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Below is the multiplication table for QR17s, including numerical approximations. Red cells represent squares, and blue cells antisquares.

The number 2, which appears among the products as an augend or minuend, is twice the multiplicative identity, and equals ⟨ 0, +2, 0, −2; 0, +2, 0, −2; 0, +2, 0, −2; 0, +2, 0, −2 ⟩ = 2C₂ − 2C₄ + 2C₆ − 2C₈ + 2C₁₀ − 2C₁₂ + 2C₁₄ − 2C₁₆.

		C1 1.991468	C2 1.965946	C3 1.923651	C4 1.864944		C5 1.790327	C6 1.700434	C7 1.596034	C8 1.478018		C9 1.347391	C10 1.205269	C11 1.052864	C12 0.891477		C13 0.722483	C14 0.547326	C15 0.367499	C16 0.184537
C1 1.991468		2 + C2 3.965946	C1 + C3 3.915120	C2 + C4 3.830891	C3 + C5 3.713978		C4 + C6 3.565379	C5 + C7 3.386361	C6 + C8 3.178452	C7 + C9 2.943426		C8 + C10 2.683287	C9 + C11 2.400256	C10 + C12 2.096746	C11 + C13 1.775348		C12 + C14 1.438803	C13 + C15 1.089982	C14 + C16 0.731863	C15 0.367499		C1 1.991468
C2 1.965946		C1 + C3 3.915120	2 + C4 3.864944	C1 + C5 3.781795	C2 + C6 3.666380		C3 + C7 3.519686	C4 + C8 3.342962	C5 + C9 3.137718	C6 + C10 2.905704		C7 + C11 2.648899	C8 + C12 2.369495	C9 + C13 2.069875	C10 + C14 1.752595		C11 + C15 1.420363	C12 + C16 1.076013	C13 0.722483	C14 − C16 0.362789		C2 1.965946
C3 1.923651		C2 + C4 3.830891	C1 + C5 3.781795	2 + C6 3.700434	C1 + C7 3.587503		C2 + C8 3.443964	C3 + C9 3.271043	C4 + C10 3.070214	C5 + C11 2.843191		C6 + C12 2.591911	C7 + C13 2.318518	C8 + C14 2.025344	C9 + C15 1.714890		C10 + C16 1.389806	C11 1.052864	C12 − C16 0.706940	C13 − C15 0.354984		C3 1.923651
C4 1.864944		C3 + C5 3.713978	C2 + C6 3.666380	C1 + C7 3.587503	2 + C8 3.478018		C1 + C9 3.338860	C2 + C10 3.171215	C3 + C11 2.976516	C4 + C12 2.756421		C5 + C13 2.512810	C6 + C14 2.247760	C7 + C15 1.963533	C8 + C16 1.662555		C9 1.347391	C10 − C16 1.020733	C11 − C15 0.685365	C12 − C14 0.344151		C4 1.864944
C5 1.790327		C4 + C6 3.565379	C3 + C7 3.519686	C2 + C8 3.443964	C1 + C9 3.338860		2 + C10 3.205269	C1 + C11 3.044333	C2 + C12 2.857423	C3 + C13 2.646135		C4 + C14 2.412270	C5 + C15 2.157826	C6 + C16 1.884971	C7 1.596034		C8 − C16 1.293481	C9 − C15 0.979892	C10 − C14 0.657943	C11 − C13 0.330381		C5 1.790327
C6 1.700434		C5 + C7 3.386361	C4 + C8 3.342962	C3 + C9 3.271043	C2 + C10 3.171215		C1 + C11 3.044333	2 + C12 2.891477	C1 + C13 2.713952	C2 + C14 2.513272		C3 + C15 2.291150	C4 + C16 2.049481	C5 1.790327	C6 − C16 1.515898		C7 − C15 1.228535	C8 − C14 0.930692	C9 − C13 0.624908	C10 − C12 0.313793		C6 1.700434
C7 1.596034		C6 + C8 3.178452	C5 + C9 3.137718	C4 + C10 3.070214	C3 + C11 2.976516		C2 + C12 2.857423	C1 + C13 2.713952	2 + C14 2.547326	C1 + C15 2.358967		C2 + C16 2.150483	C3 1.923651	C4 − C16 1.680408	C5 − C15 1.422828		C6 − C14 1.153108	C7 − C13 0.873551	C8 − C12 0.586541	C9 − C11 0.294527		C7 1.596034
C8 1.478018		C7 + C9 2.943426	C6 + C10 2.905704	C5 + C11 2.843191	C4 + C12 2.756421		C3 + C13 2.646135	C2 + C14 2.513272	C1 + C15 2.358967	2 + C16 2.184537		C1 1.991468	C2 − C16 1.781409	C3 − C15 1.556152	C4 − C14 1.317618		C5 − C13 1.067843	C6 − C12 0.808958	C7 − C11 0.543170	C8 − C10 0.272749		C8 1.478018
C9 1.347391		C8 + C10 2.683287	C7 + C11 2.648899	C6 + C12 2.591911	C5 + C13 2.512810		C4 + C14 2.412270	C3 + C15 2.291150	C2 + C16 2.150483	C1 1.991468		2 − C16 1.815463	C1 − C15 1.623969	C2 − C14 1.418620	C3 − C13 1.201168		C4 − C12 0.973468	C5 − C11 0.737462	C6 − C10 0.495165	C7 − C9 0.248643		C9 1.347391
C10 1.205269		C9 + C11 2.400256	C8 + C12 2.369495	C7 + C13 2.318518	C6 + C14 2.247760		C5 + C15 2.157826	C4 + C16 2.049481	C3 1.923651	C2 − C16 1.781409		C1 − C15 1.623969	2 − C14 1.452674	C1 − C13 1.268985	C2 − C12 1.074469		C3 − C11 0.870787	C4 − C10 0.659675	C5 − C9 0.442935	C6 − C8 0.222416		C10 1.205269
C11 1.052864		C10 + C12 2.096746	C9 + C13 2.069875	C8 + C14 2.025344	C7 + C15 1.963533		C6 + C16 1.884971	C5 1.790327	C4 − C16 1.680408	C3 − C15 1.556152		C2 − C14 1.418620	C1 − C13 1.268985	2 − C12 1.108523	C1 − C11 0.938604		C2 − C10 0.760677	C3 − C9 0.576260	C4 − C8 0.386927	C5 − C7 0.194292		C11 1.052864
C12 0.891477		C11 + C13 1.775348	C10 + C14 1.752595	C9 + C15 1.714890	C8 + C16 1.662555		C7 1.596034	C6 − C16 1.515898	C5 − C15 1.422828	C4 − C14 1.317618		C3 − C13 1.201168	C2 − C12 1.074469	C1 − C11 0.938604	2 − C10 0.794731		C1 − C9 0.644077	C2 − C8 0.487928	C3 − C7 0.327617	C4 − C6 0.164510		C12 0.891477
C13 0.722483		C12 + C14 1.438803	C11 + C15 1.420363	C10 + C16 1.389806	C9 1.347391		C8 − C16 1.293481	C7 − C15 1.228535	C6 − C14 1.153108	C5 − C13 1.067843		C4 − C12 0.973468	C3 − C11 0.870787	C2 − C10 0.760677	C1 − C9 0.644077		2 − C8 0.521982	C1 − C7 0.395434	C2 − C6 0.265512	C3 − C5 0.133325		C13 0.722483
C14 0.547326		C13 + C15 1.089982	C12 + C16 1.076013	C11 1.052864	C10 − C16 1.020733		C9 − C15 0.979892	C8 − C14 0.930692	C7 − C13 0.873551	C6 − C12 0.808958		C5 − C11 0.737462	C4 − C10 0.659675	C3 − C9 0.576260	C2 − C8 0.487928		C1 − C7 0.395434	2 − C6 0.299566	C1 − C5 0.201142	C2 − C4 0.101002		C14 0.547326
C15 0.367499		C14 + C16 0.731863	C13 0.722483	C12 − C16 0.706940	C11 − C15 0.685365		C10 − C14 0.657943	C9 − C13 0.624908	C8 − C12 0.586541	C7 − C11 0.543170		C6 − C10 0.495165	C5 − C9 0.442935	C4 − C8 0.386927	C3 − C7 0.327617		C2 − C6 0.265512	C1 − C5 0.201142	2 − C4 0.135056	C1 − C3 0.067817		C15 0.367499
C16 0.184537		C15 0.367499	C14 − C16 0.362789	C13 − C15 0.354984	C12 − C14 0.344151		C11 − C13 0.330381	C10 − C12 0.313793	C9 − C11 0.294527	C8 − C10 0.272749		C7 − C9 0.248643	C6 − C8 0.222416	C5 − C7 0.194292	C4 − C6 0.164510		C3 − C5 0.133325	C2 − C4 0.101002	C1 − C3 0.067817	2 − C2 0.034054		C16 0.184537
		C1 1.991468	C2 1.965946	C3 1.923651	C4 1.864944		C5 1.790327	C6 1.700434	C7 1.596034	C8 1.478018		C9 1.347391	C10 1.205269	C11 1.052864	C12 0.891477		C13 0.722483	C14 0.547326	C15 0.367499	C16 0.184537

