Atlas › Modular forms › Weight 4 › Level 222 › 222.4.a › 222.4.a.f

Newform orbit 222.4.a.f

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q + 4q^{2} - 6q^{3} + 8q^{4} - 4q^{5} - 12q^{6} + 17q^{7} + 16q^{8} + 18q^{9} - 8q^{10} + 47q^{11} - 24q^{12} - 31q^{13} + 34q^{14} + 12q^{15} + 32q^{16} + 75q^{17} + 36q^{18} + 187q^{19} - 16q^{20} - 51q^{21} + 94q^{22} - 39q^{23} - 48q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{97})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{5} + T_{7}$ , whose minimal polynomial is

$x^{2} - 13x - 176$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
3	$+1$
37	$-1$

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
$\operatorname{tr} a_p$	4	−6	−4	17	47	−31	75	187	−39	432	−186	74	−696	132	986	117	−958	−204	854	804	−429	−490	−457	−649	428

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.193 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, 2, 3, 6, 37, 74, 111, 222]: [1, 2, 2, 5, 11, 31, 40, 118] (certified at ell = 2147483629)
T = [(5, 1), (7, 1)]; h of degree 18 = 1 x 18 (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(222, 4)[5]   # 222.4.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 222:4 1000