Atlas › Modular forms › Weight 2 › Level 773 › 773.2.a › 773.2.a.a

Newform orbit 773.2.a.a

Trace form

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

$2q - 4q^{2} + 2q^{3} + 4q^{4} - q^{5} - 4q^{6} - 3q^{7} - 4q^{9} + 2q^{10} + 2q^{11} + 4q^{12} + 3q^{13} + 6q^{14} - q^{15} - 8q^{16} + 2q^{17} + 8q^{18} + 2q^{19} - 2q^{20} - 3q^{21} - 4q^{22} - 12q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{5})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7}$ , whose minimal polynomial is

$x^{2} + 24x - 36$

Atkin–Lehner signs

$p$	$w_p$
773	$+1$

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
$\operatorname{tr} a_p$	−4	2	−1	−3	2	3	2	2	−12	8	−11	0	−6	−13	1	4	7	−6	1	−11	−22	−10	−3	2	−7

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.297 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 773]: [0, 65] (certified at ell = 2147480893)
T = [(2, 1), (3, 3), (5, 5), (7, 7)]; h of degree 64 = 1 x 64 (phi(ord eps) x dim S_k^new); 14 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(773, 2)[0]   # 773.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 773:2 1000