Atlas › Modular forms › Weight 4 › Level 205 › 205.4.a › 205.4.a.d

Newform orbit 205.4.a.d

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q + q^{2} - 12q^{3} + 35q^{4} - 40q^{5} - 33q^{6} - 8q^{7} - 6q^{8} + 76q^{9} - 5q^{10} - 134q^{11} - 165q^{12} - 2q^{13} - 175q^{14} + 60q^{15} - 21q^{16} - 70q^{17} + 70q^{18} - 324q^{19} - 175q^{20} - 394q^{21} - 215q^{22} + 186q^{23} - 574q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{8} + 11x^{7} - 102x^{6} - 1111x^{5} + 3484x^{4} + 32189x^{3} - 53306x^{2} - 279665x + 358275$

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	−12	−40	−8	−134	−2	−70	−324	186	−224	−198	274	328	−972	−766	−1104	−2164	−344	1454	−1224	1342	−1308	−2646	88	2574

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.139 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, 41, 205]: [1, 3, 12, 66] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 40 = 1 x 40 (phi(ord eps) x dim S_k^new); 6 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, 4)[3]   # 205.4.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 205:4 1000