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

Newform orbit 530.2.a.e

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{3} - 8x^{2} - 223x + 1043$

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
$\operatorname{tr} a_p$	−3	−1	−3	3	−1	−7	−12	2	12	−4	7	17	−4	14	−11	3	1	11	1	29	7	23	−15	3	−8

$\operatorname{tr} a_n$ 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)[4]   # 530.2.a.e

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

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