Atlas › Modular forms › Weight 10 › Level 5 › 5.10.a › 5.10.a.b

Newform orbit 5.10.a.b

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 10q^{2} + 260q^{3} + 1044q^{4} + 1250q^{5} - 5336q^{6} + 1700q^{7} - 20280q^{8} + 2506q^{9} - 6250q^{10} + 23984q^{11} + 176080q^{12} + 115020q^{13} - 440352q^{14} + 162500q^{15} - 312048q^{16} + 412820q^{17} - 1061890q^{18} - 296520q^{19} + 652500q^{20} + 1084704q^{21} + 3714280q^{22} - 1049220q^{23} - 2878560q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{1009})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{2} - 250x + 14616$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$

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
$\operatorname{tr} a_p$	−10	260	1250	1700	23984	115020	412820	−296520	−1049220	−3666980	1613144	−21121940	−26957276	52889700	58412180	−39035140	−54995560	−274579716	−318580	−7130936	120858180	6877520	1402348740	830088660	638394580

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.263 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 engine's checks

dim M^+ at levels [1, 5]: [1, 5] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 3 = 1 x 3 (phi(ord eps) x dim S_k^new); 1 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(5, 10)[1]   # 5.10.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 5:10 1000