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

Newform orbit 201.4.a.d

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q + 3q^{2} - 27q^{3} + 45q^{4} + 12q^{5} - 9q^{6} + 8q^{7} + 51q^{8} + 81q^{9} + 45q^{10} + 72q^{11} - 135q^{12} - 166q^{13} + 93q^{14} - 36q^{15} + 173q^{16} + 146q^{17} + 27q^{18} + 154q^{19} + 763q^{20} - 24q^{21} + 244q^{22} + 476q^{23} - 153q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{9} - 15x^{8} - 786x^{7} + 8860x^{6} + 185381x^{5} - 1608761x^{4} - 13137808x^{3} + 97439936x^{2} + 137960192x - 785739520$

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	−27	12	8	72	−166	146	154	476	432	248	−240	406	154	494	−450	732	−914	−603	2990	−1384	2438	972	1034	−1516

$\operatorname{tr} a_n$ 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)[3]   # 201.4.a.d

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

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