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

Newform orbit 551.2.a.h

Trace form

The 18 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$18q + 2q^{2} - 2q^{3} + 26q^{4} + 7q^{5} + 12q^{6} + 11q^{7} + 6q^{8} + 34q^{9} + 5q^{11} - 13q^{12} + 10q^{13} - 20q^{14} - 10q^{15} + 46q^{16} + 7q^{17} + 16q^{18} - 18q^{19} - 4q^{20} + 14q^{21} - 12q^{22} + 12q^{23} + 4q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 18. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{18} + 4x^{17} - 455x^{16} - 1651x^{15} + 82815x^{14} + 264175x^{13} - 7822476x^{12} - 20991598x^{11} + 416972019x^{10} + 886440154x^{9} - 12730474715x^{8} - 20153860580x^{7} + 212629826642x^{6} + 251190935871x^{5} - 1710560127212x^{4} - 1901247698194x^{3} + 5371040369594x^{2} + 6113521093148x - 1982484758400$

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
$\operatorname{tr} a_p$	2	−2	7	11	5	10	7	−18	12	18	0	14	6	1	1	2	−2	37	−6	20	59	4	26	−22	16

$\operatorname{tr} a_n$ 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)[7]   # 551.2.a.h

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

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