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As I indicated in a comment on a recent MO question of Joel Bellaiche, "Primes and x^2+2y^2+4z^2" ,T2 follows from a result of Hasse on the class number of $Q(\sqrt{-2p})$, using Gauss' theorem that the number of representations of $2p$ as a sum of 3 squares is 12*(this class number). Hasse's proof is an application of the Gauss theory of genera and ambiguous forms.

As I indicated in a comment on a recent MO question of Joel Bellaiche, "Primes and x^2+2y^2+4z^2" ,T2 follows from a result of Hasse on the class number of $Q(\sqrt{-2p})$, using Gauss' theorem that the number of representations of $2p$ as a sum of 3 squares is 12*(this class number). Hasse's proof is an application of the Gauss theory of genera and ambiguous forms.

Let p be a prime that is 5 mod 8. Writing f21 as (f)(f2)(f2)(f16) we find that if R is (1/16)*(the number of representations of p by G1=xx+2yy+2zz+16tt with x,y,z and t all odd), then p is in S(f21) if and only if R is odd. Now since p is 5 mod 8, in any representation of p by G1, x,y and z must be odd. So if we set G2=xx+2yy+2zz+64tt then R=(N1-N2)/16, where N1 and N2 are the numbers of representations of p by G1 and G2 respectively. Now write p as aa+4bb with a and b congruent to 1 mod 4. Computer calculations indicate:

Conjecture 1_____N1=p+1+2a

Conjecture 2_____N2=((p+1)/2)+a+4b

If these conjectures hold then R=(p+1+2a-8b)/32. The numerator here is 4(b-1)^2 +(a+3)(a-1), which mod 64 is(a+3)(a-1). So R has the same parity as (a+3)(a-1)/32 and is odd just when a is 5 or 9 mod 16. Now mod 32, p=aa+4. So R is odd just when p is 29 or 85 mod 32, and so the conjectures imply Joel's result for S(f21).

How does one attack the conjectures? The theta series attached to G1 and G2 are modular forms for gamma_0 (64) and gamma_0 (256) respectively. If the conjectures are to hold it seems that each of these theta series should be a linear combination of Eisenstein series and cusp forms attached to Grossencharaktere for Q(i). It should be possible, using Sage, to get an explicit formulation of this, and prove the conjectures.

My proposed treatment of S(f19) is entirely similar. Suppose p is 3 mod 8. Writing f19 as(f)(f)(f)(f16) and arguing as above we find that if we take H1 and H2 to be xx+yy+zz+16tt and xx+yy+zz+64tt respectively, and let N1 and N2 be the number of representations of p by H1 and H2, then p is in S(f19) just when R=(N1-N2)/16 is odd. Now Jacobi's 4 square theorem, (see the argument in my answer to Joel's question), shows that N1 is 2(p+1). Write p as aa+2bb with a =1 or 3 mod 8. The computer suggests:

Conjecture 3_____N2=p+1+4a

So if the conjecture holds, R=(p+1-4a)/16, and one sees easily that this is odd just when p is 19 or 27 mod 32. Once again the theta series attached to H2 is a modular form for gamma_0 (256). The conjecture indicates that it should be a linear combination of Eisenstein series and cusp forms attached to Grossencharaktere for Q(root(-2)); all this should admit a proof using Sage.

Let p be a prime that is 5 mod 8. Writing $f^{21}$ as $(f)(f^2)(f^2)(f^{16})$ we find that if $R$ is (1/16)*(the number of representations of $p$ by $G_1=x^2+2y^2+2z^2+16t^2$ with $x,y,z$ and $t$ all odd), then $p$ is in $S(f^21)$ if and only if $R$ is odd. Now since $p$ is 5 mod 8, in any representation of $p$ by $G_1$, $x,y$ and $z$ must be odd. So if we set $G_2=x^2+2y^2+2z^2+64t^2$ then $R=(N1-N2)/16$, where $N_1$ and $N_2$ are the numbers of representations of $p$ by $G_1$ and $G_2$ respectively. Now write $p$ as $a^2+4b^2$ with $a$ and $b$ congruent to 1 mod 4. Computer calculations indicate:

Conjecture 1. $N_1=p+1+2a$

Conjecture 2. $N_2=((p+1)/2)+a+4b$

If these conjectures hold then $R=(p+1+2a-8b)/32$. The numerator here is $4(b-1)^2 +(a+3)(a-1)$, which mod 64 is $(a+3)(a-1)$. So $R$ has the same parity as $(a+3)(a-1)/32$ and is odd just when $a$ is $5$ or $9$ mod $16$. Now mod $32$, $p=a^2+4$. So $R$ is odd just when $p$ is $29$ or $85$ mod $32$, and so the conjectures imply Joel's result for $S(f^21)$.

