Atlas › Modular forms › Weight 4 › Level 101 › 101.4.a › 101.4.a.b

Newform orbit 101.4.a.b

Trace form

The 16 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$16q + q^{2} + 14q^{3} + 91q^{4} + 14q^{5} + 34q^{6} + 45q^{7} + 33q^{8} + 192q^{9} + 54q^{10} + 116q^{11} + 100q^{12} + 52q^{13} + 85q^{14} + 154q^{15} + 583q^{16} + 85q^{17} + 109q^{18} + 455q^{19} + 204q^{20} + 260q^{21} + 450q^{22} + 240q^{23} + 498q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 16. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{16} - 15x^{15} - 343x^{14} + 5509x^{13} + 43505x^{12} - 785215x^{11} - 2580987x^{10} + 56172021x^{9} + 77168921x^{8} - 2193438905x^{7} - 1243893859x^{6} + 46635745257x^{5} + 15893513081x^{4} - 486376478453x^{3} - 224780192032x^{2} + 1721965023226x + 1324386847700$

Atkin–Lehner signs

$p$	$w_p$
101	$+1$

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
$\operatorname{tr} a_p$	1	14	14	45	116	52	85	455	240	−213	446	566	−642	1079	168	−5	66	575	2288	1532	−1014	3536	208	1962	2935

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.103 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, 101]: [1, 27] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 25 = 1 x 25 (phi(ord eps) x dim S_k^new); 4 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(101, 4)[1]   # 101.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 101:4 1000