Atlas › Modular forms › Weight 2 › Level 619 › 619.2.a › 619.2.a.b

Newform orbit 619.2.a.b

Trace form

The 30 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$30q + 9q^{2} + q^{3} + 33q^{4} + 21q^{5} + 6q^{6} + 2q^{7} + 27q^{8} + 43q^{9} + 5q^{10} + 23q^{11} - 6q^{12} + 9q^{13} + 7q^{14} - 2q^{15} + 35q^{16} + 4q^{17} + 10q^{18} - q^{19} + 29q^{20} + 30q^{21} + 4q^{23} + 4q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 30. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{30} - 10x^{29} - 69x^{28} + 969x^{27} + 1100x^{26} - 38638x^{25} + 29988x^{24} + 837065x^{23} - 1477892x^{22} - 10941129x^{21} + 26634966x^{20} + 90458051x^{19} - 272129747x^{18} - 480732003x^{17} + 1745601319x^{16} + 1622836338x^{15} - 7271970336x^{14} - 3298093858x^{13} + 19720599786x^{12} + 3414171340x^{11} - 34108263246x^{10} - 419730055x^{9} + 36108603168x^{8} - 2501947163x^{7} - 21997160380x^{6} + 2136206797x^{5} + 6905383230x^{4} - 624395688x^{3} - 839637808x^{2} + 55995696x + 1117888$

Atkin–Lehner signs

$p$	$w_p$
619	$-1$

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
$\operatorname{tr} a_p$	9	1	21	2	23	9	4	−1	4	90	2	19	59	−4	4	34	13	16	−11	42	−4	3	−11	58	−9

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.218 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, 619]: [0, 52] (certified at ell = 2147482021)
T = [(2, 1), (3, 1)]; h of degree 51 = 1 x 51 (phi(ord eps) x dim S_k^new); 5 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(619, 2)[1]   # 619.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 619:2 1000