Atlas › Modular forms › Weight 4 › Level 201 › 201.4.a › 201.4.a.e

Newform orbit 201.4.a.e

Trace form

The 11 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$11q + 3q^{2} + 33q^{3} + 69q^{4} + 8q^{5} + 9q^{6} + 78q^{7} + 21q^{8} + 99q^{9} + 29q^{10} + 104q^{11} + 207q^{12} + 172q^{13} + 143q^{14} + 24q^{15} + 485q^{16} - 48q^{17} + 27q^{18} + 180q^{19} - 539q^{20} + 234q^{21} - 144q^{22} + 156q^{23} + 63q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 11. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{11} - 11x^{10} - 926x^{9} + 9810x^{8} + 311585x^{7} - 3285113x^{6} - 43769992x^{5} + 492984150x^{4} + 1791118440x^{3} - 28214412416x^{2} + 59618119680x + 1679511552$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
67	$-1$

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
$\operatorname{tr} a_p$	3	33	8	78	104	172	−48	180	156	−4	514	854	674	738	54	−190	18	328	737	264	330	456	−2432	−2340	1892

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.138 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, 3, 67, 201]: [1, 2, 18, 70] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 34 = 1 x 34 (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(201, 4)[4]   # 201.4.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 201:4 1000