the above table formatted in four parts
top left	top right
bottom left	bottom right

Below is an extract from the table above; it is the self-contained multiplication table for those C_n where n is even. Numerical approximations are unaffected, but the antisquares are gone.

		C2	C4	C6	C8		C10	C12	C14	C16
C2		2 + C4	C2 + C6	C4 + C8	C6 + C10		C8 + C12	C10 + C14	C12 + C16	C14 − C16
C4		C2 + C6	2 + C8	C2 + C10	C4 + C12		C6 + C14	C8 + C16	C10 − C16	C12 − C14
C6		C4 + C8	C2 + C10	2 + C12	C2 + C14		C4 + C16	C6 − C16	C8 − C14	C10 − C12
C8		C6 + C10	C4 + C12	C2 + C14	2 + C16		C2 − C16	C4 − C14	C6 − C12	C8 − C10
C10		C8 + C12	C6 + C14	C4 + C16	C2 − C16		2 − C14	C2 − C12	C4 − C10	C6 − C8
C12		C10 + C14	C8 + C16	C6 − C16	C4 − C14		C2 − C12	2 − C10	C2 − C8	C4 − C6
C14		C12 + C16	C10 − C16	C8 − C14	C6 − C12		C4 − C10	C2 − C8	2 − C6	C2 − C4
C16		C14 − C16	C12 − C14	C10 − C12	C8 − C10		C6 − C8	C4 − C6	C2 − C4	2 − C2

This trigonometric identity is key to the multiplication table:

2 cos (θ) cos (φ) = cos (θ + φ) + cos (θ − φ)

which in C_n notation works out to be:

C_m C_n = C_m+n + C_m−n

To complete some calculations, the following extensions of C_n notation are necessary:

• C₀ = 2
• C₁₇ = 0
• C_−n = C_+n
• C_17+n = −C_17−n