Atlas › Modular forms › Weight 4 › Level 41 › 41.4.a › 41.4.a.b

Newform orbit 41.4.a.b

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q + q^{2} + 4q^{3} + 43q^{4} + 10q^{5} + 18q^{6} + 48q^{7} + 33q^{8} + 99q^{9} + 82q^{10} + 34q^{11} - 218q^{12} - 60q^{13} - 304q^{14} + 46q^{15} + 59q^{16} - 82q^{17} - 563q^{18} + 144q^{19} - 410q^{20} + 84q^{21} + 82q^{22} + 204q^{23} - 86q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{7} - 5x^{6} - 199x^{5} + 1655x^{4} + 2408x^{3} - 49882x^{2} + 125528x - 82714$

Atkin–Lehner signs

$p$	$w_p$
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 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
$\operatorname{tr} a_p$	1	4	10	48	34	−60	−82	144	204	68	696	730	−287	368	−26	−892	−916	−450	−142	390	882	2890	1368	−2006	−1950

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.093 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, 41]: [1, 12] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 10 = 1 x 10 (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(41, 4)[1]   # 41.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 41:4 1000