Atlas › Modular forms › Weight 2 › Level 386 › 386.2.a › 386.2.a.d

Newform orbit 386.2.a.d

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{7} - 18x^{6} - 10x^{5} + 1560x^{4} - 3928x^{3} - 38188x^{2} + 79668x + 377584$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
193	$+1$

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	3	5	2	2	5	8	4	−8	0	−6	−1	−4	−13	−17	−5	−3	12	1	−25	28	−5	−7	4	13

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.288 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, 2, 193, 386]: [0, 1, 16, 50] (certified at ell = 2147483137)
T = [(3, 1), (5, 3)]; h of degree 17 = 1 x 17 (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(386, 2)[3]   # 386.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 386:2 1000