
0:
14/9
15
7
1:
37/45
15
7
2:
-17/45
15
7
3:
-47/45
15
7
4:
-53/45
15
7
5:
-7/9
15
7
6:
7/45
15
7
7:
73/45
15
11
0:
38/45
15
11
1:
17/45
15
11
2:
-1/45
15
11
3:
29/45
15
11
4:
17/45
15
11
5:
-37/45
15
11
6:
-43/45
15
11
7:
-1/45
15
14
0:
182/45
15
14
1:
143/45
15
14
2:
71/45
15
14
3:
11/45
15
14
4:
-37/45
15
14
5:
-73/45
15
14
6:
-97/45
15
14
7:
-109/45
16
1
0:
-35/8
16
1
1:
-7/2
16
1
2:
-15/8
16
1
3:
-1/2
16
1
4:
5/8
16
1
5:
3/2
16
1
6:
17/8
16
1
7:
5/2
16
1
8:
21/8
16
3
0:
-5/8
16
3
1:
-1
16
3
2:
-9/8
16
3
3:
0
16
3
4:
3/8
16
3
5:
0
16
3
6:
7/8
16
3
7:
1
16
3
8:
3/8
16
5
0:
5/8
16
5
1:
0
16
5
2:
-7/8
16
5
3:
-1
16
5
4:
-3/8
16
5
5:
1
16
5
6:
9/8
16
5
7:
0
16
5
8:
-3/8
16
7
0:
3/8
16
7
1:
1/2
16
7
2:
-1/8
16
7
3:
-1/2
16
7
4:
-5/8
16
7
5:
-1/2
16
7
6:
-1/8
16
7
7:
1/2
16
7
8:
11/8
16
9
0:
-3/8
16
9
1:
-1/2
16
9
2:
1/8
16
9
3:
1/2
16
9
4:
5/8
16
9
5:
1/2
16
9
6:
1/8
16
9
7:
-1/2
16
9
8:
-11/8
16
15
0:
35/8
16
15
1:
7/2
16
15
2:
15/8
16
15
3:
1/2
16
15
4:
-5/8
16
15
5:
-3/2
16
15
6:
-17/8
16
15
7:
-5/2
16
15
8:
-21/8
17
1
0:
-80/17
17
1
1:
-65/17
17
1
2:
-37/17
17
1
3:
-13/17
17
1
4:
7/17
17
1
5:
23/17
17
1
6:
35/17
17
1
7:
43/17
17
1
8:
47/17
17
2
0:
-32/17
17
2
1:
-33/17
17
2
2:
-19/17
17
2
3:
-7/17
17
2
4:
3/17
17
2
5:
11/17
17
2
6:
1
17
2
7:
21/17
17
2
8:
23/17
17
3
0:
-20/17
17
3
1:
-15/17
17
3
2:
-1
17
3
3:
-9/17
17
3
4:
9/17
17
3
5:
3/17
17
3
6:
7/17
17
3
7:
21/17
17
3
8:
11/17
17
4
0:
0
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
0:
-4/17
17
5
1:
-1/17
17
5
2:
-9/17
17
5
3:
-11/17
17
5
4:
-7/17
17
5
5:
3/17
17
5
6:
19/17
17
5
7:
7/17
17
5
8:
1/17
17
8
0:
32/17
17
8
1:
19/17
17
8
2:
-3/17
17
8
3:
-1
17
8
4:
-23/17
17
8
5:
-21/17
17
8
6:
-11/17
17
8
7:
7/17
17
8
8:
33/17
17
10
0:
4/17
17
10
1:
-3/17
17
10
2:
-7/17
17
10
3:
9/17
17
10
4:
11/17
17
10
5:
-1/17
17
10
6:
7/17
17
10
7:
1/17
17
10
8:
-19/17
17
11
0:
20/17
17
11
1:
9/17
17
11
2:
-7/17
17
11
3:
-11/17
17
11
4:
-3/17
17
11
5:
1
17
11
6:
15/17
17
11
7:
-9/17
17
11
8:
-21/17
17
16
0:
80/17
17
16
1:
65/17
17
16
2:
37/17
17
16
3:
13/17
17
16
4:
-7/17
17
16
5:
-23/17
17
16
6:
-35/17
17
16
7:
-43/17
17
16
8:
-47/17
18
1
0:
-136/27
18
1
1:
-112/27
18
1
2:
-67/27
18
1
3:
-28/27
18
1
4:
5/27
18
1
5:
32/27
18
1
6:
53/27
18
1
7:
68/27
18
1
8:
77/27
18
1
9:
80/27
18
5
0:
-8/27
18
5
1:
-14/27
18
5
2:
-5/27
18
5
3:
-8/27
18
5
4:
-23/27
18
5
5:
4/27
18
5
6:
19/27
18
5
7:
22/27
18
5
8:
13/27
18
5
9:
-8/27
18
7
0:
8/27
18
7
1:
-4/27
18
7
2:
-13/27
18
7
3:
8/27
18
7
4:
5/27
18
7
5:
-22/27
18
7
6:
-19/27
18
7
7:
14/27
18
7
8:
23/27
18
7
9:
8/27
18
17
0:
136/27
18
17
1:
112/27
18
17
2:
67/27
18
17
3:
28/27
18
17
4:
-5/27
18
17
5:
-32/27
18
17
6:
-53/27
18
17
7:
-68/27
18
17
8:
-77/27
18
17
9:
-80/27
19
1
0:
-102/19
19
1
1:
-85/19
19
1
2:
-53/19
19
1
3:
-25/19
19
1
4:
-1/19
19
1
5:
1
19
1
6:
35/19
19
1
7:
47/19
19
1
8:
55/19
19
1
9:
59/19
19
2
0:
-42/19
19
2
1:
-43/19
19
2
2:
-27/19
19
2
3:
-13/19
19
2
4:
-1/19
19
2
5:
9/19
19
2
6:
17/19
19
2
7:
23/19
19
2
8:
27/19
19
2
9:
29/19
19
3
0:
-18/19
19
3
1:
-25/19
19
3
2:
-27/19
19
3
3:
-5/19
19
3
4:
3/19
19
3
5:
-3/19
19
3
6:
15/19
19
3
7:
1
19
3
8:
9/19
19
3
9:
23/19
19
4
0:
-22/19
19
4
1:
-13/19
19
4
2:
-5/19
19
4
3:
-17/19
19
4
4:
-11/19
19
4
5:
13/19
19
4
6:
17/19
19
4
7:
1/19
19
4
8:
3/19
19
4
9:
23/19
19
6
0:
18/19
19
6
1:
5/19
19
6
2:
-15/19
19
6
3:
-23/19
19
6
4:
-1
19
6
5:
-3/19
19
6
6:
25/19
19
6
7:
27/19
19
6
8:
3/19
19
6
9:
-9/19
19
7
0:
-6/19
19
7
1:
-9/19
19
7
2:
1/19
19
7
3:
5/19
19
7
4:
3/19
19
7
5:
-5/19
19
7
6:
-1
19
7
7:
-1/19
19
7
8:
11/19
19
7
9:
17/19
19
8
0:
6/19
19
8
1:
1/19
19
8
2:
5/19
19
8
3:
-1/19
19
8
4:
-17/19
19
8
5:
-5/19
19
8
6:
-3/19
19
8
7:
-11/19
19
8
8:
9/19
19
8
9:
1
19
9
0:
42/19
19
9
1:
27/19
