Atlas › Modular forms › Weight 6 › Level 13 › 13.6.a › 13.6.a.b

Newform orbit 13.6.a.b

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q + 7q^{2} + 8q^{3} + 121q^{4} + 56q^{5} - 199q^{6} - 60q^{7} + 327q^{8} - 191q^{9} - 1291q^{10} + 556q^{11} - 1091q^{12} + 507q^{13} - 793q^{14} + 1972q^{15} + 2785q^{16} + 908q^{17} + 976q^{18} + 148q^{19} - 767q^{20} + 1390q^{21} - 3410q^{22} + 3624q^{23} - 13377q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{3} - 15x^{2} - 66x + 144$

Atkin–Lehner signs

$p$	$w_p$
13	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	7	8	56	−60	556	507	908	148	3624	−8758	−2608	−20632	−10998	2032	34260	−12570	63948	−12754	56132	77580	−43026	−61872	98092	33694	76334

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.203 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 13]: [1, 7] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 1 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(13, 6)[1]   # 13.6.a.b

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 13:6 1000