Atlas › Modular forms › Weight 6 › Level 57 › 57.6.a › 57.6.a.c

Newform orbit 57.6.a.c

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 3q^{2} + 18q^{3} + 17q^{4} - 87q^{5} - 27q^{6} - 251q^{7} - 159q^{8} + 162q^{9} - 252q^{10} + 735q^{11} + 153q^{12} - 722q^{13} - 312q^{14} - 783q^{15} - 1759q^{16} - 603q^{17} - 243q^{18} + 722q^{19} + 408q^{20} - 2259q^{21} - 5004q^{22} - 6306q^{23} - 1431q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{17})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{2} + 90x + 1413$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
19	$-1$

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 odd 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
$\operatorname{tr} a_p$	−3	18	−87	−251	735	−722	−603	722	−6306	−9444	−7544	−18572	30198	2629	2067	10008	3180	−50975	−32096	−7812	−45149	124030	29256	10836	−188900

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.242 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, 3, 19, 57]: [1, 3, 10, 36] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 4 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(57, 6)[2]   # 57.6.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 57:6 1000