How does one attack the conjectures? The theta series attached to $G_1$ and $G_2$ are modular forms for $\Gamma_0 (64)$ and $\Gamma_0 (256)$ respectively. If the conjectures are to hold it seems that each of these theta series should be a linear combination of Eisenstein series and cusp forms attached to Grossencharaktere for $\mathbb Q(i)$. It should be possible, using Sage, to get an explicit formulation of this, and prove the conjectures.

My proposed treatment of $S(f^{19})$ is entirely similar. Suppose $p$ is $3$ mod $8$. Writing $f^{19}$ as $(f)(f)(f)(f^{16})$ and arguing as above we find that if we take $H_1$ and $H_2$ to be $x^2+y^2+z^2+16t^2$ and $x^2+y^2+z^2+64t^2$ respectively, and let $N_1$ and $N_2$ be the number of representations of $p$ by $H_1$ and $H_2$, then $p$ is in $S(f^{19})$ just when $R=(N_1-N_2)/16$ is odd. Now Jacobi's 4 square theorem, (see the argument in my answer to Joel's question), shows that $N_1$ is $2(p+1)$. Write $p$ as $a^2+2b^2$ with $a =1$ or 3 mod 8. The computer suggests:

Conjecture 3. $N_2=p+1+4a$

So if the conjecture holds, $R=(p+1-4a)/16$, and one sees easily that this is odd just when $p$ is 19 or 27 mod 32. Once again the theta series attached to H2 is a modular form for $\Gamma_0 (256)$. The conjecture indicates that it should be a linear combination of Eisenstein series and cusp forms attached to Grossencharaktere for $\mathbb Q(\sqrt{-2})$; all this should admit a proof using Sage.

EDIT: Motivated by my recent simple proof of T2 (see my answer to the question of Joel cited above), I've found arguments that ought to reduce the proof of T3 to Sage calculations. The point is that forms of weight 2 are easier to deal with than forms of weight 3/2, so one should work with quadratic forms in 4 variables rather than in 3, even when the genera that arise have more than 1 class in them. Here's the idea of my argument for f21.

Let p be a prime that is 5 mod 8. Writing f21 as (f)(f2)(f2)(f16) we find that if R is (1/16)*(the number of representations of p by G1=xx+2yy+2zz+16tt with x,y,z and t all odd), then p is in S(f21) if and only if R is odd. Now since p is 5 mod 8, in any representation of p by G1, x,y and z must be odd. So if we set G2=xx+2yy+2zz+64tt then R=(N1-N2)/16, where N1 and N2 are the numbers of representations of p by G1 and G2 respectively. Now write p as aa+4bb with a and b congruent to 1 mod 4. Computer calculations indicate:

Conjecture 1_____N1=p+1+2a

Conjecture 2_____N2=((p+1)/2)+a+4b

If these conjectures hold then R=(p+1+2a-8b)/32. The numerator here is 4(b-1)^2 +(a+3)(a-1), which mod 64 is(a+3)(a-1). So R has the same parity as (a+3)(a-1)/32 and is odd just when a is 5 or 9 mod 16. Now mod 32, p=aa+4. So R is odd just when p is 29 or 85 mod 32, and so the conjectures imply Joel's result for S(f21).

How does one attack the conjectures? The theta series attached to G1 and G2 are modular forms for gamma_0 (64) and gamma_0 (256) respectively. If the conjectures are to hold it seems that each of these theta series should be a linear combination of Eisenstein series and cusp forms attached to Grossencharaktere for Q(i). It should be possible, using Sage, to get an explicit formulation of this, and prove the conjectures.

My proposed treatment of S(f19) is entirely similar. Suppose p is 3 mod 8. Writing f19 as(f)(f)(f)(f16) and arguing as above we find that if we take H1 and H2 to be xx+yy+zz+16tt and xx+yy+zz+64tt respectively, and let N1 and N2 be the number of representations of p by H1 and H2, then p is in S(f19) just when R=(N1-N2)/16 is odd. Now Jacobi's 4 square theorem, (see the argument in my answer to Joel's question), shows that N1 is 2(p+1). Write p as aa+2bb with a =1 or 3 mod 8. The computer suggests:

Conjecture 3_____N2=p+1+4a

So if the conjecture holds, R=(p+1-4a)/16, and one sees easily that this is odd just when p is 19 or 27 mod 32. Once again the theta series attached to H2 is a modular form for gamma_0 (256). The conjecture indicates that it should be a linear combination of Eisenstein series and cusp forms attached to Grossencharaktere for Q(root(-2)); all this should admit a proof using Sage.