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

Newform orbit 101.4.a.a

Trace form

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

$9q - 5q^{2} - 16q^{3} + 7q^{4} - 6q^{5} - 2q^{6} - 39q^{7} - 39q^{8} + 3q^{9} - 86q^{10} - 126q^{11} - 140q^{12} - 104q^{13} - 111q^{14} - 206q^{15} - 201q^{16} - 51q^{17} - 161q^{18} - 381q^{19} - 36q^{20} - 160q^{21} - 122q^{22} - 128q^{23} + 66q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{9} + 21x^{8} + 60x^{7} - 868x^{6} - 3569x^{5} + 7229x^{4} + 33249x^{3} - 24855x^{2} - 81988x + 68542$

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 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	−16	−6	−39	−126	−104	−51	−381	−128	71	−234	−106	−136	−513	64	−125	−396	−1385	−2998	−468	−1444	−3064	−1230	−420	−25

$\operatorname{tr} a_n$ 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)[0]   # 101.4.a.a

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

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