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

Newform orbit 41.4.a.a

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 3q^{2} - 8q^{3} - 5q^{4} - 10q^{5} - 18q^{6} - 50q^{7} - 15q^{8} - 9q^{9} + 2q^{10} - 38q^{11} + 68q^{12} + 30q^{13} + 124q^{14} + 6q^{15} + 11q^{16} + 42q^{17} + 177q^{18} - 148q^{19} + 106q^{20} + 72q^{21} + 116q^{22} + 52q^{23} + 162q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} + 11x^{2} + 33x + 29$

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 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
$\operatorname{tr} a_p$	−3	−8	−10	−50	−38	30	42	−148	52	−54	−464	−126	123	−204	424	438	−48	74	−874	−1140	1038	−2760	136	−1446	262

$\operatorname{tr} a_n$ 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)[0]   # 41.4.a.a

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

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