Atlas › Modular forms › Weight 2 › Level 530 › 530.2.a › 530.2.a.h

Newform orbit 530.2.a.h

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{7}$ , whose minimal polynomial is

$x^{5} - 7x^{4} - 123x^{3} + 482x^{2} + 3839x + 1936$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
5	$-1$
53	$-1$

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
$\operatorname{tr} a_p$	5	4	5	1	−1	0	5	−7	−1	−3	1	12	6	−8	−1	5	−31	11	19	−10	−15	−2	8	−33	13

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.337 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, 5, 10, 53, 106, 265, 530]: [0, 1, 1, 3, 5, 15, 28, 84] (certified at ell = 2147483053)
T = [(3, 1), (7, 3)]; h of degree 19 = 1 x 19 (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(530, 2)[7]   # 530.2.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 530:2 1000