Atlas › Modular forms › Weight 4 › Level 55 › 55.4.a › 55.4.a.d

Newform orbit 55.4.a.d

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q + q^{2} + 9q^{3} + 19q^{4} + 20q^{5} - 19q^{6} + 9q^{7} + 33q^{8} + 49q^{9} + 5q^{10} + 44q^{11} + 75q^{12} + 70q^{13} - 49q^{14} + 45q^{15} - 37q^{16} + 103q^{17} - 356q^{18} - 205q^{19} + 95q^{20} - 181q^{21} + 11q^{22} - 56q^{23} - 387q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{4} - 28x^{3} - 283x^{2} + 9726x - 14592$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
11	$-1$

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	9	20	9	44	70	103	−205	−56	−79	49	289	736	−152	412	1685	−842	−1097	−122	−521	−590	−1118	−122	−181	1474

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.102 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, 5, 11, 55]: [1, 3, 4, 20] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 2 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(55, 4)[3]   # 55.4.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 55:4 1000