Atlas › Modular forms › Weight 2 › Level 365 › 365.2.a › 365.2.a.d

Newform orbit 365.2.a.d

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q - q^{2} + 2q^{3} + 11q^{4} + 7q^{5} - 3q^{6} - 3q^{7} - 6q^{8} + 11q^{9} - q^{10} + 15q^{11} + 2q^{12} + 5q^{13} + 6q^{14} + 2q^{15} + 23q^{16} - 5q^{17} - 22q^{18} + 13q^{19} + 11q^{20} - 18q^{21} + 7q^{22} - 5q^{23} - 30q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{7} - x^{6} - 25x^{5} + 34x^{4} + 115x^{3} - 132x^{2} + 28x - 1$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
73	$+1$

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
$\operatorname{tr} a_p$	−1	2	7	−3	15	5	−5	13	−5	−3	11	−26	7	16	−3	−34	27	15	11	26	−7	19	−17	2	−4

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.188 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, 5, 73, 365]: [0, 1, 6, 38] (certified at ell = 2147482873)
T = [(2, 1), (3, 1)]; h of degree 25 = 1 x 25 (phi(ord eps) x dim S_k^new); 3 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(365, 2)[3]   # 365.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 365:2 1000