Atlas › Modular forms › Weight 2 › Level 205 › 205.2.a › 205.2.a.d

Newform orbit 205.2.a.d

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{13})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{2} + 7x + 9$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
41	$-1$

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
$\operatorname{tr} a_p$	−1	−6	2	−3	−6	−1	−4	−3	4	−5	−3	−3	2	−3	−19	−10	−17	−12	−7	20	−3	15	−15	−2	6

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.179 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, 5, 41, 205]: [0, 1, 4, 22] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 13 = 1 x 13 (phi(ord eps) x dim S_k^new); 2 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(205, 2)[3]   # 205.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 205:2 1000