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

Newform orbit 201.4.a.b

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q - 5q^{2} + 18q^{3} + 13q^{4} - 12q^{5} - 15q^{6} - 62q^{7} - 75q^{8} + 54q^{9} - 111q^{10} - 72q^{11} + 39q^{12} - 192q^{13} + 3q^{14} - 36q^{15} - 27q^{16} - 100q^{17} - 45q^{18} - 266q^{19} + 255q^{20} - 186q^{21} + 44q^{22} + 50q^{23} - 225q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{6} + 17x^{5} - 147x^{4} - 3245x^{3} - 1914x^{2} + 118508x + 360268$

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 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
$\operatorname{tr} a_p$	−5	18	−12	−62	−72	−192	−100	−266	50	−242	−438	−596	54	−360	−720	694	−378	−1396	−402	−964	−192	−802	2126	432	1290

$\operatorname{tr} a_n$ 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)[1]   # 201.4.a.b

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

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