Atlas › Modular forms › Weight 2 › Level 622 › 622.2.a › 622.2.a.c

Newform orbit 622.2.a.c

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{5} + 6x^{4} + 2x^{3} - 25x^{2} + x + 16$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
311	$+1$

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 odd 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
$\operatorname{tr} a_p$	−5	−1	−5	0	−5	−1	−10	0	−17	−5	6	−5	−15	1	−21	−20	−5	−3	−2	−3	−10	9	1	−22	−7

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.31 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, 2, 311, 622]: [0, 1, 27, 80] (certified at ell = 2147479121)
T = [(3, 1), (5, 1)]; h of degree 25 = 1 x 25 (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(622, 2)[2]   # 622.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 622:2 1000