19
9
2:
1/19
19
9
3:
-17/19
19
9
4:
-27/19
19
9
5:
-29/19
19
9
6:
-23/19
19
9
7:
-9/19
19
9
8:
13/19
19
9
9:
43/19
19
14
0:
22/19
19
14
1:
11/19
19
14
2:
-3/19
19
14
3:
-1/19
19
14
4:
17/19
19
14
5:
13/19
19
14
6:
-13/19
19
14
7:
-23/19
19
14
8:
-17/19
19
14
9:
5/19
19
18
0:
102/19
19
18
1:
85/19
19
18
2:
53/19
19
18
3:
25/19
19
18
4:
1/19
19
18
5:
-1
19
18
6:
-35/19
19
18
7:
-47/19
19
18
8:
-55/19
19
18
9:
-59/19
20
1
0:
-57/10
20
1
1:
-24/5
20
1
2:
-31/10
20
1
3:
-8/5
20
1
4:
-3/10
20
1
5:
4/5
20
1
6:
17/10
20
1
7:
12/5
20
1
8:
29/10
20
1
9:
16/5
20
1
10:
33/10
20
3
0:
-3/2
20
3
1:
-6/5
20
3
2:
-13/10
20
3
3:
-4/5
20
3
4:
3/10
20
3
5:
0
20
3
6:
3/10
20
3
7:
6/5
20
3
8:
7/10
20
3
9:
4/5
20
3
10:
3/2
20
9
0:
7/10
20
9
1:
4/5
20
9
2:
1/10
20
9
3:
-2/5
20
9
4:
-7/10
20
9
5:
-4/5
20
9
6:
-7/10
20
9
7:
-2/5
20
9
8:
1/10
20
9
9:
4/5
20
9
10:
17/10
20
11
0:
-7/10
20
11
1:
-4/5
20
11
2:
-1/10
20
11
3:
2/5
20
11
4:
7/10
20
11
5:
4/5
20
11
6:
7/10
20
11
7:
2/5
20
11
8:
-1/10
20
11
9:
-4/5
20
11
10:
-17/10
20
13
0:
3/2
20
13
1:
4/5
20
13
2:
-3/10
20
13
3:
-4/5
20
13
4:
-7/10
20
13
5:
0
20
13
6:
13/10
20
13
7:
6/5
20
13
8:
-3/10
20
13
9:
-6/5
20
13
10:
-3/2
20
19
0:
57/10
20
19
1:
24/5
20
19
2:
31/10
20
19
3:
8/5
20
19
4:
3/10
20
19
5:
-4/5
20
19
6:
-17/10
20
19
7:
-12/5
20
19
8:
-29/10
20
19
9:
-16/5
20
19
10:
-33/10
Definition
The unreduced Atiyah–Patodi–Singer eta invariant $\eta_u(p,q)$ measures the spectral asymmetry of the odd signature operator on even-degree complex forms on $L(p,q)$ twisted by $\chi_u$ [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 ($0\le u<p$)
Formulas
(1)
$\eta_u(p,q)=\left.\sum_{\lambda\ne0}\operatorname{sgn}(\lambda)|\lambda|^{-s}\right|_{s=0}$ by meromorphic continuation, where $\lambda$ ranges over the nonzero eigenvalues of the twisted odd signature operator with complex eigenspace multiplicities [1], Section 2.2.
(2)
$\eta_u(p,q)=-4\sum_{j=0}^{p-1}((j/p))\,(((qj+u)/p))$, where $((x))=x-\lfloor x\rfloor-\tfrac12$ for $x\notin\mathbb Z$ and $((x))=0$ for $x\in\mathbb Z$ [2], Theorem 3.3.27.
(3)
$\eta_u(p,q)=-\frac1p\sum_{j=1}^{p-1}\cot(\pi j/p)\cot(\pi qj/p)\cos(2\pi uj/p)$. This is the finite Fourier form of (2).
(4)
$\eta_{(-u)\bmod p}(p,q)=\eta_u(p,q)$.
(5)
$\eta_u(p,p-q)=-\eta_u(p,q)$.
(6)
$\eta_u(p,q)=\eta_{ur\bmod p}(p,r)$, where $1\le r<p$ and $qr\equiv1\pmod p$.
(7)
$\eta_u(p,1)=-\frac{(p-1)(p-2)}{3p}+\begin{cases}0,&u=0,\\ \frac{2u(p-u)}p-1,&1\le u<p.\end{cases}$
(8)
$\sum_{u=0}^{p-1}\eta_u(p,q)=0$.
Comments
(9)
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.
(10)
The operator sign is fixed by the negative cotangent sum (3), whose $u=0$ case is the ordinary eta invariant. Zero eigenvalues are omitted, no kernel correction is added, and the result is retained in $\mathbb Q$ rather than reduced modulo $\mathbb Z$. The twisting bundle is a complex line, with complex multiplicities [1].
(11)
Oriented lens spaces with the same $p$ are diffeomorphic exactly when $q^\prime\equiv q^{\pm1}\pmod p$ [3]. The chosen presentation fixes $g$ and therefore the labels of its characters. Coordinate exchange requires the character relabeling (6); conjugate characters have equal eta invariants. Residual automorphisms may give equal values in different rows.
(12)
Ray–Singer torsion of lens spaces uses the same quotient action and complex line characters, but measures derivatives of spectral zeta functions of Hodge Laplacians. Positive torsion ignores orientation, whereas eta changes sign on reversing it. The separately named APS rho invariant is the difference $\rho_u(p,q)=\eta_u(p,q)-\eta_0(p,q)$ [1], Section 2.2.
Programs
(P1)
sage
import numberdb.sage as numberdb
from sage.rings.rational_field import QQ

