Atlas › Modular forms › Weight 2 › Level 590 › 590.2.a › 590.2.a.k

Newform orbit 590.2.a.k

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{7}$ , whose minimal polynomial is

$x^{4} - 7x^{3} + 12x^{2} + 5x - 17$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
5	$-1$
59	$-1$

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
$\operatorname{tr} a_p$	4	3	4	4	1	2	−8	−2	1	6	14	−10	−7	10	−7	−9	4	9	−12	−1	−1	−1	8	0	−3

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.344 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, 59, 118, 295, 590]: [0, 1, 1, 3, 6, 17, 32, 94] (certified at ell = 2147482481)
T = [(3, 1), (7, 1)]; h of degree 21 = 1 x 21 (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(590, 2)[10]   # 590.2.a.k

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

cargo run --release -p sagebrush-web --example atlas -- 590:2 1000