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

Newform orbit 205.4.a.c

Trace form

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

$8q - 7q^{2} - 22q^{3} + 31q^{4} + 40q^{5} - 9q^{6} - 60q^{7} - 108q^{8} + 76q^{9} - 35q^{10} - 30q^{11} + 39q^{12} - 142q^{13} + 33q^{14} - 110q^{15} + 115q^{16} + 12q^{17} - 154q^{18} + 44q^{19} + 155q^{20} + 10q^{21} - 113q^{22} - 578q^{23} - 86q^{24} + O(q^{25})$

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

$x^{8} + 29x^{7} + 236x^{6} - 121x^{5} - 7192x^{4} - 13621x^{3} + 2136x^{2} + 1361x - 109$

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
$\operatorname{tr} a_p$	−7	−22	40	−60	−30	−142	12	44	−578	−78	−50	−994	−328	−1220	−1090	484	360	1488	−1256	−1138	−266	466	−4022	−256	−3396

$\operatorname{tr} a_n$ 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)[2]   # 205.4.a.c

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

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