def eta(p, q, u):
    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(j)*saw(q*j+u) for j in range(p)), QQ(0))

print(eta(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 and 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, Theorem 3.3.27, 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
Ray–Singer torsion of lens spaces —   A different spectral invariant of the same complex line bundles; positive torsion is unchanged by reversing orientation.
Data properties
Entries are of type: rational number
How well the digits are known: exact
Table is complete: no (it holds 637 exact eta 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 $0\le u\le\lfloor p/2\rfloor$. This covers every oriented lens-space type and every character up to conjugation within the bound, including trivial and 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 are transformed with (6) when a presentation is replaced by its inverse)
How they were obtained:

Computed by the finite sawtooth sum in exact Sage rational arithmetic. Every selected entry agrees with an independent exact calculation combining classical Dedekind reciprocity with weighted lattice-point counts from [1], equation (2.4).

more

Each value is also enclosed by the cotangent sum computed in Arb at 512 bits; all balls are finite, their radii are less than $2.64\times10^{-151}$, and nonzero values retain at least 496 bits of relative accuracy. Known zero and nonzero controls precede these checks. Conjugation, coordinate exchange, orientation reversal and the sum over all characters are checked across all 1744 unreduced labels through $p=20$. The attached generator, checks, argument-selection note and numerical evidence reproduce the computation.



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