Atlas › Modular forms › Weight 2 › Level 355 › 355.2.a › 355.2.a.d

Newform orbit 355.2.a.d

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q + 3q^{2} + 3q^{3} + 9q^{4} + 6q^{5} - 5q^{6} + 6q^{7} + 6q^{8} + 5q^{9} + 3q^{10} + 3q^{11} - 8q^{12} + 5q^{13} + 2q^{14} + 3q^{15} - q^{16} + 9q^{17} + 8q^{18} - 8q^{19} + 9q^{20} - 6q^{21} + q^{22} + 4q^{23} - 31q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{6} - 6x^{5} + x^{4} + 40x^{3} - 51x^{2} + 11x + 2$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
71	$+1$

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
$\operatorname{tr} a_p$	3	3	6	6	3	5	9	−8	4	20	−12	−4	−16	−5	5	27	−2	−7	6	−6	17	−14	8	−5	31

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.186 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, 71, 355]: [0, 1, 7, 38] (certified at ell = 2147482681)
T = [(2, 1), (3, 1)]; h of degree 23 = 1 x 23 (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(355, 2)[3]   # 355.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 355:2 1000