Atlas › Modular forms › Weight 2 › Level 551 › 551.2.a › 551.2.a.g

Newform orbit 551.2.a.g

Trace form

The 16 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 16. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{16} - 15x^{15} - 238x^{14} + 4307x^{13} + 17710x^{12} - 466979x^{11} - 345637x^{10} + 25536507x^{9} - 16692344x^{8} - 762940298x^{7} + 976014183x^{6} + 12268004321x^{5} - 18939270821x^{4} - 95472218438x^{3} + 158205848978x^{2} + 276459290012x - 483739053720$

Atkin–Lehner signs

$p$	$w_p$
19	$-1$
29	$+1$

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	4	−2	4	4	8	0	16	−8	−16	34	24	44	4	−16	−24	−8	10	26	−12	30	46	−50	28	32

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.312 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, 19, 29, 551]: [0, 2, 3, 52] (certified at ell = 2147483269)
T = [(2, 1), (3, 3)]; h of degree 43 = 1 x 43 (phi(ord eps) x dim S_k^new); 6 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(551, 2)[6]   # 551.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 551:2 1000