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

Newform orbit 201.4.a.c

Trace form

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

$7q - q^{2} - 21q^{3} + 21q^{4} + 11q^{5} + 3q^{6} - 33q^{7} - 45q^{8} + 63q^{9} - 51q^{10} - 130q^{11} - 63q^{12} + 16q^{13} + 5q^{14} - 33q^{15} + 77q^{16} + 90q^{17} - 9q^{18} - 132q^{19} - 359q^{20} + 99q^{21} - 192q^{22} - 399q^{23} + 135q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{7} - 10x^{6} - 309x^{5} + 2860x^{4} + 29946x^{3} - 240328x^{2} - 867656x + 4880952$

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
$\operatorname{tr} a_p$	−1	−21	11	−33	−130	16	90	−132	−399	−302	−555	297	−717	−245	−1072	265	−255	418	469	−1194	995	−2640	−2579	−1604	−808

$\operatorname{tr} a_n$ 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)[2]   # 201.4.a.c

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

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